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A summary and of "Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays" by Jiaxu Li, Yang Kuang and Clinton Mason[1]. The review below is written by Cody Cichowitz and Stephen Kissler. All work shown below can be assumed to originate from Li et al. unless otherwise cited.

Project Categorization

Mathematics Used: The study conducted a series of simulations to compare the proposed two-time delay model to currently existing models. In addition, the study detected bifurcation points of system parameters by determining steady state solutions. The above was accomplished by implementing MATLAB’s DDE23 routine, which numerically solves delay differential equations

Type of Model: The project considered a system of ordinary differential equations, in accordance to the mass conservation law, in the presence of two explicit time delays. The first time delay corresponded to time between elevation of glucose concentration levels and transportation of insulin to interstitial space. The second time delay denoted the time between transportation of insulin to interstitial space and change in hepatic glucose production. The model reproduced stabilized glucose and insulin oscillations, consistent with previous glucose-insulin regulatory system studies.

Biological System Studied: The glucose-insulin endocrine metabolic regulatory system, specifically in the presence of continuous enteral nutrition and constant glucose infusion

Executive summary

In order to function properly, humans need to maintain an appropriate blood plasma glucose concentration level. The glucose housed within plasma provides the necessary energy required by cells within the body. The human body has a complex glucose-insulin regulatory system that aims to maintain appropriate levels of glucose concentration within plasma. Their are several common sources of glucose with food consumption being the most commonly known. Food is eaten and then processed into more useable forms like glucose. To utilize the available glucose, the body creates insulin, a hormone that triggers glucose consumption on a cellular level. The glucose-insulin regulatory system targets and maintains a very narrow range of plasma glucose concentration, a state of euglycemia. For example, after an overnight fast the human body typically needs to have glucose concentration level within the narrow range of 70 – 110 mg/dl. When the glucose concentration falls well without of the normal range (into either a state of hyperglycemia or hypoglycemia) the body’s function is not only impaired but also put at significant risk. States of hyperglycemia and hypoglycemia can lead to retinopathy, neuropathy, peripheral neuropathy, blindness and even death. Diabetes mellitus is a disease that occurs when the body’s glucose-insulin regulatory system is not able to consistently maintain an appropriate glucose concentration.

Li, Kuang and Mason (2006) modeled a healthy glucose-insulin regulatory system by employing a system of delay differential equations that describe the concentration of glucose and insulin within the human body. A brief overview of delay differential equations can be found at http://www.scholarpedia.org/article/Delay_differential_equations. In well functioning glucose-insulin regulatory systems, there are stable oscillations of both glucose and insulin that naturally occur as the body secretes the insulin required to process the available and needed glucose. The delay differential equations accurately reproduce these stabilized oscillations that have been well observed and studied both in the body and through other proposed mathematical models. The system of delay differential equations proposed by Li, Kuang and Mason have the advantage of needing no arbitrary state variables that are often artificially introduced in systems of ordinary differential equations to capture the delayed nature of the oscillations that naturally occur. The system of delay differential equations is studied using a built-in MatLab routine, DDE23, that numerically solves delay differential equations. The solutions to these delay differential equations provide insight into the actual glucose-insulin system present within the human body. This insight is useful in the categorization and treatment of diabetic patients by helping recognize patients as healthy, pre-diabetic or diabetic, and in understanding the clinical tests currently used to diagnose diabetes.

Biological Context

In order to accurately model the glucose-insulin regulatory system, some biological context is necessary. The body has two main sources of glucose: (1) infusion of glucose from meal ingestion, oral glucose intake, eternal nutrition and constant glucose infusion and (2) internal glucose production by the liver. The primary regulatory hormones in the glucose-insulin regulatory system are insulin and glucagon, both of which originate in pancreas. As glucose enters the regulatory system from the two main sources, infusion and hepatic production, cells within the body use the available glucose. There are two types of cells that utilize the available glucose: a cell whose glucose consumption is independent of insulin (predominately brain and nervous system cells) and a cell whose glucose consumption is facilitated by insulin (predominantly muscle and fat cells). As glucose concentration levels fluctuate in the body the pancreas secretes enough glucagon and insulin to regulate the glucose levels in the plasma. Ideally the body targets and maintains a fairly stable and constant plasma glucose level. However, as the body attempts to keep glucose concentration constant, there are some experimentally-observed naturally-occurring stable oscillations in glucose and insulin concentrations. The oscillations fall into two categories: (1) rapid oscillations and (2) ultradian oscillations. Neither type of oscillation is well understood, but the rapid oscillations have been observed to have a period of five to 15 minutes and the ultradian oscillations are thought to have a period of 50 to 150 minutes. Thus, ultradian oscillations describe the upper limit of circadian rhythms, http://www.scholarpedia.org/article/Circadian_rhythms. It is thought that the rapid oscillations in glucose and insulin concentrations may occur from the 'bursting process' of insulin that occurs during pancreatic secretion. Ultradian oscillations are best observed after food consumption, glucose intake or constant enteral nutrition. Even though oscillations occur, they can still be stable and well within the appropriate range for euglycemia. The paper under review by Li et al. limits the study of the glucose-insulin regulatory system to constant enteral nutrition after an overnight fast with the hope of isolating the behavior of ultradian oscillations within system. The study of the glucose-insulin regulatory system is important because according to the NIH, diabetes impacts approximately 8.3% of the entire U.S. population (25.8 million people).

In order to maintain appropriate plasma glucose concentrations the body has two biological responses, one for low glucose levels and one for high glucose levels. When the glucose level drops, \alpha -cells in the pancreas are stimulated and secrete glucagon. The glucagon secreted from the \alpha -cells travels to the liver where it is transformed into glucose, thus elevating the body’s low glucose concentration that originally triggered the secretion of glucagon from the \alpha -cells. When the plasma glucose levels are elevated the \beta -cells in the pancreas are stimulated to lower the glucose concentration. After this stimulation there is a delay that occurs while insulin from the \beta -cells is made available for use as “remote insulin”. This total time has been estimated as a five to 15 minute delay, later quantified as \tau _{1}. Once insulin is secreted and becomes available as remote insulin it does one of three things: (1) it helps insulin-dependent cells process the available glucose (thus lowering the body’s elevated glucose concentration, which triggered the secretion of insulin), (2) it inhibits hepatic glucose production to stop the elevation of glucose concentration, or (3) it is cleared by the body through insulin degradation which primarily occurs in the liver and kidneys. Remaining insulin is cleared and removed by other tissues including muscle and adipose cells. As mentioned, the amount of insulin present directly impacts hepatic glucose production. The presence of insulin inhibits the transformation of glucagon, secreted from the \alpha -cells, to glucose. However, there is a time delay due to the lag between changes in insulin level and its impact on hepatic production, this is later quantified as \tau _{2}. This time delay is measured by examining the time that it takes for a change in insulin level to change the hepatic production by half-maximal suppression or half-maximal recovery. Quantifying this delay has been the subject of many studies and has been reported to be anywhere in the range of 4 to 50 minutes. Because there is so much uncertainty in the length of the second delay, it was important to Li et al. that their developed model work for a large range of \tau _{2}. The biological context for this model is depicted in the figure below and is translated into the mathematical model described in the Mathematical Model section of this wiki.

Web3.jpg

Figure 1- Two time delay glucose–insulin regulatory system model. The dash-dot-dot lines indicate that insulin inhibits hepatic glucose production with time delay; the dash-dot lines indicate insulin secretion from the \beta -cells stimulated by elevated glucose concentration level and the short dashed line indicates the insulin caused acceleration of glucose utilization in cells with time delay; the dashed lines indicate low glucose concentration level triggering a-cells in the pancreas to release glucagon.

History

The modern mathematical modeling of the glucose-insulin regulatory system began in 1961 when Victor Bolie employed a system of two first order ordinary differential equations to model the glucose and insulin concentration[2]. The author added a damping term in the differential equations in an attempt to capture the human body’s ability to maintain constant plasma glucose levels. However, the system of equations suggested by Boile were not able to produce the stable oscillations seen in the glucose-insulin regulatory system. The damping term added to the equations yielded a steady stable state within the expected range of plasma glucose concentration, but not it was not a solution that oscillated within the accepted range for plasma glucose concentration. The results of Bolie’s work paved the way for more rigorous and accurate modeling. Advancements in biological understanding of the glucose-insulin regulatory system inspired mathematical models aimed at better capturing the multi-organ regulatory system in a way that accounted for the physiological delays and feedback mechanisms responsible for the stable oscillations of glucose and insulin that are present within healthy human bodies. Sturis, Polonsky, Mosekilde and Cauter (1991)[3] suggested a system of six ordinary differential equations to describe the regulatory system. Their work accounted for delays in the regulatory system by introducing artificial state variables that created feedback and delay mechanisms. Instead of simply modeling the glucose and insulin levels, the approach taken by Boile, Sturis et al. tracked the glucose and insulin by also tracking artifical state variables that acted as delay functions. This work by Sturis et al. provided a system of equations that paved the way for the current mathematical exploration of diabetes mellitus. Their model produced the oscillatory behavior both in the glucose and insulin concentrations that mimicked experimentally observed oscillations. Yet the work of Sturis et al. was not complete, because the oscillations produced by the system of six equations decayed over time to a constant steady state. However, many authors have been able to use the foundation set forth by Sturis et al. to explore various issues associated with diabetes. One such paper, by Tolic et al. (2000)[4] studies the different in solutions produced for a regulatory system with oscillatory versus constant insulin supply. The paper reviewed in this wiki by Li, Kuang and Mason (2006) was able to re-frame the modeling of the glucose-insulin regulatory system using a system of delay differential equations. The use of delay differential equations reproduced the features presented in the work by Sturis et al. while simultaneously reducing the necessary system of differential equations from six to two. Having delays explicitly defined in the equations eliminates the need for physically irrelevant state variables and unnecessary and irrelevant parameters, thus increasing the accuracy with which the glucose-insulin regulatory system is modeled. Lastly, the use of delay equations extends the analysis of the glucose-insulin regulatory system by examining bifurcation points in the physical parameters present in the model. Understanding the bifurcation behavior leads to new insights into the glucose-insulin regulatory system.

Mathematical Model

The following is the guiding principle behind the model by Li et al.:

{\tfrac  {dG(t)}{dt}}= glucose production - glucose utilization

{\tfrac  {dI(t)}{dt}}= insulin production - insulin clearance

The equations above are conservation equations in the sense that they track all the glucose and insulin produced and consumed. With these in mind, the authors constructed the following two delay differential equations: G'(t)=G_{{in}}-f_{2}(G(t))-f_{3}(G(t))f_{4}(I(t))+f_{5}(I(t-\tau _{2})); I'(t)=f_{1}(G(t-\tau _{1}))-d_{i}(I(t))


Before describing the terms in the equation, it is helpful to first define the functions f_{1} through f_{6}. See Figure 2 to see the functions' shapes; the equation forms are listed below.

Figure 2 - Functional Forms, f_{1} - f_{5}

The shapes of the functions are more important than their specific equations. The first function, f_{1}(G), describes the secretion of insulin secretion as a function of glucose level; within this function there is a built-in delay which was discussed in the Biological Context section of this wiki. f_{1}(G) is an increasing sigmoidal function, physically implying that an increase glucose levels has a significant positive effect on the insulin secretion for a limited domain.f_{2}(G) describes glucose consumption by insulin independent cells, specifically the brain and nerve cells. f_{2}(G) is an increasing function that asymptotically approaches a positive value, which reflects the physiological behavior that the brain and nerve cells will consume available glucose only up to a certain finite amount. f_{3}(G) and f_{4}(I) work together in the model to account for the insulin-dependent glucose consumption by muscle and fat cells occurring within the regulatory system. The multiplication of f_{3}(G) and f_{4}(I) gives the value of insulin-dependent glucose consumption for the given values of glucose and insulin concentration. f_{3}(G) is a linearly increasing function of glucose level and f_{4}(I) is a logarithmically increasing function, so high glucose levels lead directly to high glucose uptake by these cells, but increasing insulin levels have decreasing effects on glucose uptake by these cells. The last function,f_{5}(I), describes the hepatic production of glucose as a function of insulin levels. This is a sigmoidal decreasing function, so high levels of insulin inhibit hepatic glucose production. This function also incorporates the second time delay described in the Biological Context section of this wiki. The five functions are based on experimentally determined parameters (see Sturis et al., 1991; Tolic et al., 2000). Here are those parameters:

ParametersCCSK.jpg

The equations for the functions are:

Functionaleqns.jpg


With these descriptions in place, the mathematical terms within are further described below:

  • G_{{in}}: Glucose intake, set as a constant to portray a glucose-insulin regulatory system after an overnight fast
  • f_{2}(G(t)): Glucose used by brain and nerve cells (insulin-independent consumption), as a function of glucose level.
  • f_{3}(G(t))f4(I(t)): Glucose used by muscle, fat cells, and other tissues (insulin-dependent consumption).
  • f_{5}(G(t-\tau _{2})): Hepatic glucose production, as a function of past insulin levels to account for the amount of time required for insulin to take effect on the liver.
  • f_{1}(G(t-\tau _{1})): Insulin released by \beta -cells as a function of past glucose levels, to account for the time required for insulin to pass from the inside of the \beta -cells to interstitial space and become available as remote insulin.
  • d_{i}I(t): Insulin clears the regulatory system as the body processes the excess insulin.

Here, both delays are measured in minutes, and are later examined for 0<\tau \leq 40 minutes.

By explicitly incorporating these two delays into the conservation equations, the model captures properties that have been well observed in true human glucose-insulin regulation. The model exhibits stable oscillations for a wide regime of delays, and the specific values of both \tau _{1} and \tau _{2} that produce oscillations are closely examined using bifurcation analysis.

The following are the important observations yielded by this model:

  • The observed profiles of glucose utilization agree with previous experimental results (see Cherrington et al. (2002)[5]; Luzi and DeFrenzo (1989)[6]).
  • For \tau _{2} in the regime which gives stable oscilations (> 6.75), the glucose-insulin curves have amplitude and period that match experimental data. The same is observed for \tau _{1} between 5 and 15 minutes.
  • High insulin degradation (d_{i}) leads to a faster response of insulin levels to glucose peaks, and glucose concentration increases overall as the body is able to process the produced insulin much faster.
  • Insulin peaks follow glucose peaks, which follows the physiological observation that glucose spikes trigger insulin production. Glucose levels also drop before insulin levels drop, following the physiological observation that high insulin levels inhibit glucose production. The time between the two peaks in each cycle is about 20 minutes.
  • If the insulin decay rate di is greater than approximately 0.0325, oscillatory behavior can be sustained; this matches results from Bennett and Gourley, 2004[7]
  • When either one of the delay parameters \tau _{1} or \tau _{2} are set to zero, sustained oscillations are only observed when the nonzero delay is large (\tau _{2} > 46 or \tau _{1} > 18). This confirms observations made by Sturis et al. (1991) and Tolic et al. (2000).
  • Incorporating a delay for insulin secretion (\tau _{1}) causes this model to reflect physiological observations more directly than a single delay for insulin effect on hepatic glucose production or a no-delay ordinary differential equation model.

Another example of a model incorporating two explicit time delays in the system can be found at APPM4390: Modelling the Tryptophan Operon

Classical Example

The primary goal of this model is to describe the self-sustained ultradian glucose-insulin oscillations observed in humans after an overnight fast. As noted in the History Section of this wiki, a number of different approaches have been taken to describe this same system. Common among these models is the assumption of continuous enteral nutrition and constant glucose infusion, which allows the glucose intake value to be estimated by a constant. Initial values (or in this case, initial history) for glucose and insulin are defined, and then the model is run until steady oscillatory solutions are observed (if they exist). The two-delay model produces such oscillations for a range of physiologically viable parameter values, as shown by the following two plots:

PaperGlucoseInsulin1.jpg
Figure 3- Glucose-Insulin Oscillations(1)


PaperGlucoseInsulin2.jpg
Figure 4- Glucose-Insulin Oscillations(2)


Figure 5- Oscillations by Li et al.
Figure 6- Reproduced Oscillations
Figure 7- Comparison of Existing and Two-Delay Models


Our own reproduction of the two-delay model produced similar results, though our plots use a slightly different scaling constant than Li et al. See Figure 6 for a reproduction of Figure 5.

The oscillations produced by the two time-delay model match those observed physiologically more closely than any previous attempts at modeling the same system. Previous papers failed to produce such sustained oscillations under the same biological and mathematical assumptions; Figure 7 shows insulin oscillations and glucose oscillations for five other proposed models, with the oscillations in solid black representing those produced by the two-delay model.

The correspondence between this model and physiological data is encouraging, and the next section describes additional analysis of the two-delay model that provides even more insight into the physiological system of interest.

Results and Analysis of Full Model

In addition to the glucose and insulin oscillations produced with the two time delay model, which was consistent with previous work done in the modeling of the glucose-insulin regulatory system, Li et al. were able to focus their MatLab simulations on detecting the bifurcation points of four important system parameters. Detecting the bifurcation points of these parameters was not only a mathematical exercise but also provided important biological insight into the glucose-insulin regulatory system. The four system parameters analyzed here include the delay parameters \tau _{1} and \tau _{2}, the constant glucose infusion rate, G_{{in}}, and the insulin degradation rate, d_{i}. To analyze the bifurcation of a parameter, simulations were run in DDE23 until a steady state solution appeared. As the various parameters were varied the steady states went from being stable to oscillatory stable at a specific bifurcation point. Starting with the delay parameter \tau _{2}, a bifurcation point was observed at \tau _{2}\approx 6.75 minutes. Stable oscillations were observed for \tau _{2} in the range of (6.75,40]. Both stable states fell within the appropriate levels of plasma glucose concentration (euglycemia). Moreover, the time between the peak of glucose and insulin was concentration was about 17-20 minutes. The results produced in the bifurcation of the first value was consistent with previous work done on the modeling of the glucose-insulin regulatory system.



BifPlot2.jpg
Figure 8- Bifurcations in Glucose and Insulin Limiting Values w.r.t. \tau _{2}

Examining the bifurcation of value of \tau _{1} revealed that at for \tau _{1}>3.8 minutes stable oscillations occurred and the period of the periodic solution fell within the range of 95-150 minutes. Additionally the glucose concentration was found to peak 15-28 minutes before the insulin concentration peaks. The period of the periodic solution physically implies that within the human body every hour and half to two hours the body is oscillating between two steady states of glucose concentration that both fall within the range of euglycemia.

Tau1BifPlot3.jpg
Figure 9 - Bifurcations in Glucose and Insulin Limiting Values w.r.t \tau _{1}

In this model it was assumed that the glucose intake rate was constant. Varying the glucose intake rate and looking for a bifurcation point showed that one occurred at G_{{in}}=1.25 mg/dl/min. For intake rates less than 1.25 oscillations between two steady states was sustained. However, for intake rates greater than 1.25 no oscillations were produced and single steady glucose and insulin levels were reached. This result was predicted by the human behavior observed during intra venous glucose tolerance test when large boluses of glucose were given to patients and their insulin and glucose levels were monitored. When G_{{in}}>1.25 the model no longer follows the assumption of small, constant eternal glucose intake. In a similar fashion the authors were able to show that a bifurcation point existed for the insulin degradation level at .0325. For degradation values greater than .0325 oscillatory solutions were present. By using a two time delay differential equation model, Li et. al were able to reproduce and confirm the previous and existing research on the glucose-insulin regulatory system. Additionally, the two delay model produced additional insight into the actual length of delay that occurs when insulin inhibits hepatic glucose production. One of the main benefits of the two delay equation model is that it provides very clear guidelines for what parameter values sustained oscillations of glucose and insulin levels are produce. Many of the parameters used in the mathematical modeling of the glucose-insulin regulatory system are not well quantified. However, it is well known that stable oscillations of glucose and insulin are well sustained within humans after an overnight fast. Using the bifurcation analysis, Li et. al suggest parameter values that may be more accurate than previously suggested values. One important case is that the time between the peaks of glucose and insulin is approximately 20 minutes. This was not well justified previous to this work. Using DDE23 we were able to reproduce the bifurcation plots found in Li et al. One such example is found to the right in Figure 10.

Figure 10 - Reproduced Bifurcations for \tau _{2}

The study was extended to understanding how variations of diet and mechanisms of diabetes affect the two-delay glucose-insulin regulatory model in APPM4390: Extensions to a 2-Delay Glucose-Insulin Regulatory Model

Recent Developments

Recent work that relates back to this glucose-insulin model has taken on a number of different directions. Some interesting work has been done in modeling the onset of Type II diabetes (de Gaetano et al., 2008 [8]). This is a particularly difficult problem, since the slow onset of the disease make the collection of physiological data difficult. Li has gone on to study the effect of insulin infusions on glucose-insulin regulation (see Wang et al., 2007[9] and Wang et al., 2009[10]). Smith et al. (2008)[11] and Pattaranit and Berg (2008)[12] have done further work on glucose and energy homeostasis, which is pertinent to both diabetes and obesity. This work shows a trend toward modeling disorders, now that the underlying mechanisms of the glucose-insulin regulation system seem to be sufficiently well-understood.

Further Discussion of A More Recent Study

Smith, James et al., 2009, Mathematical Modeling of Glucose Homeostasis and Its Relationship With Energy Balance and Body Fat, Obesity vol. 17:4 [11]

Due to the increasing rate of obesity in both Western society and developing countries such as India and China, many studies have been focused around understanding how obesity develops and what steps are necessary to reverse this trend. Obesity progresses from a prolonged positive energy balance, when surplus dietary energy intake builds in adipose tissue. The expansion of adipose tissue strongly correlates to glucose homeostasis, which has been widely studied using a number of mathematical modeling approaches including ordinary differential equations, delay differential equations (as achieved in the above study), partial differential equations and integrodifferential equations. The purpose of this study was to review available glucose homeostasis models to provide a future direction for modeling the onset of obesity.

The study analyzed the most prominent models of glucose homeostasis and determined that while the previous models sufficiently predicted observed data, the derivations oversimplified important details that could help identify underlying mechanisms associated with the development of obesity. It suggested that future models consider diet composition, endocrine responses in addition to insulin, stage of development, body composition, genetic predispositions, environment and most importantly the link between glucose homeostasis and availability of glucose for lipogenesis. The study adapted the model previously proposed by Sorensen (1985), that consisted of mass balance equations describing the fluxes of glucose, insulin and glucagon between six compartments: brain, heart and lungs, gut, liver, kidney, and adipose tissue and skeletal muscle. Furthermore, the model accounted for processes that increase or decrease the glucose, insulin and glucagon concentrations within each compartment. To address the link between glucose homeostasis and availability of glucose for lipogenesis, the model was expanded by an additional term describing the rate of glucose removed for storage in the adipose tissue and skeletal muscle.

After simulating three different intravenous glucose infusion protocols, the newly suggested model identified high bolus infusions over short periods of time as more significant contributors to higher peripheral glucose uptake than isoenergetic doses over longer periods of time. Although the predictions coincided with prior observations between carbohydrate type and obesity, the investigators admitted to the need of comparison to biological data. They anticipate that additional equations will be added to the model to understand how a number of metabolic processes affect body fat accretion. Yet, it must be noted that increased complexity introduces more assumptions, uncertainties and errors.

Resources

  1. Li, Jiaxu, Yang Kuang and Clinton C. Mason, 2006, Modeling the glucose–insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, Journal of Theoretical Biology vol. 242
  2. Bolie, Victor, 1961, Coefficients of normal blood glucose regulation, Journal of Applied Physiology vol. 16:5
  3. Sturis, Jeppe, Kenneth S. Polonsky, Erik Mosekilde, and Eve Van Cauter, 1991, Computer model for mechanisms underlying ultradian oscillations of insulin and glucose, Modeling Methodology Forum
  4. Tolic, Iva Marija, Erik Mosekilde and Jeppe Sturis, 2000, Modeling the Insulin}Glucose Feedback System: The Significance of Pulsatile Insulin Secretion, Academic Press
  5. Cherrington, Alan et al., 2002, Physiological Consequences of Phasic Insulin Release in the Normal Animal , Diabetes vol. 51
  6. Luzi, L. and R.A. DeFronzo, 1989, Effect of loss of first-phase insulin secretion on hepatic glucose production and tissue glucose disposal in humans, American Journal of Physiology vol. 257:2
  7. Bennett, D.L. and S.A. Gourley, 2004, Asymptotic properties of a delay differential equation model for the interaction of glucose with the plasma and interstitial insulin, Appl. Math. Comput. vol. 151
  8. de Gaetano, Andrea et al., 2008, Mathematical models of diabetes progression, American Journal of Physiology vol. 295:6
  9. Wang, Hayan, Jiaxu Li and Yang Kuang, 2007, Mathematical modeling and qualitative analysis of insulin therapies, Mathematical Biosciences vol. 210:1
  10. Wang, Hayan, Jiaxu Li and Yang Kuang, 2009, Enhanced modelling of the glucose–insulin system and its applications in insulin therapies, Journal of Biological Dynamics vol. 3:1
  11. 11.0 11.1 Smith, James et al., 2009, Mathematical Modeling of Glucose Homeostasis and Its Relationship With Energy Balance and Body Fat, Obesity vol. 17:4
  12. Pattaranit, Ratchada and Hugo Antonius van den Berg., 2008, Mathematical models of energy homeostasis, Journal of the Royal Society vol. 5:27