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A RTICLE
Flow Dynamics in Bioreactors Containing Tissue
Engineering Scaffolds
Benjamin J. Lawrence, Mamatha Devarapalli, Sundararajan V. Madihally
School of Chemical Engineering, Oklahoma State University, 423 Engineering North, Stillwater, Oklahoma 74078; telephone: 405-744-9115;
fax: 405-744-6338; e-mail: [email protected]
Received 10 April 2008; revision received 10 June 2008; accepted 22 July 2008 Published online 29 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/bit.
ABSTRACT: Bioreactors are widely used in tissue engineer- ing as a way to distribute nutrients within porous materials and provide physical stimulus required by many tissues. However, the fluid dynamics within the large porous struc- ture are not well understood. In this study, we explored the effect of reactor geometry by using rectangular and circular reactors with three different inlet and outlet patterns. Geometries were simulated with and without the porous structure using the computational fluid dynamics software Comsol Multiphysics 3.4 and/or ANSYS CFX 11 respec- tively. Residence time distribution analysis using a step change of a tracer within the reactor revealed non-ideal fluid distribution characteristics within the reactors. The Brinkman equation was used to model the permeability characteristics with in the chitosan porous structure. Pore size was varied from 10 to 200 mm and the number of pores per unit area was varied from 15 to 1,500 pores/mm 2. Effect of cellular growth and tissue remodeling on flow distribu- tion was also assessed by changing the pore size (85–10 mm) while keeping the number of pores per unit area constant. These results showed significant increase in pressure with reduction in pore size, which could limit the fluid flow and nutrient transport. However, measured pressure drop was marginally higher than the simulation results. Maximum shear stress was similar in both reactors and ranged 0.2– 0.3 dynes/cm 2. The simulations were validated experimen- tally using both a rectangular and circular bioreactor, con- structed in-house. Porous structures for the experiments were formed using 0.5% chitosan solution freeze-dried at 808 C, and the pressure drop across the reactor was monitored. Biotechnol. Bioeng. 2009;102: 935–947. ß 2008 Wiley Periodicals, Inc. KEYWORDS: bioreactor; tissue regeneration; residence time; shear stress; pore size; pore number
Introduction
Bioreactors have been widely utilized for regenerating tissues in vitro. Maturation in reactors ensures sufficient nutrient transport through cell seeded porous scaffolds (Chen and Hu, 2006; Martin et al., 2004). Some cell types respond to mechanical stress and change the tissue structure and composition to meet the functional demands. Hence, fluid flow is also utilized as a way to introduce mechanical stresses on to cells (Martin and Vermette, 2005). Using bioreactors, an improvement in the quality of the regenerated tissue is observed in some studies (Niklason et al., 1999). However, other studies show deterioration in the quality of the tissue (Heydarkhan-Hagvall et al., 2006). Flow of growth medium improves nutrient and waste transport, but alterations in the synthesis of matrix elements induced by shear stresses could deteriorate the tissues (Gooch et al., 2001). Additionally, shear stress affects the scaffold architecture as well as cellular alignment within the structures (Gray et al., 1988; Huang et al., 2005). To better understand the effect of fluid flow during tissue regeneration, a number of studies using computational fluid dynamics (CFD) have been performed (Brown and Meenan, 2007; Cioffi et al., 2006; Hutmacher and Singh, 2008; Porter et al., 2005; Sander and Nauman, 2003; Williams et al., 2002). The majority of these studies assess the flow patterns and shear stresses either within the bioreactor or around the porous construct (Bilgen and Barabino, 2007; Williams et al., 2002). Additionally, some studies do not account for porosity or transport through the scaffold (Timmins et al., 2007; Williams et al., 2002). Most studies utilize small-scale cylindrical or discoid scaffolds, focusing on wall shear stresses (Stephens et al., 2007). Flow characteristics are analyzed using either Darcy’s equation (Boschetti et al., 2006), or the Brinkman equation which is considered as an extension of Darcy’s equation (Chung et al., 2007). The Brinkman equation accounts for both viscous and drag forces in the porous medium and reduces to either the Navier–Stokes equation or Darcy’s law if either force becomes dominant (Capuani et al., 2003). However, porous
B.J. Lawrence and M. Devarapalli contributed equally to this study. Correspondence to: S.V. Madihally Contract grant sponsor: Oklahoma Center for Advancement of Science and Techno- logy Contract grant number: HR05- Contract grant sponsor: National Institutes of Health Contract grant number: 1R21DK074858-01A
ß 2008 Wiley Periodicals, Inc. Biotechnology and Bioengineering, Vol. 102, No. 3, February 15, 2009 935
characteristics change during tissue regeneration, that is, permeability of the matrix decreases due to decreased pore size. Further, many tissues (e.g., cartilage and bladder) have a high aspect ratio (large surface area relative to the thickness of the matrix). Flow within bioreactors containing large porous struc- tures with high aspect ratios has not been studied. In these scale-up systems, non-ideal fluid distribution conditions could arise from two possible factors: channeling, and dead zones. When channeling is present, some of the fluid leaves the reactor immediately without dispersing within the reactor. Dead zones reduce the effective volume of the reactor and prolong the residence time of some of the fluid elements. Non-uniform flow patterns lead to (i) poor distribution of nutrients and (ii) non-uniform shear stress distribution. These factors affect cellular colonization and the assembly of extracellular matrix elements which affect the quality of regenerated tissues. Hence, one has to understand the fluid distribution and the effect of shape of the reactor. The non-ideal fluid distribution is characterized using the residence time distribution (RTD). The RTD measures the amount of time different molecules present in the fluid spend within the reactor (Fogler, 2006; Lawrence et al., 2004). In other words, RTD is a measure of dispersal of a molecule in a flowing medium owing to the combined action of a velocity profile and molecular diffusion. Understanding these characteristics is important in design- ing the reactor shape and optimizing the location of the inlets and outlets. This study utilized a well understood parallel plate reactor (Huang et al., 2005; Macario et al., 2007) and evaluated the effect of porous structure in the flow domain. Further, a circular shape reactor with the same configuration was also studied to understand the effect of shape of the reactor on fluid flow distribution with and without the porous structure. Each reactor with three different inlet and outlet conditions was simulated using CFD packages CFX 11 (ANSYS, Inc., Canonsburg, PA) and/or Comsol Multi- physics 3.4 (COMSOL, Inc., Burlington, MA). Further experiments were performed to validate the simulation results utilizing both a rectangular and circular reactors. These results show significant increase in pressure drop when the porous characteristics are modified.
Materials and Methods
Sources of Materials
Chitosan with >310 kDa MW and 85% degree of deacety- lation, and glacial acetic acid were obtained from Sigma Aldrich Chemical Co. (St. Louis, MO). Ethanol (200 proof) was obtained from Aaper Alcohol and Chemical Company (Shelbyville, KY). All other reagents were purchased from Fisher Scientific (Waltham, MA).
Obtaining RTD Function Using CFD Two reactor geometries were used: (i) rectangular (8 cm long, 2.5 cm wide, and 0.2 cm high, with a 0.1 cm inlet and outlet diameter) and (ii) circular (10 cm diameter and 0.2 cm high, with a 0.6 cm inlet and outlet diameter). For each shape, three inlet and outlet configurations (Fig. 1) were chosen to understand the effect of inlet and outlet locations and design on flow patterns.
Rectangular Reactor Case 1: with abrupt inlet and outlet to demonstrate non- ideal mixing. Case 2: with long entrance and exit section to facilitate fully developed laminar flow. Case 3: with inlet and outlet from the top. This configuration is commonly used in a variety of cell studies such as shear effect on endothelial cell, neutrophile sequestration, and long-term bone marrow cultures (Huang et al., 2005; Tilles et al., 2001).
Circular Reactor Case 4: with inlet at the center and outlet at the periphery on the opposite side. Case 5: with inlet and outlet at the periphery and outlet on the periphery, but on the opposite sides. Case 6: was similar to case 5 except both the inlet and outlet were on the same side.
These reactor geometries were created using a CAD package (Solidworks TM^ or ANSYS Workbench 11). The CFX mesh was then created using ANSYS CAD2Mesh software. A critical challenge was overcoming problems associated with the aspect ratio, that is, very large surface area relative to the thickness of the channel. To ensure at least 10 nodes over the thickness of the reactor, the maximum element size of 0.2 mm was chosen. Preliminary simulations used a water mass flow rate that corresponds to the average wall shear stress present in human blood vessels, 2.6 g/s (156 mL/min) (Huang et al., 2005). Flow rates of 0.33 g/s (20 mL/min) and 0.0825 g/s (5 mL/min) were used for later simulations and experi- mental validation because higher flow rates compressed the porous structure. The simulation was run at steady state in order to visualize flow distributions and to generate the initial velocity profile for transient simulations. The outlet was set at atmospheric pressure and the walls were smooth with no slip condition. RTDs were obtained by introducing a tracer into the reactor feed starting at time equal to zero seconds. The tracer for the simulations was created as a duplicate fluid based on water and then renamed as tracer. This created a tracer with the same physical properties as that of water. The tracer was introduced as a step change in the concentration
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inlet and outlet diameter) was fabricated in-house from polycarbonate as shown in Figure 2A and B. The effective length and width of the reactor was based on the size of a standard microscope slide, and the height was based on the thickness of a porous matrix commonly used in our laboratory (Lawrence et al., 2007). The reactor’s height and hexagonal flow field were maintained using a 2 mm thick Teflon gasket. Likewise, a circular reactor similar to case 6 (diameter of 10 cm, a height of 2 mm, and a 0.6 cm inlet and outlet) was fabricated in-house from polycarbonate as shown in Figure 2C and D. The reactors were attached to a flow system (Fig. 2E) consisting of two fluid reservoirs, a variable speed Masterflex L/S peristaltic pump (Cole-Parmer, Vernon Hills, IL), an in-line physiological pressure transducer (Capto SP844, Capto, Skoppum, Norway), the bioreactor, and two waste reservoirs. The pressure transducer was connected to a computer via Powerlab/4SP System (ADI Instruments, Inc., Colorado Springs, CO) and data were acquired through Chart TM^ 5 for Windows. The tubing system for the reactor was designed in order to produce a step change in the reactor’s inlet concentration. Two separate tubes were run through a peristaltic pump, one from a water reservoir and another from a reservoir filled with dye tracer solution. These tubes each led to a three way stopcock (V1 and V3) that would send the fluid either towards the reactor or into a
waste reservoir. The two sides were connected at a fully open four way stopcock (V2) which functioned as a T-joint, allowing dye solution or water to enter the reactor. Initially the flow loop was primed with the valve positions (V1 to dye waste and V3 to reactor) set so as the water traveled through the reactor while the dye solution was send to a waste container. Water was run through the system until steady state was obtained. At time zero, V1 and V3 were changed simultaneously (V1 to reactor and V3 to water waste) so the dye solution traveled through the reactor and water was sent to the waste container. Fractions were collected in a 24-well plate at the reactor outlet over regular intervals. Next 100 mL of each fraction was transferred to a 96-well plate and the absorbance was measured at 490 nm; initial spectral scan over visible spectra showed that the maximum absorbance was at 490 nm wavelength. The relative concentration (C(t)/C 0 ) was calculated from the relative absorbance (A/A 0 )
CðtÞ C 0
AðtÞ A 0
where A 0 is the initial concentration of the dye solution. The RTD function (E(t)) was calculated using Equation (3) as before. Since 20 mL/min flow rate compressed the porous scaffold structure, 5 mL/min was used. For the rectangular
Figure 2. Reactor set-up used in experimentation. A: Top view of the rectangular reactor. B: Side view of the rectangular reactor. C: Top view of the circular reactor. D: Side view of the circular reactor. E: Flow-loop used for RTD analyses. [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]
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reactor 20 mL/min fractions were collected over 1 s intervals, and fractions for the 5 mL/s flow rate were collected over 5 s intervals. Since the volume of the circular reactor is larger than the rectangular reactor, samples (five drops) were collected every 5 s for 20 mL/min flow rate and at every 20 s for the 5 mL/s flow rate.
Analytical Model Derivation
The RTD function for an ideal completely developed laminar flow with no slip at the boundaries was derived as another method of verifying simulation data for rectangular reactor (Fogler, 2006). The reactor was modeled as a set of parallel plates with a width (w) and a height (h) (Truskey et al., 2004). Side and end effects were ignored with the assumption of completely developed laminar flow profile in the y-direction (direction of the reactor height). The velocity in the z-direction was calculated from the volumetric flow rate (see Appendix for detailed derivation of the equation).
V Z ¼
3 v 0 2 wh
y^2 h^2
The final external age distribution expression is
EðtÞ ¼ t^2 6 t^3
2 t 3 t
The minimum residence time was calculated from the maximum velocity equation
tmin ¼
L
Vmax
L
3 2 Vavg
2 Lwh 3 Vavgwh
2 t 3
EðtÞ ¼ 0 ; t <
2 t 3
EðtÞ ¼ t^2 6 t^3
2 t 3 t
; t 2 t 3
Equations (9) and (10) were used to generate E(t) curves for the ideal case.
Preparation of Porous Structure
Porous structures of dimensions appropriate for each reactor were generated by freezing 0.5% (wt/v) chitosan solution prepared in 0.1 M acetic acid at 808 C, then lyophilized overnight (Virtis, Gardiner, NY). The porosity and pore size distribution of these matrices in hydrated condition has been extensively characterized in our labo- ratory (Huang et al., 2005, 2006; Tillman et al., 2006). Scaffolds were stabilized in 100% ethanol and incubated in red-dye prior to use in experiments. The pore size
distribution and porous structure characteristics of the wet scaffolds were analyzed using light microscopy. Prior to viewing under light microscopy, chitosan structures were sectioned to view cross sectional architecture. Pore size and number of pores was determined from digital micrographs using Sigma Scan Pro software (Systat Software, Inc., Point Richmond, CA).
Simulation Including the Porous Structure Next, a porous medium was created within the reactor space to better understand the effect of porous structure on fluid flow, simulations were performed using COMSOL Multi- physics 3.4 (COMSOL, Inc.). Both the case 3 and case 6 geometries were created in COMSOL using configurations identical to described above. Fluid flow through the porous structures was modeled using the Brinkman equation
mr^2 us m k
us ¼ rp (11)
rus ¼ 0 (12)
where k is the permeability of the porous medium, us denotes the fluid superficial velocity vector, p is the fluid pressure, and m the effective viscosity in the porous medium (Truskey et al., 2004). Non-porous sections of the reactor were modeled as incompressible Navier–Stokes regions. The permeability (k) of the porous medium is a geometric characteristic of the porous structure at several length scales (Truskey et al., 2004). The permeability was calculated using 85 mm and 120 pores/mm 2 and the equation
k ¼ p 128
nAd^4 (13)
where nA is the number of pores per unit area and d is the pore diameter assuming the pores to be circular in shape. To understand the effect of pore size and number of pores, k values were calculated for (Tables I and II) different pore sizes and pore numbers. These values were used in the simulation. Shear stresses within the reactor were calculated using the shear stress tensor equation
t ¼ hðru þ ðruÞT^ Þ (14)
where h is the dynamic viscosity (Pa s) and u is the velocity in the open channel (m/s). The shear stress tensor is an integral part of the Navier–Stokes equations describing flow in a free channel at steady state, given by
rðurÞu ¼ r½t þ pdij (15)
ru ¼ 0 (16)
where r is the fluid’s density (kg/m 3 ), p is the pressure (Pa) and dij is the Kronecker delta function. Both the permeability
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nutrients. First, the changes in the tracer concentration across the outlet diameter were examined so that a method to determine Cmix (t) from the simulation results could be developed. For this purpose, tracer concentrations across the reactor outlet were plotted (Fig. 4). Examination of these results showed that for 1 mm outlet diameter, the tracer concentration is virtually linearly distributed over the cross section area. This suggested that for rectangular reactors approximating Cmix (t) using Equation (1) is valid. Similar profiles were obtained with 3 mm outlet diameter (date not
shown). However, increase in the outlet diameter to 6 mm showed non-linear change in tracer concentration except few initial and final time steps. Further, the concentrations were not uniform or symmetric. Since the concentration profile in the circular reactors were not linear, expressions for Cmix (t) were developed using multi-parameter regres- sion approach at each time step. Results obtained using these equations showed large errors with the RTD calculations and were not utilized further. Therefore, in circular reactors Cmix (t) was approximated using Equation (2).
Figure 3. Steady state stream line profiles in different reactors simulated with a flow rate of 20 mL/min. [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]
Figure 4. Concentration profiles at the reactor outlet for different times at the 20 mL/min flow rate. [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]
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The relative concentration (Cmix /C 0 ) was plotted against dimensionless time (Fig. 5A and C). In case 1 and case 4 the tracer exited the reactor much sooner than in other geometries. This premature exit of the tracer could be attributed to the dead volume or channeling. Next, E(t) was calculated for different time steps and plotted as a function of dimensionless time (Fig. 5B and D). All rectangular reactor peaks except case 1 aligned as expected at 2/3rd the ideal space time. Additionally, peak spreading was observed in case 1, case 2, and case 3 indicating dispersion of the tracer or flow non-idealities. Further, E(t) peak heights for case 1 and case 3 were higher than case 2. These observations suggest that there are dead-spaces within the reactor in case 1 and case 3. For case 4 the peak was higher and shifted left compared to case 5 and case 6 indicating an increase in dead volume and channeling. The bump in the E(t) curve in case 6 (Q 1.3) is further evidence for channeling. The simulation results for case 3 and case 6 were validated by experiment. The simulations and experiments had similar concentration and RTD profiles when plotted, showing that methodology is a valid approach to analyze flow properties in the reactor systems.
Effect of Porous Structure on Flow Distribution
This study utilized chitosan porous structures prepared by freeze drying at 808 C. Chitosan was selected because (i) it
is optically transparent, enabling visual monitoring of the flow distribution, (ii) the matrix will not degrade in the absence of enzymes, (iii) its mechanical properties are well described, and (iv) pore characteristics of the scaffold have been analyzed extensively (Huang et al., 2005, 2006; Tillman et al., 2006). Scaffolds formed by freezing at 808 C showed an average pore size of 85 mm ( 20 mm) and 120 pores/mm 2 , and a porosity of 85%. Porosity values were used to calculate the available reactor volume and space time. When RTD was examined for reactors containing the porous structure, results indicated (Fig. 6) that the tracer exited the reactor sooner relative to conditions without the porous structure. This could be attributed to the increased resistance to flow caused by the presence of the porous structure which could favor channeling. When fluid entered the system, the hydrodynamic forces deformed the polymer scaffold, creating a shorter path to the outlet of the reactor. This deformation was especially prevalent in the circular reactors where the inlets are located directly above the porous matrix. While performing experiments, it was observed that the scaffolds were compressed at the inlet and the flow traveled over the top of the matrix and along the walls without significantly infiltrating into the porous structure. This bypass effect was minimized in the rectangular reactor by providing space for entry and exit effects, thereby ensuring that the flow entered the porous structure instead of bypassing it.
C/C
0
E(t/
τ )
0
1
2
3
4
5
6 Case 1 (20mL/min) Case 2 (20mL/min) Case 3 (20mL/min) Theoretical (20mL/min) Experimental (20mL/min)
t/τ
t/τ
0.0 0.5 1.0 1.5 2.0 2.
E(t/
) τ
0.0 0.5 1.0 1.5 2.0 2.
C/C
0
Case 4 (20mL/min) Case 5 (20mL/min) Case 6 (20mL/min) Experimental (20mL/min)
A B
C D
E(
Θ )
E(
Θ )
Figure 5. Effect of reactor shape on the residence time distribution of the tracer at 20 mL/min. A: Concentration of the tracer at the outlet of rectangular reactor. B: E-curve for rectangular reactor. C: Concentration of the tracer at the outlet of circular reactor. D: E-curve for circular reactor. [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]
942 Biotechnology and Bioengineering, Vol. 102, No. 3, February 15, 2009
Figure 8. Shear stress distribution within the porous structure for the case with 85 mm pore size and 120 pores/mm^2. Arrows indicate the flow direction. Below the reactor are the histogram profiles for each condition. [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]
Figure 7. Effect of pore size and pore number on the pressure drop and maximum shear stress in the reactor. Square symbol represents the rectangular reactor and circular symbol represents the circular reactor. A: Effect of changing pore size with pore number on pressure drop. Gray boxes correspond to the experimental results. B: Effect of changing pore size with pore number on maximum shear stress. C: Effect of changing pore size with 120 pores/mm 2 on pressure drop. D: Effect of changing pore size with 120 pores/mm^2 on maximum shear stress. [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]
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Interestingly, shear stresses in both x and y directions were higher than that in the rectangular reactors. The maximum shear stress observed in both x and y directions were around 0.237 and 0.271 dynes/cm 2 respectively.
Effect of Changing Porosity
During the course of tissue remodeling, cells will proliferate and de novo synthesize extracellular matrix elements which are deposited in the porous structures. These processes re- duce the pore space available for fluid flow. Hence the pore size decreases but the number of pores per area does not. To understand the implications of these dynamic changes, simulations were carried out with six decreasing pore sizes (Table II) but using the number of pores per unit area that was determined experimentally (120 pores/mm 2 ). These results showed (Fig. 7C) significant increase in the pressure drop across the bed with decreased pore size. Increase in pressure drop was significantly higher in rectangular reactor relative to circular reactor, particularly at very low pore sizes. The shear stresses increased in a nonlinear manner as the pore size decreased. Similar changes were also observed in circular reactors. Further, shear stresses in x and y directions were consistently higher than rectangular reactors.
Discussion
In this study, simulations and experimental validation were performed with and without porous structures to under- stand the fluid flow distribution within the bioreactor. The rectangular reactor used in this study has been widely utilized in various studies including long-term bone marrow cultures (Koller et al., 1993), athereosclerosis and endothe- lial function (Huang et al., 2005), homing and rolling of immune cells and liver-assist bioreactor (Kim et al., 2000; Shito et al., 2001). The rectangular reactor was scaled up to circular shape to minimize the dead volumes at the corners. The reactor systems used have distinct advantages for regenerating large clinically transplantable tissues such as cartilage, bladder and skin; scaffolds can have very large surface area to volume ratios with very small thickness. Another popular bioreactor configuration involves growing cell seeded constructs in microgravity (Marolt et al., 2006). However, for growing large aspect ratio tissues, microgravity reactor would deform the structure due to bending and rolling while tumbling through the growth medium, creat- ing a random stress pattern within the developing material. Hence, microgravity reactors may not be suitable for regenerating thin cross-sectional tissues. The parallel plate reactors provide stability and support for the porous structure and developing tissue. Unlike in rectangular reactor, the circular curvature induced secondary circulations in the flow field. The RTD analyses showed increased non-ideal fluid distribution pattern in the circular reactor. Presence of porous structure further increased channeling effects in the circular reactors. The RTD analysis was based on the average concentration
profile, which is valid for smaller outlet size. However, one has to consider the nonlinear concentration profile while averaging across larger outlets. Further, the experimental results showed compression of the porous structure at 20 mL/min flow rates. The compression can be attributed to the mechanical properties of the chitosan porous structure (elastic modulus is 2–6 kPa). To minimize the compressive damage on the porous structure, flow rate was reduced to 5 mL/min. However, in the simulation results, these compressive changes were not considered and the porous medium was treated as a rigid matrix. Hence, further studies in which the elastic properties of the porous structure are coupled to the fluid force are necessary to better understand the fluid distribution. Understanding pressure drop changes is important to determine the pressure to be maintained during tissue regeneration (Sodian et al., 2000). To mimic tissue regene- ration process, the number of pores was kept constant while decreasing the pore size. The exponential increase in pressure drop as pore sizes shrink indicates that liquid flow through a porous scaffold will decrease and possibly stop unless flow system is adjusted to provide large pressure gradients. The nonlinear properties of both shear stress distributions and pressure drop suggest that there is room for optimization of the porous structure. Others have reported the effect of changing porous structure due to cell growth on the growth media distribution in cylindrical scaffolds (Chung et al., 2007). These studies used porosity based permeability values, which could change by either de- creased number of pores or decreased the pore size. Hence, it may not reflect the true regeneration characteristics. While the scaffold is subjected to the bulk forces supplied by the tissue and fluid flow, the cells experience the micromechanical properties of the individual fibers and local shear stresses within the porous structure. Flow within the reactor determines the orientation of cells within the material. Shear stresses generated by the fluid motion influence alignment of endothelial cells (Huang et al., 2005) and chondrocytes (Gray et al., 1988) grown in the parallel plate reactor. Alignment of cells and the de novo synthesized matrix elements determine the ultimate quality of the regenerated tissue. We evaluated the shear stresses within the porous structure through simulations. Shear stress in the porous structures are less compared to the estimated wall shear stresses (Gutierrez and Crumpler, 2008). The shear stress in the porous structure is 0 – 0.05 dynes/cm 2 in the rectangular reactor and are comparatively much less in the circular reactor (0.01 to 0.01 dynes/cm 2 ). Though maximum shear stresses were higher in the circular reactor, these stresses were near the inlet and the outlet where the porous structure gets compressed because of the flow. A microgravity reactor rotating at 10 rpm has an average wall shear stress of 0.2 dyne/cm 2 and a maximum shear stress of 0.3 dyne/cm 2 (Chen and Hu, 2006), comparable to parallel plate bioreactors. However, the tumbling free fall motion of the scaffolds produces a non-uniform stress distribution. Another method uses mechanical action to stretch or
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Finally the minimum residence time is calculated from the maximum velocity.
tmin ¼
L
Vmax
L
3 2 Vavg
2 Lwh 3 Vavgwh
2 t 3
(A.14)
EðtÞ ¼ 0 ; t <
2 t 3
(A.15)
EðtÞ ¼
t^2 6 t^3
2 t 3 t
; t
2 t 3
(A.16)
Equations (A.15) and (A.16) were used to generate E(t) curves for the ideal conditions.
References
Altman GH, Lu HH, Horan RL, Calabro T, Ryder D, Kaplan DL, Stark P, Martin I, Richmond JC, Vunjak-Novakovic G. 2002. Advanced bio- reactor with controlled application of multi-dimensional strain for tissue engineering. J Biomech Eng 124(6):742–749. Arrigoni C, Chitto A, Mantero S, Remuzzi A. 2008. Rotating versus perfusion bioreactor for the culture of engineered vascular constructs based on hyaluronic acid. Biotechnol Bioeng 100(5):988–997. Bilgen B, Barabino GA. 2007. Location of scaffolds in bioreactors modulates the hydrodynamic environment experienced by engineered tissues. Biotechnol Bioeng 98(1):282–294. Boschetti F, Raimondi MT, Migliavacca F, Dubini G. 2006. Prediction of the micro-fluid dynamic environment imposed to three-dimensional engineered cell systems in bioreactors. J Biomech 39(3):418–425. Brown A, Meenan BJ. 2007. Investigating the effects of fluid shear forces on cellular responses to profiled surfaces in-vitro: A computational and experimental investigation. Conf Proc IEEE Eng Med Biol Soc 5387–
- DOI: 10.1109/IEMBS.2007.4353560. Capuani F, Frenkel D, Lowe CP. 2003. Velocity fluctuations and dispersion in a simple porous medium. Phys Rev E Stat Nonlin Soft Matter Phys 67(5 Pt 2):056306. Chen H-C, Hu Y-C. 2006. Bioreactors for tissue engineering. Biotechnol Lett 28(18):1415–1423. Chung CA, Chen CW, Chen CP, Tseng CS. 2007. Enhancement of cell growth in tissue-engineering constructs under direct perfusion: Mod- eling and simulation. Biotechnol Bioeng 97(6):1603–1616. Cioffi M, Boschetti F, Raimondi MT, Dubini G. 2006. Modeling evaluation of the fluid-dynamic microenvironment in tissue-engineered con- structs: A micro-CT based model. Biotechnol Bioeng 93(3):500–510. Fogler HS. 2006. Elements of chemical reactor engineering. Upper Saddle River, NJ: Prentice Hall. Gooch KJ, Kwon JH, Blunk T, Langer R, Freed L, Vunjak-Novakovic EG.
- Effects of mixing intensity on tissue-engineered cartilage. Bio- technol Bioeng 72(4):402–407. Grashow J, Yoganathan A, Sacks M. 2006. Biaixal stress–stretch behavior of the mitral valve anterior leaflet at physiologic strain rates. Ann Biomed Eng 34(2):315–325. Gray ML, Pizzanelli AM, Grodzinsky AJ, Lee RC. 1988. Mechanical and physicochemical determinants of the chondrocyte biosynthetic response. J Orthop Res 6(6):777–792. Gutierrez RA, Crumpler ET. 2008. Potential effect of geometry on wall shear stress distribution across scaffold surfaces. Ann Biomed Eng 36(1):77–85. Heydarkhan-Hagvall S, Esguerra M, Helenius G, Sapderberg R, Johansson BR, Risberg B. 2006. Production of extracellular matrix components in tissue-engineered blood vessels. Tissue Eng 12(4):831–842.
Huang Y, Onyeri S, Siewe M, Moshfeghian A, Madihally SV. 2005. In vitro characterization of chitosan-gelatin scaffolds for tissue engineering. Biomaterials 26(36):7616–7627. Huang Y, Siewe M, Madihally SV. 2006. Effect of spatial architecture on cellular colonization. Biotechnol Bioeng 93(1):64–75. Hutmacher DW, Singh H. 2008. Computational fluid dynamics for improved bioreactor design and 3D culture. Trends Biotechnol 26(4): 166 – 172. Kim BS, Nikolovski J, Bonadio J, Mooney DJ. 1999. Cyclic mechanical strain regulates the development of engineered smooth muscle tissue. Nat Biotechnol 17(10):979–983. Kim SS, Sundback CA, Kaihara S, Benvenuto MS, Kim BS, Mooney DJ, Vacanti JP. 2000. Dynamic seeding and in vitro culture of hepatocytes in a flow perfusion system. Tissue Eng 6(1):39–44. Koller MR, Emerson SG, Palsson BO. 1993. Large-scale expansion of human stem and progenitor cells from bone marrow mononuclear cells in continuous perfusion cultures. Blood 82(2):378–384. Lawrence B, Beene J, Madihally S, Lewis R. 2004. Incorporating nonideal reactors in a junior-level course using computational fluid dynamics (CFD). Chem Eng Educ 38(2):136–141. Lawrence B, Maase E, Lin H-K, Madihally S. 2007. Multilayer composite scaffolds with properties similar to small intestinal submucosa. J Biomed Mater Res Part A (in press). PMID¼18314898. Macario DK, Entersz I, Paul Abboud J, Nackman GB. 2007. Inhibition of apoptosis prevents shear-induced detachment of endothelial cells. J Surg Res 147(2):282–289. Marolt D, Augst A, Freed LE, Vepari C, Fajardo R, Patel N, Gray M, Farley M, Kaplan D, Vunjak-Novakovic G. 2006. Bone and cartilage tissue constructs grown using human bone marrow stromal cells, silk scaf- folds and rotating bioreactors. Biomaterials 27(36):6138–6149. Martin Y, Vermette P. 2005. Bioreactors for tissue mass culture: Design, characterization, and recent advances. Biomaterials 26(35):7481–7503. Martin I, Wendt D, Heberer M. 2004. The role of bioreactors in tissue engineering. Trends Biotechnol 22(2):80–86. Niklason LE, Gao J, Abbott WM, Hirschi KK, Houser S, Marini R, Langer R.
- Functional arteries grown in vitro. Science 284(5413):489–493. Porter B, Zauel R, Stockman H, Guldberg R, Fyhrie D. 2005. 3-D computa- tional modeling of media flow through scaffolds in a perfusion bioreactor. J Biomech 38(3):543–549. Sander EA, Nauman EA. 2003. Permeability of musculoskeletal tissues and scaffolding materials: Experimental results and theoretical predictions. Crit Rev Biomed Eng 31(1):1–26. Shito M, Kim NH, Baskaran H, Tilles AW, Tompkins RG, Yarmush ML, Toner M. 2001. In vitro and in vivo evaluation of albumin synthesis rate of porcine hepatocytes in a flat-plate bioreactor. Artif Organs 25(7): 571 – 578. Sodian R, Hoerstrup SP, Sperling JS, Daebritz SH, Martin DP, Schoen FJ, Vacanti JP, Mayer JE. 2000. Tissue engineering of heart valves: In vitro experiences. Ann Thorac Surg 70(1):140–144. Stephens JS, Cooper JA, Phelan FR, Jr., Dunkers JP. 2007. Perfusion flow bioreactor for 3D in situ imaging: Investigating cell/biomaterials interactions. Biotechnol Bioeng 97(4):952–961. Tilles AW, Baskaran H, Roy P, Yarmush ML, Toner M. 2001. Effects of oxygenation and flow on the viability and function of rat hepatocytes cocultured in a microchannel flat-plate bioreactor. Biotechnol Bioeng 73(5):379–389. Tillman J, Ullm A, Madihally SV. 2006. Three-dimensional cell colonization in a sulfate rich environment. Biomaterials 27(32):5618–5626. Timmins NE, Scherberich A, Fruh J-A, Heberer M, Martin I, Jakob M. 2007. Three-dimensional cell culture and tissue engineering in a T-CUP (tissue culture under perfusion). Tissue Eng 13(8):2021–2028. Truskey GA, Yuan F, Katz DF. 2004. Transport phenomena in biological systems. Upper Saddle River, NJ: Pearson Prentice Hall. p 317–321. Williams KA, Saini S, Wick TM. 2002. Computational fluid dynamics modeling of steady-state momentum and mass transport in a bio- reactor for cartilage tissue engineering. Biotechnol Prog 18(5):951–963.
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