- Research
- Open Access
A local glucose-and oxygen concentration-based insulin secretion model for pancreatic islets
- Peter Buchwald^{1}Email author
https://doi.org/10.1186/1742-4682-8-20
© Buchwald; licensee BioMed Central Ltd. 2011
- Received: 25 April 2011
- Accepted: 21 June 2011
- Published: 21 June 2011
Abstract
Background
Because insulin is the main regulator of glucose homeostasis, quantitative models describing the dynamics of glucose-induced insulin secretion are of obvious interest. Here, a computational model is introduced that focuses not on organism-level concentrations, but on the quantitative modeling of local, cellular-level glucose-insulin dynamics by incorporating the detailed spatial distribution of the concentrations of interest within isolated avascular pancreatic islets.
Methods
All nutrient consumption and hormone release rates were assumed to follow Hill-type sigmoid dependences on local concentrations. Insulin secretion rates depend on both the glucose concentration and its time-gradient, resulting in second-and first-phase responses, respectively. Since hypoxia may also be an important limiting factor in avascular islets, oxygen and cell viability considerations were also built in by incorporating and extending our previous islet cell oxygen consumption model. A finite element method (FEM) framework is used to combine reactive rates with mass transport by convection and diffusion as well as fluid-mechanics.
Results
The model was calibrated using experimental results from dynamic glucose-stimulated insulin release (GSIR) perifusion studies with isolated islets. Further optimization is still needed, but calculated insulin responses to stepwise increments in the incoming glucose concentration are in good agreement with existing experimental insulin release data characterizing glucose and oxygen dependence. The model makes possible the detailed description of the intraislet spatial distributions of insulin, glucose, and oxygen levels. In agreement with recent observations, modeling also suggests that smaller islets perform better when transplanted and/or encapsulated.
Conclusions
An insulin secretion model was implemented by coupling local consumption and release rates to calculations of the spatial distributions of all species of interest. The resulting glucose-insulin control system fits in the general framework of a sigmoid proportional-integral-derivative controller, a generalized PID controller, more suitable for biological systems, which are always nonlinear due to the maximum response being limited. Because of the general framework of the implementation, simulations can be carried out for arbitrary geometries including cultured, perifused, transplanted, and encapsulated islets.
Keywords
- diabetes mellitus
- FEM model
- glucose-insulin dynamics
- Hill equation
- islet perifusion
- islets of Langerhans
- oxygen consumption
- PID controller
Background
In healthy humans, blood glucose levels have to be maintained in a relatively narrow range: typically 4-5 mM and usually within 3.5-7.0 mM (60-125 mg/dL) in fasting subjects [1, 2]. This is mainly achieved via the finely-tuned glucose-insulin control system whereby β-cells located in pancreatic islets act as glucose sensors and adjust their insulin output as a function of the blood glucose level. Pancreatic islets are structurally well-defined spheroidal cell aggregates of about one to two thousand hormone-secreting endocrine cells (α, β, γ, and PP-cells). Human islets have diameters ranging up to about 500 μm with a size distribution that is well described by a Weibull distribution function, and islets with diameters of 100-150 μm are the most representative [3]. Because abnormalities in β-cell function are the main culprit behind elevated glucose levels, quantitative models describing the dynamics of glucose-stimulated insulin release (GSIR) are of obvious interest [1] for both type 1 (insulin-dependent or juvenile-onset) and type 2 (non-insulin dependent or adult-onset) diabetes mellitus. They could help not only to better understand the process, but also to more accurately assess β-cell function and insulin resistance. Abnormalities in β-cell function are critical in defining the risk and development of type 2 diabetes [4], a rapidly increasing therapeutic burden in industrialized nations due to the increasing prevalence of obesity [5, 6]. A quantitative understanding of how healthy β-cells maintain normal glucose levels is also of critical importance for the development of 'artificial pancreas' systems [7] including automated closed-loop insulin delivery systems [8–10] as well as for the development of 'bioartificial pancreas' systems such as those using immune-isolated, encapsulated islets [11–13]. Accordingly, mathematical models have been developed to describe the glucose-insulin regulatory system using organism-level concentrations, and they are widely used, for example, to estimate glucose effectiveness and insulin sensitivity from intravenous glucose tolerance tests (IVGTT). They include curve-fitting models such as the "minimal model" [14] and many others [15–17] as well as paradigm models such as HOMA [18, 19]. There is also considerable interest in models focusing on insulin release from encapsulated islets [20–26], an approach that is being explored as a possibility to immunoisolate and protect transplanted islets.
The goal of the present work is to develop a finite element method (FEM)-based model that (1) focuses not on organism-level concentrations, but on the quantitative modeling of local, cellular-level glucose-insulin dynamics by incorporating the detailed spatial distribution of the concentrations of interest and that (2) was calibrated by fitting experimental results from dynamic GSIR perifusion studies with isolated islets. Such perifusion studies allow the quantitative assessment of insulin release kinetics under fully controllable experimental conditions of varying external concentrations of glucose, oxygen, or other compounds of interest [27–30], and are now routinely used to assess islet quality and function. Microfluidic chip technologies make now possible even the quantitative monitoring of single islet insulin secretion with high time-resolution [31]. We focused on the modeling of such data because they are better suited for a first-step modeling than those of insulin release studies of fully vascularized islets in live organism, which are difficult to obtain accurately and are also influenced by many other factors. Lack of vasculature in the isolated islets considered here might cause some delay in the response compared with normal islets in their natural environment; however, the diffusion time (L ^{2}/D) [32] to (or from) the middle of a 'standard' islet (d = 150 μm) is roughly of the order of only 10 s for glucose and 100 s for insulin (with the diffusion coefficients used here)-relatively small delays. Furthermore, because of the spherical structure, most of the cell mass is located in the outer regions of the islets (i.e., about 70% within the outer third of the radius) further diminishing the roles of these delays.
By using a general approach that couples local (i.e., cellular level) hormone release and nutrient consumption rates with mass transport by convection and diffusion, the present approach allows implementation for arbitrary 2D or even 3D geometries including those with flowing fluid phases. Hence, the detailed spatial distribution of insulin release, hypoxia, and cell survival can be modeled within a unified framework for cultured, transplanted, encapsulated, or GSIR-perifused pancreatic islets. While there has been considerable work on modeling insulin secretion, no models that couple both convective and diffusive transport with reactive rates for arbitrary geometries have been published yet. Most published models incorporating mass transport focused on encapsulated islets for a bioartificial pancreas [20–26]. Only very few [21, 24] included flow, and even those had to assume cylindrical symmetry. Furthermore, the present model also incorporates a comprehensive approach to account not only for first-and second-phase insulin response, but also for both the glucose-and the oxygen-dependence of insulin release. Because the lack of oxygen (hypoxia) due to oxygen diffusion limitations in avascular islets can be an important limiting [33] factor especially in cultured, encapsulated, and freshly transplanted islets [27, 28, 34, 35], it was important to also incorporate this aspect of the glucose-insulin response in the model.
In response to a stepwise increase of glucose, normal, functioning islets release insulin in a biphasic manner: a relatively quick first phase consisting of a transient spike of 5-10 min is followed by a sustained second phase that is slower and somewhat delayed [36–39]. The effect of hypoxic conditions on the insulin release of perifused islets has been studied by a number of groups [27, 28, 34, 35], and they seem to indicate that insulin release decreases nonlinearly with decreasing oxygen availability; however, only relatively few detailed concentration-dependence studies are available. Parametrization of the insulin release model here has been done to fit experimental insulin release data mainly from two studies with the most detailed concentration dependence data available: by Henquin and co-workers for glucose dependence [40] and by Dionne, Colton and co-workers for oxygen dependence [27].
Methods
Mass transport model (convective and diffusive)
where, c denotes the concentration [mol m^{-3}] and D the diffusion coefficient [m^{2} s^{-1}] of the species of interest, R the reaction rate [mol m^{-3} s^{-1}], u the velocity field [m s^{-1}], and ∇ the standard del (nabla) operator, [42]. The following diffusion coefficients were used as consensus estimates of values available from the literature: oxygen, D _{oxy,w} = 3.0 × 10^{-9} m^{2} s^{-1} in aqueous media and D _{oxy,t} = 2.0 × 10^{-9} m^{2} s^{-1} in islet tissue ([33] and references therein); glucose, D _{gluc,w} = 0.9 × 10^{-9} m^{2} s^{-1} and D _{gluc,t} = 0.3 × 10^{-9} m^{2} s^{-1}; insulin, D _{ins,w} = 0.15 × 10^{-9} m^{2} s^{-1} and D _{ins,t} = 0.05 × 10^{-9} m^{2} s^{-1} [23, 24]. Published tissue values for glucose vary over a wide range (0.04-0.5 × 10^{-9} m^{2} s^{-1}) [32, 43–46]; a value toward the higher end of this range (0.3 × 10^{-9} m^{2} s^{-1}) was used here. Very few tissue values for insulin are available (and the existence of dimers and hexamers only complicates the situation) [32, 47]; the value used here was lowered compared to water in a manner similar to glucose. For the case of encapsulated islets, the following diffusion coefficients were used for the capsule (e.g., hydrogel matrices such as alginate): D _{oxy,c} = 2.5 × 10^{-9} m^{2} s^{-1}, D _{gluc,c} = 0.6 × 10^{-9} m^{2} s^{-1}, D _{ins,c} = 0.1 × 10^{-9} m^{2} s^{-1} [23, 48].
Consumption and release rates
The three parameters of this function are R _{max}, the maximum reaction rate [mol m^{-3} s^{-1}], C _{Hf}, the concentration corresponding to half-maximal response [mol m^{-3}], and n, the Hill slope characterizing the shape of the response. This function introduced by A. V. Hill [49, 50] provides a convenient mathematical function for biological/pharmacological applications [51]: it allows transition from zero to a limited maximum rate via a smooth, continuously derivable function of adjustable width. Mathematically, the well-known two-parameter Michaelis-Menten equation [52] represents a special case (n = 1) of the Hill equation, and eq. 2 also shows analogy with the logistic equation, one of the most widely used sigmoid functional forms, being equivalent with a logarithmic logistic function, y = f(x) = R _{max}/(1 + βe ^{-n lnx }). Obviously, different parameter values are used for the different release and consumption functions (i.e., insulin, glucose, oxygen; e.g., C _{Hf,gluc}, C _{Hf,oxy}, etc.).
Oxygen consumption and cell viability
Lacking detailed data, as a first estimate, we assumed the base rate to represent 50% of the total rate possible (φ _{base} = φ _{metab} = 0.5). To maintain the previously used consumption rate at low (3 mM) glucose, a scaling factor is used, Φ _{sc} = 1.8. The metabolic component fully parallels that used for insulin secretion (n _{ins2,gluc} = 2.5, C _{Hf,ins2,gluc} = 7 mM; see eq. 6 later). With this selection, oxygen consumption increases about 70% when going from low (3 mM) to high glucose (15 mM)-slightly less than used previously in our preliminary model [33], but in good agreement with the approximately 50%-100% fold increase seen in various experimental settings [35, 36, 56–60]. As before [33], a step-down function, δ, was also added to account for necrosis and cut the oxygen consumption of those tissues where the oxygen concentration c _{oxy} falls below a critical value, C _{cr,oxy} = 0.1 μM (corresponding to p _{cr,oxy} = 0.07 mmHg). To avoid computational problems due to abrupt transitions, COMSOL's smoothed Heaviside function with a continuous first derivative and without overshoot flc1hs [61] was used as step-down function, δ(c _{oxy} > C _{cr,oxy}) = flc1hs(c _{oxy} - 1.0x10^{-4}, 0.5x10^{-4}).
Glucose consumption
These parameter values are draft first estimates only; however, changes in glucose concentrations due to glucose consumption by islets have only minimal influence on insulin release or cell survival because oxygen diffusion limitations in tissue or in media are far more severe than for glucose [55, 62]. Even if oxygen is consumed at approximately the same rate as glucose on a molar basis and has a 3-4-fold higher diffusion coefficient (i.e., D _{w}s used here of 3.0 × 10^{-9} vs. 0.9 × 10^{-9} m^{2} s^{-1}), this is more than offset by the differences in the concentrations available under physiological conditions. The solubility of oxygen in culture media or in tissue is much lower than that of glucose; hence, the available oxygen concentrations are much more limited (e.g., around 0.05-0.2 mM vs. 3-15 mM assuming physiologically relevant conditions) [62]. Glucose consumption by islet cells alters the glucose levels reaching the glucose-sensing β-cells only minimally.
Insulin release
Summary of Hill function (eq. 2) parameters used in the present model (Figure 1, eq. 3-9)
Model | Var. | C _{Hf} | n | R _{max} | Comments |
---|---|---|---|---|---|
R _{oxy}, oxygen consumption, base | c _{oxy} | 1 μM | 1 | -0.034 mol/m^{3}/s | Cut to 0 below critical value, c _{oxy} <C _{cr,oxy}. |
R _{oxy}, oxygen consumption, φ _{o,g} metabolic part | c _{gluc} | 7 mM | 2.5 | N/A | Due to increasing metabolic demand; parallels second-phase insulin secretion rate. |
R _{gluc}, glucose consumption | c _{gluc} | 10 μM | 1 | -0.028 mol/m^{3}/s | Contrary to oxygen, has no significant influence on model results. |
R _{ins,ph2}, insulin secretion rate, second-phase | c _{gluc} | 7 mM | 2.5 | 3 × 10^{-5} mol/m^{3}/s | Total secretion rate is modulated by local oxygen availability (last row). |
R _{ins,ph1}, insulin secretion rate, first-phase | ∂c _{gluc}/∂t | 0.03 mM/s | 2 | 21 × 10^{-5} mol/m^{3}/s | Modulated via eq. 8 to have maximum sensibility around c _{gluc} = 5 mM and be limited at very large or low c _{gluc}. |
Insulin secretion rate, φ _{o,g} oxygen dependence | c _{oxy} | 3 μM | 3 | N/A | To abruptly limit insulin secretion if c _{oxy} becomes critically low. |
We assumed an abrupt Hill-type (eq. 2) modulating function as φ _{ i,o } (c _{oxy}) with n _{ins,oxy} = 3 and C _{Hf,ins,oxy} = 3 μM (p _{Hf,ins,oxy} = 2 mmHg) so that insulin secretion starts becoming limited for local oxygen concentrations that are below ~6 μM (corresponding to a partial pressure of p _{O2} ≈ 4 mmHg) (Additional file 1, Figure S1). This is a somewhat similar, but mathematically more convenient function than the bilinear one introduced by Avgoustiniatos [75] and used by Colton and co-workers [76] to account for insulin secretion limitations at low oxygen (p _{O2} < 5.1 mmHg assumed by them) as it is a smooth sigmoid function with a continuous derivative (Additional file 1, Figure S1).
For a correct time-scale of insulin release, an extra compartment had to be added; otherwise insulin responses decreased too quickly compared to experimental observations (~1 min vs. ~5-10 min). Hence, insulin is assumed to be first secreted in a 'local' compartment (Figure 1) in response to the current local glucose concentration (R _{ins}, eq. 9) and then released from here following a first order kinetics [dc _{insL}/dt = R _{ins} - k _{insL}(c_{insL} - c _{ins}); k _{insL} = 0.003 s^{-1}, corresponding to a half-life t _{1/2} of approximately 4 min]. 'Local' insulin was modeled as an additional concentration with the regular convection model (eq. 1), but having a very low diffusivity (D _{insL,t} = 1.0 × 10^{-16} m^{2} s^{-1}). Throughout the entire model building process, special care was taken to keep the number of parameters as low as possible to avoid over-parameterization [77]; however, inclusion of this compartment was necessary. The model has been parameterized by fitting experimental insulin release data from two detailed concentration-dependence perifusion studies: one concentrating on the effect of glucose using isolated human islets [40] and one concentrating on the effect of hypoxia using isolated rat islets [27].
Fluid dynamics model
Here, ρ denotes density [kg m^{-3}], η viscosity [kg m^{-1} s^{-1} = Pa s], p pressure [Pa, N m^{-2}, kg m^{-1} s^{-2}], and F volume force [N m^{-3}, kg m^{-2} s^{-2}]. The first equation is the momentum balance; the second one is simply the equation of continuity for incompressible fluids. The flowing media was assumed to be an essentially aqueous media at body temperature; i.e., the following values were used: T _{0} = 310.15 K, ρ = 993 kg m^{-3}, η = 0.7 × 10^{-3} Pa s, c _{p} = 4200 J kg^{-1}K^{-1}, k _{c} = 0.634 J s^{-1}m^{-1}K^{-1}, α = 2.1 × 10^{-4} K^{-1}. As previously [33], incoming media was assumed to be in equilibrium with atmospheric oxygen and, thus, have an oxygen concentration of c _{oxy,in} = 0.200 mol m^{-3} (mM) corresponding to p _{O2} ≈ 140 mmHg. A number of GSIR perifusion studies including [40] used solutions gassed with enriched oxygen (e.g., 95% O_{2} + 5% CO_{2}; p _{O2} ≈ 720 mmHg); however, with the islet sizes used here, atmospheric oxygen already provides sufficient oxygenation so that the extra oxygen has no effect on model-calculated insulin secretion (see Results section). Inflow velocity was set to v _{in} = 10^{-4} m s^{-1} (corresponding to a flow rate of 0.1 mL/min in a ~4 mm tube), and along the inlet, a parabolic inflow velocity profile was used: 4v _{in} s(1-s), s being the boundary segment length.
Model implementation
The models were implemented in COMSOL Multiphysics 3.5 (formerly FEMLAB; COMSOL Inc., Burlington, MA) and solved as time-dependent (transient) problems allowing intermediate time-steps for the solver. Computations were done with the Pardiso direct solver as linear system solver with an imposed maximum step of 0.5 s, which was needed to not miss changes in the incoming glucose concentrations that could be otherwise overstepped by the solver. With these setting, all computation times were reasonable being about real time; i.e., about 1 h for each perifusion simulations of 1 h interval.
For visualization of the results, surface plots were used for c _{ins}, c _{oxy}, and R _{ins}. For 3D plots, c _{ins} was also used as height data. A contour plot (vector with isolevels) was used for c _{gluc} to highlight the changes in glucose. To characterize fluid flow, arrows and streamlines for the velocity field were also used. Animations were generated with the same settings used for the corresponding graphs. Total insulin secretion as a function of time was visualized using boundary integration for the total flux along the outflow boundary.
Results and Discussion
First-and second-phase insulin responses
Oxygen dependence
Accordingly, the overall experimental response to hypoxic conditions will depend on the size-distribution of the islet sample. Human islets seem to follow a Weibull distribution with the expected value of islet diameter being around 95 μm and the expected value of islet volume being 1.2 × 10^{6} μm (corresponding to an islet with d = 133 μm) [3]. In other words, most (human) islets are expected to have a diameter around 100 μm, but most of the islet mass (volume) is coming from islets with a diameter around 150 μm, which has been traditionally used as the standard islet (islet equivalent, IEQ) [81, 82]. Consequently, we chose two islets with d = 100 and 150 μm as representative for our simplified modeling.
It is important to note that even though local insulin release is becoming limited only for oxygen concentrations below 4 mmHg (≈6 μM; eq. 9), the total insulin secretion of the islets starts decreasing rapidly if surrounding oxygen levels drop below ~50 mmHg and is already half-maximal around 25 mmHg (Figure 6). The reason, of course, is that oxygen concentrations in the core of larger islets are considerable less than in the surrounding media due to diffusion limitations (see Figure 8 and 9). It is also worth noting that overall insulin response remains essentially unchanged until oxygen pressures decrease down to ~50 mmHg (Figure 6), values that are present in well vascularized tissues, and then decreases rapidly. This agrees well even with results of in vivo experiments in dogs suggesting that moderate hypoxia (p _{O2} ≈ 40 mmHg) does not affect insulin response, whereas more severe hypoxia (p _{O2} ≈ 25 mmHg) markedly inhibits it [83]. A number of GSIR perifusion studies including [40] used solutions gassed with enriched oxygen (e.g., 95% O_{2} + 5% CO_{2}; p _{O2} ≈ 720 mmHg). Compared to atmospheric oxygen (p _{O2} ≈ 140 mmHg), this does not produces any changes in the insulin profile calculated with the present model (e.g., Figure 4) since with the islet sizes used here atmospheric oxygen already provides sufficient oxygenation so that insulin secretion is not limited (Figure 8A, Figure 9A). On the other hand, transplanted islets are likely to be subject to oxygen levels below 50 mmHg [84] depending on the seeding density and the vascularization of the surrounding tissue, which can further limit their insulin secreting ability. Availability of oxygen is the main limiting factor because, under physiological conditions, oxygen concentrations are considerably lower than glucose concentrations (e.g., around 0.05-0.2 mM vs. 3-15 mM) [62], and this is well illustrated by the present calculation in Figure 8 that compares oxygen and glucose concentrations across the islets. Whereas glucose concentrations in the center of larger islets are only a few percent lower than at the periphery, oxygen concentrations in the center are considerably lower than at the periphery. R. T. Kennedy and co-workers measured somewhat larger glucose concentration decreases in the center of cultured islets (10-20%) [45], but even those are much less severe than the corresponding oxygen decreases.
With the calibrated model, detailed simulations for arbitrary inflow conditions and for arbitrary islet arrangements can be performed, and corresponding detailed graphics and animations can easily be generated. For example, calculated insulin, oxygen, and glucose concentrations along the perifusion chamber with two islets during a glucose gradient are shown in Figure 7 together with the insulin secretion rates. A set of similar results is shown in Figure 9 along a vertical cross-section through the middle of these figures. To illustrate the easy generalizability of the present approach, Additional file 1, Figure S2 shows the results of calculations obtained for a case where a supporting filter was included in the tube. While this perturbs the flow, it has essentially no effect on the overall insulin output justifying the simplifying assumptions made for the present geometry (Figure 3). Increases in the perifusion rate (e.g., up to ten-fold) also have no significant effect on calculated insulin output.
Additional file 2: Supporting Information, Video S1. Movie file showing the time-course of the insulin response of two islets to a glucose step (3 mM → 11 mM → 3 mM) under normoxic conditions (pO_{2} 140 mmHg) in a 3D representation with insulin concentration as height data and a surface color-coded for oxygen concentration (similar to Figure 10). (WMV 1 MB)
Additional file 3: Supporting Information, Video S2. Movie file showing the time-course of the insulin response of two islets to a glucose step (3 mM → 11 mM → 3 mM) under hypoxic conditions (pO_{2} 25 mmHg) in a 3D representation with insulin concentration as height data and a surface color-coded for oxygen concentration (similar to Figure 10). (WMV 2 MB)
Encapsulated islets
In patients with type 1 diabetes mellitus, the transplantation of pancreatic islet cells can normalize metabolic control in a way that has been virtually impossible to achieve with exogenous insulin, and is being explored, in a selected cohort of patients with brittle diabetes, as an experimental therapy [89, 90]. To avoid the need for life-long immunosuppression, islet encapsulation using semi-permeable membranes and various techniques has long been explored as a possible approach to develop a bioartificial pancreas-an organ capable of releasing insulin in a biomimetic manner in response to plasma glucose changes [11–13]. Many failed attempts [91] made it clear that minimizing the extra volume of encapsulating material (as well as cellular overgrowth) and the corresponding diffusional limitations are crucial for graft success. Hence, there is a considerable interest in modeling the insulin responses of such devices [20–26].
Modeling considerations
Obviously, this is still a much simplified, exploratory model; the actual mechanism of glucose-induced insulin secretion in β-cells is complex and involves various molecular-level events [1, 2, 36, 37, 39, 70, 96, 97]. The present model gives an adequate quantitative description of the main distinctive features of insulin release, but, at this stage, does not account for interspecies differences and does not incorporate a number of effects known to affect glucose-induced insulin release including, e.g., amplifiers such as glucagon-like peptide-1 (GLP-1) as well as time-dependent effects (i.e., both time-dependent inhibition and potentiation; e.g., the "glucose priming" effect) [98].
Here, we used the sigmoid direct (proportional) term to model the oxygen and glucose consumptions as well as the second phase insulin release, and the sigmoid differential term to model the first-phase insulin release (with c _{gluc} itself as the "error" signal ε; Figure 1). As always, the role of the differential term is to speed up the system; i.e., to give a large correction signal as soon as possible when the monitored value changes suddenly-exactly the role played by the first-phase insulin secretion. In the present model, we could not yet implement an integral term despite a clear need for such a term over a specified time interval to account, for example, for some inertia and/or delay in insulin secretion (integral control is part of several models, i.e., [8–10, 73, 99, 100]). However, addition of the extra compartment for delayed insulin release actually incorporates some elements usually accounted for by such an integral term.
Conclusion
In conclusion, a comprehensive insulin secretion model for avascular pancreatic islets has been implemented using Hill-type sigmoid response functions to describe both glucose and oxygen dependence. Detailed spatial distributions of all concentrations of interest are incorporated and coupled via local consumption and release functions. Following parameterization, good fit could be obtained with experimental perifusion data of human islets. Further optimization of the model is required; however, the present approach makes it relatively straightforward to couple arbitrarily complex hormone secretion and nutrient consumption kinetics with diffusive and even convective transport and run simulations with realistic geometries without symmetry or other restrictions-problems that seriously limited previous glucose-insulin modeling attempts. Because of the general framework of the implementation, the model not only helps in the elucidation of the quantitative aspects of the insulin secretion dynamics, but also allows the in silico exploration of various conformations involving cultured, perifused, transplanted, or encapsulated islets including the simulation of GSIR perifusion experiments or the study of the performance of bioartificial pancreas type devices.
Declarations
Acknowledgements
The financial support of the Diabetes Research Institute Foundation http://www.diabetesresearch.org that made this work possible is gratefully acknowledged.
Authors’ Affiliations
References
- Hallgreen CE, Korsgaard TV, Hansen RN, Colding-Jørgensen M: The glucose-insulin control system. Biosimulation in Drug Development. Edited by: Bertau M, Mosekilde E, Westerhoff HV. 2008, Weinheim: Wiley-VCH, 141-196.Google Scholar
- Suckale J, Solimena M: Pancreas islets in metabolic signaling-focus on the beta-cell. Front Biosci. 2008, 13: 7156-7171.View ArticlePubMedGoogle Scholar
- Buchwald P, Wang X, Khan A, Bernal A, Fraker C, Inverardi L, Ricordi C: Quantitative assessment of islet cell products: estimating the accuracy of the existing protocol and accounting for islet size distribution. Cell Transplant. 2009, 18: 1223-1235.View ArticlePubMedGoogle Scholar
- Kahn SE, Hull RL, Utzschneider KM: Mechanisms linking obesity to insulin resistance and type 2 diabetes. Nature. 2006, 444: 840-846.View ArticlePubMedGoogle Scholar
- Wyatt SB, Winters KP, Dubbert PM: Overweight and obesity: prevalence, consequences, and causes of a growing public health problem. Am J Med Sci. 2006, 331: 166-174.View ArticlePubMedGoogle Scholar
- Wang Y, Beydoun MA: The obesity epidemic in the United States-gender, age, socioeconomic, racial/ethnic, and geographic characteristics: a systematic review and meta-regression analysis. Epidemiol Rev. 2007, 29: 6-28.View ArticlePubMedGoogle Scholar
- Galletti PM, Colton CK, Jaffrin M, Reach G: Artificial pancreas. The Biomedical Engineering Handbook Tissue Engineering and Artificial Organs. Edited by: Bronzino JD. 2006, Boca Raton, FL: CRC Press, 71.71-71.18. 3Google Scholar
- Steil GM, Rebrin K, Janowski R, Darwin C, Saad MF: Modeling beta-cell insulin secretion-implications for closed-loop glucose homeostasis. Diabetes Technol Ther. 2003, 5: 953-964.View ArticlePubMedGoogle Scholar
- Steil GM, Panteleon AE, Rebrin K: Closed-loop insulin delivery-the path to physiological glucose control. Adv Drug Deliv Rev. 2004, 56: 125-144.View ArticlePubMedGoogle Scholar
- Hovorka R: Continuous glucose monitoring and closed-loop systems. Diabet Med. 2006, 23: 1-12.View ArticlePubMedGoogle Scholar
- Soon-Shiong P: Treatment of type I diabetes using encapsulated islets. Adv Drug Deliv Rev. 1999, 35: 259-270.View ArticlePubMedGoogle Scholar
- Silva AI, de Matos AN, Brons IG, Mateus M: An overview on the development of a bio-artificial pancreas as a treatment of insulin-dependent diabetes mellitus. Med Res Rev. 2006, 26: 181-222.View ArticlePubMedGoogle Scholar
- Teramura Y, Iwata H: Bioartificial pancreas microencapsulation and conformal coating of islet of Langerhans. Adv Drug Deliv Rev. 2010, 62: 827-840.View ArticlePubMedGoogle Scholar
- Bergman RN, Ider YZ, Bowden CR, Cobelli C: Quantitative estimation of insulin sensitivity. Am J Physiol Endocrinol Metab. 1979, 236: E667-E677.Google Scholar
- Toffolo GM, Cobelli C: Insulin modeling. Modeling Methodology for Physiology and Medicine. Edited by: Carson E, Cobelli C. 2001, San Diego: Academic Press, 305-335.View ArticleGoogle Scholar
- Makroglou A, Li J, Kuang Y: Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview. Appl Num Math. 2006, 56: 559-573.View ArticleGoogle Scholar
- Boutayeb A, Chetouani A: A critical review of mathematical models and data used in diabetology. Biomed Eng Online. 2006, 5: 43-artPubMed CentralView ArticlePubMedGoogle Scholar
- Matthews DR, Hosker JP, Rudenski AS, Naylor BA, Treacher DF, Turner RC: Homeostasis model assessment: insulin resistance and beta-cell function from fasting plasma glucose and insulin concentrations in man. Diabetologia. 1985, 28: 412-419.View ArticlePubMedGoogle Scholar
- Wallace TM, Levy JC, Matthews DR: Use and abuse of HOMA modeling. Diabetes Care. 2004, 27: 1487-1495.View ArticlePubMedGoogle Scholar
- Reach G, Jaffrin MY: Kinetic modelling as a tool for the design of a vascular bioartificial pancreas: feedback between modelling and experimental validation. Comput Methods Programs Biomed. 1990, 32: 277-285.View ArticlePubMedGoogle Scholar
- Pillarella MR, Zydney AL: Theoretical analysis of the effect of convective flow on solute transport and insulin release in a hollow fiber bioartificial pancreas. J Biomech Eng. 1990, 112: 220-228.View ArticlePubMedGoogle Scholar
- Todisco S, Calabro V, Iorio G: A lumped parameter mathematical model of a hollow fiber membrane device for the controlled insulin release. J Membr Sci. 1995, 106: 221-232.View ArticleGoogle Scholar
- Tziampazis E, Sambanis A: Tissue engineering of a bioartificial pancreas: modeling the cell environment and device function. Biotechnol Prog. 1995, 11: 115-126.View ArticlePubMedGoogle Scholar
- Buladi BM, Chang CC, Belovich JM, Gatica JE: Transport phenomena and kinetics in an extravascular bioartificial pancreas. AIChE J. 1996, 42: 2668-2682.View ArticleGoogle Scholar
- Dulong JL, Legallais C: Contributions of a finite element model for the geometric optimization of an implantable bioartificial pancreas. Artif Organs. 2002, 26: 583-589.View ArticlePubMedGoogle Scholar
- Dulong JL, Legallais C: What are the relevant parameters for the geometrical optimization of an implantable bioartificial pancreas?. J Biomech Eng. 2005, 127: 1054-1061.View ArticlePubMedGoogle Scholar
- Dionne KE, Colton CK, Yarmush ML: Effect of hypoxia on insulin secretion by isolated rat and canine islets of Langerhans. Diabetes. 1993, 42: 12-21.View ArticlePubMedGoogle Scholar
- Sweet IR, Khalil G, Wallen AR, Steedman M, Schenkman KA, Reems JA, Kahn SE, Callis JB: Continuous measurement of oxygen consumption by pancreatic islets. Diabetes Technol Therap. 2002, 4: 661-672.View ArticleGoogle Scholar
- Cabrera O, Jacques-Silva MC, Berman DM, Fachado A, Echeverri F, Poo RE, Khan A, Kenyon NS, Ricordi C, Berggren P-O, Caicedo A: Automated, high-throughput assays for evaluation of human pancreatic islet function. Cell Transplant. 2008, 16: 1039-1048.PubMed CentralView ArticlePubMedGoogle Scholar
- Bocca N, Pileggi A, Molano RD, Marzorati S, Wu W, Bodor N, Ricordi C, Buchwald P: Soft corticosteroids for local immunosuppression: exploring the possibility for the use of loteprednol etabonate in islet transplantation. Pharmazie. 2008, 63: 226-232.PubMedGoogle Scholar
- Dishinger JF, Reid KR, Kennedy RT: Quantitative monitoring of insulin secretion from single islets of Langerhans in parallel on a microfluidic chip. Anal Chem. 2009, 81: 3119-3127.PubMed CentralView ArticlePubMedGoogle Scholar
- Truskey GA, Yuan F, Katz DF: Transport Phenomena in Biological Systems. 2004, Upper Saddle River, NJ: Pearson Prentice HallGoogle Scholar
- Buchwald P: FEM-based oxygen consumption and cell viability models for avascular pancreatic islets. Theor Biol Med Model. 2009, 6: 5-PubMed CentralView ArticlePubMedGoogle Scholar
- Ohta M, Nelson D, Nelson J, Meglasson MD, Erecinska M: Oxygen and temperature dependence of stimulated insulin secretion in isolated rat islets of Langerhans. J Biol Chem. 1990, 265: 17525-17532.PubMedGoogle Scholar
- Wang W, Upshaw L, Strong DM, Robertson RP, Reems J: Increased oxygen consumption rates in response to high glucose detected by a novel oxygen biosensor system in non-human primate and human islets. J Endocrinol. 2005, 185: 445-455.View ArticlePubMedGoogle Scholar
- Hedeskov CJ: Mechanism of glucose-induced insulin secretion. Physiol Rev. 1980, 60: 442-509.PubMedGoogle Scholar
- Rorsman P, Eliasson L, Renstrom E, Gromada J, Barg S, Gopel S: The cell physiology of biphasic insulin secretion. News Physiol Sci. 2000, 15: 72-77.PubMedGoogle Scholar
- Henquin JC: Triggering and amplifying pathways of regulation of insulin secretion by glucose. Diabetes. 2000, 49: 1751-1760.View ArticlePubMedGoogle Scholar
- Henquin JC: Regulation of insulin secretion: a matter of phase control and amplitude modulation. Diabetologia. 2009, 52: 739-751.View ArticlePubMedGoogle Scholar
- Henquin JC, Dufrane D, Nenquin M: Nutrient control of insulin secretion in isolated normal human islets. Diabetes. 2006, 55: 3470-3477.View ArticlePubMedGoogle Scholar
- Comsol AB: COMSOL Multiphysics Modeling Guide, version 3.4. 2007, COMSOL ABGoogle Scholar
- Riley KF, Hobson MP, Bence SJ: Mathematical Methods for Physics and Engineering. A Comprehensive Guide. 1997, Cambridge: Cambridge University PressGoogle Scholar
- Groebe K, Erz S, Mueller-Klieser W: Glucose diffusion coefficients determined from concentration profiles in EMT6 tumor spheroids incubated in radioactively labeled L-glucose. Adv Exp Med Biol. 1994, 361: 619-625.View ArticlePubMedGoogle Scholar
- Bashkatov AN, Genina EA, Sinichkin YP, Kochubey VI, Lakodina NA, Tuchin VV: Glucose and mannitol diffusion in human dura mater. Biophys J. 2003, 85: 3310-3318.PubMed CentralView ArticlePubMedGoogle Scholar
- Kauri LM, Jung SK, Kennedy RT: Direct measurement of glucose gradients and mass transport within islets of Langerhans. Biochem Biophys Res Commun. 2003, 304: 371-377.View ArticlePubMedGoogle Scholar
- Novak MT, Yuan F, Reichert WM: Modeling the relative impact of capsular tissue effects on implanted glucose sensor time lag and signal attenuation. Anal Bioanal Chem. 2010, 398: 1695-1705.PubMed CentralView ArticlePubMedGoogle Scholar
- Søeborg T, Rasmussen CH, Mosekilde E, Colding-Jørgensen M: Absorption kinetics of insulin after subcutaneous administration. Eur J Pharm Sci. 2009, 36: 78-90.View ArticlePubMedGoogle Scholar
- Li RH, Altreuter DH, Gentile FT: Transport characterization of hydrogel matrices for cell encapsulation. Biotechnol Bioeng. 1996, 50: 365-373.View ArticlePubMedGoogle Scholar
- Hill AV: The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol. 1910, 40: iv-vii.Google Scholar
- Hill AV: The combinations of haemoglobin with oxygen and with carbon monoxide. Biochem J. 1913, 7: 471-480.View ArticlePubMedGoogle Scholar
- Goutelle S, Maurin M, Rougier F, Barbaut X, Bourguignon L, Ducher M, Maire P: The Hill equation: a review of its capabilities in pharmacological modelling. Fundam Clin Pharmacol. 2008, 22: 633-648.View ArticlePubMedGoogle Scholar
- Michaelis L, Menten ML: Die Kinetik der Invertinwirkung. Biochem Z. 1913, 49: 333-369.Google Scholar
- Buchwald P: Exploratory FEM-based multiphysics oxygen transport and cell viability models for isolated pancreatic islets. Proceedings of the COMSOL Conference 2008 Boston. Edited by: Dravid V. 2008, Boston: Comsol, Inc.Google Scholar
- Wilson DF, Rumsey WL, Green TJ, Vanderkooi JM: The oxygen dependence of mitochondrial oxidative phosphorylation measured by a new optical method for measuring oxygen concentration. J Biol Chem. 1988, 263: 2712-2718.PubMedGoogle Scholar
- Avgoustiniatos ES, Colton CK: Effect of external oxygen mass transfer resistances on viability of immunoisolated tissue. Ann NY Acad Sci. 1997, 831: 145-167.View ArticlePubMedGoogle Scholar
- Longo EA, Tornheim K, Deeney JT, Varnum BA, Tillotson D, Prentki M, Corkey BE: Oscillations in cytosolic free Ca^{2+}, oxygen consumption, and insulin secretion in glucose-stimulated rat pancreatic islets. J Biol Chem. 1991, 266: 9314-9319.PubMedGoogle Scholar
- Sweet IR, Gilbert M, Scott S, Todorov I, Jensen R, Nair I, Al-Abdullah I, Rawson J, Kandeel F, Ferreri K: Glucose-stimulated increment in oxygen consumption rate as a standardized test of human islet quality. Am J Transplant. 2008, 8: 183-192.PubMedGoogle Scholar
- Cornolti R, Figliuzzi M, Remuzzi A: Effect of micro-and macroencapsulation on oxygen consumption by pancreatic islets. Cell Transplant. 2009, 18: 195-201.View ArticlePubMedGoogle Scholar
- Hedeskov CJ, Hertz L, Nissen C: The effect of mannoheptulose on glucose-and pyruvate-stimulated oxygen uptake in normal mouse pancreatic islets. Biochim Biophys Acta. 1972, 261: 388-397.View ArticleGoogle Scholar
- Goto M, Abe H, Ito-Sasaki T, Goto M, Inagaki A, Ogawa N, Fujimori K, Kurokawa Y, Matsue T, Satomi S: A novel predictive method for assessing the quality of isolated pancreatic islets using scanning electrochemical microscopy. Transplant Proc. 2009, 41: 311-313.View ArticlePubMedGoogle Scholar
- Comsol AB: COMSOL Multiphysics User's Guide, version 3.4. 2007, COMSOL ABGoogle Scholar
- Martin Y, Vermette P: Bioreactors for tissue mass culture: design, characterization, and recent advances. Biomaterials. 2005, 26: 7481-7503.View ArticlePubMedGoogle Scholar
- Grodsky GM: A threshold distribution hypothesis for packet storage of insulin and its mathematical modeling. J Clin Invest. 1972, 51: 2047-2059.PubMed CentralView ArticlePubMedGoogle Scholar
- Purrello F, Rabuazzo AM, Anello M, Patane G: Effects of prolonged glucose stimulation on pancreatic beta cells: from increased sensitivity to desensitization. Acta Diabetol Lat. 1996, 33: 253-256.View ArticleGoogle Scholar
- Pedersen MG, Corradin A, Toffolo GM, Cobelli C: A subcellular model of glucose-stimulated pancreatic insulin secretion. Philos Transact A Math Phys Eng Sci. 2008, 366: 3525-3543.View ArticleGoogle Scholar
- Pedersen MG, Toffolo GM, Cobelli C: Cellular modeling: insight into oral minimal models of insulin secretion. Am J Physiol Endocrinol Metab. 2010, 298: E597-E601.View ArticlePubMedGoogle Scholar
- Topp B, Promislow K, deVries G, Miura RM, Finegood DT: A model of beta-cell mass, insulin, and glucose kinetics: pathways to diabetes. J Theor Biol. 2000, 206: 605-619.View ArticlePubMedGoogle Scholar
- Malaisse WJ, Sener A, Herchuelz A, Hutton JC: Insulin release: the fuel hypothesis. Metabolism. 1979, 28: 373-386.View ArticlePubMedGoogle Scholar
- Matschinsky FM, Glaser B, Magnuson MA: Pancreatic beta-cell glucokinase: closing the gap between theoretical concepts and experimental realities. Diabetes. 1998, 47: 307-315.View ArticlePubMedGoogle Scholar
- Fridlyand LE, Philipson LH: Glucose sensing in the pancreatic beta cell: a computational systems analysis. Theor Biol Med Model. 2010, 7: 15-PubMed CentralView ArticlePubMedGoogle Scholar
- Ritzel RA, Veldhuis JD, Butler PC: Glucose stimulates pulsatile insulin secretion from human pancreatic islets by increasing secretory burst mass: dose-response relationships. J Clin Endocrinol Metab. 2003, 88: 742-747.View ArticlePubMedGoogle Scholar
- Nomura M, Shichiri M, Kawamori R, Yamasaki Y, Iwama N, Abe H: A mathematical insulin-secretion model and its validation in isolated rat pancreatic islets perifusion. Comput Biomed Res. 1984, 17: 570-579.View ArticlePubMedGoogle Scholar
- Breda E, Cavaghan MK, Toffolo G, Polonsky KS, Cobelli C: Oral glucose tolerance test minimal model indexes of beta-cell function and insulin sensitivity. Diabetes. 2001, 50: 150-158.View ArticlePubMedGoogle Scholar
- Mari A, Schmitz O, Gastaldelli A, Oestergaard T, Nyholm B, Ferrannini E: Meal and oral glucose tests for assessment of beta-cell function: modeling analysis in normal subjects. Am J Physiol Endocrinol Metab. 2002, 283: E1159-E1166.View ArticlePubMedGoogle Scholar
- Avgoustiniatos ES: Oxygen diffusion limitations in pancreatic islet culture and immunoisolation. PhD Thesis. 2001, MITGoogle Scholar
- Johnson AS, Fisher RJ, Weir GC, Colton CK: Oxygen consumption and diffusion in assemblages of respiring spheres: Performance enhancement of a bioartificial pancreas. Chem Eng Sci. 2009, 64: 4470-4487.View ArticleGoogle Scholar
- Buchwald P: A general bilinear model to describe growth or decline time-profiles. Math Biosci. 2007, 205: 108-136.View ArticlePubMedGoogle Scholar
- Dufrane D, Nenquin M, Henquin JC: Nutrient control of insulin secretion in perifused adult pig islets. Diabetes Metab. 2007, 33: 430-438.View ArticlePubMedGoogle Scholar
- Buchwald P: COMSOL Multiphysics-based exploratory insulin secretion model for isolated pancreatic islets. Presented at COMSOL Conference 2010 Boston. COMSOL Conference 2010 Boston; October 7-9; Boston, MA. 2010, COMSOL, Inc.Google Scholar
- Henquin JC, Nenquin M, Stiernet P, Ahren B: In vivo and in vitro glucose-induced biphasic insulin secretion in the mouse: pattern and role of cytoplasmic Ca2+ and amplification signals in beta-cells. Diabetes. 2006, 55: 441-451.View ArticlePubMedGoogle Scholar
- Ricordi C, Gray DWR, Hering BJ, Kaufman DB, Warnock GL, Kneteman NM, Lake SP, London NJM, Socci C, Alejandro R, Zeng Y, Scharp DW, Viviani G, Falqui L, Tzakis A, Bretzel RG, Federlin K, Pozza G, James RFL, Rajotte RV, Di Carlo V, Morris PJ, Sutherland DER, Starzl TE, Mintz DH, Lacy PE: Islet isolation assessment in man and large animals. Acta Diabetol Lat. 1990, 27: 185-195.View ArticlePubMedGoogle Scholar
- Kissler HJ, Niland JC, Olack B, Ricordi C, Hering BJ, Naji A, Kandeel F, Oberholzer J, Fernandez L, Contreras J, Stiller T, Sowinski J, Kaufman DB: Validation of methodologies for quantifying isolated human islets: an islet cell resources study. Clin Transplant. 2010, 24: 236-242.PubMed CentralView ArticlePubMedGoogle Scholar
- Baum D, Griepp R, Porte D: Glucose-induced insulin release during acute and chronic hypoxia. Am J Physiol. 1979, 237: E45-E50.PubMedGoogle Scholar
- Carlsson PO, Palm F, Andersson A, Liss P: Chronically decreased oxygen tension in rat pancreatic islets transplanted under the kidney capsule. Transplantation. 2000, 69: 761-766.View ArticlePubMedGoogle Scholar
- MacGregor RR, Williams SJ, Tong PY, Kover K, Moore WV, Stehno-Bittel L: Small rat islets are superior to large islets in in vitro function and in transplantation outcomes. Am J Physiol Endocrinol Metab. 2006, 290: E771-E779.View ArticlePubMedGoogle Scholar
- Lehmann R, Zuellig RA, Kugelmeier P, Baenninger PB, Moritz W, Perren A, Clavien PA, Weber M, Spinas GA: Superiority of small islets in human islet transplantation. Diabetes. 2007, 56: 594-603.View ArticlePubMedGoogle Scholar
- O'Sullivan ES, Johnson AS, Omer A, Hollister-Lock J, Bonner-Weir S, Colton CK, Weir GC: Rat islet cell aggregates are superior to islets for transplantation in microcapsules. Diabetologia. 2010, 53: 937-945.PubMed CentralView ArticlePubMedGoogle Scholar
- Skiles ML, Fancy R, Topiwala P, Sahai S, Blanchette JO: Correlating hypoxia with insulin secretion using a fluorescent hypoxia detection system. J Biomed Mater Res B Appl Biomater. 2011, 97B: 148-155.View ArticleGoogle Scholar
- Ricordi C, Strom TB: Clinical islet transplantation: advances and immunological challenges. Nat Rev Immunol. 2004, 4: 259-268.View ArticlePubMedGoogle Scholar
- Fiorina P, Shapiro AM, Ricordi C, Secchi A: The clinical impact of islet transplantation. Am J Transplant. 2008, 8: 1990-1997.View ArticlePubMedGoogle Scholar
- King S: The evolution of islet encapsulation ventures. [http://www.solvingdiabetes.org/2009/05/21/history-of-diabetes-research]
- Garfinkel MR, Harland RC, Opara EC: Optimization of the microencapsulated islet for transplantation. J Surg Res. 1998, 76: 7-10.View ArticlePubMedGoogle Scholar
- de Vos P, Smedema I, van Goor H, Moes H, van Zanten J, Netters S, de Leij LF, de Haan A, de Haan BJ: Association between macrophage activation and function of micro-encapsulated rat islets. Diabetologia. 2003, 46: 666-673.PubMedGoogle Scholar
- Fort A, Fort N, Ricordi C, Stabler CL: Biohybrid devices and encapsulation technologies for engineering a bioartificial pancreas. Cell Transplant. 2008, 17: 997-1003.View ArticlePubMedGoogle Scholar
- Buchwald P, Bocca N, Marzorati S, Hochhaus G, Bodor N, Stabler C, Kenyon NS, Inverardi L, Molano RD, Ricordi C, Pileggi A: Feasibility of localized immunosuppression: 1. Exploratory studies with glucocorticoids in a biohybrid device designed for cell transplantation. Pharmazie. 2010, 65: 421-428.PubMedGoogle Scholar
- Rorsman P, Renström E: Insulin granule dynamics in pancreatic beta cells. Diabetologia. 2003, 46: 1029-1045.View ArticlePubMedGoogle Scholar
- MacDonald PE, Joseph JW, Rorsman P: Glucose-sensing mechanisms in pancreatic beta-cells. Philos Trans R Soc Lond B Biol Sci. 2005, 360: 2211-2225.PubMed CentralView ArticlePubMedGoogle Scholar
- Nesher R, Cerasi E: Modeling phasic insulin release: immediate and time-dependent effects of glucose. Diabetes. 2002, 51 (Suppl 1): S53-S59.View ArticlePubMedGoogle Scholar
- Renard E, Place J, Cantwell M, Chevassus H, Palerm CC: Closed-loop insulin delivery using a subcutaneous glucose sensor and intraperitoneal insulin delivery: feasibility study testing a new model for the artificial pancreas. Diabetes Care. 2010, 33: 121-127.PubMed CentralView ArticlePubMedGoogle Scholar
- Cretti A, Lehtovirta M, Bonora E, Brunato B, Zenti MG, Tosi F, Caputo M, Caruso B, Groop LC, Muggeo M, Bonadonna RC: Assessment of beta-cell function during the oral glucose tolerance test by a minimal model of insulin secretion. Eur J Clin Invest. 2001, 31: 405-416.View ArticlePubMedGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.