**Physiological Modeling**

Recall the cardiac output
problem from last quarter:

To measure cardiac output, a
physician inserts a catheter into the left atrium, injects saline at room
temperature, and measures the temperature in the aorta. The situation can be modeled as below:

You
wish to know:

- The temperature T
_{out}as a function of time, assuming that the saline inflow is constant for a given period of time (t_{in}) and zero otherwise. - The relationship between the area under the
curve (T vs. time) and the flow rate.

Consider assumptions that
are made in clinical applications:

- Saline is injected continuously at a constant
rate over a few seconds (t
_{s}). - Inflow from the vena cava is constant in time.
- Outflow to the aorta is constant in time.
- All heat loss from the heart is a result of
inflow and outflow. I.e., the
circulatory system is adiabatic.
- There is complete mixing within the heart.
- Inflow and outflow profiles are “plug flow.”
- Temperature of the blood from the vena cava is
constant.

None of these assumptions is
true. For each assumption, state why it
is not true. Then provide a reasonable
argument as to why it can be considered “approximately true.”