IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 37, NO. 12. DECEMBER 1990 1 I73

The Construction of an Anatomically Based Model of the Human Ventricular Conduction System ANDREW E. POLLARD, MEMBER IEEE, AND ROGER c. BARR, SENIOR MEMBER, IEEE

Abstract-The ventricular conduction system is a complicated net- work of specialized muscle cells responsible for the transmission of electrical activity between the atria and the ventricles of the human heart. It has been the focus of numerous electrical and anatomical studies at both the microscopic and macroscopic levels. An under- standing of its behavior at both levels is considered important, because it is primarily responsible for the spread of excitation in the ventricles. Previous computer models have been very simple ones that have been primarily adjuncts to models of the ventricles. This paper describes a strategy for the construction of conduction system models which is based on real microscopic and macroscopic features, although the model still is much simpler than reality. The model contains almost 35 000 individual cylindrical elements, each of whose physical dimen- sions approximate unit bundles of Purkinje and atrioventricular nodal cells. The model, whose physical appearance closely resembles that of the Conduction system, was generated from limited anatomical data in less than 2 min CPU time on an IBM 3090 at the Cornell National Supercomputer Facility.

I. INTRODUCTION

HE ventricular conduction system was originally dis- T covered by Tawara [ 11, who traced the Purkinje cells in the ventricles of a number of mammals through com- plicated routes along the endocardium. He postulated, on anatomical grounds, that the atrioventricular node was the electrical connection between the atria and ventricles, an observation which was proved electrophysiologically by Lewis and Rothschild [2] some years later. Since these discoveries, there has been a great deal of work in iden- tifying the gross morphology, histology, and electrical properties of the specialized conduction system because of its importance in the excitation of the ventricles. Un- derstanding of its behavior in normal and pathological conditions is essential to an understanding of the function of the whole heart.

At the gross morphology level, the major pathways of the ventricular conduction system identified by Tawara have been traced by numerous investigations [ 11-[8]. The system begins at the level of the atrioventricular node. At

Manuscript received September 5 , 1989. This work was supported in part by U.S. Public Health Service Grant HL11307 and the Cornell Na- tional Supercomputer Facility through grants from the National Science Foundation, New York State, and the IBM Corporation.

A. E. Pollard was with the Departments of Biomedical Engineering and Pediatrics, Duke University, Durham, NC 27706. He is now with the Nora Eccles Harrison Cardiovascular Research and Training Institute, Univer- sity of Utah, Salt Lake City, UT 84112.

R. C. Barr is with the Departments of Biomedical Engineering and Pe- diatrics, Duke University, Durham, NC 27706.

IEEE Log Number 9038985.

its distal end, the node connects to the bundle of His, which in turn gives off a sheet of branches to the left ven- tricle and a single branch of Purkinje fibres to the right ventricle. These two systems course toward the apex of the heart, then begin extensive ramification in both apical and basal directions, covering most portions of the en- docardium. Electrically, the conduction system is largely responsible for the excitation of the ventricles. In its prox- imal portion, it is electrically isolated from the ventricular myocardium, which allows the emergence of the activa- tion wave in the myocardium to be located in the lower half of the ventricles. In its distal portions, the system makes connections to the myocardium at numerous sites, and because the conduction velocity in Purkinje tissue is greater than that in myocardium and the electrical con- nections between Purkinje and myocardial cells are inter- spersed, the activation sequence in the myocardium gen- erally follows the sequence in the distal conduction system.

Previous models have included little of the complexi- ties in the gross morphology. Dassen et al. [9] used six reference points, designed to represent catheter stimulus and measurement sites to construct a model for conduc- tion in the whole heart. A reference point in the lower right atrium was connected to a point adjacent to the His bundle to represent the most proximal portion of the con- duction system. The His bundle point was in turn con- nected to a site in the left ventricle and one in the right ventricle. Each connection was divided into approxi- mately 100 elements, resulting in a conduction system model with 300 elements. Malik et al. [lo] developed a model with a tree structure for Purkinje elements to con- nect to a heart constructed from triangular surface ele- ments. This model included some branching, bidirec- tional connections between the conduction system and the myocardium, and its ventricular component was made from less than 200 elements. Ahlfeldt et al. [ 111 devel- oped a 2-D model of conduction in the whole heart with 120 highly interconnected elements. Of these, four were used to represent the atrioventricular node, with fast and slow conducting pathways, and 84 were used to represent the Purkinje system. More recently, Eylon et al. [12] have used a self-similar fractal structure to represent the tree- like framework of the conduction system in the construc- tion of models for the conduction system, although their presentation does not indicate a comparative size for the model.

0018-9294/90/1200-1173$01 .OO 0 1990 IEEE

1174 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 31, NO. 12. DECEMBER 1990

There are two major limitations associated with all of these models, which are related to the small number of elements. The first problem is the anatomical basis. With so few components, physical representations show little of the complicated branching or extent observed in the system. In all cases, the nodes are connected to one an- other temporally and not spatially, which leads to con- duction times on the order of the Purkinje system but not to the accurate physical characterization. The second problem is the electrophysiological basis. Since the num- ber of nodes is small, the individual elements must rep- resent large blocks of conduction tissue. The underlying basis for excitation in the Purkinje system is microscopic, and is related to the ionic properties of cell membranes. Subsequently, simulations of the electrical behavior in all of the previous models have used rules for activation and inactivation which are not membrane based.

The present project was undertaken to develop a com- puter model of the isolated ventricular conduction system from anatomical descriptions. It was our intention to build a model which incorporated the anatomical complexities of the gross morphology at the macroscopic level, but was built from elements which incorporated both the observed histology and the underlying electrophysiology of cells in the conduction system at the microscopic level. This model, however, is still much simpler than reality. To construct a model of sufficient complexity, we had to ad- dress questions at both levels. Macroscopically, we were concerned with the unique geometry of the conduction system. What was a reasonable way to place and connect the elements so they followed the major pathways of the ventricular conduction system? Would it be possible to construct these pathways in some automated fashion? How well would the gross physical dimensions of length, width and cross-sectional area of components in the model match reported values? Could we demonstrate the spatial rela- tionships between components of the system and the gross anatomy of the ventricles? Microscopically, we were con- cerned with the selection of physical dimensions for the individual elements. Specifically, what was the most rea- sonable unit for the individual elements to represent: Pur- kinje cells, unit bundles of Purkinje tissue, or some col- lection of unit bundles as in the classical Purkinje fibre? Once the selection was made, how many elements would be generated to satisfy the macroscopic requirements of the problem definition? Given the size of such a model, would it be realistic to perform the construction in a rou- tine manner? What do the steps in the construction of the model suggest about the reports on the anatomy and elec- trical behavior of the conduction system? And finally, what are the applications of such a model?

In this paper, a strategy for converting the described anatomical pathways into a network of cylinders which approximates the Purkinje network is presented, and physical representations of this network in three-dimen- sional space are included. The model contained an atrio- ventricular node made up of cylindrical elements which were entirely interconnected. At the distal end of the node,

a His bundle model was connected, which coursed toward the left and right ventricles as 35 simulation cables in par- allel. Of these 20 coursed to the left ventricle and ramified to form the left bundle branch. The remaining 15 fol- lowed a subendocardial course through the septum to the right ventricle, representing the single right bundle branch. Both components then branched extensively be- fore merging with 35 reference points on the endocardial surface of the two ventricles which represented connec- tions between Purkinje and myocardium. The reference points were then entirely interconnected to form a distal system in each ventricle which represented the terminal Purkinje network. Four of the 35 points were located near anatomical sites for the breakthrough of electrical activity on the endocardial surface, while the remaining 31 were spread throughout to allow the distal system to cover large portions of the ventricles. The strategy resulted in a single cable from the floor of the right atria to each distal ref- erence point. Procedures for following the anatomic path- ways and constructing the proximal and distal systems are described in detail.

To determine appropriate properties for the micro- scopic components of the model, specifically, the physi- cal dimensions of the individual cylinders, a formulation based on the histology of Purkinje and atrioventricular no- dal tissue is presented. From this formulation, numerical values for multicellular unit bundles, considered here as the elements of conduction, were selected, and structural differences in the arrangement of the bundles of the two types were introduced. These physical descriptors include the diameters of the individual unit bundles dbundle, the lengths of the unit bundles dx, and the ratio of membrane surface in the unit bundles to those of smooth cylinders Am.

METHODS Geometrical Framework for Construction of the Model

The model was constructed from five “component models, ” representing the atrioventricular node and His bundle, the proximal conduction system in each ventricle, and the distal conduction system in each ventricle. The “pathways” for these components were estimated from text descriptions, schematics, and pictures of the gross anatomy of the human conduction system. Text descrip- tions charted their course with regard to landmark ana- tomical structures, while schematics and pictures typi- cally demonstrated the components atop the endocardium from a laterally exposed viewpoint. Within the pictures, a number of the anatomical landmarks from the text de- scriptions were evident.

To convert the pathways from text descriptions and pic- tures to component models, several steps were involved. First, the geometry of the cardiac surfaces, such as the left endocardial surface, were specified. Then, portions of the conduction system lying on the surface were lo- cated in relation to surface points. Both steps were done in two coordinate systems, one for the surface itself, and the other in 3-D coordinates for the heart as a whole. The

POLLARD AND BARR: MODEL OF HUMAN VENTRICULAR CONDUCTION SYSTEM

coordinates in 3-D, or “heart” space, for “contour” points on the endocardial surfaces of the right and left ventricles were measured on a heart model. Each contour point then was located manually on a two-dimensional map of the endocardial surface, drawn to represent the ventricles from laterally exposed viewpoints. The contour points in each of these “lateral maps” were then auto- matically interconnected with a triangulation algorithm [131 to determine connections in heart space for construc- tion of surface representations. Pathways for the conduc- tion system were then laid on the lateral maps as lines through reference points, which resulted in a set of maps that looked like photographs and drawings from the lit- erature. Each lateral map was covered by triangles, with the apices to each of the triangles a contour point having a known heart space coordinate. Thereby, the pathways on the lateral maps could be automatically translated into heart space, and converted to “cable segments” along the surface triangle faces.

An example of the transformation of a pathway from a lateral map to heart space is presented in Fig. 1. The five contour points in the lateral map [Fig. l(a)] were con- nected to form four triangles resulting in a pyramid sur- face representation in heart space [Fig. l(b)]. When ref- erence points PI-P4 were placed on the lateral map and connected to one another, the resulting pathways fol- lowed courses through triangles 4, 3, and 2. Points along the three pathways in the lateral map were then trans- formed to five cable segments along the surface of the triangles in heart space.

This process was implemented with a number of data structures which referenced the position of the individual points for the heart surface and the conduction system model, as well as the connections between those points. Table I presents the structures for the heart surface in Fig. 1. The position in both lateral map and heart space for each contour point was contained on the “contour points coordinate lists. ” When the points were triangulated, all of the connections were maintained on two structures. Every line between two points, termed an “edge,” was referenced on the “contour points edge list.” From Table I , the line between points 1 and 4 of Fig. 1 was given edge number AB = 1. Included with the contour point numbers for each edge was the slope, which was used to evaluate line intersections during construction as de- scribed below. Each triangle was referenced by its con- stituent edges on the “contour points triangle list.” Tri- angle number, ABC = 1, was made from edges 1, 6, and 5 , which included contour points 1, 4, and 5 . The trian- gles were considered in terms of the edges because the slopes were useful in determining the location for con- duction system model points, and in the construction of pathways between those points.

For the Purkinje network, the lateral map coordinates for the reference points, and the manner in which those points connected to one another were dependent on whether each point was part of the distal or proximal por- tion of the model. The structures used to specify those

1175

2 3

(b) Fig. 1. Four reference points P , through P, were placed on the lateral map

(a) containing triangles 1 through 4 (large numbers) and contour points 1 through 5 (small numbers). When the lateral map coordinates for the triangles in (a) were converted to the heart space representation in (b), the pathways between points P , and P,, P, and P,, and P , and P,, fol- lowed the triangle faces in heart space.

connections are described in upcoming sections. For Fig. 1, the lateral map coordinates for P1-P4, and the connec- tions between those points were set. Algorithmically, the transformation began with a search of the contour points triangle list for the resident triangles of PI-P4. Then the lateral map coordinates at which the pathway between any two points intersected the edge of a surface triangle were determined, and these were taken as the endpoints to the cable segments along the three pathways. Once the lateral map coordinates for the cable segment endpoints were known, these points were transformed into heart space and discretized to build the component models.

To facilitate transformation in the component construc- tions, anatomical landmarks were located on the lateral maps built from the heart model. Fig. 2(a) presents the lateral map of the right ventricle. A total of 103 contour points and 209 triangles were used to construct the map. The distinct regions establishing borders between the free wall and the interventricular septum, and the specific structures high-lighted within these regions, allow this lateral map to be readily associated with the experimental drawings. Fig. 2(b) is the lateral map for the left ventri- cle.

Discretization of Cable Segments

To convert the pathways in the component models to microscopic elements, a set of points, termed ‘ ‘nodes, ” were identified along each of the cable segments. These nodes are the endpoints to the elements of conduction,

1176 IEEE TRANSAC :TIONS ON BIOMEDICAL ENGINEERING, VOL. 31, NO. 12, DECEMBER 1990

TABLE I DATA STRUCTURES FOR THE CARDIAC SURFACE

CONSTRUCTION

Contour points coordinate list

Contour Point 4 VI x h Y h z h

1 -1.0 1.0 -1.0 1.0 0.0 2 -1.0 -1.0 -1.0 -1.0 0.0 3 1.0 -1.0 1.0 -1.0 0.0 4 1 .o 1 .o 1 .o 1.0 0.0 5 0.0 0.0 0.0 0.0 1.0

( ulr U,) = lateral map coordinates ( x h , Y h , I , ,) = heart space coordinates

Contour points edge list

AB A B

1 1 4 2 4 3 3 3 2 4 2 1 5 1 5 6 5 4 7 5 3 8 5 2

A, B-contour points comprising the edge with number AB

Contour points triangle list

ABC AB BC CA

1 1 6 5 2 2 1 6 3 3 8 7 4 4 5 8

AB, BC, CA-edge numbers in the triangle with number ABC

which are modeled as cylinders whose physical dimen- sions approximate those of unit bundles. For simplicity, we desired a single element length dx and bundle diameter dbundle for all of the Purkinje elements. Since most of the cable segments could not be divided into an integer num- ber of elements with length dx, a procedure to insure most of the elements had length dx was developed.

This procedure is presented in Fig. 3 . The heart space coordinates for the endpoints of a cable segment were de- termined during coordinate transformation. Four nodes spaced dx apart were located along the segment, but there was a leftover distance dxleft between the fourth and fifth nodes. The leftover length was added to that of the ele- ments in the center of the cable segment, as demonstrated. Elements at the endpoints of the cable segments were then connected to the elements from other segments to con- struct the pathways. With this discretization strategy, each node was assigned a unique number i. For each i, the heart space coordinates for the node were stored alongside the element length and bundle diameter for the element be- tween i and the next node in the cable segment i + l . In addition, every node was classified as either contiguous or discontiguous where a contiguous node made connec- tions with the two sequential nodes i - 1 and i + 1. A

L F W A P W

(b)

Fig. 2. The lateral maps of the right (a) and left (b) ventricles. Anatomical regions highlighted in the right ventricle include the conus arteriosus (CA), the supraventricular crest (SVC), the lateral (LFW), anterior (AFW), and posterior (PFW) free walls, the interventricular septum (IVS), the anterior (APM) and posterior papillary muscle groups (PPM), the medial papillary complex (MPM), the origin of the moderator band (MBO) and the tricuspid value (TV). Regions highlighted in the left ven- tricle include the lateral (LFW), anterior (AFW), and posterior (PFW) free walls, the interventricular septum (IVS), the anterior (APM) and posterior papillary muscle groups (PPM), the central fibrous body (CFB), and the aortic (AV) and mitral valves (MV).

discontiguous node was one which made connections to nonsequential nodes. This condition was a result of branching in the model, and all of the connections to dis- contiguous nodes were maintained on a separate list. Fig. 3(c) demonstrates the cable arrangement at P2 from Fig. 1 where three cable segments connect at a single node. Table I1 presents the associated ‘ ‘discontiguous connec- tion list.” The first entry in Table I1 was node 1, which made only one connection, to node 2. Nonexistent con- nections on the list were referenced as zero entries. The next discontiguous node was node 5 , analogous to point P2 in Fig. 1, which made connections to nodes 4, 6, and 10. Additional discontiguous nodes in this example in- clude nodes 9, 10, and 13. As each pathway in the con- duction system model was constructed, discontiguous connections at the endpoints were referenced on the con- nection list.

POLLARD AND BARR: MODEL OF HUMAN VENTRICULAR CONDUCTION SYSTEM 1 I77

13

(C) Fig. 3. A cable segment across one of the triangles in Fig. 1 was discre-

tized into a number of elements of length dx in (a). As a result of the discretization, there were three elements of length dx and an additional element of length dx, , , . This leftover length was added to the center of the cable segment in (b). When the three cable segments across Triangle 3 of Fig. 1 were discretized in (c), a branch node was included at which three nodes connected to a single node.

Fig. 4. The distal component systems for the right (a) and left (b) ventri- cles with a number of the anatomical regions from Fig. 2 highlighted. In both maps, points ( ) are the distal reference sites and points ( 0 ) are the anatomical locations for endocardial breakthrough. The large ar- rows demonstrate the locations where border cable segments were laid between the anterior and posterior sections of the lateral free wall.

TABLE I1 THE DISCONTIGUOUS CONNECTION MAP

Key Node c, c2 c3

1 1 0 2 0 2 5 4 6 10 3 9 8 0 0 4 10 5 11 0 5 13 12 0 0

Key-used to reference the discontiguous connections Node-the global node number at which a discontiguous

connection is located C,, C,, C,-global nodes connected to node

Construction of the Distal Conduction Systems The distal component systems in each ventricle were

constructed from a number of reference sites specified as electrical connections between the conduction system and myocardium on the basis of previous anatomical and elec- trophysiological studies. Fig. 4(a) presents the reference sites for the distal component atop the lateral map in the right ventricle. From anatomical descriptions, the termi- nal Purkinje network in the right ventricle is made up of swirls of conduction tissue which cover all sections of the free wall and septum except the most basal portions of the endocardium [ 13, [2] , [7], 181. From electrophysiological descriptions, the first site of endocardial muscle activa- tion in the right ventricle is at the base of the anterior papillary muscle. Wavefronts then move along the endo- cardial surface with an invasion of the septum and adja- cent free wall [2], [ 141. To balance these descriptions, 15

reference sites were selected as connections to the myo- cardium at points throughout the ventricle, with one point located at the base of the anterior papillary muscle to rep- resent the expected endocardial breakthrough site. These points were placed on the lateral map of the right ventri- cle, and their lateral map coordinates determined. The points then were interconnected by triangulation, and the edges were made into cable segments of Purkinje ele- ments in a two-step process. Since no more than three elements were connected to any node during construction, a small ring of elements analogous to a “traffic circle” first was constructed around each of the reference sites. Then cables were laid along the pathways between the traffic circles and the endpoints to these cables were con- nected to the traffic circles by updating the discontiguous connection list. Finally, to connect the pathways along the left lateral border of Fig. 4(a) with those along the right lateral border, a number of pathways were laid be- tween the two sides at the sites marked with arrows. The most basal point on the left side was connected to the most basal point on the right, and so forth, moving apically.

In the left ventricle, twenty reference points were sim- ilarly placed on the lateral map of Fig. 2(b) to construct the distal component system of Fig. 4(b). From anatom- ical descriptions, a dense mesh of conduction tissue cov- ers all portions of the endocardium except for the basal

1178 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 31, NO. 12, DECEMBER 1990

third of the septum and the most basal portions of the free wall [ 13, [2], [7] [8]. From electrophysiological refer- ences, the first site of muscle activation is in the midsep- tum, although subsequent activation occurs at the base of the anterior and posterior papillary muscle groups [2], [ 141. From these descriptions, three points were place for endocardial breakthrough sites while the remaining 17 were selected to cover the lower two-thirds of the endo- cardium.

Construction of the Proximal Conduction Systems The proximal component models in the left and right

ventricle represent the portion of the conduction system which is intermediate between the bundle of His and the terminal Purkinje network. In the most proximal portions, these models represent the single right bundle branch and left bundle branch network anatomically. Each of these models was constructed as a group of parallel cables whose pathways were analogous to that of a tree. There was a root to the tree in the proximal portion of each ven- tricle. These roots correspond physically with the origins for the right bundle branch in the right ventricle and the emergence of the left bundle branch system in the left ventricle. From the root, branches were given off to each of the reference points in the distal conduction system. From the standpoint of construction, the proximal com- ponent model in each ventricle was made with two design objectives. First, the pathways through the lateral map had to follow the specified anatomical course. Second, a single simulation cable had to be constructed between the root of the tree and each distal reference point. To meet these two objectives, a structure analogous to a binary tree was constructed from an input list of reference point coordinates and flags. The example system in Fig. 5 was constructed from the input list in Table 111. The root was specified as node 1, and subsequent points were placed on the lateral map until a branch node was reached. At the branch, two new pathways were specified, and so forth, until the terminal nodes in the tree were reached. After the full tree was constructed, connections were made be- tween the terminal nodes and the distal reference points, which resulted in a unique pathway from the origin of the tree to each distal site. The constructed pathways were then discretized to form the component models, and con- nected to the distal systems by updating the discontiguous connection list.

The pathways for the proximal component in the right ventricle are presented in Fig. 6(a). From text descrip- tions, the right bundle branch continues from the end of the His bundle and follows a subendocardial course through the interventricular septum, passing underneath the medial leaflet of the tricuspid valve. It remains sub- endocardial within the septum, coursing beneath the me- dial papillary muscle and frequently emerging on the en- docardium distal to the base of the muscle before entering the moderator band. After entry into the moderator band complex, the system courses toward the base of the an- terior papillary muscle, and beyond the anterior papillary

Fig. 5 . An example of the proximal tree structure located atop the lateral map of Fig. 1. Points ( w ) are proximal reference points, while points ( 0) are distal reference points. Proximal points were placed on the map, and numbers were specified from top to bottom and left to right. Num- bering was stopped at the branch and terminal points. From the root (node l ) , points were added until the first branch point (node 4). Below this point, nodes were added to the left until a branch point (node 5 ) was reached, and then to the right until another branch point (node 6) was reached. When terminal points 8, 9, 10, and 12 were reached, the tree was sealed, and connections were made between the terminal nodes and the distal reference points.

TABLE 111 THE MODIFIED BINARY TREE

Node Parent Branch Terminal

1 2 3 4 5 6 7 8 9

10 11 12

0 1 2 3 * 4 * 4 * 5 I * 5 * 6 * 6

11 * Node-the proximal node number Parent-the parent node to Node Branch-indicates Node is a branch point in the tree Terminal-indicates Node is a terminal point in the tree

muscle, the right bundle branch ramifies and becomes continuous with the terminal Purkinje network [2], [5], [7 ] , [8]. This anatomical route was reproduced in Fig. 6(a), as the first points in the single right bundle branch on the lateral map follow the description closely. Fifteen cable segments follow the course to the base of the ante- rior papillary muscle, then ramify to make connections with the distal reference sites. The reported length of the human right bundle branch is 50 mm [5]. In its emerging portion, the diameter has been reported as 1 mm [5] and less than 1 mm [7] . The cross-sectional area for a 1 mm diameter cylinder is 0.78 mm2.

Pathways for the proximal component in the left ven- tricle were similarly selected for Fig. 6(b). After the His bundle courses through the central fibrous body, suben- docardial branches of the left bundle branch network

POLLARD AND BARR: MODEL OF HUMAN VENTRICULAR CONDUCTION SYSTEM 1 I79

(b) Fig. 6 . The proximal component systems for the right (a) and left (b) ven-

tricles. Points (U ) are the reference points for the proximal components, while points (U) are the distal reference sites from Fig. 3 .

emerge on the left ventricular septum adjacent to the right fibrous trigone, which is located near the central fibrous body between the aortic and mitral valves. In its emerging portion, the network fans out as a sheet, giving off branches which form groups in anterior, posterior, and septal directions. Multiple branches arise from these groups, with the posteriormost and anteriormost branches being thought of as “borders.” The anterior border heads toward the base of the anterior papillary muscle and the posterior border heads toward the base of the posterior papillary muscle. Branches in the septal portion intercon- nect with one another and the borders in a course to the midseptum which covers the area between the borders to varying degrees. Below these regions, the proximal net- work connects to numerous sites over all portions of the endocardium except the basal third of the septum and the most basal portions of the free wall [ l ] , [7], [24]. The proximal component constructed here was made from 20 cable segments. The course in Fig. 6(b) reproduces many of the characteristics of the anatomical description, al- though with 20 cables the resolution in the septal region is coarse by comparison with anatomical presentations. After the emergence of the left bundle branch distal to the central fibrous body, three distinct pathways were laid to represent the anterior and posterior borders, and the septal region. At four sites in the septal region, connections were made from the borders to represent the interconnecting mesh observed anatomically. At these sites, all cable seg- ments were merged laterally to represent the functional

interconnectivity observed electrically. Below the level of the three main pathways, the system ramified to supply the distal reference points as in the construction of the right ventricular component system. Gross physical di- mensions for the left bundle branch are highly variable. The minimum length from emergence to the distal portion has been reported at 24-45 mm 171. In hearts with fan- type systems, the width of the emerging portion ranges from 2-15.0 mm as a sheet of cells 4-25 layers thick [5], 171.

Construction of the A-V Node and His Bundle from the Right Atria

While the construction procedure was used predomi- nantly for generation of component networks in the left and right ventricles, the coordinates for 17 contour points in the lower right atrium also were included to reference the course of the atrioventricular node and His bundle component model. These two structures together are re- ferred to as the specialized atrioventricular junctional area [ 6 ] . The atrial portion of the specialized junctional area contains the proximal atrioventricular node and connec- tions between the node and atrial muscle. The atrioven- tricular portion contains the distal node and the proximal or penetrating His bundle. The ventricular portion con- tains the distal or branching His bundle and the most proximal sections of the left and right bundle branch sys- tems. The atrial portion is located near the right atrial as- pect of the central fibrous body, and is made up of both the atrial approaches to the node and the compact atrio- ventricular node. The origin for the component model in this study was at the level of the compact atrioventricular node, whose surgical landmark is the apex of the triangle of Koch. As the node courses distally, cells of the com- pact node become continuous with the Purkinje fibres of the penetrating His bundle in the atrioventricular portion of the junctional area. The triangular shaped compact node measures 5-7.5 mm [3], [5] in length and has cross-sec- tional dimensions of 1 x 3 mm with an approximate cross- sectional area of 1.5 mm2.

The cells of the proximal penetrating His bundle are a mixture of Purkinje cells and nodal cells. As it emerges from the central fibrous body, the penetrating His bundle runs anteriorly along the crest of the interventricular sep- tum giving off branches to the left bundle branch network. The region between the first offshoot and a point just dis- tal to the last offshoot is termed the branching His bundle. The length is variable, with a range of 6.5-20 mm re- ported in man 151. The diameter is 1-2 mm 131, [ 5 ] , which results in cross-sectional areas for the cylindrical bundle of 0.78-3.1 mm2. Fig. 7 demonstrates both the anatomi- cal regions included in the map of the right atria and the course for the component model. The four points in the map were used to construct the atrioventricular node, which began posterior to the apex of the triangle of Koch before descending into the central fibrous body. The model contained 35 cable segments coursing along this pathway in parallel, with 15 ultimately headed for the right

1180 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 31, NO. 12, DECEMBER 1990

Fig. 7 . A lateral map of the floor of the right atria demonstrating the an- atomical course for the atrioventricular node component model. The highlighted anatomical regions include the inferior limbus of the fossa ovalis (ILFO), the coronary sinus (CS), the triangle of Koch (TK), the central fibrous body (CFB), the tricuspid valve (TV), and the membra- nous septum (MS).

ventricle and 20 headed for the left ventricle. All elements in this portion of the model were considered atrioventri- cular node, and had physical dimensions and structural arrangement for nodal tissue. The His bundle portion be- gan at the most distal reference point in Fig. 7 and ended at the most proximal point in Fig. 6(b). The cable seg- ments at the end of the His bundle branch off along sep- arate routes to the two ventricles. All elements in this por- tion of the model had physical dimensions for Purkinje.

Selection of Physical Dimensions for the Microscopic Elements

The unit bundle was selected as the microscopic ele- ment of conduction for this study on the basis of the his- tology of Purkinje tissue. Individual Purkinje cells are ap- proximately cylindrical, with groups of cells clustered together to form unit bundles. In this closely packed ar- rangement, the cells connect to one another along lateral and longitudinal margins through low resistance path- ways. Such connections are referred to as intercalated discs or narrow clefts [15], [16]. Groups of unit bundles are surrounded by connective tissue to form classical Pur- kinje fibers. In the larger fibers, bundles join together at varying intervals through single cell connections. The groups of bundles are referred to as fascicles [17]. There is some interconnection between fascicles, although the frequency of fascicular connections is less than that of bundles and significantly less than that of cells within bundles. The unit bundle was selected because the focus electrically was on macroscopic propagation. Since the cells in bundles are tightly coupled, it has been postulated they behave like one-dimensional cylinders during uni- form propagation [ 181. Three physical descriptors were selected from a range of values for the Purkinje elements: the diameters for each individual bundle dbundle, the target lengths for the bundles dx, and a membrane surface factor A,. Values for dbundle were tabulated from the Purkinje

intracellular volume in previous reports [ 181, [ 191, as well as from reported cell diameters [20] and the definition of 2-30 cells per unit bundle [17]. From a range of 5-430 pm, an intermediate value of = 150 pm was se- lected for all Purkinje elements in the model. While the lengths of individual Purkinje cells vary from 30-100 pm, cell columns in neighboring unit bundles of cardiac mus- cle connect at 100-300 pm intervals [17]. An element length of dx = 200 pm was selected as an intermediate value to this range.

The membrane surface factor was similarly selected on the basis of histology. Since the unit bundle is a descriptor for a multicellular preparation, the surface area of mem- brane is greater than that predicted by a smooth cylinder. The ratio of membrane to cylinder surface A , was deter- mined on the basis of the number of cells in a bundle Ncell, the diameters of the individual cells dcell, and the amount of membrane folding in each cell (fold. From an Ncell range of 2-30, an intermediate value of 14 was selected. From a dcell range of 5-80 pm, an intermediate value of 40 pm was chosen. And from a tfold range of 1.4-1.8, ,$fold = 1.5 was used [19], [22], [23]. As a result, the ratio of membrane surface area when all of the cells within the bundle were considered to that of a smooth cylinder with the selected bundle diameter was determined

(Ncell)(tfold~ (dcell)

dbundle A , = 9

= 5.6 (1) Histologically, the cells of the atrioventricular node are

smaller in diameter than those of Purkinje or myocar- dium, and in the compact node the cells are tightly inter- connected to form a complicated interweaving mesh [4], [6], [ 171, [21] compared with the longitudinal orientation in groups of Purkinje cells. Despite the different arrange- ment, however, there are structural similarities between nodal and Purkinje cells. Cells of both types lack con- tractile material, contain no transverse tubules, and ex- hibit tight packing [17]. As a result of this description, it was assumed cells of the atrioventricular node collect into unit bundles for the construction of the model, and a bun- dle diameter, dbundle = 20 pm, was selected from a dcell range of 3-60 pm, [20], [21]. An element length, dx = 30 pm, was selected for a similar dX/dbundle to that used for Purkinje, and the membrane surface factor was also set to A , = 5.6.

An additional consideration in the construction of the model was the manner in which the individual elements interconnected. From descriptions of the histology, Pur- kinje unit bundles connect to one another through single cell connections along their lateral margins [ 171. Macro- scopically, the His bundle and the proximal bundle branches in their emerging portions were highly coupled, as transection of 75 % of their width resulted in no change to waveforms measured distally [25]. As a first study of this type, however, we considered only the macroscopic interconnectivity, and this was limited to a few sites by

1181 POLLARD AND BARR: MODEL OF HUMAN VENTRICULAR CONDUCTION SYSTEM

the construction process. By design, the distal systems were highly interconnected, through connections to the trap,- -!rsls> burroundLg cach of the rcfcrence points. Cables in the proximal systems were electrically isolated from one another except in the septal region of the left ventricle where lateral connections between adjacent ca- ble segments were introduced at four points. Along the His bundle, the individual cables were isolated although the atrioventricular node portion of the model was com- pletely interconnected. Each of the 35 cable segments connected with a neighboring segment every 150 pm throughout the extent of the component model. This ar- rangement was intended to present a uniform wavefront to the His bundle model, which we assumed would par- tially account for the lateral connections in the His bundle and emerging portions of the proximal bundle branches, since the function of those connections would be to make emanating wavefronts uniform.

RESULTS The five component models were combined to form the

anatomically based model of the conduction system. Fig. 8(a) is a heart space representation of the lateral maps of the left and right ventricles, and the lower right atria with a number of the anatomical regions highlighted. The viewpoint in the figure is posteroseptal, with the endocar- dial surface of the basal portion of the right ventricle ob- scured by the cavity of the lower right atria. The most basal portions of the left ventricular cavity also obscure a portion of the outflow tract. In addition to these spatial relationships, the major features expected in a represen- tation of the cavities are included. The chamber of the left ventricle has a more rounded appearance than the crescent shaped right ventricle, and the outflow tract is located basal and anterior to the left ventricle.

Four points are highlighted in the surface representa- tion to help chart the course of the component models an- atomically. Point 1 is in the lower right atria, the location for the atrioventricular node portion of the model. Point 2 is basal in the center of the interventricular septum where the branches of the system headed for the left and right ventricles separate. Point 3 is near the midseptum of the left ventricle. As described in the formulation of the distal component model in the left ventricle, the expected en- docardial breakthrough sites are in this region. Point 4 is near the base of the anterior papillary muscle in the right ventricle, which is another expected breakthrough site.

Fig. 8(b) demonstrates the constructed model atop a wireframe representation of the endocardial surfaces in Fig. 8(a). The model originates in the floor of the right atria, posterior to the tricuspid valve, then courses to a point high in the septum of the left ventricle before indi- vidual cable segments begin branching off to the two ven- tricles. The isolated reconstruction of Fig. 8(c) demon- strates the four reference points for the atrioventricular node component model from Fig. 7. There is a single group of cable segments between the last of these refer-

(c) Fig. 8 . A representation of the constructed model of the ventricular con-

duction system in heart space. The endocardial surfaces of the right (RV) and left (LV) ventricles, and the lower right atria (RA), are presented from a posteroseptal viewpoint in (a). The numbers 1-4 indicate impor- tant anatomical regions for the conduction system as described in the text. The locations of the tricuspid valve (TV) and the anterior papillary muscle groups (APM) are included in (a) anatomical references for the course of the conduction system model atop a wire frame representation of the endocardial surfaces from the same viewpoint in (b). The isolated system is presented with reference points from Figs. 4, 6 , and 7 in (c).

ventricular surface, representing the bundle of His. The three main pathways of the proximal component model in the left ventricle and cable segments which connect the outer arms to the interior are evident in the septal portion near Doint 2. Similarly. the single right bundle branch fol-

- .

ence points and the emergence of the system on the left - -

1182 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 31. NO. 12, DECEMBER 1990

TABLE IV DISTRIBUTION OF NODES IN THE CONDUCTION SYSTEM MODEL

Nnode %Nnode

dx RV LV Total RV LV Total

distal components edges traffic circles border cables

proximal components His bundle

total Purkinje total atrioventricular node

total conduction system

200 2658 200 1776 200 543 200 339

200 7489 200 1083

200 11230 30 3440

14670

3993 2738 943 312

8298 1827

14118 4180

18298

665 1 45 14 1486 65 1

15787 2910

25348 7620

32968

8 12 20 5 8 13 2 3 4 1 1 2

23 25 48 3 6 9

34 43 77 10 13 23

44 56 100

dx = target spatial step size for construction ( p m ) Nnode = number of simulation nodes %N,,,, = percentage of the total number of nodes located in the region RV = right ventricle LV = left ventricle total = combined value for left and right ventricles

lows the expected course along the endocardial surface of the right ventricular septum. The first connections be- tween the proximal and distal systems in the left ventricle are near point 3, while the breakthrough site in the right ventricle near point 4 is more apical. Below the level of the breakthrough sites , the extensive branching in the model is evident. At each of the reference sites for the distal components, a cable enters from the proximal sys- tem and numerous cables emanate from the traffic circles surrounding these points. Both ventricles are largely cov- ered with cable segments, including the lateral free walls which contain the border connections of the distal com- ponents. In the right ventricle, the lower half of the en- docardial surface is covered. In the left ventricle, the lower two-thirds are covered.

Construction statistics To build the model, 213 contour point coordinates in

lateral map and heart space for the three surfaces, 35 lat- eral map coordinates for the distal reference points, and 59 lateral map coordinates with connection information for the proximal systems were specified as input. The contour points were triangulated, resulting in 632 edges and 416 triangles. Then the pathways for the component models were laid and discretized, and the component models were connected to one another through updates to the discontiguous connection list resulting in the conduc- tion system model of Fig. 8 which contains 32 968 nodes. Of these, 1819 nodes were discontiguous. The steps from assembly to complete construction required approxi- mately 2 min CPU time on an IBM 3090 at the Cornel1 National Supercomputer Facility. Table IV presents the distribution of nodes within the different component sys- tems. In the right ventricle, the distal component system contained 2658 nodes or 8 % of the total nodes in the model. Of these, 5% were located along the edges be- tween the distal reference points, 2% were located in the

traffic circles constructed around the reference points, and 1 % were located in the border cables constructed between the arrows indicated in Fig. 4(a). In the left ventricle, the distal component system contained 3993 total nodes. The increase over the number in the right ventricle was a result of the five additional reference sites, the more basal lo- cation of those sites, and the large major diameter of the left ventricular cavity compared with that of the right ven- tricle. The distribution of nodes was similar, however. The left ventricular component system made up 12% of the total system nodes, with 8% located along the edges between reference sites, 3% located in the traffic circles, and 1 % along the border. In all, 20% of the nodes in the model were located in the distal systems.

The proximal component models contained the largest number of any of the five components. In the right ven- tricle, the proximal system was made up of 7849 or 23% of the total nodes. In the left ventricle, the model con- tained 8298 or 25%. Taken together, these two systems made up 48% of the model. Although the system in the left ventricle contained 20 simulation cables compared to 15 in the right ventricle, the number of nodes were similar to one another for two reasons. The first was the course of the His bundle in the model. From Fig. 8(b), the His bundle coursed from the floor of the right atria to the most proximal point in the left ventricular system. The right bundle branch was reflected back through the septum, re- sulting in a more circuitous route from the atrioventricular node to the base of the anterior papillary muscle than if the most proximal point in the right ventricle were con- nected directly to the atrioventricular node. The second was the location for the breakthrough site. Since the base of the anterior papillary muscle in the right ventricle was more apical than the midseptal site in the left ventricle, the pathway for the right bundle branch was longer than that for the cables in the left bundle branch system. This difference is indicated in the cable lengths from Table V.

POLLARD AND BARR: MODEL OF HUMAN VENTRICULAR CONDUCTION SYSTEM 1183

TABLE V GROSS PHYSICAL DIMENSIONS IN THE CONDUCTION SYSTEM MODEL

atrioventricular node 0.6 0.6 0.5 0.02 0.07 His bundle 1.3 1.3 5.2 0.61 2.03 right bundle branch 7.9 12.0 2.2 0.27 0.90 left bundle branch 5.1 10.5 3.0

l,,, = minimum length of a constructed cable segment (cm) l,,, = maximum length of a constructed cable segment (cm) w = width of the emerging portion for cables laid side by side (mm) A = cross sectional area of the emerging component (mm’) A, = areaoftheemergingcomponentassuming 30% is myocytevolume (mmz)

The shortest cable from the end of the His bundle to a reference site in the left ventricle was 5.1 cm compared with 7.9 cm in the right ventricle. The atrioventricular node and His bundle component model combined for 32 % of the total system nodes. Of these, 9 % were located in the His bundle portion while the remaining 23% were in the atrioventricular node. When the nodes in the model were separated into Purkinje and atrioventricular nodal elements, 77 % of the model was Purkinje while 23 % was atrioventricular node.

Comparison of the Gross Physical Dimensions Gross physical dimensions of the model for comparison

with reports are presented in Table V. These include the minimum ( Imin) and maximum (I,,,) cable length, the width for the emerging portions before any branching took place (w), the cross-sectional area of an equivalent cylin- drical bundle ( A ) , and the cross-sectional area of an equivalent cylindrical bundle assuming the calculated my- ocyte volume comprised 30% ( A ; ) of the total volume [19]. At 6 mm, the length of the atrioventricular node model corresponded well with published values. Simi- larly, the length of the His bundle at 13 mm was in the reported range of 5-20 mm. At 79 and 51 mm, the min- imum lengths for the right and left bundle branches, re- spectively, exceeded those reported. In Table v , the widths were calculated for the cables laid side by side in a parallel arrangement. Values for some components in the model were less than the reported range, while others were greater. The atrioventricular node width at 0.5 mm was less than the reported range of 1-3 mm. The His bun- dle and right bundle branch each exceeded published re- ports, and the left bundle branch width at 3.0 mm was on the low side of the reported 2.0-15.0 mm range. A values as a result of the 35 cable model were all less than those reported, although accounting for the nonmyocyte volume placed the His bundle cross-sectional area within the re- ported range and the right bundle branch close.

DISCUSSION The purpose of this study was to develop an anatomi-

cally based computer model of the human ventricular con- duction system which incorporated many of the anatomi- cal complexities within the heart. We wanted to build a

model from cylindrical elements that represent the micro- scopic basis for electrical conduction which followed the gross anatomical pathways of the system in the intact heart. To meet this goal, we addressed questions con- cerned with the unique geometry of the Purkinje network. Macroscopically, the strategy resulted in a model which followed prescribed anatomical pathways along the sur- face of the heart. It contained an atrioventricular node and a His bundle whose macroscopic lengths were within the reported range. The proximal components recreated the tree-like nature of the system, and while the minimum lengths for constituent cable segments were greater than those reported, the relationships between the right and left bundle branches were similar. The distal components formed an interconnected mesh which covered most por- tions of the endocardium. With this model, visual inspec- tion of the gross features of the system was possible for the first time.

Microscopically, the model was built from elements representing Purkinje and atrioventricular node unit bun- dles. While it was two orders of magnitudes larger than any previous model, it was still small enough to be con- structed in 2 min CPU time. This result suggests it would be reasonable to build larger and more complicated models with these techniques, since the computational aspects were not a constraint in the construction process. Proce- durally, all that was required was the characterization of the endocardial surfaces with anatomical markers, the placement of reference points for the component systems, and the selection of physical descriptors for the micro- scopic elements. The inclusion of additional elements would result in cross-sectional areas for the emerging por- tions closer to reported values, although the values for the His bundle and the right bundle branch were close when the non-myocyte volume was included. Additional ele- ments for interconnection of the major cables would also be advantageous, as the longitudinal isolation of cables in the His bundle and right bundle branch was an approxi- mation used as a starting point.

Since this is the first real strategy which has attempted to incorporate the spatial complexities of the system in a computer model, however, there were certain shortcom- ings which, if addressed, would improve future construc- tions. First and foremost is a consideration of anatomical validation. This idealized computer model was built with surface geometry from a heart model, and reference points for the conduction system from text descriptions. A log- ical extension would be the use of the same strategy in an animal model in which the surface geometry was deter- mined, then stained conduction system photographs were used to locate the reference points. While these steps were the most time consuming portion of the construction pro- cess, requiring hours or days as opposed to minutes, we expect repetition would result not only in improved meth- ods for the characterization, but in increased anatomical accuracy of the model.

With regard to the model component systems, we also suggest refinements to more accurately characterize the

1184 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 17, NO. 12, DECEMBER 1990

network. The representation of the atrioventricular node was crude, by necessity, since the histology of nodal tis- sue is complicated and not extensively documented. Structural similarities between nodal and Purkinje cells were assumed, and a more thorough examination of how atrioventricular nodes connect to His bundles would im- prove our representation. The description of the pathway for the His bundle was grossly approximated in the com- ponent model. Because pathways were contoured to the endocardial surfaces, the “crest of the intraventricular septum” course for the His hiindt= Was not characterized, as the crest is an intermediate structure to the three en- docardial surfaces. A midseptal course would affect the position of the proximal components as well, as the cut- back to the right ventricle evident in the right bundle branch in the model would be less severe if the His bundle component were more intermediate to the two ventricles.

The 1 : 1 ratio of proximal cables to distal reference sites, which was used only as an appropriate starting point, introduced limitations. The 35 : 20 : 15 distribution of ele- ments between the His bundle, left bundle branch and right bundle branch which resulted from the 1 : 1 assump- tion was not supported anatomically. This affected the septal region of the proximal left ventricular component model, in which a trifascicular framework was selected as a first approximation which fit well in the tree structure used for construction. With 20 cables, it was not possible to represent the fan type left bundle branch, in which broad regions of conduction tissue cover the area between the border branches. As a result, relatively more cables were located along the borders than were located septally , which differed from the anatomical case. Similarly, the density of Purkinje fibers present throughout the distal components was much less than that observed. This lim- itation was primarily a result of the much more highly complicated nature of the system than could be accurately represented in a model. The relative sparsity of these re- gions, however, with respect to other areas was a result, in part, of our idealized approach and the 1 : 1 ratio in construction. Small local networks attached to the exist- ing framework could be spread throughout the left bundle branch and the distal components. Such an approach would not require the addition of proximal cables to con- nect with these sites, and would remedy the relative dif- ferences in the distribution.

Implications for Future Work While this work demonstrated large and detailed models

can be constructed in a straightforward manner, our future work is primarily concerned with the numerical simula- tion of the electrical behavior during activation. Specifi- cally, how well does the sequence of excitation predicted by an anatomical construction match that established by the conduction system in the excitation of the heart? What are the effects of the macroscopic framework on this re- lationship? Are the anatomical limitations considered

above significant in this context? What effect do the mi- croscopic elements have on this relationship? Are the one- dimensional cylindrical elements sufficient to represent the microscopic behavior of the Purkinje system? And how does the model behave in response to complex stimulus patterns, such as those which result from ectopic foci and complex arrhythmias? Can these be related to the ob- served behavior in an animal model?

The described construction process, however, meets the outlined goals for this study. The model is clearly more representative of the ventricular conduction system than any previous model, and it is both large enough to include major anatomical features, and small enough to be built quickly. In addition, this procedure is straight-forward, largely automated, and has its basis in the unique geom- etry of the ventricular conduction system.

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Andrew E. Pollard (M’89) was born on April 5 , 1962. He received the B.S.E., M.S., and Ph.D. degrees from Duke University, Durham, NC, in 1983, 1985, and 1988, respectively.

He is currently a Research Associate at the Nora Eccles Harrison Cardiovascular Research and Training Institute at the University of Utah, Salt Lake City.

His research interests include the development of large scale models to represent the electrical activity in the heart.

Roger C. Barr (M’75-SM’81) is Professor of Biomedical Engineering and Associate Professor of Pediatrics at Duke University, Durham, NC, where he has been on the faculty since 1969. His research interests include problems in electrocar- diography, electrophysiology, and biomedical computing.