Extension of Murray's law using a non-Newtonian model of blood flow
© Revellin et al; licensee BioMed Central Ltd. 2009
Received: 09 April 2009
Accepted: 15 May 2009
Published: 15 May 2009
So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.
In the present paper, Murray's law which is applicable to an arterial bifurcation, is generalized to a non-Newtonian blood flow model (power-law model). When the vessel size reaches the capillary limitation, blood can be modeled using a non-Newtonian constitutive equation. It is assumed two different constraints in addition to the pumping power: the volume constraint or the surface constraint (related to the internal surface of the vessel). For a seek of generality, the relationships are given for an arbitrary number of daughter vessels. It is shown that for a cost function including the volume constraint, classical Murray's law remains valid (i.e. ΣR c = cste with c = 3 is verified and is independent of n, the dimensionless index in the viscosity equation; R being the radius of the vessel). On the contrary, for a cost function including the surface constraint, different values of c may be calculated depending on the value of n.
We find that c varies for blood from 2.42 to 3 depending on the constraint and the fluid properties. For the Newtonian model, the surface constraint leads to c = 2.5. The cost function (based on the surface constraint) can be related to entropy generation, by dividing it by the temperature.
It is demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid.
Since several decades, many studies have been carried out on the optimal branching pattern of a vascular system. Based on the simple assumption of a steady Poiseuille blood flow, the well known Murray's law  has been established. It links the radius of a parent vessel R0 (immediately upstream from a vessel bifurcation) to the radii of the daughter vessels R1 and R2 (immediately downstream after a vessel bifurcation) as R0/R1 = R0/R2 = 2-1/3. From Murray's analysis, the required condition of minimum power occurs when Q ∝ R3 where Q denotes the volumetric flow. This relation, called "cube law", is determined assuming that two energy terms contribute to the cost of maintaining blood flow in any section of any vessel: (i) the pumping power and (ii) the energy metabolically required to maintain the volume of blood which is referred to as "volume constraint". A generalization of this relation can be proposed as Q ∝ R c where c is determined from the condition of minimum power by assuming other constraints (for instance surface constraint yields Q ∝ R2.5 ). Under the condition c = 3, the shear stress on the vessel walls is uniform and independent of vessel diameter . Several studies have been carried out to determine the value of c [4–8] which usually ranges between 2 and 3. The influence of the value of c from 2 to 4 has also been investigated . The in vivo wall shear stress in an arterial system has been measured . It was found that mean wall shear stress was far from constant along the arterial tree, which implied that Murray's cube law on flow diameter relations could not be applied to the whole arterial system. According to the authors, c likely varies along the arterial system, probably from 2 in large arteries near the heart to 3 in arterioles. A method allowing for estimation of wall shear rate in arteries using the flow waveforms has been developed . This work allowed to determine the time-dependent wall shear rates occurring in fully developed pulsatile flow using Womersley's theory. They found a non-uniform distribution of wall shear rates throughout the arterial system.
Following the cubic law, Murray  proposed the optimal branching angle. Optimally, the larger branch makes a smaller branching angle than the smaller branch. This work was extended to non-symmetrical bifurcations . The arterial bifurcations in the cardiovascular system of a rat have been investigated . The results were found to be consistent with those previously reported in humans and monkeys. Murray's optimization problem has also been reproduced computationally using a three dimensional vessel geometry and a time-dependent solution of the Navier-Stokes equations .
From Murray's law, some relationships have been proposed between the vessel radius and the volumetric flow, the average linear velocity flow, the velocity profile, the vessel-wall shear stress, the Reynolds number and the pressure gradient . In the same way, based on the Poiseuille assumptions, scaling relationships have been described between vascular length and volume of coronary arterial tree, diameter and length of coronary vessel branches and lumen diameter and blood flow rate in each vessel branch [16, 17].
It is also possible to determine Murray's law using other approaches. A model have been suggested based on a "delivering" artery system of an organ characterized, (i) by the space-filling fractal embedding into the tissue and (ii) by the uniform distribution of the blood pressure drop over the artery system . The minimalist principles were not used but the result remains the same. Murray's energy cost minimization have been extended to the pulsatile arterial system, by analysing a model of pulsatile flow in an elastic tube . It is found that for medium and small arteries with pulsatile flow, Murray's energy minimization leads to Murray's Law.
Surprisingly, so far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although, the non-Newtonian approach for blood flow modeling looks more accurate. Blood is a multi-component mixture with complex rheological characteristics. Experimental investigations showed that blood exhibits non-Newtonian properties such as shear-thinning, viscoelasticity, thixotropy and yield stress [20–22]. Blood rheology has been shown to be related to its microscopic structures (e.g. aggregation, deformation and alignment of red blood cells). The non-Newtonian steady flow in a carotid bifurcation model have been investigated [23, 24]. The authors showed that in that case, viscoelastic properties may be ignored. The fact that blood exhibits a viscosity that decreases with increasing rate of deformation (shear-thining or so called pseudoplastic behavior) is thus the predominant non-Newtonian effect. There are several inelastic models in the literature to account for the non-Newtonian behavior of blood [25, 26]. The most popular models are the power-law [27, 28], the Casson  and the Carreau  fluids. The power-law model is the most frequently used as it provides analytical results for many flow situations. On the usual log-log coordinates, this model results in a linear relation between the viscosity and the shear rate. Blood viscosity have been measured by using a falling-ball viscometer and a cone-plate viscometer for shear rate from 0.1 to 400 s-1 . For both techniques, the authors found that measured values are aligned on a straight line suggesting that the power-law model fits experimental data with sufficient precision.
From the literature review, it can be established that none of the existing studies deal with the minimalist principle along with non-Newtonian models. The combination of both aspects will be studied and are presented hereafter. For a seek of generality, the relationships will be given for an arbitrary number of daughter vessels.
Non-Newtonian model of blood flow
where we have used standard cylindrical coordinates such that the z-axis is aligned with the pipe centerline. It means that the only nonzero velocity component is the axial component which is a function of the distance to the pipe centerline only.
It features two parameters: dimensionless flow index n and consistency m with units Pa.sn. On log-log coordinates, this model results in a linear relation between viscosity and shear rate. The fluid is shear-thinning like blood (i.e. viscosity decreases as shear rate increases) if n<1 and shear-thickening (i.e. viscosity increases as shear rate increases) if n > 1. When n = 1 the Newtonian fluid is recovered and in that case parameter m represents the constant viscosity of the fluid. This model is very popular in engineering work because a wide variety of flow problems have been solved analytically for it.
Extension of Murray's Law
where B k ' = Bk·π·l k and A k ' = A k ·Ψ k .
which is found to be always positive because b>0, and so are Ak' and Bk'. As a result, the extremum is a minimum.
This expression may be useful when one wants to include mass and/or heat transfer through the vessel wall. Actually, when the vessel diameter decreases, blood catch up with a non-Newtonian fluid and the heat and mass transfer through the vessel wall becomes more and more significant.
Equation (18) is the generalization of the cube law. In case of laminar Newtonian flow (a = 1 and b = 4, thus c = 3), the classical cube law is recovered. Note that whatever the value of n in the Poiseuille case, c is equal to 3.
The area ratio (expansion parameter) β which is the ratio of the combined cross-sectional area of the daughters over that of the parent vessel. Values of β greater than unity produce expansion in the total cross-sectional area available to flow as it progresses from one of the tree to the next. It can be written as:
Until now, all equations are formulated with a parent vessel that divid into a finite number of daughter vessels (1,..., j). However, since a parent vessel divide into two daughter vessels in animals and humans, further equations will be formulated with j = 2.
Meaning of the results
In this section, we will examine the meaning of the results obtained in the general case. Particularly, we will focus on the variation of each parameter with the radius of the vessel.
Velocity of flow
The maximum velocity, denoted by vmax, is attained at the center of the vessel. It can thus be obtained by setting r to 0 in Eq. (5).
As a result, the velocity profile is only function of the fluid properties and remains the same whatever the radius.
Vessel wall shear stress
In the particular case of c = 3, the vessel wall shear stress remains unchanged all along the vascular system.
If c<3, when blood flows from the parent to the daughter vessels, the vessel wall shear stress increases because the vessel radii decrease in the flow direction. On the contrary, if c>3, the vessel wall shear stress decreases because the vessel radii increase in the flow direction.
Since c is often greater than two, the Reynolds number will always increases in the direction of the blood path.
Conductance and resistance
Cross sectional area
For the Newtonian case, c = 2.5 and = 1.52. For the non-Newtonian case, n = 0.74 for instance and = 1.50. Whatever the fluid, as >1, the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, this result means that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid. This behaviour can be related to the velocity profile, which is blunter for a non-Newtonian fluid, as shown by Eq. (5).
Influence of n on the c parameter for the two different constraints.
Nominal value of n
Influence of n on different parameters
n = 1
n = 0.81
n = 0.74
Velocity of flow
Vessel wall shear stress
Cross sectional area
Blood is a multi-component mixture with complex rheological characteristics. Experimental investigations have shown that blood exhibits non-Newtonian properties such as shear-thinning, viscoelasticity, thixotropy and yield stress. Blood rheology is shown to be related to its microscopic structures (e.g. aggregation, deformation and alignment of blood cells and plattelets). Shear-thinning is the predominant non-Newtonian effect in bifurcations of blood flows.
In this study, we have proposed for the first time an analytical expression of Murray's law using a non-Newtonian blood flow model (power law model), assuming two different constraints in addition to the pumping power: (i) the volume constraint and (ii) the surface constraint. Surface constraint may be useful if one wants to include heat and/or mass transfer in the cost function, specially in capillaries. For a seek of generality, the relationships have been given for an arbitrary number of daughter vessels. Note that there is an alternative formulation of the constrained optimization problem using the Lagrange multipliers, as discussed in . However, using this approach, the results presented in this paper would not have been modified.
It has been showed that for a cost function including the volume constraint, classical Murray's law remains valid (i.e. ΣR c = cste with c = 3 is verified). In other words, the value of c is independent of the fluid properties. On the contrary, for a cost function including the surface constraint, different values of c may be calculated depending on the fluid properties, i.e. the value of n. The fluid is shear-thinning if n<1 and shear-thickening if n>1. When n = 1 the Newtonian fluid is recovered. In the present study, we have used two different blood values of n found in the literature, namely n = 0.81 and n = 0.74. In summary, it has been found that c varies from 2.42 to 3 depending on the constraint and the index n. For the particular Newtonian model, the surface constraint leads to c = 2.5.
Entropy generation has several origins: heat transfer, mass transfer, pressure drop, etc. The cost function (based on the surface constraint) can be related to entropy generation by dividing it by the temperature. It has been demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-newtonian fluid than for a Newtonian fluid. This behaviour can be related to the velocity profile, which is blunter for a non-Newtonian fluid, as shown by Eq. (5).
Based on the literature review and on our work, we propose in the following, further possible investigations:
The effect of singularities on the cost function has hardly ever been investigated. Few works exist on this aspect  and Tondeur et al. (Tondeur D, Fan Y, Luo L: Constructal optimization of arborescent structures with flow singularities. Chem. Eng. Sci. 2009, submitted.). However, the effect of the singularities on the entropy generation might not be negligible. This contribution should be added in the cost function in the future.
In reality, heat and mass might be transfered through the vessel wall, leading to resistance that should be included in the cost function. Indeed, gases, nutrients and metabolic waste products are exchanged between blood and the underlying tissue. Substances pass through the vessels by active or passive transfer, i.e. diffusion, filtration or osmosis. Moreover, pathologic states such as edema and inflammation might increase such phenomena.
It is accepted that pulsatile blood flow is more realistic than steady-state flow. The cost function should also include the effect of pulsatile flow in an elastic tube.
Blood is essentially a two-phase fluid consisting of formed cellular elements suspended in a liquid medium, the plasma. The corpuscular nature of blood raises the question of whether it can be treated as a continuum, and the peculiar makeup of plasma makes it seem different from more common fluids. In particular, when the vessel radius decreases down to the smallest capillaries, the continuum approach diverges from the reality. Treating blood as a non-continuum fluid should be a possible next step.
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