Biomaterials

Biomaterials are typically inhomogeneous, with complicated internal structure, and undergo large deformations. They are also active, comprising cells which are attached to each other and to extra-cellular matrix by filopodia which may detach and reattach. The cells themselves may contract (e.g. as muscle cells do in the heartbeat), move, or grow and divide. In addition it is unlikely that there exists a stress-free reference configuration from which deformations can be measured.

There are many areas of medicine in which accurate models of tissue mechanics would be useful. An application which will be one of our main goals is in medical imaging, and in particular the diagnosis of breast tumours(a)(b)
(c)(d)
Modelling breast tissue:
(a): skin (silver) and muscle (red) datapoints
(b): fitted cubic-Hermite mesh
(c): refined trilinear mesh
(d): refined mesh showing fibroglandular elements (green).
(Pras Pathmanathan, Thesis, Oxford, 2006) and heart problems.

The two most common imaging modalities used to screen for breast cancer are magnetic resonance imaging (MRI) and mammography (X-ray). Accurate matching between MR images and mammograms, or between mammograms taken from different angles, to localise and/or characterise potential tumours is essential for early diagnosis and improved treatment planning and monitoring. However, matching is hindered by the fact that the breast shape varies hugely between the different modalities, due to both the different positions of the patient and the fact that the breast is strongly compressed during mammography. A mathematical model that can accurately predict breast deformation under changes in body position or applied force would improve the image registration procedure.

A related problem is that of interpretation of ultrasound images of breast tissue. Recently it has been suggested that identifying (stiffer) tumour tissue with the light areas in these images may overestimate the size of the tumour. An alternative approach is to use ultrasound images under different compression to determine the displacement of tissue and infer the elastic properties (and hence the tumour position) by solving an inverse problem. At present the breast is modelled using plane-strain linear elasticity, but even in this simplified case the mathematical theory underlying the inverse problem (and its well-posedness) is lacking.

Many problems involving biomaterials are made more difficult computationally because of the existence of thin layers or small internal structures. Another problem which occurs when modelling biomaterial is that of residual stress, which is probably present even in the hypothetical “gravity free” configuration of breast tissue. Determining the residual stress from experimental measurements in vivo is nontrivial.

Residual stress can be regarded as being due to differential growth in the material. Sometimes it is important to model this growth dynamically, a particularly important application being stress adaptation of bone and osteoporosis (Taber, 1995; Cowin & Hegedus, 1976; Keller, 2001). Most models for growth of biomaterials make use of a growth tensor which must be composed with the deformation tensor when calculating the stress. Mathematical analysis of these models is in its infancy, and many questions remain. Another major area where tissue growth is important is the modelling of solid tumours (Chaplain et al., 1993, 2001).

An area where the active nature of cells arises is the contraction of individual cells in response to an external signal (e.g. the heartbeat, contraction waves in morphogenesis). Of particular interest is the coupling between configurational changes within individual cells and bulk calculations of stress and strain.

The specific projects associated with this theme are

Models for breast and heart tissue
Modelling growth, with application to colon cancer

References

L.A. Taber. Biomechanics of growth, remodelling, and morphogenesis. Applied Mechanics Reviews, 48:487–545, 1995.
S.C. Cowin and D.H. Hegedus. Bone remodelling I: Theory of adaptive elasticity. Journal of Elasticity, 6:313–326, 1976.
T.S. Keller. Finite Element Modeling of Bone Tissue Stress-Adaptation. In: Proceedings of SPIE: Complex Adaptive Structures. Edited by W.B. Spillman, Jr. Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, 4512:232–245, 2001.
M.A.J. Chaplain and B.D. Sleeman. Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory. J. Math. Biol., 31:431–473, 1993.
M.A.J. Chaplain, M. Ganesh and I.G. Graham. Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth. J. Math. Biol., 42:387–423, 2001.