Project 1: Circulation This project refers to the matlab files located at: http://www.math.nyu.edu/faculty/peskin/modsimprograms/ch1/. Model of the systemic arteries. The first thing to do is adjust the systemic arterial compliance by trial and error in order to achieve a blood pressure of 120/80 mm Hg. The adjusted compliance should be considered as the normal compliance in all subsequent studies. Next, check that the time step is small enough by reducing it by a factor of two and comparing results. (Note that one has to run through twice as many steps at the smaller time step to cover the same amount of time.) Try different initial pressures to check that the pressure settles down to the same periodic steady state independent of the initial pressure and to see how long it takes to get close enough to the periodic steady state that you don t notice any changes from one heartbeat to the next. In the following paragraphs all of the parameter changes mentioned are supposed to be made starting from a standard set of normal parameters. The student has to be careful to organize the work in this way and not to let the changes made in one computer experiment stay around and incorrectly define the reference state for the next experiment. One good way to avoid this pitfall is to run a standard initializing script that sets up normal values and then run a separate script that makes the required parameter changes for the experiment at hand. This is much better than editing the initializing script, for at least two reasons: It documents all of the changes that define a particular experiment in one place, and it avoids the task of undoing all edits to get back to the standard state. With the preliminary studies out of the way, try modeling the aging process by reducing the systemic arterial compliance (hardening of the arteries), e.g., by a factor of 2. What happens to the systolic, diastolic, mean, and pulse pressures? Can you explain the results? Next, consider normal exercise. During exercise, systemic resistance falls because of auto-regulation in the working muscles. As an example, consider a reduction in systemic resistance by a factor of 2. The response of the body is to double the cardiac output (so that the systemic arterial pressure will not fall), and this is achieved primarily by an increase in heart rate, not stroke volume. We can double the heart rate in our model by halving all three of the times T (the duration of the cardiac cycle), T S (the duration of systole), and T max (the time of peak aortic flow, relative to the onset of systole). To keep the stroke volume at its normal value in the face of these changes in timing, we must double Q max. (In reality, when heart rate changes, the corresponding changes in the duration of systole are less pronounced than those in the duration of the whole cardiac cycle.) Now consider a patient whose heart rate cannot respond to exercise. In such a patient, the needed increase in cardiac output is accomplished by increasing the stroke volume. Thus, the response to reduction in systemic resistance by a factor of 2 will be simply a doubling of Q max, with no changes is T, T S or T max. (It is interesting that the same change, doubling Q max, that was needed in the normal exercise case to maintain stroke volume is needed here to increase the stroke volume!) Try this and compare the pressure and flow waveforms to those of normal exercise. Yet another variation of the exercise theme comes about when heart rate increases 1
without any need for increased cardiac output, as in anxiety. To simulate this, leave the systemic resistance at its normal value, halve the timing parameters T, T S and T max as in normal exercise, but leave Q max at its normal value so that the stroke volume falls just enough to keep cardiac output despite the increased heart rate. How does the pulse of an anxious person compare with the pulse of an exercising person when both have the same heart rate? Finally, see what happens to the arterial pulse when the normal periodic pattern of heartbeats is interrupted. Suppose cardiac arrest occurs for some period of time, and then the heart is restarted. What happens to arterial pressure during the arrest period, and how does it recover afterwards? To simulate this, we have to modify the function QAo now (t) so that its output is zero between two specified times t arrest and t restart, and normal otherwise. (This ignores many complexities, since it assumes that the heart immediately resumes its normal rate, rhythm, and stroke volume, which is unlikely to be the case.) Let us now turn our consideration to the program that simulates the left heart coupled to the systemic arteries. Of course, one can essentially repeat the foregoing studies with this model. But consider the new opportunities opened up by the new model, and these concern the function of the valves. Recall that the left heart has an inflow (mitral) and an outflow (aortic) valve. A heart valve can malfunction in either of two distinct ways (or some combination of the two). These are stenosis in which the valve is narrowed and presents a high resistance to forward flow, and incompetence or insufficiency in which the valve is leaky and fails to prevent back flow. In our model of the left heart and systemic arteries, stenosis is easier to model than insufficiency, since we already have the necessary parameters in the model, RMi (resistance of the aortic valve) and RAo (resistance of the aortic valve). Increasing either of these resistances simulates stenosis of the corresponding valve. Try this in each case, over a range of different valve resistances, and observe the effects on stroke volume, left ventricular blood pressures, and systemic arterial blood pressure. Compare the effects of mitral stenosis to those of aortic stenosis. Note that these are the uncompensated effects of these conditions. To study mitral and/or aortic insufficiency in our model of the left heart and systemic arteries we have to modify the model by providing a parallel path for blood flow in the reverse direction through the valve. For example, the flow through the mitral valve, which was previously given by S Mi (P LA P LV )/R Mi, will now be given by (S Mi /R MiF + (1 S Mi )/R MiB )(P LA P LV ) where R MiF is the resistance of the mitral valve to forward flow and R MiB is the back flow resistance of the mitral valve. For a healthy mitral valve R MiF is very small, and R MiB is very large. Mitral stenosis is modeled by increasing R MiF and mitral insufficiency is modeled by decreasing R MiB. (Although we never have R MiF > R MiB.) The corresponding modifications in the formula for aortic flow need to be made in order to model aortic insufficiency. Once these changes are in place, one can do similar studies of insufficiency to those outlined above for stenosis. So far, we have considered only the uncompensated effects of valvular malfunction. In fact, the circulation makes various automatic adjustments which can be viewed as compensations for the effects of valve disease. Simulations that include some of these compensations will be discussed in the following paragraphs. In mitral stenosis, left atrial pressure rises to maintain stroke volume on the left side of 2
the heart. To simulate this, first do an uncompensated simulation of mitral stenosis, and observe the fall in stroke volume. Then see how much the left atrial pressure (a parameter in the model) has to be increased in order to bring the stroke volume back to normal. Plot the compensated left atrial pressure as a function of the resistance of the mitral valve. Note that left atrial pressure is essentially the same as pulmonary venous pressure, and that once pulmonary venous pressure exceeds about 28 mm Hg, fluids are pressed out into the air spaces of the lung (pulmonary edema). This puts a ceiling on the severity of mitral stenosis that can be tolerated. It would be of great interest to repeat the above study under exercise conditions, since patients with mitral stenosis may be asymptomatic at rest but may experience pulmonary edema upon attempting to exercise. In aortic stenosis, the primary compensation is an increase in thickness of the left ventricular wall, in response to the elevated left ventricular pressure. This shows up in the model as a reduction in compliance. Both the systolic and diastolic compliances will be reduced. The reduction in systolic compliance is helpful to the left ventricle during ejection, but the reduction in diastolic compliance impedes filling, and this causes a secondary compensation, which is a rise in left atrial pressure. To model this whole story, first postulate a definite relationship between ventricular compliance and ventricular pressure. For example, let both the systolic and diastolic left ventricular compliances be inversely proportional to the maximum (systolic) left ventricular pressure, the constants of proportionality being determined by the normal values. It will require an iterative process to find the right compliances for any given level of aortic stenosis, since the compliance depends on the pressure, which depends on the compliance. This iterative process involves running the program, observing the systolic left ventricular pressure, setting the diastolic compliances accordingly, re-running the program, etc. Then, for any given level of aortic stenosis, use trial and error to find the left atrial pressure that makes the stroke volume normal. Plot the compensated left atrial pressure, left ventricular pressures (systolic and diastolic), systemic arterial pressures (systolic and diastolic), and left ventricular compliances (systolic and diastolic), as functions of the resistance of the aortic valve. Finally, we consider studies that can be done with the whole circulation model. Of course, these include essentially all of the studies described above for the two partial circulation models that we have considered, except that certain changes have to be made to take into account the differences with regard to which quantities are subject to our direct control. For example, in the whole circulation model PLA is an unknown, and we are not free to vary it at will. On the other hand, at least part of the compensatory change in PLA that we have described in the case of mitral or aortic stenosis will just happen by itself in the whole circulation model, as a consequence of a shift in volume from the systemic to the pulmonary circulation. It would be interesting to see how effective this compensatory mechanism is. The rest of the change in PLA can be produced as needed by changing the blood volume, a parameter that is subject to our control in the whole circulation model. In the body, such compensatory changes in blood volume are produced by feedback mechanisms involving the kidney. Applications: To simulate the fetal circulations, it is necessary only to lay out the whole circulation model in the manner shown in Figure 1, and then to choose appropriate fetal parameters, some of which at least can be inferred from data on fetal pressures and flows such as those available in Rudolph s book. Other parameters may have to be guessed or adjusted by 3
trial and error. It would then be interesting to watch the changes in the circulation that occur at birth. These would be induced by suddenly decreasing the pulmonary resistance (first breath) and suddenly increasing the systemic resistance (constricting or clamping of the umbilical cord). Note the resulting closure of the foramen ovale and the reversal of mean flow in the ductus arteriosus. Some congenital heart diseases that can be simulated with the whole circulation model include ventricular septal defect (VSD), which is a connection between the two ventricles, atrial septal defect (ASD), which is a connection between the two atria, and patent ductus arteriosus (PDA), which is the persistence of the fetal connection between the pulmonary artery and the aorta. All of these connections are unvalved, i.e. their resistances are the same in both directions. In severe cases, these resistances are very low. All three conditions result in postnatal left-to-right shunts, with high pulmonary flow, a dangerous condition for the development of the lung. One might consider banding the pulmonary artery (increasing pulmonary resistance) as a way of reducing or eliminating the leftto-right shunt and thereby equalizing the pulmonary and systemic flows. What are the consequences of this strategy in each of the three conditions? (As a practical matter, 4
banding of the pulmonary artery would not be used in PDA, since it would be as easy to close the defect itself. In the other two cases, though, the defects lie within the heart, so an extracardiac procedure such as banding the pulmonary artery might be useful as a temporary measure. Your mission is to use computer simulation to see what the circulatory consequences of this procedure might be!) Rudolph, A.M.: Congenital Diseases of the Heart, Clinical-Physiological Considerations. 2nd Ed. Futura Publishing Company, Inc. 2001. 5