Application of Calculus: Blood Flow & Cardiac Output We all know that our blood is continually flowing through our body, or else we would be dead! However many lack the in-depth knowledge of the processes by which the body maintains its continuous blood flow through its veins and arteries. There are both numerous biological and mathematical applications, however this paper will focus primarily on the mathematical application in the movement of blood in the human body. The first mathematical application in this field is the use of calculus to determine the blood flow in an artery or a vein at a given point in time. First, the shape of the blood vessel (either a vein or artery) is assumed to be the shape of a cylindrical tube. Due to the friction
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As Δr becomes increasingly smaller, the velocity is assumed to be constant throughout the area and can be approximated as v (ri). Thus the formula for the flux per unit time can be derived as and the Riemann Sum, , (whose limit is equal to ∫2πriv(ri)dr) can be used to calculate the total volume flowing through the blood vessel. The limit of the Riemann Sum as n approaches ∞ gives the exact value of the flux, the volume of blood that passes through the blood vessel, per unit time. The process below: results in Poiseuille’s Law, which shows the direct proportionality between flux and radius of the blood vessel to the fourth power. Since the nature of this concept is defined by an equation, problems are often solved by simply plugging into the Poiseuille’s Law ( ). An example such as “Use Poisuille’s Law to calculate the rate of flow in a small human capillary when you are given ɳ=0.027, R=0.008 cm, l=2 cm, and P=4000 dynes/cm2” can be easily calculated to find the flux by substituting into formula:
F=(π(4000 dynes/(cm^2 )) (0.008)^4)/8(0.027)(2 cm)
F=1.19x 〖10〗^(-4) (cm^3)/s
The concept of blood flow can also be combined with other topics of calculus, such as the determination of average value. For
As Δr becomes increasingly smaller, the velocity is assumed to be constant throughout the area and can be approximated as v (ri). Thus the formula for the flux per unit time can be derived as and the Riemann Sum, , (whose limit is equal to ∫2πriv(ri)dr) can be used to calculate the total volume flowing through the blood vessel. The limit of the Riemann Sum as n approaches ∞ gives the exact value of the flux, the volume of blood that passes through the blood vessel, per unit time. The process below: results in Poiseuille’s Law, which shows the direct proportionality between flux and radius of the blood vessel to the fourth power. Since the nature of this concept is defined by an equation, problems are often solved by simply plugging into the Poiseuille’s Law ( ). An example such as “Use Poisuille’s Law to calculate the rate of flow in a small human capillary when you are given ɳ=0.027, R=0.008 cm, l=2 cm, and P=4000 dynes/cm2” can be easily calculated to find the flux by substituting into formula:
F=(π(4000 dynes/(cm^2 )) (0.008)^4)/8(0.027)(2 cm)
F=1.19x 〖10〗^(-4) (cm^3)/s
The concept of blood flow can also be combined with other topics of calculus, such as the determination of average value. For