The document presents a Mathematica tool for rapidly estimating flow conditions within a compression-perfusion bioreactor by calculating compression induced flow, tortuosity, interstitial velocity, shear stress using the Brinkman method, and packed bed Reynolds number from user-defined scaffold properties and perfusion parameters in under 3 minutes without expertise in fluid dynamics equations or software. The tool is intended to help bioreactor designers and could be expanded to include visualization and serve as an educational resource.

1. A Mathematica tool for rapid
estimation of flow conditions
within a compression-perfusion
bioreactor
Adam J. Taylor
Alicia El Haj, Jan Herman Kuiper

2. Background
• 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

3. Mathematica
• Previous experience
• Simple syntax and notebook format
• Power of Manipulate command
• Ease of distribution through free
Mathematica Player

4. 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).
  r“ ”2 
1√ √ √ 1 −1 +1
1−φ 4
τ = 2 1+ 2 1−φ + 1−φ+
1  √
1− 1−φ

Q ρudp
uτ Rep =
φA µ

5. Mathematics
4. Calculate shear stress using Brinkman Method via Brinkman
Constant, equivalent particle diameter and 1st (Darcian)
permeability constant
Bηu
τ= √
k1
φ3 d2
particle
k1 =
150 (1 − φ)
1−φ
dparticle = 1.5 · · dpore
φ
2 4
4 1 − 0.319285 (1 − φ) − 0.043690 (1 − φ)
B = ·
π 4 3
1 − (1 − φ) − 0.305828 (1 − φ) 1 + (1 − φ) − 0.0305828 (1 − φ)

6. Demo

7. Further work
• Improve interface/output
• Provide documentation
• Develop more relevant shear model from
first principles
ReynoldsNumber
• Include visualisation
4
5
•
3
Potential to develop into teaching tool
for bioreactor course 2
1
Porosity
0.0 0.2 0.4 0.6 0.8 1.0

8. 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

9. QA
Acknowledgements:
Alicia El Haj
Jan Herman Kuiper
EPSRC, EMDA
Download Mathematica Player: wolfram.com/player
Download the .nbp: bit.ly/b70Xlr
Contact: a.j.taylor3@lboro.ac.uk