Numerical simulation of fluid flow in a rotational bioreactor V. L. Ganimedov, E. O. Papaeva, N. A. Maslov, and P. M. Larionov
Citation: AIP Conference Proceedings 1893, 030006 (2017); View online: https://doi.org/10.1063/1.5007464 View Table of Contents: http://aip.scitation.org/toc/apc/1893/1 Published by the American Institute of Physics
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Numerical Simulation of Fluid Flow in a Rotational Bioreactor V. L. Ganimedov1, E. O. Papaeva1, 2, N. A. Maslov1, 2 and P. M. Larionov3, a) 1
Khristianovich Institute of Theoretical and Applied Mechanics Institutskaya Street 4/1, Novosibirsk, 630090 Russia 2 National Research Novosibirsk State University Pirogov Street 2, Novosibirsk, 630090 Russia 3 Federal State Novosibirsk Research Institute of Traumatology and Orthopedics Frunze Street 17, Novosibirsk, 630091 Russia a)
Corresponding author:
[email protected]
Abstract. Application of scaffold technology for the problem of bone tissue regeneration has great prospects in modern medicine. The influence of fluid shear stress on stem cells cultivation and its differentiation into osteoblasts is the subject of intensive research. Mathematical modeling of fluid flow in bioreactor allowed us to determine the structure of flow and estimate the level of mechanical stress on cells. The series of computations for different rotation frequencies (0.083, 0.124, 0.167, 0.2 and 0.233 Hz) was performed for the laminar flow regime approximation. It was shown that the Taylor vortices in the gap between the cylinders qualitatively change the distribution of static pressure and shear stress in the region of vortices connection. It was shown that an increase in the rotation frequency leads to an increase of the unevenness in distribution of the above mentioned functions. The obtained shear stress and static pressure dependence on the rotational frequency make it possible to choose the operating mode of the reactor depending on the provided requirements. It was shown that in the range of rotation frequencies chosen in this work (0.083 < f < 0.233 Hz), the shear stress does not exceed the known literature data (0.002 - 0.1 Pa).
INTRODUCTION The restoration of critical bone defects is one of the topical problems of traumatology and orthopedics. In particular, cell-mediated scaffold technologies are of significant interest. Scaffold technology for bone tissue regeneration is based on several stages: creation of a scaffold (a framework for cell seeding), seeding of the framework by mesenchymal stem cells of the patient, and then cultivation of the seeded scaffold in bioreactor. During the cultivation, the microenvironment affects the stem cells, leading to their differentiation into osteoblasts. Mechanical action of the liquid on the cells plays an important role in the process of cell differentiation on a par with osteoinductors. The shear stress of the liquid affects the cells through primary cilia (special cellular organelles associated with the cellular cytoskeleton and the functioning of ion channels) [1]. The scaffolds seeded with cell cultures were already cultivated in perfusion type bioreactors under the influence of shear stress from 0.002 to 0.1 Pa [2, 3]. The choice of reactor parameters is very important because it directly determines the efficiency of cell growth and differentiation of stem cells, the formation of extracellular bone matrix, and cell loss by scaffold. Therefore the reactor parameters for cell-seeded scaffold cultivation are still the subject of intensive research. Mathematical modeling of the process can be useful to evaluate the efficiency of scaffold cultivation in bioreactor. Numerical simulation allows one to determine the level of mechanical stress on stem cells, the structure of the fluid flow, depending on geometry of the bioreactor and working parameters.
Proceedings of the XXV Conference on High-Energy Processes in Condensed Matter (HEPCM 2017) AIP Conf. Proc. 1893, 030006-1–030006-7; https://doi.org/10.1063/1.5007464 Published by AIP Publishing. 978-0-7354-1578-2/$30.00
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MATERIALS AND METHODS The geometric model of the proposed biological reactor for bone tissue regeneration is shown in Fig. 1. It schematically consists of two coaxial cylinders with different heights filled with liquid. The inner cylinder represents metal frame with spanned scaffold sheet, which is based on polycaprolactone framework with the inclusion of gelatin, chitosan and microelements that determine osteoinduction. The outer cylinder is made from glass, allowing estimation of the osteoinduction efficiency by various optical methods like laser-induced fluorescence spectroscopy [4]. Here Fe1, Fe2, Fe3 are the surfaces of the outer cylinder and the base, and Fi1, Fi2 are the surfaces of the inner cylinder. The bottom base of the outer cylinder is replaced by the surface of the oblate ellipsoid of revolution. The inner cylinder is the segment of the cylindrical tube immersed in a liquid. The dimensions of the construction, shown in mm, were taken from the sample functioning in the clinic. The flat end faces of the cylindrical tube were replaced by the surface of the torus to facilitate the calculation conditions. The cavity of the outer cylinder was filled with water with density of 998.2 kg/m3 DQGG\QDPLFYLVFRVLW\P3DÂV All the surfaces of the outer cylinder (Fe1, Fe2, Fe3) were stationary during the reactor operation. The rotation of inner cylinder around the common axis is the only flow generator in the closed cavity of the outer cylinder.
FIGURE 1. Schematic model of the biological reactor. Fe1, Fe2, Fe3 - the surfaces of the outer cylinder, Fi1, Fi2 - the surfaces of the inner cylinder
The system of Navier–Stokes equations for an incompressible flow was used for the mathematical description of the flow. Heat exchange and gravitational forces were not taken into account in this task. External boundary conditions: immobility and attachment boundary conditions were set on the surfaces of the outer cylinder Fe1, Fe2 (Fig. 1), and also the derivative of the velocity along the normal was given 0 (8Q = 0) and immobility condition was set on the surface Fe3. Internal boundary conditions: attachment boundary conditions and the conditions of surface rotation about the axis of the cylinder were set on the surfaces of the inner cylinder Fi1, Fi2 (Fig. 1). Five rotation frequencies were simulated: 0.083, 0.124, 0.167, 0.2 and 0.233 Hz. The zero velocity field Ux = Uy = Uz = 0 was set in the whole domain of the task as initial data, and the flow regime was postulated to be laminar. The discretization of task domain was built using packet technology. A hybrid unstructured mesh was used. The boundary layer consisted of 5 layers was built in the vicinity of the inner cylinder surfaces. After several trials, all further calculations were performed using optimal discretization level of 3.1·106 elementary volumes. The numerical solution of the equations system was constructed with the help of the ANSYS FLUENT software package which is widely used in technical [5] and scientific research [6]. The created computational algorithm was tested for the task of two infinite rotating cylinders. An exact solution of this task when the condition on the Taylor number Ta < 41.3 is known as the laminar Couette flow [7]. It was shown that the solution obtained with the help of the created computational algorithm is in good agreement with Couette's analytical solution [8].
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This article presents the results of flow calculations for Taylor numbers 41.3 < Ta < 400 (in terms of rotation frequency - 0.083 < f < 0.233 Hz). In this range a laminar flow with Taylor vortices takes place in the gap between infinite coaxial cylinders. For Ta > 400 the flow passes from the cell type laminar form to the turbulent form [7].
RESULTS AND DISCUSSION Full visualization of the flow field according to the numerical solution is presented in Fig. 2 and 3. In both figures the sections “a” correspond to the rotation frequency f = 0.083 Hz (5 rpm, Ta = 144), sections “b” correspond to the frequency f = 0.167 Hz (10 rpm, Ta = 289), sections “c” correspond to the frequency f = 0.233 Hz (14 rpm, Ta = 404). The projection of the streamlines to the axial section is shown in all sections of Fig. 2 to the left of the vertical symmetry axis and the static pressure field is shown to the right. All sections of Fig. 3 to the left show the velocity magnitude and sections to the right show the vertical velocity component Uy.
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(b)
(c)
FIGURE 2. Visualization of fluid flow in the biological reactor. In each figure streamlines visualization is presented to the left, and static pressure visualization is presented to the right. Sections (a), (b) and (c) corresponds to the rotation frequencies f = 0.083, 0.167 and 0.233 Hz. Scales with a common range of parameter values are shown in section (a)
It can be seen that in the considered range of rotation frequencies the flow structure does not change, and only quantitative changes occur. Static pressure visualization shows the zones with large gradients, velocity magnitude visualization - areas of stagnant flow, streamlines visualization - direction the vortex formations rotation. For the considered task vertical velocity component visualization is the most informative about the thin structure of the flow. The distributions of the static pressure (p, Pa) and the velocity magnitude (U, m/s) in the radial direction are shown for the section y = 0.027 m and are presented in Fig. 4. This plane intersects the inner cylinder in the middle of its height, where Taylor vortices with different directions of rotation are contacted in the gap between cylinders. The results are shown for 3 rotation frequencies: 0.083, 0.167 and 0.233 Hz. It follows from Fig. 4 that force
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directed from the center to the periphery acts from the liquid on the inner moving cylinder. The distribution of the velocity magnitude in Fig. 4 (b) shows that on the reactor axis (R = 0 m) the velocity magnitude is not equal to zero and the mass transfer along the axis takes place.
(a)
(b)
(c)
FIGURE 3. Visualization of fluid flow in the biological reactor. In each figure velocity magnitude visualization is presented to the left, and vertical velocity component visualization is presented to the right. Sections (a), (b) and (c) corresponds to the rotation frequencies f = 0.083, 0.167 and 0.233 Hz. Scales with a common range of parameter values are shown in section (a)
(a)
(b)
FIGURE 4. (a) Static pressure and (b) velocity magnitude distribution in the radial direction for the section y = 0.027 m for three rotation frequencies. The solid line corresponds to the frequency f = 0.083 Hz, the dashed line corresponds to the frequency f = 0.167 Hz, the dash-dotted line corresponds to the frequency f = 0.233 Hz
The static pressure distribution (p, Pa) along the generator of the inner cylinder is given for the surface Fi1 in Fig. 5 (a) and for the surface Fi2 in Fig. 5 (b). On the graphs the value of the coordinate y is plotted along the abscissa axis. The results are presented in a one-parameter representation for the rotation frequencies f: 0.083, 0.124,
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0.167, 0.2 and 0.233 Hz. The surface Fi1 is in the zone of low pressure, and Fi2 is in the zone of high pressure. The pressure distribution on the surface Fi2 is practically symmetrical with respect to the middle of the inner cylinder and is rather uniform (Fig. 5 (b)). Taylor vortices qualitatively change the pressure distribution on the surface Fi1 (Fig. 5 (a)).
(a)
(b)
FIGURE 5. Static pressure distribution along the generator of the inner cylinder for the surface Fi1 (a) and Fi2 (b). Rotation frequency of the inner cylinder f: 0.233, 0.2, 0.167, 0.125, 0.083 Hz
(a)
(b)
FIGURE 6. Shear stress distribution along the generator of the inner cylinder for the surface Fi1 (a) and Fi2 (b). Rotation frequency of the inner cylinder f: 0.233, 0.2, 0.167, 0.125, 0.083 Hz
The distribution of the shear stress (IJ, Pa) along the generator of the inner cylinder is shown for the surface Fi1 in Fig. 6 (a), and for the surface Fi2 in Fig. 6 (b). The argument of the function and the values of the parameter with respect to the rotation frequency are identical to those shown in Fig. 5. Two sets of curves are almost symmetrical with a minimum in the middle. The clearly expressed minimum of IJ value shown in Fig. 6 (a) corresponds to the connection area of Taylor vortices. An increase in the rotation frequency leads to an increase of the unevenness in the distribution. In contrast with the surface Fi1 (Fig. 6 (a)) the shear stress on the surface Fi2 (Fig. 6 (b)) is smaller in magnitude, but have more uniform distribution. The effect of the liquid flow, averaged over the whole area of the inner rotating cylinder, is given in Fig. 7: the dependence of shear stress IJ (Fig. 7 (a)) and static pressure p (Fig. 7 (b)) on the rotation frequency f. As one can see, maximum average shear stress on the inner cylinder occurs on the side, facing the gap between the cylinders (surface Fi1). The maximum of the average static pressure takes place on the surface Fi2. Also it can be seen that in this paper in presented range of rotation frequencies (0.083 < f < 0.233 Hz) the shear stress value does not exceed
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the known literature data (0.002 - 0.1 Pa). This suggests that these rotation modes of the bioreactor can be used for cell cultivation.
(a)
(b)
FIGURE 7. Dependence of the shear stress (a) and static pressure (b) distribution along the generator of the inner cylinder on the frequency of rotation. The solid line corresponds to the surface Fi1, the dashed line corresponds to the surfaces Fi2
CONCLUSION Mathematical model and computational algorithm for calculation of flow in a rotational bioreactor has been created on the basis of package technology. The series of flow computations for various rotational frequencies (0.083 < f
