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

Il flusso sanguigno in un vaso sanguigno può essere descritto usando la legge del flusso laminare, dove la velocità varia con la distanza radiale dal centro.Il flusso sanguigno totale può essere approssimato dividendo la sezione trasversale in sottili anelli concentrici con raggi variabili.L'area di ogni anello è data dalla sua circonferenza moltiplicata per il suo piccolo spessore.Il flusso attraverso ogni anello può essere approssimato come il prodotto della sua area e della velocità a quel raggio.Sommando questi flussi per tutti gli anelli si ottiene un flusso totale approssimativo, chiamato somma di Riemann.Quando il numero di anelli tende all'infinito, la somma di Riemann si avvicina ai valori esatti, rappresentati dall'integrale definito dal centro alla parete del recipiente.La funzione di velocità nell'integrale deriva dalla legge del flusso laminare. Afferma che la velocità a una distanza r dipende dalla differenza di pressione, dalla viscosità del fluido e dalla lunghezza del recipiente.Sostituendo l'espressione della velocità nell'integrale e integrando sul raggio del vaso si ottiene il flusso sanguigno esatto.Questo risultato, chiamato Legge di Poiseuille, mostra che il flusso scala con la quarta potenza del raggio del recipiente. Ecco perché anche un piccolo restringimento dei vasi sanguigni riduce bruscamente il flusso sanguigno.

applicazioni dell’integrazione per determinare il flusso sanguigno

Applications of Integration to Find Blood Flow

Il flusso sanguigno attraverso un vaso sanguigno cilindrico pu&#xF2; essere descritto matematicamente mediante i principi del flusso laminare, un regime in cui il fluido scorre in modo regolare in strati paralleli. In questo modello, la velocit&#xE0; del sangue non &#xE8; uniforme nella sezione trasversale del vaso; al contrario, varia in funzione della distanza radiale dal centro. La velocit&#xE0; massima si osserva lungo l&#x2019;asse centrale e diminuisce progressivamente verso le pareti del vaso, dove si annulla a causa dell&#x2019;attrito viscoso.
Approssimazione del flusso sanguigno
Per stimare il flusso volumetrico totale, ossia la portata, la sezione trasversale circolare del vaso viene concettualmente suddivisa in una serie di sottili anelli concentrici. Ciascun anello contribuisce a una porzione della portata complessiva, determinata dalla sua area&#x2014;calcolata come prodotto tra circonferenza e spessore&#x2014;e dalla velocit&#xE0; del fluido a quel raggio. Sommando i contributi di tutti gli anelli si ottiene un&#x2019;approssimazione del flusso totale attraverso il vaso.
Derivazione del flusso mediante integrazione
All&#x2019;aumentare del numero di anelli e al diminuire del loro spessore, la somma dei contributi di flusso tende a un integrale definito rispetto al raggio del vaso. La legge del flusso laminare fornisce il profilo di velocit&#xE0; come funzione della distanza radiale, tenendo conto del gradiente di pressione lungo il vaso, della viscosit&#xE0; del fluido e della lunghezza del vaso. Integrando tale funzione di velocit&#xE0; dal centro alla parete esterna del vaso si ottiene il valore esatto del flusso sanguigno totale.
Questo risultato &#xE8; formalizzato nella legge di Poiseuille, secondo la quale la portata volumetrica &#xE8; direttamente proporzionale alla differenza di pressione e alla quarta potenza del raggio del vaso, e inversamente proporzionale alla viscosit&#xE0; e alla lunghezza del vaso. Questa marcata dipendenza dal raggio evidenzia l&#x2019;influenza determinante anche di piccole variazioni del diametro del vaso sull&#x2019;efficienza circolatoria.

Applicazioni dell’integrazione per determinare il flusso sanguigno

Il flusso sanguigno attraverso un vaso sanguigno cilindrico pu&#xF2; essere descritto matematicamente mediante i principi del flusso laminare, un regime ...

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

Education

Didattica

JoVE Core

Calculus

Analisi matematica

Unit

Unità

Application of Techniques of Integration

Applicazione delle tecniche di integrazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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.

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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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