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

Bir kan damarındaki kan akışı, laminar akış yasasıyla tanımlanabilir; burada hız merkezden radyal mesafeye göre değişir.Toplam kan akışı, kesitin değişen yarıçaplı ince konsantrik halkalara bölünmesiyle yaklaşık olarak hesaplanabilir.Her halkanın alanı, çevresinin küçük kalınlığı ile çarpmasıyla verilir.Her halkadan geçen akış, alan ile o yarıçaptaki hızın çarpımı olarak yaklaşık olarak hesaplanabilir.Tüm halkalar için bu akışların toplanması, Riemann toplamı olarak adlandırılan yaklaşık bir toplam akış verir.Halkaların sayısı sonsuzluğa yaklaştıkça, Riemann toplamı tam değerlere yaklaşır; bu değerler merkezden kap duvarına kadar belirlenmiş integral ile temsil edilir.İnteğaldeki hız fonksiyonu, laminar akış yasasından gelir. Bir r mesafesindeki hızın basınç farkına, sıvının viskozitesine ve kabın uzunluğuna bağlı olduğunu belirtir.Hız ifadesini integrala koyup damar yarıçapı üzerinden entegre edildiğinde, tam kan akışı elde edilir.Bu sonuç, Poiseuille Yasası olarak adlandırılır ve akışın kapının yarıçapının dördüncü kuvvetiyle ölçeklendiğini gösterir. Bu yüzden küçük bir kan damarı daralması bile kan akışını keskin şekilde azaltır.

kan akışını bulmada İntegrasyon uygulamaları

Applications of Integration to Find Blood Flow

Silindirik bir kan damar&#x131;ndan ge&#xE7;en kan ak&#x131;&#x15F;&#x131;, ak&#x131;&#x15F;kan&#x131;n paralel katmanlar h&#xE2;linde d&#xFC;zg&#xFC;n bir &#x15F;ekilde hareket etti&#x11F;i bir rejim olan laminer ak&#x131;&#x15F; ilkeleri kullan&#x131;larak matematiksel olarak tan&#x131;mlanabilir. Bu modelde kan&#x131;n h&#x131;z&#x131;, damar&#x131;n kesiti boyunca tekd&#xFC;ze de&#x11F;ildir; bunun yerine, merkezden olan radyal uzakl&#x131;&#x11F;a ba&#x11F;l&#x131; olarak de&#x11F;i&#x15F;ir. Maksimum h&#x131;z, merkezi eksen boyunca g&#xF6;r&#xFC;l&#xFC;r ve viskoz diren&#xE7; nedeniyle damar duvarlar&#x131;na do&#x11F;ru giderek azal&#x131;r; duvarlarda ise s&#x131;f&#x131;ra iner.
Kan Ak&#x131;&#x15F;&#x131;n&#x131;n Yakla&#x15F;&#x131;klanmas&#x131;
Toplam hacimsel debiyi tahmin etmek i&#xE7;in, damar&#x131;n dairesel kesiti kavramsal olarak bir dizi ince e&#x15F;merkezli halkaya ayr&#x131;l&#x131;r. Her halka, alan&#x131;na&#x2014;&#xE7;evresi ile kal&#x131;nl&#x131;&#x11F;&#x131;n&#x131;n &#xE7;arp&#x131;m&#x131; olarak hesaplanan&#x2014;ve o yar&#x131;&#xE7;aptaki ak&#x131;&#x15F;kan h&#x131;z&#x131;na ba&#x11F;l&#x131; olarak toplam ak&#x131;&#x15F;a belirli bir katk&#x131; yapar. T&#xFC;m halkalar&#x131;n katk&#x131;lar&#x131; topland&#x131;&#x11F;&#x131;nda, damardan ge&#xE7;en toplam ak&#x131;&#x15F;&#x131;n bir yakla&#x15F;&#x131;klamas&#x131; elde edilir.
&#x130;ntegrasyon Yoluyla Ak&#x131;&#x15F;&#x131;n T&#xFC;retilmesi
Halka say&#x131;s&#x131; artt&#x131;k&#xE7;a ve kal&#x131;nl&#x131;klar&#x131; azald&#x131;k&#xE7;a, ak&#x131;&#x15F; katk&#x131;lar&#x131;n&#x131;n toplam&#x131; damar&#x131;n yar&#x131;&#xE7;ap&#x131; boyunca al&#x131;nan belirli bir integrale yakla&#x15F;&#x131;r. Laminer ak&#x131;&#x15F; yasas&#x131;, damar boyunca bas&#x131;n&#xE7; gradyan&#x131;n&#x131;, ak&#x131;&#x15F;kan&#x131;n viskozitesini ve damar&#x131;n uzunlu&#x11F;unu i&#xE7;erecek &#x15F;ekilde, radyal uzakl&#x131;&#x11F;&#x131;n bir fonksiyonu bi&#xE7;iminde h&#x131;z profilini verir. Bu h&#x131;z fonksiyonunun merkezden d&#x131;&#x15F; duvara kadar integralinin al&#x131;nmas&#x131;, toplam kan ak&#x131;&#x15F;&#x131;n&#x131;n kesin de&#x11F;erini sa&#x11F;lar.
Bu sonu&#xE7;, hacimsel debinin bas&#x131;n&#xE7; fark&#x131; ve damar yar&#x131;&#xE7;ap&#x131;n&#x131;n d&#xF6;rd&#xFC;nc&#xFC; kuvveti ile do&#x11F;ru orant&#x131;l&#x131;, viskozite ve damar uzunlu&#x11F;u ile ters orant&#x131;l&#x131; oldu&#x11F;unu ifade eden Poiseuille yasas&#x131; ile form&#xFC;le edilir. Yar&#x131;&#xE7;apa olan bu belirgin ba&#x11F;&#x131;ml&#x131;l&#x131;k, damar &#xE7;ap&#x131;ndaki k&#xFC;&#xE7;&#xFC;k de&#x11F;i&#x15F;imlerin bile dola&#x15F;&#x131;m verimlili&#x11F;i &#xFC;zerinde kritik bir etkiye sahip oldu&#x11F;unu ortaya koyar.

Kan Akışını Bulmada İntegrasyon Uygulamaları

Silindirik bir kan damar&#x131;ndan ge&#xE7;en kan ak&#x131;&#x15F;&#x131;, ak&#x131;&#x15F;kan&#x131;n paralel katmanlar h&#xE2;linde d&#xFC;zg&#xFC;n ...

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

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Application of Techniques of Integration

İntegral Alma Tekniklerinin Uygulanması

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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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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