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Am J Physiol Heart Circ Physiol 292: H1336-H1339, 2007. First published November 22, 2006; doi:10.1152/ajpheart.00906.2006
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Vascular metabolic dissipation in Murray's law

Yi Liu1 and Ghassan S. Kassab1,2,3

Departments of 1Biomedical Engineering, 2Cellular and Integrative Physiology, and 3Surgery, Indiana University Purdue University, Indianapolis, Indiana

Submitted 22 August 2006 ; accepted in final form 16 November 2006


   ABSTRACT
TOP
 ABSTRACT
Glossary
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
The metabolic dissipation in Murray's minimum energy hypothesis includes only the blood metabolism. The metabolic dissipation of the vascular tree, however, should also include the metabolism of passive and active components of the vessel wall. In this study, we extend the metabolic dissipation to include blood metabolism, as well as passive and active components of the vessel wall. The analysis is extended to the entire vascular arterial tree rather than a single vessel as in Murray's formulation. The calculations are based on experimentally measured morphological data of coronary artery network and the longitudinal distribution of blood pressure along the tree. Whereas the model includes multiple dissipation sources, the total metabolic consumption of a complex vascular tree is found to remain approximately proportional to the cumulative arterial volume of the unit. This implies that the previously described scaling relations for the various morphological features (volume, length, diameter, and flow) remain unchanged under the generalized condition of metabolic requirements of blood and blood vessel wall.

vessel wall; vascular tree; scaling laws

THE CIRCULATORY SYSTEM consists of complex vascular trees that distribute and collect blood in the various organs to maintain their functions. A typical vascular tree consists of millions of vessel segments, in series and parallel, with different diameters and lengths. Although there is a great deal of heterogeneity in morphological (diameters and lengths) and hemodynamic (pressure, flow, etc.) parameters, there is a prevailing hypothesis that the design of the vascular trees obeys some simple physiological and physical principle that optimizes the operation of the system.

The best-recognized minimum dissipation principle was proposed by Murray (8), in which the metabolic consumption in a single vessel segment is proportional to the blood volume. Murray's law, which states that the cube of the radius of a parent vessel equals the sum of the cube of the radii of the daughters, has been considered in various bifurcations of different organs (5, 7, 10, 11, 13–15). On the basis of angiographic data of vascular tress, there has been important progress to establish more global relationships between morphological parameters in an entire tree and its subtrees (6). Zhou et al. (16) (ZKM model) extended Murray's cost form to stem-crown units (SCU), and they deduced a set of scaling laws that describe the optimal structure-function relationships in vascular trees. Briefly, a vessel segment was defined as a stem, and the tree distal to the stem was defined as a crown. Although the form of these scaling laws has been validated in numerous trees (3), the theory only considers the metabolic consumption of the blood. There was no consideration of the metabolic requirements of the vessel wall (passive and active).

At first glance, the volume of blood to the volume of vessel wall scales as {pi}R2L/2{pi}RHL or R/2H, where R and H are the radius of the vessel and wall thickness, respectively. For large vessels, this ratio is large, implying that the blood volume term dominates the wall volume. For smaller vessels, however, this term decreases and the volume of the vessel wall becomes significant. More importantly, vasoactive properties of the vessel wall require substantially more metabolism compared with passive properties and may be significantly larger than the blood contribution.

Taber (12) considered a cost form that includes the vessel wall basal metabolism and vasomotor tone. It was shown that the optimal geometry of a vessel segment leads to a flow shear that increases with the blood pressure, in contrast to being constant as implied by Murray's law (5). It was also shown that the metabolic cost in a vessel segment is no longer proportional to the blood volume as required in Murray's formulation; instead, it appears to depend on the local wall thickness and pressure.

Although Taber's extension of Murray's hypothesis is useful for a single vessel segment, it must be extended to an entire tree as an integrated system. Here, we extended Taber's formulation to a vascular tree and the SCUs in the framework of the ZKM model. Our results show that the relation between metabolic dissipation and volume of the tree and SCUs remains proportional as additional metabolic sources are considered. This implies that the previously validated scaling relations hold under the conditions of dissipations of blood and vessel wall. This increases the biological realism of the validated ZKM model.


   Glossary
TOP
 ABSTRACT
 Glossary
METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 

f
portion that the active contraction stress to the total wall stress

H
thickness of vessel wall

H
wall thickness-to-vessel radius ratio, H = H/R

kmeta
coefficient of the total metabolic energy in a stem- crown unit, P = kmetaV

kb
blood metabolic coefficient, Pb = kbV

kwp
coefficient of the basal metabolic energy Pwp in vessel wall, Pw = kwpV

kwa
coefficient of the active metabolic energy Pwa in vessel wall, Pwp = kwaV

L
length of a vessel segment

n
subscript, denotes quantities of the nth vessel seg ments. For example, Hn is the wall thickness of the nth vessel segment.

nv
number of vessel segments of the same order in pig LCCA

N
the number of vessel segments in a stem-crown unit

P
total metabolic consumption in a stem-crown unit

p
blood pressure

Pb
blood metabolic energy

Pw
metabolic energy in vessel wall, Pw = Pwp + Pwa

Pwp
basal metabolic energy in vessel wall

Pwa
active metabolic energy in vessel wall due to vaso motor tone

R
radius of a vessel

V
blood volume

{alpha}
passive metabolic parameter in vessel wall

beta
active metabolic parameter in vessel wall

{sigma}a
active contraction stress in the vessel wall


   METHODS
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 ABSTRACT
 Glossary
 METHODS
RESULTS
 DISCUSSION
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Blood metabolism. Murray's law considers only one source of vascular metabolism; i.e., the power required to maintain blood volume denoted as Pb. This term is assumed to be proportional to blood volume (V) of a vessel segment (6), as

Formula 1(1)
where kb is a blood metabolic coefficient. Zhou et al. (16) extended this proportionality relationship for the blood metabolism in the SCUs of a vascular tree. The extension is based on the assumption that the metabolic rate of unit volume of blood is independent of its spatial position in a vascular tree. Therefore, V in Eq. 1 becomes the cumulative crown volume of a SCU.

Metabolism in vessel wall. There are two major sources of energy cost to maintain the vessel wall (12). For a vessel segment with radius R, length L, and wall thickness H, the basal metabolic energy is assumed proportional to the volume of vessel wall; i.e., Pwp = {alpha}·2{pi}RLH, where {alpha} is a passive metabolic parameter. There are also smooth muscle cells that generate active contraction stress {sigma}a to balance the blood pressure p (9, 12). According to Laplace's law and assuming {sigma}a contributes f (0 < f ≤ 1) of the total wall stress, Taber (12) formulated the cost due to vasomotor tone as Pwa = fbetap·2{pi}R2L, where beta is an active metabolic parameter. Thus the metabolic consumption in a vessel wall segment is given by:

Formula 2(2)
in which Formula 2 = H/R denotes the thickness-to-radius ratio.

For the ZKM, the total metabolism of a SCU with N vessel segments is thus the sum over all the segments, as

Formula 3(3)
where subscript n denotes quantities of the nth segments. Note that pn should be considered as the mean pressure of the nth segment. Metabolic parameters {alpha}n, betan, and fn may vary among all the vessel segments in a SCU.

Metabolism in SCU. The total metabolic cost in a SCU is thus the sum of Pb and Pw. Whereas Pb (Eq. 1) is proportional to the crown volume V, the form of Pw (Eq. 3) is more complex and does not necessarily obey proportionality. For the passive basal metabolism, the wall thickness has been found to depend linearly on the radius along a vascular tree (1). For instance, H = 8.22 x 10–3 R + 3.2 (in µm) was reported by Guo and Kassab (2) for pig left anterior descending. Metabolic parameter {alpha} depends on the composition of the wall and may change along the tree. In the absence of experimental data, it is assumed that {alpha}n in Eq. 3 is relatively uniform over the three orders of magnitude variation in diameter over the vascular tree, i.e., {alpha}n {approx} {alpha}. Thus the basal metabolic cost of a SCU is rewritten as:

Formula 4(4)
where

Formula 4
Note that Taber (12) approximated that Formula 4 is constant, and thus kwp = 2Formula 4{alpha} is also a constant, showing that Pwp is approximately proportional to the arterial volume V. In Eq. 4, kwp may vary along a vascular tree, because the relation between thickness and radius is no longer proportional.

Next, we further consider Pwa due to active wall stress. Similarly to Eq. 4, it is rewritten as:

Formula 5(5)
where kwa depends on the exact profile of blood pressure and parameters betan and fn in a SCU. Therefore, Eq. 5 does not indicate that Pwa is proportional to the arterial volume V. In the following experimentally based calculations, however, we find that kwa is nearly constant for all SCUs in a vascular tree under certain assumptions. Thus the total metabolic consumption in a SCU is formulated as:

Formula 6(6)
where kmeta is approximately a constant only if the wall metabolic coefficient kw = kwa + kwp does not change significantly over various SCUs in a vascular tree.


   RESULTS
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 ABSTRACT
 Glossary
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 RESULTS
DISCUSSION
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Blood metabolic coefficient kb. The blood metabolism depends on the metabolic demand of the blood cells with a constant proportionality coefficient kb. Although limited information is available on its exact value, Taber (12), estimated kb as 778 (in dyn·cm–2·s–1) for rat blood.

Basal metabolic coefficient kwp. To estimate kwp, we need the value of parameter {alpha}. Taber (12) estimated {alpha} as 2.88 x 103 (in dyn·cm–2·s–1) for rat portal vein, 1.21 x 103 for bovine mesenteric vein, and 7.64 x 103 for porcine carotid artery. In general, {alpha} can be estimated in the range of 103~104. Here, we calculated kwp for the pig left common coronary artery (LCCA). The morphometric data and blood pressure are taken from our previous studies (2, 4). As in Table 1, the pig LCCA consists of 11 orders of vessel segments. For each order, the average diameter R, length L, wall thickness H, and the number of vessels nv are given, as well as the previously computed longitudinal blood pressure p. SCUs of 10 orders are considered, as in Table 2. Order 10 is the entire LCCA, order 9 is a second largest SCU, and so on, until order 1, which is a smallest SCU with only two capillary branches. We assume {alpha} = 5.0 x 103 for the entire LCCA. It is found that kwp increases from 887 (dyn·cm–2·s–1) for the entire LCCA to 4,500 for the smallest SCUs (Table 2). When compared with kb for blood, kwp is on the same or even higher order of magnitude, which means that the basal metabolism in vessel wall cannot be ignored when the structure-function relationships are considered.


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  		Table 1. Average radius, length, wall thickness, and number of vessel segments in each order of vessels in pig's left common coronary artery, and blood pressure distribution

 

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  		Table 2. Wall metabolic coefficients kwa, kwp, and kw of stem-crown units in pig left common coronary artery

 
Vasomotor metabolic tone coefficient kwa. In vasomotor tone, parameter f describes the fraction of the wall stress that is attributed to active contraction of smooth muscle cells. This parameter changes with the composition of vessel wall, becoming larger in smaller vessels. The parameter beta also depends on the type and composition of vessel wall. It is estimated as 0.158 (in s–1) for rat portal vein, 0.00872 for porcine carotid artery, and 0.0177 for bovine mesenteric vein (12). Therefore, the range of beta is very large.

We calculated kwa for the pig LCCA. Since the values of f and beta are lacking, we estimate f = 1.0 and beta = 0.01 (12) for all vessel segments. The results are given in Table 2. It is found that kwa remains nearly uniform for large SCUs (orders 6~10) at about 2,500 and then gradually decreases to about 1,000 for the smallest SCUs of order 1. The degree of variation is relatively small given the three orders of magnitude change of diameter. Furthermore, the degree of active contraction for smaller vessels is significantly higher than that of larger arteries (by as much as factor 3), which implies a significantly higher f for the smaller vessels. This could serve to further reduce the variation of kwa over the entire range of SCUs.

Total metabolic coefficient. As in Eq. 6, kmeta is the sum of blood metabolic coefficient kb and wall metabolic coefficient kw, which is the sum of active and passive coefficients kwa and kwp. Given that kb is constant in a vascular tree, kmeta is nearly constant over the entire range of SCUs only when kw is approximately constant. Whereas kwa and kwp both show variation from large SCUs to small ones (Table 2), it is interesting to determine the variation of their sum kw. In the absence of experimental data on the metabolic parameter {alpha}, we calculated kw with {alpha} = 1,000, 2,000,..., 6,000 (dyn ·cm–2·s–1), respectively. For a given {alpha}, the mean, standard deviation (SD), and coefficient of variance [CV = (SD/mean) x 100] of kw are computed over the SCUs. The variations of kwa and kwp, quantified with CV, are 30.7% and 64.2%, respectively. Note that CV for kwp does not change with {alpha}. It is found that the variations of kwa and kwp counterbalance each other when summed together. For example, CV of kw is 17.9% with {alpha} = 5,000, which is significantly lower than the CV of kwa and kwp individually. With {alpha} = 2,000 and 3,000, CV of kw further reduces to as low as 2.3% and 6.7%, respectively, meaning kwp is highly uniform for all the SCUs. If 1,000 ≤ {alpha} ≤ 5,000 for the pig LCCA, we expect that the total metabolic consumption (kmeta in Eq. 6) is approximately constant for all SCUs in a vascular tree.


   DISCUSSION
TOP
 ABSTRACT
 Glossary
 METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
The major finding of this study is that the total metabolic consumption (blood; passive and active vessel wall) is proportional to the arterial volume of the entire coronary arterial tree. This relation becomes one of the axioms of the cost function of the ZKM model that reveals a set of scaling laws of optimal design of vascular trees (3, 16, 17). Whereas the previous derivation only took into account blood metabolism (Pb is proportional to the arterial volume) of a SCU (Eq. 1), the resulting scaling laws have been tested with extensive experimental measurements of various vascular trees in different species, with excellent agreements (3). This may indicate that the proportional formula Eq. 6 is indeed a good estimate of the total metabolic consumption in the vascular tree and hence the agreement with experimental data.

It may be noted that the metabolic coefficient kmeta must be large to confer a good fit to experimental data (unpublished observations). The blood metabolic coefficient kb, with typical value around 1,000 dyn·cm–2·s–1, does not meet the requirement. The present finding that the total metabolic consumption due to multiple sources can be estimated as proportional to the arterial volume is encouraging. In fact, when basal and vasomotor tone consumptions are taken into account, the value of kmeta is significantly increased.

The present study is based on several assumptions that may be verified when more experimental data become available. Whereas we use exact wall thickness data of pig LCCA (2), instead of previous approximation of uniform Formula 6 (12), we assume uniform {alpha} along the entire vascular tree, which does not reflect the variation of composition of vessel wall in the tree. For the vasomotor tone, we also approximate that f and beta are constant. We expect a proportional relationship by summing up the passive and active metabolic consumption in the vessel wall assuming that active stress in smaller vessels contributes more to the total wall stress than in larger vessels. The latter assumption has a physiological basis as smaller vessels have a stronger tone. However, experimental measurements on the metabolic parameters {alpha}, f, and beta along a vascular tree are needed to more definitively establish the theory.

In summary, this study considers the total metabolic consumption in the SCUs of vascular trees, including blood metabolism and basal and vasomotor tone consumptions in the vessel wall. On the basis of previous experimental measurements, we estimated that the total metabolic consumption in any SCU is nearly proportional to the arterial volume, which provides further physical realism to the validated scaling laws of vascular trees.


   FOOTNOTES
 

Address for reprint requests and other correspondence: G. S. Kassab, Dept. of Biomedical Engineering, SL-174, Indiana Univ. Purdue Univ. Indianapolis, 723 West Michigan St., Indianapolis, IN 46202 (e-mail: gkassab{at}iupui.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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 ABSTRACT
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 DISCUSSION
 REFERENCES
 

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Am J Physiol Heart Circ Physiol, December 1, 2007; 293(6): H3290 - H3300.
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