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

El flujo sanguíneo en un vaso sanguíneo puede describirse mediante la ley del flujo laminar, donde la velocidad varía con la distancia radial al centro.El flujo sanguíneo total puede aproximarse dividiendo la sección transversal en anillos concéntricos delgados con radios variables.El área de cada anillo se determina por su circunferencia multiplicada por su pequeño grosor.El flujo a través de cada anillo puede aproximarse como el producto de su área y la velocidad en ese radio.Sumar estos flujos para todos los anillos da un flujo total aproximado, llamado suma de Riemann.A medida que el número de anillos tiende al infinito, la suma de Riemann se acerca a los valores exactos, representados por la integral definida desde el centro hasta la pared del recipiente.La función de velocidad en la integral proviene de la ley del flujo laminar. Establece que la velocidad a una distancia r depende de la diferencia de presión, la viscosidad del fluido y la longitud del recipiente.Sustituir la expresión de velocidad en la integral e integrar sobre el radio del vaso produce el flujo sanguíneo exacto.Este resultado, llamado Ley de Poiseuille, muestra que el flujo escala con la cuarta potencia del radio del recipiente. Por eso, incluso un pequeño estrechamiento de los vasos sanguíneos reduce bruscamente el flujo sanguíneo.

aplicaciones de la integración para calcular el flujo sanguíneo

Applications of Integration to Find Blood Flow

El flujo sangu&#xED;neo a trav&#xE9;s de un vaso sangu&#xED;neo cil&#xED;ndrico puede describirse matem&#xE1;ticamente mediante los principios del flujo laminar, un r&#xE9;gimen en el que el fluido se mueve suavemente en capas paralelas. En este modelo, la velocidad de la sangre no es uniforme a lo largo de la secci&#xF3;n transversal del vaso, sino que var&#xED;a con la distancia radial desde el centro. La velocidad m&#xE1;xima se alcanza a lo largo del eje central, disminuyendo progresivamente hacia las paredes del vaso, donde se reduce a cero debido a la fricci&#xF3;n viscosa.
Aproximaci&#xF3;n del flujo sangu&#xED;neo
Para estimar el caudal volum&#xE9;trico total, o caudal, la secci&#xF3;n transversal circular del vaso se divide conceptualmente en una serie de delgados anillos conc&#xE9;ntricos. Cada anillo aporta una porci&#xF3;n del flujo total, determinada por su &#xE1;rea (calculada como el producto de su circunferencia y su grosor) y la velocidad del fluido en ese radio espec&#xED;fico. La suma de las contribuciones de todos los anillos da una aproximaci&#xF3;n del flujo total a trav&#xE9;s del vaso.
C&#xE1;lculo del flujo mediante integraci&#xF3;n
A medida que aumenta el n&#xFA;mero de anillos y disminuye su grosor, la suma de las contribuciones al flujo se aproxima a una integral definida sobre el radio del vaso. La ley del flujo laminar proporciona el perfil de velocidad en funci&#xF3;n de la distancia radial, considerando el gradiente de presi&#xF3;n a lo largo del vaso, la viscosidad del fluido y la longitud del vaso. La integraci&#xF3;n de esta funci&#xF3;n de velocidad desde el centro hasta la pared exterior da el valor exacto del flujo sangu&#xED;neo total.
Este resultado se formaliza en la ley de Poiseuille, que establece que el caudal volum&#xE9;trico es directamente proporcional a la diferencia de presi&#xF3;n y a la cuarta potencia del radio del vaso, e inversamente proporcional a la viscosidad y a la longitud del vaso. Esta pronunciada dependencia del radio subraya la influencia crucial de cambios, incluso peque&#xF1;os, en el di&#xE1;metro del vaso en la eficiencia circulatoria.

Aplicaciones de la integración para calcular el flujo sanguíneo

El flujo sangu&#xED;neo a trav&#xE9;s de un vaso sangu&#xED;neo cil&#xED;ndrico puede describirse matem&#xE1;ticamente mediante los principios del flujo ...

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

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Application of Techniques of Integration

Aplicación de Técnicas de Integración

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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