U.S. patent number 10,385,385 [Application Number 15/136,774] was granted by the patent office on 2019-08-20 for methods and systems for volume variation modeling in digital pcr.
This patent grant is currently assigned to Life Technologies Corporation. The grantee listed for this patent is LIFE TECHNOLOGIES CORPORATION. Invention is credited to Swapnonil Banerjee, Nivedita Sumi Majumdar.
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United States Patent |
10,385,385 |
Majumdar , et al. |
August 20, 2019 |
Methods and systems for volume variation modeling in digital
PCR
Abstract
A method for performing digital polymerase chain reaction (dPCR)
is provided. The method includes partitioning a biological sample
volume including a plurality of target nucleic acids into a
plurality of partitions, where at least one partition includes at
least one target nucleic acid. The method further includes
determining a model for volume variation of the plurality of
partitions and determining a number of partitions including at
least one target nucleic acid. The method includes generating a
concentration of target nucleic acids in the biological sample
based on the model for volume variation and the fraction of
partitions including at least one target nucleic acid.
Inventors: |
Majumdar; Nivedita Sumi (San
Bruno, CA), Banerjee; Swapnonil (San Bruno, CA) |
Applicant: |
Name |
City |
State |
Country |
Type |
LIFE TECHNOLOGIES CORPORATION |
Carlsbad |
CA |
US |
|
|
Assignee: |
Life Technologies Corporation
(Carlsbad, CA)
|
Family
ID: |
55911101 |
Appl.
No.: |
15/136,774 |
Filed: |
April 22, 2016 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20160312274 A1 |
Oct 27, 2016 |
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Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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62152542 |
Apr 24, 2015 |
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62193932 |
Jul 17, 2015 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
C12Q
1/6851 (20130101); G06F 17/18 (20130101); G16B
40/00 (20190201) |
Current International
Class: |
C12Q
1/6851 (20180101); G06F 17/18 (20060101); G16B
40/00 (20190101) |
Other References
Huggett, J. F. et al. Clinical Chemistry (published online Dec.
2014) vol. 61 No. 1 pp. 79-88. cited by examiner .
Hindson B. J. High-throughput droplet digital PCR system for
absolute quantification of DNA copy number. 2011 Analytical
Chemistry vol. 83 No. 22 pp. 8604-8610. cited by examiner .
Majumdar et al. "Digital PCR Modeling for Maximal Sensitivity,
dynamic range and measurement precision." 2015 PLoS One vol. 10 No.
3 e0118833. cited by examiner .
Pinhiero, L. et al. "Evaluation of a droplet digital polymerase
chain reaction format for DNA copy number quantification" 2012
Analytical Chemistry 84 (2) 1003-1011. cited by examiner .
Kubista, Mikael et al., "DNA Diagnostics Gets Digitized", Drug
Discovery World, Dec. 31, 2011, 77-82. cited by applicant .
Majumdar, Nivedita et al., "Digital PCR Modeling for Maximal
Sensitivity, Dynamic Range and Measurement Precision", PLOS One.
vol. 10, No. 3, Mar. 25, 2015, 1-17. cited by applicant.
|
Primary Examiner: Zeman; Mary K
Attorney, Agent or Firm: Pelaez; Francois A.
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of priority of U.S. Provisional
Patent Application No. 62/152,542, filed on Apr. 24, 2015, and U.S.
Provisional Patent Application No. 62/193,932, filed on Jul. 17,
2015, which are both incorporated herein in their entireties by
reference.
Claims
What is claimed is:
1. A method for performing digital polymerase chain reaction (dPCR)
in a dPCR biological analysis system including a thermal cycler, a
processor, a detector, and a display, the method comprising:
partitioning a biological sample volume including a plurality of
target nucleic acids into a plurality of partitions, wherein at
least one partition includes at least one target nucleic acid;
amplifying, with the thermal cycler, the target nucleic acids in
the plurality of partitions; generating, by the processor, a model
for volume variation of the plurality of partitions based on a
normal distribution of effective load volumes of the biological
sample volume in the plurality of partitions; detecting, by the
detector, the amplified target nucleic acids to determine a number
of partitions including at least one target nucleic acid;
calculating, by the processor, a concentration of target nucleic
acids in the biological sample based on the model for volume
variation and the number of partitions including at least one
target nucleic acid; and displaying, on the display, the
concentration of target nucleic acids in the biological sample.
2. The method of claim 1, wherein the concentration of target
nucleic acids in the biological sample is determined by using the
equation: .times..sigma..times..times..times..function..sigma.
##EQU00024##
3. The method of claim 1, wherein the concentration of target
nucleic acids in the biological sample is determined by using the
equation:
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times. ##EQU00025##
4. The method of claim 1, further comprising: amplifying the target
nucleic acids to determine the number of partitions including at
least one target nucleic acid.
5. The method of claim 1, wherein the model for volume variation
is:
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times. ##EQU00026##
6. The method of claim 1, wherein the model for volume variation
is: P(neg)=exp(1/2.sigma..sup.2C.sup.2-C.sub.v.sub.0).
7. The method of claim 1, wherein the plurality of partitions is a
plurality of reaction sites.
8. The method of claim 1, wherein the plurality of partitions is a
plurality of throughholes.
9. The method of claim 1, wherein the plurality of partitions is a
plurality of droplets.
10. A system for performing digital polymerase chain reaction
(dPCR), the system comprising: a device configured to partition a
biological sample volume including a plurality of target nucleic
acids into a plurality of partitions, wherein at least one
partition includes at least one target nucleic acid; a thermal
cycler to amplify the plurality of target nucleic acids; a detector
for detecting the number of partitions including at least one
target nucleic acid; a memory; and a processor configured to:
generate a concentration of target nucleic acids in the biological
sample based on a model for volume variation and the number of
partitions including at least one target nucleic acid, wherein the
model is based on a normal distribution of effective load volume of
each of the biological sample volume in the plurality of
partitions; and a display to display the concentration of target
nucleic acids in the biological sample.
11. The system of claim 10, wherein the concentration of target
nucleic acids in the biological sample is determined by using the
equation: .times..sigma..times..times..times..function..sigma.
##EQU00027##
12. The system of claim 10, wherein the concentration of target
nucleic acids in the biological sample is determined by using the
equation:
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times. ##EQU00028##
13. The system of claim 10, further comprising: an amplification
apparatus configured to amplify the target nucleic acids to
determine the number of partitions including at least one target
nucleic acid.
14. The system of claim 10, wherein the model for volume variation
is:
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times. ##EQU00029##
15. The system of claim 10, wherein the model for volume variation
is: P(neg)=exp(1/2.sigma..sup.2C.sup.2-C.sub.v.sub.0).
16. The system of claim 10, wherein the plurality of partitions is
a plurality of reaction sites.
17. The system of claim 10, wherein the plurality of partitions is
a plurality of throughholes.
18. The system of claim 10, wherein the plurality of partitions is
a plurality of droplets.
Description
BACKGROUND
The great promise of digital PCR is the potential for unparalleled
precision enabling accurate measurements for genetic
quantification. Target nucleic acid molecules of a sample requiring
quantification are distributed evenly on a digital PCR consumable
with many partitions and subjected to a PCR reaction. Partitions
with template show amplification of the target nucleic acid and
partitions lacking template do not show amplification. The
observations are typically fitted with the Poisson model to predict
the number of molecules present in the sample under
measurement.
A basic assumption of the Poisson model is that molecules are
equally likely to be present in any given partition, implying that
the partitions are of equal size. For microfluidics, functioning at
sub nanoliter volumes, this assumption may easily be violated.
Recently, discussions of the detrimental effects of partition size
variation on digital PCR quantification results, particularly at
high concentrations, have come to light.
SUMMARY
A method for performing digital polymerase chain reaction (dPCR) is
provided. The method includes partitioning a biological sample
volume including a plurality of target nucleic acids into a
plurality of partitions, where at least one partition includes at
least one target nucleic acid. The method further includes
determining a model for volume variation of the plurality of
partitions and determining a number of partitions including at
least one target nucleic acid. The method includes generating a
concentration of target nucleic acids in the biological sample
based on the model for volume variation and the fraction of
partitions including at least one target nucleic acid.
A system for performing digital polymerase chain reaction (dPCR) is
provided. The system includes a device configured to partition a
biological sample volume including a plurality of target nucleic
acids into a plurality of partitions, where at least one partition
includes at least one target nucleic acid. The system further
includes a memory, and a processor configured to determine a number
of partitions including at least one target nucleic acid, and
generate a concentration of target nucleic acids in the biological
sample based on a model for volume variation and the fraction of
partitions including at least one target nucleic acid.
In various embodiments, the concentration of target nucleic acids
in the biological sample is generated by using the equation:
.times..sigma..times..times..times..function..sigma. ##EQU00001##
when using the Variable .lamda. Approximation Model Or by implicit
solution of the equation, when using the Variable .lamda. Full
Fidelity model
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times. ##EQU00002##
DESCRIPTION OF THE FIGURES
FIG. 1 illustrates that volume variation impacts higher
concentration more significantly than lower concentration according
to various embodiments described herein;
FIG. 2A-2D illustrates show increasingly higher partition size
non-uniformities created by assuming a standard deviation of 15%,
25%, 35% and 50% of the mean volumes, respectively according to
various embodiments described herein;
FIG. 3 illustrates an example of quantification results and
prediction error percent using a Poisson calculation and the volume
variation method according to various embodiment described
herein;
FIG. 4 illustrates an exemplary computing system for implementing
various embodiments described herein; and
FIG. 5 illustrates an exemplary distributed network system
according to various embodiments described herein.
DETAILED DESCRIPTION
To provide a more thorough understanding of various embodiments,
the following description sets forth numerous specific details,
such as specific configurations, parameters, examples, and the
like. It should be recognized, however, that such description is
not intended to limit the embodiments described to specific
implementations, configurations, etc. Nor do the descriptions
necessarily provide complete descriptions of the embodiments. As
such, certain aspects, features, components, etc., may be omitted
from the description of the various embodiments for ease of
explanation.
According to various embodiments described herein, a new
quantification model that can be used to accommodate for volumetric
variation and recover the quantification result at high precision
is provided. Monte Carlo simulations are used to demonstrate the
efficacy of the proposed model.
INTRODUCTION
In general, a digital PCR method distributes target molecules into
a large number of partitions such that each partition gets a number
of molecules (0, 1, 2, etc.) theoretically following a Poisson
distribution. Performing PCR on these partitions results in
amplification being detected (positives) in those partitions
containing one or more target molecules and no amplification being
detected (negatives) in those partitions containing zero target
molecules. As positives may contain more than one copy of the
target molecule, a simple summing of the number of positives will
not yield the correct number of target molecules present across the
partitions. Currently, Poisson statistics are widely employed to
estimate the total number of target molecules within the
interrogated sample. For a detailed review of standard digital PCR
modeling and characteristics, refer to [Majumdar, N., Wessel, T.,
Marks, J., "Digital PCR Modeling for Maximal Sensitivity, Dynamic
Range and Measurement Precision," in PLOS One, 2015, pp. 1-17.]
As described below, according to various embodiments described
herein, partitions may include, but are not limited to,
through-holes, sample retainment regions, wells, indentations,
spots, cavities, reaction chambers, and droplets for example.
Furthermore, as used herein, amplification may include thermal
cycling, isothermal amplification, thermal convention, infrared
mediated thermal cycling, or helicase dependent amplification, for
example.
According to various embodiments, detection of a target nucleic
acid may be, but is not limited to, fluorescence detection,
detection of positive or negative ions, pH detection, voltage
detection, or current detection, alone or in combination, for
example.
Implications of a Non-Mono-Dispersed Partition Size for Poisson
Based Quantification
Poisson statistics are founded on the notion that the probability
of any event occurring within a bounded volume depends only upon
the size of the volume itself. Digital PCR systems, by their very
nature, divide interrogated samples into a set of smaller
partitions. It is common practice to make the assumption of
mono-disperse partitioning to allow for the simplification of
assigning a common probability of acquiring any given target
molecule to each of the partitions. Mono-disperse means all of the
partitions are identically sized.
The effect of volume variation among reaction chambers on
estimating concentration was investigated with Monte Carlo
simulations. In this simulation, the average number of molecules in
a partition .lamda. is proportional to the volume of the partition.
A normal distribution of volume variation is assumed with the
standard deviation taken as a percentage of the mean volume. Data
traces 110, 108, 106, 104, and 102 show 0%, 4%, 11%, 16%, and 20%
of the mean volumes, respectively. FIG. 1 shows that volume
variation impacts higher concentration more significantly than
lower concentration. The process will underestimate at higher
concentrations proportional to the degree of variation.
FIG. 1 shows that the effect of volume variation on precision is
significant at higher concentrations. Volume variability is
simulated by assuming a normal distribution of well volumes with
the standard deviation taken as a percentage of the mean well
volume. Volumes of 865 pl for 10,000 partitions were used in the
simulation. Note that concentration at peak precision moves toward
lower concentrations (increasing negative percentage) as volume
variability increases.
Beyond the Poisson Model for Digital PCR Systems
For the Poisson model, the mean number of molecules per partition
(.lamda.) is assumed to be constant. For the volume variation
method according to various embodiments described herein, assume
that the number of average molecules in each reaction .lamda.(v) is
proportional to the volume v caught in it as given in Equation (1).
The constant of proportionality C is defined as the concentration
of the target molecules, which is the quantity of interest in this
measurement exercise. .lamda.(v)=Cv (1)
Consider the joint probability distribution of a partition being
negative and constraining the volume V. One can apply Bayes'
theorem [Bayes, Thomas, and Price, Richard, "An Essay towards
solving a Problem in the Doctrine of Chance. By the late Rev. Mr.
Bayes, communicated by Mr. Price, in a letter to John Canton, A. M.
F. R. S," in Philosophical Transactions of the Royal Society of
London 53 (0): pp. 370-418.] as given in equation (2) to arrive at
the joint probability distribution of a partition being negative
(i.e. containing no molecules) and constraining a volume v.
P(neg,v)=P(neg|v)P(v) (2)
The probability of a partition to not receive any template is given
by equation (3) according to the standard Poisson distribution,
evaluated at number of molecules=0.
.function..times..times. ##EQU00003##
Assume the partition sizes follow a truncated normal distribution
(v.sub.0, .sigma.) with the parameters v.sub.0 and .sigma. as given
by equation 4. A truncated distribution is assumed, as volumes
cannot take negative values
.function..times..times..times..sigma. ##EQU00004## (K is a
constant of proportionality, whose expression is derived in Section
A, and is given by
.sigma..times..pi..times..function..times..sigma. ##EQU00005##
Using equations (3) and (4) in (2), the joint probability of a
negative partition to constrain a volume v is given by equation
(5).
.function..times..times..times..times..sigma. ##EQU00006##
Now, the variable v may be integrated out as in equation (6).
P(neg)=.intg..sub.0.sup..infin.P(neg,v)dv (6)
The final expression for P(neg) is given as in equation (7).
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times. ##EQU00007##
Equation (7) needs to be solved implicitly in order to evaluate
concentration C. The derivation of the expression for P(neg) in
equation (7) is given in the supporting documents, Section A. The
results from this model are referred to as "Variable .lamda. Full
Fidelity" or the "Variable .lamda." model in subsequent
sections.
Alternately, an approximation may also be used, yielding a direct
closed form expression for concentration C, as given in equation
(8). The derivation for equation (8) is given in the supporting
documents, Section B. The results from this model are referred to
as "Variable .lamda. Approximation" model in subsequent
sections.
.times..sigma..times..times..times..function..sigma.
##EQU00008##
The confidence intervals around the quantification by this new
model are created by assuming that the variable log C is normally
distributed. Equation (9) describes how this formalism is valid for
computing confidence intervals.
.times..times..+-..times..times..sigma..times..times..sigma..times..times-
..function..times..function. ##EQU00009## and z is the z-score
associated with the desired confidence interval.
Now, using Equation (7) above, for P
.function..function..times..sigma..times..times..sigma..function..times..-
sigma..times..times..sigma..times..times..times..times..times..times..func-
tion..intg..infin..times..pi..times..times..times..times..times..times..ti-
mes..times..times..function..pi..times..times..times..times..times..functi-
on..times..sigma..times..times..sigma..pi..times..sigma..times..times..sig-
ma..times..sigma. ##EQU00010##
Further manipulations will show that:
.function..function..sigma..times..sigma..times..pi..times..times..sigma.-
.times..times..sigma..function..times..sigma..times..times..sigma.
##EQU00011##
Equation (12) is substituted in equation (11), which is substituted
into equation (10) and back out to equation (9) to yield the
requisite confidence intervals.
Results
Simulation Results: Impact on the Dynamic Range
Monte Carlo simulations were run to demonstrate the remedial
effects of quantification using alternate modeling that accounts
for non-uniform partition size for Poisson processes. These
simulations assume that the average number of target nucleic acids
in a partition is proportional to the volume of the partition. A
normal distribution of volume variation is assumed with the
standard deviation taken as a percentage of the mean volume. The
steps of generating the simulation data are outlined in Section
C.
An alternate model accommodating partition size variation was
proposed by Cowen [Huggett J., Cowen S., and Foy, C. Considerations
for Digital PCR as an Accurate Molecular Diagnostic Tool.sup.2 in
Clinical Chemistry 61:1, 2015, pp. 79-88]. Results from the Poisson
model, the Simon Cowen model, the variable .lamda. approximation
model and the variable .lamda. full fidelity model and are compared
in the set of FIG. 2: FIGS. 2A, 2B, 2C and 2D show increasingly
higher partition size non-uniformities created by assuming a
standard deviation of 15%, 25%, 35% and 50% of the mean volumes,
respectively.
FIG. 2: Remedial Power of Partition Size Non-uniformity Sensitive
Modeling of Poisson Processes on Precision Results from the Poisson
model 206, the Simon Cowen model 208 and the two versions 202 and
204 of the currently proposed volume variable .lamda. model are
compared. The Poisson model 206 is consistently the worst at higher
concentrations for all levels of variations under consideration.
FIGS. 2B, 2C and 2D shows that at the higher levels of standard
deviation=25%, 35% and 50%, the full fidelity volume variable
.lamda. model 202 demonstrates the best performance in recovering
from errors in quantification introduced by non-uniform partition
size.
FIG. 2A shows that at a 15% volume variation, the variable .lamda.
approx. model 204, the volume variable .lamda. full fidelity model
202 and the Simon Cowen model 208 agree closely and all outperform
Poisson model 206 consistently for concentrations above 400
copies/microliter. FIG. 2B shows that at a 25% volume variation,
the volume variable .lamda. approx. model 204, the volume variable
.lamda. full fidelity model 202 and Simon Cowen's model 208 agrees
up to a concentration of approximately 1000 copies/microliter.
Beyond this, the Simon Cowen model 208 and the two versions of the
volume variable .lamda. models 202 and 204 begin to diverge, with
the volume variable .lamda. models 202 and 204 demonstrating the
superior performance. The Poisson model 206 is consistently the
worst at all concentrations beyond 100 copies/microliter. FIGS. 2C
and 2D continues to show that at the 35% and 50% volume variations
respectively, the volume variable .lamda. models 202 and 204
demonstrate the best performance in recovering from errors in
quantification introduced by non-uniform partition size. At 50%,
the volume variable .lamda. approximate model 204 is no longer
usable.
CONCLUSIONS
Poisson based quantification is sensitive to partition size
non-uniformity, particularly at the higher concentration limits.
The impact of the non-uniformity is also felt in quantification of
the rare target as the quantification of the wild target is
impacted by it. Models that take this variation into account, as
described in various embodiments described herein, can be harnessed
to quantify accurately despite the variation. A major factor to the
success in application of these models is in the correct assessment
of the true levels of effective volume variation of the partition
size. In array based systems, the partition size meaning the
through-hole could be very uniform. Nevertheless, the loading
volume may vary across the different through-holes because of other
factors such as coating or reaction formulation. This effectively
makes the partition from the Poisson modeling stand point more
variable than the physical dimension of the through-hole. For the
volume variation modeling, what is important is the volume of
sample that is within the partition, which is the effective
volume.
Section A: Derivation of the Expression for the Probability of
Negatives in the Variable .lamda. Full Fidelity Model
We assumed that the partition sizes follow a truncated normal
distribution (v.sub.0, .sigma.) with a mean volume of v.sub.0 and a
standard deviation of .sigma. as given by equation 8. A truncated
distribution is assumed, as volumes cannot take negative values. To
evaluate the constant factor K, we integrate the probability
distribution function and equate it to 1.
.intg..infin..times..function..times..times. ##EQU00012##
.times..intg..infin..times..times..sigma..times..times.
##EQU00012.2## .times..times.'.times..sigma.'.times..sigma.
##EQU00012.3## .times..intg..infin..times.'.times..times.'
##EQU00012.4##
.sigma..pi..times..function..times..sigma..times..sigma.
##EQU00012.5## .sigma..times..pi..times..function..times..sigma.
##EQU00012.6##
Note as
>.infin.>.sigma..times..times..pi. ##EQU00013## and we obtain
the well-known constant factor associated with Gaussian
distributions between -.infin. to +.infin..
Now, using this K in equation 5, we obtain:
.function..sigma..times..pi..times..function..times..sigma..times..times.-
.times..sigma. ##EQU00014##
And following on with the integration step as outlined in equation
(6), we obtain:
.function..times..intg..infin..times..function..times..times..times..sigm-
a..times..pi..times..function..times..sigma..times..intg..infin..times..ti-
mes..times..sigma..times..times..times..sigma..times..pi..times..function.-
.times..sigma..times..times..times..times..times..intg..infin..times..time-
s..times..sigma..times..times..times..intg..infin..times..times..times..ti-
mes..sigma..times..times..times..times..sigma..times..intg..infin..times..-
times..times..times..times..sigma..times..times..times..sigma..times..intg-
..infin..times..times..sigma..times..function..sigma..times..times..times.-
.sigma..times..intg..infin..times..times..times..times..times..times..time-
s..times..sigma..sigma..times..times.'.times..intg..infin..times..times..t-
imes..times..intg..infin..times..function..times..times..times..times..int-
g..infin..times..function..times..times..times..times..times..times..times-
..times..intg..infin..times..function..times..times..times..times..times..-
times..intg..infin..times..times..sigma..times..sigma..sigma..times..sigma-
..times..times..sigma..times..times..times..intg..infin..times..times..sig-
ma..times..sigma..function..sigma..times..times..sigma..times..sigma..time-
s..times..intg..infin..times..times..sigma..times..times..times..sigma..ti-
mes..sigma..times..sigma..times..times..sigma..times..times..times..intg..-
infin..times..times..sigma..times..times..times..sigma..times..times..sigm-
a..times..sigma..times..times..times. ##EQU00015##
Using (iii) in (ii), we obtain:
.times..sigma..times..times..intg..infin..times..times..sigma..times..tim-
es..times..sigma..times..times..times..times..times..times..times.''.intg.-
.infin..times..times..sigma..times..times..times..sigma..times..times.'.ti-
mes..times..times..times..times.''.times..sigma..times..times..times..sigm-
a.''.times..intg..times..times..sigma..times..sigma..infin..times.''.times-
..times..sigma..times..times.''.times..times..sigma..times..pi..times..pi.-
.times..intg..times..times..sigma..times..sigma..infin..times.''.times..ti-
mes.''.times..times..sigma..times..pi..times..function..times..times..sigm-
a..times..sigma. ##EQU00016##
Using (iv) and (v) in (i):
.function..times..sigma..times..pi..times..function..times..sigma..times.-
.times..sigma..times..pi..times..function..times..sigma..times..times..sig-
ma..times..times..sigma..times..times..function..times..sigma..times..time-
s..sigma..function..times..sigma..times..times..sigma..times.
##EQU00017## Section B: Derivation of the Expression for the
Probability of Negatives in the Variable .lamda. Approximation
Model
If we assume that the partition sizes follow a normal distribution
(v.sub.0, .sigma.) with a mean volume of v.sub.0 and a standard
deviation of .sigma., then, by definition P(v) given in equation 4
becomes:
.function..times..sigma..times..sigma..times..times..pi.
##EQU00018##
Substituting this P(v) in equation 5,
.function..mu..times..times..times..times..sigma..times..sigma..times..ti-
mes..pi. ##EQU00019##
Now, integrating the variable v out as follows:
.function..times..intg..infin..infin..times..function..times..times..time-
s..sigma..times..times..pi..times..intg..infin..infin..times..times..times-
..sigma..times..times..times..sigma..times..times..pi..times..intg..infin.-
.infin..times..times..times..times..sigma..times..times..sigma..times..tim-
es..pi..times..times..sigma..times..intg..infin..infin..times..times..time-
s..times..sigma..times..sigma..times..times..sigma..times..times..pi..time-
s..times..sigma..times..intg..infin..infin..times..times..sigma..times..fu-
nction..sigma..times. ##EQU00020##
It is known that:
.intg..infin..infin..times..times..times..pi..times..times.
##EQU00021## So (i) becomes:
.function..sigma..times..times..pi..times..times..sigma..times..pi..funct-
ion..times..sigma..times..sigma..times..times..sigma..times..times..sigma.-
.times..times..pi..times..pi..function..times..sigma..times..sigma..times.-
.sigma..times..sigma..times..sigma..times..times..sigma..times..times..pi.-
.times..pi..function..times..sigma..times..sigma..times..sigma..times..sig-
ma..times..sigma..times..times.e.times..sigma..times..times..sigma..times.-
.times..sigma..sigma..times.e.times..sigma..times..times..times..sigma..ti-
mes..times..times..function..times..sigma..times. ##EQU00022##
Solve for concentration C as follows:
.function..function..times..sigma..times..times..times..times..times..fun-
ction..times..sigma..times..times..times..times..sigma..times..times..time-
s..function..times..times..+-..times..times..sigma..times..times..times..f-
unction..times..times..sigma..times..times..+-..times..sigma..times..times-
..times..function..sigma. ##EQU00023## Note that C can assume two
values, but the value with the negative sign in front of the square
root sign in the numerator in (iii) is used as it can be shown that
it agrees in the small .sigma. limit with the solution from using a
Poisson model for the case where variability is assumed to be
0.
With reference to FIG. 3, the improvement of the results using a
volume variation model over the traditional Poisson model is
illustrated. In this graph, a set of digital PCR experiments was
run using ERM Plasmid samples and the BCR-ABL1 Taqman assay using
standard protocol prescribed for the QuantStudio 3D Digital PCR
system. A QuantStudio 3D chip contains a plurality of partitions.
Up to 6 replicate chips were run at each concentration interrogated
in this experiment. Only chips that passed visual quality
inspection were included in the analysis. Chips were filtered out
if they showed artifacts such as bridging. The positive and
negative counts from the chips were used with both the Poisson and
the volume variation models to generate a quantification result.
The mean result from each model is reported in FIG. 3. The +- one
standard deviation around this mean value is also shown in the
figure. The bottom section shows bar graphs representing the
percent prediction error based upon annotations of what
concentration was run on the chips. The prediction error is
consistently higher for the Poisson model showing the better
performance of the Poisson Plus modeling.
Those skilled in the art will recognize that the operations of the
various embodiments may be implemented using hardware, software,
firmware, or combinations thereof, as appropriate. For example,
some processes can be carried out using processors or other digital
circuitry under the control of software, firmware, or hard-wired
logic. (The term "logic" herein refers to fixed hardware,
programmable logic and/or an appropriate combination thereof, as
would be recognized by one skilled in the art to carry out the
recited functions.) Software and firmware can be stored on
non-transitory computer-readable media. Some other processes can be
implemented using analog circuitry, as is well known to one of
ordinary skill in the art. Additionally, memory or other storage,
as well as communication components, may be employed in embodiments
of the present teachings.
FIG. 4 is a block diagram that illustrates a computer system 400
that can be employed to carry out processing functionality, and to
implement various components or subsystems of the systems described
herein according to various embodiments. For example, system 400
can comprise all or apportion of devices 540, client devices, 502,
512, or 530, servers 522, etc. Computing system 400 can include one
or more processors, such as a processor 404. Processor 404 can be
implemented using a general or special purpose processing engine
such as, for example, a microprocessor, controller or other control
logic. In this example, processor 404 is connected to a bus 402 or
other communication medium.
Further, it should be appreciated that a computing system 400 of
FIG. 4 can be embodied in any of a number of forms, such as a
rack-mounted computer, mainframe, supercomputer, server, client, a
desktop computer, a laptop computer, a tablet computer, hand-held
computing device (e.g., PDA, cell phone, smart phone, palmtop,
etc.), cluster grid, netbook, embedded systems, or any other type
of special or general purpose computing device as may be desirable
or appropriate for a given application or environment.
Additionally, a computing system 400 can include a conventional
network system including a client/server environment and one or
more database servers, or integration with LIS/LIMS infrastructure.
A number of conventional network systems, including a local area
network (LAN) or a wide area network (WAN), and including wireless
and/or wired components, are known in the art. Additionally,
client/server environments, database servers, and networks are well
documented in the art. According to various embodiments described
herein, computing system 400 may be configured to connect to one or
more servers in a distributed network. Computing system 400 may
receive information or updates from the distributed network.
Computing system 400 may also transmit information to be stored
within the distributed network that may be accessed by other
clients connected to the distributed network.
Computing system 400 may include bus 402 or other communication
mechanism for communicating information, and processor 404 coupled
with bus 402 for processing information.
Computing system 400 also includes a memory 406, which can be a
random access memory (RAM) or other dynamic memory, coupled to bus
402 for storing instructions to be executed by processor 404.
Memory 406 also may be used for storing temporary variables or
other intermediate information during execution of instructions to
be executed by processor 404. Computing system 400 further includes
a read only memory (ROM) 408 or other static storage device coupled
to bus 402 for storing static information and instructions for
processor 404.
Computing system 400 may also include a storage device 410, such as
a magnetic disk, optical disk, or solid state drive (SSD) is
provided and coupled to bus 402 for storing information and
instructions. Storage device 410 may include a media drive and a
removable storage interface. A media drive may include a drive or
other mechanism to support fixed or removable storage media, such
as a hard disk drive, a floppy disk drive, a magnetic tape drive,
an optical disk drive, a CD or DVD drive (R or RW), flash drive, or
other removable or fixed media drive. As these examples illustrate,
the storage media may include a computer-readable storage medium
having stored therein particular computer software, instructions,
or data.
In alternative embodiments, storage device 410 may include other
similar instrumentalities for allowing computer programs or other
instructions or data to be loaded into computing system 400. Such
instrumentalities may include, for example, a removable storage
unit and an interface, such as a program cartridge and cartridge
interface, a removable memory (for example, a flash memory or other
removable memory module) and memory slot, and other removable
storage units and interfaces that allow software and data to be
transferred from the storage device 410 to computing system
400.
Computing system 400 can also include a communications interface
418. Communications interface 418 can be used to allow software and
data to be transferred between computing system 400 and external
devices. Examples of communications interface 418 can include a
modem, a network interface (such as an Ethernet or other NIC card),
a communications port (such as for example, a USB port, a RS-232C
serial port), a PCMCIA slot and card, Bluetooth, etc. Software and
data transferred via communications interface 418 are in the form
of signals which can be electronic, electromagnetic, and optical or
other signals capable of being received by communications interface
418. These signals may be transmitted and received by
communications interface 418 via a channel such as a wireless
medium, wire or cable, fiber optics, or other communications
medium. Some examples of a channel include a phone line, a cellular
phone link, an RF link, a network interface, a local or wide area
network, and other communications channels.
Computing system 400 may be coupled via bus 402 to a display 412,
such as a cathode ray tube (CRT) or liquid crystal display (LCD),
for displaying information to a computer user. An input device 414,
including alphanumeric and other keys, is coupled to bus 402 for
communicating information and command selections to processor 404,
for example. An input device may also be a display, such as an LCD
display, configured with touchscreen input capabilities. Another
type of user input device is cursor control 416, such as a mouse, a
trackball or cursor direction keys for communicating direction
information and command selections to processor 404 and for
controlling cursor movement on display 412. This input device
typically has two degrees of freedom in two axes, a first axis
(e.g., x) and a second axis (e.g., y), that allows the device to
specify positions in a plane. A computing system 400 provides data
processing and provides a level of confidence for such data.
Consistent with certain implementations of embodiments of the
present teachings, data processing and confidence values are
provided by computing system 400 in response to processor 404
executing one or more sequences of one or more instructions
contained in memory 406. Such instructions may be read into memory
406 from another computer-readable medium, such as storage device
410. Execution of the sequences of instructions contained in memory
406 causes processor 404 to perform the process states described
herein. Alternatively hard-wired circuitry may be used in place of
or in combination with software instructions to implement
embodiments of the present teachings. Thus implementations of
embodiments of the present teachings are not limited to any
specific combination of hardware circuitry and software.
The term "computer-readable medium" and "computer program product"
as used herein generally refers to any media that is involved in
providing one or more sequences or one or more instructions to
processor 404 for execution. Such instructions, generally referred
to as "computer program code" (which may be grouped in the form of
computer programs or other groupings), when executed, enable the
computing system 400 to perform features or functions of
embodiments of the present embodiments described herein. These and
other forms of non-transitory computer-readable media may take many
forms, including but not limited to, non-volatile media, volatile
media, and transmission media. Non-volatile media includes, for
example, solid state, optical or magnetic disks, such as storage
device 410. Volatile media includes dynamic memory, such as memory
406. Transmission media includes coaxial cables, copper wire, and
fiber optics, including the wires that comprise bus 402.
Common forms of computer-readable media include, for example, a
floppy disk, a flexible disk, hard disk, magnetic tape, or any
other magnetic medium, a CD-ROM, any other optical medium, punch
cards, paper tape, any other physical medium with patterns of
holes, a RAM, PROM, and EPROM, a FLASH-EPROM, any other memory chip
or cartridge, a carrier wave as described hereinafter, or any other
medium from which a computer can read.
Various forms of computer readable media may be involved in
carrying one or more sequences of one or more instructions to
processor 404 for execution. For example, the instructions may
initially be carried on magnetic disk of a remote computer. The
remote computer can load the instructions into its dynamic memory
and send the instructions over a telephone line using a modem. A
modem local to computing system 400 can receive the data on the
telephone line and use an infra-red transmitter to convert the data
to an infra-red signal. An infra-red detector coupled to bus 402
can receive the data carried in the infra-red signal and place the
data on bus 402. Bus 402 carries the data to memory 406, from which
processor 404 retrieves and executes the instructions. The
instructions received by memory 406 may optionally be stored on
storage device 410 either before or after execution by processor
404.
It will be appreciated that, for clarity purposes, the above
description has described embodiments with reference to different
functional units and processors. However, it will be apparent that
any suitable distribution of functionality between different
functional units, processors or domains may be used without
detracting from the embodiments of the present teachings. For
example, functionality illustrated to be performed by separate
processors or controllers may be performed by the same processor or
controller. Hence, references to specific functional units are only
to be seen as references to suitable means for providing the
described functionality, rather than indicative of a strict logical
or physical structure or organization.
FIG. 5 is a diagram illustrating an example system 500 configured
in accordance with one example embodiment. In system 500, one or
more servers 522 can be configured to run the analysis applications
for analyzing data sets produced by one or more devices or
modalities 540. The data included in the data sets can be stored in
one or more storage devices 550. Once the data sets have been
uploaded to servers 522, then a plurality of applications running
on servers 522 can be used to manipulate, analyze and visualize the
data sets from anywhere. For example, local client devices 530 can
be used to access servers 522, e.g., through a hub or router 526.
At the same time, the data can be accessed remotely through remote
clients devices 502, which are interfaced with servers 522, e.g.,
via a gateway/hub/tunnel-server/etc. 510, which is itself connected
to the internet 508 via some internet service provider (ISP)
connection 510, or remote client servers 512, which are interfaced
with servers 522, e.g., via the internet 508 and via an ISP
connection 514.
It should also be noted that devices 540 can be directly interfaced
with servers 522, e.g., through the internet. In such embodiments,
the collection application and functionality can reside on servers
522, on devices 540, or both. In other embodiments, devices 540 can
be interfaced with client devices 502 or 512. In such embodiments,
the collection application or functionality can be included on
client devices 502 or 512, devices 540, or both.
Client devices 502, 512, and 530 can be any kind of computing
device that can be used to access servers 522. As such, these
devices can be laptop, desktop, or palmtop computers, terminals,
mobile computing devices such as smartphones or tablets, etc.
Servers 522 can comprise one or more processors, servers, routers,
co-processors, user interfaces, etc., whether co-located or located
in different locations. In short, servers 522 can comprise all of
the resources, both hardware and software, needed to perform the
functions described herein. A more detailed description of a
computer system and the resources that can be used to implement the
components illustrated in FIG. 5 is described below with respect to
FIG. 4.
Although various embodiments have been described with respect to
certain exemplary embodiments, examples, and applications, it will
be apparent to those skilled in the art that various modifications
and changes may be made without departing from the present
teachings.
* * * * *