A summary of the three-dimensional fiber structure of the McQueen/Peskin model of the heart and great vessels.
The following material has been excerpted from Appendix A of (B.E. Griffith. Simulating the blood-muscle-valve mechanics of the heart by an adaptive and parallel version of the immersed boundary method. PhD Thesis, Courant Institute of Mathematical Sciences, New York University, 2005).
The purpose of the present web page is to summarize for the convenience of the reader the major features of the three-dimensional fiber model of the heart and great vessels developed by Peskin and McQueen [Peskin89, PeskinMcQueen94, PeskinMcQueen96, McQueenPeskin00, McQueenPeskin01]. A detailed account of all of the technical details of this model is beyond the scope of the present work, and the interested reader should consult the foregoing references for a more complete description of the model. Instead, we focus here on describing the features of the model that are most relevant to the simulation results presented here.
The heart is a four chambered organ that consists of two pumps-the right and left sides of the heart-that are responsible for pumping blood through the lungs and through the peripheral organs, respectively. Each side of the heart consists of two chambers, an atrium and a ventricle, with the weaker atrium acting as a receiving chamber and as a primer pump for the more powerful ventricle. In the body, the pulmonary veins supply oxygen-enriched blood from the lungs to the left atrium, which in turn empties into the left ventricle through the mitral valve. The muscular left ventricle ejects blood through the aortic valve into the ascending aorta, through which it is distributed to all the tissues of the body (including the heart itself via the coronary arteries). The corresponding structures of the right side of the heart perform similar functions, but in this case the purpose of the heart is to return oxygen-depleted blood to the lungs to be reoxygenated. In particular, the superior and inferior vena cavae return blood from the tissues of the body to the right atrium, which in turn empties into the right ventricle through the tricuspid valve. The thin-walled right ventricle ejects blood through the pulmonic valve into the pulmonary artery, through which the blood is returned to the lungs.
The two ventricles of the heart together form a structure that is often described as possessing a somewhat conical shape, with an apex at the bottom of the heart and a base towards the top. Although the axis of this cone is not constant as it passes from apex to base, it is convenient (and traditional) to consider this axis as being vertical in anatomical discussions of the isolated heart. (Note that from this brief description of the shape of the ventricles, one might conclude that the ventricles achieve their maximum diameter at the base of the heart. This is not the case; rather, they are widest on a plane slightly below the base that is known as the equatorial plane of the heart.) The valves of the heart are nearly coplanar and essentially lie in the plane of the base. Above the base lie the ascending aorta and the main pulmonary artery as well as the left and right atria, into which are inserted the pulmonary veins and the superior and inferior vena cavae.
The three-dimensional model of the heart includes representations of each of the major structural features of the heart and nearby great vessels. These include the four chambers of the heart, i.e., the left and right ventricles and atria; the two atrioventricular (inflow) valves; the two arterial (outflow) valves; the veins that return blood to the atria, i.e., the pulmonary veins and the superior and inferior vena cavae; and finally the major arteries that carry the blood ejected by the ventricles, i.e., the ascending aorta and the main pulmonary artery. Each of these structures is modeled as a system of elastic fibers: in the case of the valves, the fibers mainly correspond to passive collagen fibers; in the great vessels, they correspond to smooth muscle tissue; and in the myocardium, they correspond to active muscle fibers that possess time-dependent contractile properties. Note that the model does not include a detailed description of the circulation. Instead, the inflow and outflow vessels all have blind ends, and fluid sources and sinks are provided to establish realistic pressure loads on each side of the heart.
The muscle fiber geometry of the model ventricles
is based on the dissections of C.E. Thomas, who chemically
removed the connective tissue between the muscle fascicles of
porcine and canine hearts and then carefully tracked the spatial
paths of small bundles of fibers isolated with tweezers
[Thomas57]. Thomas describes the ventricular muscle fibers as
being organized into a number of layers, with each fiber
beginning and ending at one of the valve rings. Each layer
is described by Thomas as consisting of two sheets: one on
which fibers spiral away from the base, and another on which they
return to the base. The model of the ventricles includes a subset
of the layers qualitatively described by Thomas, namely:
The initial configuration of the fibers of the model ventricles are taken to be geodesic curves on double-sheeted surfaces corresponding to each of the foregoing classes of fiber layers. (Note that determining the muscle fibers as geodesic curves along fiber surfaces is a procedure that is supported by the experimental work of Streeter et al. [Streeter78]. The precise manner in which these two-sheeted surfaces are defined is discussed in, e.g., [PeskinMcQueen96, McQueenPeskin01].) Generally speaking, below the equatorial plane, each individual surface is taken to be a portion of a cone, whereas above the equator, each sheet is continued to the valve rings by means of an interpolating surface.) For most of the ventricular layers, the two sheets meet along a common boundary curve that lies somewhere in the interior of the heart. Since the sheets typically are not tangent where they meet, the only way that a fiber can smoothly transition from one sheet to the other is if it aligns with the curve where the sheets intersect. Thus, the fiber structure of a particular layer of the model heart is determined by requiring that each fiber initially be tangent to the common boundary curve, and by then continuing each fiber as a geodesic curve in both directions along each of the two sheets until it eventually encounters one of the valve rings at the base of the heart.
Note that the right-inner/left-outer layer requires a slightly different treatment, since the two sheets of this layer lie side-by-side, and rather than meeting at a common boundary curve, they share a common face along the interventricular septum. To obtain the initial configuration of the fibers that make up this layer, the transition between the two sheets is taken to consist of the vertical mid-line of the shared face, and the fibers are required to be perpendicular to this mid-line. This defines an initial direction for each fiber, and so the fibers may be constructed as before, by continuing them in each direction as geodesic curves that terminate at the valve rings.
The model fibers that make up the atria and great vessels are similarly taken to be geodesic curves on surfaces, although the construction of these fiber surfaces is somewhat less involved than that of the layers that comprise the ventricles. Each of the veins is simply described as a cylinder that is hemispherically capped at one end and open at the other. The ascending aorta and pulmonary artery are described in a similar manner, except that each must be enlarged near its open end to account for the sinuses that support the arterial valve leaflets. In the case of the arteries, the open (uncapped) ends are connected to the appropriate arterial valve rings on the base of the model heart, whereas in the case of the veins, the open ends are taken to be locations at which the veins insert into the appropriate atrium. The structure of the right atrium is determined by constructing an interpolating surface that simply connects the tricuspid valve ring to the open ends of the superior and inferior vena cavae.
The construction of the left atrium is similar; however, in this case, the model includes a description of the left atrial appendage (auricle). Like the great veins and arteries, the auricle is described as a hemispherically capped tube; however, in this case, the initial surface is not a right circular cylinder but rather is a deformation of a right circular cylinder. In particular, the initial configuration of the auricle is taken to be a half-open cylinder that has been ``sheared'' so that it does not intersect the wall of the left ventricle near the equatorial plane of the model heart. The surface of the left atrium is then specified to be an interpolating surface that connects the mitral valve ring and the open end of the auricle (which both lie in the plane of the base of the model heart) to the open ends of the pulmonary veins above (all four of which lie in a plane parallel to that of the base).
The initial configuration of the muscle fibers of the model heart are displayed in Figures 1-7. Note that many of the unrealistic features that appear in the initial configuration (such as the point at the apex or the sharp edge that appears at the equatorial plane) smooth out as the heart is filled with blood during the initial part of a simulation.
The valves of the model heart are also constructed out of fibers, in this case corresponding to the passive collagen fibers that support the valve leaflets. The initial configuration of the fibers that comprise the leaflets of the atrioventricular valves are determined much as before, namely in terms of geodesic curves that wrap somewhat arbitrarily specified initial surfaces. On the other hand, the initial closed-valve configuration of the aortic valve is derived from its function, which is to support a uniform pressure load. In particular, the fibers that comprise the aortic valve are determined via the solution of a system of partial differential equations that describes the mechanical equilibrium of a one-parameter family of fibers under tension [PeskinMcQueen94]. The initial structure of the pulmonic valve is taken to be identical to that of the aortic valve, although note that the elastic properties of the two valves are different in accordance with the different pressure loads each must maintain. The initial configuration of the model heart valve leaflets are displayed in Figures 8-10.
Click on any image to view the full-sized version of that image.
Figure 1: The initial fiber structure of the model heart, as viewed from the front of the heart. On the left side of the heart (which appears on the right side of the figure), the four pulmonary veins supply blood to the left atrium, which in turn empties into the left ventricle through the mitral valve (which is obscured in the present figure). The muscular left ventricle ejects blood through the aortic valve into the ascending aorta. On the right side of the heart (which appears on the left side of the figure), the superior and inferior vena cavae return blood to the right atrium, which in turn empties into the right ventricle through the tricuspid valve, although of these only the right ventricle is readily observed in the present figure. The thin-walled right ventricle ejects blood through the pulmonic valve into the main pulmonary artery. Note that in the model, the inflow and outflow vessels all have blind ends, but sources and sinks are provided to establish realistic pressure loads on each side of the heart. Many of the unrealistic features that appear in the initial configuration (such as the point at the apex or the sharp edge that appears at the equatorial plane) smooth out as the heart is filled with blood during the initial part of a simulation. (In the present figure and all subsequent figures, only a subset of the muscle fibers are displayed, whereas all of the collagen fibers that comprise the heart valve leaflets are shown.)
Figure 2: The ventricular muscle fibers of the model heart, as viewed from the front of the heart, so that the right ventricle again appears on the left side of the figure. The present figure includes the outer/inner layer, the right-inner/left-outer layer, and the internal left-ventricular layers described above. The four coplanar valve rings are indicated by black markers and form the base of the heart. From this view, the aortic valve ring appears near the center of the figure, with the pulmonic valve ring appearing slightly below and to the left. The larger mitral and tricuspid valve rings appear respectively to the right and back of the aortic valve ring. (As before, only a subset of model muscle fibers are displayed.)
Figure 3: The four nested internal left-ventricular layers of the model heart, as viewed from the front of the heart. The valve rings are again indicated by black markers. The larger mitral valve ring is the location at which the left atrium joins the left ventricle, whereas the smaller aortic valve ring is the location at which the ascending aorta is attached. (As before, only a subset of model muscle fibers are displayed.)
Figure 4: Similar to Figure 3, but here only showing three of the nested internal left-ventricular layers.
Figure 5: Similar to Figure 3, but here only showing two of the nested internal left-ventricular layers.
Figure 6: Similar to Figure 3, but here only showing one of the nested internal left-ventricular layers.
Figure 7: The inflow structures of the model heart, viewed from the right side of the heart. The superior and inferior vena cavae appear on the left side of the figure and are connected to the right atrium. The right atrium empties through the tricuspid valve (which appears in the figure to the right of the inferior vena cava) into the right ventricle (not shown). Corresponding structures on the left side of the heart appear on the right side of the figure. They include the four pulmonary veins, the left atrium, and the left atrial appendage (auricle). The left atrium empties through the mitral valve (which appears below the left atrium and to the left of the auricle) into the left ventricle (not shown). Both the tricuspid valve and the mitral valve are supported by fans of chordae tendineae, which in turn insert into papillary muscles. (As before, except for the case of the valve leaflets, only a subset of model fibers are displayed.)
Figure 8: The four valves of the model heart viewed from above (i.e., looking from the arterial side towards the ventricles). Note that the fiber structure of both outflow (aortic and pulmonic) valves is identical, although their elastic properties differ in accordance with the different pressure loads each is required to support.
Figure 9: The four valves of the model heart, as viewed from the front of the heart. From this view, the outflow (aortic and pulmonic) valves appear above the inflow (mitral and tricuspid) valves. The pulmonic valve is located above the aortic valve, and the mitral valve is located to the right of the tricuspid valve. Note that the inflow valves are supported by fans of chordae tendineae which insert into papillary muscles, whereas the outflow valves are self-supporting.
Figure 10: A. Surface rendering of the initial closed-valve configuration of the model aortic heart valve leaflets. The structure of the model pulmonic valve is identical, although it has different elastic properties that reflect the lower pressures developed by the right ventricle. B. Similar to A, but here the curvilinear mesh that defines the initial configuration of the valve is also shown.
[Thomas57] C.E. Thomas. The muscular architecture of the ventricles of hog and dog hearts. American Journal of Anatomy, 101(1): 17-57, 1957.
[Streeter78] D.D. Streeter, W.E. Powers, A. Ross, and F. Torrent-Gusap. Three-dimensional fiber orientation in the mammalian left ventricular wall. In J. Baan, A. Noordergraaf, and J. Raines, editors, Cardiovascular System Dynamics, pages 73-84. MIT Press, Cambridge, MA, USA, 1978.
[Peskin89] C.S. Peskin. Fiber architecture of the left-ventricular wall: An asymptotic analysis. Communications on Pure and Applied Mathematics, 42(1): 79-113, 1989.
[PeskinMcQueen94] C.S. Peskin and D.M. McQueen. Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets. American Journal of Physiology-Heart and Circulatory Physiology, 266(1): H319-H328, 1994.
[PeskinMcQueen96] C.S. Peskin and D.M. McQueen. Fluid dynamics of the heart and its valves. In H.G. Othmer, F.R. Adler, M.A. Lewis, and J.C. Dallon, editors, Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology, pages 309-337. Prentice-Hall, Englewood Cliffs, NJ, USA, 1996.
[McQueenPeskin00] D.M. McQueen and C.S. Peskin. A three-dimensional computer model of the human heart for studying cardiac fluid dynamics. Computer Graphics, 34(1): 56-60, 2000.
[McQueenPeskin01] D.M. McQueen and C.S. Peskin. Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity. In H. Aref and J.W. Phillips, editors, Mechanics for a New Millennium, Proceedings of the 20th International Conference on Theoretical and Applied Mechanics (ICTAM 2000). Kluwer Academic Publishers, 2001.
Revised 15.Nov.2005 by email@example.com.