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LETTERS TO THE EDITOR
Biomedical Engineering
Indiana University Purdue University
Indianapolis, Indiana
e-mail: gkassab{at}iupui.edu
The following is the abstract of the article discussed in the subsequent letter:
The pulsatility of coronary circulation can be accurately simulated on the basis of the measured branching pattern, vascular geometry, and material properties of the coronary vasculature. A Womersley-type mathematical model is developed to analyze pulsatile blood flow in diastole in the absence of vessel tone in the entire coronary arterial tree on the basis of previously measured morphometric data. The model incorporates a constitutive equation of pressure and cross-section area relation based on our previous experimental data. The formulation enables the prediction of the impedance, the pressure distribution, and the pulsatile flow distribution throughout the entire coronary arterial tree. The model is validated by experimental measurements in six diastolic arrested, vasodilated porcine hearts. The agreement between theory and experiment is excellent. Furthermore, the present pulse wave results at low frequency agree very well with previously published steady-state model. Finally, the phase angle of flow is seen to decrease along the trunk of the major coronary artery and primary branches toward the capillary vessels. This study represents the first, most extensive validated analysis of Womersley-type pulse wave transmission in the entire coronary arterial tree down to the first segment of capillaries. The present model will serve to quantitatively test various hypotheses in the coronary circulation under pulsatile flow conditions.
To the Editor: Rogers et al. (2) provided an editorial commentary titled "Is there a need for another model on the pulsatile nature of coronary blood flow?" We thank the authors for their kind appraisal of our study (1) and appreciate the opportunity to clear up some of the misunderstandings. The model is unlike any other in that it is based on the anatomy (diameters and lengths) of the entire coronary arterial tree to the capillaries. The authors are correct that the literature is replete with lumped models of pulsatile analysis of coronary circulation. This is not "another" model, however, in that it is a distributive model anchored on vascular geometry and branching pattern unlike the previously published lumped models of pulsatile flow analysis. Regrettably, the extensive structural basis of the model went unnoticed in this commentary.
In the commentary, Rogers et al. (2) prompted, "the question that we have about such model information is the discrepancy from published results." They raise three major issues that are perceived limitations of the model that will be addressed in turn:
1) For example, two studies report significant pressure losses in vessels greater than 100–140 µm in diameter, whereas one reports minimal pressure dissipation in vessels of this size. Yet the model predicts minimal pressure losses in vessels down to70 µm in diameter, which is at odds with two reports.
To interpret a model prediction, one must understand the context. Figure 12 of our paper (1) shows the pressure profile along one particular pathway from the inlet [left anterior descending (LAD) artery] to the first capillary segment. The pathway is defined along the trunk whereby the larger branch is selected at every bifurcation. Hence, this pathway is a lower resistance path, which explains the smaller pressure drop. In other words, of the thousands of 100-µm vessels, this figure shows the one that has a smaller pressure drop.
2) Many groups have reported negative flows during systole in the coronary circulation and microcirculation; thus, there is likely an incorrect assumption made by the model calculations and an unknown variable influencing the experimental values. Another observation made is that peak flow rate in the left coronary circulation coincides with peak pressure measured at the inlet both in the absence and presence of myocardial contraction.
There is no doubt that coronary flow tends to zero or even reverses in systole during contraction. Here again lies a misunderstanding of the Huo and Kassab model. It was stated that the present model represents pulsatile flow in a diastolic heart. Although this is physiologically unrealistic at this point, it is a work in progress. The effect of muscle/vessel interaction that causes negative flows and the asynchrony of the peak pressure between the heart and the systemic circulation is certainly the next stage of model development as stated in the study.
3) Another limitation of the model should be mentioned; namely, it makes no prediction about endocardial-epicardial variations in pressures, flows, and impedances.
Again, this is an issue of the stage of the model as discussed in the study. The distribution of the arterial tree in the three-dimensional space of the heart has been recently completed, and we shall report on the physiology of the spatial distribution of blood flow shortly.
Rogers et al. (2) point out that one can find discrepancies from published experimental results with any model. Although this is certainly true, it should force us to reexamine the axioms and assumptions of the experiment and theory, which provides a real opportunity to discover something new. Theory and experiment must be integrated as this is critical to the progress of our understanding of coronary circulation, particularly in the deep layers of the heart that elude direct observations.
REFERENCES
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