Description
Analyzing a biological system modeled by an LCCDE: As described in the lecture, the relationship
between cardiac output and aortic blood pressure can be represented with a second-order LCCDE
model of the arterial system. However, blood pressure is more easily measured in a peripheral
artery than the central aorta, and the peripheral blood pressure signal (𝑝𝑝
(𝑡)) differs
substantially from the central blood pressure signal (𝑝𝑐
(𝑡)). A third-order LCCDE model of the
arterial system may be more useful in that it can represent the relationship between the left
ventricular blood flow rate signal (𝑞̇
𝑙
(𝑡) whose average gives the cardiac output) and both 𝑝𝑝
(𝑡)
and 𝑝𝑐
(𝑡). Figure 1 shows the model in electrical analog form, where voltage corresponds to
blood pressure; charge, to blood volume; and current, to blood flow rate.
Figure 1: Third-order LCCDE model of arterial system.
The parameters of the model are 𝐶𝑐 and 𝐶𝑝, which represent the central and peripheral
“compliance” or blood volume storage ability; 𝐿𝑐
, which represents the central “inertance” and
indicates the pressure required to accelerate blood; and 𝑅𝑝, which represents the peripheral
resistance to blood flow. This circuit model is governed by the following two third-order LCCDEs.
𝐿𝑐𝑅𝑝𝐶𝑐𝐶𝑝
𝑑
3𝑝𝑝
(𝑡)
𝑑𝑡
3 + 𝐿𝑐𝐶𝑐
𝑑
2𝑝𝑝
(𝑡)
𝑑𝑡
2 + 𝑅𝑝(𝐶𝑝 + 𝐶𝑐)
𝑑𝑝𝑝
(𝑡)
𝑑𝑡 + 𝑝𝑝
(𝑡) = 𝑅𝑝𝑞̇
𝑙
(𝑡)
𝐿𝑐𝑅𝑝𝐶𝑐𝐶𝑝
𝑑
3𝑝𝑐
(𝑡)
𝑑𝑡
3 + 𝐿𝑐𝐶𝑐
𝑑
2𝑝𝑐
(𝑡)
𝑑𝑡
2 + 𝑅𝑝(𝐶𝑝 + 𝐶𝑐)
𝑑𝑝𝑐
(𝑡)
𝑑𝑡 + 𝑝𝑐
(𝑡)
= 𝐿𝑐𝑅𝑝𝐶𝑝
𝑑
2𝑞̇
𝑙
(𝑡)
𝑑𝑡
2 + 𝐿𝑐
𝑑𝑞̇
𝑙
(𝑡)
𝑑𝑡 + 𝑅𝑝𝑞̇
𝑙
(𝑡)
2
1. Apply Laplace Transform techniques with “paper and pencil” to find the system functions,
𝐻𝑝
(𝑠) = 𝑃𝑝
(𝑠) 𝑄̇
𝑙 ⁄ (𝑠) and 𝐻𝑐
(𝑠) = 𝑃𝑐
(𝑠) 𝑄̇
𝑙 ⁄ (𝑠).
2. Let 𝐶𝑐 = 1.6 ml/mmHg, 𝐶𝑝 = 0.2 ml/mmHg, 𝐿𝑐 = 0.015 mmHg/(ml/s
2
), and 𝑅𝑝 = 1.1
mmHg/(ml/s). Use the built-in functions, freqs, tf, bode, and impulse, to plot the frequency
response, Bode plot, and impulse response for 𝐻𝑝
(𝑠). Does this system show overdamped,
underdamped, or critically damped characteristics? What MATLAB function can you use to
verify your answer to this question?
3. Figure 2 shows a simple sine model of one beat of the input signal 𝑞̇
𝑙
(𝑡). Create a vector to
define the sine beat signal at a sampling interval of 0.001 sec. Set 𝑆𝑉 (stroke volume, which
is the amount of blood ejected by the left ventricle per beat) to 80 ml, and 𝑇 (the beat length)
to 1 sec. Now create another vector to define a train of at least 20 unit-impulses spaced apart
by T with zeros inserted in between. Use the same sampling interval for this “impulse train”.
Finally, form a periodic or “pulsatile” input signal by convolving the sine beat signal with the
impulse train. Plot the input signal. Why does this method work?
Figure 2: Sine model of one beat of the left ventricular blood flow rate.
4. Use the built-in function lsim to determine the output 𝑝𝑝
(𝑡) of the system with system
function 𝐻𝑝
(𝑠) in response to the periodic sine flow rate waveform. Plot the output signal.
How do you think this built-in function works? What is the cause of the transient part of the
output?
5. Determine the output 𝑝𝑐
(𝑡) of the system with system function 𝐻𝑐
(𝑠) to the same input. Plot
the output signal. Compare the two steady-state outputs. Are the results what you expected?
6. Vary the model parameters (𝑆𝑉, 𝑇, 𝐶𝑐
, and 𝑅𝑝) individually by ±50% and compute the output
for each case. How do the cardiac output, mean blood pressure, and peripheral pulse pressure
(systolic minus diastolic blood pressure) change in response to each parametric variation?
Show exemplary plots to support your answers. What are biological models good for?
Deliverables: Submit a single pdf file containing answers to the questions, properly labeled
plots, and the source code.