1
Variations in Cell Density Influence Oxygen
Transport in Engineered Cardiac Tissue
Rohin K. Iyer

Abstract—The promise of creating engineered cardiac tissue
constructs for use in replacement or repair of damaged
myocardium is complicated by the challenge of delivering oxygen
to these tissues efficiently to ensure a high degree of cellularity
and viability. In particular, variation in cell density, be it at the
level of the individual groups of cells or at the macroscopic level
of the tissue may exert a significant influence on tissue oxygen
concentration since higher cell densities increase oxygen
consumption and, in turn, may exceed limitations for diffusive
transport. To test this hypothesis, a mathematical model of
oxygen concentration as a function of cell density and tissue
height was developed. A 10-fold increase in cell density resulted
in a dramatic global decrease in oxygen concentration to 0 uM at
all depths below 10 um, corresponding to just one cardiomyocyte
cell diameter. Similary, stochastic variations in cell density
modeled were modeled as a normal distribution centered about 1
x 108 cells/cm3 with a standard deviation of 1x107 cells/cm3. These
variations resulted in oxygen concentrations which deviated by
100% from the bulk value of 220 uM. Taken together, these
findings suggest a important role for cell density in oxygen
concentration in engineered tissues.
Index Terms—Cell Density, Oxygen Transport. Mathematical
Modeling, Tissue Engineering.
I.
INTRODUCTION
C
diseases are a major cause of mortality
in Canada and the United States [1]. There is currently no
cure for these debilitating diseases, and many of the
available treatments are not effective at repairing already
damaged cardiac tissue [2]. Cardiac tissue engineering holds
the promise of creating an unlimited source of tissue for repair
of damaged or diseased native heart tissue (myocardium)
during surgery. While there is great promise to tissue
engineering approaches, there are also several challenges, such
as efficient delivery of nutrients and oxygen to all areas of the
three-dimensional (3D) tissue construct. Tissue engineered
cardiac constructs are typically made by isolating cardiac
muscle cells (known as cardiomyocytes) from a patient and
cultivating these cells in vitro until thick, electrically excitable
tissues are formed. Though other cell types are present in the
cardiac milieu such as fibroblasts and endothelial cells,
cardiomyocytes consume large amounts of glucose and oxygen
ARDIOVASCULAR
R. K. Iyer is with the Laboratory for Functional Tissue Engineering at the
Institute of Biomaterials and Biomedical Engineering, Univerisity of Toronto,
Toronto, ON, Canada; (e-mail: rohin.iyer@utoronto.ca).
to fuel aerobic respiration. In the absence of oxygen (anoxic
or hypoxic conditions), cardiomyocytes produce more lactate,
which is a hallmark of poor cell health. As well, increased
supply of oxygen to engineered cardiac tissue has been shown
to improve contractile function, cell viability, and expression
of cardiac markers such as cardiac troponin I and connexin-43.
Efficient oxygen delivery to cardiac tissue is therefore of
paramount importance in tissue engineering studies.
II. THEORY
Under static culture conditions, oxygen transport within the
tissue is governed by diffusion alone since there is no
convective oxygen transport component (except possibly at the
surface of the tissue). Since the oxygen concentration in the
bulk fluid is high compared to the concentration in the tissue,
where oxygen is constantly being consumed, a concentration
gradient is established which drives the diffusion of oxygen
into the tissue. Mathematically, this can be described by
Fick’s second law of diffusion:
C
 2C
   ( DC )  D 2
t
x
(1)
where C is the concentration of oxygen entering the tissue in
umol/L, t is time, D is the diffusivity of oxygen in the tissue
space, and x is the spatial coordinate along which the diffusion
is occurring for a rectangular system.
Oxygen consumption may be assumed to follow MichaelisMenten kinetics. The rate of consumption per cell is denoted
by Vcell [(umol/cell)/s], and is given by:
Vcell 
Vmax (C )
Km  C
(2)
where C is the oxygen concentration [uM] in the tissue space
surrounding the cell, Vmax is the maximum consumption rate
per cell [(umol/cell)/s], and Km is the concentration at which
the consumption rate of the cell is half of Vmax. It is assumed
that oxygen is immediately metabolized upon entering the cell
so that oxygen consumption can be said to occur at the
maximum rate, Vmax. Values reported in the literature to for
Vmax are around 1.5 nmol/min/(106 cells), or 2.5 x 10-11
2
(mol/s)/cell. which corresponds to a value of 2.5 uM at a
physiological cell density of 1 x 108 cells/cm3[5]. Note that this
formula is only valid when looking at single cells and does not
provide info about consumption kinetics for multiple cells, as
in the case of engineered tissue. An alternative expression
must be derived to account for the effects of variation in cell
density on consumption kinetics. As discussed in the next
section, variations in cell density may influence the oxygen
concentration profile in the tissue space by increasing the
overall consumption of oxygen.
to be a normally distributed random variable with a mean of 1
x 108 cells/cm3 and a standard deviation of 1 x 107 cells/cm3
(within 10% of the mean). This assumption is in line with
reported deviations in cell density within engineered cardiac
constructs. In the same way, the global cell density will be
varied to determine its effect on global oxygen consumption.
A mathematical model of the oxygen concentration as a
function of cell density is discussed in the following section.
V. MATHEMATICAL MODEL – OXYGEN CONCENTRATION AND
CELL DENSITY
III. CELL DENSITY AS A SOURCE OF VARIABILITY
While many aspects of the engineered tissue can be
controlled, such as the construct geometry, the culture
time, and the concentrations of growth factors that are
supplemented to the culture medium, there is always a
certain degree of stochastic variability in the spatial
distribution of cells within the construct that cannot be
controlled by the experimenter. Variability in cell
density can be attributed to a number of factors,
including paracrine and autocrine gradients in
cytokines, clumping of cells during passaging, or even
cell-cell adhesion which may cause clustering of cells
into particular areas. Such variability can be
important when engineering small tissues since it
creates local gradients in oxygen concentration and
transport, thereby affecting cell viability within the
construct. At the level of the entire tissue, however,
the governing factor is overall seeding density, which
is under the control of the experimenter. Though it
desirable to seed scaffolds with cardiomyocytes at
physiologically relevant cell densities of around 108
cells/cm3, studies have shown that the measured
cardiomyocyte densities within the scaffold can vary
significantly from this value due to poor oxygen
availability far away from the bulk fluid
[3]. Also, experimental error may lead to incorrect seeding of
cells at low or high densities, leading to poor cell viability or
extremely high oxygen demand. A model of oxygen transport
that accounts for cell density variations can therefore provide
useful information on overall oxygen consumption by
engineered tissues.
IV. HYPOTHESIS
Given the potential for variability in cell density, and the
importance influence of this parameter in oxygen consumption
and diffusion, it is reasonable to hypothesize that cell density
can exert a major effect on oxygen concentration within a
tissue engineered construct, both at the level of the individual
cell and at the level of the tissue, and that this will in turn
affect the performance and properties of the tissue. Thus, it is
instructive to statistically model these variations and
incorporate the effects of cell density into the oxygen mass
balance equations to derive an expression for oxygen
concentration as a function of cell density. For simplicity, the
local cell density at any point in the scaffold will be assumed
Based on work currently being done by the author, it is
assumed that the tissue will be grown in a microbioreactor
with rectangular channels of dimension 10 mm x 1 mm x
0.1 mm (LxWxH), as shown in
below. These dimensions are physiologically relevant
since they are similar to the dimensions of real muscle fibres,
and do
not exceed
maximal
diffusional
distance
limitations
(~100
m). Since
the
geometry of
these
microwells is
Fig. 1
x=1000 m
recta
ngul
ar,
Fick’
y=10000m
s
seco
nd
z
z=L=100 m
law
in
rectangular form can be applied by treating the tissue as a slab
of height L.
3
C ( z )   n(
Fig. 1: Microbioreactor schematic and model of tissue showing rectangular
geometry
Vmax
z2
)(   Lz )  C o
D
2
(3)
VI. RESULTS
An expression for the oxygen concentration as a function of
height and cell density, C(z), can be derived by beginning with
a mass balance on oxygen (accumulation = diffusion –
consumption)[6]:
C
 2C
 D 2  nVmax
t
z
where C is the oxygen concentration in umol/L, n is the cell
density per unit volume [cells/L], z = the vertical spatial
coordinate, and Vmax, D and t = time have been previously
defined.
The following boundary conditions (BCs) must also hold:
At z = 0, C = Co (the concentration at z = 0, the top of the
well, is equal to that of the bulk, Co=220 uM)
 [3].
 At z = L, dC/dz = 0 (there is no flux at z=L, the
bottom of the well)
below shows the result of plotting the oxygen
concentration in the tissue space as a function of increasing
cell density, n. The cell density was increased linearly from 1
x 108 cells/cm3 to 1 x 109 cells/cm3. Note that lower cell
densities are not shown since these did not produce significant
effects on oxygen concentration. As can be seen in the figure,
the oxygen concentration drops from 220 uM to about 150 uM
at the physiological density of 1 x 108 cells/cm3, which is still
sufficient to maintain cell viability since the concentration is
non-zero at 100 um where the furthest cells will be found. As
the cell density increases, however, the height at which the
tissue oxygen concentration drops to zero becomes smaller and
smaller, until at 1 x 109 cells/cm3 the oxygen consumption is so
high that only 10 um of the tissue is receiving oxygen. Since
cardiomyocytes have a diameter of 10 um, this corresponds to
only one cell diameter, meaning that any cells below 10 um
will not receive any oxygen at very high cell densities. This
model can therefore aid experimenters in determining ballpark
estimates of cell densities depending on the geometry of the
tissue they are engineering.
Fig. 2
Assuming steady state,
V
C
d C
d C
 0 => 0  D 2  nVmax or
 n max
2
t
D
dz
dz
2
2
Increasing cell density from
1 to 10 times the physiological
value of 1x108 cells/cm3
Integrating with respect to z, we get:
V
dC
 n max z  C1
dz
D
Applying BC # 2, we can solve for the integration constant,
C1:
0n
Vmax
V
L  C1  C1  n max L
D
D
V
V
V
dC
 n max z  n max L  n( max )( z  L)
dz
D
D
D
Integrating again yields:
C ( z )   n(
Vmax
z2
)(   Lz )  C 2
D
2
Applying BC # 1 to solve for C2 gives the final expression:
Fig. 2: Effective oxygen concentration in Tissue Space With Increasing Cell
Density (Cell density was increased linearly in 10 steps from 1 x 108
cells/cm3 to 1 x 109 cells/cm3 in the direction of the arrow)
4
Fig. 3 Oxygen concentration in a rectangular tissue slab with a normally
distributed cell density with mean 1 x 10 8 cells/cm3 and standard deviation of
1 x 107 cells/cm3. Note that the oxygen concentration varies by as much as
100% (as denoted by the dark blue and dark red areas of the surface plot) due
to the local variations in cell density.
shows a surface plot of oxygen concentration as a
function of space for randomized local cell densities in a
representative rectangular tissue slab. The height of the
surface plot and colour gradient at a particular x and y gives a
measure of oxygen concentration at that location. It can be
seen that even small variations in cell density can produce
large deviations in oxygen concentration under the assumption
of a normally distributed cell density with a standard deviation
of 10% of the mean value of 1 x 108 cells/cm3. While these
deviations may not be controllable by the experimenter, it is
important to note that they, indeed, can influence the oxygen
concentration of the tissue. Note that none of the local oxygen
concentration values drop to zero which suggests that random
variation would probably not contribute significantly to
reduced cell viability due to hypoxia.
Fig. 3
5
VII. CONCLUSION AND DISCUSSION
It was hypothesized that variations in cell density both at the
microscopic and macroscopic level would influence the
oxygen concentration profile in a representative rectangular
tissue slab containing cardiomyocytes. By carrying out a mass
balance on oxygen, it was possible to derive a model for
oxygen concentration which incorporated a dependence on cell
density. These cell density variations were shown to exert an
influence on oxygen delivery to the construct at both the tissue
level and at the level of individual cells. In particular, Fig. 2,
which showed a dramatic drop in oxygen concentration with
increasing cell density may be used to predict appropriate
geometries for tissue engineered constructs based on cell
densities and diffusion limitations alone.
The variability in oxygen concentration depicted in Fig. 3
provides insights into potential local gradients in oxygen
concentration that may arise due to stochastic and biological
variability in cell density at the microscopic level. Indeed,
other sources of error may contribute to the variability in cell
density, such as random error in the number of cells the
experimenter may seed into a construct, cell death or
proliferation over time, which is the product of complex, yet
unknown biochemical mechanisms, and these sources of
variability are not accounted for accurately by the model
presented here. As well, the assumption of normally
distributed cell density may not be correct. Finally, the model
may be rather simplistic since it does not account for transient
changes in cell density, such as cells migrating into areas of
low cellularity in order to compensate for the low oxygen
concentration or crowding. Taken together, this suggests that
further work must be done to improve the model so that it can
be used to accurately predict the oxygen demands of the tissue.
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