A local glucose-and oxygen concentration-based insulin secretion model for pancreatic islets
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.
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.
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.
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.
Keywordsdiabetes mellitus FEM model glucose-insulin dynamics Hill equation islet perifusion islets of Langerhans oxygen consumption PID controller
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 . 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  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 , 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  including automated closed-loop insulin delivery systems [8, 9, 10] as well as for the development of 'bioartificial pancreas' systems such as those using immune-isolated, encapsulated islets [11, 12, 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"  and many others [15, 16, 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, 21, 22, 23, 24, 25, 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, 28, 29, 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 . 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)  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, 21, 22, 23, 24, 25, 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  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, 37, 38, 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  and by Dionne, Colton and co-workers for oxygen dependence .
Mass transport model (convective and diffusive)
where, c denotes the concentration [mol m-3] and D the diffusion coefficient [m2 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, Open image in new window . The following diffusion coefficients were used as consensus estimates of values available from the literature: oxygen, D oxy,w = 3.0 × 10-9 m2 s-1 in aqueous media and D oxy,t = 2.0 × 10-9 m2 s-1 in islet tissue ( and references therein); glucose, D gluc,w = 0.9 × 10-9 m2 s-1 and D gluc,t = 0.3 × 10-9 m2 s-1; insulin, D ins,w = 0.15 × 10-9 m2 s-1 and D ins,t = 0.05 × 10-9 m2 s-1 [23, 24]. Published tissue values for glucose vary over a wide range (0.04-0.5 × 10-9 m2 s-1) [32, 43, 44, 45, 46]; a value toward the higher end of this range (0.3 × 10-9 m2 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 m2 s-1, D gluc,c = 0.6 × 10-9 m2 s-1, D ins,c = 0.1 × 10-9 m2 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 : 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  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 , but in good agreement with the approximately 50%-100% fold increase seen in various experimental settings [35, 36, 56, 57, 58, 59, 60]. As before , 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  was used as step-down function, δ(c oxy > C cr,oxy) = flc1hs(c oxy - 1.0x10-4, 0.5x10-4).
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 ws used here of 3.0 × 10-9 vs. 0.9 × 10-9 m2 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) . Glucose consumption by islet cells alters the glucose levels reaching the glucose-sensing β-cells only minimally.
Summary of Hill function (eq. 2) parameters used in the present model (Figure 1, eq. 3-9)
R oxy, oxygen consumption, base
Cut to 0 below critical value, c oxy <C cr,oxy.
R oxy, oxygen consumption, φ o,g metabolic part
Due to increasing metabolic demand; parallels second-phase insulin secretion rate.
R gluc, glucose consumption
Contrary to oxygen, has no significant influence on model results.
R ins,ph2, insulin secretion rate, second-phase
3 × 10-5 mol/m3/s
Total secretion rate is modulated by local oxygen availability (last row).
R ins,ph1, insulin secretion rate, first-phase
21 × 10-5 mol/m3/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
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  and used by Colton and co-workers  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(cinsL - 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 m2 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 ; 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  and one concentrating on the effect of hypoxia using isolated rat islets .
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-1K-1, k c = 0.634 J s-1m-1K-1, α = 2.1 × 10-4 K-1. As previously , 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  used solutions gassed with enriched oxygen (e.g., 95% O2 + 5% CO2; 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.
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
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 × 106 μm (corresponding to an islet with d = 133 μm) . 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 . A number of GSIR perifusion studies including  used solutions gassed with enriched oxygen (e.g., 95% O2 + 5% CO2; 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  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) , 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%) , 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 (pO2 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 (pO2 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)
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, 12, 13]. Many failed attempts  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, 21, 22, 23, 24, 25, 26].
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) .
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, 9, 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.
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.
The financial support of the Diabetes Research Institute Foundation http://www.diabetesresearch.org that made this work possible is gratefully acknowledged.
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