Ind. Eng. Chem. Res. 1996,34, 3239-3245
3239
Importance of Nonhomogeneous Concentration Distributions Near Walls in Bioreactors for Primary Cell Cultures Ching-An Pengt and Bernhard 0. Palsson*gtp* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, and Aastrom Biosciences Znc., Ann Arbor, Michigan 481 06
Perfusion chambers of rectangular cross section are often used for cell and tissue engineering purposes. In such chambers, growth medium flows in a laminar fashion over the cell bed located on the bottom of the chamber. The secretion of physiologically active protein from the cell bed is important in tissue function. Nonuniformity in the flow distribution near the side walls leads to concentration gradients of secreted protein. The magnitude of this concentration gradient is estimated for physiological protein secretion rates, for various aspect ratios and fluid flow rates using the finite element method. The thickness of hydrodynamic boundary layer extending from the side wall (dh) is flow rate invariant, while the thickness of corresponding concentration boundary layer (6,) decreases with increasing flow rate. The ratio of 6, to 6 h always exceeds unity and varies with the flow-rate and aspect ratio. Thus, a n extended concentration gradient can emanate from the side wall of the chamber even for high aspect ratios. The estimation of the magnitude of this concentration gradient shows that it can be large enough to cause chemotactic cell motion toward the side walls. This chemotactic motion in turn can lead to nonuniform cell growth and be a detriment to effective reconstitution of tissue function ex vivo.
Introduction The ability to grow human tissues and organs ex vivo has progressed significantly during the past decade and has led t o the emergence of a field of research known as tissue engineering (see Hubbell et al., 1994;Koller and Palsson, 1993c;Langer and Vacanti, 1993;Palsson and Hubbell, 1995; Skalak and Fox, 1988). These advances have been driven on the one hand by rapidly growing information about molecular controls of cell growth, apoptosis, differentiation, and motion, and on the other hand by improved mammalian cell culture technology. However, it has proven difficult to grow primary cells in large and clinically meaningful quantities while still maintaining their specialized biological activity. Bioreactor systems for the cultivation of primary cells need to be further developed t o enable many of the promising clinical uses of cell therapies. Proper design of bioreactors demands that we understand how the physicochemical environment within bioreactors affects the cell behavior, just as it does in vivo. In general, cell culture bioreactors fall into two categories: those used for the cultivation of anchoragedependent cells and those used for the cultivation of cells in suspension. Most types of primary mammalian cells require attachment to a substrate t o be able to grow. Single-pass perfusion systems composed of parallel plates have been used for a variety of studies, such as the effect of hydrodynamic shear stresses on cells (Frangos et al., 1988;Koslow et al., 1986),cell adhesion (Lackie, 1991;Usami et al., 1993; van Kooten et al., 1992),and growth and/or maintenance of primary cells ex vivo (Koller et al., 1993a,b;Palsson et al., 1993; Rinkes et al., 1994). In spite of their extensive use, no detailed analyses of the physical rate processes that take place in such perfusion chambers have appeared. In particular, the
* To whom correspondence should be addressed. E-mail:
[email protected]. +
University of Michigan.
* Aastrom Biosciences Inc. 0888-5885/95/2634-3239$09.oo/o
concentration distributions of physiologically active compounds, such as cytokines, within bioreactors may be critical in reconstructing tissue function. Such secretion may be important where such factors are autocrine, leading to locally accelerated growth, or inductance of chemotactic motion resulting in uneven distribution of cells. Further, for cultivating tissues such as liver, bone marrow, or skin,it is well established that stroma is necessary for sustaining long-term tissue cultures (Halberstadt et al., 1994;Koller et al., 1993a,b; Naughton and Naughton, 1989;Naughton et al., 1994; Navsaria et al., 1994;Palsson et al., 1993;Rheinward and Green, 1977). The effects of growth factors and/or chemotactic factors secreted by stromal cells, leading to reconstitution of tissue function in ex vivo cultures, is expected to be significant. Therefore, the concentration distribution of secreted proteins may prove to be a critical consideration in the design and function of tissue engineering bioreactor systems. This study will use the finite element method t o solve the mass transfer problem of cytokines secreted from the cell bed located at the bottom of a laminar flow rectangular duct chamber. The effects of two key dimensionless parameters, the aspect ratio and the Graetz number, on the concentration distribution of cytokines at the bottom wall of the bioreactor are examined with particular emphasis on the determination of the length scale of any transverse concentration gradient which is extended from the side walls. Further, the magnitude of the concentration gradients formed close to side walls will be calculated to examine the possibility of cell movement induced by the uneven distribution of a chemoattractant.
Physical Model and Governing Equations A rectangular duct chamber and its coordinate system are shown in Figure 1. The origin of the coordinate system chosen is at the lefb bottom corner of the duct. The area of interest is shown in greater detail in the figure to illustrate the nonuniformity in the fluid flow and concentration distributions. The cells are located 0 1995 American Chemical Society
3240 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 Region of non-uniform matlarim
negligible compared to the convection term uz aC/& (Hubbell and McIntire, 1986). The appropriate boundary conditions are
w, 0 I
0 Iy
I
0 Ix
IL,0 Iy 5
w:
Z I
W:
O I X ~ L , O ~ ~ I W : Uniform flux from bottom wall only
0 IX
OIXIL,OIZIW:
ux ,flow velocity
3
-
eC
Y
,concentration of secreted protein at the bottom of the chamber
Domain of interest, where fluid tlow and concentmtion distributions are uneven
Figure 1. Coordinate system for a rectangular chamber with production of biological factors secreted by cells lodged on the bottom wall.
on the bottom wall of the chamber. The application of appropriate symmetry conditions a t the cross-section center line permits the restriction of the solution t o the left half of the duct chamber. The flow is assumed to be fully-developed, steady and laminar with constant fluid properties. Since the convective term in the convective-diffusion equation involves the velocity distribution, the first step in the analysis is to specify the velocity variation over the cross section. Velocity Distribution. For fully developed steady laminar flow of a Newtonian fluid, there is only one nonzero component of the velocity vector, ur(Y,z),and the momentum equation is given by
where p denotes the viscosity and dPIdx is the pressure gradient. The velocity distribution for ux(Y,z)obtained by separation of variables is given by Knudsen and Katz (1958) as u,(y*z) = - --
cosh(nn(y
w:
IL,0 Iz I
C(0YP) = 0
ac D ---(xJ,O) a2
= -K
ac
D$qy,W=O
ac
D +X,O*zZ) ?Y
ac
=0
D+x,W,z)=O ?Y
(4)
(5)
(6) (7) (8)
Equation 4 states that the medium flowing into the bioreactor does not contain the compound of interest. Equation 5 represents the mass flux of the secreted factors from the bottom wall where cells are uniformly distributed, and eqs 6 and 8 state that the top wall and left side wall are impervious to cytokines. Equation 7 gives the symmetry condition a t the cross-sectional center lines. Finally, at an outflow boundary where the fluid leaves the calculation domain, neither the value nor the flux of concentration is known. The difficulty is resolved by treating this outflow boundary condition as a free boundary condition (see detail in the Solution Strategy section below). Selection of Reference Scales. Before solving the governing equations, we select reference scales for the variables to make them dimensionless. The logical reference scales for length and depth are L and H , respectively. The reference scale used for the chamber width is H as well, since the region of interest is the boundary layer near the side wall. The reference concentration Co is derived from eq 5 . By introducing dimensionless concentration and depth variables, C = COB and z = HC, into eq 5, we obtain
Let DCdHK = 1,thus the concentration reference scale is defined as
1 X
]
- W/H)
cosh(nnW/W
ndz cos
- H)
H
(2)
Convective-Diffusion Equation. The convectivediffusion equation for the concentration of secreted compounds is given by (3) A pseudo-steady-state assumption is applied here since the biological dynamics (cell growth and motion) have much longer characteristic times than the physicochemical process considered. It has also been shown that the axial diffusion term D PC/ilx2cannot be neglected, since in the region close t o the bottom wall, where the local fluid velocities are low, the axial diffusion term is not
= (volumetric cytokine production rate) x (diffusional response time) (10)
The diffusion coefficients for secreted protein can be estimated from the Stokes-Einstein equation or from published data on protein diffusion coefficients (e.g., Young et al., 1980) to be in the range of D (5-10) x cm2/s. The value of the secretion rate K can be obtained by direct measurement of a particular protein of interest from a known number of cells, or from the cell density and the estimated maximal achievable secretion rate per cell (Savinell et al., 1989). On the basis of production rate 500-5000 moleculed(cel1.s) and cell density lo5 cells/cm2,the range of numerical values for K is 0.1- 1 pM c d s . Given H = 0.1 cm, we can estimate the reference concentration to be HKID = 0.01-0.2 pM. Scaling the Governing Equations. The dimensionless variables correspondingt o the chosen reference
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3241 scales are
Equation 2 can be integrated over the cross section, and the double integral then divided by the cross-sectional area to give the mean velocity i&(Siege1 and Savino, 1965),which can be used to nondimensionalize ux:
Figure 2. Duct chamber mesh generated by FI-GENpreprocessor. The flow domain is discretized by 16 grid lines in the x-direction, 26 in the y-direction, and 14 in the z-direction. The overall nodal points and elements used are 5824 and 4875, respectively.
(12) Introducing these reference scales into the governing equation (eq 3) and boundary conditions eqs 4-8, we obtain the following dimensionless form of the diffusion equation:
ae Gzruxux
aE
1 a2e a2e a2e - -++
E
a2aE2
aq2 ag2
(13)
nodes and elements utilized were 5824 and 4875, respectively. In each element the concentration field is approximated by a function O(&v,
