Annals of Biomedical Engineering, Vol. 25, pp. 470~76, 1997 0090-6964/97 $10.50 + .00 Printed in the USA. All rights reserved. Copyright 9 1997 Biomedical Engineering Society A Theoretical Unidirectional Valve Based on Functional Collapse of Blood Vessels in the Penis OFER BARNEA Department of Biomedical Engineering, Tel Aviv University, Tel Aviv, Israel Abstract--A model of a vessel exposed to external pressure was developed. Analytical derivation resulted in closed-form expres- sions describing pressure-flow relation in the vessel. These ex- pressions describe a behavior of a unidirectional leaky valve. The vessel model was used to represent internal arteries and veins in the penis. Together with a compliant chamber representing the corpus cavernosum, the model demonstrates the valve action of the partially collapsed vessels during penile erection. This ex- plains observations regarding arterial backflow during erectile contraction of the ischiocavernous muscles and demonstrates the differences in development of tumescence and rigidity. Keywords--Hemodynamics, model, penile erection, impotence. INTRODUCTION External pressure exerted on arteries and veins may result in either collapse of the vessel or an increase of internal pressure. Thus, flow in a vessel segment that is exposed to external pressure is reduced either due to in- creased resistance or as a result of increased back pressure. Regardless of the mechanism, the external pressure can control the flow through this segment. A "waterfall" mechanism was described in the coronary circulation, wherein intramyocardial pressure dominates the control of coronary flow rather than the pressure difference at the vessel's ends. The controversy regarding this phenomenon is related to the two previously mentioned mechanisms, namely, is the change in resistance due to reduced radius or increased intravascular pressure? In the penis, during erection, a complex interaction between arterial resistance, corporal compliance, and venous resistance determines pe- nile tumescence (volume) and rigidity (1,6). These two are the primary characteristics of erection. During the flaccid state of the penis, the compliance of both arteries and veins is low. Because both vessels are located in the corpus cavemosum, they are both exposed to corporal pressure. The common explanation to the low corporal pressure during the flaccid state is that arterial Address correspondence to Ofer Bamea, Department of Biomedical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel. (Received 4Apr96, Revised 6Aug96, Revised, Accepted 22Aug96) resistance is high in relation to venous vessels (1,3). Nor- mal flow rate is -5 ml/min (8). Erection begins with an increase in the compliance of smooth muscles surrounding arterial and venous blood vessels. This results in increased arterial flow rate to -20 ml/min (8) leading to increased corporal pressure and possible collapse of veins. In the erect state, the relaxation of the muscular layer surround- ing the blood vessels result in a corporal pressure that is close to arterial pressure. The arterial pressure gradient becomes small; and due to increased venous resistance (partially collapsed veins), venous draining flow is small and is similar to flow during the flaccid phase but can be either higher or lower. Thus, flow is similar in both steady states, but the large pressure gradient is shifted from the arterial section to the venous section, thereby increasing corporal pressure. Further increase of pressure due to con- traction of the ischiocavernous muscle (surrounding the corpus cavernosum) does not result in significant back- flow through the artery or increased flow out of the venous draining vessels. This mechanism was described in a model of the mechanics of erection (10). It is widely agreed that the vein collapses, but the de- gree of collapse is controversial. It is not clear whether there is flow through the penile veins during erection or not, despite the fact that measurements of oxygen content in the penis during erection show that it is continuously oxygenated. The measured values are Po: = 35 mm Hg during the flaccid phase, and between 60 and 70 mm Hg during erection (2,7,11). The mechanism that prevents backflow in the arterial system during penile erection has not been identified. In a previous work, we demonstrated the limiting effect of corporal pressure on venous flow (4). In this work, a more general model of a blood vessel exposed to external pressure has been developed and integrated into a simple model of the penis for both arterial and venous vessels. Using the model, we demonstrate that, in this configura- tion, the blood vessels act as unidirectional valves. Fur- thermore, we show that increased corporal pressure does not result in increased flow neither forward via venous vessels nor backward into systemic circulation through the artery. 470