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Applications of Integration to Find Blood Flow

Applications of Integration to Find Blood Flow

Blood flow through a cylindrical blood vessel can be mathematically described using the principles of laminar flow, a regime in which fluid moves smoothly in parallel layers. In this model, the velocity of the blood is not uniform across the cross-section of the vessel; rather, it varies with the radial distance from the center. The maximum velocity occurs along the central axis, decreasing progressively toward the vessel walls, where it reaches zero due to viscous drag.Approximating Blood...

Blood Flow

Blood Flow

Blood is pumped by the heart into the aorta, the largest artery in the body, and then into increasingly smaller arteries, arterioles, and capillaries. The velocity of blood flow decreases with increased cross-sectional blood vessel area. As blood returns to the heart through venules and veins, its velocity increases. The movement of blood is encouraged by smooth muscle in the vessel walls, the movement of skeletal muscle surrounding the vessels, and one-way valves that prevent backflow.

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Related Experiment Video

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A Computational Modeling Approach to Investigate the Influence of Hyperthermia on the Tumor Microenvironment

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Published on: December 1, 2023

Numerical simulation of blood and interstitial flow through a solid tumor.

¹Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA. cpozrikidis@ecs.umass.edu

Journal of Mathematical Biology

|March 12, 2009

Summary

This study models tumor blood flow and plasma leakage using a theoretical framework. Fractional plasma leakage is maximized at a specific vascular tree branching grade, influenced by interstitial and vascular permeability.

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Establishing a Physiologic Human Vascularized Micro-Tumor Model for Cancer Research

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Published on: September 15, 2023

Area of Science:

Biophysics
Mathematical Biology
Oncology

Background:

Solid tumors exhibit irregular vasculature, impacting blood flow and drug delivery.
Understanding plasma leakage is crucial for predicting tumor growth and treatment efficacy.

Purpose of the Study:

To develop a theoretical framework for blood flow and plasma leakage in solid tumors.
To analyze the influence of vascular architecture and interstitial properties on plasma extravasation.

Main Methods:

Modeling the tumor capillary bed as a bifurcating tree with deterministic and random parameters.
Applying Sterling's law for plasma leakage and Darcy's law for interstitial flow.
Solving coupled integral and differential equations using numerical discretization and iterative methods.

Main Results:

Quantified the effect of interstitial hydraulic and vascular permeability on fractional plasma leakage.
Demonstrated that fractional leakage peaks at a specific vascular tree bifurcation grade.
The model provides insights into pressure distribution within tumor vasculature and interstitium.

Conclusions:

The developed theoretical framework accurately describes blood flow and plasma leakage in tumor vasculature.
Interstitial and vascular permeability are key determinants of plasma extravasation.
Optimal vascular tree structure exists for minimizing or maximizing plasma leakage, with implications for drug delivery strategies.