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Research Papers

# The Influence of Bioreactor Geometry and the Mechanical Environment on Engineered Tissues

[+] Author and Article Information
J. M. Osborne1

Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK

R. D. O’Dea1 n2

Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UKreuben.odea@nottingham.ac.uk

J. P. Whiteley

Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK

H. M. Byrne

Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

S. L. Waters

Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, UK

1

Joint first authors.

2

Corresponding author.

J Biomech Eng 132(5), 051006 (Mar 25, 2010) (12 pages) doi:10.1115/1.4001160 History: Received July 08, 2009; Revised January 29, 2010; Published March 25, 2010; Online March 25, 2010

## Abstract

A three phase model for the growth of a tissue construct within a perfusion bioreactor is examined. The cell population (and attendant extracellular matrix), culture medium, and porous scaffold are treated as distinct phases. The bioreactor system is represented by a two-dimensional channel containing a cell-seeded rigid porous scaffold (tissue construct), which is perfused with a culture medium. Through the prescription of appropriate functional forms for cell proliferation and extracellular matrix deposition rates, the model is used to compare the influence of cell density-, pressure-, and culture medium shear stress-regulated growth on the composition of the engineered tissue. The governing equations are derived in O’Dea “A Three Phase Model for Tissue Construct Growth in a Perfusion Bioreactor,” Math. Med. Biol., in which the long-wavelength limit was exploited to aid analysis; here, finite element methods are used to construct two-dimensional solutions to the governing equations and to investigate thoroughly their behavior. Comparison of the total tissue yield and averaged pressures, velocities, and shear stress demonstrates that quantitative agreement between the two-dimensional and long-wavelength approximation solutions is obtained for channel aspect ratios of order $10−2$ and that much of the qualitative behavior of the model is captured in the long-wavelength limit, even for relatively large channel aspect ratios. However, we demonstrate that in order to capture accurately the effect of mechanotransduction mechanisms on tissue construct growth, spatial effects in at least two dimensions must be included due to the inherent spatial variation of mechanical stimuli relevant to perfusion bioreactors, most notably, fluid shear stress, a feature not captured in the long-wavelength limit.

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## Figures

Figure 1

The bioreactor system of El Haj (28)

Figure 2

The two-dimensional domain Ω and associated boundary conditions. The arrows indicate the perfusion direction in the case PU>PD.

Figure 3

The variation of the net cell proliferation rate κ(ϕ) with respect to the stimulus of interest, ϕ. (a) The step function form defined in Eq. 8 and (b) the smoothed form defined in Eq. 9. The dimensionless parameter k1ϕ represents the rate of cell proliferation, k2ϕ represents the combined rate of cell proliferation and ECM deposition, while kdϕ represents the combined rate of cell death and ECM degradation. The thresholds at which the cellular response changes are denoted ϕ1 and ϕ2.

Figure 4

Illustrative plots of (a) the cell volume fraction (θn), (b) the cell phase pressure (pn), (c) the culture medium pressure (pw), (d) the culture medium shear stress (τ), (e) the axial cell phase velocity (un), and (f) the axial culture medium velocity (uw) in the regime of uniform cell proliferation, within a channel of dimension [0,1]×[0,0.1] at time t=1. The parameters are as described in Tables  12.

Figure 5

Illustrative plots of the cell volume fraction (θn) for ((a) and (b)) constant cell proliferation and ((c) and (d)) cell density-dependent proliferation in a channel of widths h=0.2 and h=0.01 at t=1. The parameter values are given in Tables  12.

Figure 6

Illustrative plots of the cell volume fraction (θn) for ((a) and (b)) cell pressure-dependent proliferation and ((c) and (d)) culture medium shear stress-dependent proliferation in a channel of widths h=0.2 and h=0.01 at t=1. The parameter values are given in Tables  12.

Figure 7

Plots comparing the averaged (a) cell volume fraction (⟨θn⟩), (b) cell phase pressure (⟨pn⟩), (c) weighted axial cell phase velocity (⟨θnun⟩), and (d) culture medium shear stress (⟨τ⟩) at t=1 for uniform growth in a channel of dimensions of (0,1)×(0,h) for h=0.2(⋅–⋅–⋅), h=0.1 (- - -), h=0.01 (—), and the long-wavelength limit (⋯). The remaining parameter values are given in Tables  12.

Figure 8

Comparison of the averaged cell volume fraction (⟨θn⟩) for (a) cell density-dependent proliferation, (b) cell pressure-dependent proliferation, and (c) culture medium shear stress-dependent proliferation at t=1 in a channel of dimensions of (0,1)×(0,h), for h=0.2(⋅–⋅–⋅), h=0.1 (- - -), h=0.01 (—), and the long-wavelength limit (⋯). The remaining parameters are given in Tables  12.

Figure 9

Comparison of transient cell yields over time for bioreactors of widths: h=0.2(⋅–⋅–⋅), h=0.1 (- - -), h=0.01 (—), and the long-wavelength limit (⋯) for different mechanotransduction regimes. The parameter values are given in Tables  12. (a) Uniform proliferation, (b) cell density-dependent growth, (c) cell pressure-dependent growth, and (d) culture medium shear stress-dependent growth. The arrows show the direction of decreasing aspect ratio h.

Figure 10

The L2 norm ‖θn2D−θnLW‖L2 over time for h=0.2(⋅–⋅–⋅), h=0.1 (- - -), and h=0.01 (—), for different mechanotransduction regimes. The parameter values are given in Tables  12. (a) Uniform proliferation, (b) cell density-dependent growth, (c) cell pressure-dependent growth, and (d) culture medium shear stress-dependent growth. The arrows show the direction of decreasing aspect ratio h.

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