During my stay in the KTH Solid Mechanics department I had the opportunity to visit the soft tissue mechanics laboratory and to try an uniaxial tensile test on tissue from a porcine aorta.

I had the opportunity to participate in several experiments held by trainee Joey Wollfs. He is currently involved in some experimental activities in testing vascular tissues under the supervision of Marta Alloisio (PhD student) and Christian Gasser.

The following is a far cry from the high-level research that is carried out in this department at KTH. However, I tried to enclose in an article the activities that could be completed in the laboratory in a day (very little time for such activities), both from an experimental and a computational point of view.

Furthermore, the following information contains data from only one test and the results are not nearly as accurate as you would like.

## Tensile test

An uniaxial tensile test is performed by applying a displacement and reading the resulting force.

We are talking about biological tissues whose behavior is strongly non-linear. We must also take into account for the possible presence of viscoelastic effects and temperature dependence. For these reasons a relatively low strain rate is applied. In addition, several preconditioning cycles are carried out.

A CellScale BioTester is used. Ad hoc 3D printed clamps were used to fix the tissue.

Vascular tissue from a porcine aorta is tested. It is important to observe that the stiffness of young porcine aortic tissue shows good correspondence with human tissue. Furthermore, the pig tissue is much easier to find and use. From a porcine aorta a small rectangle of tissue is cutted along the longitudinal direction. The sample is clamped and prepared also with DIC markers.

## Tissue data

From the obtained data it is possibile to extract several features.

If we select a constitutive model we could try to fit the model and extract material parematers.

## Yeoh hyperelastic model

For instance, let me pick the Yeoh strain energy. This constitutive model describe an Hyperelastic behaviour. Furthermore, in literature it is showed as one the most used model for vascular tissue. I fitted the data with Wolfram Mathematica so that I could use a symbolic description of the strain energy.

For the Yeoh model the strain energy is described as:

W=\sum_{i=1}^3 C_i\left(I_1-3\right)^i

And for uniaxial extension in the 1-direction, the principal stretch are:

\lambda_1=\lambda, \lambda_2=\lambda_3

But we are considering a biological tissue that should be considered incompressible, so:

\lambda_1 \lambda_2 \lambda_3=1

So

I_1=\lambda_1^2+\lambda_2^2+\lambda_3^2=\lambda^2+\frac{2}{\lambda}

This allow us to describe the principal stress:

\sigma_{11}=-p+2 \lambda^2 \frac{\partial W}{\partial I_1}

And the engineering stress as:

T_{11}=\sigma_{11} / \lambda=2\left(\lambda-\frac{1}{\lambda^2}\right) \frac{\partial W}{\partial I_1}

So we can obtain the coefficient: c1, c2, c3.

I1 = \[Lambda]^2 + 2/\[Lambda];
W = c1 (I1 - 3) + c2 (I1 - 3)^2 + c3 (I1 - 3)^3;
T11 = 2 (\[Lambda] - 1/\[Lambda]) D[W[I1], \[Lambda]];
DWDI = c1 + 2*c2 (I1 - 3) + 3*c3 (I1 - 3)^2;
T11 = 2 (\[Lambda] - 1/\[Lambda]) DWDI
disp = Import["disp.mat"];
force = Import["force.mat"];
stretch = Flatten[disp];
stretch = (L0 + stretch*10^-3)/L0;
force = Flatten[force];
stress = (force*10^-3)/(L0*H); (*N/mm^2=MPa*)
data = Table[{stretch[[i]], stress[[i]]}, {i, 1, Length[stretch]}];
nlm = NonlinearModelFit[data, T11, {c1, c2, c3}, \[Lambda]]
nlm["ParameterTable"]

## Tissue simulation

It is also possibile to use ti coefficient to represents the model inside a simulation environment.

I used Comsol Multiphysics that use a FEM formulation to solve the PDE problem.  By inserting the material description (c1,c2,c3) and the boundary condition it is possibile to sweep around the displacements span.

The software record also images during the test and so it is possibile to use it for DIC.

Digital Image Correlation (DIC) allow us to track the movement of a surface pattern. This is done by analyzing the displacement of the pattern discretized in subsets. The maximum correlation in each window corresponds to the displacement, and this gives the vector length and direction for each window. This allow us to obtain a deformation vector field and so the complete strain field.

What is reported would require integration with further experiments and tests in order to have a correct validation. However, this article is only intended to be a description of a very interesting day spent in the solid mechanics laboratory testing biological tissues.