1 from SIAM News, Volume 34, Number 5 Modeling the Cardiovascular System— A Mathematical Adventure: Part I Mathematical and numerical investigations of the cardiovascular system, although a relatively new research area, will give rise to some of the major mathematical challenges of the coming decades, the author writes. In this first of two parts, he sketches some of the features that make the human blood circulatory system so challenging to model. The second part of the article (to appear in the next issue of SIAM News) will describe research now under way, including the development of hybrid multiscale models. Quarteroni will also be presenting research in this area as one of ten invited speakers at the first EMS–SIAM meeting, “Applied Mathematics in our Changing World,” which will be held in Berlin, September 2–6. By Alfio Quarteroni The physiology of the cardiovascular system has been elucidated only gradually, over many centuries. Among the major actors in the lengthy process have been some of the central characters in human history. Aristotle (384–322 B.C.), for example, identified the role of blood vessels in transferring “animal heat” from the heart to the periphery of the body (although he ignored blood circulation). In the third century B.C., Praxagoras realized that arteries and veins have different roles (believing that arteries transported air while veins transported blood). Galen (c. 130–200 A.D.) was the first to observe the presence of blood in arteries. Much later, in the 17th century, Sir William Harvey inaugurated modern cardiovascular research with his De Motu Cardis and Sanguinis Animalibus, in which he wrote, “When I turned to vivisection I found the task so hard I was about to think that only God could understand the heart motion.” His plaintive comment notwith- standing, Harvey observed that the morphology of valves in veins is such that they are effective only if blood is flowing toward the heart. His conclusion: “I began privately to consider if it (the blood) had a movement, as it is, it would be in a circle.” In the 18th century, the Reverend S. Hales introduced quantitative studies of blood pressure (Hemostatics, 1773). Later, Euler and D. Bernoulli both made great contributions to fluid dynamics research. In particular, Bernoulli, investigating the laws governing blood pressure as a professor of anatomy at the University of Basel in Switzerland, formulated his famous law relating pressure, density, and velocity: p + 1/2r êu ê 2 = const (vis viva equation, 1730). In the 19th century, J.P. Poiseuille, a medical doctor and a physicist, was studying the flow of blood in arteries when he derived the first simplified mathematical model of flow in a cylindrical pipe, a model that still bears his name today. T. Young later made fundamental contributions to research on elastic properties of arterial tissues and on the propagation of pressure: “The inquiry in what manner and at what degree the circulation of the blood depends on the muscular and the elastic powers of the heart and of the arteries, supposing the nature of these powers to be known, must become simply a question belonging to the most renowned departments of the theory of hydraulics” (from a lesson given by Young at the Royal Society of London in 1809). At the beginning of the 20th century, 0. Frank introduced the idea that the circulatory system is analogous to an electric network. In 1955, J. Womersley, studying vascular flows, found the analytical counterpart of Poiseuille flow in pressure gradients that vary periodically in time, a situation that more closely describes actual pressure variations during the cardiac cycle. In the second half of the 20th century, developments in mathematical modeling were limited to basic paradigms, such as flow in morphologically simple regions (e.g., Poiseuille or Womersley solutions), or to models based on electric network analogies. Exact solutions are very difficult to obtain in more general situations, because of the strong nonlinear interactions among different parts of the system and the geometric complexities of individual vascular morphologies. Flow Patterns and Arterial Disease As in many applied sciences, mathematical and numerical models are increasingly important in biology and medicine today. Most notably, mathematical and numerical investigations of the blood circulatory system, although energetically pursued for only a few years, are poised to become one of the major mathematical challenges of the coming decades. During the 1970s, in vitro or animal experiments were the main mode of cardiovascular investigations (see, for example, [9]). In recent years, advances in computational fluid dynamics, together with dramatic improvements in computer performance, have resulted in significant breakthroughs that promise to revolutionize vascular research (see, for example, [2], [3], [5], [8]). Physical quantities like shear stress, which are troublesome to measure in vivo, can be computed for real geometries with the It was in studying arterial blood flow that 19th-century physician and physicist J.P. Poiseuille developed the first simplified mathematical model of fluid flow in a cylin- drical pipe.