Cell Balance Equation Research Articles

Competing tumor cell migration mechanisms caused by interstitial fluid flow

In the seminal work by Swartz and collaborators (Shields et al., 2007) it was discovered that autologously secreted or activated (ECM-bound) chemokine forms local pericellular diffusion gradients skewed by fluid convection, and the cells subsequently chemotact up the flow-directed gradient. However, in (Polacheck et al., 2011) Kamm and collaborators found that there is a competing downstream and upstream migration transport mechanism. Their study showed that both mechanisms are present at the same time and the relative strength of these two stimuli governs the directional bias in migration for a cell population and is a function of cell density, interstitial flow rate, and CCR7 receptor availability. The main objective of this work is to give a possible explanation of these two different concurrent cell migration mechanisms by means of a theoretical model. Relying on multiphase modelling, separate momentum balance equations are formulated, respectively, for the cell phase and the interstitial fluid (IF) phase. In order to represent proteolytic activity and autologous chemotaxis a non-moving ECM component is included, as well as proteases secreted by the cancer cells and chemokine that can be released from ECM. The cell and IF momentum balance equations include cell-ECM and fluid-ECM resistance force terms (i.e., classical Darcy’s equation terms), but also a cell-fluid interaction term that can account for a more indirect effect that fluid-generated stress may have on cancer cells. We illustrate how the cancer cells can work through this term and effectively avoid being pushed in the flow direction, and even create upstream migration by controlling its magnitude and sign. We think of this as the mathematical interpretation of the experimental observation by Kamm and collaborators that the fluid generated matrix adhesion tension on the upstream side of cells activates integrin adhesion complexes, resulting in activation of focal adhesion (FA) proteins. The model predicts that generally the strength of the upstream migration mechanism is sensitive to the cell volume fraction: a lower density of cells is subject to a weaker upstream migration effect; a higher density of cancer cells can more effectively generate upstream migration. This behavior is a result of the nonlinear coupling between cell-ECM, fluid-ECM, and cell-fluid interaction terms that naturally are involved in the mathematical expression for the net cell velocity.

Simulation of the effects of oxygen carriers and scaffold geometry on oxygen distribution and cell growth in a channeled scaffold for engineering myocardium

This study proposes a mathematical model to evaluate the impact of oxygen carriers and scaffold geometry on oxygen distribution and cell growth in a 3D cardiac construct using computational fluid dynamics (CFD). Flow equations, oxygen balance equation and cell balance equation were solved using special initial and boundary conditions. The modeling results revealed that 55% increase in cardiac cell density occurred by using 6.4% perfluorocarbon oxygen carrier (PFC) compared to pure culture medium without PFC supplementation. Moreover, the effects of the scaffold geometry on cell density were examined by changing the channel numbers and the construct length. A 30% increase in the average cells density was observed by increasing the channel numbers from 37 to 145. Furthermore, the average cell density was increased 23% by decreasing the scaffold length from 0.5 to 0.2 cm length. Overall, the average cell density of cardiac cells can be increased 2-fold by using PFC oxygen carrier and optimizing the scaffold's geometry, simultaneously.

An approach for simulating the THMC process in single novaculite fracture using EPCA

An approach for simulating thermal, hydraulic, mechanical and chemical (THMC) coupled processes in single rock fractures has been developed under the framework of a self-developed numerical method, i.e., an elasto-plastic cellular automaton. The balance equations of multi-physics problems to describe the THMC process in single rock fracture are solved by using cellular automaton technique on space scale and finite difference method on time scale, respectively. Using the concept of cellular automaton, a single rock fracture surface is discretized into a system composed of cell elements. Different apertures, i.e., 0 for contact and nonzero for void, are assigned to each cell element based on the fracture surface topography. The fluid flow, stress-dependent chemical reaction and solute transport are simulated by using a cellular automaton updating rule, in which only local cell balance equation is considered. The contribution of cell elements in contact to cell’s transmissivity and convection can be ignored conveniently. The Lagrangian method is used to simulate the particle transport. Special treatment for particle transport to outer boundaries and internal boundaries is adopted. As a result, the local behaviors, such as the formation of local contact, dead ends in the fracture and the local aperture change, are conveniently updated dynamically. The approach is used to simulate the coupled THMC process in a single novaculite fracture. The behaviors of pressure dissolution caused by effective stress, free-face dissolution/precipitation, thermal-dependent fluid flow and ion transport are well reproduced by using the developed approach, subject to parameter calibration. The robustness of the general approach to such complex problems is demonstrated by comparing with experimental data.

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Cell balance equation for chemotactic bacteria with a biphasic tumbling frequency.

Alt's three-dimensional cell balance equation characterizing the chemotactic bacteria was analyzed under the presence of one-dimensional spatial chemoattractant gradients. Our work differs from that of others who have developed rather general models for chemotaxis in the use of a non-smooth anisotropic tumbling frequency function that responds biphasically to the combined temporal and spatial chemoattractant gradients. General three-dimensional expressions for the bacterial transport parameters were derived for chemotactic bacteria, followed by a perturbation analysis under the planar geometry. The bacterial random motility and chemotaxis were summarized by a motility tensor and a chemotactic velocity vector, respectively. The consequence of invoking the diffusion-approximation assumption and using intrinsic one-dimensional models with modified cellular swimming speeds was investigated by numerical simulations. Characterizing the bacterial random orientation after tumbles by a turn angle probability distribution function, we found that only the first-order angular moment of this turn angle probability distribution is important in influencing the bacterial long-term transport.

Spatial effect of tumbling frequencies for motile bacteria on cell balance equations

We performed dimensional reduction of Alt’s three-dimensional cell balance equations to one-dimensional Segel’s equations for an axisymmetrical case. The tumbling frequency of motile bacteria was assumed to switch between two different phases according to the sign of the perceived chemical gradients. Chemotactic bacteria responded to positive attractant gradients by suppressing the tumbling frequency, but appeared to be insensitive to negative attractant gradients. When both temporal and spatial attractant gradients were considered, this tumbling scenario constituted a limited swimming angle range. It was only within this angle range that the bacterial tumbling frequency was regulated according to the perceived attractant gradients. This angle range, characterized by the angle θ 0, plays a major role in determining the effective chemotactic responses. The bacterial density distribution in the velocity space was derived via a first-order perturbation analysis, and then applied to studies of two bacterial population transport parameters (the random motility coefficient and the chemotactic velocity). Two approaches different in dimensionality were presented. While the one-dimensional approach asymptotically yielded the expected, but incorrect, one-dimensional random motility coefficient as θ 0 approached zero, the three-dimensional approach always yielded the correct result. As to the chemotactic velocity, both approaches yielded the same expression. To validate our analyses and to explore possible limitations due to certain simplifying assumptions in the derivations, we further examined three numerical examples in which Galerkin finite element solutions of the Alt’s three-dimensional equation were compared with analytical solutions. The agreement between the perturbation solutions and numerical solutions for all data over the entire range of θ 0 suggested that our perturbation analysis is valid under the condition of small constant attractant gradients. For chemical gradients of arbitrary magnitudes with temporal and spatial variations, our solutions were also found efficient and robust in comparison to numerical solutions.

Mathematical Models for Motile Bacterial Transport in Cylindrical Tubes

Mathematical models considering motile bacterial transport within a geometrically restrictive cylindrical tube were developed. Two macroscopic transport parameters, the random motility coefficient as a self-diffusion coefficient of the cell population and the chemotactic velocity as a chemical-induced velocity, were derived. The three-dimensional cell balance equation was reduced to forms similar to Segel's one-dimensional phenomenological cell balance equations with additional modifications for bacteria-wall interactions. Two conceptually different approaches accounting for such interactions were presented. The first approach parallels treatments in the gas kinetic theory by viewing bacterial interactions with walls as collisions and subsequent diffusive/specular reflections, which led to the Bosanquet formula for the bacterial diffusion coefficient. Based on the experimental observation that bacterial swimming motion is guided by a straight tube, the second approach considered modifications in the bacterial swimming orientation as a consequence of various long-range interactions with the tube surface. A phenomenological turning model capable of aligning bacterial motion along a tube axis was proposed. The model predicts that under the geometrical restriction of a small cylindrical tube, the macroscopic bacterial transport resulting from the proposed turning model can exhibit behavior that ranges from dimensionally reduced diffusion to pure wave propagation, depending on the influence of the tube diameter on the reversal probability in the bacterial swimming motion. Our theoretical model provides explicit equations that explain how such a transition can occur. The predicted results were then qualitatively compared with experimental data from the literature. As a preliminary comparison, we concluded that bacterial transport in cylindrical tubes of diameter 10 μm remains in the mode of a dimensionally reduced diffusion, and shifts to a wave motion when the tube diameter decreases to 6 μm.

The Global Turning Probability Density Function for Motile Bacteria and its Applications

The angular turning probability density distribution for motile bacteria is usually measured in local coordinates and is therefore inconvenient for global analyses of the chemotactic bacterial migration. In this paper we present analytical derivations that convert the local angular turning probability density distribution into a global one. The explicit expression of a reduced global turning probability density function for motile bacteria was derived and its relevant properties were investigated. Depending on the angle variable being intergrated and the integration range, three types of cosine moments were separately defined and studied. Some statistical indices and parameters such as the directional persistence, persistence number, and one-dimensional reversal probability were found to be embedded in the various moments of the reduced global turning probability density function. Applications of the reduced global turning probability and its integrated moments to a three-dimensional cell balance equation in an axisymmetric system were also discussed.

Perturbation Expansion of Alt's Cell Balance Equations Reduces to Segel's One-Dimensional Equations for Shallow Chemoattractant Gradients

The cell balance equations of Alt are rigorously studied and perturbatively expanded into forms similar to Segel's one-dimensional phenomenological cell balance equations by considering the simplifying case of bacterial density possessing symmetry in the x and y directions responding to an attractant gradient present only in the z direction. We prove that for shallow attractant gradients the lumped integrals involving the tumbling probability frequency distribution and bacterial density distribution in the $\theta$ direction can be explicitly expressed as a product of three quantities: the mean tumbling frequency, the bacterial subpopulation density, and a reversal probability. We also derive expressions for the bacterial net flux in the Fickian form from which two macroscopic transport parameters, the random motility coefficient and the chemotactic velocity, are explicitly related to individual cell properties and chemical gradients.

An approach to a biomathematical model of lymphocytopoiesis.

Fundamental principles for the development of a biomathematical model of lymphocytopoiesis are presented in this paper. The first step in this modeling approach is the definition of appropriate anatomical compartments in order to identify dominant locations of lymphocytes in the human body, and the definition of functional compartments in order to model different maturation stages. In the second step these compartment structures are combined, and thus form the basis of a dynamical model consisting of linear differential equations. Cell balance equations are used to derive the biomathematical dynamical model which is presented using the tools of modern systems theory. As a result of intensive discussions between engineers and medical doctors, our model of lymphocytopoiesis consists of six anatomical and six functional compartments. Virtual marking technique plays a key role for the interpretation of the simulation results which are in solid agreement with biological observations. Future work is directed towards implementation of radiation damages in order to meet the final goal, namely, the evaluation of our model against the data derived from a group of chronically irradiated uranium miners.

Analysis of bacterial migration: II. Studies with multiple attractant gradients

AbstractMany motile bacteria exhibit chemotaxis, the ability to bias their random motion to ward or away from increasing concentrations of chemical substances which benefit or inhibit their survival, respectively. Since bacteria encounter numerous chemical concentration gradients simulatneously in natural surroundings, it is necessary to know quantitatively how a bacterial population responds in the presence of more than one chemical stimulus to develop predictive mathematical models describing bacterial migration in natural systems. This work evaluates three hypothetical models describing the integration of chemical signals from multiple stimuli: high sensitivity, maximum signal, and simple additivity. An expression for the tumbling probability for individual stimuli (Brown and Berg, 1974) is modified according to the proposed models and incorporated into the cell balance equation for a 1‐D attractant gradient (Ford and Cummings, 1992). Random motility and chemotactic sensitivity coefficients, required input parameters for the model, are measured for single stimulus responses. Theoretical predictions with the three signal integration models are compared to the net chemotactic response of Escherichia coli to co‐ and antidirectional gradients of D‐fucose and α‐methylaspartate in the stopped‐flow diffusion chamber assay. Results eliminate the high‐sensitivity model and favor the simple additivity over the maximum signal. None of the simple models, however, accurately predict the observed behavior, suggesting a more complex model with more steps in the signal processing mechanism is required to predict responses to multiple stimuli.

Analysis of bacterial migration: I. Numerical solution of balance equation

AbstractChemotaxis describes the ability of motile bacteria to bias their motion in the direction of increasing gradients of chemicals, usually energy sources, known as attractants. In experimental studies of the migration of chemotactic bacteria, 1‐D phenomenological cell balance equations (Rivero et al., 1989) have been used to quantitatively analyze experimental observations (Ford et al., 1991; Ford and Lauffenburger, 1991). While attractive for their simplicity and the ease of solution, they are limited in the strict mathematical sense to the situation in which individual bacteria are confined to motion in one dimension and respond to attractant gradients in one dimension only. Recently, Ford and Cummings (1992) reduced the general 3‐D cell balance equation of Alt (1980) to obtain an equation describing the migration of a bacterial population in response to a 1‐D attractant gradient. Solutions of this equation for single gradients of attractants are compared to those of 1‐D balance equations, results from cellular dynamics simulations (Frymier et al., 1993), and experimental data from our laboratory for E. coli responding to α‐methylaspartate. We also investigate two aspects of the experimentally derived expression for the tumbling probability: the effect of different models for the down‐gradient swimming behavior of the bacteria and the validity of ignoring the temporal derivative of the attractant concentration.

Cellular Dynamics simulations of bacterial chemotaxis

The temporal and spatial evolution of the density of populations of chemotactic bacteria have previously been modeled by phenomenological cell balance equations based on cell motion restricted to one dimension. These one-dimensional balance equations have been used to interpret the results of experiments involving three-dimensional motion of bacterial populations with symmetry in two of the three dimensions. We develop a computer simulation to rigorously model the movement of a large population of individual chemotactic bacteria in three dimensions. Results of the simulation are compared with results using a one-dimensional phenomenological model in order to verify the range of validity of this model under situations involving one-dimensional gradients of chemical attractants.

On the Relationship Between Cell Balance Equations for Chemotactic Cell Populations

This paper considers several cell balance equations describing the time and spatial evolution of the number density of chemotactic cell populations. The conditions are determined under which the three-dimensional cell balance equations of Alt reduce to the simpler, one-dimensional equations of Segel that have been used by Lauffenberger and colleagues to describe chemotaxis of flagellar bacteria and polymorphonuclear leukocytes, and to interpret experimental measurements on such systems.

A First-Collision Source Method That Satisfies DiscreteSnTransport Balance

The standard first collision source method is modified to include the angle-integrated, mesh cell balance equation. This modification produces a method that is fully consistent with our normal S/sub N/ balance which is essential for diagnostic purposes. The method is also second order in mesh refinements. Illustrative results are also presented in the paper. 3 refs., 5 figs., 1 tab.

Theoretical investigations of dynamic behavior of isothermal continuous stirred tank biological reactors

The dynamic behavior of an isothermal biological CSTR modelled by idealized cell and substrate balance equations has been investigated theoretically in terms of multiplicity and stability of steady states and existence and stability character of limit cycles. Various types of dynamic behavior have been classified in terms of a Damköhler number and two other system parameters. The predicted types of behavior have been illustrated by numerical computation of cell and substrate concentration trajectories.