U.S. patent application number 15/536309 was filed with the patent office on 2017-12-21 for method for calibrating a device for measuring the concentration of creatinine.
This patent application is currently assigned to Radiometer Medical Aps. The applicant listed for this patent is Radiometer Medical ApS. Invention is credited to Thomas KJAER.
Application Number | 20170363567 15/536309 |
Document ID | / |
Family ID | 54850194 |
Filed Date | 2017-12-21 |
United States Patent
Application |
20170363567 |
Kind Code |
A1 |
KJAER; Thomas |
December 21, 2017 |
METHOD FOR CALIBRATING A DEVICE FOR MEASURING THE CONCENTRATION OF
CREATININE
Abstract
A method of calibrating a device for measuring the concentration
of creatinine using one or more calibration solutions, the method
comprising: receiving concentrations at an initial time of
creatine, Cr, and/or creatinine, Crn, of the one or more
calibration solutions; receiving outputs of the measuring device at
the end time; calculating the concentration of Cr and/or Crn in the
calibration solutions at an end time using a temperature model,
wherein the temperature model indicates changes in temperature of
the calibration solutions from the initial time to the end time;
and determining a relationship between the outputs of the measuring
device and the calculated concentrations of Cr and/or Crn.
Inventors: |
KJAER; Thomas; (Smorum,
DK) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Radiometer Medical ApS |
Bronshoj |
|
DK |
|
|
Assignee: |
Radiometer Medical Aps
Bronshoj
DK
|
Family ID: |
54850194 |
Appl. No.: |
15/536309 |
Filed: |
December 17, 2015 |
PCT Filed: |
December 17, 2015 |
PCT NO: |
PCT/EP2015/080131 |
371 Date: |
June 15, 2017 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01N 27/3274 20130101;
G01N 33/70 20130101; G01N 2333/986 20130101 |
International
Class: |
G01N 27/327 20060101
G01N027/327; G01N 33/70 20060101 G01N033/70 |
Foreign Application Data
Date |
Code |
Application Number |
Dec 18, 2014 |
DK |
PA 2014 00737 |
Claims
1. A method for calibrating a device for measuring the
concentration of creatinine using one or more calibration
solutions, the method comprising: receiving concentrations at an
initial time of creatine, Cr, and/or creatinine, Crn, of the one or
more calibration solutions; receiving outputs of the measuring
device at an end time; calculating the concentration of Cr and/or
Crn in the calibration solutions at an end time using a temperature
model, wherein the temperature model indicates changes in
temperature of the calibration solutions from the initial time to
the end time; and determining a relationship between the outputs of
the measuring device and the calculated concentrations of Cr and/or
Crn.
2. The method of claim 1, wherein the measuring device includes a
sensor for measuring creatine in one or more of the calibration
solutions.
3. The method of claim 1, wherein the measuring device includes a
sensor for measuring creatinine in one or more of the calibration
solutions.
4. The method of claim 1, wherein the measuring device includes a
sensor for measuring creatine and creatinine in one or more of the
calibration solutions.
5. The method of claim 1, wherein the measuring device is an
amperometric measuring device.
6. The method of claim 1, further comprising determining the
temperature model by receiving measurements of the changes in
temperature of the calibration solutions from a temperature probe
from the initial time to the end time.
7. The method of claim 6, wherein said received measurements are
recorded measurements.
8. The method of claim 6, wherein said determining the temperature
model comprises calibrating the received measurements of the
changes in temperature with a second temperature probe.
9. The method of claim 1, wherein the determining a relationship
between the measuring device outputs and the calculated
concentrations of Cr and Crn comprises calculating sensor
sensitivities of the measuring devices.
10. The method of claim 9, further comprising receiving temperature
measurements after the end time and updating the calculated
sensitivities of the measuring devices using the temperature
measurements received after the end time.
11. The method of claim 1, wherein the end time is greater than 14
days after the initial time.
12. A computer readable medium comprising instructions which when
executed by one or more processors of an electronic device, cause
the electronic device to operate in accordance with the method as
claimed in claim 1.
13. An electronic device comprising: one or more processors; and
memory comprising instructions which when executed by one or more
of the one or more processors cause the electronic device to
operate in accordance with the method claimed in claim 1.
14. A package comprising one or more calibration solutions, the
package containing instructions for use with the method of claim
1.
15. The package of claim 14 further comprising an indication of the
initial time and the concentrations at the initial time of Cr
and/or Crn of the one or more calibration solutions.
16. The package of claim 14, further comprising a temperature probe
for measuring the temperatures of the creatinine solutions from the
initial time to the end time, and memory for storing the measured
temperatures.
17. A package comprising one or more calibration solutions, the
package containing instructions for use with the electronic device
of claim 13.
18. The package of claim 17, further comprising an indication of
the initial time and the concentrations at the initial time of Cr
and/or Crn of the one or more calibration solutions.
19. The package of claim 17, further comprising a temperature probe
for measuring the temperatures of the creatinine solutions from the
initial time to the end time, and memory for storing the measured
temperatures.
Description
FIELD OF THE INVENTION
[0001] The invention relates to methods for calibrating creatinine
measuring devices, and calibration solutions for use in those
methods.
BACKGROUND
[0002] Techniques for measuring the concentration of creatinine
(Crn) and creatine (Cr) are useful in medicine, for example in
monitoring renal disease. The concentration of Cr (cCr) and the
concentration of Crn (cCrn) in an aqueous solution can be
determined by amperometric measurement. Two sensors are used in the
measurement of cCrn: the Crea A sensor, which detects Cr; and the
Crea B sensor, which detects both Cr and Crn. The cCrn is based on
the difference between the Crea A and Crea B sensor
measurements.
[0003] In order to determine cCrn in unknown samples with
sufficient accuracy, the Crea A and Crea B sensors must be
calibrated regularly in order to determine their actual
sensitivities. The sensors can be calibrated using two aqueous
calibration solutions of known cCrn and cCr. However, a problem
with such calibration solutions is that cCr and cCrn are not fixed.
Rather, when in aqueous solution, Crn can be converted into Cr and
vice versa by water hydrolysis and elimination respectively, as
indicated by the double arrow in the reaction equation in Scheme 1,
where T is the temperature at which the reaction occurs and k.sub.1
and k.sub.2 are the rate constants for the hydrolysis and
elimination reactions respectively.
##STR00001##
[0004] After a sufficient period of time at a constant temperature
the mixture will reach a dynamic equilibrium in which cCr and cCrn
are constant. However, changes to the temperature of the solution
will shift the equilibrium ratio, and therefore the cCr and cCrn
will change.
[0005] A calibration solution having known cCr and cCrn in
equilibrium ratios, can be generated by adding the enzyme
creatinine amidohydrolase (CA) to a solution of known sum
concentration of Cr and Crn, and then allow that solution to
equilibrate by maintaining the solution at a specific temperature
for about one hour. The CA facilitates equilibration. Once
equilibrium is reached at a given temperature, the cCr and cCrn can
be determined simply by reference to the known equilibrium
constants at the given temperature.
[0006] WO 2005/052596 discloses a reference solution for quality
control on a medical analyser and a kit for holding the reference
solution. The kit comprises a container with a first and a second
compartment, wherein the first and the second compartment may be
separated by a thin wall, which may be broken upon exertion of
manual pressure, which will cause the liquids of the first and
second compartments to mix. The first compartment comprises a
buffer, and the second compartment comprises two chemical compounds
and a catalyst. When the two compunds, the catalyst and the buffer
is mixed, a thermodynamic equilibrium is reached more rapidly
allowing the solution to be used as quality control reference
solution within hours, when conditioned at 25.degree. C., whereas
the solution would otherwise have to be conditioned for much
longer, if no catalyst was used.
[0007] For sensors responding linearly to changes in both cCr and
cCrn it is required that the chosen compositions of the two
calibration solutions result in a linearly independent equation
system, i.e., cCr.sub.CalX=mcCr.sub.CalY and
cCrn.sub.CalX=ncCrn.sub.CalY with m .noteq. n is required. Hence,
only one of the calibration solutions can contain Cr and Crn at
their equilibrium concentrations unless stored at two different
temperatures. However, the best sensor calibration may be obtained
by using one solution of essentially pure Cr and one solution of
essentially pure Crn.
[0008] Calibration solutions having known cCr and cCrn in
non-equilibrium ratios can be prepared by dissolving a known
quantity of dry powder containing known quantities of Cr and Crn in
a known volume of water, or other aqueous medium.
[0009] Typically, the calibration solution is kept at between
15.degree. C. to 32.degree. C., which allows for accurate
calculations for solutions less than 14 days old, even when
approximating this to be at a constant temperature. Therefore,
existing methods are sufficiently accurate to enable the
calculation of cCr and cCrn for calibration solutions having a
short usage period (i.e. t.sub.age<14 days).
[0010] Another alternative is to produce an acidic creatinine
solution, this is thermodynamically stable and can thus be
maintained during storage. This solution can then be mixed with a
buffer-solution immediately before use to obtain a calibration
solution of essentially pure Crn with a known pH and cCrn, such as
disclosed in US 2004/0072277. A problem with this approach is that
it is difficult to produce a homogeneous mixture if the two
solutions are contained in sealed airtight pouches. Additionally,
the mixing process will also require pumping means if automatic
mixing is required and/or if the pouches are part of a closed
system, such as a cassette based solution pack. Furthermore, it is
not advisable to let the solution pass through the sensor chamber,
since the CA enzyme may leak out of the creatinine sensors, for
example, and into the solution thereby destroying the
non-equilibrium conditions. To prevent that could necessitate a
separate channel for liquid transport.
[0011] In both cases, having a 14 day time limit for the accuracy
of the calibration solution is disadvantageous in that frequent
replacement is required, which is uneconomical and inconvenient for
the end user. Furthermore, the short usage period may be reduced
even further because the end user may have to wait several days for
the solution to be delivered from the manufacturer. Alternatively
the end user may be obliged to prepare the calibration solution at
the point of use (e.g. hospital), which introduces additional work
for the end user as well as the risk of inaccuracies, for example
in weighing the Cr and Crn powder and measuring the volume of
solvent or by uneven mixing. An unmet need exists for an approach
enabling the use of calibration solutions having a t.sub.age>14
days.
SUMMARY OF THE INVENTION
[0012] In a first aspect of the present invention, the applicant
makes available a method of calibrating a device for measuring the
concentration of creatinine using one or more calibration
solutions, the method comprising: receiving concentrations at an
initial time of creatine, Cr, and/or creatinine, Crn, of the one or
more calibration solutions; receiving outputs of the measuring
device at the end time; calculating the concentration of Cr and/or
Crn in the calibration solutions at an end time using a temperature
model, wherein the temperature model indicates changes in
temperature of the calibration solutions from the initial time to
the end time; and determining a relationship between the outputs of
the measuring device and the calculated concentrations of Cr and/or
Crn.
[0013] By using a temperature model of the calibration solutions,
it is possible to accurately calibrate the sensors even when the
initial concentrations of the solutions have been measured over 14
days before calibration. The customer is not restricted to utilise
the calibration solutions within a short time span after receiving
the calibration solutions, as a temperature model allows the
customer to store the solutions at a range of temperatures, while
still providing accurate calibrations.
[0014] In some example embodiments the measuring device includes a
sensor for measuring creatine in one or more of the calibration
solutions.
[0015] In some example embodiments the measuring device includes a
sensor for measuring creatinine in one or more of the calibration
solutions.
[0016] In some example embodiments the measuring device includes a
sensor for measuring creatine and creatinine in one or more of the
calibration solutions.
[0017] In some example embodiments the measuring device is an
amperometric measuring device.
[0018] In some example embodiments, the method further comprises
determining the temperature model by receiving measurements of the
changes in temperature of the calibration solutions from a
temperature probe from the initial time to the end time. By using a
temperature probe to measure changes in temperature of the
calibration solutions over the time period between the initial time
and the end time, it is possible to accurately calculate the end
concentrations of the calibration solutions.
[0019] In some example embodiments, the received measurements are
recorded measurements.
[0020] In some example embodiments said determining the temperature
model comprises calibrating the received measurements of the
changes in temperature with a second temperature probe.
[0021] In some example embodiments the determining a relationship
between the measuring device outputs and the calculated
concentrations of Cr and Crn comprises calculating sensor
sensitivities of the measuring devices.
[0022] In some example embodiments the method further comprises
receiving temperature measurements after the end time and updating
the calculated sensitivities of the measuring devices using the
temperature measurements received after the end time.
[0023] In some example embodiments the end time is greater than 14
days after the initial time.
[0024] According to another aspect of the present invention, a
computer readable medium is provided comprising instructions which
when executed by one or more processors of an electronic device,
cause the electronic device to operate in accordance with any of
the aforementioned methods.
[0025] According to another aspect of the present invention, an
electronic device is provided comprising: one or more processors;
and memory comprising instructions which when executed by one or
more of the processors cause the electronic device to operate in
accordance with any of the aforementioned methods.
[0026] According to another aspect of the present invention, a
package is provided comprising one or more calibration solutions,
the package being suitable for use with any of the aforementioned
methods or aforementioned electronic devices.
[0027] In some example embodiments, the package further comprises
an indication of the initial time and the concentrations at the
initial time of creatine (Cr) and/or creatinine (Crn) of the one or
more calibration solutions.
[0028] In some example embodiments, the package further comprises a
temperature probe for measuring the temperatures of the creatinine
solutions from the initial time to the end time, and memory for
storing the measured temperatures.
BRIEF DESCRIPTIONS OF DRAWINGS
[0029] Examples of the present proposed apparatus will now be
described in detail with reference to the accompanying drawings, in
which:
[0030] FIG. 1 is a schematic diagram of an example of an
amperometric measuring system;
[0031] FIG. 2 is a graph showing the temperature variation of
calibration solutions and a corresponding two-temperature model
estimating the temperature variation;
[0032] FIG. 3 is a flow chart showing the derivation of the
proposed method;
[0033] FIG. 4 is a flow chart outlining the steps of the proposed
method;
[0034] FIG. 5 is a series of graphs comparing the concentrations in
the calibration solutions with the concentrations calculated in
accordance with an example embodiment of the proposed method;
and
[0035] FIG. 6 is a series of graphs comparing the concentrations in
the calibration solutions with the concentrations calculated in
accordance with a further example embodiment of the proposed
method.
DETAILED DISCLOSURE
[0036] Reference will now be made to FIG. 1 which is a schematic
diagram of a three-electrode amperometric measuring system 101. An
amperometric measuring system may have at least two electrodes: a
working electrode (WE) 110 and a combined counter and reference
electrode (CE/RE). For the three-electrode amperometric measuring
system 101, the functions of the CE/RE electrode are split into two
separate electrodes: the reference electrode (RE) 111 and the
counter electrode (CE) 112. The example amperometric measuring
system 101 also includes an ammeter 120, a voltmeter 121 and a
voltage source 122 and the electrolyte solution 140.
[0037] The WE 110 is a positively charged electrode where an
oxidation reaction occurs. The RE 111 is typically made of Ag/AgCl
and is able to maintain a stable potential, especially if no
current runs through it, thus the need for a CE 112 for passing the
current from the WE 110 back to the electrolyte solution 140. The
electrolyte solution 140/sample provides a current path between the
three electrodes. The membrane 130 selectively converts the analyte
to a substance that selectively is allowed to pass through from the
sample 150. The voltage source 122 applies the necessary potential
for maintaining the desired reduction or oxidation reaction, this
is controlled by the voltmeter 121. The ammeter 120 measures the
resulting current flowing through the electrical circuit.
[0038] The amperometric measuring system shown in FIG. 1 is an
illustrative example, and several other implementations are
envisioned. For example, the amperometric measuring system could be
a two electrode system as mentioned above.
[0039] The magnitude of an electrical current flowing through the
electrode chain is proportional to the concentration of the
substance being oxidized (or reduced) at the WE 110. Ideally, when
knowing the proportionality constant relating the electrical
current to a concentration, the concentration in any given sample
can be obtained by measuring the electrical current generated by
that particular sample.
[0040] To illustrate the measuring process in an amperometric
measuring system, we assume that: The sample 150 contains species B
which, in the membrane 130, is selectively converted to species A,
which can be oxidized at the WE 110 (WE) to A.sup.+; and the
electrolyte 140 contains species X which is reduced at the CE 112
(cathode) to X.sup.-. We assume also that the membrane 130 allows
only species A to pass from the sample into the electrolyte
solution 140.
[0041] As an appropriate potential is applied across the
electrodes, A is oxidized at the WE 110 according to the following
reaction:
A.fwdarw.A.sup.++e.sup.-
[0042] The oxidation of A produces a flow of electrons. To complete
the electrical circuit a reduction reaction where electrons are
consumed is necessary. Therefore species X is reduced at the CE 112
according to the following reaction:
X+e.sup.-.fwdarw.X.sup.-
[0043] The magnitude of the current flowing through the circuit is
proportional to the concentration of the analyte being oxidized.
The analyser can therefore automatically calculate the
concentration of the analyte in the sample given species X is in
excess.
[0044] The term sensor refers to a complete amperometric measuring
system, as shown in FIG. 1 excluding the sample 150.
[0045] Crn is not stable in aqueous solutions, e.g. blood, where it
is reversibly converted into Cr (see Scheme 1). To measure cCrn,
the present invention utilizes a two-sensor system where one sensor
(Crea A) detects Cr only, and the other sensor (Crea B) detects
both Cr and Crn. By means of a difference measurement it is
possible to obtain the cCrn value.
[0046] The sensor is protected by a multilayer membrane 130
consisting of at least three functional layers, namely the outer
membrane layer permeable to Crn and Cr; the middle enzyme layer,
and the inner membrane layer permeable to H.sub.2O.sub.2.
[0047] Crn and Cr molecules diffuse across the outer membrane
layer. The enzymes creatinine amidohydrolase, creatine
amidinohydrolase and sarcosine oxidase are immobilized between the
inner and outer membrane layers. The Crea A sensor contains only
creatine amidinohydrolase and sarcosine oxidase, and so detects
only Cr. In the Crea A sensor, the enzymatic cascade changes Cr as
follows:
Creatine+H.sub.2O.fwdarw.sarcosine+urea (creatine
amidinohydrolase)
Sarcosine+H.sub.2O+O.sub.2.fwdarw.glycine+formaldehyde+H.sub.2O.sub.2
(sarcosine oxidase)
[0048] The Crea B sensor contains all three enzymes creatinine
amidohydrolase, creatine amidinohydrolase and sarcosine oxidase,
and so detects both Crn and Cr. In the enzymatic cascade Crn/Cr
changes as follows:
Creatinine+H.sub.2Ocreatine (creatinine amidohydrolase)
Creatine+H.sub.2O.fwdarw.sarcosine+urea (creatine
amidinohydrolase)
Sarcosine+H.sub.2O+O.sub.2.fwdarw.glycine+formaldehyde+H.sub.2O.sub.2
(sarcosine oxidase)
[0049] For both the Crea A and the Crea B sensors the enzyme
reactions lead to identical end-products, one of which is
H.sub.2O.sub.2 that can diffuse across the inner membrane layer to
the WE 110 (preferably platinum). By applying a sufficiently high
electrical potential to the electrode chains of the Crea A and Crea
B sensors, H.sub.2O.sub.2 can be oxidized at the WE 110:
H.sub.2O.sub.2.fwdarw.2H.sup.++O.sub.2+2e.sup.-
[0050] To complete the electrical circuit, electrons are consumed
in reduction reactions at the CE 112 thereby maintaining a charge
balance between the WE 110 and the CE 112.
[0051] The oxidation of H.sub.2O.sub.2 produces an electrical
current (I) proportional to the amount of H.sub.2O.sub.2, which in
turn is directly related to the amount of Cr for the Crea A and the
amount of Cr and Crn for the Crea B sensors according to the sensor
response models:
I.sup.A=S.sub.Cr.sup.AcCr Equation 1
I.sup.B=S.sub.Cr.sup.BcCr+S.sub.Crn.sup.BcCrn Equation 2
[0052] Where I.sup.A and I.sup.B are the electrical currents
produced at the Crea A and Crea B sensors respectively;
S.sub.Cr.sup.A and S.sub.Cr.sup.B are the sensitivity constants
relating current (I) to Cr concentration in the Crea A and Crea B
sensors respectively and S.sub.Crn.sup.B is the sensitivity
constant relating current (I) to Crn concentration in the Crea B
sensor.
[0053] The proportionality constants, S, relating currents to
concentrations are typically referred to as sensitivities. The
constants are determined by calibrating the sensors. The current
(signal) of each sensor is measured by ammeters 120 in the
analyser. If sensor sensitivities S are known, the unknown Crn
concentration in a given sample is readily determined from the
equations above.
[0054] The changes in cCr and cCrn relative to the last known
values of cCr and cCrn can be calculated based on reaction rate
equations and rate constants described by the Arrhenius equation.
To make the calculation it is necessary to know the period of time
elapsed since the cCr and cCrn was last known (t.sub.age), and the
temperature experienced by the solution during that period.
[0055] One way of making accurate estimations of cCr and cCrn when
the calibration solution has experienced a range of temperatures,
is to record the changes in temperature since the last
concentration measurements. For example, when the calibration
solutions are shipped, the calibration solution packs may include a
temperature measurement device and a means for recording the
changes in temperature.
[0056] One example implementation would include a temperature probe
that records the temperature of the solutions at regular intervals
into computer memory, so that when the solutions need to be
calibrated, the memory can be accessed to determine the
temperatures experienced. The combination of a temperature probe
and memory for recording the recorded temperatures may be referred
to herein as a temperature logger.
[0057] Knowing the historic changes in temperature, it is possible
to accurately calculate the changes in concentration from the
initial concentrations recorded to a calculated concentration at
the time of calibration. For example, it is possible to calculate
the change in cCrn and cCr over time based on the reaction rate
equations and the rate constants described by the Arrhenius
equation, and applied to the recorded temperatures over time.
[0058] As each solution pack would require a temperature logger, to
reduce the costs it may be beneficial to use a low-cost temperature
probe. While a low-cost temperature probe may lead to inaccurate
temperature measurements, the temperature probe may be calibrated
with a more accurate temperature probe. This calibration may
involve comparing measurements by the low-cost and reference
temperature probe for one temperature or over a range of
temperatures to determine if there is a difference between the
measurements. If a difference is determined, the recorded
temperatures in the memory may be offset by this difference. The
offsetting of recorded temperatures may be applied to readings
already recorded, or may be applied to all subsequent recordings of
temperature.
[0059] Temperature loggers packaged with solution packs can be used
to provide an accurate measurement of temperature changes while the
solution pack is being stored and transported, and eventually can
be used to help calibrate sensors. Once the calibration solutions
have reached their destinations, the sensors can been calibrated
using the accompanying temperature loggers, and the temperature may
then be continued to be monitored by a more accurate temperature
logger to ensure that the sensor sensitivities are kept up to date.
The more accurate temperature logger may be part of a measurement
station or may be part of a permanent instalment of the end
user.
[0060] Surprisingly it has been found that in one embodiment
calculating the conversions in the calibrators using a temperature
logger with a temperature offset from the real temperature, and a
thermal exposure that does not quite resemble the temperature
profile experienced by the calibration solutions still can provide
an acceptable estimate of the Cr and Crn concentrations in the
calibrators. Therefore, such calibration of the temperature probes
may not be required, and the low-cost temperature probes may be
used. Using a temperature logger as an estimate of the exact
temperature profile can provide sufficient data for estimating the
Cr and Crn concentrations in the calibrators.
[0061] While using one or more temperature measurement devices and
recording the measurements can lead to very accurate calculations
for end concentrations, it may be advantageous to calculate
similarly accurate results without the use of temperature probes or
memory in order to reduce the costs of each pack of calibration
solutions. One such method involves using temperature models to
estimate the changes in temperatures experienced.
[0062] FIG. 2 is a graph showing an example temperature profile of
calibration solutions and a corresponding two-temperature model
estimating the temperature variation. The true temperature profile
210 shows the actual temperature variation from an initial time 230
(t=0) to an end time 232 (t=t.sub.age). In this example, the
temperature starts at a low 4.degree. C. for an initial period of 4
days in storage at the manufacturer 211. When the calibrations
solutions are transported, the temperature rises to a higher
temperature of 31.degree. C. for a period of roughly 14 days 212.
After transporting 212 the calibration solutions, they are stored
at a low temperature (roughly 5.degree. C.) with the customer 213
for a period of roughly 76 days. In this example, the customer
takes the calibration solutions out of storage in order to use them
over a period of 14 days 214, during which time, the calibration
solutions are kept at a higher temperature of roughly 20.degree.
C.
[0063] While the end user will be aware of the temperature of the
calibration solution if the user prepared it themselves by
combining an aqueous medium of known temperature and Cr and Crn
powder, the user will not be aware of the temperature profile from
the point of production up to usage if prepared by the
manufacturer. As shown in FIG. 2, the true temperature profile 210
can be very complicated, with large temperature fluctuations over a
long period of time. In the past, such temperature fluctuations
have made it difficult to accurately calculate the concentration
levels of the calibration solutions.
[0064] The applicant has surprisingly identified that although a
temperature profile is not necessarily known by the customer, the
complex temperature profile can be modelled by a much simpler
temperature model. An example temperature model, a two-temperature
model 220, is indicated in FIG. 2. The complex temperature profile
is modelled as having a low temperature of T.sub.1=2.degree. C. 241
for a first time period 221 from an initial time t=0 230 to an
intermediate time t=t.sub.1 231, and then having a high temperature
of T.sub.2=32.degree. C. 242 for a period 222 from the intermediate
time t=t.sub.1 231 to an end time t=t.sub.age 232. While the
two-temperature model 220 in FIG. 2 does not exactly match the true
temperature profile 210, it provides a functional estimation of the
changes in temperature.
[0065] The value of t.sub.1 can be optimised to compensate for any
discrepancies between the true temperature profile 210 and the
two-temperature model 220. For example, although the final increase
in temperature in the true temperature profile 210 occurs at about
107 days, the corresponding increase in temperature of the
two-temperature model 220 occurs at an earlier time t=t.sub.1=95
days. This earlier time for t.sub.1 is earlier than the actual
increase in temperature to compensate for the large increase in
temperature during the initial transport phase 212. Similarly, if
the temperature values T.sub.1 and T.sub.2 are chosen to be too
high or too low, the value of t.sub.1 can be adjusted to compensate
for this as well.
[0066] The true temperature profile can be simplified to a
multiple-temperature model as the pattern of storage, transport and
usage conditions for the creatinine/creatine solutions follow
similar patterns. Table 1 shows the estimated time and temperature
ranges for an example calibration solution supporting cCrn
measurements.
TABLE-US-00001 TABLE 1 Max. time [days] Temperatures [.degree. C.]
Storage at manufacturer 56 2-8 Transport from manufacturer 14 2-32
to customer Storage at customer 108 (~31/2 months) 2-8 Usage 14
15-32
[0067] Other multiple-temperature models could be used. For
example, a temperature model with three or more temperatures may be
used. Rather than discrete step changes in temperature, progressive
cooling and heating of the temperatures could be incorporated into
the model using simple exponential functions, or fluctuations could
be incorporated into the model using sinusoidal functions, for
example.
[0068] In the example embodiment provided, a two-temperature model
is used as it provides a simple illustration of the proposed
solution. In this example embodiment, the following calibrators,
Cal2 and Cal3, are used for calibrating the Crea A and Crea B
sensors:
TABLE-US-00002 TABLE 2 At time: t.sub.production Cal2 Cal3 cCr
[.mu.M] 0 500 cCrn [.mu.M] 500 0 Sensor Crea B Crea A and Crea
B
[0069] In addition to the Cr and Crn, calibrators Cal2 and Cal3 may
also contain buffers, salts, preservatives and detergents. For the
purposes of the proposed method, the concentrations of only Cr and
Crn in the calibrators will be reviewed.
[0070] In the example embodiment presented herein, the
concentrations of both Cr and Crn are being determined through the
use of two calibration solutions and two sensors. However, it is
envisioned that the proposed method may be used to calculate only
the Cr concentrations or only the Crn concentrations. If
concentrations of just one substance is determined (Cr or Crn),
then only one specific sensor (Cr or Crn) and two calibration
solutions are needed.
[0071] As shown in Scheme 1, Crn can be converted into Cr and vice
versa by water hydrolysis and elimination in a reversible reaction.
This reversible conversion reaction starts immediately when either
of the two species are dissolved in water and it will continue with
a higher rate in one direction until the system reaches
thermodynamic equilibrium, i.e., until Crn and Cr reach their
mutual equilibrium concentrations (cX.sub.eq) according to the
equilibrium constant (K.sub.eq(T)) of this reaction:
K eq ( T ) = cCr eq ( T ) cCrn eq ( T ) = k 1 ( T ) k 2 ( T ) .
Equation 3 ##EQU00001##
[0072] The equilibrium constant can also be expressed in terms of
rate constants k.sub.1(T) and k.sub.2(T) for the forward and
backward reactions (the individual arrows) involved in the
conversion reaction in Scheme 1. The rate constants are temperature
dependent, but are not necessarily equally dependent. Therefore,
the equilibrium constant K.sub.eq(T) of the conversion reaction
(and hence the equilibrium concentration of Crn and Cr) will also
be temperature dependent.
[0073] The temperature dependency of the rate constants is dictated
by the Arrhenius-equation:
k i ( T t ) = A i exp ( .beta. i T t ) = exp ( .alpha. i + .beta. i
T t ) , where .alpha. i = ln A i Equation 4 ##EQU00002##
[0074] The subscript on the temperature T is here used as a short
hand notation to indicate that it may be a function of time t.
.alpha..sub.i and .beta..sub.i are the relevant Arrhenius
parameters for the reaction in consideration, which are known with
a very high degree of accuracy.
[0075] The above reactions are spontaneously taking place in
calibrator Cal2 and Cal3 from the moment when Cr and/or Crn are
added during the production process. Therefore, the actual
concentrations of Cr and Crn in the calibrators in a solution pack
strongly depend on the time passed since production and on the
temperature profile T.sub.protiie(t) experienced by the individual
solution pack during that time span. Table 2 shows the changes in
concentration of Cr and Crn in Cal2 and Cal3 as a function of time
and the temperature profile:
TABLE-US-00003 TABLE 3 At time: t > t.sub.production Cal2 Cal3
cCr [.mu.M] .DELTA.cCrn.sub.Cal2(t, T.sub.profile(t))
500-.DELTA.cCr.sub.Cal3(t, T.sub.profile(t)) cCrn [.mu.M]
500-.DELTA.cCrn.sub.Cal2(t, T.sub.profile(t))
.DELTA.cCr.sub.Cal3(t, T.sub.profile(t))
[0076] In Table 3, .DELTA.cCrn.sub.Cal2 is the change in the Crn
concentration in Cal2 up to time t due to its one to one conversion
into Cr; and .DELTA.cCr.sub.Cal3 is the change in the Cr
concentration in Cal3 up to time t due to its one to one conversion
into Crn.
[0077] The concentration change (dcCrn) of cCrn during an
infinitesimal time span (dt) obeys the following differential
equation, also known as the rate equation:
dcCrn dt = k 2 ( T t ) cCr - k 1 ( T t ) cCrn , Equation 5
##EQU00003##
[0078] Combining the reaction rate equations (Equation 5), with the
Arrhenius equations (Equation 4) and the two-temperature model,
provides the analytic expressions cCr.sub.Cal#.sup.model(t.sub.1)
and cCrn.sub.Cal#.sup.model(t.sub.1), which are respectively the Cr
and Crn concentrations in calibrator "Cal#" as functions of the
time t.sub.1 spent at the initial temperature T.sub.1, typically
the lower temperature.
[0079] If each of the true temperature profile
(T.sub.t=T.sub.profile(t)); the relevant Arrhenius-parameters
(.alpha..sub.i and .beta..sub.i) for the above reaction; and the
starting concentrations cCr.sub.Cal#.sup.0 and cCrn.sub.Cal#.sup.0
of Cr and Crn in calibrator Cal# at time t=0, are known, then very
accurate estimates of the true Cr and Crn concentrations in that
calibrator at any time afterwards can be obtained by integration of
the rate equation.
[0080] The true temperature profile experienced by the calibration
solutions during storage, transport, storage at the customer, and
usage, may not be known by the end user if no record of temperature
variation is available. Although the exact temperature profile may
be unknown, there are several variables independent of one another
that can be measured to reveal details of the conditions.
Therefore, in order to obtain a sufficiently accurate estimate of
the unknown true temperature profile we utilize all available
degrees-of-freedom (DOFs) in the cCrn measuring system, that are
not utilized for other purposes, e.g., for calibrating the Crea A
and Crea B sensors.
[0081] The measurements with the Crea A and Crea B sensors on the
two calibrations solutions Cal2 and Cal3 result in 4 sensors
signals, i.e. 4 available DOFs. According to Table 2 and its
supporting text, 3 out of the 4 available DOFs are allocated for
sensor calibration purposes, which leave one unused DOF in surplus:
the Crea A signal on Cal2. This single DOF can be utilized to
determine the value of an "unknown" entity in the same manner as
the other 3 DOFs will be utilized for determining the 3 "unknown"
sensor sensitivities.
[0082] FIG. 3 is a flow chart showing the step-by-step derivation
for the proposed method of determining analyte levels in the
calibrators and subsequently calibrating the sensors.
[0083] As discussed above, the rate equation 310 (Equation 5) is
combined with the Arrhenius equation 311 (Equation 4) and the
temperature model 320. To choose an appropriate temperature model,
one needs to be aware of the general pattern of temperature
fluctuations that are to be expected. Table 1 provides an example
of this general pattern as it indicates the different conditions
(storage at production, transport, storage at customer, and usage)
that the calibrations solutions may undergo. The table also shows
the temperature ranges that the calibration solutions are expected
to experience, and it indicates what time periods these solutions
are expected to remain at these conditions.
[0084] The chosen temperature model may have an initial temperature
(T.sub.1) at an initial time (t.sub.production or t=0), an end
temperature (T.sub.2) at the time of calibration (t.sub.age), and
an intermediate (t.sub.1) at which the temperature changes from the
initial to the end temperature. While the initial time and end time
will be known, and the initial temperature and end temperature are
either known or estimated, the intermediate time t.sub.1 may not be
known at all. In such a scenario, the available DOF can be used to
determine this intermediate time t.sub.1.
[0085] The skilled person would look at the data available, and
construct a temperature model accordingly. Where there is one
available DOF, the temperature model would be constructed so that
there is only one unknown, such as intermediate time t.sub.1. Where
there is more than one available DOF, a temperature model can be
constructed with as many unknowns as there are available DOFs.
[0086] In the embodiment described above, the available DOF is used
to determine the variable parameter t.sub.1, in a temperature model
comprising two known (or estimated) temperatures and the time
t.sub.1 at which the temperature changes between the two. Another
possible use of the available DOF is to determine an average
conversion temperature being used throughout the whole timespan
t.sub.age-t.sub.production, in a temperature model comprising a
single unknown temperature as the variable parameter.
[0087] In the example presented herewith, the true temperature
profile T.sub.profile(t) can be modelled by a two-temperature
model, which depends on initial and end temperatures T.sub.1 and
T.sub.2, end time t.sub.age and intermediate time t.sub.1 (and
implicitly initial time t.sub.0=0). Given the general pattern
identified, the two temperature can modelled as a simple step
change from an initial temperature to an end temperature at the
intermediate time t.sub.1. This model can be represented as:
T profile model ( t ; T 1 , T 2 , t 1 , t age ) = { T 1 , 0
.ltoreq. t .ltoreq. t 1 T 2 , t 1 < t .ltoreq. t age Equation 6
##EQU00004##
[0088] Temperatures T.sub.1 and T.sub.2 can, in principle, be
chosen arbitrarily. However, to ensure that the model can encompass
all possible temperature scenarios, the temperatures T.sub.1 and
T.sub.2 may be set to match the expected temperature ranges. Using
the expected patterns set out in Table 2, the temperatures may be
set to the minimum (2.degree. C.) and maximum temperatures
(32.degree. C.) specified in any of the three preliminary
temperature intervals in Table 2. The initial temperature T.sub.1
may be representative of the expected cold storage temperature of
the calibration solutions, and the end temperature T.sub.2 may be
representative of the temperature the calibration solutions are
expected to experience during transportation and/or during use. As
such, the initial temperature and end temperature may not be the
actual temperature at the initial time t.sub.1 and end time
t.sub.age respectively.
[0089] The parameter t.sub.1 is an unknown amount of time spent at
the temperature T.sub.1. The parameter t.sub.age is the age of the
solution pack relative to the time point t=0 when the analyte
levels (cCr.sub.Cal2.sup.0, cCrn.sub.Cal2.sup.0,
cCr.sub.Cal3.sup.0, cCr.sub.Cal3.sup.0) in the calibrators are
measured using a reference method. The analyte levels and the
absolute time point at which they were measured may be stored
electronically in each individual solution pack and so that they
can be easily input upon analysis.
[0090] At step 330, the combination of the rate equation 310
(Equation 5), the Arrhenius equation 311 (Equation 4) and the
temperature model 320 is solved analytically. This is done by
solving the resultant differential equation twice, i.e., in
segments of constant temperatures T.sub.1 and T.sub.2 with
corresponding time spans t.sub.1 and t.sub.2=t.sub.age-t.sub.1,
respectively, and by using the appropriate starting conditions in
each segment. This leads to the following equations for the Cr and
Crn concentrations in a given calibrator ("Cal#") as function of
t.sub.1:
cCr Cal # model ( t 1 ; t age , T 1 , T 2 ) = k 1 ( T 2 ) K ( T 2 )
c Cal # tot + ( ( k 2 ( T 1 ) K ( T 1 ) - k 2 ( T 2 ) K ( T 2 ) ) c
Cal # tot - ( cCr Cal # 0 - k 1 ( T 1 ) K ( T 1 ) c Cal # tot ) e -
t 1 K ( T 1 ) ) e - ( t age - t 1 ) K ( T 2 ) Equation 7 cCrn Cal #
model ( t 1 ; t age , T 1 , T 2 ) = k 2 ( T 2 ) K ( T 2 ) c Cal #
tot + ( ( k 2 ( T 1 ) K ( T 1 ) - k 2 ( T 2 ) K ( T 2 ) ) c Cal #
tot - ( cCrn Cal # 0 - k 2 ( T 1 ) K ( T 1 ) c Cal # tot ) e - t 1
K ( T 1 ) ) e - ( t age - t 1 ) K ( T 2 ) Equation 8
##EQU00005##
[0091] Where
c.sub.Cal#.sup.tot=cCr.sub.Cat#.sup.0+cCrn.sub.Cal#.sup.0 and
K(T)=k.sub.1(T)+k.sub.2(T) are constants. Equation 7 gives the
concentration of Cr at end time t.sub.age. Many of the parameters
340 of the Equations 7 and 8 are already known, and the unknown
variable is t.sub.1. The values k.sub.1, k.sub.2 and K can be
determined from the Arrhenius equations, where .alpha. and .beta.
are well values that are known to a high accuracy. The
concentrations cCr.sub.Cal#.sup.0, cCrn.sub.Cal#.sup.0 and
c.sub.Cal#.sup.tot are the known concentrations from measuring the
solutions at time t=0. Temperatures T.sub.1 and T.sub.2 and known
from the temperature-model used, and t.sub.age is known from being
the time since production of the calibration solutions. Therefore,
Equations 7 and 8 are the expressions at step 331, but before the
actual concentrations can be calculated, the unknown variable
t.sub.1 needs to be determined from the steps in box 350.
[0092] From the sensor response model 351 in Equation 1, the
following universally valid relationship can readily be derived for
measurements with the uncalibrated Crea A sensor on both
calibrators at a given time, e.g. t=t.sub.age:
I Cal 2 A ( t age ) cCr Cal 2 ( t age ) = I Cal 3 A ( t age ) cCr
Cal 3 ( t age ) Equation 9 ##EQU00006##
[0093] The relationship of Equation 9 is valid for an uncalibrated
Crea A sensor as no calibration parameters, i.e. the Cr sensitivity
S.sub.Cr.sup.A, enter the equation. Equation 9 states that the
ratio between the Crea A sensor signal and the Cr concentration
measured for one calibration solution has the same value as for the
other calibration solutions at a given time.
[0094] The analytical expression for cCr.sub.Cal#.sup.model and
cCrn.sub.Cal#.sup.model in Equations 7 and 8 can now be inserted
into Equation 9 to form the "Cr-identity" 332 Equation 10.
I Cal 2 A ( t age ) I Cal 3 A ( t age ) = cCr Cal 2 model ( t 1 ; t
age , T 1 , T 2 ) cCr Cal 3 model ( t 1 ; t age , T 1 , T 2 )
Equation 10 ##EQU00007##
[0095] The raw outputs of the Crea A sensor for the two calibration
solutions at the end time can be input into the left-hand side of
Equation 10. The right-hand side of the equation is a non-linear
function of the one remaining unknown variable t.sub.1, and is a
monotonic function within most realistic temperature and time
ranges. As a monotonic function, only one solution for t.sub.1
exists, so Equation 10 can be solved by numerical methods to find
t.sub.1 at step 333. As a result, the ratio between the raw signals
of the Crea A sensor leads to one possible value of t.sub.1, and
hence one possible temperature profile approximating the true
temperature profile.
[0096] Once the value of t.sub.1, and consequently the temperature
model, are known Equations 7 and 8 for the concentrations of the
calibration solutions can be solved 334. Specifically, the Cr
concentrations cCr.sub.Cal2 and cCr.sub.Cal3, and the Crn
concentrations cCrn.sub.Cal2 and cCrn.sub.Cal3 can be determined
for the calibration solutions at the end time.
[0097] Once the concentrations of Cr and Crn have been estimated
for the calibration solutions at the end time, it is possible to
compare these concentrations with raw sensor readings of the
calibrations solutions to determine the sensitivities, and hence
calibrate 360, the sensors.
[0098] Equations 1 and 2 provide the sensor response models for the
Crea A and Crea B sensors respectively. At step 361 these sensor
response models can be used to derive the following
relationships:
S Cr A = I Cal 3 A cCr Cal 3 Equation 11 S Cr B = I Cal 2 B cCrn
Cal 3 - I Cal 3 B cCrn Cal 2 cCr Cal 2 cCrn Cal 3 - cCrn Cal 3 cCrn
Cal 2 Equation 12 S Crn B = I Cal 3 B cCr Cal 2 - I Cal 2 B cCr Cal
3 cCr Cal 2 cCrn Cal 3 - cCr Cal 3 cCrn Cal 2 Equation 13
##EQU00008##
[0099] The calibration concentrations can be determined from step
334, and the raw signals for the Crea A sensor at Cal3 and the Crea
B sensor at Cal2 and Cal3 can be determined at step 362. These
values can be substituted into Equations 11, 12 and 13 at step 363
to calculate the sensitivities of the Crea A sensor for Cr, and the
Crea B sensor for Cr and Crn. With these sensitivities, it is
possible to accurately measure the concentrations of Cr or Crn in
any given sample.
[0100] While FIG. 3 outlines the steps for deriving the proposed
method, FIG. 4 summarises the steps for carrying out an example
embodiment of the proposed method. The proposed method is not
limited to the ordering of the steps shown in FIG. 4, nor is the
method envisioned to be solely limited to this example embodiment
provided.
[0101] At step 410, the concentrations of the Cr and Crn in the
calibration solutions are measured and recorded at an initial time.
The initial time may be just after production of the solutions
and/or at any appropriate time before dispatch of the calibration
solutions. The concentrations may be measured using a range of
techniques and known sensors, such as by high-performance liquid
chromatography (HPLC). The concentrations may be recorded by any
means that allow the end user to use these recorded concentrations
in the subsequent calculations. For example, the concentrations may
be stored in writing on the calibration pack, or may be stored
electronically at a server, or may be stored electronically at the
calibration pack itself so that a calibration machine can
automatically read the stored variable. Similarly, the packs may be
time-stamped so that the initial time will be known when performing
the subsequent calculations.
[0102] At step 420 the calibration solutions are dispatched to the
end consumer. From this point, the actual temperatures the
calibration solutions are stored at may not be known.
[0103] Box 430 illustrates the steps that can be taken by an end
user attempting to calibrate the Cr and Crn sensors, starting with
receiving the calibration solutions and starting the calibration
process at an end time (also referred to as t.sub.age) 440.
[0104] At step 450, the raw Cr and Crn signals in the calibration
solutions are measured by the sensors. These raw signals are the
current outputs I.sub.Cal2.sup.A(t.sub.age),
I.sub.Cal3.sup.A(t.sub.age), I.sub.Cal2.sup.B(t.sub.age) and
I.sub.Cal3.sup.B(t.sub.age) of the sensors.
[0105] At step 460, the temperature model is determined. In the
example of a two-temperature model, this involves calculating the
intermediate time t.sub.1 by solving Equation 10 for t.sub.1. To
perform this calculation, the Arrhenius values for the relevant
reactions need to be known, along with the initial time and initial
concentrations that were recorded at step 410.
[0106] At step 470, the actual concentrations of Cr and Crn in the
calibration solutions are calculated by solving Equations 7 and 8
using the temperature model determined at step 460.
[0107] At step 480, the sensor outputs calculated in step 450
(specifically, I.sub.Cal3.sup.A(t.sub.age),
I.sub.Cal2.sup.B(t.sub.age) and I.sub.Cal3.sup.B(t.sub.age)) and
the concentrations calculated at step 470 are substituted into
Equations 11 to 13 to determine the sensor sensitivities, thereby
calibrating the sensors.
[0108] The steps performed at box 430 may be performed manually by
an end user. Alternatively, some or all of the steps at box 430 can
be automated by a system. For example, a calibration system may
take the calibration pack and automatically read electronic data
indicating the initial concentrations and initial time. The
calibration system may measure all the raw sensor outputs as
indicated in steps 450 and 470. The calibration system may contain
an electronic device with a processor for performing the
calculations of steps 460, 470 and 480. Computer software may be
supplied in a computer-readable medium that the user can install on
their own computer to automatically perform any of the calculations
of the method.
[0109] While the temperature model used to calibrate the sensors is
only an estimation of the true temperature profile of the
calibration solutions, the proposed solution provides very accurate
results. FIG. 5 shows a number of graphs comparing the actual
concentrations of the calibration solutions (using HPLC, for
example) with the concentrations calculated with the proposed
method.
[0110] The tests were performed for Cal2 and Cal3 solutions stored
at different temperatures for a variety of time spans, which
correspond to realistic time/temperature scenarios for the solution
packs, but also to extreme scenarios which exceed the specified
time/temperature ranges in Table 1.
[0111] Data points 511, 521, 531 and 541 are for calibration
solutions stored at 10.degree. C. for 53 days. Data points 512,
522, 532 and 542 are for calibration solutions stored for 53 days
at 10.degree. C., 15 days at 32.degree. C., and 34 days at
10.degree. C. Data points 513, 523, 533 and 543 are for calibration
solutions stored for 53 days at 10.degree. C., 15 days at 6.degree.
C., 29 days at 10.degree. C., and 20 days at 6.degree. C. Data
points 514, 524, 534 and 544 are for calibration solutions stored
for 53 days at 10.degree. C., 15 days at 25.degree. C., 29 days at
10.degree. C., and 20 days at 25.degree. C. Data points 515, 525,
535 and 545 are for calibration solutions stored for 53 days at
10.degree. C., 32 days at 32.degree. C., 29 days at 10.degree. C.,
and 20 days at 32.degree. C.
[0112] Graph 510 shows the calculated and actual Cr concentrations
in the Cal2 solution at different temperature conditions. The
linear trendline 516 (formula y=1.0317x+2.0703) indicates a linear
relationship between the calculated and actual concentrations, with
an R.sup.2 value of 0.98. Graph 520 shows the calculated and actual
Crn concentrations in the Cal2 solution at different temperature
conditions. The linear trendline 526 (formula y=0.9466x+24.0)
indicates a linear relationship between the calculated and actual
concentrations, with an R.sup.2 value of 0.9786. Graph 530 shows
the calculated and actual Cr concentrations in the Cal3 solution at
different temperature conditions. The linear trendline 536 (formula
y=0.9522x+17.951) indicates a linear relationship between the
calculated and actual concentrations, with an R.sup.2 value of
0.97. Graph 540 shows the calculated and actual Crn concentrations
in the Cal3 solution at different temperature conditions. The
linear trendline 546 (formula y=0.9621x+5.5329) indicates a linear
relationship between the calculated and actual concentrations, with
an R.sup.2 value of 0.97.
[0113] These graphs show that even though the temperature model is
only an estimate of the true temperature profile, the
concentrations calculated with the proposed method are consistently
close to the actual concentrations.
[0114] FIG. 6 shows a number of graphs comparing the actual
concentrations of the calibration solutions (using HPLC, for
example) with the concentrations calculated with the example
embodiment using a temperature logger to generate a temperature
model.
[0115] The tests were performed for Cal2 and Cal3 solutions stored
at different temperatures for a variety of time spans, which
correspond to realistic time/temperature scenarios for the solution
packs, but also to extreme scenarios which exceed the specified
time/temperature ranges in Table 1.
[0116] Graph 610 shows the calculated Cr concentrations in the Cal2
solution compared to the actual concentrations. The linear
trendline 611 (formula y=1.0395x-0.24446) indicates a linear
relationship between the calculated and actual concentrations, with
an R.sup.2 value of 0.9899. Graph 620 shows the calculated Crn
concentrations in the Cal2 solution compared to the actual
concentrations. The linear trendline 621 (formula y=0.9681x+15.401)
indicates a linear relationship between the calculated and actual
concentrations, with an R.sup.2 value of 0.992. Graph 630 shows the
calculated Cr concentrations in the Cal3 solution compared to the
actual concentrations. The linear trendline 631 (formula
y=0.931x+32.896) indicates a linear relationship between the
calculated and actual concentrations, with an R.sup.2 value of
0.9656. Graph 640 shows the calculated Crn concentrations in the
Cal3 solution compared to the actual concentrations. The linear
trendline 641 (formula y=0.9465x+1.5862) indicates a linear
relationship between the calculated and actual concentrations, with
an R.sup.2 value of 0.9707.
[0117] These graphs show that using temperature probes to record
historic changes in temperature can lead to very accurate
calculations for Cr and/or Crn concentrations.
[0118] It is to be understood that the present disclosure includes
permutations of combinations of the optional features set out in
the embodiments described above. In particular, it is to be
understood that the features set out in the appended dependent
claims are disclosed in combination with any other relevant
independent claims that may be provided, and that this disclosure
is not limited to only the combination of the features of those
dependent claims with the independent claim from which they
originally depend.
* * * * *