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MBW:Optimum Design of Blood Vessel Bifurcation

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Executive Summary

In this article, we consider the optimal angle at which bifurcation in blood vessels occur. By minimizing a cost function, we are able to determine the optimal radius of a blood vessel, given its length and the flow of blood through it. We then use this optimal radius to determine our minimum cost as a function of radius and length. The lengths of the blood vessels, however, depend on where the bifurcation occurs. By finding the location of the optimal bifurcation which keeps the cost function minimized, we are able to write expressions for the bifurcation angles as a function of blood vessel radius. Furthermore, using conservation of mass, we are able to eliminate the need for knowing the radius of all three vessels. With this, we can determine the angle of bifurcation of a blood vessel without knowing its radius; instead, we just need to know the radii of the other two blood vessels.

Once the expressions are derived, we can then use them to study special cases. The exemplary special case examined here occurs when the radii of the two daughter blood vessels are equal. In this case, the two blood vessels have the same bifurcation angle of 37.5^{{\circ }}.

Biological Phenomenon and Purpose

Vessels.jpg

Throughout the human body, arteries from the heart bifurcate many times in order to become capillaries. The question proposed in this article is whether or not there exists some optimal angle at which a bifurcation occurs. We shall consider a bifurcation like that shown at right.

The bifurcation occurs at the point B. To the left of B, some flow Q_{0} of blood is entering the vessel AB through A. To the right of B, Q_{0} is divided between the two vessels BC and BD, each now with flow Q_{1} and Q_{2} respectively.

The goal now is to find a cost function which can be minimized in order to determine the bifurcation pattern of the blood vessels. Then, by varying the location of B, we will be able to determine the angle of bifurcation as a function of the radius of the blood vessels.

Determination of Bifurcation Angle

Poiseuille's Law

Before trying to determine the bifurcation angle, we must first understand the flow of blood through a given vessel. This is given by Poiseuille's Law, which states that

Q={\frac  {\pi }{8}}{\frac  {(p_{1}-p_{2})}{L\mu }}a^{4}.

In this equation, \mu is the viscosity of blood, a is the radius and L is the length of the vessel and p_{1} and p_{2} represent the pressures at one end of the vessel, respectively. By analyzing Poiseuille's Law, we see that even a small decrease in vessel radius requires a large increase in pressure if flow is to be kept constant. It is possible that this could lead to hypertension.

(Fluid flow through a blood vessel is also described in MBW:Womersley Arterial Flow)

Cost Function

The cost function proposed by Murray (1926) and Rosen (1967) is the sum of the rate at which work is done on the blood and the rate at which energy is used up by the blood vessels by metabolism [1]. The rate at which work is done on the blood is equal to Q\delta p, where Q is flow and \Delta pis the change in pressure in the vessel. The rate at which energy is used by metabolism is equal to K\pi a^{2}L where K is a proportionality constant. The cost function for blood vessels is then

Costfunction=Q\Delta p+K\pi a^{2}L.

Using Poiseuille's Law, we can rewrite \Delta p and obtain

Costfunction={\frac  {8\mu L}{\pi a^{4}}}Q^{2}+K\pi a^{2}L.

To find the optimal radius for a given length and flow, the cost function is minimized with respect to a

{\frac  {\partial }{\partial a}}(costfunction)=-{\frac  {32\mu L}{\pi }}Q^{2}a^{{-5}}+2K\pi La=0

and is solved for the optimal a:

a=({\frac  {16\mu }{\pi ^{2}K}})^{{-{\frac  {1}{6}}}}Q^{{{\frac  {1}{3}}}}

At the optimal radius, the minimum cost function is equal to

Minimumcostfunction={\frac  {3\pi }{2}}KLa^{2}.

Angle Determination

System.jpg

By moving the bifurcation point B, we now aim to find the angles at which the cost function is kept minimized. Our system is shown at right. In order for the cost function of the entire system to be optimal, the cost function of the individual blood vessels must also be optimal. With this requirement, we see that the total cost function (now denoted as P) is given by

P={\frac  {3\pi K}{2}}(a_{0}^{2}L_{0}+a_{1}^{2}L_{1}+a_{2}^{2}L_{2}).

In this system, the lengths of the blood vessels are dependent on the location of the bifurcation point. Suppose there is a small change \delta in the location of the bifurcation. The change in the cost function is then

\delta P={\frac  {3\pi K}{2}}(a_{0}^{2}\delta L_{0}+a_{1}^{2}\delta L_{1}+a_{2}^{2}\delta L_{2}).

However, we want the cost function to still be minimized after this small change, i.e. we wish for \delta P to be equal to zero. This occurs at the optimal location of B. To determine the optimal location, we consider three different shifts in B.

Shift Along AB

First.jpg

Suppose that the bifurcation point is moved to location B' in the direction of the AB blood vessel, as shown at right. Then, the changes in the lengths of the vessels are

\delta L_{0}=\delta

\delta L_{1}=-\delta \cos \theta

\delta L_{2}=-\delta \cos \phi

and the change in the cost function is

\delta P={\frac  {3\pi K}{2}}\delta (a_{0}^{2}-a_{1}^{2}\cos \theta -a_{2}^{2}\cos \phi ).

In order for the cost function to remain unchanged, the following relationship must be true:

(1)a_{0}^{2}=a_{1}^{2}\cos \theta +a_{2}^{2}\cos \phi

Shift Along CB

Second.jpg

Now suppose that the bifurcation point is moved along the direction of CB, as shown at right. In this case, the changes in the lengths of the blood vessels are

\delta L_{0}=-\delta \cos \theta

\delta L_{1}=\delta

\delta L_{2}=\delta \cos(\theta +\phi )

The change in the cost function is

\delta P={\frac  {3\pi K}{2}}\delta (-a_{0}^{2}\cos \theta +a_{1}^{2}+a_{2}^{2}\cos(\theta +\phi )).

In order for the cost function to remain unchanged, the following relationship must be true:

(2)-a_{0}^{2}\cos \theta +a_{1}^{2}+a_{2}^{2}\cos(\theta +\phi )=0

Shift Along DB

Third.jpg

Finally, suppose the bifurcation point is moved along the direction of CD, as shown at right. In this case, the changes in the lengths of the blood vessels are

\delta L_{0}=-\delta \cos \phi

\delta L_{1}=\delta \cos(\theta +\phi )

\delta L_{2}=\delta

The change in the cost function is

\delta P={\frac  {3\pi K}{2}}\delta (-a_{0}^{2}\cos \phi +a_{1}^{2}\cos(\theta +\phi )+a_{2}^{2}).

In order for the cost function to remain unchanged, the following relationship must be true:

(3)-a_{0}^{2}\cos \phi +a_{1}^{2}\cos(\theta +\phi )+a_{2}^{2}=0

Expressions for Angles

We now have three equations and three unknowns. The optimal conditions (1), (2), and (3) that resulted from each of the shifts can be solved for \cos \theta , \cos \phi , and \cos(\theta +\phi ). The result is

\cos \theta ={\frac  {a_{0}^{4}+a_{1}^{4}-a_{2}^{4}}{2a_{0}^{2}a_{1}^{2}}}

\cos \phi ={\frac  {a_{0}^{4}-a_{1}^{4}+a_{2}^{4}}{2a_{0}^{2}a_{2}^{2}}}

\cos(\theta +\phi )={\frac  {a_{0}^{4}-a_{1}^{4}-a_{2}^{4}}{2a_{1}^{2}a_{2}^{2}}}

If we know the radius of each of the three vessels, we can now determine the angle of the bifurcation.

Murray's Law

By the principle of the conservation of mass, the flow of blood in the vessel AB must be equal to the sum of the flow of blood through vessels BC and BD; namely,

Q_{0}=Q_{1}+Q_{2}

If we solve our equation for optimal radius for flow, obtaining

Q=a^{3}({\frac  {16\mu }{\pi ^{2}K}})^{{-{\frac  {1}{2}}}}

and substitute this into our conservation equation, we obtain Murray's law.

a_{0}^{3}=a_{1}^{3}+a_{2}^{3}

Bifurcation Angle Equations

Using Murray's Law, we can reduce our expressions for the angles of bifurcation so that only two of the three radii need to be known. The final result is the following

\cos \theta ={\frac  {a_{0}^{4}+a_{1}^{4}-(a_{0}^{3}-a_{1}^{3})^{{{\frac  {4}{3}}}}}{2a_{0}^{2}a_{1}^{2}}}

\cos \phi ={\frac  {a_{0}^{4}-(a_{0}^{3}-a_{2}^{2})^{{{\frac  {4}{3}}}}+a_{2}^{4}}{2a_{0}^{2}a_{2}^{2}}}

\cos(\theta +\phi )={\frac  {(a_{1}^{2}+a_{2}^{2})^{{{\frac  {4}{3}}}}-a_{1}^{4}-a_{2}^{4}}{2a_{1}^{2}a_{2}^{2}}}

Special Case : a_{1}=a_{2}

In this section, we consider the case in which the daughter blood vessels have the same radius. In this case,

a_{1}=a_{2}

and our expressions for \theta and \phi reduce to

\cos \theta ={\frac  {a_{0}^{2}}{2a_{1}^{2}}}

\cos \phi ={\frac  {a_{0}^{2}}{2a_{2}^{2}}}={\frac  {a_{0}^{2}}{2a_{1}^{2}}}

which are clearly equal. This means that, when the radii of the daughter blood vessels are equal, their bifurcation angles are also equal.

Furthermore, we can determine exactly what the bifurcation angle is in this case. When the a_{1}=a_{2}, Murray's law becomes

a_{0}^{3}=2a_{1}^{3}

which can be rearranged as

{\frac  {a_{0}}{a_{1}}}=2^{{{\frac  {1}{3}}}}

Substituting this into reduced expression for \cos \theta results in

\cos \theta ={\frac  {1}{2}}(2^{{{\frac  {1}{3}}}})^{2}

When we solve for \theta , we find that the bifurcation angle is equal to 37.5 degrees.

\theta =37.5^{{\circ }}

Project Categorization

(a) Mathematics Used:

This project uses minimization and pertubation analysis to determine the optimum angles for blood vessel bifurcation.

(b) Type of Model:

Blood flow through a single vessel is modeled by Poiseuille's Law. This combined by a cost function proposed by Murray and Rosen yields a minimum-cost function which is dependent on the length and radius of the vessel.

(c) Biological System Studied

This project studies what is the optimum angle for bifurcation of a blood vessel. Doing so helps us to understand

Further Reading

Blood Vessel Branching: Beyond the Standard Calculus Problem [1]

Blood flow dynamics in microvessel bifurcations [2]

Related Work

The work by Murray is also cited in the 2011 paper "Pulse Wave Propagation in the Arterial Tree" by Vosse and Stergiopulos [3]. In this paper the effect of a pulse through a circulatory system was studied. This is important because our blood does not flow at a constant rate but it pumped from the heart in pulses. Thus by being able to study how a pulse will propagate we can form a more accurate model of our circulatory system. The model uses Murray's law to help calculate the resistance to the blood flow in an arterial network.

External Links

1. Fung, Y.C., Biomechanics: Circulation, Section 3.3. (1997).

2. Mazumadar, J., An Introduction to Mathematical Physiology and Biology, Chapter 6. (1989).

3. Murray, Cecil D. (1926). "The Physiological Principle of Minimum Work: I. The Vascular System and the Cost of Blood Volume". Proceedings of the National Academy of Sciences of the United States of America 12 (3): 207–214.

4. Murray, Cecil D. (1926). "The Physiological Principle of Minimum Work: II. Oxygen Exchange in Capillaries". Proceedings of the National Academy of Sciences of the United States of America 12 (5): 299–304.