A finite element method (FEM) based approach was used as implemented in COMSOL Multiphysics 3.4 (formerly FEMLAB) (COMSOL Inc., Burlington, MA). FEMs represent a numerical technique designed to find approximate solutions of general partial differential equations (PDE) based problems and are well-suited for complex geometries or varying domains since they rely on 'discretization' of the problem, i.e., the geometry is partitioned into small units of a simple shape (e.g., triangles for 2D and tetrahedrons for 3D subdomains) 36.

Diffusion was assumed to be governed by the generic diffusion equation in its nonconservative formulation (incompressible fluid) 37:

where, c denotes the concentration [mol·m^-3] and D the diffusion coefficient [m²·s^-1] of the species of interest (here, oxygen), R the reaction rate [mol·m^-3·s^-1], u the velocity field [m·s^-1], and ▽ the standard del (nabla) operator, 38. For oxygen consumption, a Michaelis-Menten-type consumption rate (R < 0) was assumed as customary in current literature 2739:

Here, R_max is the maximum oxygen consumption rate, the Michaelis-Menten constant corresponding to the oxygen concentration where consumption drops to 50% of its maximum, C_cr is the critical oxygen concentration below which necrosis is assumed to occur after a sufficiently long exposure, and δ a step-down function to account for the ceasing of consumption in those parts of the tissue where the oxygen concentration fell below a C_cr critical concentration. Consensus estimates of various parameters available from the literature were used. Oxygen in aqueous solutions obeys Henry's law rather well; i.e., its (mole fraction) solubility () is essentially proportional to the partial pressure of oxygen () in the surrounding media, 40. For the present exploratory calculations, c_amb = 0.200 mol·m^-3 (mM) was assumed for surfaces in contact with atmospheric oxygen. With an oxygen solubility coefficient of α = 1.45 × 10^-3 mol·m^-3mmHg^-1 (35°C) 41, this roughly corresponds to a partial pressure of 140 mmHg. A maximum oxygen consumption rate R_max (per unit islet volume) of 0.034 mol·s^-1·m^-3 was used in all calculations. With a standard islet of 150 μm diameter (and islet equivalent IEQ volume V_IEQ of 1.77 × 10^-12 m³), this corresponds to a consumption rate (per islet) of R_max = 0.06 × 10^-12 mol·s^-1/islet; both values being in the range of those measured and used in various works 20232930324243444546. As Michaelis-Menten constant, = 1.0 × 10^-3 mol·m^-3 (1 μM) was assumed, corresponding to = 0.7 mmHg – similar to the frequently used 0.44 mmHg value 23272932 or even to that determined originally for mitochondria 39. A step-down function, δ, was also added to account for necrosis and cut the oxygen consumption when the concentration fells below a critical value, C_cr = 1.0 × 10^-4 mol·m^-3 (corresponding to = 0.07 mmHg; comparable with the commonly used 0.10 mmHg 232732). COMSOL's smoothed Heaviside function with a continuous first derivative and without overshoot flc1hs 47 was used as step-down function, δ (c) = flc1hs(c-1.0 × 10^-4, 0.5 × 10^-4). In the dynamic model (perifusion chamber), oxygen consumption was allowed to also vary as a function of the local glucose concentration, c_gluc, to account for the increased metabolic demand of insulin production at higher glucose concentrations. As a first modelling attempt, this was done by introduction of an additional Michaelis-Menten-type dependency on c_gluc:

The corresponding constants were selected so as to allow an approximate doubling when going from low (3 mM) to high (11 mM) glucose concentration (C_{MM, luc} = 8 mol·m^-3, φ = 3.67). For the diffusion coefficient of oxygen in aqueous media, = 3.0 × 10^-9 m²·s^-1 was assumed; a reasonable approximation for O₂ diffusion in water at 37°C considering the commonly accepted value of 2.4 × 10^-9 m²·s^-1 at 25°C 41 and a measured value of 3.1 × 10^-9 m²·s^-1 at 45°C or fitted diffusivity equations such as the Wilke-Chang or Othmer-Thakar estimates for diffusion coefficient in aqueous solutions 48. For the diffusion coefficient of oxygen in tissue, = 2.0 × 10^-9 m²·s^-1, was assumed; slightly less than in water and the same value that was used by Radisic, Vunjak-Novakovic and co-workers 33. Avgoustiniatos and co-workers have recently determined a somewhat lower value for the effective diffusion coefficient of oxygen in rat pancreatic islets (1.3 × 10^-9 m²·s^-1) 44. The same value was used for the diffusion coefficient in silicone since it was within the range of measured values 2449.

In the more complex cases where true multiphysics models were needed, the convection and diffusion model of eq. 1 was coupled to a fluid dynamics model. For fluid dynamics, the incompressible Navier-Stokes model for Newtonian flow (constant viscosity) was used to calculate the velocity field u that results from convection 3750:

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. For cases where convective flow was also allowed in the model, an essentially aqueous media at body temperature was assumed as a first estimate: T₀ = 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.

For the present exploratory models, fully scaled realistic 2D and 3D geometries have been used with spherical islets of 100, 150, and 200 μm diameters placed in millimeter-sized device models. COMSOL's predefined 'Extra fine' and 'Fine' mesh size was used for meshing of 2D and 3D geometries, respectively resulting in meshes with 5–10,000 elements in 2D and 150,000 elements in 3D. In the convection and diffusion models, the following conditions were assumed: insulation/symmetry, n·(-D▽c+cu) = 0, for side walls, continuity for islets, and fixed concentration (c = c_amb) for liquid surfaces in contact with exterior media (top). For the case of diffusion through a membrane, a membrane/media partition coefficient K_p = c_membr/c was built into the model for oxygen through a special boundary condition using the stiff-spring method 51. An additional, separate concentration variable c₂ was added for the membrane (with a corresponding application mode), and to maintain continuous flux at the interface, an inward flux boundary condition was imposed along the membrane-fluid boundary with υ (c₂ - K_pc) and υ = 10,000 m·s^-1. In the incompressible Navier-Stokes models, no slip (u = 0) was assumed along all surfaces corresponding to liquid-solid interfaces. For the perifusion chamber, a parabolic inflow velocity profile, 4v_ins(1-s), was used on the inlet (s being the boundary segment length) and pressure, no viscous stress with p₀ = 0 on the outlet.

All models were implemented in COMSOL Multiphysics 3.4 and solved as time-dependent problems up to sufficiently long final times to reach steady state allowing free or intermediate time-steps for the solver. Computations were done with the UMFPACK direct solver as linear system solver on a Dell Precision 690 PC with a 3.2 GHz CPU running Linux.

The oxygen distribution obtained for a two-dimensional cross section of three differently sized islets in a traditional culture model, after steady state conditions are reached, is shown in Figure 1 with a corresponding animation (time-scale in seconds) shown in additional file 1. Since these are 2D cross-sections, the 'islets' here in fact correspond to strings and not spheres; hence, Figure 1 roughly corresponds to a 3D culture density of about 1,600 IEQ/cm² (i.e., a surface utilization of ~20%). Under these conditions, larger islets are predicted to have necrotic cores, a problem that is not present in smaller islets. The percent of cross-sectional areas predicted in this example to be below the critical oxygen threshold were around 25%, 5%, and 0% for the islets with diameters of 200, 150, and 100 μm, respectively. Overall, calculations are in good agreement with various experimental observations of cultured islets (see Discussion). Obviously, oxygenation can be improved by lowering the density of the consuming tissue, by reducing the diffusion path in the media, or by increasing the outside oxygen concentration. For example, the same three islets are predicted to have larger necrotic portions if the media height is larger (2 mm), and, hence, the diffusion path of the oxygen from the top is also larger (percent areas predicted to be below the critical oxygen threshold were around 50%, 30%, and 0% for the islets with diameters of 200, 150, and 100 μm, respectively) (Figure 2, additional file 2). Actual standard cultures use lower densities (100–200 IEQ/cm²) 181920, and indeed a single standard islet (ɸ = 150 μm) in the same 2D culture model, which would correspond to a lower cell density of ~500 IEQ/cm², can survive without any necrosis of its core even in this deeper media (Figure 3, additional file 3).

A true three-dimensional simulation of such a culture was also run; however, such calculations are more difficult to implement and are considerably more time consuming to run as they contain many more mesh elements. Results obtained for a representative case with randomly distributed islets at a density of approximately 600 IEQ/cm² and covered by 2 mm of media are shown in Figure 4 with a corresponding animation shown in additional file 4. In these conditions, islets up to about standard islet size (ɸ = 150 μm) show essentially no necrosis, but the larger ones show some central necrosis. For example, the larger islets here (ɸ = 200 μm) were predicted to have ≈ 10% of their volume as necrotic.

A model of a similar islet culture, but having oxygen-permeable bottom membranes was also explored as such devices are one of the possibilities being investigated to increase the oxygenation of cell cultures in general and islet cultures in particular. Calculations were performed assuming a 0.275 mm thick membrane with ten-fold higher oxygen solubility than water. As Figure 5 and additional file 5 show, much better oxygenations can indeed be achieved with such membranes even at high islet densities in agreement with experimental observations 202223. All regions of the islets considered were predicted to have oxygen concentrations well above critical levels.

Finally, a true multiphysics model incorporating both diffusion and convection due to flow was implemented to simulate oxygen consumption in a perifusion chamber model with two islets and moving media; such devices are now frequently used for the dynamic assessment of islet quality and function. As a more realistic model of the dynamics of oxygen consumption, the oxygen consumption of islets was assumed to increase with increasing glucose concentration due to the increased metabolic demand 4652535455. As a first, exploratory model, an approximate doubling of the consumption rate was assumed when going from low (3 mM) to high (11 mM) glucose concentration (eq. 3, which at high would correspond to R_max increasing from 0.034 to 0.074 mol·s^-1·m^-3). Figure 6 shows the velocity field of the flowing (incompressible) media; obviously, flow velocity has to increase were the cross section is constrained by the presence of islets to maintain a constant flux. Calculated oxygen concentrations are shown in Figure 7 at various incoming glucose concentrations (with a corresponding animation in additional file 6). At low glucose concentration (3 mM), the islets considered show no necrosis despite the relatively large seeding density because the flowing media can provide better oxygenation (Figure 7a). After the glucose concentration is increased (Figure 7b), the higher metabolic demand is predicted to result in falling of the oxygen concentrations below the critical threshold in certain regions, especially in larger islets (Figure 7c), which might result in necrosis if sufficiently prolonged. If the increased demand lasts for only a relatively limited time, part of the damage might be reversible as the glucose concentration is decreased (Figure 7d).

All models implemented here assumed that oxygen consumption takes place only within islet tissues and follows a Michaelis-Menten-type kinetics (eq. 2) that, at non-elevated glucose concentrations, plateaus at a maximum consumption rate R_max (per unit islet volume) of 0.034 mol·s^-1·m^-3. This per volume value is similar to that used by Avgoustiniatos and co-workers (0.034 mol/s/m³ 44; 0.050 mol/s/m³ 23) and Tilakaratne and co-workers (0.046 mol/s/m³) 29. As a per islet value (0.06 × 10^-12 mol·s^-1/islet), it is similar to that assumed by Dulong and Legallais (0.063 × 10^-12 mol·s^-1/islet) 3042; somewhat less than that assumed by Papas, Avgoustiniatos, and co-workers (0.127 × 10^-12 mol·s^-1/islet 2032; 0.074 × 10^-12 mol·s^-1/islet 43); and slightly larger than those measured recently in various settings by Sweet and co-workers (e.g., 0.025–0.048 × 10^-12 mol·s^-1/islet at 3 mM basal- or 20 mM high glucose 4546). Converted to a per cell value (3.0 × 10^-17 mol·s^-1/cell), it is also in general agreement with values observed with other high-demand cells 21. This consumption rate (0.034 mol·s^-1·m^-3) means that each volume unit of islet needs about 70 times its volume daily in oxygen as gas (0.034 mol/s/m³ × 24·3600 s/day × 0.02478 m³/mol). For comparison, the average respiration rate of a human (16 respiration/min each of ~0.5 L, 4% of which is oxygen consumed) gives an approximate oxygen consumption of 0.3 L/min 56, which means about 500 L/day, i.e., about 6–7 times its volume as living organism. Hence, considering that islets are metabolically high-demand cells receiving about 10 times higher blood flow than their surrounding tissue in the pancreas, this oxygen consumption rate is a realistic first estimate.

The Michaelis-Menten-type consumption rate assures that at very low O₂ concentrations, where cells only try to survive, oxygen consumption decreases with the available concentration, . Furthermore, a step-down function δ was also incorporated into the model to account for necrosis (cell death) and eliminate the oxygen consumption of those tissues where fell below a critical value, C_cr, and could cause cell death due to hypoxia after a sufficiently prolonged exposure. In general, islets seem to show a size-distribution well described by a Weibull distribution (often used as Rosin-Rammler distribution for particle size), , with most islets having smaller diameters (~50 μm), but the bulk of the volume being contributed by larger (ɸ = 2r = 100–200 μm) islets 10575859; hence, islets with diameters of 100, 150, and 200 μm were selected as representative here (especially since larger islets are of more interest as hypoxia is more likely to be a problem for them).

Results obtained here (Figure 1, 2, 3, 4) are in good overall agreement with various experimental observations indicating that when isolated islets are cultured for 24–48 h in normoxic culture conditions, large islets show central necrosis, which becomes much more severe after exposure to hypoxic culture conditions 15. As a first estimate, even the size of the necrotic core as measured for rat islets by Vasir and co-workers 15 or by MacGregor and co-workers 60 is well predicted suggesting that these exploratory models provide reasonable quantitative estimates and not just qualitative fit. Results also confirm that in traditional cultures, very low culture densities are needed to ensure viability of the core of larger islets justifying the current standard practice (100–200 IEQ/cm² corresponding to a surface utilization of only 2–3% 7181920). 3D models are considerably more difficult to implement and time-consuming to run than 2D models; nevertheless, one 3D simulation was performed (Figure 4, additional file 4) to validate the 2D simulations. Similar results were obtained – compare, for example, Figure 3 and Figure 4; in both cases, standard islets (ɸ = 150 μm) showed only very minimal central necrosis (<1%) at densities of 5–600 IEQ/cm² and a media height of 2 mm. Figure 4 also confirms that agglomeration resulting from non-uniform distribution can be a problem as necrotic regions are larger in islets that have close neighbours.

It should be noted that in all these models, instantaneous death for tissues was assumed as soon as the local values fell below the critical threshold. Hence, while these models are ultimately realistic at steady state, the time-scales, which are shown in seconds in all animations, are probably not, since, under critical conditions, real islets and cells can probably shut down their metabolism more effectively and can survive for some time before irreversible death occurs; more realistic models that can also account for the hypoxia exposure time will be developed in the future. Mammalian cells have developed various mechanisms to survive acute and even prolonged hypoxia 61. For example, a brief (10 min) ischemic preconditioning might even improve islet cell recovery after cold preservation 62. On the other hand, there are certainly additional inter-cellular danger- or death-related signals that are not taken into account by the present simplified, oxygen diffusion only models.

As illustrated by Figure 1, 2, 3, 4, devices with enhanced oxygenations are needed for more efficient islet culture. Use of cell culture devices with oxygen-permeable membrane bottoms is one of the most promising alternatives that are being explored toward this goal 2022. Silicone rubber-based membranes are a preferred choice due to their high oxygen-permeability 24. The solubility of oxygen in such silicone-based materials is also much higher than in water being, for example, around 0.3 cm³(STP)·cm^-3·atm^-1 in silicone rubber 24 compared to 0.024 cm³(STP)·cm^-3·atm^-1 in water (the latter corresponding to 0.0048 mL/L at normal air). Models as implemented here required the use of a special boundary condition using the stiff-spring method to account for the different solubilities of oxygen in the membrane and in the media; results are presented after recalibrating concentration in the membrane with the partition coefficient K_p = c_membr/c (Figure 5). They confirm that, indeed, much better oxygenation can be achieved and that the 'oxygen sandwich' designation 22 is justified as O₂ can reach the islets from both sides; in fact, under most conditions, a much larger flux is coming from the bottom through the membrane than from the top through the aqueous media. It should be noted that because in such membranes carbon dioxide tends to have an even higher permeability than oxygen 24, in certain cases, it might reach undesirably elevated or undesirably low levels (depending on the outside concentrations).

A model of a perifusion chamber was implemented because perifusion studies are now routinely used to assess islet quality and function as they allow the dynamic measurement of the glucose-stimulated insulin release (GSIR) 54636465 through the continuous monitoring of the insulin (and/or other metabolic products) released by islets placed in a perifusion column and exposed to varying levels of incoming glucose solutions. Because of the flowing media, this requires a true multiphysics approach to account for oxygen transport due to both diffusive and advective transfer. Furthermore, additional dynamics was also introduced by allowing the oxygen consumption to increase with the increasing metabolic demand imposed by the presence of higher glucose concentration, when islets are attempting to increase their insulin output. Here, an approximate doubling of the consumption rate was assumed in islets when going from low (3 mM) to high (11 mM) glucose concentrations (eq. 3) – a value in acceptable agreement with the average increase observed in rat islets by Longo and co-workers 53 or, more recently, in human islets by Sweet and co-workers 46 and also showing some correspondence to the increased proinsulin and total protein synthesis in islets in response to increasing glucose levels 66. As Figure 7 and additional file 6 show, the additional stress of increased metabolic demand might cause increased cell death, if sufficiently prolonged, due to the limited availability of oxygen; hence, straining of avascular islets by exposing them to high glucose levels for long periods of time might expose them to additional risks. Whereas the optimal glucose concentration for islet culture seems to be around 10 mM for rodent islets, it seems to be around 5 mM (90 mg/dL) for human islets 7. The relative ease of extending the present model not only to arbitrary geometries, but also to complex, multiphysics problems is an important advantage.