## Echocardiographic Ratio Indices

#### Introduction

Ratio indices are used throughout medicine to normalize clinical data with respect to one or more independent variables.  Well-known examples include urine protein/creatinine ratio, ejection fraction, valvular regurgitant fraction, and others.  Normalization for body size is a common clinical problem most dramatically represented by the range of canine breeds which span almost 2 orders of magnitude in body weight.  While not referred to as such, examples of  radiographic ratio indices include evaluation of central pulmonary vessels size by comparison to rib width, kidney size by comparison with vertebral bodies, and of course the vertebral heart score in cardiology.  I call the ratios Echocardiographic Ratio Indices (ERIs) when the measurements are derived from echocardiography.  The term  "normalization" is used here NOT to indicate that data have been transformed so as to obtain a Gaussian distribution, but rather to express data in such a way that dependency on a one or more variables has been removed or reduced.
I coined the term "ratio index" in an attempt to draw attention to a class of indices meeting a very specific criterion.  Each is a ratio of 2 values having the same physical dimensions ( or the ratio can be expressed in such a way as to fulfill this criterion).  So for example a urine protein/creatinine ratio is a ratio of 2 chemical concentrations; it actually does not matter if we express the protein concentration in mg/mL and the creatinine concentration in  moles/L as long as we agree how the ratio will be expressed.  Of course it makes much more sense to express both in mg/mL in which case the ratio is nondimensional - having no physical units.  As a counterexample, cardiologists commonly attempt to normalize ventricular volume, dividing by the estimated body surface area to obtain a so-called volume index.  This ratio has physical units of length, is NOT a ratio index, and CANNOT normalize the ventricular volume for body size although it can appear to do so over an appropriately short span of the independent variable (estimated body surface area).  Another criterion for a ratio index is that it is of Order one (O(1)) with respect to the variable(s) for which it normalizes.  This concept will mean that the index value can neither grow or decrease boundlessly with the independent variable; elaborated upon elsewhere.

#### Relative Ratio Indices / Fractional Shortening

I apply the term Relative Ratio Index 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:

$$FS = \Large\frac{LVIDd-LVIDs}{LVIDd} = 1-\frac{LVIDs}{LVIDd}$$

I use LVID to stand for the internal dimension of the left ventricle since we are also going to be talking about the outer dimension separately (LVOD). I occasionally hear cardiologists (and certainly non-cardiologists) refer to this as an index of contractility which I do not agree with.  From a purely descriptive standpoint, this is a measure of left ventricular deformation; if we could rely on a circular cross-section for the left ventricle, FS would be essentially the same as the circumferential strain at the endocardial surface.  Others refer to this as an index of "systolic function/dysfunction"; I do this too in discussions with non-cardiologists. However my preference is that if you mean "fractional shortening" you should say "fractional shortening".  It only takes a little bit more saliva to do this and avoids confusion whereas you may very well be misleading others 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 correlates 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.

The following figure suggests a range of fractional shortening where the cross sectional appearance of the left ventricle has been suggested by a highly stylized cartoon, i.e. conveying a purely circular cross-section that does not change shape with contraction.  If it really was that way, then a simple M-mode measurement through the center of the chamber would convey all the information present in the two-dimensional slice.  We know that's not the way it is in reality, but the one-dimensional measurements and indices are still among the most repeatable determinations made from echocardiography.  Please don't disregard the the ERI concept simply because I have focused on M-mode here.  I'm very particular about how the one-dimensional measurements are made and I'm not blind to the multidimensional aspects.
The left hand side of the figure displays the same diastolic short axis LV cartoon over and over again.  The proportions for that figure come from normal values in the dog but we would find that this is similar for many mammalian species.  The right hand side shows the LV cartoon at end systole with fractional shortening displayed both in terms of standard deviations ( for the dog from a specific study, i.e. mine ) and in terms of an absolute number for the fractional shortening. I remind you of the fact that FS is computed without reference to the size of the individual although we will see that there is a mild negative correlation with body size in the dog.

Cardiologists will look at this figure and surmise that something is not quite right.  That's because the experienced eye is used to the fact that changes in fractional shortening are most typically accompanied by changes in the diastolic structure of the heart as well -- dilation.  Hence the figure with a FS of 0.15 would more typically look like a more dilated heart – one with a lower fractional wall thickness.  There is probably a lot more to this than you have ever considered!  Even within the range of normal cardiac structure, certain aspects of cardiac shape are correlated with others.  (See below)
Fractional shortening is by definition a relationship between the cardiac internal dimension from diastole to systole.  There's nothing about the index relating to the external diameter of the heart and the fractional shortening of the external dimension is not at all the same thing. Consequently I had to simply invent ("fudge") something when creating the figure above.  However I did not do this arbitrarily.  The left-hand ( diastolic ) figure shows the same configuration over and over again, that is a normal external dimension in relation to the internal one  -- that is a normal fractional wall thickness.  To create the right-hand side of the figure, I had to employ some other assumption to determine the external dimension and it's this:  the cross-sectional area of the myocardium doesn't change from the left to right-hand side.  Hence all of the figures have exactly the same myocardial wall area ( the area of the gray doughnut ).  Is that a reasonable thing to assume?  Actually not – but it makes a plausible appearing figure and it's worth considering that assumption all by itself.

Myocardium is made up of tissue that is very nearly incompressible.   We know that the mass of the myocardium doesn't change between systole and diastole and so the cross sectional area of myocardium doesn't change – well, not very much.  But it's worth considering why it changes at all on a 2D echocardiogram. There are three reasons (that I know of): 1) The heart has actually moved ( i.e. in the longitudinal direction ) so that we're not slicing through the same ring of myocardium for both systole and diastole.  2) While myocardium itself is incompressible, some of the blood within the blood vessels is "wrung out" of the tissue in the process of contraction.  This relates to the so-called "erectile" properties of the tissue.  3) Because the heart also contracts in the longitudinal direction, it "gathers" myocardium into the scan plane in the process of contraction.  The LV short axis wall area increases a little bit in the process of contraction as a result. It's worth considering the latter very closely – this is how speckle tracking software is able to estimate deformation perpendicular to the scan plane when determining myocardial strain.  In fact we can readily come up with a global index of longitudinal shortening:

$$\Large\frac{WAs-WAd}{WAs}$$

Here WA stands for wall area – the cross sectional area as suggested by the gray doughnuts above. This is actually a poor man's index of longitudinal shortening that I looked at for a while to see if it was going to tell us anything we didn't already know. It's interesting to calculate but I didn't find it very useful for clinical purposes and we have much better means to evaluate longitudinal contraction now.

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

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

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:

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

Diastole

Systole

#### Normalization for Body Size

In this section, I will use up a significant quantity of your computer screen to 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.

For the next group of ratio indices, we would like to figure out a way to normalize measurements of the heart with respect to the individual's body size.  One approach to this would be to .... well, not do it at all.  We would simply say that all the hearts are the same size so we end up with the same expectation (mean value) for LVIDd; each cardiac measurement would have an expected range independent of the size of the individual. This is certainly a common approach in adult humans and veterinarians have employed it for cats, specific breeds of dogs, and other situations where we can expect that there isn't going to be a "large" range of body size. (This is called the zerothorder 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.

If we gather data from enough individuals, we see that there is a correlation between body weight (W, i.e. mass in kilograms for example ) and virtually any linear measurement of the heart ( in centimeters for example ).  Consequently investigators came up with linear regressions for each of the measurements;  data plots with a regression line typically looked something like this:

This is LVIDd (cm) plotted against body weight (W = mass in kg) for a rather large group of dogs with raw data contributed to by several veterinary cardiologists. The data set contains several specific breeds ( e.g. greyhounds, Irish wolfhounds, dachshunds, .. ) but also some generic, nonspecific breed groups.  There are over 600 individuals represented on the plot and we have an advantage that early investigators did not, that of being able to see data for an extremely wide range of body size.  This illustrates a consistent feature of all the plots, whenevera linear dimension is plotted against body weight; data are not linearly related to body mass.  The plots all suggest a curvilinear relationship that is concave downwards, i.e. with a greater ( positive ) slope at small body size than at large body size.

The plot shows the linear regression line that is obtained from these data.  Again we have the marked advantage over early investigators of having a large data set to illustrate the issue, but the regression line is highly inadequate to describe these data.  In fact we would have to say that this is a statistical failure; no statistician would allow you to get away with describing these data with a straight line because the rules of the game have been violated!  This is particularly evident towards the small dog side of the plot where the regression line misses the data entirely!  There is actually more than one statistical problem with the standard linear regression procedure for this situation.

With enough data to observe, it's clear that this linear regression procedure is inadequate ( it's just wrong ).  You would be fooled into thinking it's okay whenever you apply the procedure to a data set with a "small" range of body size.  However you would find that the slope of a linear regression line varies greatly depending on whether you are looking at a group of large dogs versus small dogs.  Without being able to see the big picture as we do here, the one big clue that this isn't working is that the regression line always has a positive y-axis intercept.  The implication is that as body size approaches zero, linear cardiac measurements approach some positive value -- the heart is bigger than the body. This is completely implausible of course, but also is a clue to the form of a statistically and physically meaningfull regression; the data MUSTapproach the origin asymptotically.  We can meaningfully have neither a body with no heart, or a heart with no body!

The basic relationship between body weight and linear dimension is apparent from a momentary consideration of geometry ( above ).  If we "double" the size of a cube, i.e. multiply the length of each edge by a factor of 2, the volume of the cube increases by a factor of 23 = 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 LW -- a characteristic length derived from the body weight.

$$L_w=(W/\rho)^{1/3}$$

If we make the first order approximation 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 Lw (from the cube root of body mass) gives us the following:

This transformation has clearly removed the great majority of nonlinearity from the plot.  The slope of the line (~1.32) clearly has physical units of cm/dm; this could obviously be expressed as 13.2 cm/cm and so is a pure number -- a ratio -- 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 standard linear least squares regression is actually invalid due to heteroscedasticity.

The above plot is from contrived data to illustrate the heteroscedasticity issue.  For small size, the relationship between the dependent and independent variables becomes more refined near the origin.  In fact what we have is a situation where each data point prescribes the slope of a line ( a tangent ) that necessarily passes through the origin.  What we need to describe this situation statistically is a simple average of the slopesto represent the slope of the regression; this turns out to be identical to a "weighted" linear regression where data points close to the origin have a greater effect on determining the outcome of the regression. 95% confidence intervals for the prediction are also quite a bit different than a standard linear regression as suggested by the figure above ( red represents the standard linear regression with 95% confidence intervals shown dashed, green is the weighted regression ).  When we carry out the weighted linear regression for the LVIDd data, a slightly different slope is obtained compared to the previous plot and confidence intervals are more realistic, finally making this a useful procedure for determining whether LVIDd measurements are within an appropriate predicted range.

This same procedure can be carried out for any of the linear(distance) measurements obtained from M-mode, 2-dimensional, or 3-dimensional echocardiography. Here's the plot for the M-mode aortic dimension from the same dogs as used for LVIDd above.

The following is the first order approximationto predict any linear (length) cardiac dimension from body mass:

$$L=k {W}^{1/3}=k L_w$$

In this expression, L is the cardiac length you wish to predict (LVIDd for example), and k is the proportionality constant to be determined from experimentation. To my way of thinking, there is NO OTHER CHOICE for the exponent of the body weight than 1/3.  ANY other exponent results in mathematical anomalies where the heart either exceeds the size of the body or becomes vanishingly small for a sufficiently small body size.
If the first-order approximation is sufficiently good, because of the reasonable degree of linearity shown, we can express these relationships as a simple ratio, e.g.:

$$\LARGE\frac{LVIDd}{BW^{1/3}}=\frac{LVIDd}{L_w}=1.343\pm0.138$$

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).

Now (please PAY ATTENTION), we are free to multiply this expression by anyarbitrary constant:

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

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 definea more convenient (intuitive) characteristic length:

$$Ao_w \equiv 0.79 L_w$$

and for LVIDd:

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

Using this arbitrary scale factor allows us to ascribe a visual interpretation to each index; the ratio of the measured dimension divided by the expected aortic diameter at the same body weight.  Please note: 1) the accuracy of the ratio index has nothing to do with the accuracy of the aortic dimension determination; the multiplication constant is 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 define the weight-based aortic area (cm2, 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

An alternative approach to the above would be to "play dumb", skip the analysis part, and go straight to an exponential regression.  The chore is to find the best regression using an equation of the following form:

$$L=A W^B$$

L is the cardiac length ( or area, or volume ) that we wish to predict, W is the body mass (e.g. in kg ), A and B are regression parameters to be found using some kind of least squares procedure.  What we're saying here is that maybe the theoretical 1/3 isn't the best number to use for B; maybe we should try to find a better one statistically. This approach has been used for a number of studies and is vastly superior to the linear regression against body weight shown at the top of the section.  I know that a subset of veterinary cardiologists have sided with this procedure as a method to obtain an improved prediction for cardiac dimensions etc.  This equation has been used in physiology relating to allometric scaling laws and you will find that the link to Wikipedia recapitulates some of the statements on this page.  Your statistician will love you for taking this approach and I certainly can't fault it from the statistical/prediction standpoint.

However I would like to point out a couple of issues with this approach. First, let's just rewrite the above as follows:

$$L=A W^B=A W^{1/3+(B-1/3)}= A L_w W^{B-1/3}$$

$$L=A L_w W^{C}$$

Here Lw = W1/3,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 Lw 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/Lw 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.

The data show that there is a mild decrease in the ratio with increasing body size.  This is telling us something important -- that larger dogs have a proportionally smaller left ventricular internal dimension to some extent ( based on this ratio ).  We can also see that the effect is mild; the plot quantifiesthis aspect of shape change.  There are other ways to get a handle on this fact, but the above is a good one.

The plot also includes a power law regression line.  This regression is the change in shape that you've implied by using the allometric scaling law, i.e. the exponent of C in the above equation is -0.043 because B didn't come out to exactly 1/3.  If this exponent is statistically different from zero then it's telling us that there is some degree of shape variation with body size.  If you work with power regressions regularly, then you would have been able to tell that the shape (the ratio of the two lengths) decreases with body size by seeing that the exponent is negative.  However lookat the form of the regression that we have chosen to represent this shape change.  I doubt that anyone would preferentially choose the curvilinear power law function to fit the data shown on the plot.  Certainly the regression line is reasonable for the data shown, but it isn't preferable to a simple linear relationship.

In fact you can see towards the left hand side of the plot where the regression line starts to turn upwards. This looks like a mild anomaly, but the regression line is actually headed for infinity as body weight approaches zero.  The implication is that the heart would be much larger than the body as body size continues to decrease. If the exponent C came out slightly positive, then the implication would be that the heart would disappear with decreasing body size.  The power law (allometric scaling) is a statistical 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:

This looks like it can be reasonably represented by a linear regression to me although there may be some heteroscedasticity issues (more data variability towards the small dog end of the plot).

Or how about this (even better).  Here the shape, y, appears to be well represented by a linear regression:

$$y=1.54-0.067 L_w$$

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.