Cardiovascular Biophysics

Vascular Resistance and Series/Parallel Networks

By analogy with Ohm's law ($V = IR$), we define vascular resistance:

$R_v = \frac{\Delta P}{Q} = \frac{8\eta L}{\pi r^4}$

The $r^{-4}$ dependence is clinically crucial: arterioles (radius ~10–50 $\mu$m) are the major site of resistance, despite their short length, because they are so narrow. A 20% decrease in arteriolar radius (e.g., from vasoconstriction) increases resistance by a factor of$(1/0.8)^4 \approx 2.4$.

Series vessels: For vessels in series (aorta → artery → arteriole → capillary):

$R_{\text{total}} = R_1 + R_2 + R_3 + \cdots$

Parallel vessels: For $N$ identical capillaries in parallel:

$\frac{1}{R_{\text{total}}} = \frac{N}{R_{\text{single}}} \quad \Rightarrow \quad R_{\text{total}} = \frac{R_{\text{single}}}{N}$

The ~10 billion capillaries in the body, despite each having enormous individual resistance, have a collective resistance that is relatively modest because they are massively parallel. The total peripheral resistance (TPR) is dominated by the arteriolar bed, typically $\text{TPR} \approx 1 \text{ mmHg} \cdot \text{min/mL}$.

The mean arterial pressure follows directly: $\text{MAP} = \text{CO} \times \text{TPR}$, where CO is cardiac output (~5 L/min), giving MAP $\approx 93$ mmHg.

Derivation: Two-Element Windkessel

Define arterial compliance $C = dV/dP$ (the volume change per unit pressure change) and peripheral resistance $R$. Conservation of volume requires:

$Q_{\text{in}}(t) = Q_{\text{out}}(t) + C\frac{dP}{dt}$

where $Q_{\text{out}} = P/R$ (flow through the peripheral resistance) and$C \, dP/dt$ is the rate of volume storage in the elastic arteries.

During diastole ($Q_{\text{in}} = 0$):

$C\frac{dP}{dt} + \frac{P}{R} = 0 \quad \Rightarrow \quad P(t) = P_{\text{sys}} \, e^{-t/(RC)}$

The time constant $\tau = RC$ determines how quickly pressure decays. With typical values$R \approx 1.0$ mmHg$\cdot$s/mL and $C \approx 1.5$ mL/mmHg, we get $\tau \approx 1.5$ s, consistent with the observed diastolic pressure decay.

Three-Element and Four-Element Windkessel

The two-element model underestimates the impedance at high frequencies (failing to capture the initial sharp rise of the pressure waveform). The three-element model adds a characteristic impedance $Z_c$ in series (representing the impedance of the proximal aorta):

$P(t) = Z_c \cdot Q_{\text{in}}(t) + P_{\text{WK}}(t)$

where $P_{\text{WK}}$ satisfies the two-element equation. In the frequency domain, the input impedance of the three-element Windkessel is:

$Z(\omega) = Z_c + \frac{R}{1 + j\omega RC}$

At $\omega = 0$: $Z(0) = Z_c + R$ (total DC resistance). At$\omega \to \infty$: $Z(\infty) = Z_c$ (characteristic impedance only).

The four-element model adds an inertance term $L$(blood inertia) in series:

$Z(\omega) = Z_c + j\omega L + \frac{R}{1 + j\omega RC}$

The inertance $L = \rho l / A$ accounts for the mass of blood in the aorta, producing a more realistic impedance phase at intermediate frequencies.

Derivation: Moens-Korteweg Equation

Consider a thin-walled elastic tube of radius $R$, wall thickness $h$, wall elastic modulus $E$, filled with fluid of density $\rho$. We derive the wave speed from conservation of mass and momentum.

Step 1 — Continuity equation: For a small segment of tube, conservation of mass gives:

$\frac{\partial A}{\partial t} + \frac{\partial (Av)}{\partial x} = 0$

Step 2 — Momentum equation: For inviscid flow:

$\rho \frac{\partial v}{\partial t} + \rho v\frac{\partial v}{\partial x} = -\frac{\partial P}{\partial x}$

Step 3 — Tube law: For a thin-walled elastic tube, the pressure-area relationship (from hoop stress balance) is:

$P - P_{\text{ext}} = \frac{Eh}{R}\left(\sqrt{\frac{A}{A_0}} - 1\right) \approx \frac{Eh}{2R_0 A_0}(A - A_0)$

Step 4 — Linearization: For small perturbations about the equilibrium state ($A = A_0 + a$, $v = v_0 + u$, $P = P_0 + p$), neglecting products of perturbations:

$\frac{\partial a}{\partial t} + A_0 \frac{\partial u}{\partial x} = 0, \quad \rho \frac{\partial u}{\partial t} = -\frac{\partial p}{\partial x}$

With $p = (Eh)/(2R_0 A_0) \cdot a$, these combine to give the wave equation:

$\frac{\partial^2 p}{\partial t^2} = c^2 \frac{\partial^2 p}{\partial x^2}$

where the Moens-Korteweg wave speed is:

$\boxed{c = \sqrt{\frac{Eh}{2\rho R}}}$

For the human aorta: $E \approx 0.4$ MPa, $h \approx 2$ mm,$R \approx 12$ mm, $\rho \approx 1050$ kg/m$^3$, giving$c \approx 5$ m/s in young adults. In elderly or atherosclerotic arteries,$E$ can increase 3–5 fold, raising PWV to 10–15 m/s — a powerful marker of cardiovascular risk.

PWV, Arterial Stiffness, and Aging

The compliance of the arterial wall is $C = dV/dP$. For a thin-walled tube of length $l$:

$C = \frac{dA}{dP} \cdot l = \frac{2\pi R^3 l}{Eh}$

This gives the Bramwell-Hill relationship linking PWV to compliance:

$c = \sqrt{\frac{V}{\rho C}} = \sqrt{\frac{A \cdot l}{\rho C}}$

With aging: collagen replaces elastin in the arterial wall, $E$ increases, $C$decreases, and PWV rises. The reflected pulse wave arrives earlier in systole (rather than during diastole), augmenting systolic pressure and reducing diastolic pressure — the hallmark of isolated systolic hypertension in the elderly.

Derivation: Frank-Starling Law from Sarcomere Mechanics

The Frank-Starling law states that the force of ventricular contraction increases with the degree of pre-systolic stretch (preload). This emerges directly from sarcomere mechanics.

Sarcomere overlap geometry: The active force generated by a sarcomere depends on the number of actin-myosin cross-bridges that can form, which in turn depends on the overlap between the thick (myosin, length $\approx 1.6\,\mu$m) and thin (actin, length $\approx 1.0\,\mu$m) filaments.

At the optimal sarcomere length $L_0 \approx 2.2\,\mu$m, overlap is maximal and force is maximum. As the sarcomere stretches beyond $L_0$, overlap decreases linearly:

$F_{\text{active}}(L) = F_{\max} \cdot \frac{L_{\max} - L}{L_{\max} - L_0} \quad \text{for } L > L_0$

where $L_{\max} \approx 3.6\,\mu$m (zero overlap). Below $L_0$, thin filaments collide in the central bare zone, also reducing force. In the normal heart, sarcomere operating length is 1.8–2.2 $\mu$m, on the ascending limb, so increased filling (stretch) increases force — the Frank-Starling mechanism.

Additionally, stretch increases calcium sensitivity of troponin C, providing a second mechanism that enhances the length-dependent activation.

Derivation: Pressure-Volume Loop and Stroke Work

The cardiac cycle traces a loop in pressure-volume space with four phases:

1. Isovolumetric contraction: Pressure rises at constant volume ($V = V_{\text{ED}}$)
2. Ejection: Aortic valve opens, volume decreases as blood is ejected
3. Isovolumetric relaxation: Pressure falls at constant volume ($V = V_{\text{ES}}$)
4. Filling: Mitral valve opens, ventricle fills

The stroke work is the area enclosed by the PV loop:

$W = \oint P \, dV$

For typical values: stroke volume SV = $V_{\text{ED}} - V_{\text{ES}} \approx 70$ mL, mean ejection pressure $\bar{P} \approx 100$ mmHg:

$W \approx \bar{P} \times \text{SV} \approx 100 \times 133 \times 70 \times 10^{-6} \approx 0.93 \text{ J}$

At 72 beats/min, cardiac power output is $\approx 1.1$ W for the left ventricle alone.

End-Systolic Pressure-Volume Relationship (ESPVR)

The end-systolic pressure-volume relationship (ESPVR) defines the maximum pressure the ventricle can generate at any given volume. It is approximately linear:

$P_{\text{ES}} = E_{\text{es}}(V_{\text{ES}} - V_0)$

where $E_{\text{es}}$ is the end-systolic elastance (a measure of contractility, typically$\approx 2.5$ mmHg/mL) and $V_0$ is the volume intercept. Increased contractility (e.g., from sympathetic stimulation) steepens the ESPVR slope, increasing stroke volume at the same preload.

The end-diastolic pressure-volume relationship (EDPVR) is nonlinear (exponential), reflecting the passive elastic properties of the myocardium:

$P_{\text{ED}} = A(e^{\beta(V_{\text{ED}} - V_0)} - 1)$

where $A$ and $\beta$ characterize myocardial stiffness. Together, the ESPVR and EDPVR bound all possible operating points of the ventricle.

Derivation: Fåhraeus-Lindqvist Effect

In 1931, Fåhraeus and Lindqvist discovered that the apparent viscosity of blood decreases as the tube diameter drops below ~300 $\mu$m. This seemingly paradoxical effect arises from the formation of a cell-free layer (CFL) near the vessel wall.

Two-fluid model: Model the vessel as two concentric layers: a central core of radius $R_c$ containing red blood cells (viscosity $\eta_c$) surrounded by a cell-free plasma layer of thickness $\delta$ (viscosity $\eta_p$):

$Q = \frac{\pi R^4 \Delta P}{8\eta_p L}\left[1 - \left(\frac{R_c}{R}\right)^4\left(1 - \frac{\eta_p}{\eta_c}\right)\right]$

The apparent viscosity is:

$\eta_{\text{app}} = \frac{\eta_p}{1 - \left(1 - \frac{R_c}{R}\right)^4\left(1 - \frac{\eta_p}{\eta_c}\right)^{-1}}$

As $R$ decreases, $\delta/R$ increases (the CFL becomes a larger fraction of the tube), so $\eta_{\text{app}}$ decreases. Below about 7 $\mu$m, red blood cells must deform to pass through single-file, and viscosity rises again.

Derivation: Casson Fluid Model for Blood

Blood exhibits a yield stress $\tau_y$ at low shear rates due to red blood cell aggregation (rouleaux formation). The Casson model captures this:

$\sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\eta_\infty \dot{\gamma}}$

where $\tau$ is shear stress, $\dot{\gamma}$ is shear rate, $\tau_y \approx 0.005$ Pa, and $\eta_\infty \approx 0.003$ Pa$\cdot$s (the high-shear-rate viscosity). Squaring:

$\tau = \tau_y + 2\sqrt{\tau_y \eta_\infty \dot{\gamma}} + \eta_\infty \dot{\gamma}$

At high shear rates ($\dot{\gamma} \gg \tau_y/\eta_\infty$), the last term dominates and blood behaves as a Newtonian fluid. At low shear rates, the yield stress term dominates, explaining why blood “gels” when stagnant (e.g., in deep veins — contributing to DVT risk).

Derivation: Turbulence and Heart Murmurs

The Reynolds number for blood flow in a vessel of diameter $D$ is:

$\text{Re} = \frac{\rho v D}{\eta}$

Turbulence occurs when $\text{Re} \gtrsim 2000$. For the aorta at peak systole:$\rho \approx 1050$ kg/m$^3$, $v \approx 1.0$ m/s,$D \approx 2.5$ cm, $\eta \approx 0.003$ Pa$\cdot$s:

$\text{Re} = \frac{1050 \times 1.0 \times 0.025}{0.003} \approx 8750$

This exceeds 2000, so brief turbulence does occur at peak systolic ejection in the aorta. However, the flow is pulsatile and the turbulent phase is transient. In pathological conditions — stenotic valves, narrowed arteries, or high cardiac output — sustained turbulence produces audible vibrations (heart murmurs and bruits). The intensity of the murmur correlates with the degree of stenosis through the pressure drop across the constriction:

$\Delta P \approx \frac{1}{2}\rho v^2 \approx \frac{\rho Q^2}{2A^2}$

where $A$ is the stenotic orifice area. By the continuity equation, a 50% reduction in area quadruples the velocity, producing a 16-fold increase in turbulent pressure fluctuations.

Applications of Cardiovascular Biophysics

• • Blood pressure measurement: The sphygmomanometer works by collapsing the brachial artery and listening for Korotkoff sounds — turbulent flow below the cuff as it is deflated. Systolic pressure = cuff pressure at first sound; diastolic = cuff pressure at disappearance.
• • Echocardiography: Doppler ultrasound measures blood velocity via $\Delta f / f_0 = 2v\cos\theta / c_{\text{sound}}$. The modified Bernoulli equation$\Delta P = 4v^2$ estimates trans-valvular pressure gradients.
• • Artificial heart valves: Mechanical valves must minimise turbulent shear stress (to avoid haemolysis) while maintaining low regurgitant volume. The wash-out volume must prevent stagnation zones where thrombi form.
• • Stent design: Drug-eluting stents must maintain laminar flow (minimising neointimal hyperplasia) while providing sufficient radial force. Wall shear stress below$\sim$0.4 Pa promotes plaque formation; stent geometry is optimized to maintain healthy WSS.
• • Atherosclerosis mechanics: Plaques form preferentially at arterial bifurcations where oscillatory shear stress occurs. The circumferential stress in the fibrous cap determines rupture risk: $\sigma = PR/h$, critically dependent on cap thickness $h$.

• • Jean-Louis-Marie Poiseuille (1840): A physician who meticulously measured the flow of distilled water through glass capillaries, establishing the $R^4$law. His motivation was understanding blood flow, though his experiments used water for reproducibility.
• • Otto Frank (1899): Proposed the Windkessel model to explain the relationship between the pulsatile cardiac output and the smoother arterial pressure waveform. His original work used the analogy of a fire engine's air chamber.
• • Ernest Starling (1914): Together with Otto Frank's earlier work, established that cardiac output is regulated by venous return through the length-tension relationship of cardiac muscle (the Frank-Starling mechanism).
• • John Womersley (1955): Extended Poiseuille flow to pulsatile conditions, introducing the Womersley number $\alpha = R\sqrt{\omega\rho/\eta}$ that characterizes the ratio of unsteady to viscous forces. For $\alpha > 1$, the velocity profile deviates significantly from parabolic.