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Bifurcation asymmetry of the porcine coronary vasculature and its implications on coronary flow heterogeneity

Department of Biomedical Engineering, University of California, Irvine, California 92697-2715

Submitted 19 April 2004 ; accepted in final form 30 July 2004

	ABSTRACT

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The branching pattern of the coronary arteries and veins is asymmetric, i.e., many small vessels branch off of a large trunk such that the two daughter vessels at a bifurcation are of unequal diameters and lengths. One important implication of the geometric vascular asymmetry is the dispersion of blood flow at a bifurcation, which leads to large spatial heterogeneity of myocardial blood flow. To document the asymmetric branching pattern of the coronary vessels, we computed an asymmetry ratio for the diameters and lengths of all vessels, defined as the ratio of the daughter diameters and lengths, respectively. Previous data from silicone elastomer cast of the entire coronary vasculature including arteries, arterioles, venules, and veins were analyzed. Data on smaller vessels were obtained from histological specimens by optical sectioning, whereas data on larger vessels were obtained from vascular casts. Asymmetry ratios for vascular areas, volumes, resistances, and flows of the various daughter vessels were computed from the asymmetry ratios of diameters and lengths for every order of mother vessel. The results show that the largest orders of arterial and venous vessels are most asymmetric and the degree of asymmetry decreases toward the smaller vessels. Furthermore, the diameter asymmetry at a bifurcation is significantly larger for the coronary veins (1.7–6.8 for sinus veins) than the corresponding arteries (1.5–5.8 for left anterior descending coronary artery) for orders 2–10, respectively. The reported diameter asymmetry at a bifurcation leads to significant heterogeneity of blood flow at a bifurcation. Hence, the present data quantify the dispersion of blood flow at a bifurcation and are essential for understanding flow heterogeneity in the coronary circulation.

asymmetry ratio; order number; flow heterogeneity; Poiseuille's resistance; coronary arteries

Our hypothesis is that the asymmetry of vascular geometry at a bifurcation is an important determinant of coronary blood flow heterogeneity. We have previously described the branching pattern (connectivity and longitudinal position matrices) and vascular geometry (diameters and lengths) of the coronary vasculature using the diameter-defined Strahler system (7, 10, 22). We found the branching pattern and vascular geometry to be quite asymmetric, which necessitated the use of the diameter-defined Strahler system. The asymmetry was most pronounced for the epicardial vessels that give rise to transmural vessels in "large trunk-small twig" patterns. The asymmetry of the branching pattern was reflected by the pattern of connectivity and longitudinal position matrices and the segments-per-element ratios.

In the present study, we describe the bifurcation asymmetry in terms of local parameters at a bifurcation called asymmetry ratios. For a given mother vessel, we defined the diameter and length asymmetry ratio as the ratio of the daughter diameters and lengths, respectively. The diameter and length ratios were computed for each mother vessel throughout the coronary vasculature (from arteries to arterioles and from venules to veins). The resulting diameter and length asymmetry ratios were used to compute the asymmetry of area, volume, resistance, and flow at various bifurcations throughout the coronary vasculature. These data allow us to predict the flow heterogeneity at a bifurcation along with other important hemodynamic parameters. Furthermore, these data are necessary for the reconstruction of coronary vascular circuits for future hemodynamic analysis.

	METHODS

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All experiments were performed in accordance with national and local ethical guidelines, including the Institute of Laboratory Animal Resources, National Institutes of Health Guide for the Care and Use of Laboratory Animals, and University of California-Irvine policies regarding the use of animals in research.

Available data. We have previously described in detail the methods of preparation, measurement, and morphometric analysis of the right coronary artery (RCA), left anterior descending (LAD) coronary artery, left circumflex (LCx) artery, coronary sinus, and thebesian veins (10, 11). The morphometric data on the coronary vessels of diameters <50 µm were obtained from histological specimens, whereas the morphometric data on vessels of diameters >50 µm were obtained from cast studies as described in Kassab et al. (10). The morphometric measurements (diameters, lengths, and node-to-node connectivity) of the microvasculature were made by changing the focal plane through the thickness of the histological sections. For cast studies, the trunks of major coronary arteries and veins were sketched, and their segments measured down to 50 µm in diameter. The raw coronary morphometric data showing node-to-node connectivity, diameters, and lengths are available on http://cvbiomech.eng.uci.edu.

Once the morphometric measurements were completed, we organized the segments and their diameters and lengths in terms of the diameter-defined Strahler's ordering scheme as described in Refs. 7 and 10. Briefly, the capillary blood vessels are defined as vessels of order 0. The smallest arterioles supplying blood to the capillaries are assigned an order number of 1. The smallest venules draining the capillaries are assigned an order number of –1. When two arterioles of order 1 meet, the confluent vessel is given an order of number 2 if its diameter exceeds the diameters of the order 1 vessels by an amount specified by a set of formulas or diameter criterion (10) or remain as order 1 if the diameter of the confluent is not larger than the amount specified by the formulas. When an order 2 artery meets another order 1 artery, the order number of the confluent is 3 if its diameter is larger by an amount specified by the diameter criterion or remains at 2 if its diameter does not increase sufficiently. This process was continued until all arterial segments were arranged in increasing diameter and assigned the order numbers 1, 2, 3, ..., n, ... Similarly, the veins are assigned the orders –1, –2, –3..., –n, ... The criterion is a value that is midway between the mean diameter of order n + 1SD and the mean diameter of the order n – 1SD for all n from 1 upward (10).

Asymmetry at a bifurcation. With the existence of a full set of ordered data on the coronary vasculature, we proceeded to define and compute the asymmetry ratios by considering each bifurcation node. The largest and the two smaller vessels meeting at a node are called the mother and daughter vessels, respectively, if the order numbers of these vessels are so arranged that n m k with diameters D_n D_m D_k. Unlike the coronary arteries, which have cylindrical cross sections and can be quantified by a diameter, the veins have approximately elliptical cross sections. We shall denote the major and minor axes diameters of the coronary veins as D_maj and D_min, respectively. For each mother vessel, n, the asymmetry ratio for the major and minor diameters of the daughter vessels, S_Dmaj and S_Dmin, are the ratios defined by

(1a)

and

(1b)

where S_Dmaj 1, by definition. Analogously, the asymmetry ratio for the length of the daughter vessels, S_l, is the ratio defined by

(2)

The asymmetry ratios of the vessel cross-sectional area, A, and volume, v, are similarly defined by

(3)

and

(4)

because the cross-sectional area of an elliptical venous vessel is A = (/4)D_majD_min, and the volume of a branch is the product of its cross-sectional area and length (11).

The flow resistance, R, for a vein with an elliptical cross section is computed as follows: assuming that the daughter vessels are long and slender, that the flow is laminar and steady, and that the flow disturbances at the entry and exit sections are negligible, then the resistance to flow in the daughter vessels is given by (Ref. 11)

(5)

where µ is the apparent viscosity. Hence, we can define an asymmetry ratio of the flow resistance by the equation

(6a)

where

(6b)

S_µ is defined as the apparent viscosity ratio of the two daughter branches (µ_m/µ_k). The data on the variation of viscosity with vessel diameter and hematocrit are given by Pries et al. (19), which was used in our calculations. They proposed a modified viscosity relationship based on a compilation of literature data on relative blood viscosity in tube flow in vitro and in vivo, which reflects the Fahraeus-Lindqvist effect as given by:

(7)

where D is the vessel diameter. This relationship was used throughout the coronary vasculature for a given vessel diameter D at a hematocrit of 0.45.

For the coronary arteries, the major and minor axes diameters are equal, and hence Eq. 1 reduces to S_D = (D_m/D_k). Similarly, Eq. 3 becomes S_A = S_D². Finally, Eq. 5 becomes Poiseuille's resistance (i.e., R = 128 µl/D), and Eq. 6 reduces to SR = S_µS_l/S_D⁴.

Horsfield et al. (6) have previously defined the difference in order number between two daughter branches at a bifurcation ( = m – k) to describe the asymmetry at a bifurcation of an airway. Additionally, we shall introduce a parameter ( = n – m), which is the difference between the order of the mother vessel and that of the larger daughter.

Data analysis. All asymmetry parameters were curve fitted by a relation of the form ln(S_x) = a + bn + cn², where ln is the natural logarithm and x corresponds to diameter (D), length (l), cross-sectional area (A), volume (v), or resistance (R). The empirical constants a, b, and c were determined by a least-squares fit. ANOVA was used to detect differences among different orders for the various parameters (i.e., S_D, S_l, S_A, S_v, S_R, , and ). Although our data are skewed, it is generally accepted that ANOVA is robust to violations of the normality assumption (4). Statistical significance was accepted at the 0.05 level.

	RESULTS

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We obtained data from the entire range of orders throughout the coronary arterial and venous trees. A total of 5,082, 4,014, and 1,946 arterial bifurcations of RCA, LAD, and LCx arteries and 13,778 and 4,505 venous bifurcations of coronary sinus and thebesian veins and their branches, respectively, was analyzed. Probability frequency distribution functions were obtained for S_D and S_l for each order of the entire coronary vasculature and were found to be right skewed. Tables 1–5 show a summary of the means ± SD of S_D and S_l in RCA, LAD, and LCx arteries and sinusal and thebesian veins, respectively. Using ANOVA, we found a significant dependence of S_D and S_l on the order number of the mother vessel. We also found that, at a given order number, the S_D for the venous trees was significantly larger than those of the corresponding arteries (orders 2–10). The S_l was larger for veins of orders 3–6 but smaller for orders 2 and 7–10 compared with arteries. The data on S_D and S_l were represented by a quadratic exponential curve fit and are summarized in Table 6.

View larger version (16K): [in this window] [in a new window]   Fig. 1. Relationship between the fraction of larger daughter vessels that also have longer lengths than the smaller daughter vessels and the order number of the mother vessel of the coronary vasculature [Flm > lk = N(lm > lk)/N(lm > lk) + N(lm < lk)], where N(lm > lk) and N(lm > lk) represent the number of bifurcations where the larger daughter diameter vessel has a longer and shorter length, respectively. RCA, right coronary artery; LAD, left anterior descending coronary artery; LCx, left circumflex artery.

View larger version (17K): [in this window] [in a new window]   Fig. 2. Relationship between the cross-sectional area ratio of daughter vessels and the order number of the mother vessel of the coronary vasculature.

View larger version (17K): [in this window] [in a new window]   Fig. 3. Relationship between the volume ratio of daughter vessels and the order number of the mother vessel of the coronary vasculature.

View larger version (19K): [in this window] [in a new window]   Fig. 4. Relationship between the Poiseuille's resistance ratio of daughter vessels and the order number of the mother vessel of the coronary vasculature.

View larger version (16K): [in this window] [in a new window]   Fig. 5. Relationship between the difference in the order between the two daughter vessels, , and the order number of the mother vessel of the coronary vasculature.

View larger version (23K): [in this window] [in a new window]   Fig. 6. Relationship between the difference in the order between the mother and larger daughter vessel, , and the order number of the mother vessel of the coronary vasculature.

View larger version (17K): [in this window] [in a new window]   Fig. 7. Relationship between the coefficient of variance, CV (SD/mean), and the order number of the mother vessel of the coronary vasculature.

	DISCUSSION

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As we have previously stated, S_D 1 by definition. S_l, on the other hand, is not bound in this way because a larger diameter daughter vessel may be shorter than the smaller daughter vessel. The fraction of larger diameter daughter vessels that have longer lengths than the smaller vessels is shown in Fig. 1. It is interesting to note that for the smallest five orders, 50% of the daughter vessels with larger diameters also have longer lengths. For the larger mother vessels, however, a higher percentage of daughter vessels of larger diameter have longer lengths.

Knowledge of S_D and S_l allows the determination of S_A, S_v, and S_R of the two daughter branches of each mother vessel. The S_A is simply the square of S_D. However, the S_A is not simply the square of mean S_D because of the skewed shape of the distribution. The S_v is the product of S_A and S_l. The S_A and S_v are greatest for the larger vessels and decrease toward the smaller vessels. The mean asymmetry ratios of area and volume at the highest orders are 150 and 560 (RCA) and 180 and 360 (sinus vein), as shown in Figs. 2 and 3, respectively. The enormity of the asymmetry is remarkable. Finally, the S_R considers the S_D, S_l, and S_µ as expressed by Eq. 6 and shown in Fig. 4. The S_R is < 1 because a larger vessel has a lower resistance than a smaller vessel. The S_D dominates over the S_l because of the fourth power exponent.

Comparison with other studies. Kassab et al. (9, 13) have previously described the bifurcation asymmetry of the right ventricular branches in normal and hypertensive right ventricles. They found that the S_D and S_l increase significantly with order number for both control and hypertensive hearts. As expected, the control data on the right ventricular branches were similar to the present data for the RCA tree.

Horsfield et al. (6) introduced the concept of to describe the branching asymmetry at a bifurcation. They showed that a fixed value of implies a property of self-similarity, i.e., each succeeding bifurcation having the branching pattern and relative dimensions of the parent vessel but on a smaller scale. In the present study, we found that the value of varies with order number. It is interesting to note that the first six orders of arterial trees have a fairly uniform value of of 1 but increase with higher orders (Fig. 5). We have added a new parameter, , which describes the order asymmetry between mother and larger daughter vessel (Fig. 6). The collective data on and can be used to reconstruct the order connectivity of the vascular tree.

Relation of asymmetry ratios to design rules. We have previously predicted several structure-structure and structure-function relationships for coronary arterial trees (29). In the process, a vessel segment was defined as a stem that perfuses a tree or a crown (25). The crown includes all vessels proximal to the stem including the first segment of capillary vessels. If A and Q represent the mean cross-sectional area and blood flow of a stem, respectively, and V and L represent the cumulative distal arterial volume and length of a crown, respectively, we predicted the following relationships:

(8)

(9)

(10)

where D_max, Q_max, V_max, and L_max correspond to the diameter and flow of the most proximal stem to the arterial tree, the volume of the entire tree of interest, and the cumulative arterial length of the entire tree. The values of , , and have been previously reported as 0.41, 1.35, and 2.42 (RCA); 0.41, 1.37, and 2.45 (LAD); and 0.40, 1.38, and 2.47 (LCx) for the arterial trees, respectively (29). These relationships have been determined and validated in the following three ways: 1) a hemodynamic analysis of coronary arterial blood flow based on detailed anatomic data yields these relationships over the entire arterial network (12); 2) a generalization of Murray's cost function and conservation of energy predict the same result over the entire tree (29); and 3) in vivo data on flow, length and volume using digital subtraction angiography (28) verified the relationships for vessels proximal to 0.5 mm in diameter as observed in an angiogram.

To relate the asymmetry ratios of stem quantities (diameter, cross-sectional area, and flow) to crown quantities (crown length and volume), we combined Eq. 8 with Eq. 1 for an artery to yield

(11a)

or

(11b)

since = 1, as shown previously (29). S_L is the crown asymmetry length compared with the segment asymmetry length S_l. The exponent of Eq. 11 has a value of 2.5. Hence, the asymmetry ratio for the crown length increases with S_D raised to the power of 2.5. Similarly, the S_V and S_Q (Eqs. 9 and 10) can be expressed in terms of S_D as:

(12)

and

(13)

respectively. The exponents of Eqs. 12 and 13 are 3.4 and 2.5, respectively. Murray's law predicts a fixed value of 3 for irrespective of the organ of interest based on the minimum energy hypothesis (18).

We have also previously verified a form for the crown resistance (R_c), i.e., equivalent resistance of vessels in a crown, as

(14)

where k_R and are constants determined from experimental data (10). The reported value of was approximately –0.55 (29). The asymmetry ratio of the crown resistance can be expressed as

(15)

Equations 11 and 12 can be combined with Eq. 15 to yield

(16)

The exponent is numerically equal to approximately –1.3, i.e., S_Rc is <1, similar to S_R. We can relate the asymmetry of flow to that of crown resistance if we combine Eqs. 13 and 16 and use = 1 as

(17a)

Because is 2.5, the exponent of Eq. 17 is –0.52. Hence, for the coronary arterial tree, Eq. 17a can be reduced to

(17b)

which is a remarkably simple relationship between flow asymmetry at a bifurcation and the crown asymmetry.

A relation between S_Rc and S_R can be determined if we combine Eq. 16 with Eq. 6 for a cylindrical vessel, i.e.,

(18a)

The exponent of Eq. 17 is obviously –0.32. Again, this can be simplified as

(18b)

In the microcirculation, at the site of vascular resistance, S_l and S_µ are 1 (Tables 1–5) and hence S_Rc S_R^(1/3). This remarkably simple relation implies that the crown resistance ratio at a bifurcation can be determined from the resistance ratio of the daughter vessels for coronary microvessels. In summary, the conclusion of the above analysis is that the crown ratios (Eqs. 11, 12, and 16) are significantly more asymmetric than the diameter asymmetry ratio.

Implications of S_D on flow heterogeneity. The bifurcation asymmetry of resistance may not be directly proportional to the flow asymmetry because the flow distribution depends on the total distal resistance (crown resistance) of the two daughter vessels, as shown in Eq. 17. Hence, it is desirable to have a more direct relationship between the geometric asymmetry ratios and the flow distribution. Such relationship is expressed by Eq. 13, which shows a power-law dependence of flow asymmetry on the diameter asymmetry. The implication of Eq. 13 is that a small increase in diameter asymmetry ratio can cause a large increase in flow asymmetry. Hence, the heterogeneity of flow at a bifurcation is very sensitive to the diameter ratio of daughter vessels.

The CV can be estimated for the asymmetry of flow. If we take the differential of Eq. 13 and divide it by Eq. 13, we obtain

(19a)

or

(19b)

Hence, the CV(S_Q) is proportional to the CV(S_D) with (a value of 2.5) as the proportionality constant. We found that CV(S_D) of veins was significantly larger than that of arteries. Hence, the CV(S_Q) at a venous bifurcation is expected to be greater than that at an arterial bifurcation. This result has important implication for coronary venous retroperfusion where blood supply to the ischemic myocardium may be provided by making an arterial anastomosis with the coronary venous system (15). It suggests that the flow delivery to the capillary vessels will be more heterogeneous than antegrade flow. We have previously shown that the capillary network homogenizes the blood flow supplied by the arterial tree (14).

Critique of methods. Although the coronary arterial tree is dominated by bifurcations (98% bifurcations, 2% trifurcations) (10), the coronary veins consists of 86%, 12.8%, 1%, and 0.2% bifurcations, trifurcations, quadrifications, and quintifurcations, respectively (11). The present data apply only to the bifurcations but can be easily extended to trifurcations, etc. Furthermore, although the arteries have a fairly uniform diameter along their segment (i.e., diameters only change upon branching), the veins have more complex variations within segments. Because the measurement of venous diameter represents an average over the segment, the asymmetry ratio for the vein also represents an average in that sense. Finally, the asymmetry ratios may change with pressure because the coronary vessels are distensible. The present data correspond to a particular distribution of pressure through the diastolic state.

The present asymmetry ratios were expressed in terms of order numbers as determined by the diameter-defined Strahler system. Numerous ordering schemes exist, however, which may result in somewhat different asymmetry ratios. The Strahler's scheme was used because it handles the asymmetry of the system very well (22). The diameter-defined modification was proposed to eliminate the overlap in the diameters between successive orders (7, 10). In a previous study (9), we considered the diameter of the mother vessel as an independent variable regardless of the order number. We found that the diameter asymmetry ratios were similar in both cases.

The asymmetry in diameters also affects the asymmetry of bifurcation angles. This issue has been examined in detail by Zamir and colleagues (27). Zamir (26) showed that in a nonsymmetrical bifurcation, the branching angle that the larger branch makes relative to the parent vessel decreases with an increase in asymmetry ratio. The angle data affect the optimality of vascular junctions as flow conduits and dictate the spatial position of the vessels relative to the myocardium.

Although asymmetry at bifurcation is an important determinant of flow distribution, it is unlikely to be the only determinant of flow heterogeneity. In the beating heart, a heterogeneous distribution of stresses can affect the perfusion pattern (1). The addition of coronary tone plays a major role in modulating regional flow (1). Although the present anatomy was obtained under dilated state, this case is applicable when the heart is stressed physiologically or in the presence of coronary artery disease. Although the coronary anatomy establishes the potential distribution of coronary flow, mechanical, metabolic (3, 21), neural (24), and signal transduction (20) dictate local perfusion of myocardium.

Significance of study. The present study shows that the large asymmetry at a bifurcation can result in significant asymmetry of flow and potentially large spatial heterogeneity of blood flow. We have previously shown that such geometric asymmetry can remodel in right ventricular hypertrophy during flow overload (9, 13). Hence, the asymmetry parameters may be important indices of flow distribution in the heart in health and disease.

The lumen cross-sectional area and length of coronary arteries and their blood volume and flow rate can be determined from an angiogram using digital subtraction angiography (8, 16, 17). Hence, the S_D, S_L, S_V, and S_Q can be quantified and their relationships (Eqs. 11–13) can be established as signatures of normal patients where coronary disease can be studied in relation to changes in these relationships. Furthermore, knowledge of S_D or S_Q allows the determination of S_Rc through Eqs. 16 or 17, respectively. The assessment of ratio of total arterial or crown resistance in a normal region compared with that in an ischemic or infarct region has obvious clinical significance.

An additional significance of the asymmetry ratios relates to the reconstruction of the full vascular arterial tree for blood flow analysis. The process of obtaining statistical morphometric data from a tree is unique, given that the same set of rules are followed each time (e.g., the diameter-defined Strahler system). The creation of a circuit from morphometric data, however, is a nonunique process. An infinite number of circuits can be created corresponding to a set of morphometric data but not uniquely specified by them. In addition to the connectivity and longitudinal position matrixes and the diameters, lengths, and number of vessels previously reported, the proposed asymmetry ratios impose additional constraints on the reconstruction of tree circuits.

	GRANTS

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This research was supported by the National Heart, Lung, and Blood Institute Grant 2 R01 HL-055554-06. G. S. Kassab is a recipient of an American Heart Association Established Investigator Award.

	ACKNOWLEDGMENTS

 
We thank Winny Tan and Marli Amin for excellent technical assistance.

	FOOTNOTES

 

Address for reprint requests and other correspondence: G. S. Kassab, Dept. of Biomedical Engineering, Univ. of California-Irvine, 204 Rockwell Engineering Center, Irvine, CA 92697-2715 (E-mail: gkassab{at}uci.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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