= End-diastolic pressure-volume relationship
{title2=$p_{\rm ed}=f(V_{\rm ed})$}
The \b[<end-diastolic pressure-volume relationship>] describes ventricular <pressure> as a function of chamber <volume> at the end of <diastole> across different filling states. It is assembled from multiple end-diastolic observations, rather than from the entire time trajectory of one filling event. It is useful for describing passive chamber behavior when the activation state, pressure reference and external constraints are sufficiently controlled.
The local inverse slope is <ventricular compliance>:
$$
C_V(V)=\left(\frac{dp_{\rm tm}}{dV}\right)^{-1},\qquad
p_{\rm tm}=p_{\rm inside}-p_{\rm outside}.
$$
An intracavity-pressure curve and a <transmural pressure> curve can differ when surrounding pressure changes. Nor does a pressure-volume curve directly specify the material elasticity of the ventricular wall: chamber geometry also enters the relation.
For an original model, take $p_{\rm tm}=A[e^{k(V-V_0)}-1]$, with $A,k>0$. Then
$$
\frac{dp_{\rm tm}}{dV}=k(p_{\rm tm}+A),\qquad
\frac{d^2p_{\rm tm}}{dV^2}=k^2(p_{\rm tm}+A)>0.
$$
The upward curvature makes the chamber locally less compliant at larger volume without requiring any change in parameters. With $A=2$ mmHg, $k=0.025\,\mathrm{mL^{-1}}$ and $V_0=60$ mL, the pressures at 100 and 120 mL are 3.437 and 6.963 mmHg. Their secant estimate $20/(6.963-3.437)\simeq5.67$ mL/mmHg describes that finite interval; the local compliance at 120 mL is about 4.46 mL/mmHg.
The pressure-volume relation was tested against imaging in a primary isolated-heart experiment. A balloon provided known ventricular volumes, a micromanometer recorded <pressure>, and automated <echocardiography> measured short-axis <area> in six perfused dog hearts. Repeated volume increments supplied multiple pressure-volume and pressure-area pairs. Their reported agreement concerned that experimental preparation, with clinical application explicitly left unvalidated.<Estimation of left ventricular compliance using on-line echocardiographic automated border detection and pressure data by John Gorcsan III and others>[Abstract, experimental apparatus, repeated filling and validation limits]{ref}
Comparing curves also requires a size convention. If a model is written $p=f(V/V_s)$ with reference volume $V_s$, then $dp/dV=f'(V/V_s)/V_s$. Two geometrically scaled chambers can therefore have different dimensional slopes even with the same normalized relation. <Volume normalization of ventricular pressure-volume curves> separates that comparison from a claim about changed wall properties. One observed end-diastolic point cannot identify an arbitrary curve or all three parameters of the exponential model.
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