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1 Rule Based Assignment of Myocardial Sheet Orientation Rolf F. Schulte, Frank B. Sachse, Christian D. Werner, Olaf Dössel email: [email protected] Abstract— Ventricles consist of an ordered laminar arrangement of myocytes. This has influence on the electrophysiological and elastomechanical properties of the heart and cannot be neglected in realistic models. Main objective was to assign this sheet orientation into any given dataset of mammalian hearts. Sheet orientation is generally radial to ventricular surfaces inside the mid-myocardium and becomes roughly tangential to it near the epi- and endocardium. The first step is to calculate wall normals, using the gradient of the Gaussian function as mask. These normals are verified through various criteria, such as the length and the direction of the normals and the distance towards boundaries of the area. Few relative sheet angles are assigned on these wall normals equidistantly, which are used as fulcra for a latter interpolation. Due to different patterns, the left and right ventricular free walls and the interventricular septum are treated separately. Finally, all relative sheet angles are transformed into pairs of absolute angles. The tool was applied on the Visible Female dataset, which is part of the Visible Human project from the National Library of Medicine (USA). Together with the fibre orientation, a complete anatomical model of myocardial sheet orientation was created. Fig. 1. Isolated mammalian myocyte; intercalated disks are located mainly at the end, but also laterally (from [3]). Keywords— myocardial sheet orientation; anatomical model I. Introduction Although the gross morphology of mammalian hearts is well investigated, its microscopical structure is still unknown in great parts. Recently, a laminar arrangement of groups of myocytes was discovered [1]. One axis of these planes runs approximately radially from endocardium towards epicardium in transmural direction. The other axis coincides with the local fibre orientation This laminar arrangement influences the electrophysiological and elastomechanical properties of the heart and cannot be neglected in realistic physiological models. Main objective of this study was to assign this sheet orientation into any given dataset of mammalian hearts with the methods of digital image processing. Due to numerous differences in the morphology between canine and human hearts, simple mapping is not feasible. Thus, a possible way is to assign myocardial sheet orientation with a set of rules. The rules used in this study are derived from [1], [2]. II. Microscopic Anatomy of Mammalian Heart Hearts are mainly build of myocytes (figure 1), cardiac muscle cells, which have a length of 50 to 120 µm and a diameter of 5 to 25 µm [6]. Myocytes are interconnected with each other via intercalated disks, Fig. 2. Visible female heart [4], including fibre orientation (from [5]). which provide mechanical and electrical coupling [7]. Intercalated disks are located mainly at the end of the cells, but also laterally. The principal axes of adjacent cells are nearly parallel, allowing to determine a mean direction, which is generally referred to as muscle fibre orientation [2]. Fibres run approximately parallel to the outer cardiac surface. The heart wall can be described as a threedimensional continuum made up essentially of cardiac muscles cells, with smooth change of direction across the wall [8]. There are no “bundles of fibres” running through the myocardium. Collagen Fibres The arrangement of cardiac myocytes is mirrored by a complex extracellular connective tissue matrix. An extensive network of collagen fibres is present inside the myocardium. According to Caulfield [9] these fibres can be divided into three different types: Fig. 3. Collagen network of the rabbit left ventricle interconnecting the myocyte (M) and the capillary (C) (×6,500; from [9]). Fig. 5. Schematic of cardiac microstructure. (a) A transmural blocks cut from the ventricular wall shows the fibre orientation inside a myocardial sheet. (b) The myocytes are shown lying three to four cells thick within a sheet. Endomysial collagen is seen linking adjacent cells within a sheet, while perimysial collagen links adjacent sheets (from [2]). Fig. 4. Two orthogonal surfaces of mid-wall specimen (from [2]). Myocyte to myocyte collagen struts, which provide equal stretch of contiguous cells, and prevent slippage of adjacent cells • Myocyte to capillary collagen struts • Meshwork of collagen fibrils surrounding groups of myocytes; few loose collagen connections inbetween permit slippage and rearrangement The latter kind of collagen fibres groups three or more myocytes into an ovoid configuration. Its main axis is parallel to the long axis of the cells, which is equivalent to the fibre axis. In the mid-myocardium the second axis is nearly perpendicular to the endo- and epicardial surface. • Muscle Sheet Arrangement LeGrice et al. [2], [1] discovered, that myocytes are ordered laminarly with relatively extensive cleavage planes inbetween muscle layers. Branching between adjacent layers is relatively sparse with muscle bridges one to two cells thick. Each of these layers consists of tightly packed groups of cardiac myocytes. This general arrangement is repeated throughout the ven- tricles. There are typically four myocytes across the thickness of a layer. One axis of the sheets is built by the myocardial fibre orientation. The other axis is measured relative to the local outer-wall normal in planes built by the principal axis of the heart and a vector radial to it. Near the equator the orientation is similar for both ventricular free walls, changing from around −90◦ at the endocardium to 30−60◦ at the epicardium. Inside the interventricular septum the orientation is changing from 90◦ in the left ventricular sub-endocardium to around −90◦ in the right ventricular sub-endocardium. In the mid-myocardium the orientation is approximately parallel to the normal. Two areas which do not follow this standard pattern are around the bases of the left ventricular papillary muscles, where there is a complex interweaving, and around the base of the left ventricular free wall, where sheets angle up into the basal skeleton [2]. In the papillary muscles, trabeculae, and the atria a laminar structure is not visible. Differences between Canine and Human Hearts All gross morphological features are similar in both mammalian hearts. The only significant differences concern their shape. Furthermore, Caulfield used various species for his investigations. The main conclusion is, that the collagen network is qualitatively similar in the rodent and human hearts, but quantitatively dissimilar [9]. Although the sheet orientation is mapped only for canine hearts up to now [2], [1], it can be assumed, that the derived rules are valid for all mammalian species, including human hearts. Reasons are the morphological similarities in their macroscopic and micro- 3 parts: Left ventricular free wall Right ventricular free wall Interventricular septum Detection of fulcra Detect starting point Calculate normal endocardium Assign angles OP mid-myocardium N epicardium γ P Correction (only LV) Interpolation Transform angles Fig. 6. Principal functioning of the developed tool. base following three areas are distinguished: left and right ventricular free wall, and the interventricular septum. The principal functioning can be observed in figure 6. left atrium P N interventricular septum left ventricluar free wall OP Fig. 8. Equidistant assignment of sheet angles (γ) relative to the outer wall normal ~ n. CG right ventricle apex Fig. 7. Sketch of left ventricle describing the basic functioning. The normal ~ n starts in P and exits in OP. The direction points towards the centre of gravity (CG). All starting points too close to the left atrium (base) and right ventricle (apex and septum) are excluded. scopic structure, common roots in evolution, and the fact, that all mammalian hearts are highly optimized and efficient in their functioning. Thus they are likely to use the same mechanisms. III. Material and Methods Requisite is a volume based anatomical model of the heart with the following classification: left and right ventricle, left and right atrium, as well as fat and other tissue. The interventricular septum is usually classified as approximately two third belonging to the left ventricle and one third to the right ventricle. Two axes are required for a complete model of ventricular sheet orientation. One axis coincides with the local myocardial fibre orientation. Models describing this orientation already exist [5] and were not subject of this study. The other axis is generally radial to the ventricular surfaces, but becomes roughly tangential to the epi- and endocardium. Two exceptions are the area near the base of the left ventricular free wall, where a correction is applied, and the bases of the left ventricular papillary muscles, which were excluded during the segmentation. The developed tool assigns few ventricular sheet angles on wall normals, which are used as fulcra for a succeeding interpolation. Due to different patterns, the Left Ventricle The first task is getting proper wall normals. Starting points are left ventricular voxels with differently classified neighbours. To these points, normals are calculated using the gradient of a Gaussian function. Now the left ventricular free wall is traversed until an exiting point to another material class is found. Several criteria apply for verifying these normals: • Distance towards right ventricle and left atrium: Removes erroneous normals near boundaries towards other cardiac areas and inside the interventricular septum. • Length of normal: Excludes abnormal sizes, e.g. caused by fat particles inside the myocardium. • Direction towards centre of gravity of the left ventricle: Due to the fact that the left ventricular free wall has roughly the shape of parts of an ellipsoid, all normals point approximately into the same direction, the centre of gravity. For this criterion the vector starting point towards centre of gravity of left ventricle is multiplied with the normal and checked. This also excludes normals starting endocardial (figure 7). On all remaining normals sheet angles are assigned equidistantly throughout the wall from the starting point P towards the exiting point OP. Due to different patterns near the basis of the left ventricular free wall, a correction for these angles is applied. The left atrium is used as an approximation for the area of the basis to avoid additional segmentation. All angles are increased towards a maximum value of 90◦ , depending on their distance to the left atrium, with the following equation: γ 0 = γ + (90◦ − γ) · scale 0 ≤ scale ≤ 1 max − distance 6 scale = max − min where distance stands for the distance of the present voxel towards the left atrium, max for the maximum distance and, min for the closest voxel. RV N LV first step P γ x N OP α α b second step OP * P* N* P Fig. 9. Normals inside the interventricular septum. In a first step the normal ~ n traverses the left ventricle (LV). In the exiting point (OP= P∗ ), the normal is turned by 180◦ (~ n∗ = −~ n) and passes through both, left and right ventricle. teria are applied: • Distance towards fat: Excludes normals near the base or apex of the heart, resembled through fat. • Length of normal • Direction of normal: The reference is the vector from the centre of gravity of the right ventricle towards the centre of gravity of the left ventricle (figure 10). left atrium base right atrium LR Fig. 11. Creation of ~b with the angle γ, the wall normal ~ n, and the principal axis p ~ CGL Normal Calculation CGR N LV RV apex Fig. 10. Sketch of interventricular septum. Starting points too close to fat (base and apex) are excluded. The direction of the normal ~ n has to match the direction of the vector LR from the the centre of gravity of the left ventricle (CGL) towards the centre of gravity of the right ventricle (CGR). In a last step these relative angles are transformed into the absolute angles ϕ and ϑ. The assignment of sheet orientation within the right ventricular free wall is similar to this, but no correction needs to be applied. Interventricular Septum Inside the interventricular septum, a different strategy is used. Because there is no separate classification for this region, the border between right and left ventricle is used for normal calculation. This border is relatively smooth. A starting point (P) is a voxel classified as left ventricle with a contiguous right ventricular voxel. First the area of the left ventricle is traversed from the starting point towards an exiting point to another material class. Now the normal is turned around and the former exiting point is set as new starting point. Then the area of both, left and right ventricle is traversed towards an exiting point to the right ventricular cavity (figure 9). For exterminating invalid normals the following cri- For performance reasons the calculation of normals on cardiac surfaces is divided into two steps. First, a normal mask is generated. In this case the gradient of a Gaussian function is used, which is defined as follows [10]: f (x) = 1 √ σ 2π x2 e− 2σ2 where σ is the standard deviation and equated with the width of the mask. By using a Gaussian function, the voxels closer to the centre are weighted stronger and thus errors caused by a rough surface are eliminated. The required gradient for the mask is: ∂f − x2 ∂x ~x ∂f σy − σ2 grad f = ∂y = f =− 2 f σ ∂f − σz2 ∂z In a second step this mask is used for a normal calculation around each eligible point. This point represents the centre of the mask. All values of the mask of neighbouring voxels with different classification are grouped in a vector, and thus result in the normal. Transformation of Relative into Absolute Angles Sheet angles are assigned in voxels of the normal relative to this normal. A transformation into the two absolute angles ϕ and ϑ is required. This is done in two steps. First, an orientation vector ~b is created as follows: ~b = ~n + p~ sin γ |~n| sin α |~ p| where γ is the relative sheet angle and α the angle between the orientation vector ~b and the first principal Fig. 12. Sheet orientation in a slice of the visible female heart [4]. The left upper part shows the left ventricular free wall, the middle is the interventricular septum, and on the right side is the right ventricular free wall axis p~. For small angles α, the orientation ~b is equated with the principal axis p~ (figure 11). Then the orientation vector is transformed into the two angles ϕ and ϑ with the following equation: x sin ϑ · cos ϕ y = r sin ϑ · sin ϕ z cos ϑ y ⇒ ϕ = arctan x r x2 + y 2 ϑ = arctan z with 0◦ ≤ ϑ ≤ 180◦ and 0◦ ≤ ϕ ≤ 360◦ . The vector needs to be normalized first, which means r = 1. Interpolation Finally, all not yet assigned points are interpolated [11]. IV. Results With the developed tool it is possible to assign myocardial sheet orientation to anatomical models of mammalian hearts. It was applied on the visible female dataset [4]. The various criteria, like the length and the direction of normals and the exclusion of normals close to critical areas, allow a stable and robust search for all valid wall normals. With enough fulcra an interpolation of angles without discontinuities is assured. Figure 12 and 13 show the results of the assignment of sheet orientation. V. Discussion A rule based assignment of myocardial sheet orientation is always as good as the data and rules it is based on. Although the microstructural properties of canine and human hearts are similar, an investigation for human hearts is recommended as well. With a phase sensitive MRT it might be possible to examine Fig. 13. Sheet orientation inside the left ventricular free wall living human hearts without negative side effects, like an impact on the health or a greater aberration of the results, due to rough investigation methods. The obtained data can be used to verify and improve the developed tool. The parameter settings can be adopted. The specimen used by LeGrice et al. [2], [1] altered during the dehydration process, making the sheet orientation more evident. An investigation method, which is not based on dehydration, could further improve gained results and minimize errors. Main conclusion is, that a rule based assignment of myocardial sheet orientation is a feasible way. The errors are depending on the quality of the derived rules. References [1] [2] [3] [4] [5] [6] [7] Ian J. LeGrice, B. H. Smaill, L. Z. Chai, S. G. Edgar, J. B. Gavin, and Peter J. Hunter, “Laminar structure of the heart: Ventricular myocyte arrangement and connective tissue architecture in the dog,” American Journal of Physiology, vol. 269, pp. H571–H582, 1995, Heart Circ Physiol. 38. Peter J. Hunter, B. H. Smaill, Poul M. F. Nielsen, and Ian J. LeGrice, Computational Biology of the Heart, chapter 6: A Mathematical Model of Cardiac Anatomy, pp. 171–215, John Wiley & Sons Ltd, 1997, ISBN 0-47196020-9. Jeffrey E. Saffitz and Kathryn A. Yamada, Cardiac Electrophysiology; From Cell to Bedside, vol. 3, chapter 21: Gap Junction Distribution in the Heart, W. B. Saunders Company, 2000, ISBN 0-7216-7811-4. M. J. Ackerman, “Viewpoint: The Visible Human Project,” Journal Biocommunication, vol. 18, no. 2, pp. 14, 1991. F. B. Sachse, R. Frech, C. D. Werner, and O. Dössel, “A model based approach to assignment of myocardial fibre orientation,” in Proc. Computers in Cardiology, 1999, pp. 145–148. W. Bargmann and W. Doerr, Eds., Das Herz des Menschen, vol. 1, Georg Thieme Verlag, Stuttgart, 1963. Robert H. Hoyt, Mark L. Cohen, and Jeffrey E. Saffitz, “Distribution and three-dimensional structure of intercellular junctions in canine myocardium,” Circulation Research, vol. 64, pp. 563–574, 1989. [8] Daniel D. Streeter, Handbook of Physiology: The Cardiovascular System, vol. 1, chapter 4: Gross Morphology and fibre geometry of the Heart, pp. 61–112, American Physiology Society, 1979. [9] J. B. Caulfield and T. K. Borg, “The collagen network of the heart,” Laboratory Investigation, vol. 40, no. 3, pp. 364–372, 1979. [10] Ilija N. Bronstein, Konstantin A. Semendjajew, Gerhard Musiol, and Heiner Mühlig, Eds., Taschenbuch der Mathematik, vol. 2, Harri Deutsch, Frankfurt am Main, 1995, ISBN 3-8171-2002-8. [11] F. B. Sachse, M. Wolf, C. Werner, and K. Meyer-Waarden, “Extension of anatomical models of the human body: Three dimensional interpolation of muscle fiber orientation based on restrictions,” Journal of Computing and Information Technology, vol. 6, no. 1, pp. 95–101, 1998.