# Applications of Computational Modelling and Simulation of Porous Medium in Tissue Engineering

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cell Culture

**Figure 1.**Structural comparison of (

**a**) Two-dimensional cell culture and (

**b**) Three-dimensional cell culture.

#### 2.1. Cell Culture under Static Conditions

_{i}represents the gradient of concentration. The conservation of mass is given as,

_{i}[12] in the literature of flow through porous medium is

_{p}is the porosity of the scaffold, and D

_{∞}is the free diffusivity of the component in water. Since determining τ of the porous medium is difficult, some use that of cells. Alternatively, Mackie-Meares relationship can be used to determine the effective diffusivity. An example of Mackie-Meares relationship [13] is

#### 2.2. Cell Culture Involving Fluid Flow

## 3. Modeling Porous Medium Properties

#### 3.1. Incorporating Permeability

_{C}represents the Kozeny constant, a dimensionless parameter that depends on the pore geometry. In human physiology, permeability of various tissues is defined using this Kozeny definition [18]. The above expression of κ does not depend on fluid viscosity and density, which helps in extrapolating the simulation results to different fluids and flow conditions. There are a number of approaches available for calculating permeability, based on fiber orientation and size or pore area and number, depending on method and materials used for porous scaffold development. When scaffolds are formed through the process of electrospinning, fibers are randomly oriented (Figure 2a). In order to calculate the permeability in randomly packed fibers, there have been many correlations. One popular equation that seem to agree with the electrospun scaffolds is:

_{A}represents the number of pores per unit area. However, some porous medium preparation techniques using leaching salt from organic solvent-based polymeric blocks such as polycaprolactone and poly lactic-co-glycolic acid (Figure 2c). One could assume the pores to be rectangular and then calculate the permeability using the expression:

_{A}is advantageous because experimental data can be used from scaffold analyses, typically reported in tissue engineering literature.

**Figure 2.**Schematic of different pore configurations in porous scaffolds used in tissue engineering. These are in the flow direction. (

**a**) Fibrous scaffolds formed by electrospinning; (

**b**) Scaffolds with nearly circular pores when formed using freeze-drying; (

**c**) Scaffolds with rectangular pores when formed using salt-leaching technique.

#### 3.2. Incorporating Scaffold Degradation

_{1}is the rate constant of the reaction which is independent of any catalyst, ${k}_{2}$ is the rate constant of a reaction catalyzed by the previously formed products, and γ is the disassociation power of the acid end group. These reactions are then related to the MW of the polymer present in the scaffold, which can be measured and then the model can be validated. The rate of scaffold degradation by hydrolysis is not restricted to the combination of Equations (23) and (24), but can also be characterized by an exponential decay or first order equation. Other research has considered enzymatic degradation according to Michaelis Menten kinetics in addition to degradation by hydrolysis [32]. Scaffold degradation can take on many forms and is highly dependent upon scaffold composition and component concentrations.

#### 3.3. Incorporating Scaffold Deformation

**F**) applied by the fluid on the scaffold. This is calculated using,

_{T}**n**represents the normal vector. In addition to the stress due to the elastic nature of the scaffold, there is a hydrodynamic stress from fluid pore pressure, which is incorporated by coupling the strain using the relationship,

_{B}represents the Biot-Willis coefficient, and P

_{f}is the fluid pore pressure. Equation (26) reduces to Hooks Law of linear elasticity when α

_{B}is zero. Poroelasticity models in earth sciences for understanding soil consolidation use similar models in conjunction with Darcy’s equation. However, these models have to use Brinkman or Forchheimer equations based on the flow regimes, as described above. Assuming the material to behave isotropically, and porosity to remain constant, elasticity matrix is obtained using the values of elastic modulus, E, and the shear modulus, G. In the linear region, shear modulus and elastic modulus can be coupled using the equation.

## 4. Validation Techniques

## 5. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Place, E.S.; Evans, N.D.; Stevens, M.M. Complexity in biomaterials for tissue engineering. Nat. Mater.
**2009**, 8, 457–470. [Google Scholar] [CrossRef] [PubMed] - Astashkina, A.; Mann, B.; Grainger, D.W. A critical evaluation of in vitro cell culture models for high-throughput drug screening and toxicity. Pharmacol. Ther.
**2012**, 134, 82–106. [Google Scholar] [CrossRef] [PubMed] - Ravi, M.; Paramesh, V.; Kaviya, S.R.; Anuradha, E.; Solomon, F.D. 3D cell culture systems: Advantages and applications. J. Cell. Physiol.
**2015**, 230, 16–26. [Google Scholar] [CrossRef] [PubMed] - Haycock, J. 3D cell culture: A review of current approaches and techniques. In 3D Cell Culture; Haycock, J.W., Ed.; Humana Press: New York, NY, USA, 2011; pp. 1–15. [Google Scholar]
- Sander, E.A.; Stylianopoulos, T.; Tranquillo, R.T.; Barocas, V.H. Image-based multiscale modeling predicts tissue-level and network-level fiber reorganization in stretched cell-compacted collagen gels. Proc. Natl. Acad. Sci. USA
**2009**, 106, 17675–17680. [Google Scholar] [CrossRef] [PubMed] - ElectrospinningCompany. Why 3d Cell Culture? Availabe online: http://www.electrospinning.co.uk/why-3d-cell-culture/ (accessed on 1 February 2016).
- Pal, A.; Kleer, C.G. Three dimensional cultures: A tool to study normal acinar architecture vs. Malignant transformation of breast cells. J. Vis. Exp.
**2014**, 86. [Google Scholar] [CrossRef] [PubMed] - Edmondson, R.; Broglie, J.J.; Adcock, A.F.; Yang, L. Three-dimensional cell culture systems and their applications in drug discovery and cell-based biosensors. Assay Drug Dev. Technol.
**2014**, 12, 207–218. [Google Scholar] [CrossRef] [PubMed] - Shamir, E.R.; Ewald, A.J. Three-dimensional organotypic culture: Experimental models of mammalian biology and disease. Nat. Rev. Mol. Cell Biol.
**2014**, 15, 647–664. [Google Scholar] [CrossRef] [PubMed] - Granot, Y.; Rubinsky, B. Mass transfer model for drug delivery in tissue cells with reversible electroporation. Int. J. Heat Mass Transf.
**2008**, 51, 5610–5616. [Google Scholar] [CrossRef] [PubMed] - Patrachari, A.R.; Podichetty, J.T.; Madihally, S.V. Application of computational fluid dynamics in tissue engineering. J. Biosci. Bioeng.
**2012**, 114, 123–132. [Google Scholar] [CrossRef] [PubMed] - Curcio, E.; Macchiarini, P.; De Bartolo, L. Oxygen mass transfer in a human tissue-engineered trachea. Biomaterials
**2010**, 31, 5131–5136. [Google Scholar] [CrossRef] [PubMed] - Podichetty, J.T.; Dhane, D.V.; Madihally, S.V. Dynamics of diffusivity and pressure drop in flow-through and parallel-flow bioreactors during tissue regeneration. Biotechnol. Prog.
**2012**, 28, 1045–1054. [Google Scholar] [CrossRef] [PubMed] - Vunjak-Novakovic, G.; Martin, I.; Obradovic, B.; Treppo, S.; Grodzinsky, A.J.; Langer, R.; Freed, L.E. Bioreactor cultivation conditions modulate the composition and mechanical properties of tissue-engineered cartilage. J. Orthop. Res.
**1999**, 17, 130–138. [Google Scholar] [CrossRef] [PubMed] - Martin, I.; Wendt, D.; Heberer, M. The role of bioreactors in tissue engineering. Trends Biotechnol.
**2004**, 22, 80–86. [Google Scholar] [CrossRef] [PubMed] - Zeng, Z.W.; Grigg, R. A criterion for non-darcy flow in porous media. Transp. Porous Media
**2006**, 63, 57–69. [Google Scholar] [CrossRef] - Pennella, F.; Cerino, G.; Massai, D.; Gallo, D.; Falvo D’Urso Labate, G.; Schiavi, A.; Deriu, M.A.; Audenino, A.; Morbiducci, U. A survey of methods for the evaluation of tissue engineering scaffold permeability. Ann. Biomed. Eng.
**2013**, 41, 2027–2041. [Google Scholar] [CrossRef] [PubMed] - Swartz, M.A.; Fleury, M.E. Interstitial flow and its effects in soft tissues. Annu. Rev. Biomed. Eng.
**2007**, 9, 229–256. [Google Scholar] [CrossRef] [PubMed] - Podichetty, J.T.; Bhaskar, P.R.; Khalf, A.; Madihally, S.V. Modeling pressure drop using generalized scaffold characteristics in an axial-flow bioreactor for soft tissue regeneration. Ann. Biomed. Eng.
**2014**, 42, 1319–1330. [Google Scholar] [CrossRef] [PubMed] - Podichetty, J.T.; Madihally, S.V. Modeling of porous scaffold deformation induced by medium perfusion. J. Biomed. Mater. Res. Part B Appl. Biomater.
**2014**, 102, 737–748. [Google Scholar] [CrossRef] [PubMed] - O'Brien, F.J.; Harley, B.A.; Waller, M.A.; Yannas, I.V.; Gibson, L.J.; Prendergast, P.J. The effect of pore size on permeability and cell attachment in collagen scaffolds for tissue engineering. Technol. Health Care
**2007**, 15, 3–17. [Google Scholar] [PubMed] - Loh, Q.L.; Choong, C. Three-dimensional scaffolds for tissue engineering applications: Role of porosity and pore size. Tissue Eng. Part B Rev.
**2013**, 19, 485–502. [Google Scholar] [CrossRef] [PubMed] - Leong, K.F.; Chua, C.K.; Sudarmadji, N.; Yeong, W.Y. Engineering functionally graded tissue engineering scaffolds. J. Mech. Behav. Biomed. Mater.
**2008**, 1, 140–152. [Google Scholar] [CrossRef] [PubMed] - Sogutlu, S.; Koc, B. Stochastic modeling of tissue engineering scaffolds with varying porosity levels. Computer-Aided Des. Appl.
**2007**, 4, 661–670. [Google Scholar] [CrossRef] - Hollister, S.J.; Lin, C.Y. Computational design of tissue engineering scaffolds. Comput. Methods Appl. Mech. Eng.
**2007**, 196, 2991–2998. [Google Scholar] [CrossRef] - Sanz-Herrera, J.A.; Garcia-Aznar, J.M.; Doblare, M. A mathematical approach to bone tissue engineering. Philos. Trans. R. Soc. A Math Phys. Eng. Sci.
**2009**, 367, 2055–2078. [Google Scholar] [CrossRef] [PubMed] - Khoda, A.K.; Ozbolat, I.T.; Koc, B. Engineered tissue scaffolds with variational porous architecture. J. Biomech. Eng.
**2011**, 133, 011001. [Google Scholar] [CrossRef] [PubMed] - Zhao, F.; Vaughan, T.; McNamara, L. Multiscale fluid-structure interaction modelling to determine the mechanical stimulation of bone cells in a tissue engineered scaffold. Biomech. Modeling Mechanobiol.
**2015**, 14, 231–243. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Zhou, S.; Li, Q. Mathematical modeling of degradation for bulk-erosive polymers: Applications in tissue engineering scaffolds and drug delivery systems. Acta Biomater.
**2011**, 7, 1140–1149. [Google Scholar] [CrossRef] [PubMed] - Shirazi, R.N.; Ronan, W.; Rochev, Y.; McHugh, P. Modelling the degradation and elastic properties of poly(lactic-co-glycolic acid) films and regular open-cell tissue engineering scaffolds. J. Mech. Behav. Biomed. Mate.
**2016**, 54, 48–59. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; Pan, J.; Han, X.; Sinka, C.; Ding, L. A phenomenological model for the degradation of biodegradable polymers. Biomaterials
**2008**, 29, 3393–3401. [Google Scholar] [CrossRef] [PubMed] - Dhote, V.; Vernerey, F.J. Mathematical model of the role of degradation on matrix development in hydrogel scaffold. Biomech. Modeling Mechanobiol.
**2014**, 13, 167–183. [Google Scholar] [CrossRef] [PubMed] - Lawrence, B.J.; Madihally, S.V. Cell colonization in degradable 3d porous matrices. Cell Adh. Migr.
**2008**, 2, 9–16. [Google Scholar] [CrossRef] [PubMed] - Naili, S.; Oddou, C.; Geiger, D. A method for the determination of mechanical parameters in a porous elastically deformable medium : Applications to biological soft tissues. Int. J. Solids Struct.
**1998**, 35, 4963–4979. [Google Scholar] [CrossRef] - Chung, C.-Y.; Mansour, J.M. Using regression models to determine the poroelastic properties of cartilage. J. Biomech.
**2013**, 46, 1921–1927. [Google Scholar] [CrossRef] [PubMed]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

German, C.L.; Madihally, S.V.
Applications of Computational Modelling and Simulation of Porous Medium in Tissue Engineering. *Computation* **2016**, *4*, 7.
https://doi.org/10.3390/computation4010007

**AMA Style**

German CL, Madihally SV.
Applications of Computational Modelling and Simulation of Porous Medium in Tissue Engineering. *Computation*. 2016; 4(1):7.
https://doi.org/10.3390/computation4010007

**Chicago/Turabian Style**

German, Carrie L., and Sundararajan V. Madihally.
2016. "Applications of Computational Modelling and Simulation of Porous Medium in Tissue Engineering" *Computation* 4, no. 1: 7.
https://doi.org/10.3390/computation4010007