Question $2[10$ points$]$: $($matlab is the program used for these questions.$)$

The Krogh tissue cylinder is a simplified model for evaluating transport of metabolites from the capillary vessels to the

surrounding tissue. You might see this model in the BME Transport course. The Krogh model geometry is shown below.

The capillary is modeled as a cylinder of constant radius $r_c,$ with wall thickness $t_m,$ oriented in length along the positive z $-$

axis. As a solute moves through the capillary lumen, it diffuses outward in the radial direction to supply the surrounding

tissue.

At some radial $($outward along $r)$ distance into the tissue $(r_{\text{c}\text{r}\text{i}\text{t}}),$ the concentration of any metabolite will have reduced to

zero. This distance gets shorter at increasing distances along the z$-$axis because the solute gets used up$.$ Solution of the

Krogh tissue cylinder model leads to a nonlinear expression describing $r_{\text{c}\text{r}\text{i}\text{t}}$ as a function of distance $z$ along the capillary:

$R^{2}ln(R^2)-R^2+1-[\frac{4D_T{C}_0}{R_0(r_c+t_m{)}^2}]+4\frac{D_T}{V{r}_c^2}[R^2-1]z+\frac{2D_T}{r_c{K}_0}[R^2-1]=0$

where $R=\frac{r_{\text{c}\text{r}\text{i}\text{t}}}{r_c+t_m}.$ This can be written in more compact form as:

$R^{2}ln(R^2)=A+E(R^2-1)$

with constants $,A=\frac{4D_T{C}_0}{R_0(r_c+t_m{)}^2},B=1-\frac{2D_T}{r_c{K}_0},D=\frac{4D_T}{V{r}_c^2},E=(B-Dz)$

$($a$)$ Write a Matlab script to calculate the critical tissue radius $(r_{\text{c}\text{r}\text{i}\text{t}})$ where glucose is no longer supplied to cells, at a

distance $z=0\mathrm{.}025cm$ under the conditions listed in the table. To make this easier, let $R^2=x$ in the equation above and

solve $f(x)=0$ to find $x,$ then convert this value $(R^2)$ to $r_{\text{c}\text{r}\text{i}\text{t}\text{.}}.$ Follow the steps below: $[5$ points$]$

$($i$)$ Plot the nonlinear function $f(x)$ from $x=-100$ to $100.$

$($ii$)$ Using an initial estimate of $x=50$ for the root $($solution$)$ value, use Matlab's built$-$in function with a tolerance

of $0\mathrm{.}01$ to solve for $x.$ Add a circle marker to your plot in $($i$)$ showing this solution.

$($iii$)$ Covert your solution for $x$ to $r_{\text{c}\text{r}\text{i}\text{t}}$ and report the result.

$($b$)$ Generate a plot of $r_{\text{c}\text{r}\text{i}\text{t}}$ versus $z$ as $z$ ranges from $0\mathrm{.}001$ cm to $0\mathrm{.}2$ cm with steps of $0\mathrm{.}001$ cm $.($All other quantities used

in $($a$)$ remain the same$).[4$ points$]$

$($c$)$ Does your plot from $($b$)$ show any unusual behavior? Describe in words $($using disp $(\text{'}\text{'}))$ what you observe and

how this can be reduced by adjusting your solution code. $[1$ point$]$