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.     

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.     

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.     

Computational fluid dynamics modeling of an inverted frustoconical shaking bioreactor for mammalian cell suspension culture

We previously developed an inverted frustoconical shaking bioreactor (IFSB) which had high mammalian cell culture performance when compared with a mechanically stirred tank reactor (STR) or a flat-bottom shaking bioreactor (FBSB). Here, we determined the mixing time (t) and volumetric oxygen transfer coefficient (k _La) of this IFSB at various speeds, and simulated the fluid hydrodynamics, including the shear stress and specific surface area, by computational fluid dynamics. The shortest mixing time was observed in a STR. The maximum kLa value of 12/h was achieved in the IFSB at an aeration rate of 4 L/h, demonstrating that our IFSB has enhanced oxygen transfer capabilities needed to meet the demands of mammalian cells. Simulation studies revealed a 3% greater specific surface area and a 21% lower shear strain in the IFSB compared to an FBSB under the same conditions. Additionally, the conical angle of the vessel, which significantly affected cell growth and recombinant protein production, was tested here. We conclude that, compared to the STR and FBSB, the IFSB has an increased liquid surface area for oxygen uptake and exhaust CO₂ stripping, an enhanced k _La for cell robust growth to a high cell density, and a lower shear stress to alleviate cell damage.     

Coupled modeling of blood perfusion in intravascular, interstitial spaces in tumor microvasculature

The coupling of intravascular and interstitial flow is a distinct feature of tumor microcirculation, due to the high vessel permeability, the low osmotic pressure gradient as well as the absence of functional lymphatic system inside tumors. In this paper, a coupled mathematical model of tumor microcirculation is developed, which provides the link between microvasculature and interstitial space perfusion through the matrices determining a neighbor point belonging to either connected vessel (matrix B) or interstitial space (matrix A), and combines the intravascular and interstitial flow by vascular leaky terms. In addition, the compliance of tumor vessels, blood rheology with hematocritic distribution at branches is also considered. The microvascular network, on which the microcirculation calculation is carried out, is generated from our two-dimensional 9-point (2D9P) model of tumor angiogenesis, improved from the previous 2D5P one. A specific coupling procedure is developed in the study to couple the intravascular and interstitial flow. It is based on the iteratively numerical simulation techniques, including local iterations at individual parameter level and one global loop to provide coupling and simulation convergence. The simulation results not only present the basic features and characteristics of tumor microcirculation, which agree with the corresponding experimental observations reported, but also predict an intimate relationship between the tumor intravascular and interstitial flow quantitatively. Among the parameters, the vascular leakiness is a key to govern the systemic flowing pattern, influence the tumor internal environment and contribute to the metastasis of tumor cells, which could not be presented by the previous uncoupled models.     

Computational fluid modeling and performance analysis of a bidirectional rotating perfusion culture system

A myriad of bioreactor configurations have been investigated as extracorporeal medical support systems for temporary replacement of vital organ functions. In recent years, studies have demonstrated that the rotating bioreactors have the potential to be utilized as bioartificial liver assist devices (BLADs) owing to their advantage of ease of scalability of cell‐culture volume. However, the fluid movement in the rotating chamber will expose the suspended cells to unwanted flow structures with abnormally high shear conditions that may result in poor cell stability and in turn lower the efficacy of the bioreactor system. In this study, we compared the hydrodynamic performance of our modified rotating bioreactor design with that of an existing rotating bioreactor design. Computational fluid dynamic analysis coupled with experimental results were employed in the optimization process for the development of the modified bioreactor design. Our simulation results showed that the modified bioreactor had lower fluid induced shear stresses and more uniform flow conditions within its rotating chamber than the conventional design. Experimental results revealed that the cells within the modified bioreactor also exhibited better cell‐carrier attachment, higher metabolic activity, and cell viability compared to those in the conventional design. In conclusion, this study was able to provide important insights into the flow physics within the rotating bioreactors, and help enhanced the hydrodynamic performance of an existing rotating bioreactor for BLAD applications. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29:1002–1012, 2013     

Computational design,structure refinement and molecular dynamics simulation of novel engineered serratiopeptidase analogs

Communicated by Ramaswamy H. Sarma     

初级视觉皮层“简单”与“复杂”神经元动力学的计算建模

本文回顾了我们在哺乳动物视觉皮层的建模工作.利用初级视觉皮层的大规模神经元网络模型,我们解释了初级视觉皮层里“简单”与“复杂”神经元现象的网络机制.所谓的“简单”细胞对视觉刺激的反应近似线性,而“复杂”细胞对视觉刺激是非线性的.我们的模型成功地再现了简单和复杂细胞分布的实验数据.     

Computational modeling of chemical reactions and interstitial growth and remodeling involving charged solutes and solid-bound molecules

Mechanobiological processes are rooted in mechanics and chemistry, and such processes may be modeled in a framework that couples their governing equations starting from fundamental principles. In many biological applications, the reactants and products of chemical reactions may be electrically charged, and these charge effects may produce driving forces and constraints that significantly influence outcomes. In this study, a novel formulation and computational implementation are presented for modeling chemical reactions in biological tissues that involve charged solutes and solid-bound molecules within a deformable porous hydrated solid matrix, coupling mechanics with chemistry while accounting for electric charges. The deposition or removal of solid-bound molecules contributes to the growth and remodeling of the solid matrix; in particular, volumetric growth may be driven by Donnan osmotic swelling, resulting from charged molecular species fixed to the solid matrix. This formulation incorporates the state of strain as a state variable in the production rate of chemical reactions, explicitly tying chemistry with mechanics for the purpose of modeling mechanobiology. To achieve these objectives, this treatment identifies the specific theoretical and computational challenges faced in modeling complex systems of interacting neutral and charged constituents while accommodating any number of simultaneous reactions where reactants and products may be modeled explicitly or implicitly. Several finite element verification problems are shown to agree with closed-form analytical solutions. An illustrative tissue engineering analysis demonstrates tissue growth and swelling resulting from the deposition of chondroitin sulfate, a charged solid-bound molecular species. This implementation is released in the open-source program FEBio (www.febio.org). The availability of this framework may be particularly beneficial to optimizing tissue engineering culture systems by examining the influence of nutrient availability on the evolution of inhomogeneous tissue composition and mechanical properties, the evolution of construct dimensions with growth, the influence of solute and solid matrix electric charge on the transport of cytokines, the influence of binding kinetics on transport, the influence of loading on binding kinetics, and the differential growth response to dynamically loaded versus free-swelling culture conditions.