Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold.

The cultivation of cartilage cells (chondrocytes) in polymer scaffolds leads to implants that may potentially be used to repair damaged joint cartilage or for reconstructive surgery. For this technique to be medically applicable, the physical parameters that govern cell growth in a polymer scaffold must be understood. This understanding of cell behavior under in vitro conditions, where diffusion is the primary mode of transport of nutrients, may aid in the scale-up of the cartilage generation process. A mathematical model of chondrocyte generation and nutrient consumption is developed here to analyze the behavior of cell growth in a biodegradable polymer matrix for a series of different thickness polymers. Recent literature has implied that the diffusion of nutrients is a major factor that limits cell growth (Freed et al., 1994). In the present paper, a mathematical model is developed to directly relate the effects of increasing cell mass in the polymer matrix on the transport of nutrients. Reaction and diffusion of nutrients in the cell-polymer system are described using the fundamental species continuity equations and the volume averaging method. The volume averaging method is utilized to derive a single averaged nutrient continuity equation that includes the effective transport properties. This approach allows for the derivation of effective diffusion and rate coefficients as functions of the cell volume fraction. The cell volume fraction as a function of time is determined by solution of a material balance on cell mass. Growth functions including the Moser, a modified Contois, and an nth-order heterogeneous growth kinetic model are evaluated through a parameter analysis, and the results are compared to experimental data found in the literature. The results indicate that cellular functions in conjunction with mass transfer processes can account partially for the general trends in the cell growth behavior for various thickness polymers. The Contois growth function appeared to describe the data more accurately in terms of the lag period at early times and the long time limits. However, all kinetic growth functions required variations in the kinetic parameters to fully describe the effects of polymer thickness. This result implies that restricted diffusion of nutrients is not the sole factor limiting cell growth when the thickness of the polymer is changed. Therefore, further experimental data and model improvements are needed to accurately describe the cell growth process.     

Analysis of cell growth in a polymer scaffold using a moving boundary approach

Two mathematical models of chondrocyte generation and nutrient consumption are developed to analyze the behavior of cell growth in a biodegradable polymer matrix. Substrate reaction and diffusion are analyzed in two regions: one consisting of cells and nutrients and the other consisting of only nutrients. A pseudo-steady state approximation for the transport of nutrients in these two regions is utilized. The rate of growth is determined by a moving boundary equation that equates the rate at which the interfacial region between the cells and the void space moves to a substrate dependent growth reaction. The change in the location of this interfacial region with time therefore depicts the rate at which the cells propagate. The two limiting cases discussed in this article represent extremes in how the cells will grow in the polymer matrix; one case assumes that cells grow inward from the external boundary, and the other case assumes that cells grow parallel to the external boundary. The results of both models are compared to experimental data found in the literature. It is found through these comparisons that the model parameters, including the unit cell spacing parameter L, the metabolic rate constant k, the growth rate constant k(G), and external mass transfer coefficient, K, may vary as the thickness of the polymer matrix is changed, however, unrealistic and large changes in the diffusion coefficients were required to account for the full range of experimental data. Furthermore, these results suggest modification of the functional form of the growth kinetics to include substrate or product inhibition, or death terms. Based upon diffusion/reaction concepts, these models for cell growth in a biodegradable polymer give bounds for the upper and lower limits of the cellular growth rate and nutrient consumption in a polymer matrix and will aid in the development of more extensive models. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 422-432, 1997.     

On periodic algal cyclostat populations

A number of useful relations for population growth rate, uptake of limiting nutrient, amount of intracellular growth limiting nutrient and extracellular limiting nutrient concentration are presented in terms of their period averaged functions. These period averages may provide “species specific” information. The period length is shown to be unique for a given dilution rate and amount of intracellular growth limiting nutrient. A plot of residual limiting nutrient concentration versus cell density describes a closed trajectory (or “phase” loop). The area of this loop may be evaluated in terms of certain period averages of the parameters of the cyclostat population. This area is characteristic for a species at any given dilution rate and growth limiting nutrient concentration entering the cyclostat vessel. Equations for describing multispecies cyclostats are presented. Also, some necessary conditions for stable coexistence in such a system are described.     

Tumor Growth Modeling from the Perspective of Multiphase Porous Media Mechanics

Multiphase porous media mechanics is used for modeling tumor growth, using governing equations obtained via the Thermodynamically Constrained Averaging Theory (TCAT). This approach incorporates the interaction of more phases than legacy tumor growth models. The tumor is treated as a multiphase system composed of an extracellular matrix, tumor cells which may become necrotic depending on nutrient level and pressure, healthy cells and an interstitial fluid which transports nutrients. The governing equations are numerically solved within a Finite Element framework for predicting the growth rate of the tumor mass, and of its individual components, as a function of the initial tumor-to-healthy cell ratio, nutrient concentration, and mechanical strain. Preliminary results are shown.     

Reduction of waste product excretion via nutrient control: Possible strategies for maximizing product and cell yields on serum in cultures of mammalian cells

Mammalian cells grown in culture excrete lactic acid and ammonium ions in quantities that may limit growth and reduce product synthesis. Frequent replenishment of the culture medium is often necessary to prevent waste product accumulation which could inhibit cell growth. Since increased medium replenishment results in increased usage of animal serum, the most expensive raw material, excessive production of waste products lowers the cell and product yield on serum, and hence increases production costs. Strategies for reducing the production of lactic acid and ammonium bymammalian cells via controlled addition of glucose and glutamine will be demonstrated. Mathematical relations coupling ammonium and glutamine kinetics will be described. Additionally, a method for automatic on-line estimation of the cell concentration was developed. This method involves calculating the ATP production rate from the oxygen uptake rate and the lactic acid production rate. Automatic online estimation of the cell concentration is critical if nutrient levels in large-scale mammaliancell cultures are to be accurately maintained via process control.     

Fermentation kinetics including product and substrate inhibitions plus biomass death: a mathematical analysis

We propose an analytical solution of the kinetic equations describing fermentations. Equations are solved in phase space, i.e. the biomass concentration is written explicitly as a function of the substrate concentration. These results hold even when cell death and an arbitrary number of substrate/product inhibitions are accounted for. Moreover, constant yield needs not be assumed.     

Linearized kinetic model of<Emphasis Type="Italic"> Listeria monocytogenes</Emphasis> biofilm growth

In this study the dynamics of biofilm formation on aluminum has been investigated. The process of cell growth has been observed using fluorescence microscopy. It has been confirmed that the process of biofilm formation can be represented as a sum of two separate processes: cell adhesion and colony proliferation. The derived set of equations describes kinetics of surface population growth and characteristic times for adsorption and combined growth processes, including characteristic time for the nutrient supply depletion. All equations contain variables based on the fundamental characteristics of bacterial population and can be easily determined from the experimental data or estimated theoretically. The developed theoretical model allows obtaining realistic values for population growth time and characteristic time for nutrient limitation occurrence during the biofilm development. Resulting equations qualitatively describe the biofilm formation process, and allow predicting microbial kinetics in the batch reactor system and determining critical values of the process parameters.     

Enhancement of cell growth in tissue-engineering constructs under direct perfusion: Modeling and simulation

Perfusion bioreactors improve mass transfer in cell-scaffold constructs. We developed a mathematical model to simulate nutrient flow through cellular constructs. Interactions among cell proliferation, nutrient consumption, and culture medium circulation were investigated. The model incorporated modified Contois cell-growth kinetics that includes effects of nutrient saturation and limited cell growth. Nutrient uptake was depicted through the Michaelis-Menton kinetics. To describe the culture medium convection, the fluid flow outside the cell-scaffold construct was described by the Navier-Stokes equations, while the fluid dynamics within the construct was modeled by Brinkman's equation for porous media flow. Effects of the media perfusion were examined by including time-dependant porosity and permeability changes due to cell growth. The overall cell volume was considered to consist of cells and extracellular matrices (ECM) as a whole without treating ECM separately. Numerical simulations show when cells were cultured subjected to direct perfusion, they penetrated to a greater extent into the scaffold and resulted in a more uniform spatial distribution. The cell amount was increased by perfusion and ultimately approached an asymptotic value as the perfusion rates increased in terms of the dimensionless Peclet number that accounts for the ratio of nutrient perfusion to diffusion. In addition to enhancing the nutrient delivery, perfusion simultaneously imposes flow-mediated shear stress to the engineered cells. Shear stresses were found to increase with cell growth as the scaffold void space was occupied by the cell and ECM volumes. The macro average stresses increased from 0.2 mPa to 1 mPa at a perfusion rate of 20 microm/s with the overall cell volume fraction growing from 0.4 to 0.7, which made the overall permeability value decrease from 1.35 x 10(-2)cm(2) to 5.51 x 10(-4)cm(2). Relating the simulation results with perfusion experiments in literature, the average shear stresses were below the critical value that would induce the chondrocyte necrosis.     

Dynamic volume-averaged model of heat and mass transport within a compost biofilter: I. Model development

Successful, long-term operation of a biofilter system depends on maintaining a suitable biofilm environment within a porous medium reactor. In this article a mathematical study was conducted of the spatial and temporal changes of biofilter performance due to interphase heat and mass transport. The method of volume averaging was used to spatially smooth the three-phase (solid, liquid, and gas) conservation equations over the biofilter domain. The packing medium was assumed to be inert, removing the solid phase mass continuity equation from the system. The finite element method was used to integrate the resulting nonlinear-coupled partial differential equations, tracking eight state variables: temperature, water vapor, dry air, liquid water, biofilm, gas and liquid phase organic pollutant, and nutrient densities, through time and space. A multiphase, gas and liquid flow model was adapted to the biofilter model from previous studies of unsaturated groundwater flow. Newton's method accelerated by an LU direct solver was used to iterate the model for solutions. Effects of packing media on performance were investigated to illustrate the utility of the model. The moisture dynamics and nutrient cycling are presented in Part II of this article.     

The kinetic effect of product instability in a Michaelis-Menten mechanism with competitive inhibition

In most kinetic studies it is assumed that both the reactant and the products are stable. However, under certain conditions spontaneous decomposition or deterioration caused by one of the participating species occurs. Studies, in which a species (the free enzyme, the enzyme-substrate complex, an inhibitor or the product of the reaction) is unstable, have appeared in the literature. However, to our knowledge, the enzymatic systems, in which a competitive inhibition and a decomposition or transformation of the products take place simultaneously, have not been studied so far. In this paper, we present a kinetic analysis of an enzyme reaction that follows a Michaelis-Menten mechanism, in which the free enzyme suffers a competitive inhibition simultaneously with the decomposition of the immediate product. In this study, we have linearised the differential equations that describe the kinetics of the process. Under the assumption of limiting concentration of enzyme, we have obtained and tested the explicit equation describing the time dependence of the product concentration using numerical calculus. With this equation and the experimental progress curve of the product, we constructed an easy procedure for the evaluation of the principal kinetic parameters of the process.     

Transient growth- and product-formation kinetics of acetic acid bacteria

Rates of growth and product formation under non-stationary conditions were measured in fermentations of industrial acetic acid bacteria. A repeated-batch process, where conditions change rapidly, and a slower shift experiment in CSTR culture were examined. Significant deviations from the steady-state kinetics determined in continuous fermentations were found for cell growth as well as for the formation of acetic acid. Algebraic functions of the inhibiting acid concentration were identified to describe the rates of reaction under stationary conditions. Transient kinetics are modeled by phenomenological differential equations. The data from both the repeated-batch experiments and the CSTR shift is consistently reproduced. Measurements and simulation results are presented in phase diagrams of the reaction rates over the concentration of acetic acid. Due to the dynamic effects, which enhance the transient rates of both growth and product formation, the repeated-batch process is superior to a continuous fermentation in terms of total volumetric productivity and final acid concentration.     

Effect of initial stresses on the wave propagation in arteries

A theoretical analysis for the problem of wave propagation in arteries is presented. Blood is treated as a Newtonian, viscous incompressible fluid. On the basis of information derived from experimental investigations on the mechanical properties of wall tissues, the arterial wall is considered to be nonlinearly viscoelastic and orthotropic. The analysis is carried out for a cylindrical artery, under the purview of membrane theory, by taking account the effect of initial stresses. The motion of the wall and that of the fluid are assumed to be axisymmetric. Particular emphasis has been paid to the propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. By employing the equations of motion of the fluid and those for the wall, together with the equations of continuity, a frequency equation is derived by exploiting the conditions of continuity of the velocity of the arterial wall and that of blood on the endosteal surface of the wall. In order to illustrate the validity of the derived analytical expressions a quantitative analysis is made for the variations of the phase velocities as well as the transmission coefficient with frequency corresponding to different transmural pressures which relate to different initial stresses. Computational results indicate that phase velocities increase with the increase of transmural pressures.     

The kinetics of nucleation and growth of sickle cell hemoglobin fibers

Polymerization of sickle cell hemoglobin (HbS) in deoxy state is one of the basic events in the pathophysiology of sickle cell anemia. For insight into the polymerization process, we monitor the kinetics of nucleation and growth of the HbS polymer fibers. We define a technique for the determination of the rates J and delay times theta of nucleation and the fiber growth rates R of deoxy-HbS fibers, based on photolysis of CO-HbS by laser illumination. We solve numerically time-dependent equations of heat conductance and CO transport, coupled with respective photo-chemical processes, during kinetics experiments under continuous illumination. After calibration with experimentally determined values, we define a regime of illumination ensuring uniform temperature and deoxy-HbS concentration, and fast (within <1 s) egress to steady conditions. With these procedures, data on the nucleation and growth kinetics have relative errors of <5% and are reproducible within 10% in independent experiments. The nucleation rates and delay times have steep, exponential dependencies on temperature. In contrast, the average fiber growth rates only weakly depend on temperature. The individual growth rates vary by up to 40% under identical conditions. These variations are attributed to instability of the coupled kinetics and diffusion towards the growing end of a fiber. The activation energy for incorporation of HbS molecules into a polymer is E(A)=50 kJ mol(-1), a low value indicating the significance of the hydrophobic contacts in the HbS polymer. More importantly, the contrast between the strong theta(T) and weak R(T) dependencies suggests that the homogenous nucleation of HbS polymers occurs within clusters of a precursor phase. This conclusion may have significant consequences for the understanding of the pathophysiology of sickle cell anemia and should be tested in further work.     

The impact of CO2 emission scenarios and nutrient enrichment on a common coral reef macroalga is modified by temporal effects

Future coral reefs are expected to be subject to higher pCO₂ and temperature due to anthropogenic greenhouse gas emissions. Such global stressors are often paired with local stressors thereby potentially modifying the response of organisms. Benthic macroalgae are strong competitors to corals and are assumed to do well under future conditions. The present study aimed to assess the impact of past and future CO₂ emission scenarios as well as nutrient enrichment on the growth, productivity, pigment, and tissue nutrient content of the common tropical brown alga Chnoospora implexa. Two experiments were conducted to assess the differential impacts of the manipulated conditions in winter and spring. Chnoospora implexa's growth rate averaged over winter and spring declined with increasing pCO₂ and temperature. Furthermore, nutrient enrichment did not affect growth. Highest growth was observed under spring pre‐industrial (PI) conditions, while slightly reduced growth was observed under winter A1FI (“business‐as‐usual”) scenarios. Productivity was not a good proxy for growth, as net O₂ flux increased under A1FI conditions. Nutrient enrichment, whilst not affecting growth, led to luxury nutrient uptake that was greater in winter than in spring. The findings suggest that in contrast with previous work, C. implexa is not likely to show enhanced growth under future conditions in isolation or in conjunction with nutrient enrichment. Instead, the results suggest that greatest growth rates for this species appear to be a feature of the PI past, with A1FI winter conditions leading to potential decreases in the abundance of this species from present day levels.     

Impacts on product quality attributes of monoclonal antibodies produced in CHO cell bioreactor cultures during intentional mycoplasma contamination events

A mycoplasma contamination event in a biomanufacturing facility can result in costly cleanups and potential drug shortages. Mycoplasma may survive in mammalian cell cultures with only subtle changes to the culture and penetrate the standard 0.2-µm filters used in the clarification of harvested cell culture fluid. Previously, we reported a study regarding the ability of Mycoplasma arginini to persist in a single-use, perfusion rocking bioreactor system containing a Chinese hamster ovary (CHO) DG44 cell line expressing a model monoclonal immunoglobulin G 1 (IgG1) antibody. Our previous work showed that M. arginini affects CHO cell growth profile, viability, nutrient consumption, oxygen use, and waste production at varying timepoints after M. arginini introduction to the culture. Careful evaluation of certain identified process parameters over time may be used to indicate mycoplasma contamination in CHO cell cultures in a bioreactor before detection from a traditional method. In this report, we studied the changes in the IgG1 product quality produced by CHO cells considered to be induced by the M. arginini contamination events. We observed changes in critical quality attributes correlated with the duration of contamination, including increased acidic charge variants and high mannose species, which were further modeled using principal component analysis to explore the relationships among M. arginini contamination, CHO cell growth and metabolites, and IgG1 product quality attributes. Finally, partial least square models using NIR spectral data were used to establish predictions of high levels (≥10⁴ colony-forming unit [CFU/ml]) of M. arginini contamination, but prediction of levels below 10⁴ CFU/ml were not reliable. Contamination of CHO cells with M. arginini resulted in significant reduction of antibody product quality, highlighting the importance of rapid microbiological testing and mycoplasma testing during particularly long upstream bioprocesses to ensure product safety and quality.     

A kinetic model describing cell growth and production of highly active, recombinant ice nucleation protein in Escherichia coli

A structured kinetic model, which describes the production of the recombinant ice nucleation protein in different conditions, was applied. The model parameters were estimated based on the variation of the specific growth rate and the intracellular product concentration during cultivation. The equations employed relate the cellular plasmid content or plasmid copy number with the cloned-gene expression; these correlations were successfully tested on the experimental data. The optimal nutrient conditions for the growth of Escherichia coli expressing the inaZ gene of Pseudomonas syringae were determined for the production of active ice nucleation protein. The kinetics of the cultures expressing the inaZ gene were studied in a bioreactor at different growth temperatures and nutrient conditions.     

Triphasic mixture model of cell-mediated enzymatic degradation of hydrogels

One critical component of engineering living tissue equivalents is the design scaffolds (often made of hydrogels) whose degradation kinetics can match that of matrix production by cells. However, cell-mediated enzymatic degradation of a hydrogel is a highly complex and nonlinear process that is challenging to comprehend based solely on experimental observations. To address this issue, this study presents a triphasic mixture model of the enzyme-hydrogel system, which consists of a solid polymer network, water and enzyme. On the basis mixture theory, the rubber elasticity theory and the Michaelis-Menton kinetics for degradation, the model naturally incorporates a strong coupling between gel mechanical properties, the kinetics of degradation and the transport of enzyme through the gel. The model is then used to investigate the particular problem of a single spherical enzyme-producing cell, embedded in a spherical hydrogel domain, for which the governing equations can be cast within the cento-symmetric assumptions. The governing equations are subsequently solved using an implicit nonlinear finite element procedure to obtain the evolution of enzyme concentration and gel degradation through time and space. The model shows that two regimes of degradation behaviour exist, whereby degradation is dominated either by diffusion or dominated by reaction kinetics. Depending on the enzyme properties and the initial hydrogel design, the temporal and spatial changes in gel cross-linking are dramatically impacted, a feature that is likely to strongly affect new tissue development.     

Triphasic mixture model of cell-mediated enzymatic degradation of hydrogels

One critical component of engineering living tissue equivalents is the design scaffolds (often made of hydrogels) whose degradation kinetics can match that of matrix production by cells. However, cell-mediated enzymatic degradation of a hydrogel is a highly complex and nonlinear process that is challenging to comprehend based solely on experimental observations. To address this issue, this study presents a triphasic mixture model of the enzyme–hydrogel system, which consists of a solid polymer network, water and enzyme. On the basis mixture theory, the rubber elasticity theory and the Michaelis–Menton kinetics for degradation, the model naturally incorporates a strong coupling between gel mechanical properties, the kinetics of degradation and the transport of enzyme through the gel. The model is then used to investigate the particular problem of a single spherical enzyme-producing cell, embedded in a spherical hydrogel domain, for which the governing equations can be cast within the cento-symmetric assumptions. The governing equations are subsequently solved using an implicit nonlinear finite element procedure to obtain the evolution of enzyme concentration and gel degradation through time and space. The model shows that two regimes of degradation behaviour exist, whereby degradation is dominated either by diffusion or dominated by reaction kinetics. Depending on the enzyme properties and the initial hydrogel design, the temporal and spatial changes in gel cross-linking are dramatically impacted, a feature that is likely to strongly affect new tissue development.     

Bistability Analysis of an Apoptosis Model in the Presence of Nitric Oxide

Bistability in apoptosis, or programmed cell death, is crucial for the healthy functioning of multicellular organisms. The aim in this study is to show the presence of bistability in a mitochondria-dependent apoptosis model under nitric oxide effects using chemical reaction network theory. The model equations are a set of coupled ordinary differential equations arising from the assumed mass action kinetics. Whether these equations have a capacity for bistability (cell survival and apoptosis) is determined using a modular approach in which the model is decomposed into modules. Each module contains only a subset of the whole model and is analyzed separately. It is seen that bistability in a module is preserved throughout the whole model after adding the remaining reactions in the pathway on these modules. It is also found that inhibitor effect of some proteins and the appearance of a reacting protein in a later stage as a product is a desired feature but not sufficient for bistability (in the absence of cooperativity effects). On the whole model, two apoptotic and two cell survival states are obtained depending on the initial cell conditions. The results suggest that the antiapoptotic effects of nitric oxide species are responsible for the bistable character of the apoptotic pathway when cooperativity is not assumed in the apoptosome formation.