The blood vascular system consists of blood vessels...

The blood vascular system consists of blood vessels (arteries. arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille's Laws gives the resistance $R$ of the blood as $$R=C \frac{L}{r^{4}}$$ where $L$ is the length of the blood vessel. $r$ is the radius, and $C$ is a positive constant determined by the viscosity of the blood. 1Poiseuille established this law experimentally. but it also follows from Equation $6.7 .2 .$ ) The figure shows a main blood vessel with radius $r$, branching at an angle $\theta$ into a smaller vessel ith radius $r_{2}$ (IMAGE CANNOT COPY) (a) Use Poiscuille's Law to show that the total resistance of the hived along the path $A B C$ is $$R=C\left(\frac{a \quad b \cot \theta}{r_{1}^{2}}+\frac{h \csc \theta}{r_{2}^{4}}\right)$$ where $u$ and $h$ are the distances shown in the figure. (b) Prowe that this resistance is minimited when $$\cos \theta=\frac{r_{2}^{4}}{r_{1}^{4}}$$ (c) Find the optimal hranching angle (correct to the nearest degrees when the radius of the smaller blood vessel is twothirds the radius of the larger vessel. (IMAGE CANNOT COPY)

The blood vascular system consists of blood vessels (arteries. arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille's Laws gives the resistance $R$ of the blood as
$$R=C \frac{L}{r^{4}}$$
where $L$ is the length of the blood vessel. $r$ is the radius, and $C$ is a positive constant determined by the viscosity of the blood. 1Poiseuille established this law experimentally. but it also follows from Equation $6.7 .2 .$ ) The figure shows a main blood vessel with radius $r$, branching at an angle $\theta$ into a smaller vessel ith radius $r_{2}$
(IMAGE CANNOT COPY)
(a) Use Poiscuille's Law to show that the total resistance of the hived along the path $A B C$ is
$$R=C\left(\frac{a \quad b \cot \theta}{r_{1}^{2}}+\frac{h \csc \theta}{r_{2}^{4}}\right)$$
where $u$ and $h$ are the distances shown in the figure.
(b) Prowe that this resistance is minimited when
$$\cos \theta=\frac{r_{2}^{4}}{r_{1}^{4}}$$
(c) Find the optimal hranching angle (correct to the nearest degrees when the radius of the smaller blood vessel is twothirds the radius of the larger vessel.
(IMAGE CANNOT COPY)

Key Concepts

Optimization Principles in Biological Systems

Biological systems, including the vascular system, are often optimized for energy efficiency. Applying optimization techniques helps in determining the ideal geometrical configurations, such as optimal branching angles and vessel dimensions, that minimize flow resistance and reduce the energy required for circulation.

Trigonometric Methods in Geometric Analysis

Trigonometry is used to relate the geometric parameters of branches and angles in vascular systems. Functions like sine, cosine, and cotangent are crucial for accurately expressing the relationships between different segments of a vessel, and for integrating these relationships into formulas for resistance, ultimately aiding in the optimization process.

Resistance in Fluid Dynamics

Resistance in a fluid system represents how the vessel geometry and fluid properties impede flow. Since resistance is highly sensitive to changes in radius due to the fourth power dependency in Poiseuille's Law, even minor adjustments in the vessel’s diameter can lead to substantial changes in the overall resistance, affecting the efficiency of fluid transport.

Poiseuille's Law

A fundamental principle in fluid dynamics, Poiseuille's Law quantifies the resistance to laminar flow in a cylindrical tube. It shows that the resistance is directly proportional to the length of the vessel and inversely proportional to the fourth power of its radius, with a constant that depends on the fluid's viscosity. This law is essential for understanding how changes in vessel dimensions affect the energy required to pump fluids.

Vascular Geometry in Fluid Flow

The configuration and dimensions of blood vessels, including diameter, length, and branching angles, have a significant impact on blood flow and resistance. Studying vascular geometry helps in optimizing the properties of blood vessels to minimize energy expenditure, which is especially important in the design of efficient circulatory systems.

Transcript

00:01 And this problem, we're told that we have a blood vessel here that is branched.

00:13 So, sure, it's to minimize the energy expended by the heart and pumping with blood.

00:17 In particular, this energy is reduced when the resistance of the blood is lowered.

00:21 So we have persoil's relation is that the resistance of a flow to a circular tube is the resistance is c a constant times l over r to the fourth, where r is the radius of the tube or the pipe.

00:41 And l is the distance that you're measuring the resistance between those.

00:47 And then c depends on the fluid.

00:55 So we have this relationship.

00:57 And so we have this case where we have a branching blood vessel.

01:05 So we have a major, one blood vessel here and then another one comes off of it.

01:09 And obviously that's how, you know, you branch out and get oxygen, you know, oxygen to the vascular system spreads out to your body.

01:23 So they want us to figure out what is the resistance between here, between a and c.

01:30 So as this comes out this way, there's one thing in here that isn't, i think simplified this a little bit is that there should be flow rate should be somewhere in here.

01:47 As a mechanical engineer, you know, the faster you push something through here, the more resistance you'll get.

01:58 So i think what they're basically assuming here is that you have the same flow rates everywhere here.

02:05 But that because of conservation of mass that's not going to be true because you're the flow rate here is going to have to sum up to the flow rates here because any material blood coming in here gets split and it goes out here and here so anyway there's more to this problem but i think they've simplified it just to to make it a simpler so a couple a couple things we need to do we need to find these ls so we can split this up in a b and bc so l1 is b c and if we know this is this distance here is b then l2 here um i'll guess this is l2 i want l2 is b times the co -sequent of theta or theta is this branching angle so that means r2 the resistance along this section of tubing or of blood vessel is c times b cosecant of theta all over r2 to the 4.

03:08 And then this distance here, ab, is just a minus this distance, and this distance is b cotangent of theta.

03:18 So this distance is a minus b cotangent of theta.

03:21 So the resistance in the, this part of the blood vessel is c times quantity a minus b coatangent of theta all over r1 to the fourth.

03:32 So the total resistance is simply the sum of those two.

03:37 And that is, that is, you know, some function of theta.

03:42 So we got in terms of theta.

03:44 Now, they want us to figure out the theta that would minimize that, given these r as being constant...