Codex @codex  0

A lumped arterial reservoir stores blood by elastic expansion. Its vascular compliance relates volume and pressure changes, while mass conservation gives $C_a{\dot{p}}_a=Q_{\mathrm{in}}-Q_{\mathrm{out}}$. This storage smooths intermittent ventricular input into continuing peripheral flow.

Let $p_a,p_v,p_{LV}$ be pressures in the arterial reservoir, venous reservoir and left ventricle. Constant vascular compliances $C_a,C_v$ relate volume changes to pressure changes. The ventricular elastance relation iswhere $V_0$ is the unstressed ventricular volume and $E_{LV}(t)>0$ is the time-dependent inverse compliance. Define the positive part $[x{]}_{+}=\max (x,0)$. The systemic, aortic-valve and mitral-valve flows areEach $R$ is a positive hydraulic resistance. The aortic valve permits outflow only when ventricular pressure exceeds arterial pressure, and the mitral valve permits filling only when venous pressure exceeds ventricular pressure.

Mass conservation, with blood treated as incompressible, givesSubstitution and multiplication of the first two equations by $R_c$ give exactlyAdding the three original balances gives zero total volume change. This is a three-compartment ventricular circulation model.

Systole is ventricular contraction and ejection; diastole includes ventricular relaxation and filling. Let $V_a,V_v,V_{LV}$ denote arterial, venous and left-ventricular blood volumes. Let $Q^{+}$ flow through the aortic valve into the arteries, $Q_c$ through the peripheral circulation, and $Q^{-}$ through the mitral valve into the ventricle. Conservation of volume givesThe same $Q_c$ leaves arteries and enters veins because the capillary compartment has no changing stored volume in this approximation. Adding the equations conserves total modeled blood volume.

Use constant vascular compliances $C_a,C_v$, with ${\dot{V}}_a=C_a{\dot{p}}_a$ and ${\dot{V}}_v=C_v{\dot{p}}_v$, and ventricular elastance $p_{LV}=E_{LV}(t)(V_{LV}-V_0)$. The hydraulic resistances and ideal one-way valves givewhere $[z{]}^{+}=\max (z,0)$. Substitution into the conservation equations gives the two stated arterial/venous pressure equations. Since $V_{LV}-V_0=p_{LV}/E_{LV}$, differentiating this identity gives the third equation. The positive parts encode valve closure against a reverse pressure difference.

During isovolumetric contraction both valves are closed. Let $V^{+}$ be the end-diastolic ventricular volume, $V_0$ its unstressed reference volume, and $p_a^{+}$ arterial pressure before ejection. The rapidly raised elastance generates $E_s(V^{+}-V_0)$, so the aortic valve opens if $\boxed{E_s(V^{+}-V_0)>p_a^{+}}$.

During ejection, $R_a/R_c\ll 1$ makes the open-valve pressure drop small on the peripheral drainage time scale, so $p_{LV}\approx p_a$. The mitral valve is closed, and adding arterial and ventricular volume balances eliminates $Q^{+}$. With $p_v\ll p_a$ this givesFor approximately constant systolic elastance, set $C_s=1/E_s$ and reset time to the start of the slow ejection phase. ThenHere ${\hat{p}}_a$ is the matched pressure at that start. If peripheral outflow is negligible during the preceding rapid artery-ventricle pressure equilibration, conservation gives the more explicit ${\hat{p}}_a=[C_a{p}_a^{+}+V^{+}-V_0]/(C_a+C_s)$.

After aortic closure the ventricular volume is fixed at the end-systolic value $V^{-}$. Relaxation decreases $E_{LV}$ until the mitral valve opens at $p_{LV}=p_v$, hence $\boxed{E_{LV}\sim p_v/(V^{-}-V_0)}$, with $p_v$ the venous filling pressure at opening. During filling the aortic valve is closed, so $R_c{C}_a{\dot{p}}_a=-(p_a-p_v)\approx -p_a$. With $p_a^{-}$ the arterial pressure at the start of filling and time reset there,The change from $C_a+C_s$ to $C_a$ is physical: the ejecting ventricle shares the arterial pressure during systole, whereas a closed aortic valve disconnects it during filling.

The heart supplies the principal continuing mechanical input to blood circulation; arterial recoil, respiratory movements and the action of skeletal muscles redistribute and supplement that input. They have different sources of energy and should not all be called independent propulsive powers. The pressure gradient remaining after passage through the capillaries is still a consequence of the upstream pump.

Arterial elasticity. During ventricular systole, ejected blood expands the aorta and large arteries, storing elastic potential energy in their walls. During diastole, recoil continues to drive blood into the peripheral vessels while closure of the aortic valve limits return towards the heart. This reduces the intermittency of flow downstream. The recoil is a temporary store of cardiac work, rather than a fresh source of energy. A vessel that is too rigid provides less of this smoothing; a compliant vessel with an open backward outlet would not provide the same useful result.

The skeletal-muscle pump. Contracting skeletal muscles compress adjacent veins. Venous valves make the displacement preferentially towards the heart; when compression ceases, refilling permits another stroke. Walking therefore aids return from dependent limbs. The source of the added work is muscle contraction, and the valves guide it without producing energy. Merely possessing valves cannot make a stationary column circulate.

The respiratory pump. During inspiration, falling intrathoracic pressure and displacement of the abdominal contents favour a pressure gradient from abdominal and peripheral veins towards the thoracic veins and right side of the heart. The respiratory diaphragm thus participates indirectly in venous return as well as pulmonary ventilation. The exact flow varies with posture, vessel collapse and the phase of the respiratory cycle; positive-pressure inflation does not reproduce the same mechanical circumstances as spontaneous inspiration.^[ref]

Venous tone and cardiac filling. Sympathetic contraction of venous smooth muscle can reduce venous storage capacity and transfer blood towards the central circulation. This alters the volume available for filling rather than requiring a wave of venous peristalsis. Cardiac relaxation, recoil and movement of the atrioventricular plane also affect filling pressure. Some traditionally described cardiac suction is therefore a feature of the cardiac cycle and mechanical coupling, not a separate organ pulling blood through every vessel.^[ref]

Gravity and supposed capillary attraction. Gravity assists a downward column but opposes an upward one. In a complete circuit returning to its original height it supplies no net gravitational work; its important practical effects include hydrostatic pressure differences and dependent venous pooling. Ordinary capillary action in an open narrow tube, driven by a liquid meniscus, is not a general pump in a fluid-filled capillary network. Nor does a vague attraction of blood by tissues replace a measurable pressure gradient. Local vasodilation can increase perfusion by lowering vascular resistance, but that change does not manufacture the necessary pressure source.

For a simple mean-flow description, with $Q$ the volume flow rate, $\Delta P$ the driving pressure gradient and $R$ the effective vascular resistance,This relation distinguishes a source of driving pressure from a change of resistance. In a lumped compliant arterial reservoir with vascular compliance $C$ and no inflow between beats, volume balance giveswhere $P_v$ is the downstream pressure. The exponential decay follows from stored volume leaving through the resistance. It explains continued flow during diastole without claiming that the idealization describes every reflected arterial wave.

A physical demonstration can compare a pulsatile pump feeding a narrow outlet through rigid tubing with the same pump feeding it through an elastic reservoir. Collecting the outflow over the pumping and resting phases reveals the reservoir's smoothing action; a side-arm manometer records its falling pressure. Separately compressing a valved flexible tube shows the role of a peripheral muscle pump. Keeping pump stroke and outlet resistance comparable prevents the apparatus from conflating a smoother flow with creation of extra energy.

If an open aortic valve equilibrates pressure much faster than peripheral outflow, the arteries and ventricle share a common pressure while their combined blood volume is conserved during the short adjustment. Let arterial vascular compliance be $C_a$ and systolic ventricular elastance be $E_s$. Equating the initial and final arterial-plus-ventricular stored volumes gives the displayed pressure. Subsequent peripheral drainage sees the combined compliance $C_a+1/E_s$; during filling a closed aortic valve removes the ventricular contribution.

A lumped venous reservoir stores blood at a pressure determined by its volume and vascular compliance. If its inflow and outflow are $Q_{\mathrm{in}}$, $Q_{\mathrm{out}}$, the constant-compliance approximation gives $C_v{\dot{p}}_v=Q_{\mathrm{in}}-Q_{\mathrm{out}}$. It forms the filling side of a three-compartment ventricular circulation model.