= Ventricular compliance
{title2=$C_V$}
\b[<Ventricular compliance>] describes how much a ventricle's <volume> changes for a small change of distending <pressure>, with the mechanical state and other relevant conditions specified. For a passive chamber relation $V(p_{\rm tm})$, its local value is
$$
C_V=\frac{dV}{dp_{\rm tm}},\qquad p_{\rm tm}=p_{\rm inside}-p_{\rm outside}.
$$
The distinction between intracavity and <transmural pressure> matters when external pressure changes. A large local compliance means a relatively large volume increment for a small pressure increment; its reciprocal is the local chamber stiffness. It is a chamber property and need not equal a material property of the muscular wall.
For a linear model $p_{\rm tm}=(V-V_0)/C_V$, the derivative is constant. The quotient $V/p_{\rm tm}$ generally is not that compliance, because it also contains the reference volume $V_0$. Nor does a single pressure-volume point determine a slope. The <end-diastolic pressure-volume relationship> instead assembles measurements at a specified phase over different filling states.
An original nonlinear model makes this distinction visible. Take
$$
p_{\rm tm}=A\left[e^{k(V-V_0)}-1\right],\qquad A>0,\quad k>0.
$$
Direct <differentiation> gives
$$
\boxed{C_V=\frac{1}{k(p_{\rm tm}+A)}}.
$$
With stipulated $A=2$ mmHg and $k=0.025\,\mathrm{mL^{-1}}$, the local compliances at pressures 6 and 14 mmHg are 5 and 2.5 mL/mmHg. The second operating point is less compliant even though the model parameters are unchanged. These are explanatory values, not human reference ranges or a diagnosis.
\Image[/ventricular-compliance-operating-point.png]
{title=Different local ventricular compliance on the same stipulated passive pressure-volume curve}
{description=Original nonlinear model with local tangent lines at pressures 6 and 14 mmHg. Their inverse slopes give compliances 5 and 2.5 mL/mmHg. This illustrates operating-point dependence, not patient data or reference ranges.}
A primary experiment made the measurement problem concrete. Six excised dog hearts were perfused outside the body. An intraventricular balloon supplied a known volume, a micromanometer catheter recorded <pressure>, and <echocardiography> with automated boundary detection measured a mid-ventricular short-axis <area>. Repeated volume increments allowed pressure-area estimates to be compared with the balloon's pressure-volume reference. The reported agreement supported that experimental estimation method, while the authors explicitly left clinical validation open.<Estimation of left ventricular compliance using on-line echocardiographic automated border detection and pressure data by John Gorcsan III and others>[Abstract, apparatus, incremental filling and comparison of pressure-area and pressure-volume relations]{ref}
A pressure-volume trajectory during <left ventricular filling> also reflects time-dependent relaxation, valve flow and external constraints. Its instantaneous slope should not automatically be interpreted as the derivative of one fixed passive curve. Specifying the phase, pressure reference and conditions makes <ventricular compliance> a testable mechanical quantity rather than a synonym for filling volume.
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