Biomedical engineers often design instrumentation to measure physiological processes, such as blood pressure. To do this they must develop mathematical models of the process. The following equation is a specific case of one model used to describe the blood pressure in the aorta during systole (the period following the closure of the heart’s aortic valve). The variable 1 represents time in seconds, and the dimensionless variable y represents
the pressure difference across the aortic valve, normalized by a constant reference pressure.

y(1) = e-St sin (9.71 + i)

Plot this function for 1 2: O.

Solution

Note that if t is a vector, the’ MATLAB functions exp ( – S*t) and sin (9 . 7*t + pi/2) will also be vectors the same size as t. Thus we must use element-by-element multiplication to compute y(1).

In addition, we must decide on’ the proper spacing to use for the vector t and its upper limit. The sine function sin(9.71 +n/2) oscillates with a frequency of 9.7 rad/sec, which is 9.7 (27r) = 1.5 Hz. Thus its period is 1/1.5 = 2/3 sec. The spacing of t should be a small fraction of the period .in order to generate enough ‘points to plot
the curve. Thus we select a spacing of 0.003 to give approximately 200 points per period.

The amplitude of the sine wave decays’ with time because the sine is multiplied by the decaying exponential e-8r The exponential’s initial value is eO = I, and it will be 2 percent of its initial value at 1 = 0.5 (because e-8(o.S) = 0.02) ..Thus we select the upper limit of t to be 0.5. The session is:

The plot is shown in Figure 2.3-4. Note that we do not see much of an oscillation despite the presence of a sine wave. This is because the period of the sine wave is greater than the time it takes for the exponential e-St to become essentially zero .

Figure 23-4 Aortic pressure response for Example 2.3-3.