U.S. patent application number 13/962962 was filed with the patent office on 2014-06-19 for predicting pharmacokinetic and pharmacodynamic responses of a component of a system to an input into the system.
This patent application is currently assigned to Arrapoi, Inc.. The applicant listed for this patent is Arrapoi, Inc.. Invention is credited to GLENN A. WILLIAMS.
Application Number | 20140172765 13/962962 |
Document ID | / |
Family ID | 49262579 |
Filed Date | 2014-06-19 |
United States Patent
Application |
20140172765 |
Kind Code |
A1 |
WILLIAMS; GLENN A. |
June 19, 2014 |
PREDICTING PHARMACOKINETIC AND PHARMACODYNAMIC RESPONSES OF A
COMPONENT OF A SYSTEM TO AN INPUT INTO THE SYSTEM
Abstract
Non-mechanistic, differential-equation-free approaches for
predicting a particular pharmacokinetic and pharmacodynamic
responses of a system to a given input are provided in the form of
systems, methods, and devices. These approaches are generally
directed to a non-compartmental method of predicting a
time-dependent pharmacokinetic response, or pharmacodynamics
response, of a component of a system to an input into the system.
The systems, methods, and devices provide the ability to (i) reduce
the cost of research and development by offering an accurate
modeling of heterogeneous and complex physical systems; (ii) reduce
the cost of creating such systems and methods by simplifying the
modeling process; (iii) accurately capture and model inherent
nonlinearities in cases where sufficient knowledge does not exist
to a priori build a model and its parameters; and, (iv) provide
one-to-one relationships between model parameters and model
outputs, addressing the problem of the ambiguities inherent in the
current, state-of-the-art systems and methods.
Inventors: |
WILLIAMS; GLENN A.; (Redwood
City, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Arrapoi, Inc. |
Redwood City |
CA |
US |
|
|
Assignee: |
Arrapoi, Inc.
Redwood City
CA
|
Family ID: |
49262579 |
Appl. No.: |
13/962962 |
Filed: |
August 9, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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13717644 |
Dec 17, 2012 |
8554712 |
|
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13962962 |
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Current U.S.
Class: |
706/46 |
Current CPC
Class: |
G16B 5/00 20190201; G06N
5/02 20130101; G06F 17/17 20130101; G06F 30/20 20200101; G06F
2111/10 20200101; G16C 20/30 20190201 |
Class at
Publication: |
706/46 |
International
Class: |
G06N 5/02 20060101
G06N005/02 |
Claims
1. A non-compartmental method of predicting a time-dependent,
pharmacokinetic response of a component of a mammalian system to an
input into the system, the method comprising: selecting a component
of the system, the component selected from the group consisting of
a body fluid; selecting a set of actual inputs, the set of actual
inputs having an element selected from the group consisting of a
DNA, a virus, a protein, an antibody, a bacteria, a chemical, a
dietary supplement, a nutrient, and a drug; obtaining a set of
time-dependent actual pharmacokinetic responses of the component to
the set of actual inputs; using the set of actual inputs and the
set of time-dependent actual pharmacokinetic responses to provide a
model for predicting a test pharmacokinetic response to a test
input, the model comprising the formula C ( t ) = [ M 0 0 + M 0 1 (
kernel ) ] + [ M 1 0 + M 1 1 ( kernel ) ] { 1 - - [ N 1 0 + N 1 1 (
kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 + N 1 1 ( kernel ) ] t } + + [
M n 0 + M n 1 ( kernel ) ] { 1 - - [ N n 0 + N n 1 ( kernel ) ] t 1
+ ( K - 2 ) - [ N n 0 + N n 1 ( kernel ) ] t } ( 9 ) ##EQU00056##
wherein, M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, .
. . , M.sup.1.sub.n are overall scaling parameters; N.sup.0.sub.1,
. . . , N.sup.0.sub.n and N.sup.1.sub.1, . . . , N.sup.1.sub.n are
exponential scaling parameters; n ranges from 1 to 4; K is an
overall shifting parameter; and, C(t) is the time-dependent
pharmacokinetic response to the test input at time t; and, kernel
.ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0 ;
##EQU00057## wherein, C.sub.0 is the initial amount of the test
input; K.sub.p is a shifting parameter related to C.sub.0; and,
.alpha..sub.p is shifting and scaling parameter related to C.sub.0;
and, using the model to obtain the time-dependent test
pharmacokinetic response to the test input.
2. The method of claim 1, wherein the component is blood, blood
plasma, or blood serum.
3. The method of claim 1, wherein the component is urine.
4. The method of claim 1, wherein the component saliva.
5. The method of claim 1, wherein the test pharmacokinetic response
is a bacterial load.
6. The method of claim 1, wherein the test pharmacokinetic response
is a viral load.
7. The method of claim 1, wherein the test pharmacokinetic response
is a drug concentration.
8. The method of claim 1, wherein the set of actual inputs includes
a set of dosages of a drug.
9. The method of claim 1, wherein the set of actual inputs includes
a set of drugs.
10. A device for predicting a time-dependent, pharmacokinetic
response of a component of a mammalian system to an input into the
system, the device comprising: a processor; a database for storing
a set of actual input data, a set of time-dependent actual
pharmacokinetic response data of a component comprising a body
fluid, test input data, and time-dependent test pharmacokinetic
response data of the component on a non-transitory computer
readable medium; an enumeration engine on a non-transitory computer
readable medium to parameterize a non-compartmental model for
predicting a test pharmacokinetic response of the component to a
test input, the non-compartmental model comprising the formula C (
t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel ) ] {
1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 + N 1
1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N n 0
+ N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel ) ]
t } ( 9 ) ##EQU00058## wherein, M.sup.0.sub.0, . . . ,
M.sup.0.sub.n and M.sup.1.sub.0, . . . , M.sup.1.sub.n are overall
scaling parameters; N.sup.0.sub.1, . . . , N.sup.0.sub.n and
N.sup.1.sub.1, . . . , N.sup.1.sub.n are exponential scaling
parameters; n ranges from 1 to 4; K is an overall shifting
parameter; and, C(t) is the time-dependent pharmacokinetic response
of the component to the test input at time t; and, kernel .ident. 1
- - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0 ; ##EQU00059##
wherein, C.sub.0 is the initial amount of the test input; K.sub.p
is a shifting parameter related to C.sub.0; and, .alpha..sub.p is
shifting and scaling parameter related to C.sub.0; and, a
transformation module on a non-transitory computer readable medium
to transform the test data into the time-dependent pharmacokinetic
response data of the component using the non-compartmental
model.
11. The device of claim 10, wherein the component is blood, blood
plasma, or blood serum.
12. The device of claim 10, wherein the component is urine.
13. The device of claim 10, wherein the component saliva.
14. The device of claim 10, wherein the test pharmacokinetic
response is a bacterial load.
15. The device of claim 10, wherein the test pharmacokinetic
response is a viral load.
16. The device of claim 10, wherein the test pharmacokinetic
response is a drug concentration.
17. The device of claim 10, wherein the set of actual inputs
includes a set of dosages of a drug.
18. The device of claim 10, wherein the set of actual inputs
includes a set of drugs.
19. A non-compartmental method of predicting a time-dependent,
pharmacodynamic response of a component of a mammalian system to an
input into the system, the method comprising: selecting a component
of the system, the component selected from the group consisting of
a cell, a tissue, an organ, a DNA, a virus, a protein, an antibody,
a bacteria; selecting a set of actual inputs, the set of actual
inputs having an element selected from the group consisting of a
DNA, a virus, a protein, an antibody, a bacteria, a chemical, a
dietary supplement, a nutrient, and a drug; obtaining a set of
time-dependent actual pharmacodynamic responses of the component to
the set of actual inputs; using the set of actual inputs and the
set of time-dependent actual pharmacodynamic responses of the
component to provide a model for predicting a test pharmacodynamic
response of the component to a test input, the model comprising the
formula C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 (
kernel ) ] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [
N 1 0 + N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1
- - [ N n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1
( kernel ) ] t } ( 9 ) ##EQU00060## wherein, M.sup.0.sub.0, . . . ,
M.sup.0.sub.n and M.sup.1.sub.0, . . . , M.sup.1.sub.n are overall
scaling parameters; N.sup.0.sub.1, . . . , N.sup.0.sub.n and
N.sup.1.sub.1, . . . , N.sup.1.sub.n are exponential scaling
parameters; n ranges from 1 to 4; K is an overall shifting
parameter; and, C(t) is the time-dependent pharmacodynamic response
to the test input at time t; and, kernel .ident. 1 - - .alpha. p C
0 1 + ( K p - 2 ) - .alpha. p C 0 ; ##EQU00061## wherein, C.sub.0
is the initial amount of the test input; K.sub.p is a shifting
parameter related to C.sub.0; and, .alpha..sub.p is shifting and
scaling parameter related to C.sub.0; and, using the model to
obtain the time-dependent test pharmacodynamic response to the test
input.
20. The method of claim 19, wherein the component is a bodily fluid
selected from the group consisting of blood, blood plasma, blood
serum, saliva, and urine.
21. The method of claim 19, wherein the component is a tumor
cell.
22. The method of claim 19, wherein the component is a virus.
23. The method of claim 19, wherein the component is a
bacteria.
24. The method of claim 19, wherein the test pharmacodynamic
response is a bacterial load.
25. The method of claim 19, wherein the test pharmacodynamic
response is a viral load.
26. The method of claim 19, wherein the test pharmacodynamic
response is a tumor marker.
27. The method of claim 19, wherein the test pharmacodynamic
response is a blood chemistry.
28. The method of claim 19, wherein the test pharmacodynamic
response is a biomarker.
29. The method of claim 19, wherein the set of actual inputs
includes a set of dosages of a drug.
30. The method of claim 19, wherein the set of actual inputs
includes a set of drugs.
31. The method of claim 19, wherein the input is a diabetes drug,
and the time-dependent pharmacodynamic response is glucose in the
bloodstream.
32. A device for predicting a time-dependent pharmacodynamic
response of a component of a mammalian system to an input into the
system, the device comprising: a processor; a database for storing
a set of actual input data, a set of time-dependent actual
pharmacodynamic response data of a component, test input data, and
time-dependent test pharmacodynamic response data of the component
on a non-transitory computer readable medium; an enumeration engine
on a non-transitory computer readable medium to parameterize a
non-compartmental model for predicting a test pharmacodynamic
response of the component to a test input, the non-compartmental
model comprising the formula C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ]
+ [ M 1 0 + M 1 1 ( kernel ) ] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ]
t 1 + ( K - 2 ) - [ N 1 0 + N 1 1 ( kernel ) ] t } + + [ M n 0 + M
n 1 ( kernel ) ] { 1 - - [ N n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2
) - [ N n 0 + N n 1 ( kernel ) ] t } ( 9 ) ##EQU00062## wherein,
M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . . . ,
M.sup.1.sub.n are overall scaling parameters; N.sup.0.sub.1, . . .
, N.sup.0.sub.n and N.sup.1.sub.1, . . . , N.sup.1.sub.n are
exponential scaling parameters; n ranges from 1 to 4; K is an
overall shifting parameter; and, C(t) is the time-dependent
pharmacodynamic response of the component to the test input at time
t; and, kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) -
.alpha. p C 0 ; ##EQU00063## wherein, C.sub.0 is the initial amount
of the test input; K.sub.p is a shifting parameter related to
C.sub.0; and, .alpha..sub.p is shifting and scaling parameter
related to C.sub.0; and, a transformation module on a
non-transitory computer readable medium to transform the test data
into the time-dependent pharmacodynamic response data of the
component using the non-compartmental model.
33. The device of claim 32, wherein the component is a bodily fluid
selected from the group consisting of blood, blood plasma, blood
serum, saliva, and urine.
34. The device of claim 32, wherein the component is a tumor
cell.
35. The device of claim 32, wherein the component is a virus.
36. The device of claim 32, wherein the component is a
bacteria.
37. The device of claim 32, wherein the test pharmacodynamic
response is a bacterial load.
38. The device of claim 32, wherein the test pharmacodynamic
response is a viral load.
39. The device of claim 32, wherein the test pharmacodynamic
response is a tumor marker.
40. The device of claim 32, wherein the test pharmacodynamic
response is a blood chemistry.
41. The device of claim 32, wherein the test pharmacodynamic
response is a biomarker.
42. The device of claim 32, wherein the set of actual inputs
includes a set of dosages of a drug.
43. The device of claim 32, wherein the set of actual inputs
includes a set of drugs.
44. The device of claim 32, wherein the input is a diabetes drug,
and the time-dependent pharmacodynamic response is glucose in the
bloodstream.
Description
BACKGROUND
[0001] 1. Field of the Invention
[0002] The teachings generally relate to a non-mechanistic,
differential-equation-free approach for predicting a particular
pharmacokinetic and pharmacodynamic responses of a system to a
given input.
[0003] 2. Description of the Related Art
[0004] Research and development has historically relied on physical
modeling to develop new technologies. Given the speed at which
computers can perform computations, and the vast amount of computer
memory available, computer modeling allows us to speed-up and
reduce costs of research by facilitating the creation of a large
number of simulations over a wide range of physical scales very
quickly. As with physical modeling, computer modeling and
simulation deals with first characterizing and then predicting
input-response type relationships. What type of reaction will occur
between two chemicals? What is the flow response when a given
amount of water is introduced into a particular porous media? How
will the components of a watershed--rivers, reservoirs, aquifers,
etc.--react when subjected to a given rainfall or contamination
event? How will a person's blood glucose level respond to a given
meal? How will a diseased tissue respond to a drug regimen? These
are all input-response-type questions that can be addressed through
mathematical/computational modeling and simulation. Generally
speaking, this can be referred to as "input-response modeling". In
the field of drug design, this can also be referred to as
"dose-response modeling." An accurate model will give researchers a
way of running simulations to quickly observe and test a large
number of complex input-response phenomena that might be too costly
and time-consuming to observe and test in a real-world setting.
[0005] The reliance on physical modeling can be very expensive,
which makes the use of computer modeling an attractive way to
reduce costs. For example, the average drug developed by a major
pharmaceutical company costs at least $4 billion, and it can be as
much as $11 billion. The range of money spent is quite wide, for
example, as AstraZeneca has spent about $12 billion in research
money for every new drug approved; Eli Lilly spent about $4.5
billion per drug; and, Amgen has spent about $3.7 billion per drug.
The costs are so high, at least in part, because single clinical
trial can cost $100 million, and the combined cost of manufacturing
and clinical testing for some drugs can add up to $1 billion.
Computer modeling of drugs, if improved such that it can be done
efficiently and effectively, can cut costs and help make the
business of drug discovery more attractive. Other industries, of
course, can also benefit from such efficient and effective computer
modeling methods.
[0006] State-of-the-art systems and methods, however, typically use
mechanistic computer models to try and avoid the costs of physical
modeling. Unfortunately, such models can be very complex,
insufficient and ambiguous, and moreover, lacking in accuracy. Such
models use established empirical formulas as "first principles"
that provide the framework to make "mechanistic" predictions.
Complex biological systems can be modeled, for example, using
laboratory experiments to establish such first-principle-type
relationships between components of the system. For example,
laboratory experiments can be used to determine the ways in which a
certain disease progresses in the human body, and this can be used
to help predict how effective a drug might be in stopping, or
slowing down, the progression of a disease.
[0007] Unfortunately, the current, state-of-the-art approaches have
some serious limitations. There are problems, for example, in
dealing with heterogeneous and complex systems, in that the models
fail by insufficiently characterizing the systems. Predicting the
flow of rainfall through the ground to an adjacent stream, for
example, involves a complex and heterogeneous combination of media
types in the ground. The variations throughout the media make it
difficult-to-impossible to apply Darcy's Law accurately in such a
complex system. And, although possible in theory, accurately
identifying and modeling such complex and heterogeneous media
throughout the system is often considered cost prohibitive, as well
as time prohibitive in many cases. As the systems become more
mechanistically complex, of course, we need more empirical
relationships and a more complex model. Hydraulic conductivity
mechanisms may not be enough, for example, as there can also be
chemical reaction mechanisms affecting the movement of the fluids.
Human biological systems are examples of highly complex systems
that are difficult to scale from the lab to the human body, as
measurements that can be taken in the lab may not be obtainable in
the human body, for example. In predicting the response of a tumor
to a drug, for example, measuring in vitro or ex vivo tumor size
and growth in small time scales is one thing, but getting such in
vivo measurements can be difficult-to-impossible. In addition, a
system may have nonlinearities that need to be addressed, requiring
further and often futile attempts at adjusting the mechanistic
model. Moreover, current models often cannot map input properties
to model parameters. This is because they lack the necessary
one-to-one relationships between model parameters and model output.
This lack of specificity results in an ambiguity between model
parameters and output that makes it impossible to get unique
input-response relationships, such that the same input can produce
a wide range of responses, or many different inputs could produce
the same response.
[0008] Accordingly, one of skill will appreciate a data-based,
non-mechanistic, differential-equation-free approach for predicting
a particular response of a system to a given input. In particular,
one of skill will appreciate having the ability to (i) reduce the
cost of research and development by offering an accurate modeling
of heterogeneous and complex physical systems; (ii) reduce the cost
of creating such systems and methods by simplifying the modeling
process; (iii) accurately capture and model inherent nonlinearities
in cases where sufficient knowledge does not exist to a priori
build a model and its parameters; and, (iv) provide one-to-one
relationships between model parameters and model outputs,
addressing the problem of the ambiguities inherent in the current,
state-of-the-art systems and methods.
SUMMARY
[0009] The teachings generally relate to a non-mechanistic,
differential-equation-free approach for predicting a particular
response of a system to a given input. In some embodiments, the
teachings are generally directed to a non-compartmental method of
predicting a time-dependent response of a component of a system to
an input into the system. The method can comprise identifying the
system, the component, the input, and the time-dependent response;
wherein, the input includes a set of actual inputs and a test
input, and the time-dependent response includes a set of
time-dependent actual responses and a test response; obtaining the
set of time-dependent actual responses of the component to the set
of actual inputs; and, using the set of actual inputs and the set
of time-dependent actual responses to provide a model for
predicting the test response to the test input, the model
comprising the formula:
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 2 + N / ( kernel ) ] i 1 + ( K - 2 ) - [ N 1 0 + N
1 1 ( kernel ) ] i } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N n
0 + N n 1 ( kernel ) ] i 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel )
] i } ( 9 ) ##EQU00001##
[0010] wherein,
[0011] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0012] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0013] n ranges from 1 to 4;
[0014] K is an overall shifting parameter; and,
[0015] C(t) is the time-dependent response to the test input at
time t;
[0016] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00002## [0017] wherein, C.sub.0 is the initial amount of the
input; K.sub.p is a shifting parameter related to C.sub.0; and,
.alpha..sub.p is shifting and scaling parameter related to
C.sub.0.
[0018] The teachings include a non-compartmental method of
predicting a time-dependent response of a component of a mammalian
system to an input into the system. And, it should be appreciated
that the response can be measured in vivo, in vitro, or ex vivo, in
some embodiments. The methods can also comprise selecting a
component of the system, the component selected from the group
consisting of a cell, a tissue, an organ, a DNA, a virus, a
protein, an antibody, a bacteria; selecting a set of actual inputs,
the set of actual inputs having an element selected from the group
consisting of a DNA, a virus, a protein, an antibody, a bacteria, a
chemical, a dietary supplement, a nutrient, and a drug; obtaining a
set of time-dependent actual responses of the component to the set
of actual inputs; and, using the set of actual inputs and the set
of time-dependent actual responses to provide a model for
predicting a test response to a test input, the model comprising
the formula
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 2 + N / ( kernel ) ] i 1 + ( K - 2 ) - [ N 1 0 + N
1 1 ( kernel ) ] i } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N n
0 + N n 1 ( kernel ) ] i 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel )
] i } ( 9 ) ##EQU00003##
[0019] wherein,
[0020] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0021] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0022] n ranges from 1 to 4;
[0023] K is an overall shifting parameter; and,
[0024] C(t) is the time-dependent response to the test input at
time t;
[0025] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00004## [0026] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0.
[0027] In some embodiments, the teachings are directed to a device
for predicting a time-dependent response of a component of a
physical system to an input into the system. In these embodiments,
the device can comprise a processor; a database for storing a set
of actual input data, a set of time-dependent actual response data,
test input data, and time-dependent test response data on a
non-transitory computer readable medium; an enumeration engine on a
non-transitory computer readable medium to parameterize a
non-compartmental model for predicting a test response to a test
input, the non-compartmental model comprising the formula
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00005##
[0028] wherein,
[0029] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0030] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0031] n ranges from 1 to 4;
[0032] K is an overall shifting parameter; and,
[0033] C(t) is the time-dependent response to the test input at
time t;
[0034] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00006## [0035] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0; and, a transformation module on a non-transitory computer
readable medium to transform the test data into the time-dependent
response data using the non-compartmental model.
[0036] The systems can be virtually any physical or non-physical
system known to one of skill in which that person of skill may want
to predict a particular response of the system to a given input. In
some embodiments, the system can be an environmental system, and
the component can be selected from the group consisting of air,
water, and soil. In some embodiments, the system can be a mammal,
and the component can be selected from the group consisting of a
cell, a tissue, an organ, a DNA, a virus, a protein, an antibody, a
bacteria. In some embodiments, the system can be a chemical system,
a biological system, a mechanical system, an electrical system, a
financial system, a sociological system, a political system, or a
combination thereof. As such, the teachings provided herein include
general methods of predicting a particular response of any such
system to a given input. For example, a biological system can have
a biological input, a mechanical system can have a mechanical data
input, an electrical system can have a relative electrical data
input, a financial system can have a relative financial data input,
a sociological system can have a relative sociological data input,
a political system can have a relative political data input, and
the like.
[0037] In some embodiments, the teachings are directed to a device
for predicting a time-dependent response of a component of a
mammalian system to an input into the system. In these embodiments,
the device can comprise a processor; a database for storing a set
of actual input data, a set of time-dependent actual response data,
test input data, and time-dependent test response data on a
non-transitory computer readable medium; an enumeration engine on a
non-transitory computer readable medium to parameterize a
non-compartmental model for predicting a test response to a test
input, the non-compartmental model comprising the formula
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00007##
[0038] wherein,
[0039] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0040] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0041] n ranges from 1 to 4;
[0042] K is an overall shifting parameter; and,
[0043] C(t) is the time-dependent response to the test input at
time t;
[0044] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00008## [0045] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0; and, a transformation module on a non-transitory computer
readable medium to transform the test data into the time-dependent
response data using the non-compartmental model.
[0046] Any desired component known to one of skill can be used, in
which the desired component is a component of interest to the
person of skill. In some embodiments, the component can be blood, a
tumor cell, a virus, a bacteria, or a combination thereof.
[0047] Any desired test response known to one of skill can be used,
in which the desired test response is a response of interest to the
person of skill. In some embodiments, the test response is a
bacterial load, a viral load, a tumor marker, a blood chemistry, or
a combination thereof.
[0048] Any desired set of actual inputs known to one of skill can
be used, in which the desired set of actual inputs are of interest
to the person of skill. In some embodiments, the set of actual
inputs can include a set of dosages of a drug, a set of drugs, or a
combination thereof.
[0049] Any desired input known to one of skill can be used, in
which the desired input is of interest to the person of skill. In
some embodiments, the input is a diabetes drug, and the
time-dependent response can be glucose in the bloodstream.
[0050] The systems, methods, and devices taught herein transform
input data into response data and, as such, can be used to obtain
the time-dependent test response to the test input. And, the
devices taught herein can be in any form, whether handheld,
desktop, intranet, internet, or otherwise cloud-based. In some
embodiments, the device can be a handheld device including, but not
limited to, a PDA, a smartphone, an iPAD, a personal computer, and
the like, including devices that are not intended for any other
substantial use.
BRIEF DESCRIPTION OF THE DRAWINGS
[0051] FIG. 1 shows a general technology platform for systems that
can be used in the practice of the methods taught herein, according
to some embodiments.
[0052] FIG. 2 illustrates a processor-memory diagram to describe
components of a system, according to some embodiments.
[0053] FIG. 3 is a concept diagram illustrating a system taught
herein, according to some embodiments.
[0054] FIG. 4 shows an example of a prior art, simple
two-compartment linear model, with forward (k.sub.f) and reverse
(k.sub.r) reactions between the two compartments as well as
elimination (k.sub.e) from the second compartment, according to
some embodiments.
[0055] FIG. 5 illustrates a flowchart for a non-compartmental
method of predicting a time-dependent response of a component of a
system to an input into the system, according to some
embodiments.
[0056] FIG. 6 shows how a network may be used for the systems and
methods taught herein, in some embodiments.
[0057] FIG. 7 shows a prior art, two-compartment linear model that
was constructed to model the PK behavior of a particular drug,
according to some embodiments.
[0058] FIG. 8 shows the data used to calibrate this model (find
optimal parameter values) a two-compartment linear model that was
constructed to model the PK behavior of a particular drug,
according to some embodiments.
[0059] FIG. 9 shows a linear two-compartment model solute on for
C.sub.p(t) compared to data for the pharmacokinetic modeling,
according to some embodiments.
[0060] FIG. 10 shows the C.sub.p(t) response function compared to
the data for each of the 25 mg, 100 mg, and 400 mg cases, according
to some embodiments.
[0061] FIG. 11 shows the P(t) response function compared to data
for the enzyme reaction modeling, according to some
embodiments.
[0062] FIGS. 12A and 12B illustrate the pharmacokinetic and
pharmacodynamic model as used in predicting viral loads in response
to administration of tenofovir, according to some embodiments.
[0063] FIG. 13 shows a plot of the responses provided using the
systems and methods taught herein as compared to the large-scale
compartment model, according to some embodiments.
[0064] FIG. 14 shows a three-compartment model that is used as a
simple representation for the absorption of a compound between the
intestines and bloodstream for a dosing study, according to some
embodiments.
[0065] FIG. 15 shows the prediction of the bloodstream
concentration vs. time profile for a 1000 mg dose, using both the
linear and systems and methods taught herein, according to some
embodiments.
DETAILED DESCRIPTION
[0066] Non-mechanistic, differential-equation-free approaches for
predicting a particular response of a system to a given input are
provided in the form of systems, methods, and devices. These
approaches are generally directed to a non-compartmental method of
predicting a time-dependent response of a component of a system to
an input into the system. The systems, methods, and devices provide
the ability to (i) reduce the cost of research and development by
offering an accurate modeling of heterogeneous and complex physical
systems; (ii) reduce the cost of creating such systems and methods
by simplifying the modeling process; (iii) accurately capture and
model inherent nonlinearities in cases where sufficient knowledge
does not exist to a priori build a model and its parameters; and,
(iv) provide one-to-one relationships between model parameters and
model outputs, addressing the problem of the ambiguities inherent
in the current, state-of-the-art systems and methods.
[0067] FIG. 1 shows a general technology platform for systems that
can be used in the practice of the methods taught herein, according
to some embodiments. The computer system 100 may be a conventional
computer system and includes a computer 105, I/O devices 110, and a
display device 115. The computer 105 can include a processor 120, a
communications interface 125, memory 130, display controller 135,
non-volatile storage 140, and I/O controller 145. The computer
system 100 may be coupled to or include the I/O devices 150 and
display device 155.
[0068] The computer 105 interfaces to external systems through the
communications interface 125, which may include a modem or network
interface. It will be appreciated that the communications interface
125 can be considered to be part of the computer system 100 or a
part of the computer 105. The communications interface 125 can be
an analog modem, isdn modem, cable modem, token ring interface,
satellite transmission interface (e.g. "direct PC"), or other
interfaces for coupling the computer system 100 to other computer
systems. In a cellular telephone or PDA, for example, this
interface can be a radio interface for communication with a
cellular network and may also include some form of cabled interface
for use with an immediately available personal computer. In a
two-way pager, the communications interface 125 is typically a
radio interface for communication with a data transmission network
but may similarly include a cabled or cradled interface as well. In
a personal digital assistant, for example, the communications
interface 125 typically can include a cradled or cabled interface
and may also include some form of radio interface, such as a
BLUETOOTH or 802.11 interface, or a cellular radio interface.
[0069] The processor 120 may be, for example, a conventional
microprocessor such as an Intel Pentium microprocessor or Motorola
power PC microprocessor, a Texas Instruments digital signal
processor, or a combination of such components. The memory 130 is
coupled to the processor 120 by a bus. The memory 130 can be
dynamic random access memory (DRAM) and can also include static ram
(SRAM). The bus couples the processor 120 to the memory 130, also
to the non-volatile storage 140, to the display controller 135, and
to the I/O controller 145.
[0070] The I/O devices 150 can include a keyboard, disk drives,
printers, a scanner, and other input and output devices, including
a mouse or other pointing device. The display controller 136 may
control in the conventional manner a display on the display device
155, which can be, for example, a cathode ray tube (CRT) or liquid
crystal display (LCD). The display controller 135 and the I/O
controller 145 can be implemented with conventional well known
technology, meaning that they may be integrated together, for
example.
[0071] The non-volatile storage 140 is often a FLASH memory or
read-only memory, or some combination of the two. Any non-volatile
storage can be used. A magnetic hard disk, an optical disk, or
another form of storage for large amounts of data may also be used
in some embodiments, although the form factors for such devices
typically preclude installation as a permanent component in some
devices. Rather, a mass storage device on another computer is
typically used in conjunction with the more limited storage of some
devices. Some of this data is often written, by a direct memory
access process, into memory 130 during execution of software in the
computer 105. One of skill in the art will immediately recognize
that the terms "machine-readable medium," "computer-readable
storage medium," or "computer-readable medium" includes any type of
storage device that is accessible by the processor 120 and also
encompasses a carrier wave that encodes a data signal. Objects,
methods, inline caches, cache states and other object-oriented
components may be stored in the non-volatile storage 140, or
written into memory 130 during execution of, for example, an
object-oriented software program. In some embodiments, these media
can include modules or engines, for example, in which the modules
or engines are complete, in that they can include the software,
hardware, software/hardware combinations, and any other components
recognized by one of skill that enable their operability in their
functions as taught herein.
[0072] The computer system 100 is one example of many possible
different architectures. For example, personal computers based on
an Intel microprocessor often have multiple buses, one of which can
be an I/O bus for the peripherals and one that directly connects
the processor 120 and the memory 130 (often referred to as a memory
bus). The buses are connected together through bridge components
that perform any necessary translation due to differing bus
protocols.
[0073] In addition, the computer system 100 is controlled by
operating system software which includes a file management system,
such as a disk operating system, which is part of the operating
system software. One example of an operating system software with
its associated file management system software is the family of
operating systems known as Windows CE.RTM. and Windows.RTM. from
Microsoft Corporation of Redmond, Wash., and their associated file
management systems. Another example of operating system software
with its associated file management system software is the LINUX
operating system and its associated file management system. Another
example of an operating system software with its associated file
management system software is the PALM operating system and its
associated file management system. The file management system is
typically stored in the non-volatile storage 140 and causes the
processor 120 to execute the various acts required by the operating
system to input and output data and to store data in memory,
including storing files on the non-volatile storage 140. Other
operating systems may be provided by makers of devices, and those
operating systems typically will have device-specific features
which are not part of similar operating systems on similar devices.
Similarly, WinCE.RTM. or PALM operating systems may be adapted to
specific devices for specific device capabilities. Other examples
include Google's ANDROID, Apple's IOS, Nokia's SYMBIAN, RIM's
BLACKBERRY OS, Samsung's BADA, Microsoft's WINDOWS PHONE,
Hewlett-Packard's WEBOS, and embedded Linux distributions such as
MAEMO and MEEGO, and the like.
[0074] The computer system 100 may be integrated onto a single chip
or set of chips in some embodiments, and typically is fitted into a
small form factor for use as a personal device. Thus, it is not
uncommon for a processor, bus, onboard memory, and display/I-O
controllers to all be integrated onto a single chip. Alternatively,
functions may be split into several chips with point-to-point
interconnection, causing the bus to be logically apparent but not
physically obvious from inspection of either the actual device or
related schematics.
[0075] FIG. 2 illustrates a processor-memory diagram to describe
components of a system, according to some embodiments. The system
200 shown in FIG. 2 can include, for example, a processor 205 and a
memory 210 (that can include non-volatile memory), wherein the
memory 210 includes a subject-profile module 215, a database 220,
an offering module 225, a solutions module 230, an integration
engine 235, and an instruction module 240. And, as shown in the
figure, other components can be included.
[0076] The system includes an input device (not shown) operable to
allow a user to enter a personalized subject-profile into the
computing system. Examples of input devices include a keyboard, a
mouse, a data exchange module operable to interact with external
data formats, voice-recognition software, a hand-held device in
communication with the system, and the like.
[0077] The offering module 225 can be embodied in a non-transitory
computer readable storage medium and operable for offering an
opportunity for members of a network community to provide a
submission of input data, response data, or the like, to the
network community. The instruction module 240 can be embodied in a
non-transitory computer readable storage medium and operable for
providing instruction to a member of the network community
regarding a criteria for making a submission of any type, or
interacting within the community in any way.
[0078] The player/challenge database 220 can be embodied in a
non-transitory computer readable storage medium and operable to
store a library of users, user-submissions, input data, response
data, and the like, wherein the database can include any text or
any other media, including data compilations, statistics, and the
like, or whatever other information may be considered useful to the
network community.
[0079] The subject-profile module 215 can be embodied in a
non-transitory computer readable storage medium and operable for
receiving the personalized subject-profile and converting the
personalized subject profile into a user profile. The user profile
can comprise a set of personal statistics for the user, along with
a tracking of the user's participation in the network community, as
well as data regarding the same. As such, this provides a way for
users of similar interests to identify one another and target
community groups, subgroups, and even one-on-one communications.
The input device can allow a user to enter a personalized
subject-profile into a computing system. And, the personalized
subject-profile can comprise a questionnaire designed to obtain
information to be used to produce a personalized file for the
user.
[0080] The transformation module 230 can be embodied in a
non-transitory computer readable storage medium and operable for
parsing input data, response data, other such data, and the like in
the database into categories for use in user analyses. The
enumeration engine 235 can be embodied in a non-transitory computer
readable storage medium and operable to parameterize, for example,
a non-compartmental model for predicting a test response to a test
input.
[0081] It should be appreciated that any of the modules or engines
can have additional functions, and additional modules or engines
can be added to further provide even more functionality. Of course,
the system will have a processor 205. And, the graphical user
interface (not shown) can be used for displaying video, audio,
and/or text to the user.
[0082] In some embodiments, the system further comprises a
parameterization module operable 245 to derive select parameters
such as, for example, display-preference parameters from the user
profile, and the graphical user interface displays select data from
the database 220 in accordance with the user's display preferences
and in the form of the customized set of information subset
options. Select parameters may include user selections,
administrator selections, or some combination thereof. For example,
the user may prefer a select combination of shapes, colors, sound,
and any other of a variety of screen displays and multimedia
options. Furthermore, the selections can be used to personalize and
change the display-preference parameters easily and at any
time.
[0083] In some embodiments, the system further comprises a data
exchange module 250 operable to interact with external data formats
obtained from another database or source, such as a remote memory
source, including any external memory or file known to one of
skill, including other user databases within the network
community.
[0084] In some embodiments, the system further comprises a
messaging module (not shown) operable to allow users to communicate
with other users. The users can email one another, post blogs, or
have instant messaging capability for real-time communications. In
some embodiments, the users have video and audio capability in the
communications, wherein the system implements data streaming
methods known to those of skill in the art.
[0085] The systems taught herein can be practiced with a variety of
system configurations, including personal computers, multiprocessor
systems, microprocessor-based or programmable consumer electronics,
network PCs, minicomputers, mainframe computers, and the like. The
teachings can also be practiced in distributed computing
environments where tasks are performed by remote processing devices
that are linked through a communications network. As such, in some
embodiments, the system further comprises an external computer
connection and a browser program module 270. The browser program
module 270 can be operable to access external data through the
external computer connection.
[0086] FIG. 3 is a concept diagram illustrating a system taught
herein, according to some embodiments. The system 300 contains
components that can be used in a typical embodiment. In addition to
the subject-profile module 215, database 220, the offering module
225, the transformation module 230, the enumeration engine 235, and
the instruction module 240 shown in FIG. 2, the memory 210 of the
device 300 also includes parameterization module 245 and the
browser program module 270 for accessing the external database 320.
The system can include a speaker 352, display 353, and a printer
354 connected directly or through I/O device 350 connected to I/O
backplane 340.
[0087] It should be appreciated that, in some embodiments, the
system can be implemented in a stand-alone device, rather than a
computer system or network, such that the device functions as a
virtual system as provided herein, but does not perform any other
substantially different functions. In figure FIG. 3, for example,
the I/O device 350 connects to the speaker (spkr) 352, display 353,
and microphone (mic) 354, but could also be coupled to other
features. Other features can be added such as, for example, an
on/off button, a start button, an ear phone input, and the like. In
some embodiments, the system can turn on and off through motion.
And, in some embodiments, the systems can include security measures
to protect the user's privacy, integrity of data, or both.
State-of-the-Art Modeling is Complex, Insufficient, and
Ambiguous
[0088] Input-response computer modeling is typically formulated
mathematically by relating the rates of change of species within
the system to amounts of species present in the system. Rates of
change are expressed as first-order derivatives; therefore the
resulting formulation is a system of first-order differential
equations. Running a simulation, or running the model, is simply
solving the system of differential equations. The output of the
simulation are the concentration vs. time curves of each of the
species in the system. The coefficients of the terms in the
differential equations are often referred to as the parameters of
the model. An example of such a system is given below:
.differential. C 1 .differential. t = k 11 C 1 + k 12 C 2 + + k 1 n
C n ##EQU00009## .differential. C 2 .differential. t = k 21 C 1 + k
22 C 2 + + k 2 n C n ##EQU00009.2## ##EQU00009.3## .differential. C
n .differential. t = k n 1 C 1 + k n 2 C 2 + + k nn C n ;
##EQU00009.4##
where, in this example, C.sub.1, C.sub.2, . . . , C.sub.n represent
the concentrations of the n different species in the system and
k.sub.11, k.sub.12, . . . , k.sub.nn are the parameters of the
model. Changing the values of the parameters will change the output
of the model. Proper adjustment of the parameters will yield the
desired output; i.e., concentration curves that match a desired set
of available data. This adjustment of parameters to produce desired
output is referred to as parameter optimization or model
calibration.
[0089] If all of the k.sub.ij's are real-valued constants, then the
system is said to be a linear system of differential equations.
Many physical systems are modeled using linear systems of
differential equations, but there are often cases where a linear
model is insufficient and a nonlinear model is required. In a
nonlinear system of differential equations, at least one of the
k.sub.ij's is a function of one of the C'.sub.is. For a linear
system, the solution for the C'.sub.is as functions of time will be
of the form:
C.sub.i(t)=M.sub.i.sub.1e.sup..beta.i.sup.1.sup.t+M.sub.i.sub.2e.sup..be-
ta.i.sup.2+ . . . +M.sub.i.sub.ne.sup..beta.i.sup.n.sup.t (1)
where, the number of terms n is the same as the number of species
being modeled. Each of the solution variables M.sub.ij and
.beta..sub.ij is a function of the model parameters k.sub.11, . . .
, k.sub.nn. For a linear system, each of the solution functions,
C.sub.i(t), will be a linear function of all the initial values,
C.sub.i(0).
[0090] This approach of setting up a model (system of differential
equations, etc.) with associated parameters that affect the output
(solution functions) is called a mechanistic approach to modeling.
In a mechanistic approach, the model species and parameters can be
constructed to represent actual physiological components
(physiologically-based modeling) or can simply serve as a
sufficient number of mathematical degrees of freedom to allow for
accurate model fits to given data.
[0091] In order to formulate a system of differential equations in
the modeling process, a compartmental approach is often used. That
is, a network of compartments is set up, with connections between
each that specify the rate at which species are transferred between
compartments. Compartmental models can be constructed using linear
or nonlinear reactions between compartments. In linear models,
parameter values are constants. FIG. 1 shows an example of a simple
two-compartment linear model, with forward (k.sub.f) and reverse
(k.sub.r) reactions between the two compartments as well as
elimination (k.sub.e) from the second compartment. In this linear
model, k.sub.f, k.sub.r, and k.sub.e are all real-valued
constants.
[0092] FIG. 4 shows an example of a prior art, simple
two-compartment linear model, with forward (k.sub.f) and reverse
(k.sub.r) reactions between the two compartments as well as
elimination (k.sub.e) from the second compartment, according to
some embodiments.
[0093] The resulting differential equations are:
V 1 .differential. C 1 .differential. t = - k f C 1 + k r C 2
##EQU00010## V 2 .differential. C 2 .differential. t = k f C 1 - (
k r + k e ) C 2 ; ##EQU00010.2##
where, V.sub.1 and V.sub.2 represent the physical volumes of
compartments 1 and 2, respectively. These volumes are often not
known and have to be either physically or mathematically estimated.
The compartmental modeling approach can be, but is not always,
physiologically-based. In a physiologically-based model, each
compartment represents an actual physiological entity, and the
reactions between compartments are based on expert knowledge of the
interactions between the included physiological entities.
[0094] FIG. 4 is an example of a mechanistic approach to
input-response modeling, and the vast majority of input-response
modeling is done using a mechanistic approach. In this approach,
the components of the model--nodes, connections, differential
equations, parameters, etc.--are set up based on knowledge of the
underlying physical mechanisms present in the system. Parameter
values are initially set based on expert knowledge of how certain
components of the system should behave with respect to other
related components. This provides a very useful tool in exploratory
research, where one can examine the effects that result from
turning certain `knobs` or `handles` (parameters) in the model.
There are two serious limitations of this mechanistic approach. The
first is one of sufficiency and the other is one of ambiguity.
[0095] Mechanistic models often lack sufficient content to provide
accurate predictions of input-response relationships, and this is
because current expert knowledge is often lacking in its ability to
fully characterize a system or all of the interactions within a
system. This lack of knowledge might manifest itself in not having
enough compartments in a compartmental model, or in having linear
transfer rates between compartments when in fact the underlying
process is nonlinear. What is often done in these cases is to go
back to the model and arbitrarily add compartments or make certain
reactions nonlinear, in an attempt provide the necessary
mathematical foundation to allow for sufficiently accurate fits to
given data. In this way, many models become
non-physiologically-based when the intent was to build a
physiologically-based model.
[0096] Another significant limitation to the mechanistic approach
comes from the fact that in mechanistic models, the model
parameters are serving as an intermediary between the model inputs
and outputs. The parameters are useful in serving as handles to
affect output, but there is often not a unique mapping between
model inputs and output. That is, there may be more than one way
(or even an infinite number of ways) to achieve a certain output
from a given set of inputs. This ambiguity can be very problematic
when attempting to do things like map the properties of the input
to the output. For example, in a dose-response model, it would be
extremely valuable to be able to map molecular properties of a drug
compound to a particular response within the body. Using a
mechanistic dose-response model, this mapping would have to go from
input to model parameters to output. If there are many different
sets of model parameters that can produce the same output, then it
becomes very difficult, or impossible, to use the parameters as an
intermediary in constructing an effective mapping from input
properties (molecular properties of dose compound) to output
(response within the body).
The Systems and Methods Set-Forth Herein are Simple, Sufficient,
and Unambiguous
[0097] To address the limitations of the current, state-of-the-art,
the teachings set-forth herein include a novel system of modeling
that uses a data-based, non-mechanistic, differential-equation-free
approach for predicting a particular response of a system to a
given input. There is no system of differential equations, yet the
form of the response function is similar to a solution function
obtained from a system of differential equations. Because there is
no system of differential equations, there are no associated "model
parameters." The only unknowns that need to be optimized are the
variables in the response function. This eliminates the potential
ambiguity that is present in using differential equation parameters
as the intermediary between input and output, as is the case in a
mechanistic approach. The response function for this new approach
is an extension of the solution function for a system of linear
differential equations, Equation 1, where the exponential terms are
replaced by terms containing rational functions of exponentials.
The basic form is given by:
C ( t ) = M 0 + M 1 [ 1 - - .alpha. 1 t 1 + ( K - 2 ) - .alpha. 1 t
] + + M n [ 1 - - .alpha. n t 1 + ( K - 2 ) .alpha. n t ] . ( 2 )
##EQU00011##
[0098] If K=ln(2), then the response function (2) reduces to a form
that is equivalent to the linear solution function (1).
[0099] One of the characteristics of a solution function for a
nonlinear system is that the variables M.sub.0, M.sub.1, . . . ,
M.sub.n and .alpha..sub.1, . . . , .alpha..sub.n are all functions
of the initial input condition, or dose. That is, if we define dose
as C.sub.0, then M.sub.0, M.sub.1, . . . , M.sub.n and
.alpha..sub.1, . . . , .alpha..sub.n are all functions of C.sub.0.
In this new formulation, the functions M.sub.0(C.sub.0),
M.sub.1(C.sub.0), . . . , M.sub.n(C.sub.0) and
.alpha..sub.1(C.sub.0), . . . , .alpha..sub.n(C.sub.0) are also
defined using the formulation of Equation (2). These functions are
given by:
M 0 ( C 0 ) = M 0 0 + M 0 1 [ 1 - ( - .alpha. M 0 1 ) C 0 1 + ( K M
0 - 2 ) ( - .alpha. M 0 1 ) C 0 ] + + M 0 a [ 1 - ( - .alpha. M 0 q
) C 0 1 + ( K M 0 - 2 ) ( - .alpha. M 0 q ) C 0 ] ##EQU00012##
##EQU00012.2## M n ( C 0 ) = M n 0 + M n 1 [ 1 - ( - .alpha. M n 1
) C 0 1 + ( K M n - 2 ) ( - .alpha. M n 1 ) C 0 ] + + M n q [ 1 - (
- .alpha. M n q ) C 0 1 + ( K M n - 2 ) ( - .alpha. M n q ) C 0 ]
##EQU00012.3## .alpha. 1 ( C 0 ) = N 1 0 + N 1 1 [ 1 - ( - .alpha.
.alpha. 1 1 ) C 0 1 + ( K .alpha. 1 - 2 ) ( - .alpha. .alpha. 1 1 )
C 0 ] + + N 1 q [ 1 - ( - .alpha. .alpha. 1 q ) C 0 1 + ( K .alpha.
1 - 2 ) ( - .alpha. .alpha. 1 q ) C 0 ] ##EQU00012.4##
##EQU00012.5## .alpha. n ( C 0 ) = N n 0 + N n 1 [ 1 - ( - .alpha.
.alpha. n 1 ) C 0 1 + ( K .alpha. n - 2 ) ( - .alpha. .alpha. n 1 )
C 0 ] + + N n q [ 1 - ( - .alpha. .alpha. n q ) C 0 1 + ( K .alpha.
n - 2 ) ( - .alpha. .alpha. n q ) C 0 ] . ##EQU00012.6##
[0100] The full implementation of this formulation would require
the estimation of a large number of parameters. In many
embodiments, however, a reduced form will be sufficient for
providing accurate models of input-response relationships. In some
embodiments, a less reduced form, or even the full implementation,
may be used.
[0101] The reduced form makes two assumptions. The first is that
the number of terms in the M.sub.0(C.sub.0), M.sub.1(C.sub.0), . .
. , M.sub.n(C.sub.0) and .alpha..sub.1(C.sub.0), . . . ,
.alpha..sub.n(C.sub.0) functions is truncated at 1; i.e., q=1. The
second is that only one a parameter and only one K parameter is
used for all of the M.sub.0(C.sub.0), M.sub.1(C.sub.0), . . . ,
M.sub.n(C.sub.0) and .alpha..sub.1(C.sub.0), . . . ,
.alpha..sub.n(C.sub.0) functions; i.e.,
.alpha..sub.M.sub.0.sup.1=.alpha..sub.M.sub.t.sup.1= . . .
=.alpha..sub.M.sub.n.sup.1=.alpha..sub..alpha..sub.1.sup.1= . . .
=.alpha..sub..alpha..sub.n.sup.1.ident..alpha..sub.p (3)
K.sub.M.sub.0.sup.1=K.sub.M.sub.t.sup.1= . . .
=K.sub.M.sub.n.sup.1=K.sub..alpha..sub.1.sup.1= . . .
=K.sub..alpha..sub.n.sup.1.ident.K.sub.p (4)
[0102] Substituting the relationships (3) and (4) into the
functions M.sub.0(C.sub.0), M.sub.1(C.sub.0), . . . ,
M.sub.n(C.sub.0) and .alpha..sub.1(C.sub.0), . . . ,
.alpha..sub.n(C.sub.0), and truncating those functions at q=1
yields:
M 0 ( C 0 ) = M 0 0 + M 0 1 [ 1 - - .alpha. p C 0 1 + ( K p - 2 ) -
.alpha. p C 0 ] ( 5 ) M n ( C 0 ) = M n 0 + M n 1 [ 1 - - .alpha. p
C 0 1 + ( K p - 2 ) - .alpha. p C 0 ] ( 6 ) .alpha. 1 ( C 0 ) = N 1
0 + N 1 1 [ 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0 ] (
7 ) .alpha. n ( C 0 ) = N n 0 + N n 1 [ 1 - - .alpha. p C 0 1 + ( K
p - 2 ) - .alpha. p C 0 ] ( 8 ) ##EQU00013##
[0103] To simplify, define a kernel function, which is a function
of initial input condition, or dose, C.sub.0
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
##EQU00014##
[0104] Substituting Equations (5)-(8) into Equation (2) yields:
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00015##
[0105] This new form for the response function allows for nonlinear
model behavior as well as time-lagged effects. Theoretically, this
allows for accurate characterization and model description of
complex physical phenomena. The new response function contains n
terms, where n is an arbitrary number and can be set to achieve
desired accuracy.
[0106] The modeling approach using this new formulation is to
estimate the values of K, K.sub.p, .alpha..sub.p, M.sub.0.sup.0, .
. . , M.sub.n.sup.0, M.sub.0.sup.1, . . . , M.sub.n.sup.1,
N.sub.1.sup.0, . . . , N.sub.n.sup.0, and N.sub.1.sup.1, . . . ,
N.sub.n.sup.1 that yield the best fit of response function to
available data. In this approach, there will be much less (ideally
not any) ambiguity between values of response function variables
and goodness of fit between model and data. In other words, if you
define error as the difference between available data and model
prediction, then error as a function of response function variables
will be more convex and contain fewer local minima than the error
as a function of model parameters in the case of a mechanistic
modeling approach.
[0107] A great deal of system information is condensed into the
response function variables of the new formulation. Complex
phenomena such as nonlinear behavior can be described using much
fewer degrees of freedom than is the case in a mechanistic approach
where a large number of model parameters is typically used. This
will reduce the redundancy that often occurs in mechanistic models
using a large number of model parameters. The response variables in
the new formulation can even take into account information that is
not known prior to building a model, but shows up in the form of
response data. Thus, the new formulation avoids the insufficiency
that is often seen in mechanistic models.
[0108] Essentially, the variables in the response function
(Equation (9)) will all be unique functions of the model
parameters, but the reverse is not true. That is, the model
parameters are not necessarily unique functions of response
variables (as will be demonstrated in detail in Example 1).
Therefore, the response variables in the systems and methods taught
herein represent some (unknown) function of model parameters, if
there were model parameters. But because the systems and methods
taught herein allow for nonlinear behavior, for which there are not
analytical solutions, the response variables represent complicated
functions of many different potential model parameters, and
therefore provide sufficiency in the case where sufficient
knowledge does not exist to a priori build the model and its
parameters. This new formulation also removes the ambiguity that
exists in mechanistic modeling approaches, where model parameters
are not unique functions of response variables.
[0109] The optimization of the response variables in the systems
and methods taught herein is even more complicated than the
optimization of the variables in the linear response function given
by Equation (1), and requires a series of unconstrained and
constrained linear and nonlinear optimization procedures (which are
described in more detail in Example 9). It should be noted that if
a linear response function is sufficient, then the optimization of
the response variables in the systems and methods taught herein can
be used, in some embodiments, to yield a response function that is
equivalent to the linear response function given by Equation (1).
Therefore, the systems and methods taught herein will accurately
describe both linear and nonlinear phenomena. Once optimal values
of response variables are obtained for a given system, the model
can be used to yield an accurate prediction of the system's
response to the introduction of an input of interest. The goal of
this method is to provide accurate input-response predictions over
a wide range of scale. For example, this algorithm could be used to
make accurate predictions of responses on the tissue/organ-scale in
the human body based solely on the molecular properties of input
compounds. This could have significant impact in areas such as
absorption-distribution-metabolism-excretion (ADME) prediction in
drug design, as well as drug development in personalized
medicine.
[0110] FIG. 5 illustrates a flowchart for a non-compartmental
method of predicting a time-dependent response of a component of a
system to an input into the system, according to some embodiments.
The method can comprise identifying 505 the system and the
component, identifying 510 the input, and identifying 515 the
time-dependent response; wherein, the input includes a set of
actual 520 inputs and a test 525 input, and the time-dependent
response includes a set of time-dependent actual 530 responses and
a test 535 response; obtaining the set of time-dependent actual
responses of the component to the set of actual inputs; and, using
the set of actual inputs and the set of time-dependent actual
responses to provide 540 a model for predicting the test response
to the test input, the model comprising the formula:
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00016##
[0111] wherein,
[0112] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0113] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0114] n ranges from 1 to 4;
[0115] K is an overall shifting parameter; and,
[0116] C(t) is the time-dependent response to the test input at
time t;
[0117] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00017## [0118] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0.
[0119] The last step in FIG. 5 is using 550 the model for
predictions.
[0120] Non-compartmental methods of predicting a time-dependent
response of a component of a mammalian system to an input into the
system are also provided. In these embodiments, the methods can
comprise selecting a component of the system, the component
selected from the group consisting of a cell, a tissue, an organ, a
DNA, a virus, a protein, an antibody, a bacteria; selecting a set
of actual inputs, the set of actual inputs having an element
selected from the group consisting of a DNA, a virus, a protein, an
antibody, a bacteria, a chemical, a dietary supplement, a nutrient,
and a drug; obtaining a set of time-dependent actual responses of
the component to the set of actual inputs; and, using the set of
actual inputs and the set of time-dependent actual responses to
provide a model for predicting a test response to a test input, the
model comprising the formula
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00018##
[0121] wherein,
[0122] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0123] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0124] n ranges from 1 to 4;
[0125] K is an overall shifting parameter; and,
[0126] C(t) is the time-dependent response to the test input at
time t;
[0127] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00019## [0128] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0.
[0129] Devices for predicting a time-dependent response of a
component of a physical system to an input into the system are
provided. In these embodiments, the device can comprise a
processor; a database for storing a set of actual input data, a set
of time-dependent actual response data, test input data, and
time-dependent test response data on a non-transitory computer
readable medium; an enumeration engine on a non-transitory computer
readable medium to parameterize a non-compartmental model for
predicting a test response to a test input, the non-compartmental
model comprising the formula
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00020##
[0130] wherein,
[0131] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0132] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0133] n ranges from 1 to 4;
[0134] K is an overall shifting parameter; and,
[0135] C(t) is the time-dependent response to the test input at
time t;
[0136] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00021## [0137] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0; and, a transformation module on a non-transitory computer
readable medium to transform the test data into the time-dependent
response data using the non-compartmental model.
[0138] The systems can be virtually any physical or non-physical
system known to one of skill in which that person of skill may want
to predict a particular response of the system to a given input. In
some embodiments, the system can be an environmental system, and
the component can be selected from the group consisting of air,
water, and soil. In some embodiments, the system can be a mammal,
and the component can be selected from the group consisting of a
cell, a tissue, an organ, a DNA, a virus, a protein, an antibody, a
bacteria. In some embodiments, the system can be a chemical system,
a biological system, a mechanical system, an electrical system, a
financial system, a sociological system, a political system, or a
combination thereof. As such, the teachings provided herein include
general methods of predicting a particular response of any such
system to a given input. For example, a biological system can have
a biological input, a mechanical system can have a mechanical data
input, an electrical system can have a relative electrical data
input, a financial system can have a relative financial data input,
a sociological system can have a relative sociological data input,
a political system can have a relative political data input, and
the like.
[0139] In some embodiments, the input into the system can cause a
substantial effect or a negligible effect. The term "negligible
effect" can be used, for example, to mean that the activity does
not increase or decrease more than about 10% when compared to any
one or any combination of the compounds of interest, respectively,
without the other components. In some embodiments, the term
"negligible effect" can be used to refer to a change of less that
10%, less than 9%, less than 8%, less than 7%, less than 6%, less
than 5%, less than 4%, and less than 3%. In some embodiments, the
term "negligible effect" can be used to refer to a change ranging
from about 3% to about 10%, in increments of 1%.
[0140] The effects of the input can be biological, such as in drug
testing or the testing of compositions used in treating a subject.
The compositions tested, for example, can be referred to as
extracts, compositions, compounds, agents, active agents, bioactive
agents, supplements, drugs, and the like. In some embodiments, the
terms "composition," "compound," "agent," "active", "active agent",
"bioactive agent," "supplement," and "drug" can be used
interchangeably and, it should be appreciated that, a "formulation"
can comprise any one or any combination of these. Likewise, in some
embodiments, the composition can also be in a liquid or dry form,
where a dry form can be a powder form in some embodiments, and a
liquid form can include an aqueous or non-aqueous component.
Moreover, the term "bioactivity" can refer to the function of the
compound when administered in any way known to one of skill,
including parenterally or non-parenterally, including orally,
topically, or rectally to a subject. In some embodiments, the term
"target site" can be used to refer to a select location on or in a
subject that could benefit from an administration of a compound. In
some embodiments, a target can include any site of action in which
the agent's activity, such as any therapeutic activity including
anti-hyproliferative activity, antioxidant activity,
anti-inflammatory activity, analgesic activity, and the like, can
serve a benefit to the subject. The target site can be a healthy or
damaged tissue of a subject. As such, the teachings include a
method of administering one or more compounds taught herein to any
healthy or damaged tissue, such as epithelial, connective, muscle,
or nervous tissue, including hematopoietic, dermal, mucosal,
gastrointestinal or otherwise.
[0141] The systems and methods herein can determine the stability
of a composition in a system. In some embodiments, a composition or
formulation can be considered as "stable" if it loses less than 10%
of its original activity. In some embodiments, a composition or
formulation can be considered as stable if it loses less than 5%,
3%, 2%, or 1% of its original activity. In some embodiments, a
composition or formulation can be considered as "substantially
stable" if it loses greater than about 10% of its original
activity, as long as the composition can perform it's intended use
to a reasonable degree of efficacy. In some embodiments, the
composition can be considered as substantially stable if it loses
activity at an amount greater than about 12%, about 15%, about 25%,
about 35%, about 45%, about 50%, about 60%, or even about 70%. The
activity loss can be measured by comparing activity at the time of
packaging to the activity at the time of administration, and this
can include a reasonable shelf life. In some embodiments, the
composition is stable or substantially stable, if it remains useful
for a period ranging from 3 months to 3 years, 6 months to 2 years,
1 year, or any time period therein in increments of about 1
month.
[0142] Moreover, the systems and methods provided herein can be
used in predicting the efficacy of therapeutic treatments. The
terms "treat," "treating," and "treatment" can be used
interchangeably in some embodiments and refer to the administering
or application of the compositions and formulations taught herein,
including such administration as a health or nutritional
supplement, and all administrations directed to the prevention,
inhibition, amelioration of the symptoms, or even a cure of a
condition in a subject. The terms "disease," "condition,"
"disorder," and "ailment" can be used interchangeably in some
embodiments. The term "subject" and "patient" can be used
interchangeably in some embodiments and refer to an animal such as
a mammal including, but not limited to, non-primates such as, for
example, a cow, pig, horse, cat, dog, rat and mouse; and primates
such as, for example, a monkey or a human. As such, the terms
"subject" and "patient" can also be applied to non-human biologic
applications including, but not limited to, veterinary, companion
animals, commercial livestock, and the like.
[0143] In some embodiments, the methods further comprise orally
administering an effective amount of an oral dosage form of a
composition to a subject to systemically treat a disease or
disorder, including any disease or disorder taught herein. In some
embodiments, the methods further comprise orally administering an
effective amount of an oral dosage form of a composition to a
subject as a dietary supplement. In some embodiments, the methods
further comprise orally administering an effective amount of an
oral dosage form of a composition to a subject in combination with
a second drug. In some embodiments, the teachings are directed to a
method of treating an inflammation of a tissue of subject, the
method comprising administering an effective amount of a
composition to a tissue of the subject. In some embodiments, the
teachings are directed to treating a wounded tissue, the method
comprising administering an effective amount of a composition to a
tissue of the subject. In some embodiments, the teachings are
directed to treating a hyperproliferative disorder, such as cancer,
either liquid or solid, the method comprising administering an
effective amount of a composition to a subject in need thereof.
[0144] An "effective amount" of a compound can be used to describe
a therapeutically effective amount or a prophylactically effective
amount. An effective amount can also be an amount that ameliorates
the symptoms of a disease. A "therapeutically effective amount" can
refer to an amount that is effective at the dosages and periods of
time necessary to achieve a desired therapeutic result and may also
refer to an amount of active compound, prodrug or pharmaceutical
agent that elicits any biological or medicinal response in a
tissue, system, or subject that is sought by a researcher,
veterinarian, medical doctor or other clinician that may be part of
a treatment plan leading to a desired effect. In some embodiments,
the therapeutically effective amount should be administered in an
amount sufficient to result in amelioration of one or more symptoms
of a disorder, prevention of the advancement of a disorder, or
regression of a disorder. In some embodiments, for example, a
therapeutically effective amount can refer to the amount of an
agent that provides a measurable response of at least 5%, at least
10%, at least 15%, at least 20%, at least 25%, at least 30%, at
least 35%, at least 40%, at least 45%, at least 50%, at least 55%,
at least 60%, at least 65%, at least 70%, at least 75%, at least
80%, at least 85%, at least 90%, at least 95%, or at least 100% of
a desired action of the composition.
[0145] In cases of the prevention or inhibition of the onset of a
disease or disorder, or where an administration is considered
prophylactic, a prophylactically effective amount of a composition
or formulation taught herein can be used. A "prophylactically
effective amount" can refer to an amount that is effective at the
dosages and periods of time necessary to achieve a desired
prophylactic result, such as prevent the onset of a sunburn, an
inflammation, allergy, nausea, diarrhea, infection, and the like.
Typically, a prophylactic dose is used in a subject prior to the
onset of a disease, or at an early stage of the onset of a disease,
to prevent or inhibit onset of the disease or symptoms of the
disease. A prophylactically effective amount may be less than,
greater than, or equal to a therapeutically effective amount.
[0146] In some embodiments, a therapeutically or prophylactically
effective amount of a composition may range in concentration from
about 0.01 nM to about 0.10 M; from about 0.01 nM to about 0.5 M;
from about 0.1 nM to about 150 nM; from about 0.1 nM to about 500
.mu.M; from about 0.1 nM to about 1000 nM, 0.001 .mu.M to about
0.10 M; from about 0.001 .mu.M to about 0.5 M; from about 0.01
.mu.M to about 150 .mu.M; from about 0.01 .mu.M to about 500 .mu.M;
from about 0.01 .mu.M to about 1000 nM, or any range therein. In
some embodiments, the compositions may be administered in an amount
ranging from about 0.005 mg/kg to about 100 mg/kg; from about 0.005
mg/kg to about 400 mg/kg; from about 0.01 mg/kg to about 300 mg/kg;
from about 0.01 mg/kg to about 250 mg/kg; from about 0.1 mg/kg to
about 200 mg/kg; from about 0.2 mg/kg to about 150 mg/kg; from
about 0.4 mg/kg to about 120 mg/kg; from about 0.15 mg/kg to about
100 mg/kg, from about 0.15 mg/kg to about 50 mg/kg, from about 0.5
mg/kg to about 10 mg/kg, or any range therein, wherein a human
subject is often assumed to average about 70 kg. Moreover, the
systems and methods taught herein can use micro-dosing, which can
include the administration of dosages that are one, two, or perhaps
three orders of magnitude less than the dosages described above, in
some embodiments.
[0147] Any drug activity can be investigated using the systems and
methods taught herein. In some embodiments, the activity can
include, for example, free radical scavenger and antioxidant,
inhibiting lipid peroxidation and oxidative DNA damage;
anti-inflammatory activity; neurological treatments for Alzheimer's
disease (anti-amyloid and other effects), Parkinson's disease, and
other neurological disorders; anti-arthritic treatment;
anti-ischemic treatment; treatments for multiple myeloma and
myelodysplastic syndromes; psoriasis treatments (topically and
orally); cystic fibrosis treatments; treatments for liver injury
and alcohol-induced liver disease; multiple sclerosis treatments;
antiviral treatments, including human immunodeficiency virus (HIV)
therapy; treatments of diabetes; cancer treatments; and, reducing
risk of heart disease; to name a few.
[0148] Any response can be investigated using the systems and
methods taught herein. For example, the amounts of the agents can
be reduced, even substantially, such that the amount of the agent
or agents desired is reduced to the extent that a significant
response is observed from the subject. A "significant response" can
include, but is not limited to, a reduction or elimination of a
symptom, a visible increase in a desirable therapeutic effect, a
faster response to the treatment, a more selective response to the
treatment, or a combination thereof. In some embodiments, the other
therapeutic agent can be administered, for example, in an amount
ranging from about 0.1 .mu.g/kg to about 1 mg/kg, from about 0.5
.mu.g/kg to about 500 .mu.g/kg, from about 1 .mu.g/kg to about 250
.mu.g/kg, from about 1 .mu.g/kg to about 100 .mu.g/kg from about 1
.mu.g/kg to about 50 .mu.g/kg, or any range therein. Combination
therapies can be administered, for example, for 30 minutes, 1 hour,
2 hours, 4 hours, 8 hours, 12 hours, 18 hours, 1 day, 2 days, 3
days, 4 days, 5 days, 6 days, 7 days, 8 days, 9 days, 10 days, 2
weeks, 3 weeks, 4 weeks, 6 weeks, 3 months, 6 months 1 year, any
combination thereof, or any amount of time considered desirable by
one of skill. The agents can be administered concomitantly,
sequentially, or cyclically to a subject. Cycling therapy involves
the administering a first agent for a predetermined period of time,
administering a second agent or therapy for a second predetermined
period of time, and repeating this cycling for any desired purpose
such as, for example, to enhance the efficacy of the treatment. The
agents can also be administered concurrently. The term
"concurrently" is not limited to the administration of agents at
exactly the same time, but rather means that the agents can be
administered in a sequence and time interval such that the agents
can work together to provide additional benefit. Each agent can be
administered separately or together in any appropriate form using
any appropriate means of administering the agent or agents. One of
skill can readily select the frequency, duration, and perhaps
cycling of each concurrent administration.
[0149] As such, in some embodiments, the teachings are directed to
a device for predicting a time-dependent response of a component of
a mammalian system to an input into the system. In these
embodiments, the device can comprise a processor; a database for
storing a set of actual input data, a set of time-dependent actual
response data, test input data, and time-dependent test response
data on a non-transitory computer readable medium; an enumeration
engine on a non-transitory computer readable medium to parameterize
a non-compartmental model for predicting a test response to a test
input, the non-compartmental model comprising the formula
C ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] + [ M 1 0 + M 1 1 ( kernel )
] { 1 - - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N 1 0 +
N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - - [ N
n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) - [ N n 0 + N n 1 ( kernel
) ] t } ( 9 ) ##EQU00022##
[0150] wherein,
[0151] M.sup.0.sub.0, . . . , M.sup.0.sub.n and M.sup.1.sub.0, . .
. , M.sup.1.sub.n are overall scaling parameters;
[0152] N.sup.0.sub.1, . . . , N.sup.0.sub.n and N.sup.1.sub.1, . .
. , N.sup.1.sub.n are exponential scaling parameters;
[0153] n ranges from 1 to 4;
[0154] K is an overall shifting parameter; and,
[0155] C(t) is the time-dependent response to the test input at
time t;
[0156] and,
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
; ##EQU00023## [0157] wherein, C.sub.0 is the initial amount of the
test input; K.sub.p is a shifting parameter related to C.sub.0;
and, .alpha..sub.p is shifting and scaling parameter related to
C.sub.0; and, a transformation module on a non-transitory computer
readable medium to transform the test data into the time-dependent
response data using the non-compartmental model.
[0158] Any desired component known to one of skill can be used, in
which the desired component is a component of interest to the
person of skill. In some embodiments, the component can be blood, a
tumor cell, a virus, a bacteria, or a combination thereof.
[0159] Any desired test response known to one of skill can be used,
in which the desired test response is a response of interest to the
person of skill. In some embodiments, the test response is a
bacterial load, a viral load, a tumor marker, a blood chemistry, or
a combination thereof.
[0160] Any desired set of actual inputs known to one of skill can
be used, in which the desired set of actual inputs are of interest
to the person of skill. In some embodiments, the set of actual
inputs can include a set of dosages of a drug, a set of drugs, or a
combination thereof.
[0161] Any desired input known to one of skill can be used, in
which the desired input is of interest to the person of skill. For
example, the systems, methods, and devices can be used in drug
screening. In some embodiments, the input is a diabetes drug
candidate, and the time-dependent response can be glucose in the
bloodstream. In some embodiments, the input is a cancer drug
candidate, and the time-dependent response can be a cell apoptosis,
tumor size reduction, reduced metastasis. In some embodiments, the
input is an antibiotic drug candidate, and the time-dependent
response can be a bacterial load. In some embodiments, the input is
an antiviral drug candidate, and the time-dependent response can be
a viral load. In some embodiments, the input is an immunomodulatory
drug candidate, and the time-dependent response can be a measure of
an immune response. In some embodiments, the input is an
anti-inflammatory drug candidate, and the time-dependent response
can be an inflammatory response. In some embodiments, the input is
an analgesic drug candidate, and the time-dependent response can be
a pain response.
[0162] The systems, methods, and devices taught herein transform
input data into response data and, as such, can be used to obtain
the time-dependent test response to the test input. And, the
devices taught herein can be in any form, whether handheld,
desktop, intranet, internet, or otherwise cloud-based. In some
embodiments, the device can be a handheld device including, but not
limited to, a PDA, a smartphone, an iPAD, a personal computer, and
the like, including devices that are not intended for any other
substantial use.
[0163] FIG. 6 shows how a network may be used for the systems and
methods taught herein, in some embodiments. FIG. 6 shows several
computer systems coupled together through a network 605, such as
the internet, along with a cellular network and related cellular
devices. The term "internet" as used herein refers to a network of
networks which uses certain protocols, such as the TCP/IP protocol,
and possibly other protocols such as the hypertext transfer
protocol (HTTP) for hypertext markup language (HTML) documents that
make up the world wide web (web). The physical connections of the
internet and the protocols and communication procedures of the
internet are well known to those of skill in the art.
[0164] Access to the internet 605 is typically provided by internet
service providers (ISP), such as the ISPs 610 and 615. Users on
client systems, such as client computer systems 630, 650, and 660
obtain access to the internet through the internet service
providers, such as ISPs 610 and 615. Access to the internet allows
users of the client computer systems to exchange information,
receive and send e-mails, and view documents, such as documents
which have been prepared in the HTML format, for example. These
documents are often provided by web servers, such as web server 620
which is considered to be "on" the internet. Often these web
servers are provided by the ISPs, such as ISP 610, although a
computer system can be set up and connected to the internet without
that system also being an ISP.
[0165] In some embodiments, the system is a web enabled application
and can use, for example, Hypertext Transfer Protocol (HTTP) and
Hypertext Transfer Protocol over Secure Socket Layer (HTTPS). These
protocols provide a rich experience for the end user by utilizing
web 2.0 technologies, such as AJAX, Macromedia Flash, etc. In some
embodiments, the system is compatible with Internet Browsers, such
as Internet Explorer, Mozilla Firefox, Opera, Safari, etc. In some
embodiments, the system is compatible with mobile devices having
full HTTP/HTTPS support, such as IPHONE, ANDROID, SAMSUNG,
POCKETPCs, MICROSOFT SURFACE, video gaming consoles, and the like.
Others may include, for example, IPAD and ITOUCH devices. In some
embodiments, the system can be accessed using a Wireless
Application Protocol (WAP). This protocol will serve the non HTTP
enabled mobile devices, such as Cell Phones, BLACKBERRY devices,
etc., and provides a simple interface. Due to protocol limitations,
the Flash animations are disabled and replaced with Text/Graphic
menus. In some embodiments, the system can be accessed using a
Simple Object Access Protocol (SOAP) and Extensible Markup Language
(XML). By exposing the data via SOAP and XML, the system provides
flexibility for third party and customized applications to query
and interact with the system's core databases. For example, custom
applications could be developed to run natively on APPLE devices,
Java or .Net-enabled platforms, etc. One of skill will appreciate
that the system is not limited to any of the platforms discussed
above and will be amenable to new platforms as they develop.
[0166] The web server 620 is typically at least one computer system
which operates as a server computer system and is configured to
operate with the protocols of the world wide web and is coupled to
the internet. Optionally, the web server 620 can be part of an ISP
which provides access to the internet for client systems. The web
server 620 is shown coupled to the server computer system 625 which
itself is coupled to web content 695, which can be considered a
form of a media database. While two computer systems 620 and 625
are shown in FIG. 6, the web server system 620 and the server
computer system 625 can be one computer system having different
software components providing the web server functionality and the
server functionality provided by the server computer system 625
which will be described further below.
[0167] Cellular network interface 643 provides an interface between
a cellular network and corresponding cellular devices 644, 646 and
648 on one side, and network 605 on the other side. Thus cellular
devices 644, 646 and 648, which may be personal devices including
cellular telephones, two-way pagers, personal digital assistants or
other similar devices, may connect with network 605 and exchange
information such as email, content, or HTTP-formatted data, for
example. Cellular network interface 643 is coupled to computer 640,
which communicates with network 605 through modem interface 645.
Computer 640 may be a personal computer, server computer or the
like, and serves as a gateway. Thus, computer 640 may be similar to
client computers 650 and 660 or to gateway computer 675, for
example. Software or content may then be uploaded or downloaded
through the connection provided by interface 643, computer 640 and
modem 645.
[0168] Client computer systems 630, 650, and 660 can each, with the
appropriate web browsing software, view HTML pages provided by the
web server 620. The ISP 610 provides internet connectivity to the
client computer system 630 through the modem interface 635 which
can be considered part of the client computer system 630. The
client computer system can be, for example, a personal computer
system, a network computer, a web TV system, or other such computer
system.
[0169] Similarly, the ISP 615 provides internet connectivity for
client systems 650 and 660, although as shown in FIG. 6, the
connections are not the same as for more directly connected
computer systems. Client computer systems 650 and 660 are part of a
LAN coupled through a gateway computer 675. While FIG. 6 shows the
interfaces 635 and 645 as generically as a "modem," each of these
interfaces can be an analog modem, isdn modem, cable modem,
satellite transmission interface (e.g. "direct PC"), or other
interfaces for coupling a computer system to other computer
systems.
[0170] Client computer systems 650 and 660 are coupled to a LAN 670
through network interfaces 655 and 665, which can be ethernet
network or other network interfaces. The LAN 670 is also coupled to
a gateway computer system 675 which can provide firewall and other
internet related services for the local area network. This gateway
computer system 675 is coupled to the ISP 615 to provide internet
connectivity to the client computer systems 650 and 660. The
gateway computer system 675 can be a conventional server computer
system. Also, the web server system 620 can be a conventional
server computer system.
[0171] Alternatively, a server computer system 680 can be directly
coupled to the LAN 670 through a network interface 685 to provide
files 690 and other services to the clients 650, 660, without the
need to connect to the internet through the gateway system 675.
[0172] Through the use of such a network, for example, the system
can also provide an element of social networking, whereby users can
contact other users having similar subject-profiles, or user can
contact anyone in the public to forward the personalized
information. In some embodiments, the system can include a
messaging module operable to deliver notifications via email, SMS,
TWITTER, FACEBOOK, LINKEDIN, and other mediums. In some
embodiments, the system is accessible through a portable, single
unit device and, in some embodiments, the input device, the
graphical user interface, or both, is provided through a portable,
single unit device. In some embodiments, the portable, single unit
device is a hand-held device.
[0173] Regardless of the information presented, the system includes
a broader concept of a platform for the research community, whether
corporate, academic, private, or not-for-profit, for example, to
communicate in an engaging way, whether confidential or public. For
example, the systems and methods taught herein can enable
researchers to use a computer/mobile network mobile interface to
propose problems and solutions, offer data, request data, and
otherwise communicate regarding issues of common interest. The
systems and methods presented herein can be considered a
"game-changer" in art of research and development using computer
modeling.
[0174] It should be also appreciated that the methods and displays
presented herein, in some embodiments, are not inherently related
to any particular computer or other apparatus, unless otherwise
noted. Various general purpose systems may be used with programs in
accordance with the teachings herein, or it may prove convenient to
construct a specialized apparatus to perform the methods of some
embodiments. The required structure for a variety of these systems
will be apparent to one of skill given the teachings herein. In
addition, the techniques are not described with reference to any
particular programming language, and various embodiments may thus
be implemented using a variety of programming languages.
Accordingly, the terms and examples provided above are illustrative
only and not intended to be limiting; and, the term "embodiment,"
as used herein, means an embodiment that serves to illustrate by
way of example and not limitation. The following examples are
illustrative of the uses of the present invention. It should be
appreciated that the examples are for purposes of illustration and
are not to be construed as limiting to the invention.
Example 1
Pharmacokinetics Modeling
[0175] The systems and methods taught herein can be used in
pharmacokinetic (PK) models. In this example, a compartmental
approach was used in a PK model to show the advantages of using the
non-mechanistic formulations and modeling approaches taught
herein.
[0176] PK models are often used to describe the fate of substances
administered externally to a living organism. In drug development,
they are typically used to model the concentration of a drug in the
bloodstream after oral, intravenous, or subcutaneous introduction
into the body. PK analysis is performed by non-compartmental or
compartmental methods. Non-compartmental methods estimate the
exposure to a drug by estimating parameters such as area under the
concentration-time curve (AUC), mean residence time, clearance,
elimination half-life, elimination rate constant, peak plasma
concentration (C.sub.max), time to reach C.sub.max, and minimum
inhibitory concentration (MIC). Compartmental methods estimate the
concentration-time graph using kinetic models. The advantage of
compartmental over some non-compartmental analyses is the ability
to predict the concentration at any time. The disadvantage is the
difficulty in developing and validating the proper model.
[0177] 1.1 Compartmental Pharmacokinetics
[0178] FIG. 7 shows a prior art, two-compartment linear model that
was constructed to model the PK behavior of a particular drug,
according to some embodiments. In this example, the first
compartment represents the gastro-intestinal (GI) region and the
second represents plasma.
[0179] The resulting differential equations are:
V i .differential. C i .differential. t = - k f C i + k r C p
##EQU00024## V p .differential. C p .differential. t = k f C i - (
k r + k e ) C p ; ##EQU00024.2##
[0180] where, C.sub.i and C.sub.p are the concentrations of the
drug in the GI and plasma compartments, respectively; V.sub.i and
V.sub.p are the volumes of distribution for the GI and plasma
compartments, respectively; and k.sub.f, k.sub.r, and k.sub.e are
the reaction rate constants. The initial conditions for this model
are C.sub.i(0)=initial dose=C.sub.0, C.sub.p(0)=0.
[0181] The species of interest in this example is the plasma
concentration, C.sub.p. The solution to this system of differential
equations for C.sub.p is:
C p ( t ) = M C 0 ( .beta. 1 t - .beta. 2 t ) ; .beta. 1 >
.beta. 2 where , .beta. 1 = - k f V i - k r + k e V p + ( k f V i +
k r + k e V p ) 2 - 4 k f k e V i V p 2 ( 10 ) .beta. 2 = - k f V i
- k r + k e V p - ( k f V i + k r + k e V p ) 2 - 4 k f k e V i V p
2 ( 11 ) M = k f V p ( k f V i + k r + k e V p ) 2 - 4 k f k e V i
V p ( 12 ) ##EQU00025##
[0182] Note that, regardless of the parameter values, the solution
for C.sub.p(t) is linear with respect to the initial dose; i.e.,
solutions for different initial doses are simply scalar multiplies
of one another.
[0183] FIG. 8 shows the data used to calibrate this model (find
optimal parameter values), a two-compartment linear model that was
constructed to model the PK behavior of a particular drug,
according to some embodiments. Doses of 25 mg, 100 mg, and 400 mg
were administered orally. See, for example, Bergman, A., et al.
Biopharm. Drug Dispos., 28: 307-313 (2007), which is hereby
incorporated herein by reference in its entirety.
[0184] When the solution variables .beta..sub.1, .beta..sub.2 and
Mare optimized to yield the best fit for all of the data, the
resulting optimal values are: [0185] .beta..sub.1=-0.0025 [0186]
.beta..sub.2=-0.0165 [0187] M=13; which gives the solution for any
initial dose as
[0187] C.sub.p(t)=13C.sub.0(e.sup.-0.0025t-e.sup.-0.0165t).
[0188] In this case, the optimized solution variables give the best
fit for the middle dose data, while overestimating the lower-dose
data and underestimating the higher-dose data.
[0189] FIG. 9 shows a linear two-compartment model solute on for
C.sub.p(t) compared to data for the pharmacokinetic modeling,
according to some embodiments. In particular, the model solution
for C.sub.p(t) is compared to the data for each of the 25 mg, 100
mg, and 400 mg cases. It shows that the model provides a good fit
to the 100 mg data, but there is an overestimation of the 25 mg
data and a significant underestimation of the 400 mg data.
[0190] One limitation of the mechanistic modeling approach--the
inability of the linear two-compartment model to accurately model
the fate of the drug over the entire range of dose values; i.e., an
insufficiency. The mechanistic approach lacks the necessary
structure to adequately model the PK of this drug over the entire
range of dose values. In this case, an insufficiency is that one of
reaction rates is non-linear rather than linear. Adding
compartments in this case will not improve the results.
[0191] Another limitation of the mechanistic modeling
approach--ambiguity of model parameters, as shown by the following
analysis: Equations (10)-(12) give expressions for solution
variables (.beta..sub.1, .beta..sub.2, and M) in terms of model
parameters (k.sub.f, k.sub.r, k.sub.e, V.sub.i, and V.sub.p). In
order to find the values of model parameters that correspond to a
given set of optimal values for solution variables, we must find
expressions for model parameters in terms of solution variables.
These expressions are found by enforcing the constraints that all
model parameters be greater than zero.
- .beta. 1 .ltoreq. k f V i .ltoreq. - .beta. 2 ( 13 ) V p = k f M
( .beta. 1 - .beta. 2 ) ( 14 ) k e = .beta. 1 .beta. 2 V i V p kf (
15 ) k r = ( - .beta. 1 - .beta. 2 ) V p - k f V p V i - k e ( 16 )
##EQU00026##
[0192] By choosing any combination of k.sub.f and V.sub.i that
satisfies condition (13), one can then solve for the remaining
parameters V.sub.p, k.sub.e, and k.sub.r using the given solution
variable values (.beta..sub.1, .beta..sub.2, and M) and Equations
(14)-(16). Therefore, in this particular example with
.beta..sub.1=-0.0025, .beta..sub.2=-0.0165, and M=13, the optimal
values for k.sub.f, V.sub.i, V.sub.p, k.sub.e, and k.sub.r are any
that satisfy the following conditions:
0.0025 .ltoreq. k f V i .ltoreq. 0.0165 ( 17 ) V p = 5.49 * k f (
18 ) k e = V i V p 24242 * k f ( 19 ) k r = 0.019 * V p - k f V p V
i - k e ( 20 ) ##EQU00027##
[0193] By choosing any combination of k.sub.f and V.sub.i that
satisfies condition (17), one can then solve for the remaining
parameters V.sub.p, k.sub.e, and k.sub.r using Equations (18)-(20).
Thus, while there is only one set of solution variable values that
result from a given set of model parameter values (Equations
(10)-(12)), there are an infinite number of model parameter values
that can result from a given set of solution variable values
(Equations (13)-(16)). This non-unique solution to
parameter-mapping illustrates the ambiguity that is present in a
mechanistic approach to modeling, where model parameters are used
as intermediaries between inputs and model outputs (solution
functions). This ambiguity makes it difficult-to-impossible to map
input properties to output solutions by way of model
parameters.
[0194] 1.2 Non-Compartmental Pharmacokinetics
[0195] The systems and methods taught herein are non-compartmental
in design. Non-compartmental PK analysis fits concentration-time
curves to available data, and then uses these curves to estimate
parameters such as AUC, half-life, C.sub.max, and time to reach
C.sub.max. The PK parameters can then be used, for example, to
describe the behavior of a drug after it is introduced into the
body.
[0196] The systems and methods taught herein are different than
traditional non-compartmental PK approaches for at least the reason
that traditional approaches use a mathematical formulation similar
to Equation (1), which describes a linear system. The systems and
methods taught herein, for example, are also able to automatically
describe non-linearities in the system and give more accurate fits
to the data. Moreover, there is the problem of non-unique mappings,
which is also an issue with current non-compartmental PK analyses.
Different concentration-time curves can have the same AUC but
different C.sub.max, or the same C.sub.max, but different AUC, for
example. And, different concentration-time curves can have the same
AUC but different shapes, resulting in the time above minimum
concentration being different (different clearance rates). One of
skill will appreciate that such ambiguities make it difficult to
map properties of an input compound to its PK parameters,
significantly impacting the value of PK properties in making
predictions of the behavior of potential drug compounds in a
system.
[0197] The Systems and Methods Taught Herein Yield Predictions that
are More Accurate than Current State-of-the-Art Methods
[0198] Using the systems and methods taught herein, we can
construct a model that yields accurate predictions of the fate of
the drug over the entire range of dose values. The systems and
methods taught herein provide equation (9), as taught herein for
example, which is a three-term model that worked well for this
particular PK example. Optimization of the response variables for
the C.sub.p(t) function gives the following optimized values of the
variables in the response function of equation (9) for the PK
example:
TABLE-US-00001 TABLE 1 K K.sub.p .alpha..sub.p term i M.sub.i.sup.0
M.sub.i.sup.1 N.sub.i.sup.0 N.sub.i.sup.1 2.283 0.150 0.0010 0
-0.028 0.081 -- -- 1 -3.430 -10.433 0.0040 0.0037 2 4.432 10.121
0.0667 0.0274
[0199] It should be appreciated that the systems and methods taught
herein provided a simplified modeling approach, as regardless of
how many compartments or nonlinear reactions might have been
attempted to achieve sufficient accuracy from a mechanistic
approach to this problem, the systems and methods provided herein
were sufficient with only the 13 values shown in Table 1.
[0200] FIG. 10 shows the C.sub.p(t) response function compared to
the data for each of the 25 mg, 100 mg, and 400 mg cases, according
to some embodiments. As seen in FIG. 10, the systems and methods
taught herein use the C.sub.p(t) response function to fit the data
very well, illustrating that the systems and methods taught herein
can accurately capture the inherent nonlinearity and, therefore,
accurately model the fate of the drug over the entire range of dose
values.
[0201] The additional degrees of freedom in the systems and methods
taught herein provided a model that was more accurate than the
compartmental model. In this example, the mechanistic compartment
model contains only five model parameters and therefore involves
fewer degrees of freedom than the systems and methods taught
herein. In contrast, the two compartments, linear reactions, and
five parameters in the compartmental model were not sufficient, as
they did not adequately model the fate of the drug over the entire
range of dose values. One of skill will appreciate that such
current, state-of-the-art models can easily become large and
involve hundreds of parameters. The systems and methods taught
herein, however, provided sufficient accuracy using much fewer
degrees of freedom, reducing the ambiguity that is otherwise
present in the mechanistic approach with its large number of
parameters.
Example 2
Enzyme Reaction Modeling (Non-Linear Kinetics)
[0202] This example models enzymatic reactions, which are
inherently nonlinear in nature. Many input-response models are
constructed using an assumption of linear reaction kinetics, which
is often insufficient, particularly for large-scale and complex
phenomena. One of skill will appreciate that, as shown in the
previous PK example, a linear model may not accurately describe the
fate of a drug over a wide range of input doses.
[0203] Enzyme kinetics is the study of the chemical reactions that
are catalyzed by enzymes. The effects of reaction conditions on
reaction rate are investigated which can reveal the catalytic
mechanism of the enzyme, its role in metabolism, how its activity
is controlled, and how a drug or an agonist might inhibit the
activity. Typically, an enzymatic reaction involves an enzyme E
binding to a substrate S to form a complex ES, which in turn is
converted to a product P and the enzyme. This is represented
schematically as:
E + S k f k r ES .fwdarw. k cat E + P ##EQU00028##
where k.sub.f, k.sub.r, and k.sub.at denote the rate constants.
[0204] Applying the law of mass action, which states that the rate
of a reaction is proportional to the product of the concentrations
of the reactants, gives a system of four non-linear differential
equations that define the rate of change of reactants with time
t:
.differential. [ S ] .differential. t = - k f [ E ] [ S ] + k r [
ES ] ( 21 ) .differential. [ E ] .differential. t = - k f [ E ] [ S
] + k r [ ES ] + k cat [ ES ] ( 22 ) .differential. [ ES ]
.differential. t = k f [ E ] [ S ] - k r [ ES ] - k cat [ ES ] ( 23
) .differential. [ P ] .differential. t = k cat [ ES ] . ( 24 )
##EQU00029##
[0205] In this mechanism, the enzyme E is a catalyst, which only
facilitates the reaction, so its total concentration, free plus
combined, [E]+[ES]=[E].sub.0, is a constant. This conservation law
can also be obtained by adding Equations (22) and (23). This system
is nonlinear because of the products [E][S] that appear.
[0206] If you make the assumption that the concentration of the
intermediate complex does not change on the time-scale of product
formation, then
.differential. [ ES ] .differential. t = 0 k f [ E ] [ S ] = k r [
ES ] + k cat [ ES ] . ##EQU00030##
[0207] Combining this with the enzyme concentration law gives:
[ ES ] = [ E ] 0 [ S ] k r + k cat k f + [ S ] ##EQU00031##
[0208] From Equation (24),
.differential. [ P ] .differential. t = k cat [ ES ] = k cat [ E ]
0 [ S ] k r + k cat k f + [ S ] ##EQU00032##
[0209] If we define the following constants,
V max = k cat [ E ] 0 the maximum reaction velocity K m = k r + k
cat k f the Michaelis constant ; ##EQU00033##
then, we arrive at the Michaelis-Menten model of enzyme
kinetics
.differential. [ P ] .differential. t = V max [ S ] K m + [ S ] , (
25 ) ##EQU00034##
which relates the rate of product formation to the concentration of
substrate.
[0210] State-of-the-Art Michaelis-Menten Models Create Error Due to
Invalid Assumptions
[0211] Michaelis-Menten-type rates are not only used to model
enzyme kinetics but are also used to model other saturable,
nonlinear phenomena. Michaelis-Menten-type rates are often used in
mechanistic compartment modeling to describe the nonlinear rate at
which one species in a system is produced as a function of the
concentration of some other species in the system. For this reason,
it provides a very useful and practical example for comparing the
systems and methods taught herein to typical mechanistic approaches
to modeling nonlinear phenomena. The problem with using
Michaelis-Menten-type rates for applications other than those for
which it was derived is that the assumptions used to derive the
approximation might not be applicable. For example, two assumptions
used in deriving the Michaelis-Menten approximation are 1)
k.sub.cat<<k.sub.f, and 2) E.sub.0 (the initial enzyme
concentration)<<S.sub.0 (the initial substrate
concentration). But these assumptions might not always be valid
when attempting to use a Michaelis-Menten-type rate between two
compartments in a systems biology model, which is often done.
[0212] To illustrate the error involved in using
Michaelis-Menten-type rates when the underlying assumptions might
not be valid, consider a system where k.sub.f=k.sub.r=k.sub.cat=0.2
and E.sub.0=10.0. A Michaelis-Menten approximation (Equation (25))
of this system would have V.sub.max=2.0 and K.sub.m=2.0. It was
found that using the systems and methods taught herein, using
Equation (9) in some embodiments, a two-term model was sufficient
for this particular enzymatic reaction example. Optimization of the
solution variables for the P(t) function gives optimized values of
the variables in Equation 9 as shown in Table 2 for the enzyme
reaction modeling:
TABLE-US-00002 TABLE 2 K K.sub.p .alpha..sub.p term i M.sub.i.sup.0
M.sub.i.sup.1 N.sub.i.sup.0 N.sub.i.sup.1 1.065 2.798 0.1986 0
-0.002 -0.038 -- -- 1 0.992 0.057 0.2352 -0.0866
[0213] FIG. 11 shows the P(t) response function compared to data
for the enzyme reaction modeling, according to some embodiments. As
shown in FIG. 11, the solution for P(t) using the system of
differential equations (Equations (21)-(24)) can be compared to
P(t) obtained using the Michaelis-Menten approximation and to that
obtained using the systems and methods taught herein, for S.sub.0
values of 5.0, 10.0, and 20.0. The symbols represent the solution
for P(t) from the system of differential equations, the dashed line
represents the solution for P(t) using the Michaelis-Menten
approximation, and the solid line represents the solution for P(t)
using the systems and methods taught herein. The lowest lines are
the solutions for the S.sub.0=5.0 case, the middle set of lines are
the solutions for the S.sub.0=10.0 case, and the top lines are the
solutions for the S.sub.0=20.0 case. As can be readily seen from
FIG. 11, the state-of-the-art method of using the Michaelis-Menten
approximation shows a substantially inferior predictive power than
the systems and methods taught herein.
[0214] One of skill will appreciate that the systems and methods
taught herein provide a much more accurate representation for the
solution of the system of differential equations over the entire
range of initial substrate concentrations.
Example 3
Pharmacodynamic Modeling
[0215] This example compares the results of a published
pharmacodynamic model to a model constructed using the systems and
methods taught herein. From this example, one of skill will
appreciate that the systems and methods taught herein provide a
more accurate viral load response prediction than that obtained
using the published, state-of-the-art large-scale compartmental
model which contains many compartments, differential equations,
nonlinear reactions, and parameters.
[0216] While PK models are used to describe the fate of substances
administered externally to a living organism, pharmacodynamic (PD)
models are used to describe the response of some system entity to
the introduction of a substance administered externally. It is
often said that PK models describe what the body does to a drug,
whereas PD models describe what the drug does to the body. In terms
of input-response, PK models describe the response of the input
compound upon introduction into the body, while PD models describe
the response of some other system entity after introduction of a
certain compound. Both are input-response models, but in PD
modeling, the response of interest is a system component that is
different than the input compound. For example, a PD model might
describe the amount of a certain type of infectious bacteria that
is present over time after introduction of a specific antibiotic;
whereas, a PK model would describe the fate of the antibiotic over
time.
[0217] The published model is a PD model designed to predict HIV
viral load response to the administration of the drug tenofovir in
oral doses of 75, 150, 300, and 600 mg. See Duwal, S., et al. PLoS
One, 7(7):e40382 (2012), which is hereby incorporated herein by
reference in its entirety. The published PD model is coupled to a
pharmacokinetic model, a four-compartment model containing both
linear and nonlinear Michaelis-Menten kinetics, and it consists of
a nonlinear system of eight differential equations. As such, the
coupled pharmacokinetic-pharmacodynamic model is a mechanistic
model containing 12 species and 31 free parameters. As a virus
dynamics model, it was used to predict viral loads following
tenofovir treatment in HIV-infected patients.
[0218] FIGS. 12A and 12B illustrate the pharmacokinetic and
pharmacodynamic model as used in predicting viral loads in response
to administration of tenofovir, according to some embodiments. FIG.
12A is a drawing of a pharmacokinetic model of the system, and FIG.
12B is a drawing of a virus dynamics model. In FIG. 12A, D refers
to an input dose of tenofovir disoproxil fumurate (TDF), an
antiviral pro-drug, in a subject. With respect to plasma PK,
C.sub.1 is a compartment that resembles plasma pharmacokinetics,
and C.sub.2 is a compartment for the poorly perfused (peripheral)
tissues in the pharmacokinetic model. With respect to cell PK,
C.sub.cell resembles the concentrations of tenofovir disphosphate
(TFV-DP) in peripheral blood mononuclear cells. Parameters k.sub.12
and k.sub.21 are the rate constants for influx and outflux to/from
the peripheral compartment C.sub.2, and k.sub.a and k.sub.e are the
rates of TFV uptake for the elimination into/out-of C.sub.1,
respectively. F.sub.bio is bioavailability. V.sub.max and k.sub.m
are Michaelis-Menten kinetics parameters, and k.sub.out is the
cellular elimination rate constant of TFV-DP. See, for example,
pages 2 and 3 of Duwal, S., et al. PLoS One, 7(7):e40382
(2012).
[0219] FIG. 12B is coupled to FIG. 12A in that the .beta..sub.T,
.beta..sub.M, CL.sub.T, and CL.sub.M parameters in the
pharmacodynamics model are functions of the C.sub.cell
concentration from the pharmacokinetics model. In FIG. 12B, In
brief, the virus dynamics model comprises T-cells, macrophages,
free non-infectious virus (T.sub.U,M.sub.U,V.sub.NI, respectively),
free infectious virus V.sub.1, and four types of infected cells:
infected T-cells and macrophages prior to proviral genomic
integration (T.sub.1 and M.sub.1, respectively) and infected
T-cells and macrophages after proviral genomic integration (T.sub.2
and M.sub.2, respectively). .lamda..sub.T and .lamda..sub.M are the
birth rates of uninfected T-cells and macrophages, and
.delta..sub.T and .delta..sub.M denote their death rate constants.
The parameters .delta..sub.PIC,T and .delta..sub.PIC,M refer to the
intracellular degradation of essential components of the
pre-integration complex, e.g., by the host cell proteasome, which
return early infected T-cells and macrophages to an uninfected
stage, respectively. Parameters .beta..sub.T and .beta..sub.M
denote the rate of successful virus infection of T-cells and
macrophages in the presence of TFV-DP, respectively, while the
parameters CL.sub.T and CL.sub.M denote the clearance of virus
through unsuccessful infection of T-cells and macrophages in the
presence of TFV-DP. Parameters k.sub.T and k.sub.M are the rate
constants of proviral integration into the host cell's genome and
N.sub.T* and N.sub.M* denote the total number of released
infectious and non-infectious virus from late infected T-cells and
macrophages and N.sub.T and N.sub.M are the rates of release of
infectious virus. The parameters .delta..sub.T1, .delta..sub.T2,
.delta..sub.M1 and .delta..sub.M2 are the death rate constants of
T.sub.1, T.sub.2, M.sub.1, and M.sub.2 cells, respectively. The
free virus (infectious and non-infectious) gets cleared by the
immune system with the rate constant CL. See, for example, pages 3
and 4 of Duwal, S., et al. PLoS One, 7(7):e40382 (2012).
[0220] The complicated modeling shown by FIGS. 12A and 12B can be
simplified using the systems and methods taught herein. It was
found, for example, that using the systems and methods taught
herein, which includes using Equation (9), a three-term model was
sufficient. Optimization of the response variables for the viral
load function gives the following optimized values of the variables
as shown in Table 3:
TABLE-US-00003 TABLE 3 K K.sub.p .alpha..sub.p term i M.sub.i.sup.0
M.sub.i.sup.1 N.sub.i.sup.0 N.sub.i.sup.1 0.233 5.010 0.0211 0 0.76
0.14 -- -- 1 26.21 7.63 0.0007 0.0054 2 -5.24 2.68 -0.0004
0.0131
[0221] FIG. 13 shows a plot of the responses provided using the
systems and methods taught herein as compared to the large-scale
compartment model, according to some embodiments. The published
model (PM) was taken from Duwal, et al. See Duwal, S., et al. PLoS
One, 7(7):e40382 (2012), as described herein. The systems and
methods taught herein are the new model (NM) and are compared to
PM. Dashed and dotted lines represent PM, the predicted median
viral kinetics, using the model of Duwal. The symbols represent
actual data points from the observed viral kinetics, and the solid
lines represent predicted responses using the systems and methods
taught herein, NM. Once daily 75 mg TDF dosing and once daily 300
mg TDF dosing are shown.
[0222] As shown in FIG. 13, the new model is able to accurately
capture the same input-response behavior that is produced by the
larger mechanistic model. This increased level of accuracy is
important not only in dosing studies of tenofovir but also in
creating more accurate predictions of viral load response to test
input compounds other than tenofovir. Surprisingly, the systems and
methods taught herein functioned very well with only 13 response
variables rather than the 31 model parameters used by the
state-of-the-art model. As such, the systems and methods taught
herein are less prone to the ambiguity in model parameter to
solution mapping that is present in a mechanistic model. One of
skill will appreciate this surprising and unexpected control over
such ambiguities, particularly if one were to try to make
predictions of the pharmacodynamic response based on properties of
input compounds.
Example 4
Quantitative Structure-Activity Relationship (QSAR) Predictions
[0223] This example shows that the systems and methods taught
herein can be used to determine quantitative structure-activity
relationships (QSAR), the mapping of molecular structure properties
of an input compound to a response, or activity, within a given
system. QSAR allows one of skill, for example, to (i) summarize a
relationship between chemical structures and biological activity in
a dataset of chemicals; and (ii) predict the activities of new
chemicals. It is this same type of characterization and prediction
that can be obtained with the systems and methods taught herein,
significantly impacting a wide variety of fields, including drug
design and personalized medicine. One of skill will appreciate that
the systems and methods taught herein can be used to relate
properties of an input to a particular response profile and address
the desire to relate the variables of an input-response model (the
model parameters in a mechanistic model or the response function
variables in the systems and methods taught herein, for example) to
properties of the input. Moreover, one of skill will also
appreciate the systems and methods taught herein for their ability
to relate parameters of a dose response model in drug design to the
molecular properties of a proposed drug (input compound). The
accurate mapping of input molecular properties to model parameters
allows the art to input compounds covering a wide range of
molecular properties and get an accurate description of the
resulting response for each. Accordingly, the systems and methods
taught herein provide the basis for an `in silico` screening
process, where one could select an input compound that yields the
most desirable response.
[0224] Mechanistic models lack the necessary one-to-one
relationships between model parameters and model output. As
demonstrated in previous examples, this is why such mechanistic
models are often unable to produce sufficient maps of input
properties to model parameters. This is a problem of "a lack of
specificity," in that it is possible to achieve the same output
from many different sets of model parameters. Unfortunately, this
lack of specificity between parameters and output is a serious
problem in that it becomes impossible to expose unique
input-response relationships. For example, by way of ambiguous
parameters, the same input could produce a wide range of responses,
or many different inputs could produce the same response. The
systems and methods taught herein, however, can reduce or even
eliminate this ambiguity, and allow for more accurate mappings
between input properties and output (response) profiles via the
response function variables.
[0225] Using Molecular Properties to Select Drug Candidates
[0226] Molecular properties are often used to determine if a
chemical compound with a certain pharmacological or biological
activity has properties that would make it a likely orally active
drug in humans. Such properties can include, but are not limited
to, number of hydrogen bond donors, number of hydrogen bond
acceptors, molecular weight, octanol-water partition coefficient,
electrostatic potential, surface charge, surface potential,
density, ionization energy, H.sub.vaporization, H.sub.hydration,
lipophilicity parameter, pK.sub.a, boiling point, refractive index,
dipole moment, reduction potential, ovality, HOMO energy,
polarizability, molecular volume, vdW surface area, molecular
refractivity, hydration energy, surface area, LUMO energy, charges
on individual atoms, solvent accessible surface area, maximum + and
- charge, hardness, Taft's steric parameter, 3D configuration of
atoms, and secondary structure such as helices, beta strands, beta
sheets, coils, and loops. Molecular properties that are more
geometrical in nature are used, for example, to determine if a
chemical compound meets the essential, or desired, structural
parameters for binding with a receptor. Because the systems and
methods taught herein can remove much of the ambiguity between
input properties and response profiles, they will be more likely to
make accurate mappings from biological activity and structural
properties of candidate drug molecules to response profiles. As
such, the systems and methods taught herein can provide an
extremely valuable tool for pre-clinical modeling and prediction of
activity against a given target, or PK-ADME (absorption,
distribution, metabolism, and excretion) properties of candidate
drug compounds.
[0227] The Problem of Ambiguity in Current, State-of-the-Art
Modeling
[0228] To demonstrate the ambiguity that would arise in attempting
to map molecular properties of an input compound to variables in
the response function, consider the pharmacokinetic modeling
problem presented in Example 1. As was shown in that example, there
were an infinite number of model parameter values (k.sub.f,
k.sub.r, k.sub.e, V.sub.i, and V.sub.p) that could yield the
desired values for the variables .beta..sub.1, .beta..sub.2, and M
in the solution function C.sub.p(t) when using a linear,
mechanistic, compartmental modeling approach. A typical QSAR study
of this problem would attempt to map molecular properties of an
input compound to model parameter values. For example, if molecular
weight (W) and partition coefficient (log P) were the predominant
factors in the pharmacokinetic properties of a compound, then one
would attempt to describe the model parameters k.sub.f, k.sub.r,
and k.sub.e as functions of W and log P (it is assumed that
V.sub.i, and V.sub.p are parameter values that would have to be
estimated but would be independent of W and log P). Once such
functions are constructed, the values of the response function
variables (.beta..sub.1, .beta..sub.2, and M) would be directly
determined by the molecular weight and partition coefficient of the
input compound. This is shown mathematically below:
.beta..sub.1=F.sub.1(k.sub.f,k.sub.r,k.sub.e)k.sub.f=G.sub.1(W,log
P)
.beta..sub.2=F.sub.2(k.sub.f,k.sub.r,k.sub.e)k.sub.r=G.sub.2(W,log
P)
M=F.sub.3(k.sub.f,k.sub.r,k.sub.e)k.sub.e=G.sub.3(W,log P)
.beta..sub.1=H.sub.1(W,log P)=F.sub.1(G.sub.1(W,log
P),G.sub.2(W,log P),G.sub.3(W,log P))
.beta..sub.2=H.sub.2(W,log P)=F.sub.2(G.sub.1(W,log
P),G.sub.2(W,log P),G.sub.3(W,log P))
M=H.sub.3(W,log P)=F.sub.3(G.sub.1(W,log P),G.sub.2(W,log
P),G.sub.3(W,log P))
[0229] Therefore, given the molecular weight and partition
coefficient of an input compound, the values of response function
variables could be computed directly, thus giving a complete
time-course pharmacokinetic profile of that compound. Examples of
F.sub.1, F.sub.2, and F.sub.3 functions were given in Example 1,
Equations (10)-(12). Attempting to compute accurate molecular
property to model parameter functions (G.sub.1, G.sub.2, and
G.sub.3 functions) demands a set of input-response data for input
compounds covering a range of molecular properties. This data would
be used to find the optimal function types and function values for
the molecular property to model parameter functions.
[0230] The limitation of this approach comes from the ambiguity
that is present in attempting to construct the molecular property
to model parameter functions. For the sake of simplicity, consider
the case where molecular weight is the only property that affects
response. And consider the same pharmacokinetic problem from
Example 1, where the values .beta..sub.1=-0.0025,
.beta..sub.2=-0.0165, and M=13 were found to provide the best fit
to the given observations of response (the data sets of responses
to given inputs). In that example, expressions were derived for
model parameter values as functions of solution variable values
(Equations (13)-(16)). These expressions showed that for a given
set of .beta..sub.1, .beta..sub.2, and M values, there are an
infinite number of model parameter values that can result. These
expressions also provided bounds for the model parameter values.
Thus, the molecular property to model parameter functions must be
bounded in this case. There are many types of functions that can
provide such bounds, but consider the functional form given in
Equation (2) using only two terms:
G ( W ) = M 0 + M 1 ( 1 - - .sigma. W 1 + c - .sigma. W )
##EQU00035##
[0231] This function is bounded by M.sup.0+M.sup.1 and
M.sup.0-M.sup.1/c. Using this form to define the parameter values
as functions of W gives:
k f = G 1 ( W ) = M 1 0 + M 1 1 ( 1 - - .sigma. 1 W 1 + c 1 -
.sigma. 1 W ) ##EQU00036## k r = G 2 ( W ) = M 2 0 + M 2 1 ( 1 - -
.sigma. 2 W 1 + c 2 - .sigma. 2 W ) ##EQU00036.2## k e = G 3 ( W )
= M 3 0 + M 3 1 ( 1 - - .sigma. 3 W 1 + c 3 - .sigma. 3 W )
##EQU00036.3##
[0232] Equation (13) from Example 1 gives the allowable range for
k.sub.f as a function of the given .beta..sub.1 and .beta..sub.2,
and the calculated V.sub.i.
- .beta. 1 .ltoreq. k f V i .ltoreq. - .beta. 2 - .beta. 1 .ltoreq.
G 1 ( W ) V i .ltoreq. - .beta. 2 ##EQU00037##
[0233] Since G.sub.1(W)/V.sub.i is bounded by
(1/V.sub.i)(M.sub.1.sup.0+M.sub.1.sup.1) and
(1/V.sub.i)(M.sub.1.sup.0-M.sub.1.sup.1/c.sub.1), then
- .beta. 1 .ltoreq. 1 V i ( M 1 0 + M 1 1 ) .ltoreq. - .beta. 2
##EQU00038## and - .beta. 1 .ltoreq. 1 V i ( M 1 0 - M 1 1 c 1 )
.ltoreq. - .beta. 2 M 1 1 ( 1 + 1 c 1 ) < ( .beta. 1 - .beta. 2
) V i - .beta. 1 V i - M 1 1 .ltoreq. M 1 0 .ltoreq. - .beta. 2 V i
+ M 1 1 c 1 ( if M 1 1 < 0 ) - .beta. 1 V i + M 1 1 c 1 .ltoreq.
M 1 0 .ltoreq. - .beta. 2 V i - M 1 1 ( if M 1 1 > 0 ) ;
##EQU00038.2## [0234] Where, .beta..sub.1<0, .beta..sub.2<0,
V.sub.i>0, M.sub.1.sup.0>0, and c.sub.1>0. There are no
constraints placed on (i.e.,
-.infin..ltoreq..sigma..sub.1.ltoreq..infin.).
[0235] Thus, the allowable values for M.sub.1.sup.0, M.sub.1.sup.1,
and c.sub.1, are given by the .beta..sub.1, .beta..sub.2, and
V.sub.i values obtained from fitting the data. Thus, there are an
infinite number of values for the variables (M.sub.1.sup.0,
M.sub.1.sup.1, and c.sub.1) that describes the relationship between
the molecular property W and the model parameter k.sub.f. This will
also be true of the variables describing the relationship between
the molecular property W and the model parameters k.sub.r and
k.sub.e. Depending on the values of .beta..sub.1, .beta..sub.2, and
V.sub.i, the range of allowable values for M.sub.1.sup.0,
M.sub.1.sup.1, and c.sub.1 could be quite large.
[0236] There are, of course, other types of nonlinear functional
forms that could be used for the G(W) functions, but all will
introduce additional parameters and the same type of ambiguity will
result. Therefore, the non-unique mappings that exist between model
parameters and response functions in a mechanistic model will
extend to the mappings between input molecular properties and model
parameters in a QSAR study. This will result in a non-unique
mapping between input molecular properties and output response
functions. Such a non-unique mapping will make it prohibitively
difficult to obtain accurate and effective QSAR predictions.
[0237] Using the Systems and Methods Taught Herein; Eliminating
Mechanistic Modeling Parameters to Reduce Ambiguity
[0238] The approach for QSAR prediction using the systems and
methods taught herein is to start with input-response data for
input compounds having a wide range of molecular properties. For
each compound, various doses would be tested and a model can be
built using the new formulation; i.e., optimal values would be
found for the response function variables K, K.sub.p,
.alpha..sub.p, M.sub.0.sup.0, . . . , M.sub.n.sup.0, M.sub.0.sup.1,
. . . , M.sub.n.sup.1, N.sub.1.sup.0, . . . , N.sub.n.sup.0, and
N.sub.1.sup.1, . . . , N.sub.n.sup.1. A mapping can then be
constructed between the molecular properties of the input compounds
and the optimal values of the response function variables. Once
this mapping, or set of functions, is found, then predictions can
be made as to what type of response will result from introduction
of a given compound into the system. All that would be required as
input is the specific values of the molecular properties of a
compound. These values would then uniquely determine the values of
the response function variables in the systems and methods taught
herein, which would give a time-course profile of the desired
response. Using that time-course profile, one could evaluate the
effectiveness of the input compound in achieving a desired
response. The mapping from molecular properties to response
functions will contain less ambiguity because it eliminates the
intermediate step of mechanistic model parameters. It will be much
more likely to obtain accurate mappings between input molecular
properties and response function variables because of the reduction
in ambiguity. With such mappings, virtual screenings can be
performed to assess the likelihood that a particular input compound
will produce a desired response. A mathematical description of the
QSAR process using the systems and methods taught herein is given
below.
[0239] Using available data, models would be set up for each input
compound based on the observed responses due to various doses. Each
of these models would contain optimal values of the response
function variables K, K.sub.p, .alpha..sub.p, M.sub.0.sup.0, . . .
, M.sub.n.sup.0, M.sub.0.sup.1, . . . , M.sub.n.sup.1,
N.sub.1.sup.0, . . . , N.sub.n.sup.0, and N.sub.1.sup.1, . . . ,
N.sub.n.sup.1. From these models, functions would be fit that map
molecular properties of the input to the variables in the response
function. These functions are analogous to the H functions that
were composed in the mechanistic case (Equations (26)). For
example, if molecular weight, W, and partition coefficient log P
were the only molecular properties considered, there would be 4n+5
functions, where n is the number of the final term in the response
function. Using the optimal values of response function variables
that were derived for each input compound, and the molecular weight
and partition coefficient of each compound, the following functions
(mappings) would be estimated:
K = H 1 ( W , log P ) K p = H 2 ( W , log P ) .alpha. p = H 3 ( W ,
log P ) M 0 0 = H 4 ( W , log P ) N 1 0 = H 2 n + 6 ( W , log P ) M
n 0 = H n + 4 ( W , log P ) N n 0 = H 3 n + 5 ( W , log P ) M 0 1 =
H n + 5 ( W , log P ) N 1 1 = H 3 n + 6 ( W , log P ) M n 1 = H 2 n
+ 5 ( W , log P ) N n 1 = H 4 n + 5 ( W , log P ) ##EQU00039##
[0240] In some embodiments, there would typically be more than two
molecular properties considered, and thus the construction of the H
functions would require higher-dimensional approximations. The
extension to higher dimensions does not significantly alter the
basic approach, but it would require additional computational
cost.
[0241] Once the H functions are established, a direct connection is
made from molecular weight and partition coefficient of a compound
to values of the response function variables. Based on this
connection, when a new compound is considered, its molecular weight
and partition coefficient are used to calculate values of the
variables in the response function. After calculating the values of
the variables in the response function, the result is a full
time-course profile of the response. This profile can then be used
to assess properties such as maximum concentration, time to maximum
concentration, time above a minimum concentration, clearance,
permeability, size of solid tumor, etc.--all of which are very
valuable in systems biology and drug design modeling. These
predictions of response provide an extremely valuable tool by which
large numbers of compounds can be screened very quickly using
high-speed and large-storage computers.
Example 5
Micro-Dosing Studies
[0242] Micro-dosing is a technique for studying the behavior of
drugs in humans through the administration of doses so low
("sub-therapeutic") that they are unlikely to produce whole-body
effects, but high enough to allow the cellular response to be
studied. This allows us to see the PK of the drug with almost no
risk of side effects. This is usually conducted before clinical
Phase I trials to predict whether a drug is viable for that phase
of testing. Human micro-dosing aims to reduce the resources spent
on non-viable drugs and the amount of testing done on animals. As
only micro-dose levels of the drug are used, analytical methods are
limited and extreme sensitivity is needed. Accelerator mass
spectrometry (AMS) is the most common method for micro-dose
analysis. Many of the largest pharmaceutical companies have now
used micro-dosing in drug development, and the use of the technique
has been provisionally endorsed by both the European Medicines
Agency and the Food and Drug Administration. It is expected that
human micro-dosing will gain a secure foothold at the
discovery-preclinical interface driven by early measurement of
candidate drug behavior in humans.
[0243] There are many reasons for potential drug candidates to be
dropped from the pharmaceutical pipeline. A suitable compound must
demonstrate efficacy in the target patient population and have an
acceptable safety profile, requirements which are themselves
extremely demanding. One property of a compound that influences
these and other factors is its PK profile. That is, how efficiently
the compound is absorbed from the site of administration into the
body, how well it is distributed to various sites within the body,
including the site of action, and how rapidly and by what
mechanism(s) it is eliminated, by excretion and metabolism
(ADME--absorption, distribution, metabolism and excretion).
Furthermore, the vast majority of compounds are metabolized,
therefore the fate of the newly formed metabolites must be taken
into account, as many of these are active and some have adverse
side effects. It has been estimated that between 10% and 40% of
potential drugs fail during early clinical trials because of
unsuitable PK features. A poor PK profile may render a compound of
so little therapeutic value as to be not worth developing. For
example, very rapid elimination of a drug from the body would make
it impractical to maintain a compound at a suitable level to have
the desired effect. Clearly, the ideal is to only test in humans
those compounds that have desirable PK properties. However, this is
no trivial task. The problem is that despite significant progress
to date generally, we are still unable to predict the PK profile in
humans of many drug classes from in vitro and computer-based
methods. We are therefore reliant on information gained in animals,
which is based on past experience and has been the most predictive,
to help screen the compounds for those with an appropriate PK
profile. One commonly-applied approach to predicting a human PK
profile based on animal data is allometric scaling, which scales
the animal data to humans, assuming that the only difference among
animals and humans is body size. While body size is an important
determinant of PK, it is certainly not the only feature that
distinguishes humans from animals and, therefore, this simple
approach has been estimated to have less than 60% predictive
accuracy.
[0244] This is where micro-dosing comes in. Clinical testing phases
1 to 3 involve evaluating pharmacological doses generally first in
human volunteers and then in patients for efficacy and safety. The
hypothesis is that micro-dosing will help reduce or replace the
extensive testing in animals of the many compounds that do not have
desirable PK properties in humans and subsequently would be
rejected. But what is a micro-dose, and how could it help? A
micro-dose is so small that it is not intended to produce any
pharmacologic effect when administered to humans and therefore is
also unlikely to cause an adverse reaction. For practical purposes
this dose is defined as 1/100th of that anticipated to produce a
pharmacological effect, or 100 micrograms, whichever is the
smaller. The interest in giving such a micro-dose to humans early
in the drug development process is centered on the view that many
of the processes controlling the PK profile of a compound are
independent of dose level. Therefore, a micro-dose will provide
sufficiently useful PK information to help decide whether it is
worth continuing compound development, which includes, for example,
toxicity testing in animals.
[0245] Computer models can provide valuable analytical tools in the
area of micro-dosing, although there are serious practical hurdles
that must be resolved. As we have seen in the previous examples, a
computer model that does not accurately capture all of the linear
and nonlinear effects within a system will not yield accurate
extrapolations of low-dose results to higher-doses. This is where
the systems and methods herein will have significant positive
impact, where in some embodiments they will include a dose-response
model using several different micro-doses, and then extrapolate
that model to higher, therapeutic doses.
[0246] Testing of the systems and methods taught herein has shown
that in true micro-dosing studies, if the low-dose data used to
construct the model covers a wide enough range, then accurate
predictions can be made for doses that are roughly one order of
magnitude higher. To illustrate this point, consider the case of
intestinal drug absorption. The absorption of drugs via the oral
route is a subject of intense and continuous investigation in the
pharmaceutical industry since good bioavailability implies that the
drug is able to reach the systemic circulation by mouth. The
intestine is an important tissue that regulates the extent of
absorption of orally administered drugs, since the intestine is
involved in first-pass removal. A simple model of intestinal drug
absorption focuses on the permeation of a drug compound across the
epithelial cells that separate the blood vessels and intestines.
The ability of a compound to permeate the cell layer is governed by
diffusive processes as well as cell membrane transporters that can
actively move compounds in the opposite direction of a
concentration gradient. These transporters counteract the
permeation of a compound that would occur by diffusion alone, due
to a concentration gradient.
[0247] The simple model of intestinal drug absorption can be
represented as a three-compartment model, where one compartment
represents the intestine, one the cell layer, and the other the
bloodstream. Forward and reverse diffusion rates can be set up
between the compartments, and the cell membrane transporters can be
represented by a non-reversible rate between the cell and the
intestine. Because the capacity of the cell membrane transporters
is limited, it is a "saturable" process. That is, once the
transporters have become saturated with a particular compound, they
can no longer accept any more and will then continue to transport
at a constant rate. This type of saturable process is nonlinear and
is typically modeled using Michaelis-Menten kinetics. The
compartment model and associated differential equations are given
below.
[0248] FIG. 14 shows a three-compartment model that is used as a
simple representation for the absorption of a compound between the
intestines and bloodstream for a dosing study, according to some
embodiments. The compartment modeling can include the following
equations:
V i .differential. C i .differential. t = - k 1 C i + k 2 C e + ( V
m k m + C e ) C e ##EQU00040## V e .differential. C e
.differential. t = k 1 C i - ( k 2 + k 3 ) C e + k 4 C b - ( V m k
m + C e ) C e ##EQU00040.2## V b .differential. C b .differential.
t = k 3 C e - k 4 C b ; ##EQU00040.3##
where, V.sub.i, V.sub.e, and V.sub.b represent the volumes of
distribution for the intestinal, epithelial cell, and bloodstream
compartments, respectively; k.sub.1, k.sub.2, k.sub.3, and k.sub.4
represent the diffusion rates; and k.sub.m, V.sub.m are the
Michaelis-Menten rate constants for the active transport. For the
purpose of this example, V.sub.i=V.sub.e=V.sub.b=1.0,
k.sub.1=k.sub.2=1.0, k.sub.3=k.sub.4=5.0, k.sub.m=1.0, and
V.sub.m=5.0. The initial concentrations are all 0 except for the
intestinal compartment whose initial condition is equal to the
input dose, C.sub.0.
[0249] In order to perform a dosing study, C.sub.0 values of 1, 10,
and 100 mg were used to construct a model of the absorption of a
compound between the intestines and bloodstream using the new
formulation and a linear model that does not take into account the
nonlinear transport effect. It would be reasonable to expect that,
given these initial values, a linear model might be chosen since
that would provide a fairly accurate fit to the data. The two
models were then used to predict the concentration profile in the
blood compartment that results from an input dose of 1000 (one
order of magnitude higher than the highest dose used to construct
the model). The model results were then compared to the numerical
solution of the system of differential equations (referred to as
the "data"). It was found that using the systems and methods taught
herein (using Equation (9)), a three-term model was sufficient for
this particular example. Optimization of the response variables for
the C.sub.b(t) function gives the following optimized values of the
variables in the response function used by the systems and methods
taught herein for the dosing study, as shown in Table 4:
TABLE-US-00004 TABLE 4 K K.sub.p .alpha..sub.p term i M.sub.i.sup.0
M.sub.i.sup.1 N.sub.i.sup.0 N.sub.i.sup.1 1.684 1.259 0.1225 0
0.0006 -0.0049 -- -- 1 0.0211 0.1927 2.0551 -0.0427 2 0.1049
-0.0003 5.5767 0.0497
[0250] FIG. 15 shows the prediction of the bloodstream
concentration vs. time profile for a 1000 mg dose, using both the
linear and systems and methods taught herein, according to some
embodiments. Both (i) the linear model and (ii) the model of the
systems and methods taught herein are compared to the `data,` or
numerical solution. Both the linear model and the systems and
methods taught herein provide accurate fits to the C.sub.0=1, 10,
and 100 mg data sets (discussed as observed, but not plotted, for
purposes of clarity). But, when you consider the use of the model
to predict the C.sub.0=1000 mg data set, FIG. 15 shows that the
systems and methods taught herein provide a significantly more
accurate fit to the data. This is because the systems and methods
taught herein were able to pick up the nonlinear behavior due to
the saturable membrane transport phenomena. It could be argued that
one could adjust the mechanistic model to reflect the nonlinearity,
but it may not be known a priori where the nonlinear phenomena
occurs and precisely what the nonlinear kinetic rate(s) should be.
The systems and methods taught herein pick up the nonlinearity
automatically and are able to extend that to make accurate
predictions of response due to higher-dose initial conditions.
Example 6
The Use of Surrogates in Modeling: Biomarkers and Metabolomics
[0251] This example shows how the use of surrogates for response
data in modeling to predict a response. Surrogates can include, for
example, biomarkers and metabolomics. If the generation of response
data is prohibitively expensive or time-consuming, for example,
then the use of biomarkers or metabolites allows for the
construction of a model that might otherwise be impossible to
build. For example, if the response of interest is the size of a
solid tumor and we would like to have observations over a
relatively short time scale (minutes-hours), then we would have to
obtain images of the tumor every few minutes or hours, and the cost
of imaging technology in itself could be prohibitive.
[0252] In some embodiments, the term "biomarker" can be used to
refer a biological molecule found in blood, other body fluids, or
tissues that is (i) a sign of a normal or abnormal process, or of a
condition or disease; or, (ii) used to see how well the body
responds to a treatment for a disease or condition. In some
embodiments, A biomarker can also be called "a molecular marker" or
"a signature molecule." In some embodiments, a biomarker can be
diagnostic, for example, to help diagnose a cancer, perhaps before
it is detectable by conventional methods. In some embodiments, a
biomarker can be prognostic, for example, to forecast how
aggressive the disease process is and/or how a patient can expect
to fare in the absence of therapy. And, in some embodiments, a
biomarker can be predictive, for example, to help identify which
patients will respond to which drugs. For example, biomarker can be
used as a surrogate indication of the progression of a tumor, for
example, the measurement of which can be less time-consuming and
costly than the measurement of the tumor size. The
prostate-specific antigen (PSA) is an example of a protein produced
by cells of the prostate gland that can be measured in blood
samples, as prostate cancer can increase PSA levels in the blood,
making PSA a biomarker for prostate tumors. Other examples of
biomarkers include, but are not limited to, C reactive protein
(CRP) for inflammation; high cholesterol for cardiovascular
disease; S100 protein for melanoma; HER-2/neu gene for breast
cancer; BRCA genes for breast and ovarian cancers (BRCA1 and
BRCA2); CA-125 for ovarian cancer; BNP in heart failure, CEA in
colorectal cancer; creatine levels in renal failure; cerebral blood
flow for Alzheimer's disease, stroke, and schizophrenia; high body
temperature for infection; and, the size of brain structures for
Huntington's disease.
[0253] Metabolomics uses metabolites as the intermediates and
products of metabolism, and metabolomics can be used in
input-response modeling, for example, in the area of drug toxicity
assessment. In some embodiments, metabolic profiling of a body
fluid can be used as a surrogate. In some embodiments, metabolic
profiling of urine or blood plasma can be used as a surrogate, for
example, to detect the physiological changes caused by toxic insult
of a chemical. Pharmaceutical companies can use metabolomics in
modeling, for example, to test the toxicity of potential drug
candidates: if a compound can be eliminated before it reaches
clinical trials on the grounds of adverse toxicity, it saves the
enormous expense of the trials. In some embodiments, the metabolite
that is profiled can be an endogenous metabolite produced by the
subject, an exogenous metabolite, or a xenometabolite produced by a
foreign substance such as a drug. In some embodiments, a metabolite
can include, but are not limited to, a lipoprotein or albumin.
[0254] In some embodiments, phenyalanine and tyrosine
concentrations can be used for diagnosing inborn errors of
metabolism (IEM), as they are considered as potentially the most
clinically applicable metabolic biomarkers in combination with
glucose for diabetes diagnosis.
[0255] In some embodiments, metabolites can be used in cancer
studies. For example, a subset of six metabolites (sarcosine,
uracil, kynurenine, glycerol-3-phosphate, leucine and proline) have
shown to be significantly elevated upon disease progression from
benign to clinically localized prostate cancer and metastatic
prostate cancer. One metabolite, sarcosine, has been identified as
a potential candidate for future development in biomarker panels
for early disease detection and aggressivity prediction in prostate
cancer. Components of a mammalian system that can be used in such
studies include, for example, plasma, tissue and urine. Blood serum
can be used, for example, as the component in studies of renal
cancer colorectal cancer, pancreatic cancer, leukemia, ovarian
cancer, and oral cancer. Urine can be used, for example, as the
component in studies of breast cancer, ovarian cancer, cervical
cancer, hepatocellular carcinoma, and bladder cancer. And, saliva
can be used, for example, as the component in studies of oral
cancer, pancreatic cancer, and breast cancer, as well as
periodontal disease.
[0256] In some embodiments, metabolites can be used in
cardiovascular studies. For example, pseudouridine, citric acid,
and the tricarboxylic acid cycle intermediate 2-oxoglutarate can be
used in some embodiments as serum biomarkers. Cardiovascular
conditions can include myocardial ischemia and coronary artery
disease. In some embodiments, dicarboxylacylcarnitines can be used
to predict death/myocardial infarction outcomes. And, in some
embodiments, plasma levels of asymmetric dimethylarginine can be
used to predict major adverse cardiac events in patients with acute
decompensated heart failure and with chronic heart failure.
[0257] All of the previous examples--PK modeling (Example 1),
enzyme reaction modeling (Example 2), PD modeling (Example 3), QSAR
predictions (Example 4), and micro-dosing studies (Example 5)--rely
on response data in order to build a model. Accordingly, surrogates
such as biomarkers and metabolomics can be used as a means to
obtain response data to build a useful model, particularly where
the generation of response data is prohibitively expensive or
time-consuming.
Example 7
Ex Vivo Testing and Personalized Medicine
[0258] Ex vivo testing results can be used to build the models for
use with the systems and methods taught herein. The term "ex vivo"
can be used to refer to experimentation or measurements done in or
on tissue in an environment outside the organism with minimum
alteration of natural conditions. Ex vivo conditions allow
experimentation under more controlled conditions than is possible
in in vivo experiments (in the intact organism), at the expense of
altering the "natural" environment. A primary advantage of using ex
vivo tissues is the ability to perform tests or measurements that
would otherwise not be possible or ethical in living subjects.
Examples of ex vivo testing would be studying the growth of
bacteria in human cells and the associated antimicrobial activity
of potential antibiotics; or, studying the chemosensitivity of
fresh human hematopoietic cells, as well as malignant cells, in
order to select drugs with preferential toxicity to malignant
cells.
[0259] As such, the results of ex vivo testing can be used to
construct input-response models of a particular subject and, based
on that model, make predictions as to what types of therapeutic
compounds might be effective in yielding a desired response within
that subject. These models would have to be able to capture the
complex, nonlinear behavior that is present in cell-, tissue-, and
organ-scale processes. The ability of the systems and methods
taught herein to quickly provide accurate and robust models of
complex, nonlinear phenomena, as demonstrated in the previous
examples, makes them useful in the application of ex vivo testing.
One of skill will appreciated the significant impact in the area of
personalized medicine made possible by the systems and methods
taught herein; i.e., developing drug therapies at a dosage that is
most appropriate for an individual patient.
Example 8
Demand Forecasting
[0260] The systems and methods taught herein have many potential
applications outside of systems biology and drug design. For
example, an important area of application is demand forecast
modeling, where the input could be an individual consumer and the
response is a product or service that individual might choose or
require in the future. These products or services could be, for
example, retail consumer products, health care services, or
internet web sites.
[0261] In the case of QSAR modeling for biological applications, a
model is built using available data and the molecular properties of
an input compound are mapped to the parameters of the model. This
mapping is then used to predict a certain response of interest
based on the molecular properties of the input. In the case of
demand forecasting, a model would be built using available data on
individuals and their observed demand for products and services.
The specific attributes of those individuals could then be mapped
to the parameters of the demand model.
[0262] Using demand forecasting for mapping, one could predict a
future demand for products and services based solely, for example,
on one or more specific attributes of an individual. This type of
modeling, and the predictions they would allow, would be very
valuable for consumer products manufacturers, health care service
providers, and those trying to reach potential customers through
online web services.
Example 9
Implementation of the Algorithms and Optimization of Response
Function Variables
[0263] This example shows the implementation of the algorithms and
optimization of response function variables for use in the systems
and methods taught herein.
[0264] 9.1 Algorithm
[0265] Take the following steps: [0266] 1) Read in data: t.sub.i,
f.sub.i; i=1, . . . , npts, where npts is the total number of
points in all the data sets; [0267] 2) Normalize all data values:
f.sub.i*=f.sub.i/scale, where:
[0267] fscale = { C 0 , if response species is the same as input
species 1 , otherwise ; ##EQU00041## [0268] 3) Transform data:
[0268] t ^ i = t i t max , where t max is the largest t i value (
smallest t i value is assumed to be 0 ) ##EQU00042## f ^ i = f i *
+ f max * - 2 f min * f max * - f min * , where f min * and f max *
are the smallest and largest f i * values ; ##EQU00042.2## [0269]
4) Fit data to the equation:
[0269] M ^ 0 0 + M ^ 0 1 ( kernel ) + [ M ^ 1 0 + M ^ 1 1 ( kernel
) ] { 1 - [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ 1 + ( K - 2 ) [ N ^
1 0 + N ^ 1 1 ( kernel ) ] t ^ } + + [ M ^ n 0 + M ^ n 1 ( kernel )
] { 1 - [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ 1 + ( e K - 2 ) [ N ^
n 0 + N ^ n 1 ( kernel ) ] t ^ } = f ^ ; ( 27 ) ##EQU00043##
where kernel is defined as:
kernel .ident. 1 - - .alpha. p C 0 1 + ( K p - 2 ) - .alpha. p C 0
##EQU00044##
and this is done by minimizing the following objective function for
K, K.sub.p, .alpha..sub.p, M .sub.0.sup.0, . . . , M .sub.n.sup.0,
M .sub.0.sup.1, . . . , M .sub.n.sup.1, N .sub.n.sup.0, . . . , N
.sub.n.sup.0, and N .sub.1.sup.1, . . . , N .sub.n.sup.1(see
section 9.2 for details of the minimization procedure):
F = i = 1 npts { [ M ^ 0 0 + M ^ 0 1 ( kernel ) ] + [ M ^ 1 0 + M ^
1 1 ( kernel ) ] { 1 - [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i 1 + (
K - 2 ) [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i } + + [ M ^ n 0 + M
^ n 1 ( kernel ) ] { 1 - [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i 1 +
( K - 2 ) [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i } - f ^ i } 2 ( 28
) ##EQU00045## [0270] 5) Transform response function variables back
to original space (t.sub.i, f.sub.i):
[0270] M ^ 0 0 = M 0 0 + f max * - 2 f min * f max * - f min *
.revreaction. M 0 0 = M ^ 0 0 ( f max * - f min * ) - f max * + 2 f
min * ##EQU00046## M ^ j 0 = M j 0 f max * - f min * .revreaction.
M j 0 = ( f max * - f min * ) M ^ j 0 ; j = 1 , , n ##EQU00046.2##
M ^ j 1 = M j 1 f max * - f min * .revreaction. M j 1 = ( f max * -
f min * ) M ^ j 1 ; j = 0 , , n ##EQU00046.3## N ^ j 0 = N j 0 t
max .revreaction. N j 0 = N ^ j 0 t max ; j = 1 , , n
##EQU00046.4## N ^ j 1 = N j 1 t max .revreaction. N j 1 = N ^ j 1
t max ; j = 1 , , n ##EQU00046.5##
[0271] This will yield a final response function in terms of time
and initial dose,
f ( t ) = [ M 0 0 + M 0 1 ( kernel ) ] ( fscale ) + [ M 1 0 + M 1 1
( kernel ) ] ( fscale ) { 1 - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + (
K - 2 ) [ N 1 0 + N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 (
kernel ) ] ( fscale ) { 1 - [ N n 0 + N n 1 ( kernel ) ] t 1 + ( K
- 2 ) [ N n 0 + N n 1 ( kernel ) ] t } ; ( 29 ) ##EQU00047##
and, this function will serve as the prediction for a full
time-course response for a given dose. Transforming Equation 27
into Equation 29:
[0272] Substituting relationships from [3.] and [5.] into Equation
(27) gives:
M 0 0 + f max * - 2 f min * f max * - f min * + M 0 1 ( kernel ) f
max * - f min * + [ M 1 0 f max * - f min * + M 1 1 ( kernel ) f
max * - f min * ] { 1 - [ N 1 0 ( t max ) + N 1 1 ( t max ) (
kernel ) ] t t max 1 + ( K - 2 ) [ N 1 0 ( t max ) + N 1 1 ( t max
) ( kernel ) ] t t max } + + [ M n 0 f max * - f min * + M n 1 (
kernel ) f max * - f min * ] { 1 - [ N n 0 ( t max ) + N n 1 ( t
max ) ( kernel ) ] t t max 1 + ( K - 2 ) [ N n 0 ( t max ) + N n 1
( t max ) ( kernel ) ] t t max } = f * + f max * - 2 f min * f max
* - f min * ##EQU00048##
[0273] Cancelling the f*.sub.max-f*.sub.min and t.sub.max terms
gives:
M 0 0 + f max * - 2 f min * + M 0 1 ( kernel ) + [ M 1 0 + M 1 1 (
kernel ) ] { 1 - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) [ N 1
0 + N 1 1 ( kernel ) ] t } + + [ M 1 0 + M 1 1 ( kernel ) ] { 1 - [
N n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) [ N n 0 + N n 1 ( kernel
) ] t } = f * + f max * - 2 f min * ; ##EQU00049##
[0274] The f*.sub.max and 2f*.sub.min terms cancel out, and
f*=f/fscale, which gives:
M 0 0 + M 0 1 ( kernel ) + [ M 1 0 + M 1 1 ( kernel ) ] { 1 - [ N 1
0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 ) [ N 1 0 + N 1 1 ( kernel ) ]
t } + + [ M n 0 + M n 1 ( kernel ) ] { 1 - [ N n 0 + N n 1 ( kernel
) ] t 1 + ( K - 2 ) [ N n 0 + N n 1 ( kernel ) ] t } = f f scale
##EQU00050##
[0275] Multiplying both sides by fscale gives:
[ M 0 0 + M 0 1 ( kernel ) ] ( f scale ) + [ M 1 0 + M 1 1 ( kernel
) ] ( f scale ) { 1 - [ N 1 0 + N 1 1 ( kernel ) ] t 1 + ( K - 2 )
[ N 1 0 + N 1 1 ( kernel ) ] t } + + [ M n 0 + M n 1 ( kernel ) ] (
f scale ) { 1 - [ N n 0 + N n 1 ( kernel ) ] t 1 + ( K - 2 ) [ N n
0 + N n 1 ( kernel ) ] t } = f ; ##EQU00051##
which is equivalent to Equation (29).
[0276] 9.2 Optimization of Response Function Variables
[0277] The optimization procedure consists of a set of nested
optimizations for the response function variables K, K.sub.p,
.alpha..sub.p, and the N .sub.j.sup.0 and N .sub.j.sup.1's: [0278]
Perform a one-dimensional bounded search to find the K value (note:
there is only one K value across multiple data sets within a given
experiment) that minimizes a function whose value is determined by
[0279] Performing a one-dimensional bounded search to find the
K.sub.p value that minimizes a function whose value is determined
by [0280] Performing a one-dimensional bounded search to find the
.alpha..sub.p value that minimizes a function whose value is
determined by [0281] Cycling through a series of two-dimensional
bounded, adaptive grid-refinement searches to find the N
.sub.j.sup.0 and N .sub.j.sup.1 values that minimize the objective
function F, Equation (28)
[0282] To calculate N .sub.j.sup.0 and N .sub.j.sup.1's,
[0283] (i) start with the objective function, Equation (28),
F = i = 1 npts { [ M ^ 0 0 + M ^ 0 1 ( kernel ) ] + [ M ^ 1 0 + M ^
1 1 ( kernel ) ] { 1 - [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i 1 + (
K - 2 ) [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i } + + [ M ^ n 0 + M
^ n 1 ( kernel ) ] { 1 - [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i 1 +
( K - 2 ) [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i } - f ^ i } 2 ;
##EQU00052##
and,
[0284] (ii) solve the system of 2(n+1) linear equations that
results from setting
.differential. F .differential. M ^ j 0 = 0 ; j = 0 , , n
##EQU00053## and ##EQU00053.2## .differential. F .differential. M ^
j 1 = 0 ; j = 0 , , n ##EQU00053.3## .differential. F
.differential. M ^ 0 0 = 2 i = 1 npts { } = 0 ##EQU00053.4##
.differential. F .differential. M ^ 0 1 = 2 i = 1 npts ( kernel ) {
} = 0 .differential. F .differential. M ^ 1 0 = 2 i = 1 npts { } [
1 - [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i 1 + ( K - 2 ) [ N ^ 1 0
+ N ^ 1 1 ( kernel ) ] t ^ i ] = 0 .differential. F .differential.
M ^ 1 1 = 2 i = 1 npts ( kernel ) { } [ 1 - [ N ^ 1 0 + N ^ 1 1 (
kernel ) ] t ^ i 1 + ( K - 2 ) [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^
i ] = 0 ##EQU00053.5## ##EQU00053.6## .differential. F
.differential. M ^ n 0 = 2 i = 1 npts { } [ 1 - [ N ^ n 0 + N ^ n 1
( kernel ) ] t ^ i 1 + ( K - 2 ) [ N ^ n 0 + N ^ n 1 ( kernel ) ] t
^ i ] = 0 .differential. F .differential. M ^ n 1 = 2 i = 1 npts (
kernel ) { } [ 1 - [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i 1 + ( K -
2 ) [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i ] = 0 ; ##EQU00053.7##
where , { } = { [ M ^ 0 0 + M ^ 0 1 ( kernel ) ] + [ M ^ 1 0 + M ^
1 1 ( kernel ) ] { 1 - [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i 1 + (
K - 2 ) [ N ^ 1 0 + N ^ 1 1 ( kernel ) ] t ^ i } + + [ M ^ n 0 + M
^ n 1 ( kernel ) ] { 1 - [ N ^ n 0 + N ^ n 1 ( kernel ) ] t ^ i 1 +
( K - 2 ) [ N n 0 + N n 1 ( kernel ) ] t ^ i } - f i } .
##EQU00053.8##
[0285] Rearranging yields:
M ^ 0 0 i = 1 npts 1 + M ^ 0 1 i = 1 npts ( ) + M ^ 1 0 i = 1 npts
[ 1 ] + M ^ 1 1 i = 1 npts ( ) [ 1 ] + + M ^ n 0 i = 1 npts [ n ] +
M ^ n 1 i = 1 npts ( ) [ n ] = i = 1 npts f ^ i ##EQU00054## M ^ 0
0 i = 1 npts ( ) + M ^ 0 1 i = 1 npts ( ) 2 + M ^ 1 0 i = 1 npts (
) [ 1 ] + M ^ 1 1 i = 1 npts ( ) 2 [ 1 ] + + M ^ n 0 i = 1 npts ( )
[ n ] + M ^ n 1 i = 1 npts ( ) 2 [ n ] = i = 1 npts ( ) f ^ i
##EQU00054.2## M ^ 0 0 i = 1 npts [ 1 ] + M ^ 0 1 i = 1 npts ( ) [
1 ] + M ^ 1 0 i = 1 npts [ 1 ] 2 + M ^ 1 1 i = 1 npts ( ) [ 1 ] 2 +
+ M ^ n 0 i = 1 npts [ 1 ] [ n ] + M ^ n 1 i = 1 npts ( ) [ 1 ] [ n
] = i = 1 npts [ 1 ] f ^ i ##EQU00054.3## M ^ 0 0 i = 1 npts ( ) [
1 ] + M ^ 0 1 i = 1 npts ( ) 2 [ 1 ] + M ^ 1 0 i = 1 npts ( ) [ 1 ]
2 + M ^ 1 1 i = 1 npts ( ) 2 [ 1 ] 2 + + M ^ n 0 i = 1 npts ( ) [ 1
] [ n ] + M ^ n 1 i = 1 npts ( ) 2 [ 1 ] [ n ] = i = 1 npts ( ) [ 1
] f ^ i ##EQU00054.4## ##EQU00054.5## M ^ 0 0 i = 1 npts [ n ] + M
^ 0 1 i = 1 npts ( ) [ n ] + M ^ 1 0 i = 1 npts [ 1 ] [ n ] + M ^ 1
1 i = 1 npts ( ) [ 1 ] [ n ] + + M ^ n 0 i = 1 npts [ n ] 2 + M ^ n
1 i = 1 npts ( ) [ n ] 2 = i = 1 npts [ n ] f ^ i ##EQU00054.6## M
^ 0 0 i = 1 npts ( ) [ n ] + M ^ 0 1 i = 1 npts ( ) 2 [ n ] + M ^ 1
0 i = 1 npts ( ) [ 1 ] [ n ] + M ^ 1 1 i = 1 npts ( ) 2 [ 1 ] [ n ]
+ + M ^ n 0 i = 1 npts ( ) [ n ] 2 + M ^ n 1 i = 1 npts ( ) 2 [ n ]
2 = i = 1 npts ( ) [ n ] f ^ i ; ##EQU00054.7## where , ( ) = (
kernel ) ##EQU00054.8## and [ j ] = [ 1 - [ N ^ j 0 + N ^ j 1 (
kernel ) ] t ^ i 1 + ( K - 2 ) [ N ^ j 0 + N ^ j 1 ( kernel ) ] t ^
i ] , j = 1 , , n ##EQU00054.9##
[0286] This yields the following system of equations, in matrix
form:
[ .SIGMA.1 .SIGMA. ( ) .SIGMA. [ 1 ] .SIGMA. ( ) [ 1 ] .SIGMA. [ n
] .SIGMA. ( ) [ n ] .SIGMA. ( ) .SIGMA. ( ) 2 .SIGMA. ( ) [ 1 ]
.SIGMA. ( ) 2 [ 1 ] .SIGMA. ( ) [ n ] .SIGMA. ( ) 2 [ n ] .SIGMA. [
1 ] .SIGMA. ( ) [ 1 ] .SIGMA. [ 1 ] 2 .SIGMA. ( ) [ 1 ] 2 .SIGMA. [
1 ] [ n ] .SIGMA. ( ) [ 1 ] [ n ] .SIGMA. ( ) [ 1 ] .SIGMA. ( ) 2 [
1 ] .SIGMA. ( ) [ 1 ] 2 .SIGMA. ( ) 2 [ 1 ] 2 .SIGMA. ( ) [ 1 ] [ n
] .SIGMA. ( ) 2 [ 1 ] [ n ] .SIGMA. [ n ] .SIGMA. ( ) [ n ] .SIGMA.
[ 1 ] [ n ] .SIGMA. ( ) [ 1 ] [ n ] .SIGMA. [ n ] 2 .SIGMA. ( ) [ n
] 2 .SIGMA. ( ) [ n ] .SIGMA. ( ) 2 [ n ] .SIGMA. ( ) [ 1 ] [ n ]
.SIGMA. ( ) 2 [ 1 ] [ n ] .SIGMA. ( ) [ n ] 2 .SIGMA. ( ) 2 [ n ] 2
] [ M ^ 0 0 M ^ 0 1 M ^ 1 0 M ^ 1 1 M ^ n 0 M ^ n 1 ] = [ .SIGMA. f
^ i .SIGMA. ( ) f ^ i .SIGMA. [ 1 ] f ^ i .SIGMA. ( ) [ 1 ] f ^ i
.SIGMA. [ n ] f ^ i .SIGMA. ( ) [ n ] f ^ i ] . ##EQU00055##
[0287] When a solution to this system of equations is required, the
C.sub.0, K, K.sub.p, .alpha..sub.p, N .sub.j.sup.0 and N
.sub.j.sup.1 values are known. Therefore, the C.sub.0, K.sub.p, and
.alpha..sub.p values are used to calculate a (kernel) value, and
the (kernel), K, N .sub.j.sup.0 and N .sub.j.sup.1 values are used
to calculate the [j] values, (j=1, . . . , n). Given a (kernel)
value and the [j] values, the linear system of equations (shown
above) can be solved to give all of the M .sub.j.sup.0 and M
.sub.j.sup.1 values (j=0, . . . , n).
* * * * *