Isec Manual

ISEC User's Manual(working version)

Version 3.4a

Roman Hovorka

September 1994

READ FIRST

ISEC, which stands for "I(nsulin-)SEC(retion)", is a computer program which has beendesigned and developed by Roman Hovorka during his postdoctoral fellowship at theCentre for Measurement & Information in Medicine, City University, funded by GlaxoResearch and Development, Ltd., and is available, at no charge, to interestedinvestigators. A condition of this availability is that any manuscript and report basedupon the ISEC software acknowledges the source of ISEC with a statement that ISEC canbe obtained from the author, Roman Hovorka, PhD, Metabolic Modelling Group, Centrefor Measurement and Information in Medicine, Department of Systems Science, CityUniversity, Northampton Square, London EC1V OHB, United Kingdom.

DISCLAIMER

THIS SOFTWARE IS PROVIDED ``AS IS'' AND ANY EXPRESS OR IMPLIEDWARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIESOF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE AREDISCLAIMED. IN NO EVENT SHALL Roman Hovorka BE LIABLE FOR ANY DIRECT,INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIALDAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTEGOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESSINTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDINGNEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THISSOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

AIM OF ISEC

ISEC estimates insulin secretion from plasma C-peptide concentrations.

AIM OF MANUAL

The manual shows how to use ISEC. An overview of methods adopted by ISEC is given.No special knowledge is required except for familiarity with the PC environment.

NOTE

Due to a demand for a revised version of the manual a working version has been produced

which does not include technical and implementation details. Such details will bereported separately or in an updated version of the manual.

ISEC V3.4a User's Manual (working version) Page 3

CONTENT

1. INSTALLING ISEC ............................................................................................................ 3

2. HARDWARE REQUIREMENTS ....................................................................................... 4

3. RUNNING ISEC ................................................................................................................. 53.1 First Input File...................................................................................................... 53.2 Second Input File .................................................................................................. 53.3 Output File ............................................................................................................ 5

4. INSULIN SECRETION...................................................................................................... 64.1 Introduction........................................................................................................... 64.2 Co-secretion of Insulin and C-peptide.................................................................. 64.3 Plasma C-peptide as Indicator of Insulin Secretion............................................ 64.4 Measurement Error .............................................................................................. 6

5. ISEC OVERVIEW............................................................................................................... 75.1 Regularisation Method of Deconvolution Constrained to Non-Negative

Values .............................................................................................................. 75.2 C-peptide Kinetics ................................................................................................. 75.3 Segmentation of Study Period.............................................................................. 75.4 Study Sub-Periods................................................................................................. 75.5 Basal Secretion...................................................................................................... 8

6. EXAMPLE RUN OF ISEC.................................................................................................. 96.1 First Input File TEST ......................................................................................... 106.2 Second Input File MODE.................................................................................... 116.3 Output File RESULTS........................................................................................ 12

7. REMARKS......................................................................................................................... 147.1 Assumptions ........................................................................................................ 147.2 Point-Area Deconvolution and ISEC.................................................................. 147.3 Execution Time.................................................................................................... 147.4 Implementation and Portability......................................................................... 147.5 Missing Values .................................................................................................... 147.6 User's Estimation of the Measurement Error .................................................. 147.7 Format Notes....................................................................................................... 15

APPENDIX A ........................................................................................................................ 16A.1 Run-Time Errors................................................................................................. 16

ACKNOWLEDGMENT ........................................................................................................ 18

REFERENCES...................................................................................................................... 19


1. INSTALLING ISEC

To install the program, copy ISEC.EXE from the distribution disk into an arbitrarydirectory on the hard disk of your computer.


2. HARDWARE REQUIREMENTS

The version of ISEC provided by the author requires an IBM PC compatible computerwith MS-DOS version 4.0 or higher, an 80386 processor or higher, and at least 340 kBRAM. A numerical co-processor is not required although it is recommended as heavynumerical calculations are involved.


3. RUNNING ISEC

Choose the directory where ISEC is installed as the default directory, type ISEC followedby the names of two input files and the name of an output file. Once started, ISEC readsdata from the two input files and reports its results in the output file. No interactionbetween the program and the user is required during the execution of the program. ISECdoes not have a graphical output.

3.1 First Input File

The first input file contains identification of the data set, the subject's sex, classification,age, height, weight and the time course of C-peptide concentrations. Classificationindicates whether the subject is normal, obese (weight > 115% ideal body weight) or hasnon-insulin dependent diabetes mellitus (NIDDM).

3.2 Second Input File

The second input file indicates the error of the C-peptide measurements and thesegmentation of the study period. The error of C-peptide measurements is represented asa coefficient of variation (CV).

3.3 Output File

The results of the calculations are reported in the output file. The file containsidentification of the data set, error assessment, and the time course of insulin secretionand C-peptide concentrations.


4. INSULIN SECRETION4.1 Introduction

Insulin is secreted by beta-cells in the pancreas. Direct measurement of the rate of insulinsecretion requires catheterisation of the hepatic portal vein and is therefore not suitablefor typical clinical conditions. Indirect estimation employing plasma insulin concentrationfails due to large and variable hepatic extraction of insulin.

In normal subjects, insulin secretion varies greatly during the day and is modulated bymany plasma-born substances among which plasma glucose is the major determinant.

Insulin secretion is impaired in patients with non-insulin dependent diabetes mellitus(NIDDM). Normalisation of insulin secretion is beneficial to patients with NIDDM.Quantification of insulin secretion allows an assessment of pancreatic responsiveness tobe made. It is therefore possible to test the effect of various treatments on insulinsecretion. Naturally, many other clinical/research situations exist in which thequantification of insulin secretion is useful.

4.2 Co-secretion of Insulin and C-peptide

In 1969, Rubenstein et al (1) showed that insulin and a polypeptide known as C-peptideare secreted by beta-cells in a one-to-one molar ratio. Unlike insulin, C-peptide is notcleared by the liver (2) and its plasma concentration is a good indicator of secretion. Inthis manual, C-peptide secretion and insulin secretion are interchangeable as they arenumerically equal in molar terms.

4.3 Plasma C-peptide as Indicator of Insulin Secretion

Plasma C-peptide concentration is an indirect indicator of insulin secretion. As thekinetics of C-peptide are linear (3), secretion is proportional to plasma concentrationduring steady-state conditions. Under non-steady-state conditions, however, this simplerelationship does not hold. In such a case, C-peptide plasma concentration is affected notonly by 'immediate' secretion but also 'previous' secretion. Plasma concentration at anygiven time results from the accumulation of C-peptide in plasma due to immediate andpast secretion. This accumulation process is usually denoted by the mathematical term'convolution'.

To reconstruct secretion from plasma concentration, an inverse operation has to beadopted. Not surprisingly, the operation is called de(-)convolution and ISEC adopts onerepresentation of this inverse operation named the regularised method of deconvolution(4,5).

4.4 Measurement Error

In the case of C-peptide, the measurement error has a great impact on the deconvolutionof the secretion profile. When the measurement error is not taken into consideration, e.g.when point-area deconvolution method is employed, deconvolution under frequentsampling of plasma C-peptide (1 to 30 minutes sampling frequency) gives unrealisticoscillations in the time course of insulin secretion. The frequency of oscillations usuallyreflects the sampling frequency. Often, negative secretion values are calculated.


5. ISEC OVERVIEW5.1 Regularisation Method of Deconvolution Constrained to Non-NegativeValues

ISEC implements a regularisation method of deconvolution (4,5) constrained to non-negative values.

ISEC produces a non-negative secretion profile which satisfies the two followingconditions. First, plasma concentration calculated from the secretion profile differs fromthe measured concentration by a pre-defined quantity. Second, the secretion profile hasminimum oscillations, as quantified by a norm of second differences, among those profileswhich satisfy the former condition.

The first condition indicates that the secretion profile does not have to result in plasmaconcentrations which exactly match the measured concentrations. The difference ishowever pre-defined and usually represents the measurement error.

In general, there are many (in fact an infinite number of) secretion profiles which satisfythe former condition. A preference favouring profiles with low oscillations is adopted byISEC.

This results in peaks and troughs in the time course of insulin secretion, caused by themeasurement error, to be 'smoothed out'. Note that 'true' secretion peaks and troughs arenot smoothed out under the condition that the measurement error is small whencompared to the increase (decrease) in plasma concentration resulting from the change insecretion. Naturally, changes resulting in deviation of plasma concentration within themeasured error cannot be identified.

5.2 C-peptide Kinetics

Plasma C-peptide concentration results from two simultaneously occurring processes, C-peptide appearance (secretion) and C-peptide disappearance (disposal). In principle,plasma concentration alone is not sufficient to estimate appearance without theknowledge of the disappearance and vice versa. ISEC determines the C-peptidedisappearance model (kinetic model) from subject's individual data. The determination isbased on a population-based study (6) which has shown that the parameters of the C-peptide kinetic model can be approximated from the subject's weight, height, age, sex andclassification (normal, obese, NIDDM). The use of approximated parameters doesintroduce an error though. However, it has been shown that this error is comparable tothe variations due to a duplicate experimental assessment of the parameters (6).

5.3 Segmentation of Study Period

ISEC requires the study period to be divided into distinct segments. Insulin secretion iscalculated as a step-wise function given the segmentation. The segmentation does nothave to follow the sampling schedule although it usually will have the form of the refinedsampling schedule. A fine segmentation should be adopted when rapid changes in insulinsecretion are expected, a crude segmentation can be adopted when insulin secretion isexpected to change slowly.

5.4 Study Sub-Periods


A study often includes landmark points which divide the study into two or more distinctsub-periods. Most studies include basal and post-intervention periods, for example.Smoothing of the secretion between the two sub-periods is often an unwanted feature asour a priori expectation is that there might be a sudden change in insulin secretion. ISECutilises this a priori knowledge of the study design. Oscillations in C-peptide secretion arequantified separately for each study sub-period. In consequence, the calculated timecourse of C-peptide secretion loses its 'smoothness' at landmark points.

No limit to the number of sub-periods is imposed by ISEC.

5.5 Basal Secretion

Basal secretion is often of importance. It can be defined as a constant insulin secretionwhich results in basal C-peptide concentration.

Smoothing sometimes interferes with the calculation of the basal secretion. To avoid thisinterference, ISEC includes a feature which allows the basal secretion to be calculatedcorrectly without the interference of smoothing. To employ this feature, a 'basal' C-peptidemeasurement must be specified. Such a measurement is then treated by ISEC as a'perfect' measurement with the measurement error approaching zero. The constraineddeconvolution method is then forced to fit the basal C-peptide concentration and producesthe correct basal secretion.

Note that in the current version of ISEC only one C-peptide measurement can be specifiedas basal. Should several basal C-peptide concentration been measured experimentally, amean value should be calculated and supplied to ISEC.


6. EXAMPLE RUN OF ISEC

Select the directory where ISEC is installed to be the default directory. Copy files TESTand MODE which are included on the distribution disk to this default directory (TESTcontains the time course of C-peptide concentration measured in a normal subjectfollowing a meal at time zero).

Type

ISEC TEST MODE RESULTS

Allow about 40 seconds execution time on a 486DX 33 MHz PC-AT compatible computerduring which CODE prints on the screen various symbols indicating the progress of thecalculations. Results will then be stored in the file RESULTS.


6.1 First Input File TEST

ID: TEST-DATASEX: MSUBJECT: normalAGE: 30 yearsHEIGHT: 179 cmWEIGHT: 81.1 kgTIME (min) C-PEPTIDE (pmol/l)0 700 basal15 80030 117045 201060 155075 151090 2170120 1240140 1020180 950210 720

The first line indicates the name of the data set to beTEST-DATA. An arbitrary string with no blanks isaccepted. The string should not be longer than 100characters.

The second line indicates the subject to be M(ale). Toindicate the opposite sex, use F(emale).

The third line indicates the patient's classification tobe normal. On of the three categories must bespecified: normal, obese or niddm, which stands fornormal subjects, obese subjects (body weight > 115%ideal body weight), and patients with NIDDM,respectively.

The next three lines indicate age, height and weight,respectively.

The time course of C-peptide measurements follows.Each line contains the time of a measurement andthe measured C-peptide concentration in pmol/l.Missing values are specified using a -999.0 value inthe C-peptide measurement column.

The basal keyword indicates that the measurementat time 0 should be considered 'perfect', i.e. with themeasurement error approaching zero. basal can bespecified only for the first measurement.


6.2 Second Input FileMODE

Error-CV: 5.0 %Constraint: non-negativeReporting: 5 minSegmentation0 break5101520253035404550556065707580859095100105110115120125130135140145150155160165170175180185190195200205210

The first line indicates that the measurement erroris 5%.

The second line is included for historical reasons.

The third line indicates the time course of C-peptideconcentration calculated by ISEC should be reportedevery 5 minutes.

The specification of segmentation follows. There are43 segments defined:prior to 0 min,interval 0 to 5 min,interval 5 to 10 min,......interval 205 to 210 min.

A landmark point dividing the study into two sub-periods is defined using the break keyword at timezero.


6.3 Output File RESULTS

ISEC V3.4a June 1994ID: TEST-DATADate: Thu Oct 20 20:52:05 1994CV wanted: 5 %CV achieved: 5.00 %Constraint: LINEAR

Estimated insulin secretion:

time secretion(min) (pmol/kg/min)

from to0.0 2.16 break

0.0 5.0 1.925.0 10.0 2.5810.0 15.0 3.2615.0 20.0 4.0320.0 25.0 5.0625.0 30.0 6.3630.0 35.0 7.7035.0 40.0 8.4340.0 45.0 8.1645.0 50.0 6.9350.0 55.0 5.4955.0 60.0 4.4560.0 65.0 4.1665.0 70.0 4.6070.0 75.0 5.6475.0 80.0 6.9680.0 85.0 7.9285.0 90.0 8.0990.0 95.0 7.3295.0 100.0 6.01100.0 105.0 4.50105.0 110.0 3.08110.0 115.0 1.99115.0 120.0 1.39120.0 125.0 1.30125.0 130.0 1.50130.0 135.0 1.84135.0 140.0 2.18140.0 145.0 2.43145.0 150.0 2.61150.0 155.0 2.72155.0 160.0 2.77160.0 165.0 2.77165.0 170.0 2.72170.0 175.0 2.63175.0 180.0 2.51180.0 185.0 2.36185.0 190.0 2.20190.0 195.0 2.01195.0 200.0 1.82200.0 205.0 1.61205.0 210.0 1.41

Fit to C-peptide concentration:

time calculated measured diff. CV(min) (pmol/l) (pmol/l) (pmol/l) (%)0.0 700.0 700.0 -0.0 0.0 basal5.0 682.410.0 719.815.0 792.3 800.0 -7.7 1.020.0 894.825.0 1036.930.0 1226.0 1170.0 56.0 4.835.0 1451.040.0 1657.245.0 1782.7 2010.0 -227.3 11.350.0 1790.355.0 1710.060.0 1600.5 1550.0 50.5 3.365.0 1522.370.0 1511.075.0 1581.2 1510.0 71.2 4.780.0 1722.085.0 1882.3

90.0 2000.0 2170.0 -170.0 7.895.0 2027.0

The first line indicates the version of ISEC andits release date.

The second line repeats the name of the dataset.

The third line specifies the date and time of therun.

The fourth line repeats the CV specified in thesecond input file (5%).

The fifth line reports the CV achieved by ISEC(also 5%). The two values should be equal. Ifdifferent, the user has either underestimated oroverestimated the error in the C-peptidemeasurements, or ISEC could not fit themeasured C-peptide concentrations using thespecified segmentation. The latter case isprobably due to the segmentation being too'crude'. As a rule of thumb, the segmentationshould include at least one segment per C-peptide measurement.

The sixth line is included to support furtherdevelopment of ISEC.

A list of insulin secretion rates follows.

The first rate indicates secretion of2.16 pmol/kg/min prior to 0 min.

The rate on the next line indicates secretion inthe segment 0 to 5 min. Secretion of1.92 pmol/kg/min was calculated in thissegment.

Rates on the following lines should beinterpreted in a similar way.

The break keyword is included to confirm theinput specification. The same applies for thekeyword basal in the list below.

A second list follows. It contains the timecourse of C-peptide concentration reportedevery five minutes. The report frequency is as


specified in the second input file. The list canbe regarded as the 'fit' to the measured C-peptide concentration.

100.0 1959.4105.0 1816.9110.0 1630.5115.0 1435.5120.0 1264.7 1240.0 24.7 2.0125.0 1140.6130.0 1064.7135.0 1027.1140.0 1013.8 1020.0 -6.2 0.6145.0 1011.2150.0 1011.9155.0 1012.0160.0 1009.1165.0 1002.0170.0 990.0175.0 972.6180.0 950.0 950.0 0.0 0.0185.0 922.2190.0 889.5195.0 852.4200.0 811.4205.0 767.1210.0 720.1 720.0 0.1 0.0


The list also contains measuredconcentrations, differences between measuredand calculated concentrations, and thedifferences represented as the coefficient ofvariation (CV)

The average CV is indicated by theCV achieved heading. It is calculatedaccording to the formula

where N is the number of measurements. The 'perfect' basal measurement is notincluded in the calculation of CV achieved.

The results, measured C-peptide concentrations, ISEC calculated C-peptideconcentrations, and estimated insulin secretion are summarised in Fig. 1.

% 100 x measured

|calculated-measured|= CV

% x100measured

) calculated- measured( N1=achieved CV 2

i

2ii

N

=1i��

Figure 1. Calculated insulin secretion. Plot includes C-peptide measurements (�),insulin secretion (solid line) and model fit to C-peptide concentration (dotted line).


7. REMARKS7.1 Assumptions

The following set of assumptions is adopted by ISEC in calculating insulin secretion:

1) C-peptide kinetics are described by a two-compartment model (7).

2) The parameters of the model of C-peptide kinetics are approximated from the subject'sweight, height, age, sex and classification (6).

3) The measurement errors are uncorrelated, normal with zero mean, and with a constantcoefficient of variation.

7.2 Point-Area Deconvolution and ISEC

ISEC can be used to calculate insulin secretion using the point-area deconvolutionmethod. To carry out the point-area deconvolution, specify a zero value for Error-CV inthe second input file and indicate a segmentation equal to the sampling schedule. Thecalculated time course of insulin secretion then results in C-peptide plasmaconcentrations which closely match measured concentrations.

7.3 Execution Time

The number of segments is the major determinant of the execution time. As a rule ofthumb, the calculation time is linearly related to the number of segments, e.g. doublingthe number of segments results in an two-fold increase in the execution time.

In the example run, there are 43 segments and the calculations took about 40 seconds ona 486DX 33 MHz PC-AT compatible computer. Halving the segment length would resultin a two-fold increase in the number of segments and a corresponding a two-fold increasein the execution time.

7.4 Implementation and Portability

ISEC is written in C++ language for the Borland C++ compiler Version 3.1. No testing hasbeen performed to investigate the portability to other C++ compilers. The source code isthe property of the author and no modifications or permanent changes should be madewithout the written consent of the author.

7.5 Missing Values

ISEC allows missing values to be specified. To indicate a missing value, use a -999.0 asthe value in the C-peptide measurement column in the first input file.

7.6 User's Estimation of the Measurement Error

The intra-assay error associated with C-peptide measurements, expressed as thecoefficient of variation (CV), is at best around 4%. It is recommended to increase this errorby at least 1% considering errors from other error sources, e.g. the error in sampledilution and unreported sampling time deviations. A value of at least 5% (depending onthe C-peptide assay adopted) should be therefore used in the Error-CV data item in the


second input file.

7.7 Format Notes

The format of the two input files is described in the example. The following additionalnotes apply, namely:1) Any number of spaces (" ") is allowed whenever a single space is acceptable.2) Capital and lower case letters are interchangeable.


APPENDIX AA.1 Run-Time Errors

During execution, ISEC may terminate with an error message. A list of error messages (initalics) is listed below together with a short explanation (normal text) of the nature of theerror.

Use: ISEC <input_file_1 name> <input_file_name 2> <output_file name>ISEC was called with less than three parameters

Cannot open input file <file_name>A file with the name specified cannot be opened for input. Check the existence of the file.

Cannot open output file <file_name>A file with the name specified cannot be opened for output. Check the availability of spaceon your computer or existence of a file path, if this is used to specify the name of theoutput file.

ID section missingThe ID section of the first input file is missing/incorrect. Check the first input file.

SEX section missingThe SEX section of the first input file is missing/incorrect. Check the first input file.

Bad SEX descriptionA wrong identifier of sex. Use M(ale) or F(emale) in the first input file.

SUBJECT section missingThe SUBJECT section of the first input file missing/incorrect. Check the first input file.

Bad SUBJECT descriptionA wrong identifier in the SUBJECT section of the first input file. Use normal, obese, orniddm.

AGE section missingThe AGE section of the first input file is missing/incorrect. Check the first input file.

HEIGHT section missingThe HEIGHT section of the first input file is missing/incorrect. Check the first input file.

WEIGHT section missingThe WEIGHT section of the first input file is missing/incorrect. Check the first input file.

TIME C-PEPTIDE section missingThe label line TIME C-PEPTIDE is missing/incorrect in the first input file. Check the firstinput file.

TIME C-PEPTIDE section:"BASAL" can be specified only for the first measurementOnly the first measurement can be specified as 'basal'. Check the first input file.


Nonnumeric values in TIME C-PEPTIDE sectionA non-numeric value occurred in the time course of C-peptide concentration. Check thefirst input file.

ERROR section missing/incorrectThe ERROR section of second input file is missing/incorrect. Check the second input file.

CONSTRAINT section missing/incorrectThe CONSTRAINT section of the second input file is missing/incorrect. Check the secondinput file.

REPORTING section missing/incorrectThe REPORTING section of the second input file is missing/incorrect. Check the secondinput file.

SEGMENTATION section missingThe SEGMENTATION section of the second input file is missing/incorrect. Check thesecond input file.

Nonnumeric values in SEGMENTATION sectionA non-numeric value occurred in the SEGMENTATION section of the second input file.Check the second input file.

No memory in heapNo additional memory available on the program heap. Mostly due to too many segmentsspecified in the second input file. As a rule of thumb, having 600 kB of continuous RAMavailable, it should be possible to specify up to 1000 segments.

Root must be bracketed in RTSAFEThe initialisation of non-linear regression analysis failed probably due to too few (orwrongly spaced) segments. Try to modify the segmentation in the second input file.

Maximum number of iterations exceeded in RTSAFENon-linear regression analysis could not converge with a sufficient precision. Try tomodify segmentation (decrease the number of segments).

ANY OTHER ERROR e.g.Bad dimensions in "+"Bad dimensions in "-"Bad dimensions in "*"etc.

Internal errors which should not occur. If they do, try to modify segmentation. If it doesnot help please send the two input files which caused the run-time error to the author. Aneffort will be made to remove the potential bug from the next release.


ACKNOWLEDGMENT

Thanks to my colleagues Drs Malcolm A Young, Paul Soons, and Tom R Hennessy fortheir help during the preparation of this manual and for stimulating ideas which, Ibelieve, have improved the performance and the user-acceptance of ISEC. Though it ishard for me to say I am nevertheless the only person responsible for all the errors and theomissions present in the user manual and in the program. Special thanks to Dr David JAEckland for critical but constructive discussions which stimulated further developmentsof ideas adopted by ISEC. Thanks also to Prof Ewart R Carson and Dr Alan Bye forcontinuous support during my work on ISEC.


REFERENCES

1. Rubenstein, A. H., J. L. Clark, F. Melani, and D. F. Steiner. 1969. Secretion ofproinsulin C-peptide by pancreatic beta cells and its circulation in blood. Nature224:697-699.

2. Polonsky, K. S., J. B. Jaspan, W. Pugh, D. Cohen, M. Schneider, T. Schwartz, A. R.Moossa, H. Tager, and A. H. Rubenstein. 1983. Metabolism of C-peptide in the dog: in vivodemonstration of the absence of hepatic extraction. J. Clin. Invest. 72:1114-1123.

3. Polonsky, K. S., B. H. Frank, W. Pugh, A. Addis, T. Karrison, P. Meier, H. Tager, andA. H. Rubenstein. 1986. The limitations to and valid use of C-peptide as a marker of thesecretion of insulin. Diabetes 35:379-386.

4. Twomey, S. 1965. The application of numerical filtering to the solution of integralequations encountered in indirect sensing measurements. J. Franklin Inst. 279:95-109.

5. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. 1992. NumericalRecipes in C. Cambridge University Press, Cambridge.

6. Van Cauter, E., F. Mestrez, J. Sturis, and K. S. Polonsky. 1992. Estimation of insulinsecretion rates from C-peptide levels: Comparison of individual and standard kineticparameters for C-peptide clearance. Diabetes 41:368-377.

7. Eaton, R. P., R. C. Allen, D. S. Schade, K. M. Erickson, and J. Standefer. 1980.Prehepatic insulin production in man: kinetic analysis using peripheral connectingpeptide behaviour. J. Clin. Endocrinol. Metab. 51:520-528.

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