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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...

Physiological Pharmacokinetic Models: Blood Flow-Limited Versus Diffusion-Limited Models

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Physiological pharmacokinetic models, often called flow-limited or perfusion models, typically assume a swift drug distribution between tissue and venous blood, creating a rapid drug equilibrium. This premise is based on the idea that drug diffusion is extremely fast, and the cell membrane presents no barrier to drug permeation. In this scenario, where no drug binding occurs, the drug concentration in the tissue equals that of the venous blood leaving the tissue. This greatly simplifies the...

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.

Autoregulation of Blood Flow

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Autoregulation mechanisms are characterized by their inherent capacity for self-regulation without necessitating specific nervous stimulation or endocrine control. These mechanisms facilitate the adjustment of blood flow and, therefore, perfusion specific to each tissue region. This self-regulation encompasses chemical signals and myogenic controls.
Chemical Signaling in Autoregulation
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Bernoulli's Equation for Flow Along a Streamline

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:

Capillarity in Fluid

Capillarity in Fluid

Capillarity describes the movement of liquid in small spaces without external forces acting on it. The capillarity is driven by surface tension and adhesive interactions between the liquid and surrounding solid surfaces. This effect is often seen in narrow tubes, porous materials, and fine particles.
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Numerical Simulation of Unsteady Blood Flow through Capillary Networks.

J M Davis¹, C Pozrikidis

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

Bulletin of Mathematical Biology

|November 10, 2010

Summary

Unsteady blood flow in capillary networks can spontaneously oscillate due to cell partitioning dynamics at bifurcations. These oscillations depend on network structure and cell distribution, influencing blood flow behavior.

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Spatial Temporal Analysis of Fieldwise Flow in Microvasculature

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Published on: November 18, 2019

Area of Science:

Fluid dynamics
Biophysics
Computational biology

Background:

Capillary networks are crucial for blood transport.
Understanding blood flow dynamics in capillaries is complex due to cellular interactions and network geometry.

Purpose of the Study:

To develop and implement a numerical method for computing unsteady blood flow in branching capillary networks.
To investigate the conditions leading to spontaneous oscillations in capillary blood flow.

Main Methods:

A one-dimensional convection equation was integrated using a finite-difference method to track discharge hematocrit.
A partitioning law with a dimensionless exponent (q) governed cell distribution at bifurcations.
Simulations were performed on tree-like capillary networks with varying generations (m).

Main Results:

A supercritical Hopf bifurcation was identified at a critical value of q, leading to self-sustained oscillations.
A phase diagram (m-q plane) was presented, outlining conditions for unsteady flow.
Oscillations were induced by increasing viscosity, decreasing vessel diameter ratios, or decreasing terminal vessel diameters, and inhibited by increasing network generations (m).

Conclusions:

The study demonstrates that complex blood flow dynamics, including spontaneous oscillations, can emerge in capillary networks.
The findings highlight the critical role of cell partitioning laws and network architecture in governing blood flow behavior.
The numerical model accurately predicted results compared to discrete cell models.