5.4 Basics of Flow, Pressure & Resistance

Pressure, Flow, and Resistance

Quick Answer: Flow through a vessel is directly proportional to the pressure gradient and inversely proportional to resistance. Poiseuille's law shows resistance is dominated by vessel radius (raised to the fourth power), so vasoconstriction dramatically reduces flow. The PA-CAT Bulletin of Information, rev. 20240815, includes this under Circulation.

The Core Relationship

Q = ΔP / R, where Q is flow, ΔP is the pressure difference between two points, and R is resistance. This single equation links the entire cardiovascular system: cardiac output is the flow, the pressure gradient is roughly MAP − venous pressure, and total peripheral resistance is set mainly by arteriolar tone.

Poiseuille's Law

For laminar flow of a Newtonian fluid in a rigid tube:

Q = (π ΔP r⁴) / (8 η L)

• r = vessel radius — the dominant term because it is raised to the fourth power. Doubling radius increases flow 16-fold; halving radius cuts flow to 1/16.
• η = viscosity — resistance rises linearly with viscosity.
• L = vessel length — longer vessels add resistance linearly.
• ΔP = pressure gradient.

The corresponding expression for resistance is R = 8 η L / (π r⁴). In practice, L is constant (vessels don't grow or shrink acutely), so physiologic changes in resistance come from radius (vasoconstriction/dilation) and viscosity (hematocrit).

Viscosity and Hematocrit

Blood is a non-Newtonian suspension, but at high shear it approximates Newtonian behavior. Viscosity is determined mainly by:

• Hematocrit (the biggest factor). Polycythemia (Hct > 55%) raises viscosity, raising resistance and afterload; anemia lowers viscosity, increasing flow and cardiac workload.
• Plasma proteins — fibrinogen and immunoglobulins increase viscosity; multiple myeloma can cause hyperviscosity syndrome.
• Temperature — viscosity rises in cold extremities, contributing to sluggish flow.

Laminar vs. Turbulent Flow

Laminar flow is smooth, layered, and parabolic — the center moves fastest, the wall layer is stationary. It is silent. Turbulent flow is chaotic and produces vibrations audible as murmurs or bruits. The Reynolds number (Re) predicts turbulence:

Re = (ρ v D) / η, where ρ is density, v is velocity, D is diameter, η is viscosity.

Turbulence increases when:

• Velocity rises (exercise, stenotic valves).
• Diameter changes abruptly (valve stenosis, aneurysms).
• Viscosity falls (anemia — explains flow murmurs in severe anemia).
• Density is high (minor effect).

A bruit over the carotid suggests stenosis causing turbulence; anemia-related murmurs typically disappear when hematocrit is corrected.

Mean Arterial Pressure

At rest, the heart spends more time in diastole (~2/3 of the cycle) than systole (~1/3), so the time-weighted average pressure is closer to diastolic than systolic. The approximation:

MAP ≈ DBP + 1/3 (SBP − DBP)

Example: 120/80 mmHg → MAP ≈ 80 + 1/3 (40) = 93 mmHg.

This formula fails at high heart rates (where systole and diastole approach equal duration) and with very narrow or wide pulse pressures. Pulse pressure (SBP − DBP) is determined largely by stroke volume and arterial compliance — a wide pulse pressure suggests low compliance (aging, atherosclerosis) or high stroke volume (aortic regurgitation, fever, anemia).

Series and Parallel Vessels

The systemic circulation is a parallel network of organs, each branching from the aorta. Parallel resistances add reciprocally: 1/R_total = 1/R₁ + 1/R₂ + …. Adding more parallel vessels decreases total resistance — the basis of recruiting capillaries during exercise. Within an organ, vessels are largely in series (artery → arteriole → capillary → venule → vein); resistances add linearly. Arterioles contribute the largest single share (~60% of total peripheral resistance) because they have the greatest smooth-muscle control of radius.

Clinical Applications

• Hemorrhage lowers driving pressure (MAP), reducing flow to organs unless compensated by vasoconstriction (raises R, defending central pressure).
• Aortic stenosis creates a high-velocity jet through a narrowed valve — turbulent, producing a systolic murmur.
• Anemia lowers viscosity, increasing flow and producing a flow murmur; the heart compensates with higher cardiac output.
• Local arteriolar dilation (e.g., exercising muscle) dramatically increases local flow through the r⁴ term.
• Epidural anesthesia blocks sympathetic vasoconstrictor tone, lowering TPR and MAP.

Critical Closing Pressure

Vessels can collapse when external pressure exceeds internal pressure. The critical closing pressure is the transmural pressure below which a vessel snaps shut. In severe hemorrhage, arterioles may collapse, halting flow even if some pressure remains. This concept explains why very low blood pressure can produce ischemia disproportionate to the residual pressure reading.

Understanding these relationships frames the next sections on microcirculation and cardiac output — the same equations describe capillary exchange and venous return.

Applied Hemodynamics: MAP, Laplace, and Compliance

The core flow equation Q = ΔP / R becomes clinically useful once you identify the systemic equivalents: cardiac output is Q, the driving pressure is roughly MAP − right atrial pressure, and the resistance is total peripheral resistance (TPR). Rearranged, MAP ≈ CO × TPR (plus venous pressure, usually small). This is why vasodilators that lower TPR (e.g., hydralazine, calcium channel blockers) reduce MAP even when cardiac output is unchanged, and why a fall in CO during hemorrhage is initially offset by sympathetic vasoconstriction raising TPR, defending MAP until decompensation. PA-CAT questions often give two of the three values and ask for the third — keep units consistent and remember that TPR is normally about 1000 dynes·s·cm⁻⁵ at rest.

The r⁴ relationship is the single most tested hemodynamic fact. Consider an arteriole constricting from radius 1 to radius 0.8: flow changes by (0.8)⁴ = 0.41, a 59% reduction from only a 20% radius decrease. Conversely, a 10% radius increase raises flow by (1.1)⁴ ≈ 1.46, a 46% gain. This is why arterioles, despite being only a small fraction of total vessel length, contribute ~60% of TPR — they are the smooth-muscle-controlled radius gate. When a question states that a vasoconstrictor doubles arteriolar radius impact, translate that to a 16-fold flow change and check whether the answer choices reflect that magnitude rather than a linear intuition.

Wall tension and the law of Laplace explain aneurysm risk and cardiac remodeling. For a cylindrical vessel, T = P × r / h (tension proportional to pressure and radius, inversely proportional to wall thickness). An aneurysm increases radius while wall thickness often thins, so tension rises on both counts — a positive-feedback loop that favors rupture. This is why abdominal aortic aneurysms are repaired electively once diameter exceeds ~5.5 cm in men. In the heart, dilated cardiomyopathy raises wall tension at any given pressure, increasing oxygen demand and reducing mechanical efficiency; conversely, left ventricular hypertrophy thickens the wall (h) to normalize tension against chronically elevated pressure (afterload reduction), though at the cost of reduced compliance.

Compliance and pulse pressure link vascular stiffness to the systolic-diastolic gap. Arterial compliance C = ΔV / ΔP falls with aging and atherosclerosis as elastin fragments and collagen stiffens the vessel wall. With lower compliance, the same stroke volume produces a larger pressure rise, widening pulse pressure (SBP − DBP) — the classic isolated systolic hypertension of older adults. Conversely, aortic regurgitation widens pulse pressure through a different mechanism: diastolic run-off into the ventricle lowers DBP while high stroke volume raises SBP, producing the characteristic bounding pulse (Corrigan pulse) and a water-hammer quality. Distinguishing mechanism (stiff vessel vs. diastolic run-off) from the numeric pattern is a common PA-CAT reasoning step.

Finally, viscosity and length shift resistance over longer time scales. Chronic hypertension stimulates vascular remodeling — arteriolar wall thickening and modest lengthening — which raises resistance independently of acute radius changes, explaining why TPR remains elevated even after a single vasodilator dose. Polycythemia (Hct > 55%) raises viscosity and therefore resistance, increasing afterload and thrombosis risk; anemia lowers viscosity, producing high-flow murmurs and a hyperdynamic circulation. Multiple myeloma, by raising paraprotein levels, can cause hyperviscosity syndrome with bleeding, visual disturbance, and neurologic symptoms even at normal hematocrit — a reminder that viscosity is more than hematocrit alone.