Name: using integration to model blood flow
Uploaded: 2026-01-12T12:24:36.000Z
Duration: 1 min 27 s
Channel: JoVE

Blood flow in a blood vessel can be described using the law of laminar flow, where velocity varies with radial distance from the center.The total blood flow can be approximated by dividing the cross-section into thin concentric rings with varying radii.Each ring&#8217;s area is given by its circumference multiplied by its small thickness.The flow through each ring can be approximated as the product of its area and the velocity at that radius.Summing these flows for all rings gives an approximate total flow, called the Riemann sum.As the number of rings approaches infinity, the Riemann sum approaches the exact values, represented by the definite integral from the center to the vessel wall.The velocity function in the integral comes from the law of laminar flow. It states that the velocity at a distance r depends on the pressure difference, the fluid&#8217;s viscosity, and the vessel&#8217;s length.Substituting the velocity expression into the integral and integrating over the vessel radius yields the exact blood flow.This result, called Poiseuille&#8217;s Law, shows that flow scales with the fourth power of the vessel&#8217;s radius. This is why even a small narrowing of blood vessels sharply reduces blood flow.

toepassingen van integratie om de bloedstroom te bepalen

Applications of Integration to Find Blood Flow

De bloedstroom door een cilindrisch bloedvat kan wiskundig worden beschreven aan de hand van de principes van laminaire stroming, een stromingsregime waarbij de vloeistof zich soepel in evenwijdige lagen voortbeweegt. In dit model is de snelheid van het bloed niet uniform over de dwarsdoorsnede van het vat; deze varieert met de radiale afstand tot het middelpunt. De maximale snelheid treedt op langs de centrale as en neemt geleidelijk af in de richting van de vaatwanden, waar deze als gevolg van viskeuze wrijving nul wordt.
Benadering van de bloedstroom
Om de volumestroom te schatten, wordt de cirkelvormige dwarsdoorsnede van het bloedvat conceptueel opgedeeld in een reeks dunne concentrische ringen. Elke ring levert een bijdrage aan de totale volumestroom, die wordt bepaald door zijn oppervlak &#x2014; berekend als het product van de omtrek en de dikte &#x2014; en door de vloeistofsnelheid op de betreffende straal. Door de bijdragen van alle ringen te sommeren, verkrijgt men een benadering van de totale stroming door het vat.
Afleiding van de stroming met behulp van integratie
Naarmate het aantal ringen toeneemt en hun dikte afneemt, nadert de som van de afzonderlijke bijdragen een bepaalde integraal over de straal van het bloedvat. De wet van laminaire stroming beschrijft het snelheidsprofiel als functie van de radiale afstand, waarbij het drukverschil langs het bloedvat, de viscositeit van de vloeistof en de lengte van het vat worden betrokken. Door deze snelheidsfunctie te integreren van het middelpunt tot de buitenwand van het bloedvat wordt de exacte waarde van de totale bloedstroom verkregen.
Dit resultaat wordt geformaliseerd in de wet van Poiseuille, die stelt dat de volumestroom recht evenredig is met het drukverschil en met de vierde macht van de straal van het bloedvat, en omgekeerd evenredig is met zowel de viscositeit als de lengte van het bloedvat. Deze uitgesproken afhankelijkheid van de straal benadrukt de cruciale invloed die zelfs kleine veranderingen in de vaatdiameter kunnen hebben op de effici&#xEB;ntie van de bloedsomloop.

Toepassingen van integratie om de bloedstroom te bepalen

De bloedstroom door een cilindrisch bloedvat kan wiskundig worden beschreven aan de hand van de principes van laminaire stroming, een stromingsregime ...

applications-of-integration-to-find-blood-flow

Education

JoVE Core

Calculus

Unit

Eenheid

Application of Techniques of Integration

Toepassing van integratietechnieken

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

using integration to model blood flow

Blood flow in a cylindrical blood vessel can be modeled with integration and laminar flow. Laminar flow is smooth motion in parallel layers. In this model, blood speed changes with distance from the center of the vessel.

The blood moves fastest along the central axis. As the distance from the center increases, the velocity decreases. At the vessel wall, the velocity drops to zero because of viscous drag. This changing speed across the cross-section is the key idea behind the calculation.

To estimate the total volumetric flow, or flux, the circular cross-section is divided into thin concentric rings. Each ring has an area found from its circumference and thickness. The flow from each ring depends on that area and the blood speed at that radius. Adding the contributions from all of the rings gives an approximation of the total flow through the vessel.

When the rings become thinner and more numerous, the sum approaches a definite integral over the vessel radius. The velocity profile comes from the law of laminar flow, which includes the pressure gradient, the viscosity of the fluid, and the length of the vessel. Integrating from the center to the outer wall gives the exact total blood flow.

This result is expressed by Poiseuille’s Law. The law shows that volumetric flow rate increases with pressure difference and with the fourth power of the vessel radius. It also decreases as viscosity and vessel length increase. This strong radius dependence shows why even small changes in vessel diameter can greatly affect circulatory efficiency.

Using Integration to Model Blood Flow

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applications of integration to find blood flow

Blood flow in a cylindrical blood vessel can be modeled with integration and laminar flow. Laminar flow is smooth motion in parallel layers. In this model, blood speed changes with distance from the center of the vessel. The blood moves fastest along the central axis. As the distance from the center increases, the velocity decreases. At the vessel wall, the velocity drops to zero because of viscous drag. This changing speed across the cross-section is the key idea behind the calculation. To...

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key terms and concepts

Explore how integration and laminar flow are used to calculate blood flow in a cylindrical vessel, from concentric rings to Poiseuille’s Law.

Using Integration to Model Blood Flow | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Using Integration to Model Blood Flow

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