# C5.9 Mathematical Mechanical Biology - Material for the year 2018-2019

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Fluid Mechanics: Part A Waves and Fluids and Part B Viscous flow is recommended.

Solid mechanics: One of the Part C courses (Solid Mechanics or Elasticity/Plasticity is recommended).

Some mathematical biology or physiology is desirable but not necessary as the material for a particular biological system will be part of the course.

16 lectures

### Assessment type:

- Written Examination

The course will be motivated by outstanding problems in physiology and biology but the emphasis is on the mathematical tools needed to answer some biologically relevant problems. The course is divided into modules and three modules will be given during a term but these modules can change from one year to the next.

The goal of this course is to learn the physical background and mathematical methods behind many problems arising in mechanical biology from the cellular level and upwards. Students will familiarise themselves with key notions used in modern research in bio-physics and mechano-biology.

1D Biological Mechanics. Bio-Filaments (2 1/2 weeks)

(a) Introduction: bio-molecules (actin, microtubules, DNA,...)

(b) Randomly fluctuating chains (statistical mechanics)

(c) Continuous filaments (neurons, stems, roots, plants)

(d) Differential geometry of curves: Kirchhoff rod theory and beam theory

2D Biological Mechanics. Bio-Membranes (2 1/2 weeks)

(a) Introduction: lipid bilayer, cell membranes

(b) Differential geometry of surfaces: curvatures, Gauss--Bonnet theorem

(c) Fluid membranes: shape equation, fluctuating membranes

(d) Solid membranes: hyperelastic isotropic materials, shells. Application to the cell, plants and microbes.

Aspects of 3D Biological Mechanics:

Low Reynolds number flows: Scallop theorem, Cell Motility, Ciliary Pumping.

The following modules will not be taught in 2016-17.

Bio-Fluids (3 weeks)

(a) Low Reynolds Number: Motility, Scallop theorem.

(b) Complex biofluids: active and non-Newtonian fluids

(c) Circulation: Blood flow, microcirculation, networks

Multiphase/Multiphysics methods (3 weeks)

(a) Coupling fluids and solids: poro-elastic tissue

(b) Coupling fluid, solids and chemistry: tissue swelling

(c) A general thermodynamics approach

(d) Application to tissue engineering, wound healing.

Bio-solids and growth (3 weeks)

(a) Introduction: nonlinear elasticity for soft tissues

(b) one-dimensional growth theory

(c) volumetric growth: multiplicative decomposition

(d) application to neuronal growth, tumour

- Physical Cell Biology, second ed. Rob Phillips et al. Garland Science.
- Cardiovascular solid mechanics. Cells, tissues, and organs, Humpherey, 2002, Springer.
- Nonlinear Solid Mechanics: A Continuum Approach for Enineering: A Continuum Approach for Engineering, G. Holzapfel, 200, Wiley.
- The hydrodynamics of swimming microorganisms, Lauga and Powers, Rep Prog Phys 72, 2009.