Problem $1$: Modeling a Physiological System $(40$ pts$)$

You are at a family gathering and one of your family members, who you happen to know has congestive heart failure, is complaining about swollen feet and is saying that they don't understand how a problem with their heart is causing their feet to swell. Being the smart and savy biomedical engineering student that you are, you whip out your laptop $($which you always have on hand for situations like this$)$ and open Matlab and Simulink. Thinking carefully about the relationship between fluid consumption and cardiac function, you buld a simple closed loop model to show your relative how the two are related. You decide to base your model off of the Guyton$-$Coleman Fluid Balance, using the equations below. To make your model more accurate, you ask your family member what they had to drink that day and include it in your model's input $($see Fluid Consumption table below$).$ When you explain the relationship by showing the model to your family member they are very impressed and thank you for helping them understand what was going on with such a clear model!

A$.$ Using Simulink, create a closed loop model system based on the equations below. Set the simulation time to $24$ seconds where $1$ second of simulation time is $1$ hour of real time. Use a $1$D lookup table block driven by a clock to generate your model input, $V_{in},$ based on the table information below. Use a mux to send the simulated data from $V_{in},CO,$MAP, and $V_{\text{o}\text{u}\text{t}}$ into a To Workspace block. Be sure to select a data type that you are comfortable working with on this block as well as a unique variable name.

B$.$ In your script, open and run your Simulink model. Print an image of your model and include it in a text section of your code. Create a set of four subplots showing Vin, CO$,$ MAP, and Vout Vs$.$ time.

C$.$ Copy and paste your model from part A into a new model file. Adjust it so that you can model the system as if fluid is retained and there is no fluid output. In your script, open and run your Simulink model. Print an image of your model and include it in a text section of your code. Create a set of three subplots showing $V_{in},CO,$ and MAP vs$.$ time.

D$.$ In a text section of your script, explain what happened to the system when fluid was retained as compared to the healthy system. How does this relate to the pathology of congestive heart failure? What is the clinical term for fluid retention?

$\frac{d{V}_{in}}{at}=0\mathrm{.}06(V_{in}-V_{\text{o}\text{u}\text{t}})$

$V_{\text{e}\text{x}\text{t}\text{r}\text{a}\text{c}\text{e}\text{l}\text{l}\text{u}\text{l}\text{a}\text{r}}={\displaystyle \underset{0}{\overset{t}{}}}\frac{d{V}_{in}}{dt}dt+V_{\text{e}\text{x}\text{t}\text{r}\text{a}\text{c}\text{e}\text{l}\text{l}\text{u}\text{l}\text{a}\text{r}}(0)$

$V_{\text{B}\text{l}\text{o}\text{o}\text{d}}=0\mathrm{.}33V_{\text{e}\text{x}\text{t}\text{r}\text{a}\text{c}\text{e}\text{l}\text{l}\text{u}\text{t}\text{a}\text{r}}$

$MSP=3\mathrm{.}5(V_{\text{B}\text{l}\text{o}\text{o}\text{d}}-3)$

$CO=\frac{(MSP-P_{RA})}{R_{VR}}=\frac{(MSP-P_{RA})}{0\mathrm{.}07TPR}$

MAP$=CO**TPR$

$V_{\text{o}\text{u}\text{t}}=0\mathrm{.}05(MAP-80)$

Where

Fluid Consumption Information

$\backslash$table$[[$Time$,$Amount,Average Intake Rate$],[8$am$-9$am$,$Coffee and Orange Juice,$8$ oz of each,$8m\frac{L}{m}in$