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

血管中的血流可用层流定律来描述，其中流速随距中心的径向距离而变化。总血流量可通过将横截面分割为若干具有不同半径的薄同心环来估算。每个环的面积等于其周长乘以较小的厚度。每个环的流速可近似为其面积与该半径处流速的乘积。对所有环的流量求和可得到一个近似的总流量，称为黎曼和。当环的数量趋近于无穷大时，黎曼和趋近于从中心到血管壁的定积分所表示的精确值。积分中的速度函数来源于层流定律。该定律指出，距离为 r 处的速度取决于压力差、流体的黏度以及血管的长度。将速度表达式代入积分中，并对血管半径进行积分，可得到精确的血流量。这一结果被称为泊肃叶定律，表明流体流量与血管半径的四次方成正比。因此，即使血管发生轻微狭窄，也会显著减少血流量。

利用积分法计算血流的应用

Applications of Integration to Find Blood Flow

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

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Application of Techniques of Integration

积分技术的应用

KEY TERMS AND CONCEPTS

关键术语与概念

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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Consumer Surplus from Demand and Integration

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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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Finding Probability with Area Under a PDF

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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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