Estimation of the Arterial Wall Material Properties by Using Strain Energy Function and

Inflation$-$Extension Experiment....

cubic polynomial strain energy function:

$W(E_{},E_{zz})=A(E_{}{)}^2+B{E}_{}{E}_{zz}+C(E_{zz}{)}^2+D(E_{}{)}^3+$

$E(E_{}{)}^2{E}_{zz}+F{E}_{}(E_{zz}{)}^2+G(E_{zz}{)}^3$

where $E_{}$ and $E_{zz}$ are the components of Cauchy$-$Green strain tensor and $A,$dots,$G$ are the material

parameters to be estimated.

Derive the corresponding components of the second Kirchhoff stress tensor, $s_{}$

and $S_{zz}$ as functions of $A,.,G.$

Use the data of inflation$-$extension experiment that provides two arrays, $S_{}^{\text{e}\text{x}\text{p}}[i]$ and $S_{zz}^{\text{e}\text{x}\text{p}}[i]$

where $S_{}^{\text{e}\text{x}\text{p}}[i]=S_{}[E_{}^{\text{e}\text{x}\text{p}}[i],E_{zz}^{\text{e}\text{x}\text{p}}[i]]$ and $S_{zz}^{\text{e}\text{x}\text{p}}[i]=S_{zz}[E_{}^{\text{e}\text{x}\text{p}}[i],E_{zz}^{\text{e}\text{x}\text{p}}[i]]($see the tables below$)$

Table Carotid

Table Left Iliac

Estimate the $A,$dots,$G$ coefficients by minimizing the error function

$S_i[S_{}[i]-S_{}^{\text{e}\text{x}\text{p}}[i]{]}^2+[S_{zz}[i]-S_{zz}^{\text{e}\text{x}\text{p}}[i]{]}^2.$

Use the Levenberg$-$Marquardt minimization algorithm.

To estimate the quality of fit, compute the correlation coefficients $(S_{}vs.S_{}^{\text{e}\text{x}\text{p}}$ and $S_{zz}vs.S_{zz}^{\text{e}\text{x}\text{p}})$

and plot points $S_{}[i],S_{}^{\text{e}\text{x}\text{p}}[i]($the same figure$)$ and $S_{zz}[i],S_{zz}^p[i]($the same figure$).$ Do it for

both arteries.

Matlab Instructions:

First, enter the data and relevant equations into Matlab.

$*$Tip: it might be easier to put the respective data set into just one matrix with multiple

rows$/$columns; for example: combine the E$-$values in a nx $2$ or $2$ xn matrix$).$

Next, set each of the coefficient values $($A$,$ B$,$ C$,$ etc.$)$ as units of one matrix, so that when

solving, it outputs one large matrix with all the coefficients included.

$*$Note: make sure that you double check that dimensions to avoid an error.

In order to utilize the Levenberg$-$Marquardt algorithm in Matlab, use the following:

options $=$ optimoptions$(\text{'}$Isqcurvefit$\text{'},$'Algorithm','levenberg$-$marquardt'$)$;

Solve for the coefficients using isqcurvefit.

You may then use Excel, or Matlab, to calculate and plot the respective theoretical vs$.$

experimental data sets for $s_{}$ and $s_{zz}.$

Lastly, solve for the correlation coefficients comparing theoretical vs$.$ experimental $s_{}$ and $s_{zz}.$