## Cardiac Size

Determination and interpretation of cardiac size is a subject that has been covered in many sources and in many different ways.  No doubt many will regard the following as archaic since more accurate methods are becoming available to determine cardiac volume and ejection fraction ( e.g. MRI, 3D echocardiography).   As usual, the purpose here is to explore underlying geometric and mathematical principles that govern the subject, not necessarily to devise the optimal volume formula.  Nevertheless, an understanding of these principles remains pertinent and impacts clinical interpretation.

$$\Large V_S=\Large\frac{4 \pi r^3}{3}$$

The volume of a sphere has been represented by VS.  A sphere has a single, constant radius, r.   Omitting the derivation, the volume of an ellipsoid (VE) is entirely analogous:

$$\Large V_E=\Large\frac{4 \pi r_1r_2r_3}{3}$$

Now there are three radii, all potentially different, corresponding to perpendicular axes of orientation of the ellipsoid.  As usual, there is a dimensionless shape factor, 4π/3 in this case, that multiplies a dimensional component that has physical units of volume where each radius has units of length.  The shape factor for the ellipsoid is identical to the sphere since, of course, there is the potential for the ellipsoid to actually revert to the purely spherical geometry if all the radii were the same.  It is also a simple matter to rewrite the volume of the ellipsoid in terms of a single radius (or any other characteristic length of the ellipsoid) by modifying the expression for the shape factor:

$$\Large V_E=\left [\Large\frac{4 \pi}{3}\left (\Large\frac{r_2}{r_1}\right ) \left (\Large\frac{r_3}{r_1}\right )\right ]r_1^3$$

Be sure to recognize that this formula is identical to the previous one.  The term enclosed in the brackets is the nondimensional shape factor; while ratios of measured dimensions appear in the formula, the physical units cancel in each ratio.  It will be obvious that it is arbitrary which of the radii is represented by r1 and so the formula could be written in terms of any one of the radii.  Subsequent steps in approximating left ventricular volume as an ellipsoid are misrepresented in countless cardiology texts and so each step is shown in the following.  First, the two short axis radii are assumed to be equal and are expressed in terms of the measured diameter, r = D/2.  The third radius is the length of the ventricle, measured from apex to mitral annulus:

$$\Large V_E=\Large\frac{4 \pi r_1r_2r_3}{3}=\Large\frac{4 \pi}{3}L \left [\Large\frac{D}{2}\right ]^2 =\Large\frac{\pi}{3}L D^2$$

We now make a critical assumption/approximation about the geometry, that the length is twice the diameter; L = 2D:

$$\Large V_E=\Large\frac{4 \pi}{3}2D \left [\Large\frac{D}{2}\right ]^2=\Large\frac{4\pi}{3} \Large\frac{D^3}{2}$$

The left ventricle is one half of this full ellipsoid figure:

$$\Large V_{E_{1/2}}=\Large\frac{2\pi}{3} \Large\frac{D^3}{2} = \Large\frac{\pi}{3}D^3 \approx D^3$$

This is the formula most commonly shown, but mathematical errors are typically included in the derivation.  Alternatively, if we assume that L = D, then the V ≈ D3 formula corresponds to the full ellipsoid as shown below.

The volume of these two figures is mathematically identical, but the geometric assumptions are not.  Another misinterpretation that appears occasionally is that the D3 formula (shape factor = 1) assumes that the ventricle is a sphere.  The volume of a sphere is 4πr3/3 = 4π(D/2)3/3 = 4π/8/3 D3 = π/6 D3 The shape factor multiplying D3 for a sphere is π/6 ≈ 1/2 (not 1).

Area-Length Method

It may not be apparent that the so-called area-length method also relies on prolate-ellipsoid geometry for its derivation.  From the above example, the volume of the full ellipsoid is as follows:

$$\Large V_E =\Large\frac{\pi}{3}L D^2$$

Then a half ellipsoid is:

$$\Large V_{E_{1/2}}=\Large\frac{\pi}{6}L D^2$$

The cross-sectional area of a half ellipsoid is:

$$\Large A_{E_{1/2}}=\Large\frac{\pi}{2}L \Large\frac{D}{2}$$

This would be the area traced out on the 2D image of the left ventricle.  The area squared:

$$\Large [A_{E_{1/2}}]^2=\Large\frac{\pi^2}{4}L^2 \Large\frac{D^2}{4}=\Large\frac{\pi^2D^2L^2}{16}$$

A simple division and algebraic manipulation follows:

$$\Large \Large\frac{V_{E_{1/2}}}{[A_{E_{1/2}}]^2}=\Large\frac{8}{3\pi L}$$

$$\Large V_{E_{1/2}}= [A_{E_{1/2}}]^2 \Large\frac{8}{3\pi L} \approx 0.85 \Large\frac{[A_{E_{1/2}}]^2}{L}$$

While the area has been measured directly from the long axis 2D image ( no geometric approximations ), it's assumed in the derivation that this area comes from a prolate ellipsoid, specifically a 1/2 ellipsoid with radii D/2 and L.  We see that the physical units are squared area divided by length ( [L2]2/L ) which has the appropriate physical units of volume (L3) with a shape factor of 8/(3π).

The well-known Teichholz formula is the result of applying the prolate ellipsoid formula while evaluating the L/D ratio as a function of cardiac dilation.  The half ellipsoid was employed so that the volume formula is:

$$\Large V_{E_{1/2}}=\Large\frac{\pi}{6}L D^2=\Large\frac{\pi}{6}\left [\Large\frac{L}{D}\right ] D^3$$

For people with dilatative remodeling of the left ventricle, it was found that L/D changes significantly with cardiac dilation so that the ratio becomes smaller, i.e. the left ventricle becomes more globoid with dilation.  A curve fitting procedure yielded the following formula:

$$\Large \Large\frac{L}{D}\approx \Large\frac{13\Large\frac{1}{3}}{D+2.4}\$$

Substituting this function into the ½ ellipsoid formula yields the Teichholz formula:

$$\Large \Large\frac{\pi}{6} \Large\frac{L}{D}D^3\approx\Large\frac{\pi}{6} \Large\frac{13\Large\frac{1}{3}}{D+2.4}D^3\approx \Large\frac{7}{D+2.4}D^3$$

The shape factor function, $$\Large\frac{7}{D+2.4}$$ , is shown below:

Note that this formula has the usual format with a shape factor that multiplies a dimensional component, D3.   To be consistent, the shape factor must be nondimensional and so it is assumed that the constants that appear in the formula must have units of length, specifically centimeters.

Notice in particular that the expression for the shape factor, L/D, is a function specifically of D, the short axis diameter of the left ventricle.  A normal value for D  in this function should  return a normal value for the shape, L/D.  Consequently, the Teichholz  shape function is specific to human sized left ventricles.  As written, the Teichholz formula is misapplied to nonhuman left ventricles and its application to veterinary species is entirely unjustified.  On inspection of the formula, it is apparent that the shape factor is a direct function of the measured diameter.  This is a meaningful way to convey ventricular dilation for a human sized left ventricle only.   If D were 5.4 cm, for example, this would be a normal size (and shape) for a human ventricle, but gross dilation for almost any dog or smaller-than-human mammal.  The following figure conveys the shape that is depicted by the Teichholz formula for a range of relevant mammals, i.e. with approximate expectations for ventricular diameter inserted into the Teichholz formula.  The misrepresentation is rather dramatic and, of course, more extreme for species with body size unlike the human.  The formula overestimates ventricular volume for species smaller than human, and underestimates it for larger species.

Use of the Teichholz formula for left ventricular volume estimation in nonhuman species is wholly unjustified and is an example of failure of the veterinary peer review system.  If forced to review your manuscript where this formula is inappropriately applied, I will reject it.

In the hope of further understanding the most essential inconsistency of this formula (there is more than one), let's simply fix it:

$$\Large V \approx \Large\frac{7}{\Large\frac{\bar{D}_{H}}{\bar{D}_{S}}D+2.4}D^3$$

In this corrected formula, $$\bar{D}_{H}$$ stands for the normal or expected value of D in humans, e.g. approximately 5.4 cm depending on the source, and $$\bar{D}_{S}$$ is the expected value in the species of interest.  Observe that if the human is the species of interest, then the ratio of $$\bar{D}_{H}/ \bar{D}_{S}$$ that appears in the denominator is 1.0 and the original formula is recovered.  This ratio rescales the formula so that a baseline normal value for the shape factor is returned whenever  $$D=\bar{D}_{S}$$ .  However, the value of $$\bar{D}_{S}$$ is entirely dependent on the size and species of the animal being studied.  It's a different value of $$\bar{D}_{S}$$ for every single body size!   We would need to replace $$\bar{D}_{S}$$ in this formula with a formula that returns the appropriate value of $$\bar{D}_{S}$$ as a function of body size, e.g. $$\bar{D}_{S}(W)$$ where W is body weight or some other suitable measure of body size.

The above serves to reveal  the most significant inadequacy of this formula, but  its application to veterinary species has never been validated in any way.  It seems reasonable that a canine or feline heart undergoing dilatative remodeling might change its shape in a way that is geometrically similar to the human.  However this is a point for investigation and the application of the formula to animals serves no purpose, clinical or scientific.

### Clinical Sidelight

Even if applied only to the correct species, the Teichholz formula uses only a single length measurement (LVID) to estimate ventricular volume.  That means there is a one to one relationship between the volume determined by the formula and any other volume that employs just the one measurement.  Here's the relationship between the Teichholz volume and the D3 volume:

One of these volumes might be a more accurate prediction of the actual ( depending on whose ventricle you're examining ), but it's unlikely that one is going to be clinically superior to the other; they both contain the same limited information!  For a particular clinical "cutoff" value of the Teichholz formula, there is an exact corresponding cutoff value for the D3 formula (coming from the same value for LVID) and you haven't accomplished anything by going through the gyrations.  Because the two volumes are nonlinearly related, you might find a slight statistical advantage of one relative to the other in predicting clinical outcome, but your money will be better spent elsewhere, i.e. to include other information into your predictions.   Similarly, I don't see any clinical advantage to calculating the volume at all -- just use the measured value of LVID. ( but normalize it properly!)

From my perspective, the only application of this formula is for educational purposes – to illustrate mathematical misconceptions.