Physics in vessel physiology

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Physics in Vessel Physiology

Vascular physiology is governed by a small set of physical laws that explain how blood flows, how vessels resist or accommodate that flow, and how vessel walls bear the mechanical loads imposed by pressure. Below is a structured breakdown of each core principle.

1. The Fundamental Flow Equation

Blood flow (Q) through any vessel follows the same logic as Ohm's law in electrical circuits:

Q = ΔP / R

Q = flow (mL/min)
ΔP = pressure gradient (driving pressure; difference between upstream and downstream pressure)
R = resistance

Flow is directly proportional to the pressure gradient and inversely proportional to resistance. The direction of flow always moves from high to low pressure - left ventricle to aorta during systole, vena cava to right atrium at all times.

Total peripheral resistance (TPR/SVR) is calculated by substituting cardiac output for Q and the aorta-to-vena cava pressure difference for ΔP. The primary mechanism by which the body controls blood flow is by changing arteriolar resistance.

Costanzo Physiology 7th Edition, p. 128-129

2. Poiseuille's Law - Resistance to Flow

The physical determinants of resistance are captured in the Poiseuille equation:

$$R = \frac{8\eta l}{\pi r^4}$$

Variable	Meaning
η	Viscosity of blood
l	Length of the vessel
r	Radius (raised to the 4th power)

The most important concept: the r⁴ relationship. A 50% reduction in radius does not double resistance - it increases it by 16-fold (2⁴). A 75% reduction in radius (radius cut to 1/4 of original) increases resistance by 256-fold (4⁴), reducing flow to less than 0.4% of baseline. This is why partial arterial occlusion (e.g., carotid stenosis, coronary artery disease) produces catastrophic changes in flow.

Blood viscosity (η) is determined largely by hematocrit. Polycythemia raises viscosity and therefore resistance; severe anemia lowers viscosity. Viscosity also rises at very low flow rates (non-Newtonian behavior of blood).

Costanzo Physiology 7th Edition, p. 129-130

3. Series vs. Parallel Resistance

The arrangement of vessels determines total resistance:

Series Resistance (within one organ)

Blood flows sequentially: artery → arteriole → capillary → venule → vein

$$R_{total} = R_{artery} + R_{arterioles} + R_{capillaries} + R_{venules} + R_{vein}$$

Total resistance is the sum of all individual resistances
Arterioles contribute the largest portion - hence the greatest pressure drop occurs here
Total flow is equal at every level in series

Parallel Resistance (between organs)

Cardiac output distributes simultaneously among cerebral, coronary, renal, GI, skeletal muscle, and skin circulations:

$$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...$$

Total resistance is less than any single component resistance
Adding more parallel vessels lowers overall resistance - this is why the systemic vasculature can distribute large volumes of blood

Series and parallel resistance diagram showing vessels within an organ (series) and the organ circulations (parallel)

Costanzo Physiology 7th Edition, p. 130

4. Laminar Flow and the Reynolds Number

Under normal conditions, blood flow is laminar (streamlined):

A parabolic velocity profile develops: velocity = 0 at the vessel wall (blood adheres to wall), and is maximal at the center
Layers of blood slide past each other in concentric "sheets" with no mixing between layers
Laminar flow is silent

When the flow pattern is disrupted (at valves, stenotic segments, branch points), flow becomes turbulent:

Blood streams mix radially and axially
Kinetic energy is wasted in radial and axial motion, so more driving pressure is needed to maintain the same flow
Turbulent flow produces audible sounds - the physiological basis of Korotkoff sounds (blood pressure measurement) and cardiac/vascular murmurs

The Reynolds number (N_R) predicts whether flow will be laminar or turbulent:

$$N_R = \frac{\rho \cdot d \cdot v}{\eta}$$

Variable	Meaning
ρ	Density of blood
d	Vessel diameter
v	Mean velocity of flow
η	Viscosity

N_R < 2000 → laminar flow
N_R > 2000 → increasing probability of turbulence
N_R > 3000 → turbulent flow always occurs

Key triggers for turbulence in clinical practice:

Decreased viscosity (anemia, low hematocrit) - raises N_R
High-velocity flow at stenotic segments - even though diameter is smaller, the velocity increase predominates and raises N_R
This explains why anemia produces "flow murmurs" and why stenotic vessels create audible bruits

Costanzo Physiology 7th Edition, p. 131-132

5. Law of Laplace - Wall Tension

A vessel wall must withstand the outward force of intravascular pressure. The Law of Laplace (Young-Laplace, early 1800s) describes this relationship:

$$T = \Delta P \cdot r$$

Or, for thick-walled vessels (incorporating wall thickness h):

$$T = \Delta P \cdot \frac{r}{h}$$

Where:

T = wall tension (force per unit length, dynes/cm)
ΔP = transmural pressure (intravascular minus extravascular)
r = vessel radius
h = wall thickness

Vessel wall tension and shear stress diagram showing transmural pressure forces and endothelial shear

Clinical implications of Laplace's Law:

Vessel	Radius	Pressure	Wall Tension	Structural adaptation
Aorta	Large	~100 mmHg	Very high	Thick wall with elastin and collagen
Arteriole	Small	~60 mmHg	Moderate	Thick smooth muscle layer
Capillary	Very small	~25 mmHg	Very low	Single endothelial cell layer sufficient
Aneurysm	Progressively larger	Same	Progressively higher	Wall rupture risk increases as diameter expands

This is why capillaries (with their tiny radii) can withstand significant pressures despite having a single-cell wall - their small radius keeps wall tension low. Conversely, an aortic aneurysm becomes increasingly unstable as it dilates: larger radius → higher wall tension → greater rupture risk, in a vicious self-reinforcing cycle.

Shear stress (a separate mechanical force) is the frictional drag that flowing blood exerts on the endothelial surface. It is proportional to flow velocity and viscosity, and inversely related to r³. Endothelial mechanosensors detect shear stress and regulate vascular remodeling throughout life.

Guyton and Hall Textbook of Medical Physiology, p. 186-187
Medical Physiology (Boron & Boulpaep), p. 662-663

6. Vascular Distensibility and Compliance

All blood vessels are distensible - they expand when pressure rises. This property differs markedly between vessel types:

Vascular distensibility = fractional increase in volume per mmHg pressure rise:

$$\text{Distensibility} = \frac{\Delta V}{\Delta P \times V_0}$$

Vascular compliance (capacitance) = total volume stored per mmHg:

$$\text{Compliance} = \frac{\Delta V}{\Delta P}$$

Note: compliance = distensibility × volume. A vein may be 8× more distensible than a corresponding artery AND have 3× the volume, giving it ~24× greater compliance.

Functional consequences:

Arteries: Low compliance (relatively stiff walls) → pressure rises sharply with volume addition → act as pressure reservoirs that maintain driving force during diastole (Windkessel effect)
Veins: High compliance → enormous volumes can be accommodated with minimal pressure change → act as volume reservoirs (capacitance vessels). Up to 500 mL can be shifted into veins with almost no change in venous pressure

Sympathetic tone shifts the pressure-volume curve: Sympathetic stimulation increases smooth muscle tone, raising pressure at any given volume. Inhibition does the opposite. This is how the body rapidly redistributes blood volume during hemorrhage or exercise.

Volume-pressure curves of the arterial and venous systems showing effects of sympathetic stimulation and inhibition

Guyton and Hall Textbook of Medical Physiology, p. 187-188

Summary Table

Physical Principle	Key Equation	Primary Site of Importance	Clinical Relevance
Flow-pressure-resistance	Q = ΔP/R	Entire vasculature	Basis of CO, BP, TPR calculations
Poiseuille (radius⁴)	R = 8ηl/πr⁴	Arterioles	Stenosis, vasodilation/constriction
Series resistance	R_total = R₁ + R₂ + ...	Within organs	Arterioles dominate organ resistance
Parallel resistance	1/R_total = 1/R₁ + 1/R₂ + ...	Between organ systems	Why adding beds lowers total R
Reynolds number	N_R = ρdv/η	Large vessels, stenoses	Murmurs, bruits, Korotkoff sounds
Law of Laplace	T = ΔP·r/h	All vessels, aneurysms	Aneurysm risk, capillary integrity
Compliance	C = ΔV/ΔP	Arteries & veins	Windkessel effect, volume buffering

These physical laws work together. Arterioles control resistance (via r⁴ in Poiseuille's law), veins buffer volume (via compliance), arterial walls withstand tension (via Laplace), and the entire system maintains laminar flow to minimize energy losses (governed by Reynolds number).

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