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Ann Thorac Surg 1999;68:1442
© 1999 The Society of Thoracic Surgeons
a Department of Cardiac and Vascular Surgery, German Heart Center Munich, Technical University of Munich, Lazarettstr 36, 80636 Munich, Germany
b Laboratories of Experimental Cardiac Surgery, University of Heidelberg, 69120 Heidelberg, Germany
c German Heart Center Munich, 80636 Munich, Germany
e-mail: bauernschmitt{at}dhm.mhn.de
To the Editor
We appreciate Dr Poullis’ interest in our article and his discussion remarks. Modeling physiologic systems like the cardiovascular system may always raise questions, as there are fundamentally different approaches to solve the problem. There are several points stressed by the author on which we would like to comment.
Modeling the arterial tree using the basic transmission line equations has a long tradition in understanding arterial hemodynamics. Wormersley and Taylor [1,2] solved and proved first the Navier–Stokes equations with respect to the transmission line theory we used in our paper. Over the years Westerhof, Noordergraf, O’Rourke, and Avolio [3–6] developed and refined this theory for use with computer models. We can see no evidence that the difference equations that we used, which are based on a fundamental and basic theory found in any electrical engineering textbook, break any fundamental law of electronic circuit analysis.
The model does account for matched impedances, which can be proved easily by calculating the resonance frequencies and impedances of the segments by the well-known equations. In addition, a huge impedance mismatch would always lead to instabilities and oscillations in the whole system, which is not the case in our situation.
From the basic transmission line theory it is well known that a transmission line has to be terminated by its characteristic impedance to avoid power reflections. If not, more or less strong reflections are the result. Modeling the arterial transmission line, the steady state of the simulation shows the expected pressure and flow time curves that result from superimposed reflection waves. The comparison with experimental data shows a very good correspondence as well, so the reflection phenomena are visible in the results.
For us it is not clear what is meant by an open circuit model or a model that is isolated from the subsequent states, because our model includes several feedback and feedforward mechanisms. However, including a C component in the first equation means that the nonlinear axial component would not be neglected. This, of course, is possible, but the axial effect is so small that there seems to be no benefit for simulation of cardiovascular dynamics. The same is true for the inertia of the mass pushed toward the radial direction. It is very small and can be neglected as well.
On the other hand, we are convinced that it is possible to perform the simulation with the software package Dr Poullis seems to prefer, and maybe it will provide additional information. So it would surely be a good idea to do so and to compare the results with those we found in our simulation.
In addition, we consider Dr Poullis’ closing remark extremely important. Optimization of the patients’ heart rate with respect to the fundamental frequency of systemic vascular impedance indeed may have major benefit in the postoperative treatment strategies of any cardiac surgical patient. There is no doubt that this new approach justifies performing large clinical studies to prove its benefit.
References
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