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Sara and Bev are studying for a biology exam, but wishing they could get back to their favorite homework - calculus.
Sara and Bev are studying for a biology exam, but wishing they could get back to their favorite homework - calculus. This chapter is on the system of blood vessels in the body, which is made up of arteries, arterioles, capillaries, and veins. Sara wonders out loud if there is a reason for the branching patterns she sees in the textbook diagrams. Candice happens to walk by and overhear the conversation. She suggests that the reason might be that the transport of blood from the heart through all organs of the body and back to the heart should be as efficient as possible. She suggests to Sara and Bev that one way this can be done is by having large enough blood vessels to avoid turbulence and small enough blood cells to minimize viscosity. Then Sara suggests they use calculus to derive the angleθθfor branches in blood vessels such that total resistance to the flow of blood is minimized. She draws the picture below and says they could assume that a main vessel of radius 7 mm runs along a horizontal line. We want to carry blood to a point 500 mm from the start of the main vessel and 140 mm away from the vessel. A side artery, of radius 5 mm, can head for this point branching at any point along the vessel. Bev points out that in order to solve the problem, they would also have to know how the resistance of blood flow is related to the size of the vessel. Fortunately, Candice remembers that they can use Poiseuille’s law for that. Specifically, the resistanceRRin the system is proportional to the lengthLLof the vessel andLLinversely proportional to the fourth power of the radiusrr. That is,R=k⋅Lr4R=k⋅Lr4, wherek=0.16k=0.16is a constant determined by the viscosity of the blood. Labeling the distance that blood would travel through the larger vesselxxand the distance through the smaller arterydd, Sara notes that the total resistance along the route is the sum of the resistance on each segment. What angleθθminimizes this
