## Echocardiographic Ratio Indices

#### Introduction

#### Relative Ratio Indices / Fractional Shortening

*when there is nothing about the index that relates to the intrinsic size of the individual. Rather than trying to explain the characteristics of the class, examples include fractional shortening, ejection fraction, and regurgitant fraction. In fact the ones I employ are almost always a fraction of something; I include the word "fraction" or "fractional" in the name to try to set them apart. Here's the formula for the fractional shortening:*

**Relative Ratio Index**

*and yourself*by calling it something else. I was heartened to hear Dr. Jay Cohn give a presentation where he stated that he felt that indices like FS and EF are probably more specific for left ventricular

*remodeling*than "systolic function". I would embellish this by saying that the interpretation of EACH index is

**disease specific.**(e.g. a particular value for EF means something entirely different for chronic MR than for DCM.) Whatever the clinical finding/abnormality, these are

*of disease; it's up to us to determine how an index or finding is related to disease, prognosis, or treatment. I know this is obvious but we continue to goof this up so it's a lesson worth keeping right in your first 64K of RAM.*

**correlates**

*correlated*with others. (See below)

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

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

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 the 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:

#### Fractional Wall Thickness

Diastole

#### Normalization for Body Size

*justify*the ratio index method as a means of normalizing for body size. Click HEREto skip to the figures that will allow you to just start using the method. If you think the ratio index approach is slipshod, error-prone, non-analytical, or if you are still normalizing cardiac lengths or volumes by body surface area, then I suggest you delve into what follows.

*order solution as the exponent of the dependent variable is 0.) However when we start talking about dogs as a species, we are definitely going to have to do some analysis to arrive at a meaningful explanation for the variation of heart measurements with body size.*

**zeroth**

**MUST**approach the origin asymptotically. We can meaningfully have neither a body with no heart, or a heart with no body!

^{3}= 8! Body mass is most directly related to body volume, but we can't find the cube root of a mass; the physical units make this impossible. However we can find the cube root of a volume and we can calculate a volume from mass if we know the density of the object; the exact expression is \( M=\rho V\) where \( M\) is mass, \( \rho\) is density (mass/volume) and \( V\) is volume so \( V=M/\rho\) and the length of the cube edge is \( L=(M/\rho)^{1/3}\) ; this is the relation we will use to determine a representative length from the body mass. I don't know about you but my density is pretty close to the same as water and my volume varies a bit depending on whether I take a lung full of air. I float in water with a lungfull and sink if I exhale forcibly. Sure we vary in density, but the number can't stray very far. Water has a mass of 1 kg per cubic decimeter (1 decimeter = 10 centimeters ). Hence the length we obtain from taking the cube root of body mass (in kg) has real physical units. It is the edge length (in dm) of a cube of water having the same mass as the individual in question. I'm going to call this length L

_{W }-- a characteristic length derived from the body weight.

*first order***that dogs of variable size are geometrically similar, then we are forced to the same conclusion: linear dimensions of the heart are most directly proportional to the cube root of body mass. Plotting the same LVIDd data against L**

*approximation*_{w}(from the cube root of body mass) gives us the following:

*-- without physical units. We've also eliminated the unacceptable positive y intercept. This regression is through the origin so that we have 0 body size associated with 0 linear dimension as it must be. This last statement has a statistical implication also; if we are certain that all data must approach the origin asymptotically ( for very small body size, and yes we are sure ), then the confidence intervals must become narrower for small body size. The*

**a ratio***standard*linear least squares regression is actually invalid due to heteroscedasticity.

*to predict any linear (length) cardiac dimension from body mass:*

**first order approximation**

Over a short enough range of body size, cardiac length measurements are approximately proportional to the cube root of body mass. The latter has the interpretation of a length - the edge length (in dm) of a cube of water having the same mass as the individual. Hence a cardiac length divided by the cube root of body mass constitutes a ratio of 2 lengths. Such a ratio would be a constant if all individuals exhibited perfect geometric similarity. Variation and deviation from geometric similarity means that each ratio has a a range of expected values (e.g. mean, standard deviation) that can be determined statistically from normal individuals. Conformity with this expectation is an indication that the measured length is appropriate for the individual's body mass (assuming that the body mass is representative of the individual's intrinsic size).

\( k \Large\frac{LVIDd}{BW^{1/3}}=k \Large\frac{LVIDd}{L_w}=k(1.343\pm0.138)\)

AS AN AID to conceptualization, I chose k = 1.0/0.79 =1.27, the reciprocal of the slope obtained for the aorta (in the dog). For this particular choice we have (for the aorta):

\( \Large\frac{Ao}{0.79 L_w}\approx(1.0\pm0.113)\)

In consequence we can ** define**a more convenient (intuitive) characteristic length:

\( \Large\frac{LVIDd}{0.79 L_w}\equiv \frac{LVIDd}{Ao_w}\approx(1.716\pm0.177)\)

*arbitrary*! Its sole purpose is to provide a pleasing, intuitive interpretation for the ratio indices. Investigators had been using the measured aortic diameter as a length scale previously and I thought this would make it more intuitive to transition to weight-based ratio indices. I can tell you that this choice makes it a good deal easier to explain to people. Note that exact numerical values shown here are slightly different from those in my original publication; it's an entirely different data set! Nevertheless the value for k above, the slope for the aortic size conversion constant is the same number. The value for this constant is very consistent and does not vary with body size in the dog. Similarly I will the

**the weight-based aortic**

*define***(cm**

*area*^{2}, dog) as follows:

\( AoA_w\equiv\pi[\Large\frac{Ao_w}{2}]^2=\pi [\frac{0.79 L_w}{2}]^2\)

#### The Second-Order Approximation

_{w}= W

^{1/3},the characteristic length from the body weight, and C = B-1/3 above (I'm playing a bit fast and loose here, leaving the density out of the equation from the definition of Lw). If B is not equal to 1/3, then the value of C will be other than zero and we have a situation where L and L

_{w}are not proportional but exhibit a curvilinear relationship dependent on the weight, W; the function still passes through the origin. The first problem with the allometric approach is that B and C can't logically be anything but pure whole numbers from a purely physical argument; we can't find the square root of a length or exponentiate a mass to anything other than an integer number.

Next, let's look at some real data where LVIDd/L_{w} has been plotted against body weight. We are plotting a ratio of 2 lengths against body mass and this is one way to tell if there is any dependence on body size for this particular aspect of shape.

*expedient*that gives good (quite good) predictions for estimating dimensions but in general is not a good choice for describing changes in shape that occur with size.

Suppose we try a different approach and START with a plot of the ratio, LVIDd/Lw against body mass:

\( y=1.54-0.067 L_w\)

However the independent variable in this case, y, is LVIDd/Lw:

\( \Large\frac{LVIDd}{L_w}=1.54-0.067 L_w\)

\( LVIDd=1.54 L_w-0.067 L_w^2\)

Finally we substitute the previously obtained expression for Lw (including the density this time):

\( LVIDd=1.54(W/\rho)^{1/3}-0.067(W/\rho)^{2/3}\)

Or generically,

\( LVIDd=A (W/\rho)^{1/3}+B (W/\rho)^{2/3}\)

Note the following: 1)The expression is dimensionally sound. W/ρ is a *volume* so we're not trying to do any sort of impossible exponentiation. 2) A and B are both physically realizable with A being nondimensional (pure number) and B has units of 1/length. We can't ascribe any physical attributes to the regression parameters in the power regression. 3) The first term is an isometric term - the expected LVIDd you'd get if all dogs had the same shape. 4) The second term approximates the change in shape with body size.

That looks good except for one flaw: like the power regression, the equation predicts that the shape will go to 0 for large dogs and that the LVIDd will actually go negative. Clearly this approach too is limited in its range of application. To completely fix these anomalies we would need to use a function that is O(1) (Order one) to describe the shape. I'll leave that for another day.

#### Left Ventricular Dimension

#### Ratio Index Changes with Disease