Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана

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vessel order number was selected as the independent variable in order to partition the anatomical data into three regimes (Kaimovitz, Lanir, & Kassab, 2005 ):

1. Epicardial (vessel orders 8–11, which mainly exist in the epicardium)

2. Transmural (orders 5–8, which perforate the myocardium)

3. Perfusion sub-networks (orders 1–5, which are within the sheets of myocardium

ratio was formulated as asymmetry ratio mat rix (ARM) since the ratio is a function of order of mother and daughter segments (see Appendix 5; Tables 2.23, 2.24, and

viewed as an index of vessel taper, while the latter (inter-element) represents branching asymmetry.

Figure 2.11a shows the relationship between D LCx, and RCA arteries, respectively, where D

and order number, whi ch shows a monotonic

trend where the larger orders are more asymmetric than lower orders. Finally, Fig. 2.11c shows the relationship between AER and order number. The data is partitioned into two sub-regimes: Epicardial and transmural sub-networks (orders 5–11), and perfusion sub-networks (orders 1–5). In the epicardial and transmural sub-networks, the AER is relatively uniform (no significant change based on order number) but the AER increases significantly in the perfusion sub-networks from order 5 towards the capillaries. These findings (Fig. 2.11a, c) imply a structure– function relation at order 5 that may mark a hemodynamic transition from transmural to perfusion sub-networks.

Many arcading venules and veins are found within the epicardial surface of the porcine heart (Fig. 2.13 ). The arcades can be classified as intra-connecti ons (between parts of the same vein) and inter-connections (between different veins). The intra­connecting arcades may have multiple feeding vessels but a single draining vessel. Consequently, the morphometry of these vessels can be described as a function of the order of the draining vessels (Kassab et al., 1994b). The inter-connecting arcades are either on the epicardial surface, or endocardial surface connecting sinusal to Thebesian veins. These anastomotic veins may have multiple feeding vessels but only two draining vessels. The morphometric features of the inter-connecting vessels have been summarized in the form of a tree/anastomoses matrix as described by Kassab et al. (1994b) and are summarized in Appendix 7. These anastomoses are very important when considering retrograde perfusion when blood is delivered through the coronary venous system where the arterial counterparts are too diseased to deliver oxygen and nutrients to the myocardium (see review in Kassab, Navia, March, and Choy (2008)).

2.3 Reduction of Coronary Vasculature 49

Fig. 2.15 Reconstructed spatial distribution of arterioles and venules from a histological section taken 1.7 mm from the epicardial surface of right ventricle. Solid vessels, arterioles; open vessels, venules. Shaded regions, arteriolar zones (“islands”); unshaded regions, venular zones (“ocean”). A and B indicate origin in the arteriolar zone and ending in the venular zone, respectively, of a vessel. Reproduced from Kassab and Fung (1994) with permission

length between arterioles and venules can be much longer than the average segmen­tal length.

An overriding characteristic of the coronary capil laries is that most of them are parallel to the myocardial musc le fibers. Therefore, a capillary originating from a point in an arteriolar zone will carry blood to a point in the venular zone in a flow that is parallel to the muscle fiber. An actual reconstructed case is shown in Fig. 2.15,in which the muscle fibers are horizontal. The topological study of the arteriolar and venular zones by Kassab and Fung (1994) has yielded the result that, in plane cross sections (Fig. 2.15), the arteriolar zones can be concep tualized as islands in an ocean consisting of the venular zone. The ocean (venular zone) surrounding the islands (arteriolar zones) appear as non-overlapping regions, each of which lies in an immediate neighborhood of an island. As can be seen in Fig. 2.15, capillary blood vessel AB has an origin A in an island and an ending B in the ocean next to that island. If it is assumed that every point in the island has an equal chance of being the point A, and that every point in the ocean in a direction parallel to the muscle fiber has an equal chance of being point B, then the average length of capillaries AB is exactly one-half of the average length of the straight-line segments intercepted by the island and its immediate ocean. The linear distance between the centers of mass of adjacent arteriolar and venular domains of a capillary bed was measured, and the average value of these measurements is defined as the mean functional capillary length. The theoretical mean functional capillary length is the length of a straight line.

adjacent arterial and venular domains, the results of which were 501 μm for right ventricle and 512 μm for the left ventricles. The ratios of the two measurements are

1.09 and 1.07 for right and left ventricles, respectively. These data suggest that most capillary pathways followed the shortest path from terminal arteriole to collecting venule.

The reductionist process outlined above for obtaining statistical morphometric data from a vascular tree is unique, given that the same set of rules are followed each time (e.g., Weibel’s generation system, Strahler’s order scheme, or the diameter-defined system). The integration process of creation of a circuit from statistical morphomet­ric data, however, is a non-unique process. An infinite number of circuits can be created corresponding to a set of morphometric data, but not uniquely specified by them. Hence, it is necessary to apply a set of anatomical and physiological con­straints to ensure the fidelity of the reconstruction as demonstrated below for two approaches: (1) Node-to-node reconstruction, and (2) Statistical 3D reconstruction of coronary vasculature.

Since it is experimentally difficult to reconstruct the entire vascular circuit of any organ because of the enormous number of vessels, Mittal et al. (2005) introduced a method for the reconstruction of the full coronary vascular tree from partial mea­surements. The method includes the use of data on those parts of the tree that are measured to extrapolate the data to those parts that are missing. Specifically, a two-step approach was employed in the reconstruction of the entire coronary arterial tree down to the capillary level. Vessels >40 μm were reconstructed from cast data, while vessels < 40 μm were reconstructed from histological data. The cast data were reconstructed one-bifurcation at a time, while histological data were reconstructed one-subtree at a time by “cutting” and “pasting” of data from measured to missing vessels. The reconstruction algorithm yielded a full arterial tree down to the first capillary bifurcation with 1.9 million vessel segments for the RCA tree, 2.04 million vessel segments for the LAD tree, and 1.15 milli on vessel segments for the LCx tree. The node-to-node connectivity along with the diameter and length of every vessel segment was noted (Fig. 2.16). Once the full tree was reconstructed, the assignment of order numbers to every vessel segment in the tree was automated, according to the diameter-defined Strahler system. Consequently, the diam eters, lengths, number of vessels, segments-per-element ratio, connectivity and longitudinal matrices were

Fig. 2.16 A schematic of an arteriolar tree with the nodes labeled numerically and the segment diameters and lengths specified. The first and second numbers in the parentheses correspond to the diameter and length of the segment, respectively. The “2” denotes a broken vessel whose length is unknown. The heavy line corresponds to the trunk of the arteriole tree. The partly “solid” and partly “dashed” segments correspond to broken vessels >8 μm which were extrapolated using the proposed algorithm. The entirely “dashed” segments correspond to extrapolated vessels. Reproduced from Mittal et al. (2005) with permission

determined for every order number and were found to be consistent with measured statistical data. The node-to-node network model lacked the 3D spatial distribution, however, which is addressed in the model below.

A large-scale 3D stochastic reconstruction of the asymmetric coronary arterial and venous trees (RCA, LAD, LCx, and sinusal veins) of the porcine heart has been carried out by Kaimovitz et al. (2005) and Kaimovitz, Lanir, and Kassab (2010). The model spans the entire coronary arterial tree and venous trees down to the capillary vessels. The 3D tree structure was reconstructed initially in rectangular slab geom­etry by means of global geometrical optimization using parallel simulated annealing (SA) algorithm which is a probabilistic technique for approximating the global

optimum of a given function. The SA optimization was subject to constraints

An asymmetrical, non-dichotomous coronary vascular network was reconstructed based on the statistical morphological data obtained by Kassab, Lin, and Fung (1994a), Kassab et al. (1993, 1994b), and Kassab, Pallencaoe, et al. (1997) for the coronary arterial trees (RCA, LAD and LCx), venous trees (Thebesian and sinusal), and capillary network (Fig. 2.17). Additional data of the coronary arterial tree were obtained from measurements by VanBavel (1989) and VanBavel and Spaan (1992)

2.4 Integration of 3D Coronary Vasculature 53

positioned on the epicardial surface.