Graduate Course in Biomechanical Modelling


Jonas Stålhand

Joakim Holmberg

Anders Klarbring

Course objectives

After completing the course, the participant is able to:
  • set up constitutive models for physiological systems using fundamental principles in mechanics, and
  • use the constitutive model to analyse and simulate normal and diseased physiological conditions as well as the effect of drugs.

Course organisation

The course is structured in two blocks with a total of 22 lectures (45 minutes each). Block A concerns more general concepts in biomechanics while block B is focused on constitutive modelling for mechanobiological processes. The course is given in Swedish/English.


The examination consists of 5 hand-in assignments (3 block A + 2 block B). Participants are encouraged to discuss problems and solutions with each other but the assignments are to be solved and reported individually. Plagiarism is not permitted. Examiner is Jonas Stalhand, Division of Mechanics, e-mail:

Lecture plan (preliminary)

Lecture plan block 1
1-4Introduction to continuum mechanicsBasic consepts, kinematics, stress, balance laws, frame invariance, principle of virtual, dissipation inequality and constitutive.Spencer (1980)
5-8Wave propagation in arteries.Transmission line equations, arterial tree, boundary conditions, cardiovascular system, hypertesion, medication.Zamir (2000)
9-12Passive soft tissuesPhysiology, arterial walls, constituents, normal function, diseases. Constitutive laws, nonlinear materials, strain-energy, material convexity, incompressibility, anisortopy.Holzapfel (2000), Holzapfel et al. (2000)
13-14Motion biomechanics.Modelling bodies in motion, muscle models, underdetermined systems, solution techniques. Nikravesh (2008), Zajac (1989)

Lecture plan block 2
15-18Muscle modellingCell, filaments, activation, classical models, mechano-chemical coupling, kinematic model, constitutive relations, evolution laws.Stålhand et al. (2008), Murtada et al.(2010)
19-22Soft tissue remodellingBone remodelling, topology optimization techniques, soft tissue remodelling, multiplicative decomposition, mixture theory.Klarbring and Torstenfelt (online)


6/12 hp. Participants my choose to take the whole course (blocks A and B) for 12 credits or only block A for 6 credits.


The course material is based on research articles, relevant books in the field, and other materials. Examples of relevant literature are:
  • Holzapfel G.A. (2000) Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley & Sons. Chichester
  • Holzapfel G.A., Gasser T.C., Ogden R.W. (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity 61: 1-48
  • Humphrey J.D. (2002) Cardiovascular Solid Mechanics. Cells, Tissues, and Organs. Springer, New York
  • Humphrey J.D., Rajagopal K.R. (2002) A constraind mixture model for growth and remodeling of soft tissues, Mathematical Models and Methods in Applied Sciences 12: 407-430
  • Klarbring A., Torstenfelt B. Dynamical systems, SIMP, bone remodeling and time dependent loads, Structural and Multidisciplinary Optimization Structural and Multidisciplinary Optimization, on line
  • McMahon T.A. (1984) Muscles, Reflexes, and Locomotion. Princeton University Press, New Jersey
  • Murtada S.-I., Holzapfel G.A., Arner A. (2011) Experiments and mechanochemical modeling of smooth muscle contraction: Significance of filament overlap. J. Theor. Biol., doi:10.1016/j.jtbi.2011.11.012
  • Nikravesh P.E. (2008) Planar multibody dynamics: formulation, programming and applications. CRC Press, Boca Raton, Florida
  • Olsson T., Klarbring A. (2008) Residual stresses in soft tissue as a consequence of growth and remodeling: application to an arterial geometry. Eur. J. Mech. A/Solids 27: 959-974.
  • Stålhand J., Klarbring A., Holzapfel G.A. (2008) Smooth muscle contraction: Mechanochemical formulation for homogeneous finite strains. Prog. Biophys. Mol. Biol. 96: 465-481
  • Zajac F.E. (1989) Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit. Rev. Biomed. Eng. 17: 359-411
  • Zamir M. (2000) The physics of pulsatile flow. Springer-Verlag, New York

Page responsible: Jonas Stålhand
Last updated: 2012-09-24