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

Der Blutfluss in einem Blutgefäß kann mit dem Gesetz des laminaren Flusses beschrieben werden, bei dem die Geschwindigkeit mit dem radialen Abstand vom Zentrum variiert.Der gesamte Blutfluss lässt sich annähern, indem der Querschnitt in dünne konzentrische Ringe mit unterschiedlichen Radien unterteilt wird.Die Fläche jedes Rings wird durch seinen Umfang multipliziert mit seiner geringen Dicke gegeben.Der Durchfluss durch jeden Ring kann als Produkt seiner Fläche und der Geschwindigkeit an diesem Radius angenähert werden.Die Summe dieser Ströme für alle Ringe ergibt eine ungefähre Gesamtströmung, die Riemannsche Summe genannt wird.Wenn die Anzahl der Ringe sich dem Unendlichen nähert, nähert sich die Riemannsche Summe den exakten Werten, dargestellt durch das bestimmte Integral vom Zentrum bis zur Gefäßwand.Die Geschwindigkeitsfunktion im Integral ergibt sich aus dem Gesetz der laminaren Strömung. Es besagt, dass die Geschwindigkeit in einer Entfernung r vom Druckunterschied, der Viskosität der Flüssigkeit und der Länge des Behälters abhängt.Durch das Einsetzen des Geschwindigkeitsausdrucks in das Integral und die Integration über den Gefäßradius ergibt sich der exakte Blutfluss.Dieses Ergebnis, genannt Poiseuilles Gesetz, zeigt, dass der Fluss mit der vierten Potenz des Radius des Gefäßes skaliert. Deshalb verringert schon eine kleine Verengung der Blutgefäße den Blutfluss stark.

anwendungen der integration zur bestimmung des blutflusses

Applications of Integration to Find Blood Flow

Der Blutfluss durch ein zylindrisches Blutgef&#xE4;&#xDF; l&#xE4;sst sich mathematisch mithilfe der Prinzipien der laminaren Str&#xF6;mung beschreiben, einem Str&#xF6;mungsregime, bei dem die Fl&#xFC;ssigkeit gleichm&#xE4;&#xDF;ig in parallelen Schichten verl&#xE4;uft. In diesem Modell ist die Blutgeschwindigkeit nicht &#xFC;ber den gesamten Gef&#xE4;&#xDF;querschnitt konstant, sondern variiert mit dem radialen Abstand vom Zentrum. Die maximale Geschwindigkeit tritt entlang der Mittelachse auf und nimmt stetig zu den Gef&#xE4;&#xDF;w&#xE4;nden hin ab, wo sie aufgrund der viskosen Reibung null erreicht.
Approximation des Blutflusses
Zur Absch&#xE4;tzung des gesamten Volumenstroms wird der kreisf&#xF6;rmige Gef&#xE4;&#xDF;querschnitt gedanklich in eine Reihe d&#xFC;nner, konzentrischer Ringe unterteilt. Jeder Ring tr&#xE4;gt einen Anteil zum Gesamtfluss bei, der sich aus seiner Fl&#xE4;che &#x2013; berechnet als Produkt von Umfang und Dicke &#x2013; und der Str&#xF6;mungsgeschwindigkeit an diesem spezifischen Radius ergibt. Die Summe der Beitr&#xE4;ge aller Ringe liefert eine N&#xE4;herung des Gesamtflusses durch das Gef&#xE4;&#xDF;.
Herleitung des Blutflusses mittels Integration
Mit zunehmender Anzahl der Ringe und abnehmender Dicke n&#xE4;hert sich die Summe der einzelnen Flussbeitr&#xE4;ge einem bestimmten Integral &#xFC;ber den Gef&#xE4;&#xDF;radius. Das Gesetz der laminaren Str&#xF6;mung liefert das Geschwindigkeitsprofil als Funktion des radialen Abstands unter Ber&#xFC;cksichtigung des Druckgradienten entlang des Gef&#xE4;&#xDF;es, der Viskosit&#xE4;t des Fluids und der Gef&#xE4;&#xDF;l&#xE4;nge. Die Integration dieser Geschwindigkeitsfunktion vom Zentrum bis zur Gef&#xE4;&#xDF;wand ergibt den exakten Wert des Blutflusses.
Dieses Ergebnis wird durch das Gesetz von Poiseuille formalisiert, das besagt, dass der Volumenstrom direkt proportional zur Druckdifferenz und zur vierten Potenz des Gef&#xE4;&#xDF;radius sowie umgekehrt proportional zur Viskosit&#xE4;t und L&#xE4;nge des Gef&#xE4;&#xDF;es ist. Diese starke Abh&#xE4;ngigkeit vom Radius verdeutlicht den entscheidenden Einfluss selbst kleiner &#xC4;nderungen des Gef&#xE4;&#xDF;durchmessers auf die Effizienz des Kreislaufsystems.

Anwendungen der Integration zur Bestimmung des Blutflusses

Der Blutfluss durch ein zylindrisches Blutgef&#xE4;&#xDF; l&#xE4;sst sich mathematisch mithilfe der Prinzipien der laminaren Str&#xF6;mung beschreiben, ...

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

Education

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JoVE Core

Calculus

Analysis

Unit

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Application of Techniques of Integration

Anwendung von Integrationstechniken

KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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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