stress and strain field reconstruction

Whenever living cells are actively migrating through biological tissue, they are exerting forces onto the surrounding material. Depending on the local mechanical properties, the cellular traction forces lead to complex deformations of the material.

An important step towards a biophysical understanding of cancer cell migration is a quantitative measurement of these material deformations. For this purpose, the material is directly observed with optical microscopy and an image of the region around the cell is taken.

Next, we switch off the force generating apparatus of the cell by applying suitable chemicals. Once the cell relaxes, we take a second image, showing the deformation state of the material without the cellular forces.

By comparing the strained and relaxed images, it is possible to extract a quantitative deformation map, connecting each point of image 1 with the corresponding point of image 2 by a shift vector. This procedure of matching points between an original and a deformed image is known as "image registration".

The image registration procedure becomes easier if the material contains landmark objects of known shape. For this purpose we attach fluorescent markers to the material (spherical beads in random distribution). The picture below shows a pair of images containing such micro-beads:

Since each micro-bead is large enough to affect several neighboring pixels of the camera, a shift of the bead position can be detected with sub-pixel resolution, using techniques such as center-of-mass algorithms. In fact, we achieve a resolution in the nanometer range with our purely optical setup. Nevertheless, the registration procedure occasionally fails, for example in cases of clusters of close-by beads.

In the future, we would like to become independent from artificial markers and instead use the natural features of the material itself as landmarks. This would require an image registration method which can deal with complex shapes, such as fibres:

Another ongoing project is the reconstruction of the cellular forces from the known deformation field of the material. This, of course, requires a model of the mechanical properties of the material. For this purpose, we are using two-dimensional essays, in which the cell is placed onto a surface of an artificial, linear elastic material with a well-defined Youngs modulus and Poisson ratio. However, in the case of a realistic biological medium, such as a three-dimensional random collagen fibre network, the mechanical properties are extremely complex.

It is an open question to which extent the linear elastic model of a homogeneous, isotropic medium is applicable to such fibre networks, even on length scales much larger than the typical mesh size.


We are now working on the development of a robust, fast registration method, taylor-made for the above applications. It will be suitable for a multigrid solution and eventually is to be implemented on a graphics processing unit (GPU). We are seeking collaborators for this project.