A Mathematica tool for rapid estimation of flow conditions within a compression-perfusion bioreactor

Adam J. TaylorAlicia El Haj, Jan Herman Kuiper

• Compression-perfusion bioreactor

• History of use and widely described

• Interest in flow conditions induced by compression, in particular surface shear stresses

• FE/CFD vs Navier Stokes vs Shortcut methods

“As a novice it will take two to three weeks to start to understand some of the simple worked examples. To construct a basic model could take six to twelve months.Even then you will need significant help from an experienced user.”

— Prof Stan Kolaczkowski. Transport Phenomena 4, Nov 2008. University of Bath

BackgroundBackground

Mathematica

• Previous experience

• Simple syntax and notebook format

• Power of Manipulate command

• Ease of distribution through free Mathematica Player

Mathematics

1. Set inputs: Scaffold height, diameter, porosity, average pore size, perfusion rate, media and viscosity, compression time and ratio

2. Calculate compression induced flow and integrate to perfusion

3. Calculate tortuosity (Yu’s method) and interstitial velocity to give packed bed reynolds number (confirm Darcian flow, Re < 1).

τ = 12

1 +�

12

√1− φ

�+

√1− φ +

r“1√1−φ

−1”2

+ 14

1−√

1−φ

u � τQ

φARep =

ρudp

µ

4. Calculate shear stress using Brinkman Method via Brinkman Constant, equivalent particle diameter and 1st (Darcian) permeability constant

B =4π

· 1− 0.319285 (1− φ)2 − 0.043690 (1− φ)4��1− (1− φ)− 0.305828 (1− φ)4

� �1 + (1− φ)− 0.0305828 (1− φ)3

�

dparticle = 1.5 · 1− φ

φ· dpore

k1 =φ3d2

particle

150 (1− φ)

τ =Bηu√

k1

Mathematics

Demo

Further work

• Improve interface/output

• Provide documentation

• Develop more relevant shear model from first principles

• Include visualisation

• Potential to develop into teaching tool for bioreactor course

0.0 0.2 0.4 0.6 0.8 1.0Porosity

1

2

3

4

5ReynoldsNumber

Literature

Baas et al. In vitro bone growth responds to local mechanical strain in three-dimensional polymer scaffolds. Journal of biomechanics (2010) vol. 43 (4) pp. 733-9

Milan et al. Computational modelling of the mechanical environment of osteogenesis within a polylactic acid-calcium phosphate glass scaffold. Biomaterials (2009) vol. 30 (25) pp. 4219-26

Vossenberg et al. Darcian permeability constant as indicator for shear stresses in regular scaffold systems for tissue engineering. Biomech Model Mechanobiol (2009) vol. 8 pp. 499-507

Whittaker et al. Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor. J Theor Biol (2009) vol. 256 (4) pp. 533-46

Boschetti et al. Prediction of the micro-fluid dynamic environment imposed to three-dimensional engineered cell systems in bioreactors. J Biomech (2006) vol. 39 (3) pp. 418-25

Yu and Li. A geometry model for tortuosity of flow path in porous media. Chinese Physics Letters (2004)

Innocentini et al. Prediction of ceramic foams permeability using Ergun's equation. Materials Research (1999) vol. 2 pp. 283-289

Q&A

Acknowledgements:Alicia El HajJan Herman KuiperEPSRC, EMDA

Download Mathematica Player: wolfram.com/playerDownload the .nbp: bit.ly/b70Xlr

Contact: a.j.taylor3@lboro.ac.uk