Stress-modulated growth, residual stress, and vascular heterogeneity.

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.     

Direction change of plant root growth by the impenetrable boundary

Spatial boundary conditions must be considered when utilizing mathematical modeling of plant root growth in the container or along with the imbedding solid obstacle. Using basic root growth principles and the geometry of the boundary surface, a mathematical model can be designed to keep all root elements inside the container or outside the obstacle without passing through the boundary after the minimum deflection of growth direction, and it is based on the minimum friction between root tips and soil and energy saving principles. Such a mathematical method is used to simulate the spatial distribution of root growth and the morphological architecture of the root system near the boundary. The validity of this model is supported by experimental observations that confirm some typical characteristics predicted by the simulations. This model can be widely used in resolving boundary condition complications where water and nutrients are consumed by plants in a spatially limited or heterogeneous resource field.     

A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis.

Using the biphasic theory for hydrated soft tissues (Mow et al., 1980) and a transversely isotropic elastic model for the solid matrix, an analytical solution is presented for the unconfined compression of cylindrical disks of growth plate tissues compressed between two rigid platens with a frictionless interface. The axisymmetric case where the plane of transverse isotropy is perpendicular to the cylindrical axis is studied, and the stress-relaxation response to imposed step and ramp displacements is solved. This solution is then used to analyze experimental data from unconfined compression stress-relaxation tests performed on specimens from bovine distal ulnar growth plate and chondroepiphysis to determine the biphasic material parameters. The transversely isotropic biphasic model provides an excellent agreement between theory and experimental results, better than was previously achieved with an isotropic model, and can explain the observed experimental behavior in unconfined compression of these tissues.     

Model of growth and ergot alkaloid production by Claviceps purpurea

A growth model for Claviceps purpurea in submerged batch culture is presented. In developing the model, the basic principles of the growth and the morphological properties of C. purpurea are considered. The growth of C. purpurea is assumed to occur in a three-step manner; the first step involves the assimilation and the growth of cells; the second one involves cell division, and the third one involves transformation of the mature cells to a state where they have no ability to divide but do have the ability to produce ergot alkaloids and then they gradually die. Inorganic phosphate is assumed to be the limiting substrate for the first and the second steps in conditions of carbon source being in excess. The model constants are determined by model simulation and graphical searching techniques to find the minimum value of the absolute difference between the experimental and the simulated curves for biomass, alkaloids, and sucrose.     

Storage and growth of denitrifiers in aerobic granules: part II. model calibration and verification

A mathematical model to describe the simultaneous storage and growth activities of denitrifiers in aerobic granules under anoxic conditions has been developed in an accompanying article. The sensitivity of the nitrate uptake rate (NUR) toward the stoichiometric and kinetic coefficients is analyzed in this article. The model parameter values are estimated by minimizing the sum of squares of the deviations between the measured and model-predicted values. The model is successfully calibrated and a set of stoichiometric and kinetic parameters for the anoxic storage and growth of the denitrifiers are obtained. Thereafter, the model established is verified with three set of experimental data. The comparison between the model established with the ASM1 model and ASM3 shows that the present model is appropriate to simulate and predict the performance of a granule-based denitrification system.     

A model for the competitive growth of two diatoms

A parallel between an experimental situation in marine microecology and its analytical model is presented. Two diatoms are cultured in a continuous flow apparatus and are studied for their competitive performance. A differential equations model is proposed to represent their growth behavior. The parameters of the model are evaluated from the experimental data and it is then identified from additional data.     

Growth models of cultures with two liquid phases. IV. Cell adsorption, drop size distribution, and batch growth

In this work, a mathematical model which can be used to describe butch growth in fermentations with two liquid phases present is developed for systems in which the growth limiting substrate is dissolved in the dispersed phase. The model takes into account the drop size distribution, the rate of adsorption of cells on the drop surface, the rate of desorption of cells from the drop surface, substrate transport between phases, phase equilibrium, and growth kinetics. The model also considers the effect, of coalescence and redispersion of oil drops in the system. It is assumed that the composition of the dispersed phase is such that substrate utilization from it causes little or no change in the interfacial area. A discrete uniform distribution and a discrete normal distribution which is obtained from an experimental distribution curve are used as drop size distributions. Simulation results are obtained for a wide range of parameter values using the IBM S/360 Continuous System Modeling Program.     

The estimation of growth yield and maintenance parameters for photoautotrophic growth

Methods are presented for examining the consistency of experimental data for microbial growth where light energy is converted to chemical energy through photosynthesis. True growth yield and maintenance parameters are estimated for several sets of available experimental data. Methods of parameter estimation are presented which allow all of the measured variables to be used simultaneously for parameter estimation. The results show that a wide range of values have been found for the true growth yield and maintenance parameters. Values of the true growth yield range from 0.04 to values above those predicted by the Z-scheme model for photosynthesis.     

A simple, leaky cell growth model for plant cell aggregates

A simple growth model is proposed for plant cell aggregates which accounts for leakage of a single intermediate metabolite from the aggregates to the medium. This model predicts a lag phase in the growth curve whose extent is determined by the intermediate metabolite leakage coefficient and its equilibrium distribution coefficient between the medium and the cell aggregates, the size of the inoculum relative to the system total water content, and the initial intermediate metabolite content in the medium. The model thus provides for an interaction between growing plant cells and their environment in a way that has heretofore been unquantified. Preliminary validation of the model has been made against literature data of Dioscorea deltoidea grown in batch suspension cell culture on sucrose, yielding a correlation coefficient of 0.997. The predicted glucose + fructose concentration in the medium agrees reasonably well with experimental measurements after ca, 3.5 days of culture, although a discrepancy exists between model prediction and experiment immediately after startup. Further validation of the model is suggested on this and other plant species.     

Prediction of effects of amino acid supplementation on growth of E. coli B/r

A mathematical model for the growth of a single cell of E. coli on medium containing amino acid is presented. A mixture of purified amino acids (glutamate, aspartate, serine, tyrosine, and leucine) combined in the ratios found in a natural digest (casein) were employed as the nitrogen source. Each of these amino acids is the representative of a different family of amino acids. The transport mechanisms and assimilation routes for each amino acid were inserted into the prototype model. The enzyme activities and saturation constants used in the model were based on literature data. The maximum velocities for uptake systems were calculated from experimental data. The formation and homeostasis of amino acid pools were regulated through cross-control of the activities of biosynthetic enzymes and of membrane transport of exogenous nutrients. The size of each amino acid pool was determined with mass balance equations that included terms for a transport system, a biosynthesis system, a transaminase enzyme system for interchange between the amino acid families, and a consumption system. The predictions of the extended model with regard to nutrient concentrations and growth rates compared well with the experimental data.     

Airlift column photobioreactors for Porphyridium sp. culturing: Part II. verification of dynamic growth rate model for reactor performance evaluation

Dynamic growth rate model has been developed to quantify the impact of hydrodynamics on the growth of photosynthetic microorganisms and to predict the photobioreactor performance. Rigorous verification of such reactor models, however, is rare in the literature. In this part of work, verification of a dynamic growth rate model developed in Luo and Al-Dahhan (2004) [Biotech Bioeng 85(4): 382-393] was attempted using the experimental results reported in Part I of this work and results from literature. The irradiance distribution inside the studied reactor was also measured at different optical densities and successfully correlated by the Lambert-Beer Law. When reliable hydrodynamic data were used, the dynamic growth rate model successfully predicted the algae's growth rate obtained in the experiments in both low and high irradiance regime indicating the robustness of this model. The simulation results also indicate the hydrodynamics is significantly different between the real algae culturing system and an air-water system that signifies the importance in using reliable data input for the growth rate model.     

Analysis of cell growth and diffusion in a scaffold for cartilage tissue engineering

Developments in tissue engineering over the past decade have offered promising future for the repair and reconstruction of damaged tissues. To regenerate three dimensional and weight-bearing implants, advances in biomaterials and manufacturing technologies prompted cell cultivations with natural or artificial scaffolds, in which cells are allowed to proliferate, migrate, and differentiate in vitro. In this article, we develop a mathematical model for cell growth in a porous scaffold. By treating the cell-scaffold construct as a porous medium, a continuum model is set up based on basic principles of mass conservation. In addition to cell growth kinetics, we incorporate cell diffusion in the model to describe the effects of cell random walks. Computational results are compared to experimental data found in the literature. With this model, we are able to investigate cell motility, heterogeneous cell distributions, and non-uniform seeding for tissue engineering applications. Results show that random walks tend to enhance uniform cell spreads in space, which in turn increases the probabilities for cells to acquire nutrients; therefore random walks are likely to be a positive contribution to the overall cell growth on scaffolds.     

An investigation of non-steady-state algal growth. II. Mathematical modelling of co-nutrient-limited algal growth

The ability of mathematical models to simulate competition for nutrients between three algal species, the diatom Thalassiosira pseudonana, a marine raphidophyte Heterosigma carterae and the dinoflagellate Alexandrium minutum, was investigated. Transient growth models were parameterized and tested using a number of closely controlled laboratory data sets including batch monocultures, batch competition experiments and semi-continuous culture competition experiments. The cell quota model of algal growth was found to be adequate to simulate growth of both the raphidophyte and the dinoflagellate. Batch monoculture data for diatom growth obtained under either nitrogen (N) or silicon (Si) limitation could also be simulated with a quota-style model, which in this case included feedback inhibition of nutrient uptake. However, to simulate both batch and semi-continuous culture experiments (and competition between the species), it was necessary to consider diatom Si-N metabolism. A model was derived which contains a representation of both intracellular N ad Si, and of the interaction of these nutrients within the cell. The model used a co-nutrient limitation based on the perceived functional and structural role of N and Si, respectively, within the cell. Simulations indicated that models capable of adequately representing monoculture growth in batch culture may produce erroneous results when incorporated into models of competition. The co-nutrient model is a first step to producing tractable algal growth models which will represent multiple nutrient stress in transient growth conditions.      

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.     

Transport of bacteria in porous media: II. A model for convective Transport and growth

A model is presented for the coupled processes of bacterial growth and convective transport of bacteria has been modeled using a fractional flow approach. The various mechanisms of bacteria retention can be incorporated into the model through selection of an appropriate shape of the fractional flow curve. Permeability reduction due to pore plugging by bacteria was simulated using the effective medium theory. In porous media, the rates of transport and growth of bacteria, the generation of metabolic products, and the consumption of nutrients are strongly coupled processes. Consequently, the set of governing conservation equations form a set of coupled, nonlinear partial differential equations that were solved numerically. Reasonably good agreement between the model and experimental data has been obtained indicating that the physical processes incorporated in the model are adequate. The model has been used to predict the in situ transport and growth of bacteria, nutrient consumption, and metabolite production. It can be particularly useful in simulating laboratory experiments and in scaling microbial-enhanced oil recovery or bioremediation processes to the field. (c) 1994 John Wiley & Sons, Inc.     

Water deficits uncouple growth from photosynthesis, increase C content, and modify the relationships between C and growth in sink organs

In plants, carbon (C) molecules provide building blocks for biomass production, fuel for energy, and exert signalling roles to shape development and metabolism. Accordingly, plant growth is well correlated with light interception and energy conversion through photosynthesis. Because water deficits close stomata and thus reduce C entry, it has been hypothesised that droughted plants are under C starvation and their growth under C limitation. In this review, these points are questioned by combining literature review with experimental and modelling illustrations in various plant organs and species. First, converging evidence is gathered from the literature that water deficit generally increases C concentration in plant organs. The hypothesis is raised that this could be due to organ expansion (as a major C sink) being affected earlier and more intensively than photosynthesis (C source) and metabolism. How such an increase is likely to interact with C signalling is not known. Hence, the literature is reviewed for possible links between C and stress signalling that could take part in this interaction. Finally, the possible impact of water deficit-induced C accumulation on growth is questioned for various sink organs of several species by combining published as well as new experimental data or data generated using a modelling approach. To this aim, robust correlations between C availability and sink organ growth are reported in the absence of water deficit. Under water deficit, relationships weaken or are modified suggesting release of the influence of C availability on sink organ growth. These results are interpreted as the signature of a transition from source to sink growth limitation under water deficit.     

Model of copepod growth influenced by the food carbon:nitrogen ratio and concentration, under the hypothesis of strict homeostasis

This study describes a model which addresses the processes of ingestion, assimilation, respiration, excretion and growth of copepods as a function of the concentration of food and its elemental composition in terms of carbon and nitrogen (N). Two experimental data sets are used to estimate several parameters of the model concerned with the influence of food quality. The results of the model suggest that the concentration of food and its quality (i.e. the C:N ratio) largely determine copepod growth. Both the experimental data sets and the model output show that low carbon relative to the nitrogen content of food does not limit the production of copepods. Comparing the results of the model to those of a previous model on bacteria suggests large differences between bacterial and copepod physiological responses to a variable quality of the substrate or food. The results of these models suggest that the regeneration of ammonium performed by copepods always favors regenerated primary production, whereas that performed by bacteria, depending on the quality of assimilated substrates, can favor or limit regenerated production.