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Applied Mathematics and Computation 196 (2008) 185–192

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Improving the precision of model parameters using modelbased signal enhancement and the linear minimalmodel following an IVGTT in the healthy man

Margarita Fernandez *, Dionisio Acosta, Adolfo Quiroz

Dpto. de Computo Cientifico y Estadistica, Universidad Simon Bolivar, Apartado Postal 89000, Caracas 1080-A, Venezuela

Abstract

The problem of signal enhancement has been addressed by several authors in the past and continues to be of particularinterest in many applications. In this respect, the present authors have been exploring the effect of the model based signalenhancement (MBSE) approach to recover the signal of blood glucose dynamics from noise contaminated measurementscollected from seven healthy patients after an intravenous glucose tolerance test (IVGTT). These observations correspondto a system with an impulse–response behaviour for which it is often hypothesized that a sum of exponential signals can beused for modeling the data. The exponential model order has been derived from the singular value decomposition analysisof these data set. A linear version of the classic minimal model, known as the linear minimal model (LMM), has been usedto model the patient’s behaviour. After fitting the LMM first to the experimental data and then to the MBSE signalobtained from the exponential modelling approximation, the effect on the precision of the LMM parameters has been sta-tistically assessed. A non-parametric test has been devised to evaluate the significance of the differences between the pre-cision obtained when no MBSE is applied and the precision after MBSE is performed. The results obtained suggest that theprecision of the LMM parameters can be improved by more than 50% (p-value < 0.01) for all the model parameters. Inparticular, the insulin sensitivity SI and glucose effectiveness SG parameters that are useful diagnostic indices in Type 2Diabetes Mellitus are improved by 50% and 62% respectively.� 2007 Elsevier Inc. All rights reserved.

Keywords: Minimal model; Signal processing; Sensitivity index; Non-parametric testing; Intravenous glucose tolerance test; Insulin;Glucose

1. Introduction

The problem of signal enhancement has been addressed by several authors in the past and continues to beof particular interest in many applications. These algorithms propose different methods to recover the signalfrom noise contaminated measurements to achieve a signal that is of better quality.

0096-3003/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2007.05.044

* Corresponding author.E-mail address: margarita@cesma.usb.ve (M. Fernandez).

186 M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192

Among these methods, the model based signal enhancement approach was presented by Cadzow in 1988 [1]based on the fact that the underlying signal is known or is hypothesized to possess well defined attributes orproperties that can be used in the recovery process. The modified signal will possess these theoretical proper-ties and will intuitively provide a better representation of the underlying process being measured than did theoriginal noise contaminated signal [1]. Its true utility however is dependent on the user’s ability in selectingsignal properties which characterize a given application [1].

In this respect, the present authors have been exploring the effect of this signal enhancement approach torecover the signal of blood glucose dynamics from noise contaminated measurements. These observations cor-respond to a system with an impulse–response behaviour for which it is often hypothesized that a sum of expo-nential signals can be used for modeling the data [2]. The use of the signal enhancement algorithm forexponential models is presented by Cadzow in [1], where the theoretical background concerning exponentialmodels for analyzing experimentally obtained data is detailed.

The first study we have performed so far corresponds to data taken from Type I diabetic patients after anoral glucose tolerance test (OGTT) [3]. After recovering the signal by using the above described model basedsignal enhancement (MBSE) technique, a simple model of the blood glucose dynamics, namely the linear min-imal model (LMM), was fitted to the enhanced signal and the effect on the model parameters precision wasanalysed. The results obtained suggest that the use of this technique might be useful for improving the preci-sion of the model parameters, although the statistical significance of these results could not be properlyassessed due to the small size of the sample.

The aim of the present study has been therefore to perform the analysis on a larger data set that includes theblood glucose from seven healthy patients after an intravenous glucose tolerance test (IVGTT). In contrast tothe previous study, this data set also includes the plasma insulin measurements so there is no need to model thedynamics of plasma insulin.

As in [3], the LMM has been used to model the patient’s behaviour. This model is formally equivalent to thenonlinear minimal model presented by Bergman et al. in 1979 [4] and has been successfully validated in pre-vious studies, including Type I [3,5] and Type II [6,7] diabetic patients. It has the advantage of its linear struc-ture which makes it simpler and a good candidate for the closed loop control of glucose.

2. Methods

2.1. Data

Data from seven healthy patients (i 6 7) was collected following an standard IVGTT. The exogenous doseof glucose is known to be 300 mg/kg of body weight and it was administered at time 0 during 60 s. Sampleswere taken after overnight fast at �30, �15, 0, 2, 3, 4, 5, 6, 8, 10, 15, 20, 25, 30, 40, 60, 80, 100, 120, 140, 160,180, and 240 min. The body weight was also recorded for all patients under study.

2.1.1. Exponential model order selection

The data to be analyzed are specified by xð1Þ; xð2Þ; . . . ; xðNÞ over the time interval 1 6 n 6 N, which can intheory be represented as a pth-order exponential signal as follows:

xðnÞ ¼Xq

k¼1

ak

Ymk�1

m¼0

ðnÞmðzkÞn; ð1Þ

where the ak and zk have in general complex values and m1 þ m2 þ � � � þ mq ¼ p is the exponential modelorder.

The exponential model order was identified by setting p equal to the number of nonzero singular values thatresulted from the singular value decomposition of the forward data matrix [1]. Moreover, we plotted the nor-malized singular values of the whole data set and chose p equal to the number of singular values that had thelargest magnitude, that is, where the curve showed a noticeable drop. This means that from that pointonwards the contribution of the remaining singular values was not significant. The rationale behind thisapproach is that signal related singular values will have the largest magnitude and so will contribute the most,

M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192 187

whereas noise related singular values will have a small contribution, hence having small magnitudes(references).

Since this data preprocessing algorithm expects an equally spaced vector of samples, glucose data from theIVGTT had to be interpolated beforehand.

2.2. Model of the patient

The LMM was chosen to describe the dynamics of glucose after an IVGTT. This model is mathematicallydescribed by the following set of equations:

_q1 ¼ �SGq1 � xþ p0; q1ð0þÞ ¼ q10þ ; ð2Þ_x ¼ �p2½x� SIq2ðtÞ�; xð0Þ ¼ x0; ð3Þy1 ¼ q1=V 1; ð4Þ

where x (g/min) is a lumped expression describing the control action of insulin and q1 (g) and q2 (U) representthe quantity of plasma glucose and the quantity of plasma insulin respectively.

The model parameters are: p2 (1/min) which describes the insulin action, p0 (g/min) is the extrapolatedhepatic glucose production at zero glucose quantity and SG (1/min) and SI (1/min per U) are parametersof glucose effectiveness and insulin sensitivity respectively.

The initial value q10þ represents the quantity of glucose q1 at time t = 0+ and includes the effect of glucoseinjection by assuming immediate mixing.

The model output is given by the concentration of glucose in blood y1 (mg/dl) calculated from x1 and thevolume of distribution of glucose V1, which is taken as 20% of the patient’s body weight [8]. The output of themodel y1 corresponds to the glucose data obtained experimentally as described in the data section.

Also, the profile of q2 that is supplied as a model input, is calculated through the relationship q2 = y2V11,where y2 is the plasma insulin concentration measured experimentally and V11 is the volume of distribution ofinsulin in plasma, approximated as 4.5% of the body weight.

Initial estimates of the model parameters have been obtained from previous studies where the LMMhas been identified following an IVGTT in the healthy man [9,6]. These values are as follows: SG = 0.0418(1/min), SI = 0.2889 (g/U min), and p2 = 0.0200, with p0 and xb determined from the steady state constraints.The extrapolated value of glucose q10þ is calculated through the relationship q10þ ¼ q10 þ D, where q10 can bedetermined from the first glucose measurement y1(0) = y10 and D is the glucose dose known a priori from theIVGTT experiment.

The model was subsequently fitted to the noise contaminated glucose experimental data using nonlinearleast squares with SG, SI, p2 and q10 defined as the unknown parameters. The precision of these optimal esti-mates was assessed by means of the coefficient of variation (CV) calculated as the ratio between the standarddeviation and the mean value of each parameter [2].

The signal of glucose was recovered from the noise contaminated experimental data using MBSE and theLMM again fitted as described above. The order of the exponential model was calculated as described inthe previous section. The precision of the parameter estimates was also calculated by means of the CV andthe differences with the CV’s obtained before MBSE assessed as explained in the next section.

The goodness of fit of the LMM after MBSE was evaluated by means of the R2 statistic which is interpretedas the fraction of the total variance of the data that is explained by the each model.

2.2.1. Statistical analysis

For each model parameter, and the ith data set included in our study (i 6 7), we have two values, namely,CV(i) and CVMBSE(i) for the parameter coefficient of variation. We are interested in the ratio:

rðiÞ ¼ CVMBSEðiÞCVðiÞ : ð5Þ

In order to assess the effect due to the application of MBSE, we set as our null hypothesis, H0, that the ob-

served CV values are independent of each other, with CVðiÞ ¼ðdÞCVMBSEðiÞ (here ¼ðdÞ means equal in distribution).

188 M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192

The alternative of interest, H1, is that CVMBSE(i) is stochastically smaller than CV(i) (for stochastic orderingand its applications see [10]). Under H0, the quantities:

Fig. 1decrea

ZðiÞ ¼ logCVMBSEðiÞ

CVðiÞ

� �; i 6 7 ð6Þ

are independent random variables, symmetric around zero, and small values of:

T ¼ 1

n

Xi

ZðiÞ ð7Þ

can be considered evidence against H0 and in favor of the alternative hypothesis that the application of MBSEreduces (stochastically) the CVs. We evaluate the value of T by means of the non-parametric procedurefor symmetry around zero proposed by Fisher (see the description in [11]). Conditionally on jZð1Þj;jZð2Þj; . . . ; jZð7Þj, the null distribution of T is given by the 27 values obtained in Eq. (7) by changing in all pos-sible ways the signs of the Z(i)’s. From this null distribution, the p-value of the observed T can be computedexactly in the present case. Additionally, when the value of T is considered significantly small and we reject H0

in favor of H1, we get, through a standard procedure for permutation tests, an estimate of the typical (median)percentage reduction in CV due to the application of MBSE, DCV, as

DCV% ¼ 100� ð1� expðT ÞÞ ð8Þ

10 20 30 40 50 60 70 80 90 100 110 1200

10

20

30

40

50

60

70

80

90

100

Singular Values (Ordinal Values)

Nor

mal

ized

Sin

gula

r Val

ues

(%)

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

. Plot of the normalized singular values for the seven data sets and detail of the first 10. For most data sets, the curve stopssing at the third singular value.

M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192 189

and, if we denote by q1�a/2 the 1 � a/2 quantile of the conditional null distribution of T, then we have a non-parametric confidence interval for DCV% given by:

100� ð1� expðT þ q1�a=2Þ; 1� expðT � q1�a=2ÞÞ: ð9Þ

The residuals of the MBSE approximation were tested for randomness by means of the Anderson–Darling test[12] and plotted against time to detect possible outliers or systematic deviations. The presence of bias was as-sessed by studying the skewness coefficient of the residuals with respect to both its asymptotic gaussian quan-tiles and its finite sample quantiles estimated by Montecarlo for the sample size under study [13]. Thecorrelation of errors was also studied by computing the Pearson’s correlation coefficient betweeneð1Þ . . . eðN � 1Þ and eð2Þ . . . eðNÞ for each data set, where e(i) is the ith residual and N is the total numberof points, and performing a t-test on the transformed coefficients [14].

This analysis of residuals was also followed to assess the goodness of fit of the LMM to the enhanced signal.

3. Results and discussion

In Fig. 1 the plot of normalized singular values is shown for the seven data sets. It can be observed that thecurve stops exhibiting a significant decrease at the third singular value. In consequence, we chose to set theexponential model order as p = 3.

The fitting of the LMM to the MBSE signal obtained from the exponential modelling approximation canbe observed in Fig. 2. In this plot the LMM follows the dynamics of each patient with a value of R2 close to 1in all cases confirming the quality of the fit. Also, the MBSE signal seems to approximate the raw data veryclosely. This qualitative analysis has been complemented by the analysis of residuals that follows.

-100 0 100 200 30050

100

150

200

250

300

350

Time (mins)

y 1 (m

g/dl

)

Glucose DataLMM, R2= 0.98MBSE Signal

-100 0 100 200 30050

100

150

200

250

300

350

Time (mins)

y 1 (m

g/dl

)

Glucose DataLMM, R2= 0.99MBSE Signal

-100 0 100 200 30050

100

150

200

250

300

350

400

Time (mins)

y 1 (m

g/dl

)

Glucose DataLMM, R2= 0.99MBSE Signal

-100 0 100 200 30050

100

150

200

250

300

Time (mins)

y 1 (m

g/dl

)

Glucose DataLMM, R2= 0.97MBSE Signal

Fig. 2. Ability of the LMM to follow the MBSE signal. Only four out of seven data sets are shown.

0 50 100 150 200-3

-2

-1

0

1

2

3

Time (mins)

Res

idua

ls (u

nitle

ss)

Fit of LMM to MBSE signal

Percentil 2.5Percentil 97.5Median

Fig. 3. Plot of the standardized residuals for all data sets (i = 7), showing the error of the LMM when approximating the MBSE signal.

190 M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192

The residuals of the LMM exhibit a random pattern as shown in Fig. 3. This result is achieved whenneglecting the first 10 min of the curve. In this respect, the Anderson–Darling test provided evidence of ran-domness between t = 10 and t = 240 min for 6 out of 7 (86%) data sets under study (p-value < 0.05). There isno evidence of bias for any of the data sets when considering the 95% confidence interval and residuals showedno correlation according to the Pearson’s test of correlation.

In contrast, no evidence of randomness was found for any of the data sets when considering the time inter-val t ¼ ½0; 240�. This result suggests that the LMM is not able to account for the dynamics of glucose duringthe first minutes of the IVGTT, which is in line with what have been observed in the past with the classic min-imal model due to the single compartmental representation of glucose [15]. A two compartmental modelapproach seems to be required to account for the rapid changes in glucose concentration that result afteran intravenous glucose injection during the first minutes [16]. Further research should be carried out to deter-mine if the LMM would also benefit from the two compartmental approximation.

In Fig. 4 the residuals of the MBSE approximation also show a random pattern only when the first 10 minof the curve are discarded. The Anderson–Darling test provided evidence of randomness between t = 10 andt = 240 min for 7 out of 7 (100%) data sets under study (p-value < 0.05). At the same time, there was no evi-dence of bias for any of the data sets when considering the 95% confidence interval around the expected value.Finally, the correlation test did not found evidence that the errors were correlated. According to this analysis,it can be confirmed that the MBSE algorithm is not removing valuable data from the original data set and thatis reproducing the information of interest with the added benefit that noise has been eliminated from the signalunder study.

0 50 100 150 200-3

-2

-1

0

1

2

3

Time (mins)

Sta

ndar

ized

Err

orMBSE approximation to raw data

Percentil 2.5Percentil 97.5Median

Fig. 4. Plot of the standardized residuals for all data sets (i = 7), showing the difference between the data and the MBSE approximation.

Table 1Mean and standard deviation of coefficients of variation (in %) of the LMM before and after MBSE (i = 7)

Parameter SG SI p2 q10þ

Before MBSE

Mean 13.97 5.19 30.23 1.92Std 5.33 2.48 16.03 0.26

After MBSE

Mean 6.48 3.44 14.57 0.96Std 5.17 3.12 9.00 0.51

Table 2Permutation CV ratio test results for the ratio between the CVs of the LMM before and after signal preprocessing

Parameter p-Value Reduction (DCV %) 95% CI

SG 0.00781 61.86 14.58–82.97SI 0.00781 50.19 12.98–71.48p2 0.00781 53.31 13.48–74.80q10 0.00781 53.94 14.26–75.27

The null hypothesis is rejected when p-value < 0.01.

M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192 191

192 M. Fernandez et al. / Applied Mathematics and Computation 196 (2008) 185–192

Moreover, the mean CV of the model parameters after MBSE is applied shows a significant reduction ofCV for all parameters in the LMM, as shown in Table 1. The reduction is of 62%, 50% , 53% and 54% forparameters SG, SI, p2 and q10 respectively, as shown in Table 2. Of particular interest would be the improve-ment obtained for SI and SG given that these parameters are useful diagnostic indices in Type 2 DiabetesMellitus.

However, as with the LMM the MBSE approximation is also unable to approximate the first portion of thecurve suggesting that the order of the exponential model p is not adequate to account for the glucose dynamicsat the early stage of the test. However, increasing the size of p would result in data overfitting at the end of thecurve that is not desirable. This undermodelling issue should be addressed in future investigations.

4. Conclusions

This investigation has shown that the application of the MBSE approximation would be able to signifi-cantly improve by more than 50% the precision of the LMM parameters when fitting data of the IVGTTin the healthy man. Of particular interest would be the improvement obtained for SI and SG given that theseparameters are useful diagnostic indices in Type 2 Diabetes Mellitus. Further studies should be carried out toverify if this improvement can also be achieved for patients with diabetes.

Acknowledgement

The authors would like to thank Dr. Angelo Avogaro (University of Padova School of Medicine, Padova,Italy) for providing the data that made this study possible.

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