Dimensional analysis of MINMOD leads to definition of the disposition index of glucose regulation and improved simulation algorithm
 Aparna Nittala^{1},
 Soumitra Ghosh^{1},
 Darko Stefanovski^{2},
 Richard Bergman^{2} and
 Xujing Wang^{1}Email author
https://doi.org/10.1186/1475925X544
© Nittala et al; licensee BioMed Central Ltd. 2006
Received: 07 May 2006
Accepted: 14 July 2006
Published: 14 July 2006
Abstract
Background
Frequently Sampled Intravenous Glucose Tolerance Test (FSIVGTT) together with its mathematical model, the minimal model (MINMOD), have become important clinical tools to evaluate the metabolic control of glucose in humans. Dimensional analysis of the model is up to now not available.
Methods
A formal dimensional analysis of MINMOD was carried out and the degree of freedom of MINMOD was examined. Through reexpressing all state variable and parameters in terms of their reference scales, MINMOD was transformed into a dimensionless format. Previously defined physiological indices including insulin sensitivity, glucose effectiveness, and first and second phase insulin responses were reexamined in this new formulation. Further, the parameter estimation from FSIVGTT was implemented using both the dimensional and the dimensionless formulations of MINMOD, and the performances were compared utilizing Monte Carlo simulation as well as real human FSIVGTT data.
Results
The degree of freedom (DOF) of MINMOD was found to be 7. The model was maximally simplified in the dimensionless formulation that normalizes the variation in glucose and insulin during FSIVGTT. In the new formulation, the disposition index (Dl), a composite parameter known to be important in diabetes pathology, was naturally defined as one of the dimensionless parameters in the system. The numerical simulation using the dimensionless formulation led to a 1.5–5 fold gain in speed, and significantly improved accuracy and robustness in parameter estimation compared to the dimensional implementation.
Conclusion
Dimensional analysis of MINMOD led to simplification of the model, direct identification of the important composite factors in the dynamics of glucose metabolic control, and better simulations algorithms.
Background
To mathematically model a physiological mechanism, the factors that govern its modus operandi can be taken as independent or dependent variables in the mathematical model and the behavior of the whole system be described by a set of simple or complex, differential or integral equations. The minimal model of glucose regulation, MINMOD, is one such model. First formulated and introduced by Richard Bergman and colleagues, it describes the kinetics of plasma glucose and insulin during a Frequently Sampled Intravenous Glucose Tolerance Test (FSIVGTT), and allows dissection of the composite effects of insulin secretion and insulin sensitivity on glucose tolerance and risk for diabetes [1–3]. The model was termed as "minimal" for it was the least complex mathematical representation that is capable of accounting for the observed dynamic relationship between insulin and glucose disappearance [1, 2]. In FSIVGTT, after overnight fasting, individual subjects are given an initial infusion of glucose bolus of 300 mg/kg body weight at the beginning of the experiments. At periodic time intervals afterwards, blood samples for glucose and insulin measurement will be taken for up to 180 minutes normally, every 2–5 minutes within the first 30 minutes, every 5–10 minutes for 30–60 minutes, and every 30 minutes from 60 to 180 minutes. Mathematically, the regulation of plasma glucose concentration G(t) was formulated to be [1–3]:
where G _{ b }is the baseline glucose concentration, p _{ 1 }is the insulinindependent glucose disappearance rate, and X(t) is an auxiliary function representing insulinexcitable tissue glucose uptake activity. It is proportional to the insulin concentration in a "remote" interstitial compartment (which was later shown to be the interstitial fluid [4]), and is described by
where l _{ b }is the baseline insulin concentration, p _{ 2 }is the rate constant of the spontaneous decrease in X(t), and p _{ 3 }is the rate of insulindependent increase in tissue X(t). The change in insulin is given by
Parameters in MINMOD.
Quantity  Description  Value and Unit  Dimensions  

M  L  T  
p _{1}  Insulinindependent glucose disappearance rate. Also known as glucose effectiveness (S_{G}).  ~10^{2} min^{1}  0  0  1 
p _{2}  Rate constant expressing the spontaneous decrease of tissue glucose uptake ability.  ~10^{2} min^{1}  0  0  1 
p _{3}  Insulindependent increase in tissue glucose uptake ability, per unit of insulin concentration excess over baseline insulin.  ~10^{5} min^{2} (μU/ml)^{1}  1  3  2 
n  Disappearance rate of endogenous insulin.  ~10^{1} min^{1}  0  0  1 
γ  Rate of second phase endogenous insulin secretion  ~10^{2}–10^{3} (μU/ml) min^{2}  0  0  2 
G _{ b }  Baseline plasma glucose  ~100 mg/dl  1  3  0 
G _{0}  Initial glucose concentration during FSIVGTT  ~300 mg/dl  1  3  0 
h  A threshold value (higher than basal) for plasma glucose above which the second phase insulin secretion is stimulated  ~100 mg/dl  1  3  0 
l _{ b }  Baseline plasma insulin  ~10 μU/ml  1  3  0 
l _{0}  Initial insulin concentration  ~30 μU/ml  1  3  0 

First phase pancreatic responsiveness , where AIR = (I _{0}  I _{ b })/n (Acute Insulin Response) is the total insulin release during first phase [6], and φ _{1} measures first phase insulin release per unit rise of glucose above basal.

Second phase pancreatic responsiveness φ _{2} = ∂^{2} (IDR _{2})/∂G∂t = γ, is the dependence of rate of rise of the 2^{nd} phase insulin secretion on glucose.
S _{ I }later led to the definition of the disposition index DI = S _{ I }* AIR [7, 8] (notice that it is dimensionless). Bogardus and colleagues have demonstrated that Dl was an excellent predictor (prognostic index) for which individual will develop type 2 diabetes mellitus (T2DM) in the Pima native American population [9]. In addition, S _{ I }was found to possess greater heritability than indices defined by other models including the homeostasis or the fasting insulin model assessments [4, 10]. The model is now the basis for a large number of laboratory and clinical investigations (~50 reports/year) [4, 11]. According to the American Diabetes Association Consensus Development Conference on insulin resistance [12], it is one of the only two methods (the other one is the euglycemic insulin clamp) that are recommended for assessing peripheral insulin resistance due to their satisfactory, consistent performance.
Since the publication of MINMOD, its mathematical structure, system properties, as well as simulation techniques of its computer program have also been studied [11, 13, 14]. However, up to now, a dimensional analysis is still lacking. How many degrees of freedom (DOF) does a system described by MINMOD have? How many independent indices can be defined from the model to characterize the system? How many indices are needed to differentiate different pathological states of glucose metabolic control? These questions have not been examined in depth. In addition, in a mathematical model, each variable or parameter has an associated dimension or unit that reflects its influence on the system behavior. One could carry out model analysis and computer simulation by taking the parameters as they are, with their respective units, as has been done with MINMOD up till now. There are certain disadvantages with such approaches. The parameters do not necessarily have the same range of values. While some vary within a narrow scale, others may span a wide range such that the absolute values of these parameters can be extremely different from each other. This makes it difficult to compare the relative importance of the parameters in controlling the system properties, as there is no uniform scale – a reference scale – based on which all the parameters can be studied [15].
Dimensional analysis is a process to simplify mathematical models expressed in differential equations [16, 17]. The technique rescales every variable and parameters in terms of their intrinsic reference quantities so that the equations can be expressed in terms of dimensionless variables and parameters whose typical scales are all ~O(1) [16]. An intrinsic reference quantity is one that reflects the intrinsic value scale of the variable; it can be its basal or maximal value, or its dynamic range of variation, for example. This process reduces the number of variables by removing redundant degrees of freedom. Using it, one can analyze the behavior of the system regardless of the units used to measure the variables. The dimensionless formulation helps to identify the dominant terms in the equations, as well as their interactions in the model behavior and their influence on the solution structure. It reveals which variables, or rates of change, can be thought of as small, or even 'negligible' relative to others. In addition, the process often reveals that some of the parameters do not affect the system's dynamic behavior independently, and they can be combined into dimensionless indices that reflect their collective effect. Such indices are usually the most effective predictors of system behavior. Examples include the Reynolds number (the ratio of the inertial force to the viscous force) in fluidics, the R number in epidemiology of infections, and the ratio of tumor growth to normal growth in oncology. Furthermore, with dimensional analysis, one can rescale models to duplicate the behavior of the original system provided that the governing dimensionless parameters have the same values in the two systems [18]. This will allow the identification of scaleinvariant parameters, and the translation between animal models and human studies.
In this paper we carry out dimensional analysis of MINMOD. We show that it leads to the direct identification of important pathological indices. Further, we implement the computer simulation of MINMOD using the dimensionless formulation of glucose regulation. Utilizing Monte Carlo simulation as well as real human FSIVGTT data, we compare the new implementation to the original dimensional implementation in terms of model fit, speed of convergence, accuracy of parameter estimation and robustness against noise.
Methods
Dimensional analysis of MINMOD
The dimensions of the state variables and parameters
State variables, their meanings, values ranges in humans, units, and dimensions in terms of the fundamental units of mass (M), length (L) and time (T).
Quantity  Description  Value and Unit  Dimension  

M  L  T  
G(t)  Plasma glucose concentration at time t  ~100–300 mg/dl  1  3  0 
l(t)  Plasma insulin concentration at time t  ~10–30 μU/ml  1  3  0 
X(t)  An auxiliary function proportional to insulin concentration in the interstitial compartment.  ~0–10^{3}min^{1}  0  0  1 
There are totally 10 parameters (p _{ 1 }, p _{ 2 }, p _{ 3 }, n, γ, h, G _{ b }, G _{ 0 }, l _{ b }, l _{ 0 }) in MINMOD, expressed in a total of 3 fundamental units. According to Buckingham's Pi theorem [17, 19], the DOF of this system is 7, implying that the model can be described in a simplified form with 7 free dimensionless parameters. Usually there is more than one way to nondimensionalize a multiparameter model, and it is worthwhile to make a choice that is meaningful and also maximally simplifies the equations. For example, it would be worthwhile to carefully define the intrinsic reference scale for each quantity, such that all rescaled dimensionless quantities vary within the same order of magnitude in value.
Among the three state variables, only G(t) and l(t) can be measured. X(t), which reflects the amount of insulin in the interstitial compartment, cannot be measured directly. In addition, equation 1B can be solved with . In FSIVGTT X(0) = 0, it follows that . Therefore we will focus on the reference choices for G and l, and explore and compare several means of nondimensionalization of MINMOD.
Nondimensionalization choice a, rescale time by the glucose disappearance rate
One natural choice to rescale G and l is with their baseline values and define . In the model fitting of FSIVGTT with MINMOD, we usually start with the second time point when the plasma glucose and insulin concentrations have peaked after the initial glucose bolus. Therefore the initial values of G _{0} and I _{0} are also the maxima for the glucose and insulin secretion (during the second phase of endogenous insulin release, the insulin concentration can peak again, but normally with a much lower peak value than in the first phase release). Under this choice, the rescaled glucose and insulin concentration vary between and , respectively. It might be argued that in the effort to uptake the exogenous glucose, the system can overcorrect and reach a concentration lower than G _{ b }before it reaches the equilibrium value of G _{ b }. De Gaetano et al have carried out a steady state analysis of MINMOD, and showed that likely the system approaches the steady state G _{ b }from below [13]. Even if this is the case, the minimal value is still very close to G _{ b }, with .
There are two natural time scales in this system, 1/p _{1} the time scale of the glucose disappearance on its own, and 1/n the time scale of the insulin disappearance. If we chose to use 1/p _{1} to rescale time, equations 1A–1C become:
with τ = p _{1} t, , and . The initial conditions become: , and . All the barred variables and parameters are unitless. The number of free parameters in the system indeed reduces to 7: , and [ _{0}].
Choice b, rescale time by the insulin disappearance rate
Similarly, we can rescale time by 1/n, the disappearance rate of endogenous insulin. In this choice the dimensionless parameters are τ = nt, , and , and the differential equations become
Choice c, rescale to normalize the variation in glucose and insulin
Computer simulation of glucose disappearance in the dimensionless formulation
A computer program to simulate the glucose disappearance component of MINMOD is developed by us in Matlab using both the dimensional and the dimensionless formulations. For dimensionless implementation, the formulation in choice c was adopted. The minimization function Isqnonlin.m available in the optimization toolbox of Matlab was utilized for parameter estimation. The program flow is similar as described in figure 1 of [3], and the same weighting scheme of the measured data points is adopted. In the initial computer implementation of MINMOD reported in [3], both glucose and insulin profiles were fitted by the mathematical model (equations 1A–1C). FSIVGTT and MINMOD were first developed utilizing a dog model. Later, it was found that as humans are more insulin resistant [8], and more endogenous insulin secretion is needed to generate sufficient insulin action in order for the computer program to accurately estimate the model parameters. It was found that this demand was more imperative for assessment of insulin sensitivity in diabetic (both type 1 and type 2 diabetes) subjects, as these individuals exhibit diminished or absent insulinsecretory response and/or higher insulin resistance. To overcome this problem, modified FSIVGTT protocol was introduced where a second injection of 100–300 mg tolbutamide [20] or 30 mU/kg body weight insulin [21] was infused 20 min after the initial glucose bolus. In these protocols, while the glucose regulation can still be adequately described by equations 1A–1B, the insulin regulation will require modifications of the original model given in equation 1C. In view of the variations in FSIVGTT protocols presently used in clinical practice, later computer simulations of MINMOD mainly focused on the glucose regulation, and the estimation of parameters p _{1}, p _{2} and p _{3} [5]. For the same reason and for simplicity, we implement our simulation algorithms only for the glucose regulation component. The relevant equations are and .
Monte Carlo simulation
The program flow chart for the Monte Carlo simulation is given in figure 1. Briefly, we first generated 100 simulated glucose and insulin temporal profiles using equation 1A–1C of MINMOD, with parameters randomly distributed within the following ranges: p _{1} ∈ [1 4]·10^{2} min^{1}, p _{2} ∈ [1.5 2]·10^{2} min^{1}, p _{ s }∈ [0.32 1.6]·10^{5} min^{2} (mU/l)^{1}, n ∈ [1 4]·10^{1} min^{1}, γ ∈ [1 4]·10^{3} (μU/ml) min^{2}, h ∈ [92 125] mg/dl, G _{0} ∈ [210 310] mg/dl, and I _{0} ∈ [100 125] μU/ml. The basal levels of glucose and insulin were fixed at G _{ b }= 90 mg/dl, and I _{ b }= 14 μU/ml. We then used the generated profiles to fit for the parameters [p _{1} p _{2} p _{3} G _{0}] in the dimensional formulation, and for [ _{1} _{2} _{3} _{1}] in the dimensionless formulation. The fitting stringencies were set at ['TolFun', 1e4, 'ToIX', 1e4; 'MaxFunEvals', 1e6]. More details of these options of the Isqnonlin function can be found from Matlab's web site [22]. By default we adopted the LevenbergMarquardt leastsquare minimization algorithm, though we have also tried the GaussNewton method for error minimization and obtained similar results.
To assess the robustness of our algorithm against noise we have also added Gaussian noise at the levels of 1–10% to the glucose profile of these 100 data sets, and examined the ability of our program to recover the true values of the parameters.
Human FSIVGTT data
FSIVGTT data of 20 human subjects were kindly provided to us by the FUSION (FinlandUnited States Investigation Of Noninsulindependent diabetes mellitus) study group [23, 24]. In this study healthy nondiabetic offspring of T2DM patients were recruited, and the tolbutamide modified FSIVGTT protocol [20] was administered to the individuals. Glucose and insulin were measured at 14 time points: 0, 2, 4, 8, 19, 22, 25, 30, 40, 50, 70, 100, 120, and 180 min. The initial values of the parameters were determined using the same method as described in the Monte Carlo simulation flowchart in figure 1.
Results
The dimensionless physiological indices in MINMOD
Choice a
In the new formulation, all rate constants are normalized with the time scale of glucose facilitated glucose uptake, and measures of insulin secretion are normalized with respect to glucose uptake.
Choice b
Choice c
This is by far the most natural choice to nondimensionalize MINMOD. It normalizes the variation in glucose and insulin. Out of the three choices, it leads to the most simplified expressions of model equations, initial conditions and steady state solution. Importantly, the new formulation directly leads to the definition of Dl, and points out that it is an important index for the dynamic regulation of glucose.
Comparison of the Monte Carlo simulations
Simulation performance
We have found that the simulation in dimensionless formulation is able to converge 20% faster, with mean number of iterations 6.4 ± 2.7 (Dim'less) versus 8.0 ± 1.4 (Dim'nl), p < 0.001 before converging. On average, each iteration in the dimensionless program also takes ~20% less time than one iteration in the dimensional program. On an Dell OptiPlex GX620 PC with Pentium^{®} dual 3 GHz CPU and 2 GB of RAM, the average CPU times are 0.122 s/per iteration in the dimensionless implementation versus a 0.159 s/per iteration in the dimensional implementation. Therefore the dimensionless implementation takes ~50% less simulation time.
Comparison between the dimensional and the dimensionless implementation of MINMOD. Results are from 100 Monte Carlo simulations.
Comparisons  Dimensional  Dimensionless  Significance 

Number of iterations  8.0 ± 1.4  6.4 ± 2.7  P < 0.001 
R ^{2}  0.97 ± 0.02  0.97 ± 0.03  N.S 
p _{1} % difference from true value  (53.2 ± 7.6)%  (53.2 ± 9.6)%  N.S. 
p _{1}, FSD of estimation  (14.1 ± 2.7)%  (14.9 ± 6.0)%  N.S 
p _{2}, % difference from true value  (188 ± 160)%  (187 ± 155)%  N.S. 
p _{2}, FSD of estimation  (11.6 ± 1.6)%  (12.1 ± 3.5)%  N.S 
p _{3}, % difference from true value  (513 ± 324)%  (505 ± 317)%  N.S. 
p _{3}, FSD of estimation  (11.6 ± 1.7)%  (12.1 ± 3.5)%  N.S 
G _{0}, % difference from true value  (1.3 ± 0.4)%  (1.3 ± 0.4)%  N.S. 
G _{0}, FSD of estimation  (0.1 ± 0.04)%  (0.1 ± 0.06)%  N.S 
S _{ I }, % difference from true value  (52.4 ± 25.5)%  (52.1 ± 25.5)%  N.S 
Robustness against noise
When there is noise in the data, we observed that the program failed to converge in parameter estimation for some datasets within the default tolerance. The number of failed fitting goes up with increasing amount of noise. In real clinical/laboratory set up, noise in measurements are unavoidable, and it may have been a major factor that contributed to failure of parameter fitting some investigators have experienced with MINMOD [25, 26]. Using a threshold of R^{2} > 0, the failure rate is plotted against the noise level in figure 4B. Clearly, the simulation using the dimensionless formulation of MINMOD is significantly more robust. We have also calculated the mean R^{2} after removing the failed fittings. The performance of dimensionless method is still better (figure 4A).
After removing data sets with erroneous parameter estimations, we have calculated the mean ratio of the estimated S _{ G }and S _{ I }to their true values, and plotted their values against noise levels in the right panels of figure 6 (normalized by the first data points). The estimation in S _{ G }is more robust with the dimensionless method, whilst no significant difference was observed for the S _{ I }estimation.
Human FSIVGTT data
Overall it takes many more iterations to fit the human FSIVGTT data than the simulated data. The improvement in speed with the dimensionless implementation is more significant in human data: 14.8 ± 8.8 iterations versus 36.3 ± 74.8 iterations, a 2.5 fold reduction with p = 0.02. Interestingly, the gain in CPU time per iteration is also more significant. Using the same PC, it is 0.115 s versus 0.210 s per iteration, leading to an almost 5 fold gain in CPU time.
The estimations in S _{ G }and S _{ I }by the two approaches correlate highly (figures 9C–9D), with correlation coefficient 0.991 and 0.999 respectively. There is a consistent small difference in S _{ G }estimation, with the values derived using the dimensionless implementation ~11.9% higher on average than the dimensional implementation (p = 0.002). There is no difference in S _{ I }estimation (p = 0.33). The FSD in the parameter estimation also correlate well (figures 9E–9F). The FSD in S _{ G }estimation in the dimensionless implementation exhibits a significant (p = 0.007) albeit small improvement over the dimensional counterpart, suggesting that the higher S _{ G }value derived by the dimensionless implementation could be more accurate. The FSD in S _{ I }is not different (p = 0.26). All p values were obtained using the paired ttest on the logarithm transformed data, as the distribution of the transformed data is much less skewed (which can be seen from figure 9).
Discussion
In diabetes research and clinical practice, it is very important to assess β cell function and insulin sensitivity, so as to evaluate the pathological status and risk of an individual. Many tests have been designed, and numerous indices have been defined. Using insulin sensitivity as an example, over a dozen have been experimented [27, 28], none has been deemed best at predicting disease risk. These indices were all associated with certain units by definition. Some (Cederholm index [29] for example) would include several constants in the definition to accommodate the conversion between different units in measuring glucose or insulin concentration. Some indices are very similar by nature, like the Gutt index [30] and the Cederholm index [29], but can appear to be quite different as the same quantities were defined in different units. Therefore, when using these indices one must follow exactly their proposed forms and the glucose/insulin concentration units in order to obtain meaningful results. These make the utilization of and the comparison between different indices extremely clumsy. A dimensional analysis and definition of dimensionless parameters will eliminate most of the problems. In addition, it has been found that a combination of several measures could predict disease risk better than individual ones [31]. However, it is not clear how many measures are needed for best prediction.
The fitting failure rate in the Monte Carlo simulation is high. This is expected to improve if we implement more sophisticated initial parameter value estimation and baseline correction algorithms, as found in [5]. In addition, the model fitting to the 20 FUSION subjects seems to be much better than the fitting of the Monte Carlo data sets at any noise levels, with higher R^{2} values, less failure rate and better agreements between the two methods. This is likely due to the fact that the FUSION used the modified FSIVGTT protocol, which is known to lead to better parameter estimation. As insulin secretion mechanism in modified protocol is not known, it is not possible to run a Monte Carlo simulation for the modified protocol.
Conclusion
In this work we performed dimensional analysis of MINMOD. We found that with a nondimensionalization choice that normalizes the variation in glucose and insulin during FSIVGTT, the pathologically important Dl index is naturally defined in the model and it has the meaning of the insulin sensitivity at unit first phase pancreas response. Several additional dimensionless indices were also defined, which potentially offer means to better characterize the metabolic control of glucose and its dysregulation under pathological conditions. In addition, we have explored the advantages of computer simulation of MINMOD using its dimensionless formulation. Using simulated data as well as real human FSIVGTT data, we found that whilst the new approach gives highly correlative (correlation coefficients all above 0.96) parameter estimations to the original dimensional formulation, it led to significantly improved simulation speed and is much more robust against noise.
Declarations
Acknowledgements
This work is supported in part by a special fund from Children's Hospital Foundation, Children's Research Institute of Wisconsin and Children's Hospital of Wisconsin. We thank FUSION group, specially, Dr. Heather Stringham, for providing the FSIVGTT data.
Authors’ Affiliations
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