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

يمكن وصف تدفق الدم في الوعاء الدموي باستخدام قانون التدفق الطبقي، حيث تتغير السرعة مع المسافة الشعاعية من المركز.يمكن تقريب تدفق الدم الكلي بتقسيم المقطع العرضي إلى حلقات رقيقة متحدة المركز بأنفساق متغيرة.تعطى مساحة كل حلقة بمطارها مضروبا في سمكها الصغير.يمكن تقريب التدفق عبر كل حلقة كحاصل ضرب مساحتها والسرعة عند ذلك نصف القطر.جمع هذه التدفقات لجميع الحلقات يعطي تدفقا تقريبيا كليا، يسمى مجموع ريمان.مع اقتراب عدد الحلقات من اللانهاية، يقترب مجموع ريمان من القيم الدقيقة، الممثلة بالتكامل المحدد من المركز إلى جدار الوعاء.دالة السرعة في التكامل تأتي من قانون التدفق الطبقي. تنص على أن السرعة عند مسافة r تعتمد على فرق الضغط، ولزوجة السائل، وطول الوعاء.استبدال تعبير السرعة في التكامل والتكامل عبر نصف قطر الوعاء ينتج تدفق الدم بالضبط.تظهر هذه النتيجة، المسماة قانون بوازوي، أن التدفق يتدرج مع القوة الرابعة لنصف قطر السفينة. لهذا السبب حتى تضيق بسيط في الأوعية الدموية يقلل بشكل حاد من تدفق الدم.

تطبيقات التكامل لإيجاد تدفق الدم

Applications of Integration to Find Blood Flow

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

Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Application of Techniques of Integration

تطبيقات تقنيات التكامل

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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

arc length of a curve

Measuring Curved Paths with Calculus

Measuring curved paths with calculus is important in engineering projects. Roller coaster tracks, cables, and railway lines often follow smooth curves instead of straight lines. Accurate path length helps with planning and design. A curve is first split into small segments. Each short segment is treated as a straight line between two nearby points on the curve. Adding these line segments gives an estimate of the total distance along the path. The estimate becomes better when the segments are...

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Catenary Cable Length in Power Lines

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight. A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the...

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The arc length function measures the distance traveled along a smooth curve from a fixed starting point to a moving endpoint. It is useful when a straight-line estimate is not accurate enough. For curves that are continuous and differentiable, arc length gives a precise way to track length along the curve. To find this distance, the curve is broken into many small pieces. Each piece is treated like a short straight line. Its length depends on the horizontal change and the vertical change over...

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Surfaces of revolution are created when a two-dimensional curve is rotated around an axis. The rotation produces a three-dimensional shape. In engineering, this calculation can help determine the surface area of a rocket nozzle. That information matters when a heat-resistant coating must be applied evenly. When a curve is revolved about the x-axis, it sweeps out a continuous surface. To find its area, the curve is first broken into small straight-line pieces over short intervals. Each rotated...

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applications-of-integration-to-find-hydrostatic-pressure

Hydrostatic Force on Dam Walls with Integration

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth. A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall. Integration helps measure this changing force. Pressure in a still fluid rises...

applications of integration to find centers of mass

applications-of-integration-to-find-centers-of-mass

Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

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Pappus Theorem for Volume and Surface Area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

applications of integration to find consumer surplus

applications-of-integration-to-find-consumer-surplus

Consumer Surplus from Demand and Integration

Consumer surplus measures the economic gain buyers receive when they pay less than the highest price they are willing to pay. In microeconomics, this idea comes from the demand function, which links quantity to the price consumers will pay. Because willingness to pay usually falls as quantity rises, the demand curve slopes downward. When a product is sold at a fixed price, some buyers would have paid more for the first units they bought. The difference between that willingness to pay and the...

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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Normal distribution helps explain how standardized test scores can be spread across a group. It is a key idea in statistics for describing symmetry, center, and variability in large sets of data. Test score data often start as a histogram. In a histogram, each bar shows how many scores fall within a score range. To turn that count-based graph into a continuous probability model, the histogram is normalized by dividing each bar height by the total number of observations and by the bin width. <br...

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applications-of-integration-to-probability-density-functions

Finding Probability with Area Under a PDF

Continuous probability distributions describe random variables that can take any real value within a set range. Height is one example, since it can be measured with growing precision, such as 163.5 or 165.25 centimeters. That makes height a continuous random variable. The pattern of a continuous random variable is shown with a probability density function, or PDF. A PDF shows how probability is spread across the possible values. In a continuous distribution, probabilities are assigned to...

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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