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Chapter 7
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Scaling Laws of Coronary Vasculature
7.1 Introduction
As demonstrated in Chap. 2, the coronary vasculature is complex. A natural question
arises as to whether the construction of the vasculature is random or if it follows
some design principles. This question has been pondered for nearly a century and
numerous investigations have attempted to explain the design of vascular trees based
on various principles, including cost functions of energy (Murray, 1926; Rosen,
1967), surface area and volume (Kamiya & Togawa, 1972), tensile stress (Kurz &
Sandau, 1997), and shear force (Zamir, 1977). The major impetus for these investigations was the desire to understand the structure/function of the coronary vasculature relation under homeostatic conditions. Pathological states of the vasculature
may then be understood in relation to perturbations in homeostasis.
Although the morphogenesis of the vascular tree is likely to be determined by a
preprogrammed genetic algorithm, a number of physical, chemical, and biological
factors related to the functional needs of a particular tissue may determine the
subsequent growth and remodeling during postnatal development. The final vascular
tree structures reflect the changes brought about by natural selection and adaptation
to the environment, which subsequently lead to the survival value of the structure.
The design determinants of the coronary vasculature are undoubtedly multi-factorial
and an analysis of their effects on structure and function is often non-trivial.
In this chapter, we provide a simplified yet comprehensive analysis of the design
of coronary vasculature. The structural data is used to understand the function of the
coronary vasculature (e.g., structure/function relationship), including the validated
models of coronary blood flow described in Chaps. 5 and 6. Specific attention is
placed o n the ZKM model, which expands Murray’s original law (1926) from an
individual vessel branch to a more generalized model that considers the entire
vasculature, by using the minimum energy hypothesis along with conservation of
energy. Experimental and computation validations of the ZKM model are provided.
A more generalized theory is also considered that invokes the fractal nature of the
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coronary vascular trees that leads to scaling laws of flow resistance and volume,
along with an experimental validation of scaling of myocardial mass.
7.2 Murray’s Law
The most well-recognized structure–function relation based on a design principle of
the vasculature was first formulated by Murray in 1926, based on a minimum
operation cost hypothesis (Murray, 1926). Energy expenditure is required for circulation because of frictional losses sustained during conduction of fluid through a
vascular network. The reduction in frictional losses by the use of larger caliber
vessels must be balanced with the cost of metabolic maintenance of the blood
volume (Murray, 1926) as well as passive and active aspects (e.g., media and smooth
muscle cells) of the blood vessel wall (Liu & Kassab, 2007b). Murray proposed a
cost function which consisted of a compromise between the frictional flow and
metabolic blood cost, leading to a principle of economy or minimum energy
expenditure. He derived an optimal condition for the relation between diameter
(structure) and flow rate (function) in a vessel segment. Specifically, Murray’s law
states that the flow rate is proportional to the cube of diameter, i.e., Q ¼ k
Q and D are the volumetric flow rate and diameter of a vessel segment, respectively,
and k
is a blood constant (see derivation in Appendix 1). Murray’s diameter–flow
b
relation in conjunction with conservation of mass at a bifurcation leads to the wellknown diameter relation: the cube of the diameter of a mother vessel equals the sum
of the cubes of the diameters of the daughters (Murray, 1926).
Murray’s law predicts a universal exponent which is invariant (3.0) for all trees
whose internal flows obey laminar conditions. Fifty years after the formulation of
Murray’s law, Uylings (1977) argued that the exponent can vary in the range of
2.33–3.0 depending on whether the flow is turbulent (2.33) or laminar (3.0).
Numerous papers have been published in the past 90 years on Murray’s law and
on the validation of the exponent (see review in Kassab (2006) and Sherman (1981)).
The studies show some support but with significant scatter. Sherman (1981)
reviewed the literature on Murray’s law and concluded that it is not obeyed in the
most proximal bifurcations of aorta, the pulmonary trunk, the vena cavae, or the
pulmonary veins. In the section below, we review studies by our group which show
that Murray’s exponent of 3 only holds for the small coronar y arteriolar vessels
(<50 μm; orders <5) but does not hold for the larger coronary vasculature. These
results prompted a more generalized formulation for the entire coronary vasculature,
as described in the following section.
D3where
b
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