An Investigation on the Influence of Pore Size and Porosity of Sponge on
Maximum Cell Concentration
Jiajie Peng and Chi-Hwa Wang
Department of Chemical and Biomolecular Engineering
National University of Singapore, 4 Engineering Drive 4, Singapore 117576
Abstract
This study investigates the effect of pore size and porosity on the maximum cell
concentration in sponge through their influence on nutrient diffusion and surface area.
Brownian dynamics simulation is employed to analyze the effect of pore size and porosity
exerted on effective diffusivity of sponge. Only porosity is found to have prominent impact
on effective diffusivity of nutrient. Simple geometrical relation is used to estimate the impact
of pore size and porosity on surface area per unit volume of sponge. Pore size shows an
inverse relationship with surface area while porosity is positively related to surface area.
Since both nutrient diffusion and surface area are positively related to maximum cell density,
a sponge with smaller pore size and larger porosity may probably acquires the capacity of
sustaining a higher maximum cell concentration.
Introduction
Tissue engineering has revealed innumerous potential in finding an alternative for
tissue regeneration for medical purpose (Davis and Vacanti, 1996; Mooney and Mikos, 1999).
For this purpose, a scaffold with desired feature is thought to have great significance in
accommodating cell, guiding their growth and tissue regeneration in three dimensions (Yang
et al., 2001; Zhu et al., 2008). Porous sponge is often employed as scaffold in tissue
engineering for growing cells (Kose et al., 2003). Also, for the success of transplantation of
regenerated tissue, a large number of living cells is sometimes desired to be maintained in
scaffold to provide the biological function of tissue when implanted (Davis and Vacanti,
1996). In other words, the maximum living cell concentration in sponge is of great interest.
The cell concentration is influenced by many factors, such as the nature of cell, the
culturing condition, the culturing medium, the nutrient and waste metabolite diffusion rate
and available space (Butler, 2004). Among these factors, the nutrient and waste particle
diffusion rate and available space are considerably influenced by the microstructure of sponge,
specifically pore size and porosity. Firstly, the effective diffusivity of nutrient and waste
particle is significantly affected by the microstructure of sponge (Yang et al., 2001). Secondly,
the available space for cell growth is usually dictated by surface area of sponge, which in turn
controlled by the microstructure of sponge. Normal animal cells are anchorage-dependent,
which means the cell needs to adhere to a solid substratum before proliferation and the
substratum is provided by the surface of pore wall (Butler, 2004). Surface area per unit
volume of sponge is controlled by pore size and porosity. Since pore size and porosity of
sponge exert large influence on nutrient diffusion and available space, they may have certain
relation with the maximum living cell concentration.
The objective of this work is to analyze qualitatively how pore size and porosity affect
the maximum living cell concentration in sponge through their influence on nutrient diffusion
1
and surface area. The effect of pore size and porosity on effective diffusivity of nutrient is
investigated by Brownian Dynamics simulation and the influence of pore size and porosity on
surface area is examined by simple geometrical relationship between them.
Description of Methods
Diffusion:
The inward nutrient and outward waste metabolite diffusion can be regarded as small
particle diffuse inside the sponge, since both of them are molecules and thus the size is very
small comparing to the scale of pore diameter. This is reasonable because the diameter of
sponge pore should not be smaller than the diameter of cell in suspension (Yang et al., 2001).
Therefore the sponge pore size is huge comparing to nutrient or waste metabolite particles,
given the fact that the size of cell is normally much larger than nutrient or waste particles.
Brownian dynamics simulation is implemented to study this issue. In our simulation, several
assumptions are made in order to reduce the computational cost. Firstly, the pores are of
uniform size. Secondly, the particle representing nutrient and waste metabolite is considered
as point and occupy no volume. Since many traditional scaffold fabrication techniques for
tissue engineering application only produce scaffolds with random porous architectures, the
scaffold investigated in the simulation is also a random porous sponge to match back the
realistic porous scaffold (Buckley & Kelly, 2004). The scaffold is simulated by a cubic box
containing N pores of uniform size. In order to generate a random porous scaffold, the N
pores are moved randomly inside the box until the desired porosity is obtained. Afterward the
whole structure is fixed and subjected to further investigation.
Firstly, the interconnectivity of sponge is tested since interconnectivity also
considerably influences the nutrient and waste diffusion inside sponge (Yang et al., 2001).
The result is presented by the distribution of neighboring pores. The distribution of
neighboring pores shows the occurring probability of a pore (called target pores hereafter)
which possesses a certain size of a pore cluster in the sponge. The size of pore cluster can be
simply understood as the number of neighboring pores that connecting to the target pores. The
distribution of neighboring pores does not take into consideration the size of each aperture
created on the target pores’ wall. Nevertheless, the pore cluster distribution serves a good
indication to the interconnectivity of sponge.
Subsequently, the effective diffusivity of nutrient is investigated. In the simulation, all
the length scale is normalized by the radius of pore, σ, and time by σ2/D0, where D0 is the free
diffusivity of particle in pure solvent. Langevin equation is employed to describe the dynamic
motion of nutrient, which is shown below:
U ( r )
dr


R
dt
r
(1)
where r and ζ are the position vector and the friction coefficient , R is the random force due to
the incessant collision of the solvent molecules with the particle, and U(r)is the interaction
potential. The particle inertial effect has been neglected.Moreover, the excluded volume effect
between nutrient and pore wall is considered by implementing the reflection boundary
condition. However, the hydrodynamic interaction between nutrient and solvent is neglected
here. Then the dimensionless mean-square displacement (MSD) of tracer is calculated with
2
about 6*10^6 time step. Afterward, the normalized diffusivity, Deff/D0, is determined by
Deff
D0

1d
2
 r ( t )  r ( 0) 
6 dt
(2)
Where, r is the position vector of nutrient particle. Deff is the effective diffusivity of particles
in sponge. The long-time diffusivity is approximated by the normalized diffusivity after an
enough time period.
For our purpose, three cases with porosity of 0.84, 0.74 and 0.64 are selected. In each
case, two different normalized pore diameters are tested in the simulation. The larger pore
size is d=2, while the small pore size is d=1. d is the diameter of pore. In order to obtain
statistically unbiased results, ten individual runs are performed for each simulation and mean
results of the ten runs will be accepted as results and further analyzed.
Surface area:
The surface area per unit volume is determined by the porosity and pore size of the
sponge. The surface area per unit volume can be derived from surface to volume ratio S/V.
For sphere, the surface area S can be written as
S  4r 2
(3)
And the volume V can be expressed as
4
V  r 3
3
(4)
Where r is the radius of the sphere. Therefore, the surface to volume ratio can e obtained by
dividing Eq. 1 over 2 (Rhodes, 2008). Thus the ratio is
S 3

V r
(5)
or in terms of diameter of sphere, d, it can be expressed as
S 6

V d
(6)
Let’s consider a sponge with volume V, a porosity of βand round shape pores of
uniform diameter d. The volume possessed by pore is Vpore = βV. Assume that the apertures do
not significantly reduce the surface area of pore wall, which means the actual surface area of
pore wall is similar to the surface area of a sphere of the same size, then the surface area of
pore can be approximated by Eq (6), S 
6V pore
d
. In terms of sponge volume, V, it can be
expressed as
S
6V
d
(7)
Results and Discussion:
Nutrient diffusion
1) Interconnectivity:
The effect of pore size on the pore cluster distribution is shown in Fig. 1 in appendix. It
3
can be observed that the pore cluster distribution of small pore (d=1) and that of large pore
(d=2) collides on each other in all the three cases. The results suggest that the pore size exerts
little influence on the pore cluster distribution and therefore on interconnectivity.
The effect of porosity on the pore cluster distribution is illustrated in Fig. 2 in appendix.
Since pore size has little influence on pore cluster distribution, only the data from small pore
(d=1) in each case are compared and a significant difference appears. The distribution
function basically retains its bell shape in all three cases, but shift rightward from the case
with porosity of 0.64 to that with porosity of 0.84. The shift apparently demonstrates that
average pore cluster size increases with increased porosity, which means that the average
aperture number on wall of every pore rise up with lifted porosity. This indicates the
interconnectivity is clearly affected by the changing porosity. Therefore, among pore size and
porosity, only porosity exerts a prominent impact on the interconnectivity and they are
positively related.
2) Diffusivity
The influence of pore size and porosity on normalized diffusivity, Deff/D0 is illustrated
in table 1.
Table 1. summary of normalized diffusivity
Normalized diffusivity under different conditions
normalize ddiffu-sivity
porosity
0.84
0.74
0.64
normalized diameter
d=1
0.78 ± 0.01
0.61 ± 0.01
0.50 ± 0.01
d=2
0.81 ± 0.01
0.63 ± 0.02
0.54 ± 0.01
It is clear that the variation of pore size does not create a significant impact on normalized
diffusivity. The difference between the normalized diffusivities of larger pore and small pore
do not exceed 0.04 for all three cases. On the contrary, the normalized diffusivity rises up
considerably from 0.50 to 0.78 when the porosity is increased from 0.64 to 0.84. This
observation is consistent with the previous conclusion that porosity is positively related to
interconnectivity in the sponge. It has been reported that high interconnectivity can promote
the effective diffusivity of nutrient and wastes particles in sponge (Moone, et al., 1996).
Therefore the positive relation between porosity and effective diffusivity is expected.
Additionally, it has been noted that highly porous sponge could lead to easy diffusion of
nutrient and waste particles inside (Yang et al., 2001). This observation further confirms our
findings.
Then it is apparent that among pore size and porosity, only porosity significantly affects
the effective diffusivity of nutrient and waste in sponge and they are positively associated.
Under the same environment, larger effective diffusivity leads to greater nutrient and waste
diffusion rate. It is also known that greater nutrient diffusion rate elevate the maximum cell
density (Butler, 2004). Hence, it may be concluded that when only nutrient diffusion is the
determining factor for maximum cell concentration, pore size exerts little influence on
4
maximum cell concentration, while porosity and maximum cell concentration is positively
related.
Surface Area
The effect of pore size and porosity on the surface area of sponge can be investigated
through Eq (7), S 
6V
derived previously, and for a unit volume of sponge, V equals to 1.
d
Therefore, it can be observed that the surface area of a unit volume of sponge decreases with
increased pore diameter and increases with increased porosity. From the fact that larger
surface area could support more cells, it can be deduced that higher surface area per unit
volume can elevate the maximum cell density (Butler, 1996; Davis and Vacanti, 1996; Butler,
2004). Thus, when only surface area is the determining factor for maximum cell concentration,
pore diameter is inversely related to maximum cell concentration while porosity is again
positively related to it.
It should be noted that the correlation between pore size and porosity with the surface
area may probably not follow a simple proportional relation as suggested by Eq. (7). It is
because the equation does not take into consideration that the apertures created on pore wall
may reduce the surface area. With Eq. (7), it is only possible to do a rough estimation and
more investigation should be conducted as recommended in ‘future study’ section.
In an experimental study, two sponges, namely PLGA 85:15 and PLGA 50:50 sponges,
were employed for culturing Hep3B cells. PLGA 85:15 sponge had a mean pore size of 34.13
μm±6.84μm, with an overall porosity of 76.4±3.9%. While PLGA 50:50 sponge had a mean
pore size of 59.7 μm±7.4μm, with an overall porosity of 84.4±4.1 % (Zhu et al., 2008). It
deserves some attentions that Hep3B cell is used for culture. Unlike normal cell, Hep3B cell
is cancerous. Cancer cell may not even need a surface to attach before proliferation can occur
(Hokari et al., 2007). Consequently the actual available space for cell growth is not dictated
by surface area and the correlation between surface area and maximum cell concentration
may not prevail. Despite this, it was observed in the study that ‘the cell grew along the wall of
the pores’ (Zhu et al., 2008). As the cell behaved just like normal cell, our hypothesis may
still applicable for this case. Since the culturing environment, such as culturing condition and
culture medium is identical for the two sponges, the difference in nutrient diffusion and
surface area should be primarily responsible for the different maximum cell concentration in
sponges. With a little larger porosity, the nutrient diffusion in PLGA 50:50 sponge should be
a little higher than that in PLGA 50:50 sponge as suggested by our conclusion for effective
diffusivity. Thus the maximum cell concentration in PLGA 50:50 sponge is expected to be
higher. However, the opposite situation is reported. This interesting observation may be
attributed to different surface area inside the two sponges. Set the surface area per unit
volume of PLGA 85:15 as S1, and that of PLGA 50:50 as S2, from Eq. (7), the ratio of S1 to
S2 is
S1 1d 2

S 2  2 d1
(8)
Where β1 and d1 are porosity and mean pore diameter of PLGA 85:15 sponge, while β2 and
d2 are porosity and mean pore diameter of PLGA 50:50 sponge. Since β1= 0.76, d1= 34.1μm,
5
and β2= 0.84, d2= 59.7μm, it can be calculated that the surface area of PLGA 85:15 sponge is
approximately 1.6 times larger than that of PLGA 50:50 sponge. Interestingly, it is reported
from the study that the MTT assay revealed that the maximum living cell concentration in
PLGA 85:15 was also about 1.6 times larger than that in PLGA 50:50 sponge. Despite the
inaccuracy of Eq. (6) discussed before, the very close result from surface area ratio and MTT
assay serves as a clear evidence that our hypothesis about the impact of pore size and porosity
on maximum cell concentration via controlling the surface area is valid.
Combining the results obtained from nutrient diffusion and surface area, the following
conclusion may be drawn. When the microstructure of sponge is the solely determining factor
for maximum cell concentration, pore size of sponge is inversely related to maximum cell
concentration, while porosity is positively related to it. Therefore, smaller pore size and
higher porosity may grant a sponge the capacity of maintaining a higher maximum cell
concentration. However, it worth our attention that in order to achieve the ultimate objective
of tissue regeneration, the variation of pore size and porosity is not dictated by the
relationship investigated in this work. The pore size should be larger than the cell to be seeded
inside, and very porous sponge may suffer from poor mechanical strength (Yang et al., 2001).
These factors are beyond the scope of this work and therefore not considered.
Future study
Experimental determination of actual surface area of sponge
As mentioned previously, Eq. (6), the one used to estimate the surface area of sponge
has its drawback. In order to obtain a more precise data on surface area, another approach to
obtain the surface area of sponge is desired. Among various methods, mercury porosimetry is
a convenient way for obtaining surface area of micropores. The basic working principle of
mercury porosimetry is as following. The intrusion volume of mercury is recorded at different
external pressure. Then Washburn equation is employed to find the relationship between the
applied pressure and pore size into which mercury can intrude under that pressure. The
Washburn equation used in this purpose is often given in this form:
P
2
cos 
r
(9)
where P is the applied pressure, γis the surface tension of mercury, r is the pore radius and 
is the contact angle between mercury and pore wall (Nagy & Vas, 2005). After the mercury
porosimetric data is obtained, several methods are available to calculate the surface area, such
as Boer’s t-method and Brunauer’s MP method (Lowell, 1991). A more accurate result on
surface area may help us evaluating the error of our rough estimation and further investigating
the actual impact of pore size and porosity on surface area for a certain sponge.
Conclusion
In this study, the influence of pore size and porosity on nutrient diffusion, surface area
and hence on the maximum cell concentration in sponge is investigated. Brownian dynamics
simulation is employed to analyze the effect of pore size and porosity exerted on effective
diffusivity of sponge, and simple geometrical relation are used to estimate the impact of pore
size and porosity on surface area per unit volume of sponge. Pore size shows little influence
on effective diffusivity of nutrient but has a inverse relationship with surface area. Porosity is
6
positively related to both effective diffusivity and surface area. In conclusion, a sponge with
smaller pore size and larger porosity may probably acquire the capacity of sustaining a higher
maximum cell concentration.
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Figures
a)pore cluster distribution for
0.16
porosity 0.84
0.14
probability
0.12
0.1
d=1
d=2
0.08
0.06
0.04
0.02
0
-0.02 0
5
10
15
Nc
20
25
30
b) pore cluster distribution for porosity 0.74
0.3
probablility
0.25
d=1
d=2
0.2
0.15
0.1
0.05
0
-0.05 0
5
10
Nc
15
20
c) pore cluster distribution for porosity 0.64
0.2
probability
0.15
d=1
d=2
0.1
0.05
0
0
-0.05
5
10
15
20
Nc
Fig 1 comparison of pore cluster distribution functions of different pore size for
porosity: a) 0.84; b) 0.74; c) 0.64
8
pore cluster distribution of different porosity for d=1
probability
0.2
porosity=0.84
porosity=0.64
porosity=0.74
0.15
0.1
0.05
0
-0.05 0
5
10
15
20
25
Nc
Fig 2 comparison of pore cluster distribution functions of different porosity
9