## Ejection Fraction

Armed with concepts from the foregoing, it's now possible to discuss some commonly employed cardiovascular parameters in a new light.  The ejection fraction (EF) has been a much touted indication of cardiac health with a great deal of effort devoted to its accurate determination.  We are now in a position to express the cardiac volume in terms of other physical measurements of its size:

$$\large EDV = SFd (LVIDd)^3$$
$$\large ESV = SFs (LVIDs)^3$$

EDV and ESV stand for LV end-diastolic and end-systolic volume.  As usual, LVID is used here to represent the left ventricular internal dimension with subscripts d and s to indicate diastole and systole respectively.  It's necessary to cube LVID to derive a volume from a length. SF in each case is a shape factor with subscripts d and s for diastole and systole.  Shape factors are pure numbers ( dimensionless ) and each equation has physical units of volume.     Let it be clear that there is no approximation going on here; each expression is an exact equation for the cardiac volume.  In other words:

$$\large SFd\equiv\frac{EDV}{(LVIDd)^3}$$
$$\large SFs\equiv\frac{ESV}{(LVIDs)^3}$$

where EDV and ESV are the actual volumes, however you got them.  We are now in a position to rewrite the ejection fraction (exactly) from the LVID measurement:

$$\large EF= \frac{EDV-ESV}{EDV}=\frac{SFd (LVIDd)^3-SFs (LVIDs)^3}{SFd (LVIDd)^3}$$

One of the ventricular dimensions can be expressed in terms of the other and the fractional shortening:

$$\large FS=\frac{LVIDd-LVIDs}{LVIDd}$$

$$\large LVIDs=LVIDd(1-FS)$$

$$\large EF=\frac{SFd(LVIDd)^3-SFs(LVIDd)^3(1-FS)^3}{SFd(LVIDd)^3}$$

$$\large EF=1-\frac{SFs(1-FS)^3}{SFd}$$

Here we see the ejection fraction expressed exactly as a function of the fractional shortening and 2 shape factors. The point of this exercise is to show that the inability of simplified formulas to accurately assess the ejection fraction is not due to the irregular shape of the ventricle. It's due specifically to changing shape of the ventricle between diastole and systole.  (Certainly there are technical contributions in actually making the measurements as well.)  If the ventricle were the same shape in systole and diastole, then the 2 shape factors would be the same (regardless of the shape) and the formula would reduce to:

$$\large EF=1-(1-FS)^3=3 FS-3 FS^2+FS^3$$

While on this subject, I'll point out that calculating the ejection fraction is no way to evaluate the accuracy of cardiac volume formulas.  The shape factors in this equation can be wildly inaccurate and still yield a believable ejection fraction as long as the shape doesn't change drastically so that the shape factors cancel in the numerator and denominator (approximately).   An accurate ejection fraction is no justification for the Teichholz formula in veterinary medicine.

It's certainly true that the more measured information is included, the better chance there is for an accurate volume determination.  As an illustrative exercise, I'll repeat the process above where we've measured a cross-sectional area instead of the simpler ventricular diameter; you get to choose long axis or short axis slice.

$$\large EDV = SFd_2 (LVAd)^{3/2}$$
$$\large ESV = SFs_2 (LVAs)^{3/2}$$

Subscript 2 for the shape factors is to distinguish them as completely different from the previous example.  LVA here represents left ventricular cross-sectional area; note again that there is no mathematical anomaly by raising an area to the $$\large 3/2$$ power.

$$\large SFd_2\equiv\frac{EDV}{(LVAd)^{3/2}}$$
$$\large SFs_2\equiv\frac{ESV}{(LVAs)^{3/2}}$$

$$\large EF = \frac{SFd_2 (LVAd)^{3/2}-SFs_2 (LVAs)^{3/2}}{SFd_2 (LVAd)^{3/2}}$$

$$\large F\Delta A\equiv \frac{LVAd-LVAs}{LVAd}$$
$$\large LVAs=LVAd (1-F\Delta A)$$

$$\large EF = 1-\frac{SFd_2}{SFs_2}(1-F\Delta A)^{3/2}$$

$$\large F\Delta A$$ here is to represent the fractional change in area as defined by the equation shown. Once again we see that it is the changing shape, as defined by the ratio of the 2 shape factors, that prevents us from obtaining the exact ejection fraction from the fractional cross-sectional area change.

If this shape factor concept is a little bit too obtuse for you, consider the example of a prolate ellipsoid of revolution where the LV cross-sectional area is measured from  the long axis image. The volume and area of the half ellipsoid:

$$\large V=\frac{2\pi}{3}r^2 L$$

$$\large A=\frac{\pi}{2}r L$$

The shape factor is the thing we multiply $$\large A^{3/2}$$ by to get the volume:

$$\large SF=\frac{V}{A^{3/2}}=\sqrt{\frac{2}{\pi}} \frac{4Lr^2}{3 (Lr)^{3/2}}$$

With D=2r, the shape factor simplifies to:

$$\large SF=\frac{4}{3\sqrt{\pi L/D}}$$

Substituting this into the previously determined formula for EF:

$$\large EF=1-\sqrt{\frac{Dd Ls}{Ds Ld}}(1-F\Delta A)^{3/2}$$

The term under the square root sign comes from the ratio of shape factors where subscripts d and s refer to diastole and systole as usual.  Other constants in the shape factor ratio cancel out when you do the math. Note that you would measure both LV area and length for the area-length method, but this determines the L/D ratio also.

$$\large A=\frac{\pi r L}{2}=\frac{\pi D L}{4}$$
$$\large D=\frac{4A}{\pi L}$$
$$\large \frac{L}{D}=\frac{\pi L^2}{4A}$$

#### Clinical Sidelight

Ejection fraction is an index that is sometimes used euphemistically for "systolic function" or even "contractility".    I'm borrowing some credibility from Jay Cohn M.D. in stating that EF may be more of an index of ventricular (dilatative) remodeling (not an exact quote from a recent conference I attended).   An engineer might say that the EF is the thing you multiply EDV by to get the stroke volume - not very imaginative but precisely correct.

EF is clearly related to disease severity for some conditions ( ischemic heart disease, dilated cardiomyopathy) but not others (e.g.  hypertrophic cardiomyopathy, chronic mitral regurgitation).  We now have a new plethora of indices with the advent of diastology and strain imaging.  These also will have a specific and limited range of applicability --  whatever they mean in a particular context will not necessarily extend to other situations.   What does it mean to have an abnormal (or normal) EF?  The answer to a question like this is always "It depends".