End-diastolic pressure-volume relationship ($p_{\mathrm{ed}}=f(V_{\mathrm{ed}})$)

The 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: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 $p_{\mathrm{tm}}=A[e^{k(V-V_0)}-1]$, with $A,k>0$. ThenThe upward curvature makes the chamber locally less compliant at larger volume without requiring any change in parameters. With $A=2$ mmHg, $k=0.025\mathrm{m}{\mathrm{L}}^{-1}$ and $V_0=60$ mL, the pressures at 100 and 120 mL are 3.437 and 6.963 mmHg. Their secant estimate $20/(6.963-3.437)\simeq 5.67$ 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.^[ref]

Comparing curves also requires a size convention. If a model is written $p=f(V/V_s)$ with reference volume $V_s$, then $dp/dV=f^{\prime }(V/V_s)/V_s$. 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.

Volume normalization of ventricular pressure-volume curves expresses chamber volume relative to a stated reference scale before comparing pressure-volume relations. It can separate the effect of chamber size from differences in the shape of an end-diastolic pressure-volume relationship. It does not automatically remove effects of wall thickness, geometry, external pressure or active state.

Let $v=V/V_s$ for fixed positive reference volume $V_s$, and suppose $p=F(v)$. The chain rule givesThus equal normalized curves can have different dimensional ventricular compliances. The scaled compliance $C_V/V_s=1/F^{\prime }(v)$ has units of inverse pressure and compares the same normalized operating point.

For an original example, take $F(v)=2[e^{3(v-1)}-1]$ mmHg. At $v=1.2$, pressure is 1.644 mmHg and $F^{\prime }(v)=6e^{0.6}\simeq 10.933$ mmHg. Chambers with $V_s=100$ and 200 mL then have volumes 120 and 240 mL and compliances approximately 9.147 and 18.294 mL/mmHg. Their compliances differ by twofold while $C_V/V_s\simeq 0.09147\mathrm{mmH}{\mathrm{g}}^{-1}$ agrees. These are chosen model chambers, not reference values.

An affine convention $v=(V-V_r)/V_s$ must state both its origin $V_r$ and scale $V_s$. Choosing each curve's reference from the same data can also conceal the very shift being compared. Normalizing every single observation by its own volume makes every normalized volume one and destroys information about curve shape.

An experimental comparison needs independent volume information and a specified pressure reference. The 1994 isolated-heart study used known intraventricular balloon volumes, micromanometer pressure and image-derived ventricular area to compare pressure-volume and pressure-area measurements.^[ref] That apparatus illustrates why a geometric surrogate must be checked before it becomes the denominator of a normalization.

A transparent report identifies the reference volume, whether pressure is intracavity or transmural pressure, the cardiac phase and the operating points compared. A size-normalized chamber curve is still a chamber measurement; it is not by itself a material stress-strain relation for myocardium.