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Research Papers

# Continuum Modeling of Biological Tissue Growth by Cell Division, and Alteration of Intracellular Osmolytes and Extracellular Fixed Charge Density

[+] Author and Article Information
Gerard A. Ateshian1

Department of Mechanical Engineering, and Department of Biomedical Engineering, Columbia University, New York, NY 10027ateshian@columbia.edu

Kevin D. Costa, Evren U. Azeloglu

Cardiovascular Research Center, Mount Sinai School of Medicine, New York, NY 10029

Barclay Morrison, Clark T. Hung

Department of Biomedical Engineering, Columbia University, New York, NY 10027

In the special case when a fluid cannot flow, due to being entirely trapped within pores or otherwise bound to the solid, it may be modeled as part of the solid matrix.

In mixture theory, the effective stress $Teσ$ in a constituent is defined such that its associated traction vector on a surface represents the force acting on that constituent, normalized by the elemental area of the mixture. It is thus an apparent stress; apparent stresses can be added together to provide a net apparent stress, and its associated traction vector on an elemental mixture area.

The values of $p0$, $ψ0$, $c0$, and the corresponding reference chemical potentials $μ0α(θ)$ at a given $θ$ are prescribed standard states that remain invariant (30).

If the FCD results from a homogeneous molecular species $σ$ of concentration $cσ$ and charge number $zσ$, then $zσcσ=zFcF$.

Let the interface divide the domain into two regions, denoted by “$+$” and “−,” then $[[f]]n≡f+n++f−n−=(f+−f−)n$, where $n$ is the unit outward normal to the $+$ side.

Using monovalent counterions yields a closed-form solution for the Donnan osmotic pressure in the ECM, as shown in Sec. 3; multivalent ions can be analyzed similarly but require numerical solutions. Analyzing multiple neutral solutes (whether membrane-permeant or -impermeant) requires a straightforward superposition of the analysis of a single solute.

It is common to let $p∗=0$ such that $pm$ and $pc$ represent gauge pressures; it is similarly convenient to let $ψ∗=0$.

In this treatment where we explicitly model the ambient bath pressure $p∗$, traction-free should be understood to mean that the applied traction is only that of the ambient pressure.

However, there may be other physicochemical factors, such as osmotic pressure resulting from configurational entropy of the matrix-bound charged molecular species, that may not vanish under hypertonic conditions. Thus, as often encountered in continuum mechanics, a true stress-free reference configuration may not be achievable experimentally, but only mathematically.

For example, $c∗r=0.15M$ to reproduce common physiological conditions, though any other value may be chosen.

Recall that the porous solid matrix is compressible because its pores may change their volume during deformation, even though the matrix skeleton and the interstitial fluid are modeled as intrinsically incompressible. Thus, it is acceptable to have Poisson’s ratio equal to zero in this framework.

1

Corresponding author.

J Biomech Eng 131(10), 101001 (Sep 01, 2009) (12 pages) doi:10.1115/1.3192138 History: Received November 11, 2008; Revised May 04, 2009; Published September 01, 2009

## Abstract

A framework is formulated within the theory of mixtures for continuum modeling of biological tissue growth that explicitly addresses cell division, using a homogenized representation of cells and their extracellular matrix (ECM). The model relies on the description of the cell as containing a solution of water and osmolytes, and having a porous solid matrix. The division of a cell into two nearly identical daughter cells is modeled as the doubling of the cell solid matrix and osmolyte content, producing an increase in water uptake via osmotic effects. This framework is also generalized to account for the growth of ECM-bound molecular species that impart a fixed charge density (FCD) to the tissue, such as proteoglycans. This FCD similarly induces osmotic effects, resulting in extracellular water uptake and osmotic pressurization of the ECM interstitial fluid, with concomitant swelling of its solid matrix. Applications of this growth model are illustrated in several examples.

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## Figures

Figure 1

Effect of growth of fixed charge density on opening angle of rat aorta. Due to symmetry, only one-quarter of a cut ring is displayed. The opening angle increases from 13 deg to 95 deg as c̃mrF increases from 40 mEq/L to 120 mEq/L in the intima and media.

Figure 2

Effect of growth of intracellular membrane-impermeant solute concentration on opening angle of rat aorta. Due to symmetry, only one-quarter of a cut ring is displayed. The opening angle decreases from 13 deg to −37 deg as c̃cri decreases from 210 mEq/L to 175 mEq/L in the intima and media.

Figure 3

Growth of hyaline cartilage model in long bone morphogenesis. Due to symmetry, only one octant of the model is shown. In this analysis growth occurs by cell division. Sixfold increases in φcrs and c̃cri lead to a sixfold increase in tissue volume (χ=1).

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