Allometric, Isometric, Ratio Indices, Shape Factors, Etc.

Straight to the Heart

  • Methods (in veterinary medicine) to estimate echocardiographic length measurements typically employ some function of the body mass (weight \(\equiv W\) ) as the predictor, the independent variable of a regression formula.  
  • To predict a length, both \(W^1\) (weight) and \(W^{2/3}\) (body surface area) are inadequate as (linear) independent variables and lead to invalid regression formulae.  The approach violates a physical/mathematical principle; these predictors are not of the same order as the dependent variable and are not linearly related to it. 
  • \(W^{1/3}\) has the same order as length and \(1/3\) is the only exponent of \(W\) that achieves this.  As a physical quantity however, \(W\) cannot be raised to a fractional power.  An implicit step is necessary for mathematical and physical consistency, i.e. division of \(W\) by density ( \( \rho \)  ) which yields a volume, \( W / \rho \).  With physical dimensions of length cubed, the volume can be raised to any exponent \(n/3\) where \(n\) is an integer.  
  • A ratio of 2 lengths, 2 areas, or 2 volumes within an object defines a shape factor.  If only the size of an object is changed, a rescaled version results that is geometrically similar to the first but with all length, area, and volume ratios unchanged.  Shape factors are intrinsically independent of size; a change in size alone does not/cannot change a shape factor.  A collection of objects of differing size but the same shape are said to exhibit an isometric scaling law. A ratio is nondimensional and can be raised to any exponent without violating physical/mathematical principles.
  • \(W^{1/3}\) and \(W^{B}\) as predictor variables lead to improved regression results for prediction of a length where \(B\) is left as a regression parameter.  \( y = A \; W^{B}\), the allometric scaling formula, leads to improved regression results for prediction length \(y\) compared with \(y = A \; W^{1/3}\)    \( \Large \text{if} \)    \(B\) is statistically different from 1/3.  This occurs specifically if shape \( \large \frac{y}{W^{1/3}}  \) is correlated with (changes with) size.   
  • Implications of the allometric equation can be appreciated by rewriting the formula as \( A \; W^{1/3} \; W^C\) where \(C = B-1/3\).  Hence there is a part of the formula due exclusively to size change ( \(W^{1/3}\) ) and another that relates to changes in shape that are correlated with size (\( W^C\) ). This allows us to see that that the allometric formula also specifies the functional form of the shape change, \(W^C\), the allometric formula (again). It indicates that the shape factor decreases with size, if \(C <0\), ( \(B < 1/3\) ), and the reverse if \(C>0\) ( \(B > 1/3\) ).  It specifies that the shape factor changes allometrically, however, and this is not plausible.  For nonzero \(C\), the formula indicates that the shape approaches either \(0\) OR \(\infty\) for small size, and the opposite relationship for large size.  (The implication is that the measurement of interest is either infinitely larger or smaller than the individual.  Anomalies result because the formula does not accurately/appropriately represent the functional form of shape changes:
    • The value of \(C\) is entirely dependent on the range of body sizes included in a study. Consequently the allometric formula is not appropriate for comparing groups of different body size.  
    • This will be obscured (not obvious) by the fact that the value of \(B\) is "anchored" near to the isometric value (\(1/3\) for length measurements); it can't stray far from this number.
    • The allometric formula raises a physical quantity to a fractional power and so gives a non-physical description of size and shape relationships in the context of echocardiography.
    • The allometric formula is unsuited for determining regression parameters unless individual data are included from a subpopulation with a sufficient range in body size; the range required will depend on the amount of variation in the measurement.  It does a remarkably good job of representing the relationship between body size ( \(W\) ) and echocardiographic length, area, and volume measurements (considering that it misrepresents the functional form of the shape changes).  It should be kept in mind, however, that this is a statistical expedient.  Other (improved) approaches are suggested below.

Background

If you've been paying attention, you'll know that there has been some discussion in the past decade or so about how to normalize various types of measurements, particularly echocardiographic dimensions, for body size.  Some of the preceding articles at this website were intended to give some background on the subject.  In this article I'll try to be a little more direct, explain differences between approaches ( in my view of course ) and why I've advocated for an understanding of ratio indices.  

From a practical standpoint this is about estimating whether a measurement you've made pertaining to the cardiovascular system ( length, area, volume, velocity, etc., etc.) is appropriate for the individual in question.  The immense problem in cardiology is that each such measurement may depend on multiple independent variables ( size, age, heart rate, preload, afterload, etc., etc.).  We are focusing in this article on the body size issue which is enough of a problem to be getting on with in veterinary medicine, particularly in dogs where there is such a wide range of body size.

Whatever measurement you make, you may find that the measurement is correlated with body size, and this is certainly true of the length measurements made of the heart.  At the dawn of echocardiography, length measurements (e.g. LVDd, IVSs) were all made from M-mode recordings with physical units of centimeters or millimeters.  A zero(eth) order approximation or prediction for these measurements is simply that each measurement is NOT dependent on body size.  Then the prediction depends simply on some measure of central tendency ( mean, median ) and some measure of variability of the measurement ( standard deviation, quartiles, range, etc.).  This is a procedure that is still widely employed in human cardiology ( for some purposes ) and also in veterinary medicine if the range of body size is sufficiently restricted ( e.g. cats, specific breeds of dogs, horses, etc. ).  The problem comes in whenever we attempt to apply this approach to any group where body size varies sufficiently, in which case the value \(\pm\) range approximation is no longer an adequate description.  This would obviously occur whenever we look at changes that occur in growing animals, regardless of species, since we all develop from a considerably smaller size than the one we finally achieve.

If we look at a species that varies widely in size (dogs of course), we'll see that any length, area, or volume determination is correlated with body size; body weight for example.  But this will also be true for any group where body size ranges sufficiently; we have only to include a wide enough range of individuals to demonstrate this. It is immediately obvious that each length measurement made from an M-mode echocardiogram is correlated with body weight and this led early investigators to devise (compute) linear regressions of each length measurement against body weight.  The next figure shows an example plot of left ventricular internal dimension against body weight with a linear regression line and regression parameters included.

This is now a first order approximation 

\(\large y = m \;W + b\)

where body mass (\(W\)) has been used as the independent (prediction) variable.  Unfortunately, \(W\) is the wrong independent variable.  It may seem like we've accomplished something good here, but the linear regression is in fact invalid as are the confidence intervals for the predictions (not shown).  Notice particularly that the regression line has a nonzero y-intercept ( \( b > 0\) ) so that individuals with small body size always fall below the prediction line.  This is seen for every single regression of a linear dimension against body weight.  In fact these data must approach the origin asymptotically i.e. for very small animals; any regression model that does not approach a length  of 0 (or area, or volume) for a body weight of 0 is not physically plausible.  Sure you can dream up some grossly pathologic case that skirts this law, but it is of no importance to the discussion.

The essential failing of the approach is that the relationship between ANY body length and body weight is intrinsically nonlinearwe have only to include a sufficient range of body size for the predictions to fail.  In consequence we would see also that the regression parameters are completely dependent on the range of sizes included in the study; the slope of the line is large for a group of small individuals (with a small intercept) and the slope is smaller for a group of larger individuals (with a larger intercept).  Subsequently several investigators applied the age-old practice ( in cardiology ) of regressing the linear measurements against body surface area. These seemed to exhibit improvement relative to the regressions against body weight, but in fact exhibit the same old problem albeit to a lesser extent.

 

The Similarity Principle

The problem here is that these approaches are empirical and have no physical basis.  We'll try to get it right this time by starting with a physical conception to be used as a guiding principle.  If we start with a square box of a particular size ( on the left in the figure below ) and double all its linear measurements, we obtain the box on the right.

 

 

Although each length of the box has been doubled, we would find readily that each area has been multiplied by a factor of 4 (22) and the volume has increased by a factor of 8  (23).  Elsewhere on this website I have belabored these points extensively and I'll only refer you to that information rather than reiterate it here.  Take a moment to convince yourself that

  • Each/every length measurement on the large box is twice as large as the small box.  The RATIO of ANY 2 lengths of the large box is THE SAME as the RATIO of THE SAME 2 lengths in the small box.
  • Each/every area measurement on the large box is 4 times as large as the small box.  The RATIO of ANY 2 areas of the large box is THE SAME as the RATIO of THE SAME 2 areas in the small box.
  • Each/every volume measurement on the large box is 8 times as large as the small box.  The RATIO of ANY 2 volumes of the large box is THE SAME as the RATIO of THE SAME 2 volumes in the small box.

The principal I've just illustrated applies to all objects of the same shape.  Square boxes are pretty simple from a geometric standpoint, but we can readily imagine an object of any complexity being scaled up or down in size without changing any of the relative geometry.  Two such objects are said to be self similar and obey a similarity principle.  Using this principle, I'll simply state a few more facts without proof.

  • \( \LARGE \text{IF} \) 2 objects are self-similar, then the ratio of any 2 lengths, areas, or volumes within one will be the same (identical) in the other. (For complicated structures like the heart, this pure concept is of course corrupted by the technical issues of making the measurements and even defining measurements that are unambiguous.)
  • For a collection of objects that are self-similar, ANY such ratios are absolutely, intrinsically independent of size.  To say this another way: ratios of (definable) lengths, areas, or volumes cannot change with size; they can only change with shape.  Each aspect of shape is defined by such a ratio. (There may be an infinite number of appropriate ratios to define an object that's more complex than a box; it's up to us to define and incorporate useful ones into cardiology practice.)
  • For a collection of objects that are self-similar, any length IS  (exactly) proportional to the square root of ANY area; length IS  (exactly) proportional to the cube root of ANY volume; area IS  (exactly) proportional to volume raised to the 2/3 power;
  • I will indicate in passing that the reflexive practice of normalizing cardiac data by dividing it by body surface area is nonsensical and not scientific.  (It's atrocious math unless you're trying to normalize an area.)

The importance of the second bulleted item above cannot be overstated.  Ratios (of length, area, or volume) are absolutely, intrinsically independent of size.  This provides the basis by which we can make sense of our data.  If a ratio changes with size (is correlated with size), then that indicates a change in shape with size.  If a particular ratio is uncorrelated with size, then that particular aspect of shape is uncorrelated with size. You must not lose track of these facts!  If you wish to compare groups of different size on some aspect of the heart, you can ONLY compare ratios between the groups because that is the thing that is intrinsically independent of size.  

Understand that I am not implying that ratios will be the same between groups of animals (e.g. dog breeds) or that the ratios won't change with animal size.   But I am stating as fact that comparing ratios is a meaningful way to determine a change in shape, perhaps the only way.  

Length, Area, Volume from Weight (Mass)

In the process of deriving estimates of echo measurements we're going to use the body weight (mass) as the independent variable to represent body size, and raise it to various fractional powers.  Let's examine this practice.  If you've ever read any of my junk on ratio indices, you might remember coming across something like this:

\(\Large   y = \frac{x}{W^{1/3}} \)

where we're trying to index a linear measurement \(\large x\) by dividing it by body mass ( \( \large W\) ) raised to the 1/3 power.  Now take out your calculator and try to compute the cube root of a weight.........

What happened?  Did it work?  No it didn't!!  If you actually got something, it was by calculating the cube root of a pure number.  I'm betting there was no place on you device to enter a weight, because if you did that you probably just broke your calculator!  NOBODY can calculate the cube root of a weight. Weight is a quantity with physical dimensions of force i.e. mass multiplying acceleration or  M L/T2 where M is mass, L is length, and T is time.    In all my junk on this topic I've made it clear that we are using body mass to estimate an intrinsic volume of the individual.  This is only possible due to an approximation of constant density among the individuals under study.

\(\large V = \Large \frac{M}{\rho} \)

Volume has physical dimension of L3 and we can take the cube root of it and get a length.  So there is an implicit step in this calculation to derive a volume from the body mass; it's only because of this step that we can legally find the cube root of it since the cube root of a volume is a length.  It's a real (physical) volume and a real length!   If you're measuring body mass in kg, the length we get from this calculation is the edge length (in decimeters) of a cube of water having the same mass as the individual (a cubic decimeter of water has a mass of 1.0 kg so \( \rho = 1.0\) kg/dm.  Since it is a volume (from weight), we can also raise it to the 2/3 power (an area), 4/3, 5/3, -2/3, etc. without causing any harm.

For phase 2, however, we'll also be looking at formulae where we take this weight (volume) and raise it to some power that we obtained from a regression, say 0.271.  And here, you have overstepped your authority.  The only physical quantity that will take this kind of a beating is a pure number -- a dimensionless one -- and that's the only kind of an input your calculator will take.  If you've raised a weight or volume to a fractional power, you've entered the non-physical realm. 

Now if anyone actually reads this, I'm sure there will be some eye rolling going on.  Of course you can raise a weight to any power you want; it's all over the literature! Your statistician told you to do it!  Doesn't make it right.  I come from a culture (engineering = applied mathematics) where this sort of thing is frowned upon.   There are good reasons for adhering to this code!  That's why we have ultrasound machines, airplanes, bridges, .... \(\infty\).

Allometric vs Isometric

Isometric means "same measure" and allometric means "different measure" and we'll soon see how this distinction arises.  To jump right into it, the allometric formula  to predict a linear echo measurement from body weight (mass), looks like this:

\(\large y = A \; W^B \)  (Allometric scaling)

\(\large A\) and \(\large B\) are regression parameters, determined usually from a least squares algorithm; \( \large y\) is the physical measurement we're trying to predict, i.e. length (e.g. LVDd).  Alternatively we would be comparing a measured value of \(\large y\) to confidence intervals predicted by the regression at a specific value of \(\large W\).  As we''ll see, the corresponding isometric regression would be one where we claim to already "know" that B = 1/3 (if we're predicting a length).

\(\large y = A \; W^{1/3}\) (Isometric scaling; the value 1/3 pertains specifically to the situation where \(y\) is a  length)

This is actually the true first order approximation for some length, \(\large y\), not the linear regressions against \(W\) that were brought up at the top of the page.  This is obvious enough when written as:

\(\large y = A \hat{X}\)

with or without telling you that the independent (prediction) variable is:

\(\large \hat{X} = W^{1/3} \)

Other choices for \( \large \hat{X}\) are possible of course (e.g. height, femur length, crown-rump length), but the essential feature is that for a linear regression we must use a length as the independent variable to predict a length, an area to predict an area, etc.  Otherwise a nonlinear relationship between \(\large \hat{X}\) and \(\large y\) is assured.  We all know that there are potential problems with using W as the independent variable having to do with the body condition of the individual (from cachexia to obesity, chopped off a leg, etc.); we'll assume that we're using some sort of corrected body weight that appropriately represents to size of the individual.  \(W\) is a measure of global body size and is typically the most readily available data. 

Here again are the scaling laws:

\(\large y = A \; W^B\) (Allometric)

\(\large y = A \; W^{1/3}\) (Isometric for a length)

Both formulas have the inherent property that they approach the origin asymptotically, 0 length at 0 weight; that's a good thing as we've already seen. The allometric formula has 2 parameters to be determined from a regression, the isometric only one.  It's no problem having your computer do a little extra work, but this requirement could be either an advantage or a disadvantage depending on the situation or your point of view. The allometric formula requires a range of data in the independent variable \(W\) to allow a meaningful regression, the isometric  formula does not.  The latter "assumes" the way in which the data are related to the independent variable.   Consequently we don't need a large amount of data and we can determine the value of the regression parameter \(A\) from a limited number of data points.  This is ideal if, for example, we're looking at a group with a limited size range (adult cats, humans, or a specific canine breed).   The allometric formula has 2 parameters and might (should?) be expected to do a better job of predicting \(\large y\), that is if regression parameter \(\large B\) is statistically different from 1/3.  But now let's look at this in other terms; we'll divide each formula by \(\large W^{1/3} \):

\(\Large \frac{y}{W^{1/3}} \equiv \large \hat{Y} = A \; W^{(B- 1/3)} = A \; W^{C}\) (Allometric for a length)

\(\Large \frac{y}{W^{1/3}} \equiv \large \hat{Y} = A\) (Isometric for a length)

Here we've defined a new dependent variable, \(\large \hat{Y}\), that is the ratio of a measured length, \(\large y\), to \(\large W^{1/3}\).  \(\large \hat{Y}\) is the ratio of 2 lengths and we've determined previously that this is intrinsically independent of size.  The isometric formula indicates that this is equal to \(A\), a constant.  So the isometric formula is the zero(eth) order approximation of \(\large \hat{Y}\) the ratio - the shape.  It indicates that the ratio is a constant (it is independent of \(W\)).  The allometric formula on the other hand indicates that the shape, \(\large \hat{Y}\), changes with W.  Furthermore, the functional form of the relationship is indicated; it's the Allometric formula all over again but with a different value for the exponent, \(\large C = B - 1/3\).  In other words, we could start over again and simply write the Allometric formula (for a length) as:

\(\large y = A \; W^{1/3} \; W^C\)

To my way of thinking this shows off the formula for what it is. The isometric part ( \( A \; W^{1/3}  \)  ) describes the  change in \(y\) due to changing size;  the \(W^C\) part describes the change due to changing shape.  (The value of \(A\) actually affects both shape and size in the formula.)

Describing the Shape Change

But the question arises whether the allometric formula does a good job of describing the shape change.  Data for the following plots are contrived to exhibit features to be explained. We'll start with a baseline example that conforms to the isometric law; data have been specified with minimal variation of the independent variable initially so that issues will be more obvious.  On the left, a length measurement ( \(y\) ) has been plotted against \(W\) which gives the now familiar sideways cubic parabola.  There are 2 groups with different size ranges, but they both have precisely the same shape with regard to the particular measurement.  This is evident in the right hand figure where we now plot the shape itself, \(L/W^{1/3}\), against \(W^{1/3}\); the ratio was contrived to be precisely 2.0.  Regression results for the allometric formula are also shown on the left and the procedure correctly picks off the correct exponent of \(1/3\) for both, i.e. \(B = 1/3\).   On the right, the allometric formula also generates correct regression parameters; the value of \(C\) is \(B - 1/3\) i.e. pretty close to zero.

 

If the shape is allowed to vary with size however, the allometric regression gives unexpected results.  Once again the data on the left show the length measurement plotted against \(W\).  The plot on the right however shows that the shape is correlated with size (negatively correlated); in this case there is a linear relationship between the shape factor ( \(L/W^{1/3}\)  ) and \(W^{1/3}\) (by design).  The allometric regression generates parameters that adequately fit the data for both the length and the shape factor plots, but an essential problem is now evident; the regression parameters depend on the size inclusion range.  We get 2 entirely different set of regression parameters depending on the group. Note that the linear dependence of shape on \(W^{1/3}\) was contrived to demonstrate this fact and the degree of dependence (the slope the shape factor against \(W^{1/3}\) ) is much greater than would be typical than for data I've encountered.

The reason for the anomaly is obvious enough when plotted this way and harks back to the top of the page.  We're attempted to use an inappropriate functional form to fit the data.  Early investigators tried to use a straight line to fit a curve ( length plotted against \(W\) ). Here we've tried to use a curve ( \( A W^C\)  ) to fit a straight line.  The allometric formula indicates that the shape is headed off for \(\infty\) (or 0 ) at low values of \(W\).  This is mighty unlikely; it means that heart is going to be larger than the individual at a sufficiently small size (the ratio goes to \(\infty\) ), even though both are headed for 0.  Both the \(A\) and \(B\) parameters are affected so as to accommodate the data.  (Note: The regression parameters for the curves on the right side are available from the left. The red curve on the left is \( y = A \; W^B = 2.2831 \; W^{0.1846}\); so the red curve on the right is \( y = A \; W^{B-1/3} = 2.2831 W^{-0.1487} \). )

Admittedly, the data were contrived so that the shape factor varies linearly with \(W^{1//3}\) for illustrative purposes, and this ain't necessarily so for real data (but it is for data I've seen).  But would anyone intentionally choose an allometric function (\( A W^{C}\)  ) to fit the data points shown on the right above??  (Nope, data are linear).   So the worst case scenario is that the regression parameters of the allometric formula in this situation are an artifact of the range of data under study.   If the shape change is not represented appropriately as \(A W^C\), then the regression parameters do not represent intrinsic properties of the relationship and will depend strongly on the data range (the form of the regression is inappropriate). 

The next plots are just a recap of this situation but with more realistic data (still contrived).  All that's been done was to add some (Gaussian) variation to the data to show that outcome just described was not due to the lack of data variation in the above.  Here's the isometric case: 

 

And here's the case where the shape varies linearly with \(W^{1/3}\) :

 

How would we fix this issue?  One should actually plot the shape factor against \(W\) or \(W^{1/3}\), just as shown, in an attempt to determine the functional relationship between the predictor and the the shape.  In this case one would "guess" that there is a linear relationship between the shape factor and \(W^{1/3}\)   (except that I know exactly how the data were created).

\(\large \frac{y}{\hat{X}} = A + B \hat{X}\) where \(\hat{X} = W^{1/3}\) as before.

\(\large y = A \hat{X} + B \hat{X}^2 = A \; W^{1/3} + B \; W^{2/3}\)

This then would be the true second order approximation for these data and we would find that the regression parameters \(A = 2\) and \(B = -0.2\) represent the data irrespective of the data range chosen.  The point is you need to look at a plot of the shape factor to decide whether the allometric formula is a good idea or not.  It's also necessary to test the value of \(C\) of the allometric formula (\(B-1/3\)) statistically to determine whether the shape is significantly dependent on size.  This is simply accomplished by testing the shape factor data for correlation with \(W\) (or \(W^{1/3}\) ).

 My advice on this:

  • For comparing echo measurements between groups of animals where there is a "small" range of body size within each group, the ratio index concept is preferred (isometric scaling); it may be the only meaningful mechanism for comparison.  Note: the "vertebral heart score" in veterinary medicine is a ratio index.
  • Regressions using the allometric formula (2 regression parameters) may give improved predictions for echocardiographic length measurements predicted from body mass as compared with an isometric approach (1 regression parameter)   \( \large  \text{if} \)  the value of the allometric exponent is statistically different from \(1/3\).  Note: an allometric "vertebral heart score" would involve fractional powers of the vertebrae count.  If this doesn't sound rational to you, you're a ratio guy.
  • Regression parameters from allometric scaling cannot be used to compare groups of differing size; the regression parameter values will depend on the range of the independent variable.  The regression parameters do not represent a meaningful aspects of the relationships.
  • Reflexive indexing by body surface area is an abominable practice.  Go study your algebra.

 


 

 

 

 

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