Cross Sectional Area and Velocity Relation

Cross Sectional Area and Velocity Relation is the physical principle, derived from conservation of mass applied to an incompressible fluid, that the velocity of blood flow through any segment of the vasculature is inversely proportional to the total cross sectional area available for flow at that segment, so that blood moves fastest where total vascular cross sectional area is smallest and slowest where total vascular cross sectional area is largest. This relationship explains why blood velocity varies dramatically across the circulatory system even though the total volumetric flow rate, equal to cardiac output, remains the same at every level of the circuit under steady state conditions.

Physical Basis of the Relationship

Conservation of Mass in an Incompressible Fluid

Because blood is effectively incompressible under physiological conditions, the volume of blood entering any continuous segment of the circulation per unit time must equal the volume of blood leaving that segment per unit time, a principle known as continuity. This continuity requirement holds regardless of how many individual vessels the segment contains, or how their combined cross sectional area compares to that of the segment before or after it.

Mathematical Expression of the Continuity Principle

The continuity principle is expressed mathematically by stating that volumetric flow rate, Q, is equal to the product of cross sectional area, A, and mean velocity, v.

Rearranging this relationship shows that velocity is equal to flow divided by cross sectional area.

Because total flow, Q, remains constant across any set of vessels arranged in series along the circulatory pathway under steady state conditions, this expression shows directly that velocity must decrease wherever cross sectional area increases, and must increase wherever cross sectional area decreases.

Application to the Vascular Tree

Aggregate Cross Sectional Area Versus Individual Vessel Diameter

Although individual vessels become progressively smaller in diameter moving from the aorta toward the capillaries, the total number of parallel vessels at each successive branching generation increases substantially, so that the aggregate cross sectional area summed across all vessels at a given level of the vascular tree actually increases moving from the aorta to the capillaries, reaching its maximum at the level of the capillary bed, before decreasing again through the converging venous system.

Velocity Profile Across the Circulatory System

Following directly from the inverse relationship between cross sectional area and velocity, blood velocity is highest in the aorta, where aggregate cross sectional area is smallest, decreases progressively through the arterial tree as aggregate cross sectional area increases, reaches its lowest value within the capillary bed, where aggregate cross sectional area is greatest, and then increases again through the venules and veins as the vascular network converges and aggregate cross sectional area decreases toward the vena cava.

Visual Representation of the Cross Sectional Area to Velocity Relation

Physiological Significance of Reduced Capillary Velocity

Sufficient Transit Time for Exchange

The dramatic reduction in blood velocity that occurs within the capillary bed as a direct consequence of the large aggregate cross sectional area present at this level is physiologically essential, since it provides sufficient transit time for red blood cells traversing the capillary to allow diffusional exchange of oxygen, carbon dioxide, nutrients, and metabolic waste products to reach near equilibrium with the surrounding interstitial fluid before the blood exits into the venous system.

Consequence of Insufficient Aggregate Capillary Area

If capillary density within a tissue is reduced, whether through vascular rarefaction associated with disease or through insufficient capillary recruitment during periods of high metabolic demand, aggregate capillary cross sectional area falls short of the level required to slow blood velocity adequately, reducing the transit time available for exchange and potentially impairing the delivery of oxygen and nutrients to the tissue despite an unchanged total blood flow rate.

Relationship to Other Hemodynamic Variables

Distinction From the Pressure and Resistance Relationship

The cross sectional area to velocity relationship is a consequence of the geometric and volumetric properties of flow, distinct from the relationship between pressure, flow, and resistance described by the fundamental hemodynamic equation, though both relationships apply simultaneously within the same vascular segments. Cross sectional area determines how a given flow rate translates into velocity at any point in the circuit, while resistance determines what pressure gradient is required to sustain that same flow rate, so that a complete description of blood movement through any vessel segment requires consideration of both relationships together rather than either one in isolation.