How to Calculate Blood Flow Rate

Blood Flow Rate Calculator (Poiseuille's Law)

Unit: mmHg (millimeters of mercury)

Unit: mm (millimeters)

Unit: cm (centimeters)

Unit: cP (centipoise). Normal range: 3.0 – 4.0

Understanding hemodynamics—the dynamics of blood flow—is crucial in physiology and medicine. The rate at which blood flows through a vessel is determined by physical laws that relate pressure, vessel dimensions, and fluid properties. This calculator uses Poiseuille's Law, the standard physics model for laminar flow in cylindrical vessels.

The Formula: Poiseuille's Law

The flow rate ($Q$) of a fluid through a cylindrical pipe (like a blood vessel) is directly proportional to the pressure difference and the fourth power of the radius, and inversely proportional to the viscosity and length of the vessel.

Q = (π · ΔP · r⁴) / (8 · η · L)

Where:

• Q = Blood Flow Rate
• ΔP = Pressure Difference between the two ends of the vessel (Pressure Gradient)
• r = Radius of the blood vessel
• η (eta) = Viscosity of the blood
• L = Length of the vessel

Key Variables Explained

1. Pressure Difference (ΔP)

Blood flows from high pressure to low pressure. The greater the pressure gradient (for example, between the arteries and veins), the faster the flow. In our calculator, this is measured in millimeters of mercury (mmHg), the standard medical unit.

2. Vessel Radius (r) – The "Power of 4"

The radius of the vessel is the most critical factor in regulating blood flow. Because the radius is raised to the fourth power ($r^4$) in the equation, even a tiny change in vessel width causes a massive change in flow.

Example: Doubling the radius of a vessel increases the flow rate by 16 times ($2^4 = 16$). This is why vasodilation (widening of blood vessels) is such an effective mechanism for the body to increase oxygen delivery to tissues.

3. Vessel Length (L)

The longer the vessel, the more friction the blood encounters, and the slower the flow rate. Length is inversely proportional to flow. However, in the human body, vessel length is generally constant, unlike radius which changes frequently.

4. Blood Viscosity (η)

Viscosity refers to the "thickness" of the blood. Thicker fluid flows more slowly. Blood viscosity is primarily determined by hematocrit (the concentration of red blood cells). Conditions like polycythemia (high red blood cell count) increase viscosity and resistance, thereby reducing blood flow.

Calculating Vascular Resistance

While flow rate tells you how much blood is moving, Resistance (R) tells you how hard it is to push that blood through. It can be derived from Darcy's Law ($Q = \Delta P / R$) or Poiseuille's equation:

R = (8 · η · L) / (π · r⁴)

This calculator automatically computes the resistance in standard hemodynamic units (mmHg·min/ml).

Clinical Applications

Physicians and physiologists use these calculations to understand conditions such as:

• Hypertension: High blood pressure often results from increased vascular resistance (narrowing of vessels).
• Stenosis: A narrowing of an artery (e.g., carotid stenosis) drastically reduces radius, plummeting flow rates unless pressure increases significantly to compensate.
• Shock: In circulatory shock, the body attempts to maintain flow to vital organs by altering resistance (vasoconstriction) and heart rate.

Example Calculation

Let's say we have a small artery with the following characteristics:

• Pressure Difference: 50 mmHg
• Radius: 1.5 mm
• Length: 5 cm
• Viscosity: 3.5 cP

Using the calculator above, we find that the blood flow rate is approximately 1,725 ml/min. If the vessel constricts slightly to a radius of 1.0 mm (a 33% reduction in size), the flow drops to roughly 340 ml/min—an 80% drop in flow!