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

Le flux sanguin dans un vaisseau sanguin peut être décrit à l’aide de la loi de l’écoulement laminaire, où la vitesse varie avec la distance radiale au centre.Le flux sanguin total peut être approximé en divisant la section transversale en fines anneaux concentriques de rayons variables.La surface de chaque anneau est donnée par sa circonférence multipliée par sa petite épaisseur.Le flux à travers chaque anneau peut être approximé comme le produit de sa surface et de la vitesse à ce rayon.La somme de ces flux pour tous les anneaux donne un flux total approximatif, appelé somme de Riemann.Lorsque le nombre d’anneaux tend vers l’infini, la somme de Riemann tend vers les valeurs exactes, représentées par l’intégrale définie du centre jusqu’à la paroi du vaisseau.La fonction de vitesse dans l’intégrale provient de la loi de l’écoulement laminaire. Elle indique que la vitesse à une distance r dépend de la différence de pression, de la viscosité du fluide et de la longueur du récipient.En substituant l’expression de vitesse dans l’intégrale et en intégrant sur le rayon du vaisseau, on obtient exactement le flux sanguin.Ce résultat, appelé loi de Poiseuille, montre que l’écoulement évolue avec la quatrième puissance du rayon du vaisseau. C’est pourquoi même un léger rétrécissement des vaisseaux sanguins réduit fortement le flux sanguin.

applications de l’intégration pour déterminer le débit sanguin

Applications of Integration to Find Blood Flow

Le d&#xE9;bit sanguin dans un vaisseau cylindrique peut &#xEA;tre d&#xE9;crit math&#xE9;matiquement &#xE0; l&#x2019;aide des principes de l&#x2019;&#xE9;coulement laminaire, un r&#xE9;gime dans lequel le fluide s&#x2019;&#xE9;coule en couches parall&#xE8;les de fa&#xE7;on r&#xE9;guli&#xE8;re. Dans ce mod&#xE8;le, la vitesse du sang n&#x2019;est pas uniforme sur toute la section transversale du vaisseau ; elle varie plut&#xF4;t avec la distance radiale au centre. La vitesse maximale est atteinte le long de l&#x2019;axe central et diminue progressivement vers les parois du vaisseau, o&#xF9; elle devient nulle en raison des frottements visqueux.
Approximation du d&#xE9;bit sanguin
Pour estimer le d&#xE9;bit volumique total, ou flux volumique, la section circulaire du vaisseau est divis&#xE9;e conceptuellement en une s&#xE9;rie de fins anneaux concentriques. Chaque anneau contribue au d&#xE9;bit total, en fonction de sa surface &#x2014; calcul&#xE9;e comme le produit de sa circonf&#xE9;rence et de son &#xE9;paisseur &#x2014; et de la vitesse du fluide &#xE0; ce rayon. La somme des contributions de tous les anneaux donne une approximation du d&#xE9;bit total &#xE0; travers le vaisseau.
Calcul du d&#xE9;bit par int&#xE9;gration
Lorsque le nombre d&#x2019;anneaux augmente et que leur &#xE9;paisseur diminue, la somme des contributions au d&#xE9;bit tend vers une int&#xE9;grale d&#xE9;finie portant sur le rayon du vaisseau. La loi de l&#x2019;&#xE9;coulement laminaire fournit le profil de vitesse en fonction de la distance radiale, en tenant compte du gradient de pression le long du vaisseau, de la viscosit&#xE9; du fluide et de la longueur du vaisseau. En int&#xE9;grant cette fonction de vitesse du centre &#xE0; la paroi externe, on obtient la valeur exacte du d&#xE9;bit sanguin total.
Ce r&#xE9;sultat est exprim&#xE9; par la loi de Poiseuille, qui stipule que le d&#xE9;bit volumique est directement proportionnel &#xE0; la diff&#xE9;rence de pression et &#xE0; la quatri&#xE8;me puissance du rayon du vaisseau, et qu&#x2019;il est inversement proportionnel &#xE0; la viscosit&#xE9; et &#xE0; la longueur du vaisseau. Cette forte d&#xE9;pendance au rayon souligne l&#x2019;influence critique de variations m&#xEA;me faibles du diam&#xE8;tre du vaisseau sur l&#x2019;efficacit&#xE9; circulatoire.

Applications de l’intégration pour déterminer le débit sanguin

Le d&#xE9;bit sanguin dans un vaisseau cylindrique peut &#xEA;tre d&#xE9;crit math&#xE9;matiquement &#xE0; l&#x2019;aide des principes de ...

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

Education

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JoVE Core

Calculus

Calcul

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Application of Techniques of Integration

Application des techniques d'intégration

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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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