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

Przepływ krwi w naczyniu krwionośnym można opisać za pomocą prawa przepływu laminarnego, gdzie prędkość zmienia się wraz z promieniową odległością od środka.Całkowity przepływ krwi można przybliżyć, dzieląc przekrój na cienkie koncentryczne pierścienie o zmiennych promieniach.Powierzchnia każdego pierścienia jest dana przez jego obwód pomnożony przez niewielką grubość.Przepływ przez każdy pierścień można przybliżyć jako iloczyn jego powierzchni i prędkości w tym promieniu.Sumowanie tych przepływów dla wszystkich pierścieni daje przybliżony całkowity przepływ, zwany sumą Riemanna.Gdy liczba pierścieni zbliża się do nieskończoności, suma Riemanna zbliża się do dokładnych wartości, reprezentowanych przez całkę określoną od środka do ściany naczynia.Funkcja prędkości w całce pochodzi z prawa przepływu laminarnego. Stwierdza ona, że prędkość w odległości r zależy od różnicy ciśnień, lepkości płynu oraz długości naczynia.Podstawienie wyrażenia prędkości do całki i całkowanie przez promień naczynia daje dokładny przepływ krwi.Ten wynik, zwany prawem Poiseuille'a, pokazuje, że przepływ skaluje się z czwartą potęgą promienia naczynia. Dlatego nawet niewielkie zwężenie naczyń krwionośnych gwałtownie zmniejsza przepływ krwi.

zastosowania całkowania w wyznaczaniu przepływu krwi

Applications of Integration to Find Blood Flow

Przep&#x142;yw krwi przez cylindryczne naczynie krwiono&#x15B;ne mo&#x17C;na opisa&#x107; matematycznie, wykorzystuj&#x105;c zasady przep&#x142;ywu laminarnego, czyli przep&#x142;ywu, w kt&#xF3;rym ciecz porusza si&#x119; p&#x142;ynnie w r&#xF3;wnoleg&#x142;ych warstwach. W tym modelu pr&#x119;dko&#x15B;&#x107; krwi nie jest jednolita w ca&#x142;ym przekroju naczynia, lecz zmienia si&#x119; wraz z odleg&#x142;o&#x15B;ci&#x105; promieniow&#x105; od &#x15B;rodka. Maksymalna pr&#x119;dko&#x15B;&#x107; wyst&#x119;puje wzd&#x142;u&#x17C; osi centralnej, stopniowo malej&#x105;c w kierunku &#x15B;cian naczynia, gdzie osi&#x105;ga zero z powodu oporu lepko&#x15B;ci.
Przybli&#x17C;anie przep&#x142;ywu krwi
Aby oszacowa&#x107; ca&#x142;kowite nat&#x119;&#x17C;enie przep&#x142;ywu obj&#x119;to&#x15B;ciowego, ko&#x142;owy przekr&#xF3;j naczynia jest koncepcyjnie podzielony na szereg cienkich koncentrycznych pier&#x15B;cieni. Ka&#x17C;dy pier&#x15B;cie&#x144; odpowiada za cz&#x119;&#x15B;&#x107; ca&#x142;kowitego przep&#x142;ywu, okre&#x15B;lon&#x105; przez pole &#x2013; liczone jako iloczyn obwodu i grubo&#x15B;ci &#x2013; oraz pr&#x119;dko&#x15B;&#x107; cieczy w danym promieniu. Zsumowanie sk&#x142;adowych wszystkich pier&#x15B;cieni daje przybli&#x17C;enie ca&#x142;kowitego przep&#x142;ywu przez naczynie.
Wyznaczanie przep&#x142;ywu za pomoc&#x105; ca&#x142;kowania
Wraz ze wzrostem liczby pier&#x15B;cieni i zmniejszeniem ich grubo&#x15B;ci, suma sk&#x142;adowych przep&#x142;ywu zbli&#x17C;a si&#x119; do ca&#x142;ki oznaczonej wzgl&#x119;dem promienia naczynia. Prawo przep&#x142;ywu laminarnego przedstawia profil pr&#x119;dko&#x15B;ci jako funkcj&#x119; odleg&#x142;o&#x15B;ci promieniowej, uwzgl&#x119;dniaj&#x105;c gradient ci&#x15B;nienia wzd&#x142;u&#x17C; naczynia, lepko&#x15B;&#x107; cieczy i d&#x142;ugo&#x15B;&#x107; naczynia. Ca&#x142;kowanie tej funkcji pr&#x119;dko&#x15B;ci od &#x15B;rodka do zewn&#x119;trznej &#x15B;ciany daje dok&#x142;adn&#x105; warto&#x15B;&#x107; ca&#x142;kowitego przep&#x142;ywu krwi.
Wynik ten jest sformalizowany w prawie Poiseuille&#x2019;a, kt&#xF3;re stwierdza, &#x17C;e obj&#x119;to&#x15B;ciowe nat&#x119;&#x17C;enie przep&#x142;ywu jest wprost proporcjonalne do r&#xF3;&#x17C;nicy ci&#x15B;nienia i czwartej pot&#x119;gi promienia naczynia, a odwrotnie proporcjonalne do lepko&#x15B;ci i d&#x142;ugo&#x15B;ci naczynia. Ta wyra&#x17A;na zale&#x17C;no&#x15B;&#x107; od promienia podkre&#x15B;la krytyczny wp&#x142;yw nawet niewielkich zmian &#x15B;rednicy naczynia na wydajno&#x15B;&#x107; kr&#x105;&#x17C;enia.

Zastosowania całkowania w wyznaczaniu przepływu krwi

Przep&#x142;yw krwi przez cylindryczne naczynie krwiono&#x15B;ne mo&#x17C;na opisa&#x107; matematycznie, wykorzystuj&#x105;c zasady przep&#x142;ywu ...

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

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Application of Techniques of Integration

Zastosowanie technik całkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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-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...

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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,...

theorem of pappus

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...

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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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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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