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

O fluxo sanguíneo em um vaso sanguíneo pode ser descrito usando a lei do fluxo laminar, onde a velocidade varia com a distância radial ao centro.O fluxo sanguíneo total pode ser aproximado dividindo a seção transversal em anéis concêntricos finos com raios variados.A área de cada anel é dada por sua circunferência multiplicada por sua pequena espessura.O fluxo através de cada anel pode ser aproximado como o produto de sua área e a velocidade nesse raio.Somar esses fluxos para todos os anéis obtém um fluxo total aproximado, chamado soma de Riemann.À medida que o número de anéis se aproxima do infinito, a soma de Riemann se aproxima dos valores exatos, representados pela integral definida do centro até a parede do vaso.A função de velocidade na integral vem da lei do fluxo laminar. Ele afirma que a velocidade a uma distância r depende da diferença de pressão, da viscosidade do fluido e do comprimento do recipiente.Substituir a expressão de velocidade na integral e integrar sobre o raio do vaso produz o fluxo sanguíneo exato.Esse resultado, chamado Lei de Poiseuille, mostra que o fluxo escala com a quarta potência do raio do recipiente. Por isso, até mesmo um pequeno estreitamento dos vasos sanguíneos reduz drasticamente o fluxo sanguíneo.

aplicações da integração para calcular o fluxo sanguíneo

Applications of Integration to Find Blood Flow

O fluxo sangu&#xED;neo atrav&#xE9;s de um vaso sangu&#xED;neo cil&#xED;ndrico pode ser descrito matematicamente utilizando os princ&#xED;pios do fluxo laminar, um regime no qual o fluido se move suavemente em camadas paralelas. Nesse modelo, a velocidade do sangue n&#xE3;o &#xE9; uniforme em toda a se&#xE7;&#xE3;o transversal do vaso; em vez disso, varia com a dist&#xE2;ncia radial a partir do centro. A velocidade m&#xE1;xima ocorre ao longo do eixo central, diminuindo progressivamente em dire&#xE7;&#xE3;o &#xE0;s paredes do vaso, onde atinge zero devido ao arrasto viscoso.
Aproxima&#xE7;&#xE3;o do fluxo sangu&#xED;neo
Para estimar a vaz&#xE3;o total, a se&#xE7;&#xE3;o transversal circular do vaso &#xE9; conceitualmente dividida em uma s&#xE9;rie de an&#xE9;is conc&#xEA;ntricos finos. Cada anel contribui com uma por&#xE7;&#xE3;o do fluxo total, determinada por sua &#xE1;rea &#x2014; calculada como o produto de sua circunfer&#xEA;ncia e espessura &#x2014; e pela velocidade do fluido naquele raio espec&#xED;fico. A soma das contribui&#xE7;&#xF5;es de todos os an&#xE9;is fornece uma aproxima&#xE7;&#xE3;o do fluxo total atrav&#xE9;s do vaso.
Determina&#xE7;&#xE3;o do fluxo por integra&#xE7;&#xE3;o
&#xC0; medida que o n&#xFA;mero de an&#xE9;is aumenta e sua espessura diminui, a soma das contribui&#xE7;&#xF5;es do fluxo se aproxima de uma integral definida sobre o raio do vaso. A lei do fluxo laminar fornece o perfil de velocidade em fun&#xE7;&#xE3;o da dist&#xE2;ncia radial, incorporando o gradiente de press&#xE3;o ao longo do vaso, a viscosidade do fluido e o comprimento do vaso. Integrando essa fun&#xE7;&#xE3;o de velocidade do centro at&#xE9; a parede externa, obt&#xE9;m-se o valor exato do fluxo sangu&#xED;neo total.
Esse resultado &#xE9; formalizado na Lei de Poiseuille, que afirma que a vaz&#xE3;o volum&#xE9;trica &#xE9; diretamente proporcional &#xE0; diferen&#xE7;a de press&#xE3;o e &#xE0; quarta pot&#xEA;ncia do raio do vaso, e inversamente proporcional &#xE0; viscosidade e ao comprimento do vaso. Essa acentuada depend&#xEA;ncia do raio ressalta a influ&#xEA;ncia cr&#xED;tica de at&#xE9; mesmo pequenas altera&#xE7;&#xF5;es no di&#xE2;metro do vaso na efici&#xEA;ncia da circula&#xE7;&#xE3;o.

Aplicações da integração para calcular o fluxo sanguíneo

O fluxo sangu&#xED;neo atrav&#xE9;s de um vaso sangu&#xED;neo cil&#xED;ndrico pode ser descrito matematicamente utilizando os princ&#xED;pios do fluxo ...

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

Education

JoVE Core

Calculus

Cálculo

Unit

Unidade

Application of Techniques of Integration

Aplicação de Técnicas de Integração

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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