Computational modeling of combined cell population dynamics and oxygen transport in engineered tissue subject to interstitial perfusion

This work presents a computational model of tissue growth under interstitial perfusion inside a tissue engineering bioreactor. The model accounts both for the cell population dynamics, using a model based on cellular automata, and for the hydrodynamic microenvironment imposed by the bioreactor, using a model based on the Lattice–Boltzmann equation and the convection-diffusion equation. The conditions of static culture versus perfused culture were compared, by including the population dynamics along with oxygen diffusion, convective transport and consumption. The model is able to deal with arbitrary complex geometries of the spatial domain; in the present work, the domain modeled was the void space of a porous scaffold for tissue-engineered cartilage. The cell population dynamics algorithm provided results which qualitatively resembled population dynamics patterns observed in experimental studies, and these results were in good quantitative agreement with previous computational studies. Simulation of oxygen transport and consumption showed the fundamental contribution of convective transport in maintaining a high level of oxygen concentration in the whole spatial domain of the scaffold. The model was designed with the aim to be computationally efficient and easily expandable, i.e. to allow straightforward implementability of further models of complex biological phenomena of increasing scientific interest in tissue engineering, such as chemotaxis, extracellular matrix deposition and effect of mechanical stimulation.     

Computational fluid dynamics modeling of steady-state momentum and mass transport in a bioreactor for cartilage tissue engineering

Computational fluid dynamics (CFD) models to quantify momentum and mass transport under conditions of tissue growth will aid bioreactor design for development of tissue-engineered cartilage constructs. Fluent CFD models are used to calculate flow fields, shear stresses, and oxygen profiles around nonporous constructs simulating cartilage development in our concentric cylinder bioreactor. The shear stress distribution ranges from 1.5 to 12 dyn/cm(2) across the construct surfaces exposed to fluid flow and varies little with the relative number or placement of constructs in the bioreactor. Approximately 80% of the construct surface exposed to flow experiences shear stresses between 1.5 and 4 dyn/cm(2), validating the assumption that the concentric cylinder bioreactor provides a relatively homogeneous hydrodynamic environment for construct growth. Species mass transport modeling for oxygen demonstrates that fluid-phase oxygen transport to constructs is uniform. Some O(2) depletion near the down stream edge of constructs is noted with minimum pO(2) values near the constructs of 35 mmHg (23% O(2) saturation). These values are above oxygen concentrations in cartilage in vivo, suggesting that bioreactor oxygen concentrations likely do not affect chondrocyte growth. Scale-up studies demonstrate the utility and flexibility of CFD models to design and characterize bioreactors for growth of tissue-engineered cartilage.     

Computational fluid dynamics modeling of momentum transport in rotating wall perfused bioreactor for cartilage tissue engineering

In this study, computational fluid dynamics (CFD) analysis of a rotating-wall perfused-vessel (RWPV) bioreactor is performed to characterize the complex hydrodynamic environment for the simulation of cartilage development in RWPV bioreactor in the presence of tissue-engineered cartilage constructs, i.e., cell-chitosan scaffolds. Shear stress exerted on chitosan scaffolds in bioreactor was calculated for different rotational velocities in the range of 33-38 rpm. According to the calculations, the lateral and lower surfaces were exposed to 0.07926-0.11069 dyne/cm(2) and 0.05974-0.08345 dyne/cm(2), respectively, while upper surfaces of constructs were exposed to 0.09196-0.12847 dyne/cm(2). Results validate adequate hydrodynamic environment for scaffolds in RWPV bioreactor for cartilage tissue development which concludes the suitability of operational conditions of RWPV bioreactor.     

Analysis of oxygen transport in a diffusion-limited model of engineered heart tissue

Cardiac tissue engineering has made notable progress in recent years with the advent of an experimental model based on neonatal cardiomyocytes entrapped in collage gels and purified basement membrane extract, known as "engineered heart tissues" (EHTs). EHTs are a formidable display of tissue-level contractile function and cellular-level differentiation, although they suffer greatly from mass transport limitations due to the high density of metabolically active cells and the diffusion-limited nature of the hydrogel. In this report, a mathematical model was developed to predict oxygen levels inside a one-dimensional, diffusion-limited model of EHT. These predictions were then compared to values measured in corresponding experiments with a hypoxia-sensitive stain (pimonidazole). EHTs were cast between two plastic discs, which allowed for mass transfer with the culture medium to occur in only the radial direction. EHTs were cultured for up to 36 h in the presence of pimonidazole, after which time they were snap-frozen, histologically sectioned, and stained for bound pimonidazole. Quantitative image analysis was performed to measure the distance from the culture medium at which hypoxia first occurs under various conditions. As tested by variation of simple design parameters, the trends in oxygen profiles predicted by the model are in reasonable agreement with those obtained experimentally, although a number of ambiguities related to the specific model parameters led to a general overprediction of oxygen concentrations. Based on the sensitivity analysis in the present study, it is concluded that diffusion-reaction models may offer relatively precise predictions of oxygen concentrations in diffusion-limited tissue constructs.     

Computational modeling of nutrient utilization in engineered cartilage

This study presents a mathematical model for simulating cartilaginous culture of chondrocytes seeded in scaffolds and for investigating the effects of glucose and oxygen concentration and pH value on cell metabolic rates. The model can clearly interpret the unexplained experimental observation (Sengers BG, Heywood HK, Lee DA, Oomens CWJ, Bader DL. Nutrient utilization by bovine articular chondrocytes: A combined experimental and theoretical approach. J Biomech Eng. 2005;127:758–766.), which showed that the oxygen concentration within the scaffold may increase instead of continuously decreasing in static cartilaginous culture of chondrocytes. Results from simulation demonstrate that when cells metabolize glucose and form lactate under high glucose concentration conditions, the acidity in the culture environment increases, inhibiting cell metabolic rates in the process. Consequently, the rate of oxygen consumption decreases in later stages of cell culture. As oxygen can be replenished through the free surface of the culture medium, oxygen concentration within the scaffold increases rather than decreases over time in the acidic environment. Different initial glucose concentration yields different results. In low glucose concentration conditions, oxygen concentration basically keeps decreasing with culture time. This is because the pH in the environment does not significantly change because of slower glycolysis rate in low glucose concentration cases, forming less lactic acid. From the simulation results, additional information regarding in vitro culture of chondrocytes is obtained. The correlations between nutrient consumption, lactate secretion, and pH changes during cell culture are also understood and may serve as a reference for in vitro cell culture research of tissue engineering. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29: 452–462, 2013     

Engineered cartilage constructs subject to very low regimens of interstitial perfusion

We have studied an in vitro engineered cartilage model, consisting of bovine articular chondrocytes seeded on micro-porous scaffolds and perfused with very low regimens of interstitial flow. Our previous findings suggested that synthesis of sulphated glycosaminoglycans (sGAG) was promoted in this model, if the level of shear generated on cells was maintained below 10 mPa (0.1 dyn/cm2). Constructs were stimulated with a median shear stress of 1.2 and 6.7 mPa using two independent culture chambers. Quantification of the applied stresses and of oxygen consumption rates was obtained from computational modelling. Experimentally, we set a time zero reference at 24 hours after cell seeding and total culture time at two weeks. The cell metabolic activity, measured by MTT, was significantly lower in all constructs at two weeks (-73% in static controls, -66% in the 1.2 mPa group and -60% in the 6.7 mPa group) vs. the time zero group, and significantly higher (+33%) in the 7 mPa group vs. static controls. The ratio between synthesis of collagen type II/type I, measured by Western Blot, was significantly higher in the 1.2 mPa constructs (+109% vs. the 6.7 mPa group, +120% vs. the time zero group and +286% vs. static controls). A trend of decreased alpha-actin expression was observed with increased ratio of type II to type I collagen, in all groups. These results reinforce the notion that, at early time points in culture, hydrodynamic shear below 10 mPa may promote formation of extra-cellular matrix specific to hyaline cartilage in chondrocyte-seeded constructs.     

Non-linear phenomena in oxygen transport to tissue

A non-linear partial differential equation is analyzed using multiple scale techniques and similarity transformations in order to examine the role of hemoglobin and myoglobin in facilitating oxygen transport to tissue.Supported by NSF Grant DCB 8902472     

Mathematical modelling of oxygen transport to tissue

 The equations governing oxygen transport from blood to tissue are presented for a cylindrical tissue compartment, with blood flowing along a co–axial cylindrical capillary inside the tissue. These governing equations take account of: (i) the non–linear reactions between oxygen and haemoglobin in blood and between oxygen and myoglobin in tissue; (ii) diffusion of oxygen in both the axial and radial directions; and (iii) convection of haemoglobin and plasma in the capillary. A non–dimensional analysis is carried out to assess some assumptions made in previous studies. It is predicted that: (i) there is a boundary layer for oxygen partial pressure but not for haemoglobin or myoglobin oxygen saturation close to the inflow boundary in the capillary; (ii) axial diffusion may not be neglected everywhere in the model; (iii) the reaction between oxygen and both haemoglobin and myoglobin may be assumed to be instantaneous in nearly all cases; and (iv) the effect of myoglobin is only significant for tissue with a low oxygen partial pressure. These predictions are validated by solving the full equations numerically and are then interpreted physically. Received: 13 October 2000 / Revised version: 12 June 2001 / Published online: 17 May 2002     

The effect of hydrodynamic shear on 3D engineered chondrocyte systems subject to direct perfusion

Bioreactors allowing direct-perfusion of culture medium through tissue-engineered constructs may overcome diffusion limitations associated with static culturing, and may provide flow-mediated mechanical stimuli. The hydrodynamic stress imposed on cells within scaffolds is directly dependent on scaffold microstructure and on bioreactor configuration. Aim of this study is to investigate optimal shear stress ranges and to quantitatively predict the levels of hydrodynamic shear imposed to cells during the experiments. Bovine articular chondrocytes were seeded on polyestherurethane foams and cultured for 2 weeks in a direct perfusion bioreactor designed to impose 4 different values of shear level at a single flow rate (0.5 ml/min). Computational fluid dynamics (CFD) simulations were carried out on reconstructions of the scaffold obtained from micro-computed tomography images. Biochemistry analyses for DNA and sGAG were performed, along with electron microscopy. The hydrodynamic shear induced on cells within constructs, as estimated by CFD simulations, ranged from 4.6 to 56 mPa. This 12-fold increase in the level of applied shear stress determined a 1.7-fold increase in the mean content in DNA and a 2.9-fold increase in the mean content in sGAG. In contrast, the mean sGAG/DNA ratio showed a tendency to decrease for increasing shear levels. Our results suggest that the optimal condition to favour sGAG synthesis in engineered constructs, at least at the beginning of culture, is direct perfusion at the lowest level of hydrodynamic shear. In conclusion, the presented results represent a first attempt to quantitatively correlate the imposed hydrodynamic shear level and the invoked biosynthetic response in 3D engineered chondrocyte systems.     

Comprehensive assessment and mathematical modeling of T cell population dynamics and homeostasis

Our current view of T cell differentiation and population dynamics is assembled from pieces of data obtained from separate experimental systems and is thus patchy. We reassessed homeostasis and dynamics of T cells 1) by generating a mathematical model describing the spatiotemporal features of T cell differentiation, and 2) by fitting this model to experimental data generated by disturbing T cell differentiation through transient depletion of dividing T cells in mice. This specific depletion was obtained by administration of ganciclovir to mice expressing the conditional thymidine kinase suicide gene in T cells. With this experimental approach, we could derive quantitative parameters describing the cell fluxes, residence times, and rates of import, export, proliferation, and death across cell compartments for thymocytes and recent thymic emigrants (RTEs). Among other parameters, we show that 93% of thymocytes produced before single-positive stages are eliminated through the selection process. Then, a postselection peripheral expansion of naive T cells contributes three times more to naive T cell production than the thymus, with half of the naive T cells consisting of dividing RTEs. Altogether, this work provides a quantitative population dynamical framework of thymocyte development, RTEs, and naive T cells.     

Emergent cell and tissue dynamics from subcellular modeling of active biomechanical processes

Cells and the tissues they form are not passive material bodies. Cells change their behavior in response to external biochemical and biomechanical cues. Behavioral changes, such as morphological deformation, proliferation and migration, are striking in many multicellular processes such as morphogenesis, wound healing and cancer progression. Cell-based modeling of these phenomena requires algorithms that can capture active cell behavior and their emergent tissue-level phenotypes. In this paper, we report on extensions of the subcellular element model to model active biomechanical subcellular processes. These processes lead to emergent cell and tissue level phenotypes at larger scales, including (i) adaptive shape deformations in cells responding to slow stretching, (ii) viscous flow of embryonic tissues, and (iii) streaming patterns of chemotactic cells in epithelial-like sheets. In each case, we connect our simulation results to recent experiments.     

Mathematical modeling of oxygen transport in skeletal muscle

Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.     

On the dynamics of a population subject to slowly changing vital rates

Approximate closed-form expressions are given for the birth and death rates, the age-specific growth rates, and the age distribution of a population subject to slowly changing vital rates. The slower these rates have changed in the recent past, the better the approximations will be. These results, which are reminiscent of stable population theory, are tested by projecting the female population of the United States subject to realistically changing levels of fertility. The closed-form expressions yield quite accurate results and show that the dynamics of a population subject to slowly changing vital rates are to a large extent independent of the particular patterns of fertility. As in stable population theory, the asymptotic behavior of such a population is essentially determined by current survival rates and recent values of the intrinsic growth rate.     

Computational modelling of wounded tissue subject to negative pressure wound therapy following trans-femoral amputation

Proof-of-concept computational models were developed and applied as tools to gain insights into biomechanical interactions and variations of oxygen gradients of wounded tissue subject to negative pressure wound therapy (NPWT), following trans-femoral amputation. A macro-scale finite-element model of a lower limb was first developed based on computed tomography data, and distributions of maximum and minimum principal stress values we calculated for a region of interest (ROI). Then, the obtained results were applied iteratively as new sets of boundary conditions for a specific spatial position in a capillary sub-model. Data from coupled capillary stress and mass- diffusion sub-models were transferred to the macro-scale model to map the spatial changes of tissue oxygen gradients in the ROI. The ?70 mmHg NPWT resulted in a dramatic change of a wound surface area and the greatest relative contraction was observed at ?150 mmHg. Tissue lateral to the depth of the wound cavity revealed homogenous patterns of decrease in oxygenation area and the extent of such decrease was dependent on the distance from the wound surface. However, tissue lateral to the width of the wound demonstrated heterogeneous patterns of change, as evidenced by both gradual increase and decrease in the oxygenation area. The multiscale models developed in the current study showed a significant influence of NPWT on both macro-deformations and changes of tissue oxygenation. The patterns of changes depended on the depth of the tissue, the geometry of the wound, and also the location of tissue plane.     

Predictions for oxygen supply control to enhance population stability of engineered production strains

The simultaneous growth and product formation in a microbial culture is an important feature of several laboratory, industrial, and environmental bioprocesses. Metabolic burden associated with product formation in these bioprocesses may lead to growth advantage of a nonproducing mutant leading to a loss of the producing population over time. A simple population dynamics model demonstrates the extreme sensitivity of population stability to the engineered productivity of a strain. Here we use flux balance analysis to estimate the effects of the metabolic burden associated with product secretion on optimal growth rates. Comparing the optimal growth rates of the producing and nonproducing strains under a given processing condition allows us to predict the population stability. In order to increase stability of an engineered strain, we determine processing conditions that simultaneously maximize the growth rate of the producing population while minimizing the growth rate of a nonproducing population. Using valine, tryptophan, and lysine production as specific examples, we demonstrate that although an appropriate choice of oxygenation may increase culture longevity more than twofold, total production as governed by economic criterion can be increased by several orders of magnitude. Choice of optimal nutrient and oxygen supply rates to enhance stability is important both for strain screening as well as for culture of engineered strains. Appropriate design of the culture environment can thus be used to enhance the productivity of bioprocesses that use engineered production strains. (c) 1994 John Wiley & Sons, Inc.     

Comparison of occupancy modeling and radiotelemetry to estimate ungulate population dynamics

Radiotelemetry and unmarked occupancy modeling have been used to estimate animal population growth, but have not been compared for ungulates. We compared white-tailed deer (Odocoileus virginianus) population growth estimates from radiomarked individuals and occupancy modeling of unmarked individuals and evaluated advantages and disadvantages of each method. Estimates of population growth were obtained using remote camera (N = 54/year) detection/non-detection occupancy surveys of unmarked deer and from survival and recruitment data of radiomarked adult females (N = 87) and neonate fawns (N = 127) in a predominantly forested region of the Upper Peninsula of Michigan, USA, 2009–2011. We hypothesized that occupancy models and radiotelemetry data would have similar population growth trends because both methods sampled the same temporally closed population. Percent changes in camera trap data generally reflected finite population growth (λ) of radiomarked deer which increased (λ = 1.10 ± 0.01) from 2009 to 2010, but decreased (λ = 0.87 ± 0.02) from 2010 to 2011. Also, unmarked adult female abundance and fawn:adult female ratios generally reflected trends in radiomarked deer survival and recruitment. Royle–Nichols occupancy model abundance estimates had wide confidence intervals, which may preclude using this method from accurately estimating deer population growth. Radiotelemetry provided more precise population growth estimates, while allowing collection of vital rates and location data. However, the Royle–Nichols occupancy model may be preferred to radiotelemetry because it reflected yearly variation in population growth with reduced labor and no invasive marking. Researchers should consider the objectives and logistics of their study when choosing a specific method.     

Determination of oxygen gradients in engineered tissue using a fluorescent sensor

Nutrient and oxygen supply of cells are crucial to tissue engineering in general. If a sufficient supply cannot be maintained, the development of the tissue will slow down or even fail completely. Previous studies on oxygen supply have focused on measurement of oxygen partial pressures (pO(2)) in culture media or described the use of invasive techniques with spatially limited resolution. The experimental setup described here allows for continuous, noninvasive, high-resolution pO(2) measurements over the cross-section of cultivated tissues. Applying a recently developed technique for time-resolved pO(2) sensing using optical sensor foils, containing luminescent O(2)-sensitive indicator dyes, we were able to monitor and analyze gradients in the oxygen supply in a tissue over a 3-week culture period. Cylindrical tissue samples were immobilized on top of the sensors. By measuring the luminescence decay time, two-dimensional pO(2) distributions across the tissue section in contact with the foil surface were determined. We applied this technique to cartilage explants and to tissue-engineered cartilage. For both tissue types, changes were detected in monotonously decreasing gradients of pO(2) from the surface with high pO(2) to minimum pO(2) values in the center of the samples. Nearly anoxic conditions were observed in tissue constructs ( approximately 0 Torr) but not in excised cartilage discs ( approximately 20 Torr) after 1 day. Furthermore, the oxygen supply seemed to strongly depend on cell density and cell function. Additionally, histological analysis revealed a maximum depth of approximately 1.3 mm of regular cartilage development in constructs grown under the applied culture conditions. Correlating analytical and histological analysis with the oxygen distributions, we found that pO(2) values below 11 Torr might impair proper tissue development in the center. The results illustrate that the method developed is an ideal one to precisely assess the oxygen demand of cartilage cultures.     

Computational modeling of blood component transport related to coronary artery thrombosis in Kawasaki disease

Coronary artery thrombosis is the major risk associated with Kawasaki disease (KD). Long-term management of KD patients with persistent aneurysms requires a thrombotic risk assessment and clinical decisions regarding the administration of anticoagulation therapy. Computational fluid dynamics has demonstrated that abnormal KD coronary artery hemodynamics can be associated with thrombosis. However, the underlying mechanisms of clot formation are not yet fully understood. Here we present a new model incorporating data from patient-specific simulated velocity fields to track platelet activation and accumulation. We use a system of Reaction-Advection-Diffusion equations solved with a stabilized finite element method to describe the evolution of non-activated platelets and activated platelet concentrations [AP], local concentrations of adenosine diphosphate (ADP) and poly-phosphate (PolyP). The activation of platelets is modeled as a function of shear-rate exposure and local concentration of agonists. We compared the distribution of activated platelets in a healthy coronary case and six cases with coronary artery aneurysms caused by KD, including three with confirmed thrombosis. Results show spatial correlation between regions of higher concentration of activated platelets and the reported location of the clot, suggesting predictive capabilities of this model towards identifying regions at high risk for thrombosis. Also, the concentration levels of ADP and PolyP in cases with confirmed thrombosis are higher than the reported critical values associated with platelet aggregation (ADP) and activation of the intrinsic coagulation pathway (PolyP). These findings suggest the potential initiation of a coagulation pathway even in the absence of an extrinsic factor. Finally, computational simulations show that in regions of flow stagnation, biochemical activation, as a result of local agonist concentration, is dominant. Identifying the leading factors to a pro-coagulant environment in each case—mechanical or biochemical—could help define improved strategies for thrombosis prevention tailored for each patient.