ORIGINAL ARTICLE Artificial Skin, Muscle, Bone / Joint, Neuron

Computational study of culture conditions and nutrient supplyin a hollow membrane sheet bioreactor for large-scale bone tissueengineering

Ramin Khademi • Davod Mohebbi-Kalhori •

Afra Hadjizadeh

Received: 8 March 2013 / Accepted: 5 September 2013 / Published online: 27 September 2013

� The Japanese Society for Artificial Organs 2013

Abstract Successful bone tissue culture in a large

implant is still a challenge. We have previously developed

a porous hollow membrane sheet (HMSh) for tissue engi-

neering applications (Afra Hadjizadeh and Davod Moh-

ebbi-Kalhori, J Biomed. Mater. Res. Part A [2]). This study

aims to investigate culture conditions and nutrient supply

in a bioreactor made of HMSh. For this purpose, hydro-

dynamic and mass transport behavior in the newly pro-

posed hollow membrane sheet bioreactor including a

lumen region and porous membrane (scaffold) for sup-

porting and feeding cells with a grooved section for

accommodating gel-cell matrix was numerically studied. A

finite element method was used for solving the governing

equations in both homogenous and porous media. Fur-

thermore, the cell resistance and waste production have

been included in a 3D mathematical model. The influences

of different bioreactor design parameters and the scaffold

properties which determine the HMSh bioreactor perfor-

mance and various operating conditions were discussed in

detail. The obtained results illustrated that the novel scaf-

fold can be employed in the large-scale applications in

bone tissue engineering.

Keywords Hollow membrane sheet � Bioreactor �Scaffold � 3D mathematical model � Bone tissue

List of symbols

Ci;in mol/m3� �

Inlet substrate concentration

CECSi mol/m3

� �Local nutrient concentration in the

ECS

Dil m2=sð Þ Diffusion coefficient of species i in

the lumen

Die;s m2=sð Þ Effective diffusion coefficient in the

scaffold

Dieff m2=sð Þ Effective diffusion coefficient of

species i within the ECS

Dicell m2=sð Þ Molecular diffusivity of species i in

the cell phase

Die;ECS m2=sð Þ Effective diffusivity of cell-free gel

matrix for species i

H mð Þ Length of the bioreactor

I Unit tensor

i Species i (for oxygen i = 1, glucose

i = 2, lactate i = 3)

j Region j (for lumen ¼ 1, scaffold

j ¼ 2, ECS j ¼ 3)

Kmi mol/m3� �

Michaelis–Menten parameter

L mð Þ Geometric length

Mi kg/molð Þ Molecular weight of an arbitrary

species i

Mn kg/molð Þ Number-averaged molecular weight

of the mixture

n Unit normal vector at the interface

Nij mol/m2 s� �

Mass flux of species i in region j

p (Pa) Pressure

R mð Þ Inner lumen radius

Ri kg/molð Þ Metabolic rate

R. Khademi � D. Mohebbi-Kalhori (&)

Chemical Engineering Department, University of Sistan

and Baluchestan, P.O. Box 98164-161, Zahedan, Iran

e-mail: d.mohebbi@usherbrooke.ca; davoodmk@eng.usb.ac.ir

A. Hadjizadeh

Chemical Engineering Department, Ecole Polytechnique de

Montreal, 2900 Boulevard Edouard-Montpetit, Montreal,

QC H3T 1J4, Canada

A. Hadjizadeh (&)

Faculty of Biomedical Engineering, Center of Excellence on

Biomaterials, AmirKabir University of Technology, Tehran, Iran

e-mail: afra.hajizadeh@aut.ac.ir

123

J Artif Organs (2014) 17:69–80

DOI 10.1007/s10047-013-0732-2

Re Reynolds number

t mð Þ Thickness of the scaffold

T Transpose operator

u m/sð Þ Velocity vector of the fluid in the bulk

u0 m s�1ð Þ Average inlet velocity of the fluid in

the lumen

Vmaxi mol/cells sð Þ Michaelis–Menten constant

Vcell m3=cellð Þ Specific cell volume

w mð Þ Distance between centers of two

lumens set side by side

W mð Þ Geometric width

x mð Þ Horizontal coordinate

y mð Þ Vertical coordinate

z mð Þ Applicate coordinate

Greek letters

cECS Porosity of the CS region

cs Porosity of the scaffold

e Cell volume fraction ¼ qcellVcell� �

jECS m2ð Þ Hydraulic permeability of the ECS

region

js m2ð Þ Hydraulic permeability of the scaffold

jHD;ECS Hindrance coefficient for the lumen-ECS

diffusion

jHD;s Hindrance coefficient for the lumen-

scaffold diffusion

l kg/m sð Þ Dynamic viscosity of the culture medium

q kg/m3� �

Mass density of the culture medium

qcell cells/m3� �

Cell density

u Partition coefficient of the nutrients

xi Weight fraction of an arbitrary species i

Introduction

Nowadays one prevalent way to resolve tissue defects is

grafting. Blood type, size match, and immune system

rejections are problems that may occur in the traditional

methods. Tissue engineering offers significant promise as a

viable alternative to current clinical strategies for repair

and replacement of damaged tissue or organs by trans-

planting functional tissue engineering constructs (TECs)

that are grown in in vitro 3D scaffold materials [1–3].

In recent years, there has been a great need for clinical-

scale TECs [2–4]. Providing an alternative solution by

creating engineered bone tissue of desired size and shape is

done in the bone tissue engineering (BTE) field [4, 5].

However, in ex vivo, it is difficult to supply nutrients for

cells adequately without a capillary network because nat-

ural bones are highly vascularized, and they rely on blood

vessels to deliver nutrients to cells situated deep in the

mineralized bone matrix [3, 5, 6]. The scaffold in a bio-

reactor is not only a physical support for cells, but it also

affects the cell metabolism, differentiation, and morpho-

genesis [7].

Sampling and direct monitoring transport phenomena in

the tissue engineering bioreactor are almost impossible. It

is why noninvasive methods [4] or computational analysis

have been previously proposed [3]. Computational fluid

dynamics (CFD) simulations are useful tools to gain insight

in local flow field and nutrient transport in the bioreactors.

In the design step of the porous biomaterials that serve the

implants in tissue engineering, a central concern is con-

trolling the fluid hydrodynamics and mass transport [8].

Until now, a significant number of studies have been done

in order to improve the performance of the bioreactors. In

brief, mathematical models have been developed for cell

population growth [3, 7–10] and nutrient transport in a 2D

model [6, 11–13]. The mathematical models described

above neglect either catabolic production [3, 9, 11, 12] or

cell resistance against mass transfer [3, 7–10].

Hadjizadeh and Mohebbi-Kalhori [2] devised and fab-

ricated a novel HMSh scaffold for growing bone and viable

tissue constructs on a laboratory scale. The purpose of their

work was to construct a biocompatible, biodegradable and

permeable scaffold with sufficient mechanical strength and

elasticity for large-scale tissue engineering applications.

Simplicity of the fabrication and cell seeding are the most

important advantages of the HMSh scaffold. The systems

can be seen in the form of a hollow membrane sheet bio-

reactor (HMShB). The HMShB consists of: (1) lumens for

flowing culture media, (2) extracapillary space (ECS) filled

with a gel-cell matrix, (3) the scaffold, which is a porous

hollow membrane sheet that provides an appropriate bed to

supply nutrients for growing cells.

In this work, a modeling study was performed to iden-

tify the regions of the bioreactor where minimal nutrients

occur to find whether cultured cells are under hypoxic

conditions or not in order to indicate the novel fabricated

scaffold capability. A mathematical solution in a 3D model

was presented for the bioreactor containing the HMSh

scaffold. Oxygen [9] and glucose [3] are the main nutrients

that limit cell growth, along with the waste products, such

as lactate, that the cells are predicted to produce. Therefore,

including those species in the model could make it more

realistic than other previously reported mathematical

models. In addition, the influences of physical properties

and geometric parameters of the scaffold and various

operating conditions in HMShB were analyzed.

Materials and methods

A simple method to design and fabricate the HMSh was

applied to provide a capable bed to support tissue growth.

For fabrication of such a scaffold by a new method, which

70 J Artif Organs (2014) 17:69–80

123

we have described previously [2], a polymer solution as a

primary material was prepared and then cast on a special

tray (Fig. 1a). The tray was molded to make cylindrical

channels (lumens) for the nutrient flow; these were

removed after the processes. The next steps were

immersing precipitation and subsequently air casting. In

the last operation, the solvent was vaporized, and a porous

matrix remained. Then a mixture of the collagen gel and

cells were placed on the fabricated scaffold (Fig. 1b); after

sandwiching the scaffold (Fig. 1c), it was placed in the

bioreactor for feeding the cells and growing tissue

(Fig. 1d). When the scaffold was tweaked, the lumens were

counterpoised irregularly (Fig. 1e). There may be numer-

ous lumens in the large and complex system.

Geometry and mathematical model

In this work, it has been assumed that any lumen is set

between its two opposite lumens exactly (Fig. 1f). It is an

appropriate pattern as an element (Fig. 1g) from the system

which is representative of the whole sandwich. The values

of the geometric parameters such as scaffold length (H) and

lumen radius (R) were selected from the experimental

system previously fabricated by Hadjizadeh and Mohebbi-

kalhori [2]. As shown in Fig. 1f and g, w is the distance

between centers of two lumens set side by side, and t is the

thickness of the scaffold. L and W are the length and width

of the selected element, respectively. All simulations were

done based on aforementioned dimensions that are sum-

marized in Table 1.

Governing model equations

In some mathematical models which have been proposed

previously [3, 6], the convection term was ignored in the

scaffold and ECS regions, whereas in a perfusion biore-

actor, the transport phenomena occur mainly via convec-

tion, so metabolite supply to cells was more uniform [7]

(a) (b)

(c) (d)

(e) (f) (g)

Fig. 1 Schematic depiction of

a the solvent casting process,

b mixture of collagen gel and

cells, c sandwiching the

scaffold, d hollow membrane

sheet bioreactor, e cross-section

of the bioreactor, f a pattern of

six layers of the HMSh during

feeding of the cells, g enlarged

drawn element as computational

domain

J Artif Organs (2014) 17:69–80 71

123

and not diffusion-limited [14, 15]. Therefore, the convec-

tive term was not eliminated in this study.

In previous studies, for obtaining of the velocity

field, a coupled Navier-Stokes and Darcy’s law (NS-

DL) has been applied. In this study, a steady state

laminar flow ðRe� 2100Þ and coupled NS-Brinkman

were used in the homogenous regions (lumens) and the

porous media (scaffold and ECS), respectively. Further,

Fick’s law was applied for prediction of the nutrient

profile.

Table 1 Values and typical

ranges of the parameter used in

the present model

Parameters Values Ranges Units References

Operating parameters

u0 7:45� 10�3 0:25; 0:5; 1ð Þ � u0 m s�1 [4, 9–11]

CO2 ;in 0:22 � mol m�3 [2]

CGl;in 5:55 � mol m�3 [2]

T 310 � K [2]

Re 10:9 0:25; 0:5; 1ð Þ � Re Dimensionless Based on u0

l 6:78� 10�4 � kg m�1 s�1 [4, 9–11]

q 1000 � kg m�3 [2]

Molecular weights

Lactate 0:090 � kg mol�1 [2]

Glucose 0:180 � kg:mol�1 [2]

Oxygen 0:032 � kg:mol�1 [2]

Water 0:018 � kg:mol�1 [2]

Bioreactor parameters

H 40 30�50 mm [1]

L 1:1 1:0�1:2 mm [1]

R 0:5 0:4�0:6 mm [1]

t 0:75 0:65�0:75 mm [1]

w ¼ 2Lð Þ 2:2 2:0�2:4 mm

Diffusivities

DO2

l3:29� 10�9 � m2 s�1 [2]

DGll 5:4� 10�10 � m2 s�1 [2]

DLal 1:1� 10�7 � m2 s�1 [2]

DO2

cell1:59� 10�9 � m2 s�1 [2]

DGlcell 1:0� 10�10 � m2 s�1 [2]

DLacell 1:0� 10�10 � m2 s�1 [2]

Porous membrane parameters

u 1 � � [2]

cs 0.8 0:6�0:95 � [21]

js 1.0 9 10-10 10-12–10-9m2 [21]

jHD;s 0:94 � � [2]

ECS

cECS 0:73 0:92�0:73 � [21]

jECS 1:0� 10�12 10�13�10�10 m2 [21]

jHD;ECS 0:94 � � [2]

Metabolic rate parameters

qcell 150� 1012 1=3; 2=3; 1ð Þ � qcell cells m�3 In this work

Vcell 2:8� 10�15 � m3 cell [2]

KmO2

0:01105 � mol m�3 [2]

VmaxO2

1:75� 10�17 0:5�3:3ð Þ � 10�17 mol s�1 cells�1 [2]

KmGl 6:3� 10�5 � g m�3 [2]

VmaxGl 1:25� 10�17 1:25�1:4ð Þ � 10�17 mol cm�3 s�1 [2]

72 J Artif Organs (2014) 17:69–80

123

Lumen region

The circulating culture medium was taken as continuous

phase, and the fluid was assumed incompressible and

Newtonian with the constant physical properties. The

governing equations are shown as follows.

Continuity equation:

q � ru ¼ 0 ð1Þ

Momentum conservation equation:

qðu � rÞu ¼ r � �pI þ l ruþ ruð ÞT� �� �

ð2Þ

where q, u, p and l are the density, bulk velocity, pressure,

and dynamic viscosity of the fluid, respectively.

Species concentration equations:

r � qDilrxi þ qxiD

il

rMn

Mn

� �¼ q u � rð Þxi ð3Þ

The symbols Dil and xi are diffusion coefficient in the

fluid and weight fraction of an arbitrary species i. The Mn is

the average molecular weight of the mixture according to

the following equation:

Mn ¼X xi

Mi

� ��1

ð4Þ

where Mi is the molecular weight of an arbitrary species i.

Porous membrane region (scaffold)

The convection term was considered using the Brinkman

equation, but there was no metabolic/cellular reaction in this

region because the cell migration was neglected from the ECS

to the porous membrane. Hence, the velocity field and nutrient

distributions were obtained from the following equations:

Continuity equation:

q � ru ¼ 0 ð5Þ

Momentum conservation equation:

qcs

u � rð Þ u

cs

� �¼ r � �pI þ l

cs

ruþ ruð ÞT� �

� 2l3cs

r � uð ÞI� �

� ljs

u

ð6Þ

where cs and js are the porosity and permeability of the

porous membrane, respectively.

Species concentration equation:

r � qDie;srxi þ qxiD

ie;s

rMn

Mn

� �¼ q u � rð Þxi ð7Þ

in which Die;s is the effective diffusivity of the porous

membrane for the species i through the porous membrane

that was obtained from:

Die;s ¼ Di

l � cs � jHD;s � u ð8Þ

where jHD;s is the hindrance coefficient and u is the par-

tition coefficient of the species i [3].

Extracapillary space region (ECS)

The ECS (gel-cell matrix) was considered to be an iso-

tropic porous medium saturated with the culture solution. A

uniform cell seeding within the region was assumed. Since

the cells and gel were both in this region, mass transport

behavior was provided by a coupled convection–diffusion

and a metabolic/cellular reaction. The governing equations

are defined as follows.

Continuity equation:

q � ru ¼ 0 ð9Þ

Momentum conservation equation:

qcECS

u � rð Þ u

cECS

� �

¼ r � �pI þ lcECS

ruþ ruð ÞT� �

� 2l3cECS

r � uð ÞI� �

� ljECS

u ð10Þ

Species concentration equation:

q u � rð Þxi�r � qDieffrxiþ qxiD

ieff

rMn

Mn

� �¼ Ri ð11Þ

where Dieff is the effective diffusion coefficient of the

species i, which is defined as [3]:

Dieff

Die;ECS

¼2

Dicell

þ 1Di

e;ECS

� 2eECS1

Dicell

� 1Di

e;ECS

� �

2Di

cell

þ 1Di

e;ECS

þ eECS1

Dicell

� 1Di

e;ECS

� �

2

664

3

775 ð12Þ

in which Dicell is the molecular diffusion of the species i in

cells and Die;ECS is the effective diffusivity of cell-free gel

matrix for the species i as follows:

Die;ECS ¼ Di

l � cECS � jHD;ECS � u ð13Þ

Eqs. (12) and (13) consider the cell and porous matrix

resistances against the mass transport, respectively [3].

Reaction rate

In this study, the nutrients were assumed to follow the

Michaelis–Menten kinetics [16].

Ri ¼ Vmaxi

CECSi

Kmi þ CECS

i

e;

e ¼ qcellVcell ðfor oxygen i ¼ 1 and glucose i ¼ 2Þ ð14Þ

J Artif Organs (2014) 17:69–80 73

123

The required parameters of cells were obtained from the

literature as given in Table 1.

As mentioned before, it has been assumed that the gel

matrix was seeded uniformly; therefore, e doesn’t have

spatial dependency.

The lactate production rate was given by the following

expression [17]:

R3 ¼ �2R2 þ1

3R1 ð15Þ

Equation (15) shows the local production rate which is

dependent on the uptake rates.

Boundary conditions

For solving Eqs. (1) to (15) the appropriate boundary

conditions had to be used only at the computational

domain, because the continuity of the velocity and mass

were imposed at the interfaces, which can be expressed:

ujlumen ¼ ujscaffold¼ ujinterface ð16aÞ

ujscaffold ¼ ujECS¼ ujinterface ð16bÞ

where ujj is the bulk velocity, i.e. region j (j is 1, 2, and 3

for lumen, scaffold, and ECS regions, respectively).

n � Ni1 � Ni

2

� �¼ 0 ð16cÞ

n � Ni2 � Ni

3

� �¼ 0 ð16dÞ

Nij ¼ q ujj � r

� �xi�r � qDi

jrxiþ qxiDij

rMn

Mn

� �ð16eÞ

where symbol n is the normal vector at the interfaces and N

denotes the mass flux. In the lumen, scaffold and ECS

regions, Dij is Di

l, Die;s and Di

eff , respectively.

The computational domain is limited to six faces

x ¼ 0; L; y ¼ 0;W ; z ¼ 0;Hð Þ where the appropriate

boundary conditions should be imposed. Since all regions

are separated at x ¼ 0; L (Fig. 1g), the boundary conditions

can be used in the boundaries:

n � u ¼ 0 x ¼ 0; L ð17aÞ

(a) (b)

(c) (d)

Fig. 2 a Schematic 3D grid

topology of computational

domain, b effect of grid

distribution on oxygen

concentration profile in diagonal

direction for the two different

cross sections, c a grid topology

of the cross section, d oxygen

concentration along the axial

direction

74 J Artif Organs (2014) 17:69–80

123

K � K � nð Þn ¼ 0; K ¼ l ruþ ruð ÞT� �� �

n

n � quxi � qDijrxi þ qxiD

ij

rMn

Mn

� �¼ 0 x ¼ 0; L ð17bÞ

The face at y ¼ 0;W is the porous scaffold and ECS

interface. A symmetry condition was applied for the

momentum equations:

n � u ¼ 0 y ¼ 0;W ð18Þ

K � K � nð Þn ¼ 0; K ¼ l ruþ ruð ÞT� �� �

n

whereas the species consumption and production only

occurred in the ECS region; therefore, the symmetric

condition was not valid for these boundaries. It is clear

from Fig. 1f that the computational domain is repeated at

the top y ¼ Wð Þ and bottom y ¼ 0ð Þ regularly, hence the

periodic condition was imposed in the boundaries.

xi;y¼0 ¼ xi;y¼W y ¼ 0;W ð19Þ

Two other boundaries are in the inlet and outlet of the

bioreactor. These surfaces divide into three distinct sub-

boundaries as inlet and outlet as indicated in Fig. 1g: (1)

inlet/outlet section of the top lumen, (2) inlet/outlet section

of the bottom lumen and (3) up-/downstream section of the

scaffold and ECS.

Inlet z ¼ 0ð Þ condition for the top and bottom sub-

boundaries assumed to be fully developed flow, which can

be expressed as:

u x; yð Þ ¼ u0 1� x

R

� �2

� y� t

R

� �2� �

z ¼ 0;x

R

� �2

þ y� t

R

� �2

� 1 ð20aÞ

uðx; yÞ ¼ u0 1� x� L

R

� �2

� y� W=2þ tð ÞR

� �2 !

z ¼ 0;x� L

R

� �2

þ y� W=2þ tð ÞR

� �2

� 1 ð20bÞ

For both inlet lumens, it was assumed that the nutrient

concentration is uniform at the inlet with a value of Ci;in:

Ci;0 ¼ Ci;in i ¼ 1; 2ð Þ z ¼ 0 ð20cÞ

and the boundary conditions at the outlet z ¼ Hð Þ of the

lumen regions were:

p ¼ pout z ¼ H ð21aÞ

n � qDijrxi

� �¼ 0 z ¼ H ð21bÞ

The outlet gauge pressure pout was set equal to 0.

A no-slip and a no-flux condition were used for the inlet

and outlet of the third subsection as follows:

u ¼ 0 z ¼ 0;H ð21cÞ

n � quxi�qDijrxiþqxiD

ij

rMn

Mn

� �¼ 0 z¼ 0;H ð21dÞ

Grid dependency test

The computational domain is shown in Fig. 1c, g. The sets

of algebraic and partial differential equations subjected to

appropriate boundary conditions were solved based on the

finite element method. Several different grids were tested

for finding the most suitable grid; finally, the whole domain

in the bioreactor was divided into the discrete hexagonal

sub-domains as shown in Fig. 2a, b. Because of a high

concentration gradient in the x� y directions, the maxi-

mum size of a single mesh element was set to 0.11 mm and

the size of the element in the z direction was set to

0.28 mm. The results at a maximal cell density and Rey-

nolds number in two directions of the scaffold are shown in

Fig. 2c, d. The model converges when the weighted rela-

tive residual norm was less than 10�6. Under such condi-

tions, with a further increase in the number of mesh

elements ð190000Þ, no significant difference between the

results was found.

Results

Effect of the bioreactor length, hydrodynamic behavior,

and cell density

It is seen in Fig. 3a that nutrient concentration decreases by

increasing the length of the bioreactor. Since the cells are

only seeded in the ECS region, nutrient consumption only

takes place in this zone. Figure 3a shows that the oxygen

uptake rate is more than the rate of glucose consumption;

therefore, oxygen plays a major role in influencing the cell

growth. This result was confirmed by previous studies [7, 9].

Generally, the regime of fluid flow in the lumen region

of the HMShB is laminar. A maximum Reynolds number

of 10:9 was applied based on inlet velocity and the domain

dimensions presented in Table 1. Figure 3b shows a con-

tour plot of the oxygen concentration in the cross section at

z ¼ H. The lowest (blue) and highest (red) levels of oxygen

occur in the center of the ECS and lumen region. A critical

line x ¼ 0ð Þ was selected to discuss the effect of Reynolds

number and cell density on oxygen distribution (Fig. 3c).

By increasing the cell density, the oxygen uptake rate is

J Artif Organs (2014) 17:69–80 75

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increased, hence the culture medium maintains at a low

level of nutrient concentration. Therefore, we find that the

relative minimum and maximum of the curves in Fig. 3c

refer to the center of the gel-cell matrix and lumen region,

respectively.

Appropriate surface area and enough residence time are

two important parameters that affect the mass flux.

Decreasing Re increases residence time, hence mass

transfer takes place resulting in a decrease of oxygen

concentration in the lumen. For the case of high cell

(a)

(b)

(c)

×

Fig. 3 a Nutrient concentration profile in two arbitrary lines of the

bioreactor i = Oxygen, Glucose (in), Lactate (out), qcell ¼ 150�1012cells m�3; Re ¼ 10:9, L ¼ 1:1 mm and t ¼ 0:75 mm, b oxygen

distribution in cross section (z = 40) of the bioreactor for

qcell ¼ 150� 1012 cells m�3; Re ¼ 10:9, c effect of Reynolds num-

ber and cell density on oxygen profile

76 J Artif Organs (2014) 17:69–80

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density, increasing of the flow rate has a noticeable effect

on the oxygen distribution at the first stage, while further

increasing the flow rate will not improve the nutrient

transport significantly. This result indicates that for a high

Reynolds number ðRe ¼ 10:9Þ, mass transfer is controlled

by the diffusion resistance in the porous matrixes, and for a

low Reynolds number ðRe ¼ 2:72Þ the radial diffusion in

the lumen is a determinant parameter. It shows that the

nutrients are sufficient in the entire domain under the

applied operating conditions. Therefore, the reaction rate

controls the mass transport in the regions. The lowest curve

in Fig. 3c relates to a maximum cell density

(q ¼ 150� 1012cells m�3) and a minimum Reynolds

number ðRe ¼ 2:72Þ. Hypoxia for the bone cells has been

evaluated around 0:02 mol m�3 by Salim et al. [18], which

can affect the long-term metabolism of cells such as in

extracellular matrix production. With respect to Fig. 3c,

the scaffold not only provides an appropriate bed for

growing bone cells, but may be designed for engineering of

large-scale bone tissue.

Effect of the physical properties

In general, for the scaffolds without cells, the porosity may

vary from 0:6�0:95 [19] and the permeability of the

scaffold is in the range of 10�12�10�9m2 [20]. During the

process of cell culture, the porosity may drop from 0:97 to

0:73 and the permeability may decrease to 10 % of the

initial value as well [21]. A high Reynolds number was set

in the lumen region of the bioreactor to be able to observe

the effect of scaffold parameters on the oxygen

distribution.

Four porosity values (0.6, 0.7, 0.8, and 0.9) of the

scaffold have been tested at a constant permeability of

10-10 m2. According to Fig. 4a, the oxygen concentration

increases with increased porosity of the scaffold due to an

(a)

(c) (d)

(b)

Fig. 4 a Effect of the porosity of scaffold on oxygen concentration

profile, b effect of the permeability of scaffold on oxygen concen-

tration profile, c oxygen concentration for different values of

geometric parameters of the scaffold: inter-lumen spacing ðwÞ and

lengthðHÞ, d oxygen concentration for the radius of the lumen ðRÞ and

thickness ðtÞ of the scaffold

J Artif Organs (2014) 17:69–80 77

123

enhancement of the effective diffusion coefficient. In

contrast, it is not affected by those parameters in the lumen

region.

According to the Fig. 4b the oxygen concentration

increases when the scaffold permeability increases at a

cs = 0.8. But the enhancements are not as significant as the

effect of porosity, because the permeability of the scaffold

is indicated only in the momentum equation; therefore, its

effect is not significant.

Effects of the geometric parameters

Inter-lumen spacing wð Þ is an important design parameter

that determines the size of the engineered bone tissue. The

effect of the lumen spacing of the scaffold on the cell

growth behavior accounts for the interstitial space between

the lumens’ walls. In addition, the length Hð Þ of the bio-

reactor may affect the nutrient transport in HMShB. Nine

pairs of inter-lumen spacing and bioreactor length values

have been chosen. The gel-cell matrix is placed between

the inter-lumen spacing, thereby w widening the gel-cell

matrix. It is evident that by increasing the length of bio-

reactor, lower nutrient concentration occurs. Hence, for

engineering of long-bone tissue, decreasing the distance

between the lumens is preferred.

In Fig. 4c the lowest curve pertains to a maximum of the

two parameters w ¼ 2:4; H ¼ 50 mmð Þ. According to the

figure the value of relative minimum in the curve is 0:57,

that is fivefold more than hypoxia for cells 0:02 mol m�3ð Þ[18].

Figure 4d shows oxygen concentration for two thick-

nesses of the lumen wall and three different inner lumen

radii. The six studies were carried out for the same Rey-

nolds number Re ¼ 10:9ð Þ. Decreasing the thickness of the

lumen wall leads to decreased mass-transport resistance,

hence the maximum nutrient concentration is referred to

the smallest thickness t ¼ 0:65 mmð Þ and the largest radius

of lumen R ¼ 0:6 mmð Þ. Figure 4d indicates that the rela-

tive maximum of the oxygen distribution in different

conditions is dislocated due to variation of t and w. How-

ever, the result of the model demonstrates that the fabri-

cated scaffold w ¼ 2:2; t ¼ 0:75; R ¼ 0:5; z ¼ 40ðmmÞð Þcan provide an appropriate bed for growing of the cells/

tissue; therefore, decreasing of the lumen wall thickness is

not necessary and not advised, because a thin lumen wall

scaffold loses its mechanical strength.

Distribution of the species concentration

Figure 5 displays the species concentration in the cross

sections of the bioreactor for oxygen, glucose and lactate,

respectively. Figure 5a, b shows that the maximum

concentration of nutrients is in the lumen regions. The

glucose concentration has decreased less than that of the

oxygen due to low uptake rate. Therefore, oxygen con-

centration can be an important parameter for the cell pro-

liferation when the scaffold size increases. As it can be

seen from the Fig. 5c, the lactate produced by cells diffuses

from the gel-cell matrix region to the lumen through the

lumen wall then is washed out by the fluid flow and is

removed from the scaffold region; otherwise, the decrease

in pH associated with lactate accumulation significantly

can inhibit both cell proliferation and metabolism in the

region.

Discussion

The experimental strategy in our previous work [2] and

mathematical modeling outlined in this study enable these

techniques to guide HMShB design based on the oxygen

and glucose requirements and lactate production of the cell

population. A rigorous modeling approach was developed

and employed to provide design and operating data that

ensure the oxygen and glucose concentration throughout

the HMShB are held above a prescribed bone cell-specific

minimum value that ensures the growth of a functional

bone tissue construct and eliminates the hypoxic regions in

the bioreactor.

Highest oxygen concentrations in the ECS region are

located near the border of ECS and scaffold. In regions

with limited culture media velocity (scaffold and ECS),

nutrient supply is mostly dependent on diffusion, which

causes a large nutrient gradient.

In contrast, the oxygen concentration in the lumen

regions for a constant cell density almost remains constant

along the scaffold axis, so the zone around the lumens

which is oxygenated has a constant radius throughout the

entire simulation domain. Meanwhile, by increasing the

cell density or decreasing the Reynolds number (fluid

flow rate), the oxygen concentration decreases in the

lumens, which results in a significant oxygen decrease in

the ECS.

Concerning the oxygen transport, which is recognized as

a limiting nutrient, we can conclude that the oxygen supply

in the lumen is fully adequate to provide all bone cells with

sufficient oxygen. However, as shown by Komarova et al.

[22], glucose plays an essential role in the energy genera-

tion of mature osteoblasts. The results of this study dem-

onstrate that under controlled conditions, the glucose

concentration was available to maintain bone cell growth

throughout the ECS region.

It is expected that the study described in this paper will

help to enhance our understanding of the design and

developmental requirements of more effective HMSh

78 J Artif Organs (2014) 17:69–80

123

based bioreactors, and thus to bring practical bone tissue

generation a step closer to clinical usage.

Conclusion

The present work reports a modeling framework to char-

acterize species transport in a novel HMShB for the bone

tissue growth. An approach based on a 3D mathematical

model was proposed and numerically solved with the

appropriate boundary conditions for the equations of

momentum and mass conservation. This model involves

cell resistance and metabolite production (lactate) as well

as porous matrix resistance and nutrient consumption,

respectively. The effects of geometric parameters, physical

properties of the scaffold and process parameters on the

z/L=0.1

x0.0 0.2 0.4 0.6 0.8 1.0

y

0.0

0.5

1.0

1.5

2.0

2.5

z/L=0.5

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

z/L=0.9

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

x0.0 0.2 0.4 0.6 0.8 1.0

y

0.0

0.5

1.0

1.5

2.0

2.5

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

x0.0 0.2 0.4 0.6 0.8 1.0

y

0.0

0.5

1.0

1.5

2.0

2.5

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

x0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

(a)

(b)

(c)

Fig. 5 Concentration profile of a oxygen, b glucose, c lactate in three sections of the bioreactor, qcell ¼ 150� 1012cells m�3; Re ¼ 10:9

J Artif Organs (2014) 17:69–80 79

123

nutrient distribution in different cell densities were studied.

Among different biochemical and mechanical stimuli, the

oxygen transport plays a major role in influencing the

bioreactor performance. The results illustrated that the

scaffold not only satisfies our goal but that it can be used as

a large-scale scaffold in the bioreactor for accommodating

a high cell density tissue. This re-emphasizes the fact that

the HMShB has indeed the potential to produce large-scale

3D bone tissue constructs for clinical applications.

The model can be further developed by considering

time-dependent biological phenomena such as cell growth,

death, and migration in addition to the degradation rate of

the biodegradable scaffold material during cell/tissue

cultivation.

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