The Order of Things

WAIT! This article addresses a topic that is not well understood by many clinicians! It's concerned with the behavior of functions and how they relate to the physical world. You will likely need a foundation in these issues to understand the circulation, many indices used in clinical cardiology, math in general, and certainly other articles on this website. We use many functions and formulas in cardiology and we'll be exploring some problematic examples subsequently. However we will need to cover some background material first.

Mathematics is a language and, like any other, you will become proficient in it by reading, speaking, and writing it. Part of this article is simply a primer on how to start looking at mathematical expressions - to evaluate their meaning and form - to analyze them. Mathematical functions are (often) expressions of the way in which a dependent variable(s) changes with one or more independent variables. Here comes one now:

\( \Large y_1(x) = m_1 x + b_1 \)

 

In the foregoing, \( y_1\) , is the dependent variable whose value depends on the independent variable, \( x\) . Including \( (x)\) after \( y_1\) is a way of indicating the dependency explicitly; this is sometimes omitted if the form of the function makes it obvious enough. \( y_1(x)\) also depends on the values of \( m_1\) and \( b_1\) , but these are intended to be constants for the given situation; they are function parameters, not variables.


It's readily apparent that \( y_1(x)\) grows in an unbounded fashion (without limit) with increasing value of \( x\) . Not only that, it grows in a particular way i.e. linearly; an incremental increase in the value of \( x\) results in a linear increase in the value of \( y_1(x)\) through the slope parameter, \( m_1\) . Clinical researchers tend to be overly expectant of linear relationships whereas researchers in the physical sciences rely much more heavily on physical arguements to determine the mathematical form of functions.

We say that \( y_1(x)\) is of Order \( x\) (for large \( x\) ) in the above situation and this is represented notationally as \( y_1 = O(x)\) where the latter is known variably as big-O, big-Oh, Landau notation, Bachmann-Landau notation, and others. It DOES NOT matter what the values of \( m_1\) and \( b_1\) are in this context. The statement \( y_1 = O(x)\) captures something very important about the grand scheme of this function's behaviour. It's a kind of approximation but it's still a rigorous statement! The concept expressed is paramount throughout the physical sciences and relates to everything from determining whether a system is stable (global warming?) to whether the universe will eventually collapse into a singularity (the Big Squish).


Consider now a second function:

\(\Large y_2(x) = m_2 x^2 + b_2\)

Examples of both functions are plotted above for arbitrarily chosen parameters. This second function is of Order \( x\) squared, i.e. \( y_2(x) = O(x^2)\) in Landau notation and it also grows monotonically ( in one direction only ) with increasing \( x\) if \( x > 0\) . However, it does so in an entirely different way than \( y_1(x)\) did. For positive values of the constants (\( m_1, m_2, b_1, b_2\) ), \( y_2(x)\) will eventually outstrip \( y_1(x)\) with increasing \( x\) , and it will do so spectacularly!

We'll explore the size relationship of the 2 functions by considering the ratio, \( y_2(x)/y_1(x)\) which yields, well, another function that we'll call \( y_3(x)\) .

\( \Large y_3(x)=\Large\frac{y_2(x)}{y_1(x)}=\Large\frac{m_2 x^2 + b_2}{m_1 x + b_1}=O(x)\) as \( x\rightarrow \infty\)


If \( x\) gets large enough (much larger than either \( b_1/m_1\) or \( b_2/m_2\) ), then the function starts to look like the following:

\( \Large y_3(x)=\Large\frac{m_2 x^2 + \epsilon}{m_1 x + \epsilon}\approx \Large\frac{m_2 x^2}{m_1 x} \approx \Large\frac{m_2}{m_1}x=O(x) \)

where \( \epsilon\) is a value that's a lot smaller than the terms involving \( x\) . The ratio of the functions is plotted below.

Saying that \( y_3(x)=O(x)\) as \( x\rightarrow \infty\) means that the ratio \( y_2(x)/y_1(x)\) grows with increasing \( x\) and goes to infinity for large \( x\) . In other words, \( y_2(x)\) is INFINITELY greater than \( y_1(x)\) for large \( x\) , even though \( y_1(x)\rightarrow \infty\) . \( y_3(x)\) goes to infinity in a particular fashion, i.e. linearly (\( y_3(x)=O(x)\) ). While both functions go to infinity, one is infinitely greater than another! This may be your first exposure to the idea that one kind of infinity can be different from another.

If the above plot of \( y_3(x)\) looks like a parabola to you, then GOOD! You're wrong, and you've learned something important, i.e. that you may not be able to guess the Order of a function by looking at a plot (or data). The slope of \( y_3(x)\) reaches a limit with increasing \( x\) (that slope is \( m_2/m_1\) ) and the function starts looking like a straight line for large \( x\) . In contrast, the slope of a parabola like \( y_2(x)\) increases forever with increasing \( x\) .

For non-zero values of all the constants, the ratio of the two functions approaches a finite limit as \( x \rightarrow 0\) ; in this case the value is simply \( b_2/b_1\) . We say that the ratio of the two functions is Order 1 ("order one" or \( O(1)\) ) as \( x \rightarrow 0\) . In English, what is meant here is that the value of the function is reasonably confined in the vicinity of \( x=0\) (and there is a rigorous definition of "reasonable" that you can explore if you like). What is NOT meant is that the ratio = 1.0 or that it is necessarily constant as the independent variable changes. The ratio of the two functions is \( O(1)\) (as \( x \rightarrow 0\) ) without specifying the values of \( b_1\) or \( b_2\) ; i.e. the actual ratio could be 1, 10,000, 1 billion, or anything else.This part of the deal is often even more confusing for folks than having ratios go to infinity or zero but is, nonetheless, the way of the world. Also note that we have just finished looking at a function that is \( O(x)\) in one location ( large \( x\) ) and \( O(1)\) in another (\( x\approx 0\) ). There's nothing wrong with this sort of thing; happens all the time.

Consider now a new case where both \( b_1\) and \( b_2\) happen to be zero. Then the ratio of the two functions reduces to:

\[ \Large y_3(x)=\frac{y_2(x)}{y_1(x)}=\frac{m_2 x^2}{m_1 x}=O(x) \] as \( x\rightarrow 0\)

Even though we can't allow \( x\) to go all the way to 0 (because we would have 0 in the denominator), the ratio of the two functions goes to 0 in the vicinity of \( x=0\) . \( y_2(x)\) is infinitely smaller than \( y_1(x)\) as \( x \rightarrow 0\) , even though both functions are heading for 0! We started evaluating this situation by arbitrarily looking at the ratio of \( y_2(x)/y_1(x)\) . If we had decided to look at the reciprocal instead (\( y_1(x)/y_2(x)\) ), we would see that the ratio goes to 0 as \( x \rightarrow \infty\) and that it goes to \( \infty\) as \( x \rightarrow 0\) (i.e. \( y_1(x)/y_2(x) = O(x^{-1})\) ). This new ratio has a singularity at \( x=0\) and it is a singularity of \( O(x^{-1})\) which tells us something about how quickly it goes to \( \infty\) . On the other hand, the function \( y(x) = x^{-n}\) has a singularity of order n at \( x=0\) and goes to \( \infty\) faster than \( O(x^{-1})\) for \( n>1\) .

We'll now observe in passing that there is a complete hierarchy of functions, defined by this Order thing, which has to do with how large one is with respect to another. It's likely already obvious to you that \( x^n\) is a much smaller function than than \( x^{n+1}\) (for large \( x\) ), but this concept extends to virtually ALL functions including \( \ln(x)\) , \( cos(x)\) , \( \arctan(x)\) , and everything else. It's like poker, see? You need to know which hand beats another! And FYI the ACE of functions is \( e^x\) . It turns out that \( e^x = O(x^{\infty})\) as \( x \rightarrow \infty\) ! Alternatively, \( e^{1/x}\) has an infinite Order singularity at \( x=0\) . It goes to infinity faster than any other function in the vicinity of \( x=0\) (as far as I know).

We can also have functions that depend on multiple variables; \( g(x,y)=y^2/x\) = \( O(y^2,x^{-1})\) . No problem, eh?

What IS the point?

We'll go over some examples shortly where I hope it will become quite clear that this is an important topic. First I want to enumerate some quick points on how this relates to practical research.

  • Over a large enough interval of the independent variable, functions of different order behave dramatically different. One of the problems in clinical research is that we often don't explore a large range of the independent variable (we don't have data for it). This can make it appear that data are linearly related to the independent variable whereas this is far from the truth in actuality. This is one of the reasons your statistician wants you to use a logarithmic transformation in many situations. This type of transformation allows the dataset to "seek its own order". Maybe the order of a data relation comes out to \( y(x) = O(x^{1.273})\) (or whatever) when you let things fall where they may using statistical methods. There's nothing wrong with functions having fractional or even irrational order as we'll soon see. But Beware! This isn't appropriate in all situations. Again, physical scientists have a tendency to use physical arguments to determine the order of a function and your statistician may not be geared into this thought process. You can't, for example, legally exponentiate anything but a pure number (no physical units) to the power 1.273 so you might get a visit from the math police if you're doing this to data with physical units such as mL, kg, etc.
  • Functions of different Order can't be equal to each other except at a point (single value for the independent variable) or a relatively "few" number of points, i.e. specific values of the independent variable. In the initial example above, there are no values you can plug in for \( m_1, m_2, b_1, b_2\) so that \( y_1(x)=y_2(x)\) for a large range of\( x\) . In mathematics, this fact is exploited extensively, even within one equation. We know apriori that terms on one side of an equation have to be balanced by terms of the same order on the other side. ( Otherwise it's not really an the equation at all.)
  • Quantities that are not \( O(1)\) (with respect to the relevant variables) are not normalized! The culture of medicine is that the expert can spout off a value or range of values that can be considered normal for a particular parameter or index. An index that is not \( O(1)\) is not an index - it's a function that depends on (varies with) important aspects of the individual's state.

Examples

Finally! Consider first several quantities of medical interest - creatinine, blood density, red blood cell count, hematocrit, and total red blood cell mass (i.e. per individual). Which one of these things is not like the others in terms of its functional Order?

All of these quantities are \( O(1)\) except for the last. I did this intentionally to help get you used to the \( O(1)\) concept. We hope your creatinine is somewhere around 1 (mg/dL) but of course this could vary with a whole lot of things. Blood density is actually very close to 1.0 (gm/cm3) but RBC is liable to be something like 5-6 x 106 (/?L) if you're in good shape. Nevertheless, all these things are \( O(1)\) ! The function Order concept doesn't have anything to do with actual numerical values, but how a quantity changes with independent variables. In fact, the above quantities all have one thing in common ( except total RBC mass); they are all examples of "concentrations", quantities expressed per unit of volume or fractional volume ( hematocrit ). Keep in mind that we could always change the physical units we're using so that the actual numbers associated with each quantity would come out numerically close to the number 1. \( O(1)\) does not mean that the quantity is a constant either. Offhand, I can't actually think of a concentration that isn't \( O(1)\) .

Total red blood cell mass, on the other hand, changes with body size and is of the same order as body mass or volume. Without getting too involved in proving this, we could say simply that the red blood cell mass is some fraction ( ranging from 0.0 - 1.0 ) of the total body mass. The fraction is \( O(1)\) and so RBC mass is the same order as body mass. We determined in another article (see The Shape of Things) that volume is proportional to length cubed ( \( V = O(L^3)\) ) and this is going to be important in upcoming examples.


Pure Math

We learn at a tender age that the slope of a line is the "rise over the run", \( \Delta y/\Delta x\) . That's fine and dandy if the slope is a constant - i.e. the slope doesn't vary. In calculus, we work with functions that might wiggle and curve all over the place. \( f(x)=x^2\) , for example, is a function whose slope changes with the value of \( x\) . In calculus, we define the slope of a function as follows:

\( \Large\frac{df}{dx}=\Large\frac{f(x+\Delta x)-f(x)}{\Delta x}\) , taking the limit as \( \Delta x \rightarrow 0\) .

The part on the left is read "the derivative of (function) \( f\) with respect to \( x\) ". On the right, we're looking at the ratio of the "rise" (\( f(x+\Delta x)-f(x)\) ), over the "run" (\( \Delta x\) ) in the vicinity of \( x\) , whatever actual value you choose it to be. While it's mathematically illegal for \( \Delta x\) to actually reach \( 0\) , we want to see what happens if it gets very dang close! Suppose for example that the function of interest is \( y=x^2\) that we already worked with above. We have:

\( y(x+\Delta x) = (x+\Delta x)^2 = x^2 + 2 x \Delta x + (\Delta x)^2\) (exactly!) and

\( y(x) =x^2\)

The entire expression is \( \Large\frac{dy}{dx}=\Large\frac{(x^2 + 2 x \Delta x + (\Delta x)^2) - x^2}{\Delta x} = \Large\frac{2 x \Delta x + (\Delta x)^2)}{\Delta x }\) and we find the limit of this expression as \( \Delta x \rightarrow 0\) . Now here's the point in Re function Order: We learned above that \( \Delta x^2\) is infinitely smaller than \( \Delta x\) ! Consequently we can neglect the \( \Delta x^2\) term in the vicinity of \( x=0\) and the whole thing reduces to:

\( \Large\frac{dy}{dx}=2x\)

What happened? We took the derivative of a function (\( y(x) = x^2\) ) and got .... another function! You give me a value for \( x\) , and I'll tell you the slope of \( y(x) = x^2\) at that value of \( x\) ! \( dy/dx\) of \( y=x^2\) is \( 2x\) (so if you gave me x=4.2, the slope there is 8.4). Taking this a few steps further, here's a series of similar functions:

\( y(x) = x^3\) : \( y(x+\Delta x) = x^3+3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3\) : \( \Large\frac{dy}{dx}=3x^2\)

\( y(x) = x^4\) : \( y(x+\Delta x) = x^4+4x^3\Delta x+6x^2(\Delta x)^2+4x(\Delta x)^3+(\Delta x)^4\) : \( \Large\frac{dy}{dx}=4x^3\)

\( y(x) = x^n\) : \( y(x+\Delta x) = x^n+n x^{n-1}\Delta x +\) higher Order terms : \( \Large\frac{dy}{dx}=n x^{n-1}\)

The final result on the right hand side (the derivative function) is arrived at simply from the definition given above, plugging in the expressions on each line for \( y(x)\) , \( y(x+\Delta x)\) , dividing by \( \Delta x\) , and taking the limit as \( \Delta x \rightarrow 0\) in each case. The pattern is obvious by the time we get to the last line and we have the generalized method for finding the derivative function of \( y(x)=x^n\) (which, by the way, works for non-interger values of \( n\) also). For increasing value of the exponent, we end up with \( y(x+\Delta x) = x^n+n x^{n-1}\Delta x\) (whatever n happens to be) plus additional terms that are completely negligible as \( \Delta x\) approaches \( 0\) ("higher order terms") in the final expression (after \( x^n\) has been subtracted). The \( nx^n\Delta x\) stuff would be referred to a the "leading order term" in the literature; it's the biggest thing (the Only thing) in the final expression that amounts to anything as \( \Delta x \rightarrow 0\) .

This Order thing we've been discussing is at the root of a lot of mathematics! This is a concept that's worth understanding (and there are quite a few math faux pas in the medical literature that result from not understanding).


REAL Cardiology

In the next examples, we're switching from pure math to real life, clinical research situations. In this setting, we are typically attempting to explain data variation, i.e. between dependent and independent variables. This often comes down to statistical methods where the form of a regression equation has been estimated or guessed. Can it cause problems to use a formula with the wrong Order when performing a regression? (Well, YES!)

The Teichholz formula appears as:

\( V=\Large\frac{7}{D+2.4}D^3\)

where \( V\) is left ventricular volume and \( D\) is the short axis diameter. For small \( D\) , this formula is \( O(D^3)\) because the \( D\) in the fraction is negligible with respect to the 7 and 2.4; NOTE that the physical units associated with the 7 and 2.4 must be the same as \( D\) (cm specifically!). For large \( D\) , this formula is \( O(D^2)\) ; it acts like an area, not a volume. To be perfectly legitimate, we would prefer to have the fraction multiplying \( D^3\) to be \( O(1)\) , not \( O(D^{-1})\) . This is a forgivable faux pas under the circumstances as long as the application of the formula is limited to an appropriate range of the independent variable. Veterinarians, however, have not done this. In fact this formula is completely inappropriate for use in veterinary cardiology for another reason. Click the above link to explore further.

For purposes of illustration ( only ), let's consider what this formula would look like with it's Order fixed:

\( V=\Large\frac{k D +7}{D+2.4}D^3\) and I propose the value of \( k=\pi/6\) . With this minor change, the shape factor ( the fraction that multiplies \( D^3\) ) is now \( O(1)\) ; it varies with the value of \( D\) but it doesn't stray very far. It acts just like the old Teichholz formula for small values of \( D\) , but approaches a limiting value (\( k=\pi/6\) ) for very large values of \( D\) . The value of \( k\) was chosen to give a reasonable shape factor for an extremely dilated ventricle; it's the shape factor for a sphere. The formula for the volume of a sphere is \( V=4\pi r^3/3 = 4\pi (D/2)^3/3 =\pi/6 D^3\) . The new formula is shown ONLY to illustrate the Order principal. I've never seen a ventricle so dilated that the shape actually became a sphere, but I think that would be a logical limiting value for the shape. Suggestion: try hauling out your spreadsheet and plot the Teichholz shape factor next to the new one.


Indexing Cardiac Size

It seems that the medical profession(s) will normalize just about anything by dividing by body surface area, thereby creating an "indexed" version of the quantity. Hence we have the end-systolic volume index, end-diastolic volume index, stroke volume index, cardiac index, valve area index, body mass index, and so on. Which of the above is not like the others in terms of function Order?

From my point of view, the valve area index is the standout; it's the only one that is \( O(1)\) and the only one that's correctly normalized for body size. With volume of Order length cubed (\( V=O(L^3)\) ) and body surface area of Order length squared (\( V=O(L^2)\) ), all of the volume indices above are (\( O(L^3/L^2) = O(L)\) ). Each of these indices can be expected to approach 0 for small body size and grow in an unbounded fashion in proportion to body length or the cube root of body weight (mass) (\( O(W^{1/3})\) ). This will not be apparent for a small range of body size and I suspect this is largely responsible for continued satisfaction with the approach in human cardiology. However there is no basis for this practice and there are certainly physicians who have recognized it ( check out the appendix in Principles and Practice of Echocardiography, Weyman). It will also "work" for animals if the practice is restricted to a narrow range of size, e.g. within a breed such as Dobermans, in the sense that the "indexed" volume will show less body size dependence than the raw volume. In one sense, this is because area (\( O(L^2)\) ) and volume (\( O(L^3)\) ) are "close" to the same Order. Try normalizing a linear dimension (\( O(L^1)\) ) by body surface area (\( O(L^1/L^2)=O(L^{-1})/\) ) or, worse, body weight (\( O(L^1/L^3)=O(L^{-2})\) ) and you'll begin to see how bad the practice is. These functions tend to infinity for small body size and it's clear that this is an unacceptable method of normalization.

Got Data?

We're going to try to determine whether the above theory has any practical clinical application by considering some canine echocardiographic data. The plots that follow derive from echocardiographic LV IDd measurements in 626 normal dogs by several expert contributors. Ventricular volumes were determined from the \( D^3\) formula and plotted against an estimate of body surface area (\( k W^{2/3}\) , m2) computed from the body weight (\( W\) , kgs); for comparison, the volume is also plotted against the body weight directly. My claim is that ventricular volume is the same order as \( W\) , \( O(L^3)\) where \( L\) is length; if this is true, then we should see problems with nonlinearity plotting ventricular volume against body surface area.

A weighted linear regression line and 95% prediction intervals are also shown on each plot; the regression line corresponds to the simple average of the indexed volume from the entire data set, i.e. the average of either \( EDV/(k W^{2/3})\) (lower plot) or \( EDV/W\) (upper plot). At first inspection, these regressions seem equally poor, exhibiting a large amount of data scatter on either side of the linear prediction. There doesn't appear to be anything to recommend indexing cardiac volume by body weight rather than surface area. (I simply HATE working with real data! Especially when it doesn't support my pet theory!)

While there are problems with both plots, the second (lower, volume vs surface area) is a statistical train wreck. We can see that more easily by focusing on the data segment ( below ) closest to the origin where we see that the regression line simply misses the data points entirely! That's because the relationship between ventricular volume (\( D^3\) ) and body surface area IS NOT linear; the data relationship, like the functions, are of different Order.

Here are some implications:

  • For a sufficiently wide range of the independent variable, individuals will begin to depart from an assumed straight-line relationship between ventricular volume and body surface area. The departure is as shown; individuals will fall below the regression line with decreasing body size. That's because the ratio of volume to surface area is not a constant but a function of \( O(L^3/L^2) = O(L) = O(V^{1/3}) = O(A^{1/2})\) (L is length, A is area, V is volume which is the same order as body mass). Each of these is a function that goes to \( 0\) for small body size (data fall below the V:BSA linear regression) and \( \infty\) for large body size (data lie above the V:BSA linear regression).
  • Conversely, the nonlinearity will be statistically undetectable for a sufficiently narrow range of body size ( full-grown humans or cats, single breeds of dogs, other size-restricted groups).
  • If the "indexed" volume is correlated with the independent variable, you may be exceeding the statistical limits of applicability; the range of body size may be too great.
  • "Indexed" cardiac volumes (volume / BSA) ARE NOT INDEXED. They CANNOT be used to compare cardiac volumes for groups with widely varying body size. You cannot compare cardiac volumes of large breed dog to small. You cannot follow growing animals with this method. (Examples of both faux pas appear in the literature.)

You might be inclined to "fix" some of these inconsistencies by performing a linear regression with a nonzero intercept. This creates a new set of issues.

  • First of all, it can be stated with certainty that the data approach the origin asymptotically. This may not seem like a reliable fact, but it is the most reliable thing about the whole discussion. Consider the alternative, i.e. that the intercept is nonzero. For a positive y-intercept, we have a situation where there's a positive ventricular volume and no body to go with it (0 body surface area)! A negative y-intercept implies 0 heart volume at a positive body size. While the latter is certainly feasible for certain highly pathological situations, it isn't really of interest for understanding the normal relationship between body size and heart size. While we can never reach a body size of 0, you can take this thought experiment to the nth degree and consider the smallest imaginable mammal that still has a heart (embryo ?). That's a mighty small body surface area and ventricular volume! (and the data point will be adjacent to the origin).
  • The bottom line is that we KNOW that the data pass through the origin. Consequently any function used to "fit" (regress) the data that does not pass through the origin will fail to represent the data in some region sufficiently close to the origin.
  • More accurate determination of the ventricular volume ( area-length, Simpsons rule, MRI, etc.) will not alter this outcome! It's due to the nonlinear relationship between area and volume, not inaccuracies of either determination.

We can also play dumb and perform a logarithmic regression of the data. This will fit the data to a function of the form \( y(x) = A x^B\) where \( A\) and \( B\) are the 2 regression parameters; \( y(x)\) is end diastolic volume, \( x\) is body weight (kgs). This is really what your statistician wanted you to do in the first place. The function passes through the origin which we've established is a necessity for consistency. The result for the parameter \( B\) for the canine data above is 0.9018; not quite the expected value of 1.0 predicted by theory alone (but better than 2/3 which is the exponent implied by the standard "end-diastolic volume index"). If we look at the end-systolic volume (plotted below for the same dogs), parameter \( B\) is 0.9779 (very close to the expected 1.0). The plot of end-systolic volume against body surface area (lower plot below) exhibits the same problems seen for end diastolic volume noted above. There is no physical basis for indexing cardiac volume to body surface area; data sets of sufficient range support the principle of indexing cardiac volume by body volume (body mass).

At one time I attempted to unearth the origins of the curious practice of indexing cardiac volume by body surface area. I could get no further than an article by JM Tanner from the Journal of Applied Physiology, volume 2, July 1949 titled "Fallacy of per-weight and per-surface area standards, in their relation to spurious correlation". Here are some points about the investigation worth keeping in mind:

  • This was an investigation to normalize cardiac output, not cardiac volume. I don't know if this article is the source of the volume "indexing" mindset that persists today.
  • Tanner explicitly rejected the notion that stroke volume would go to 0 with negligible body size. In so doing, the one reliable data point of the study was eliminated (... in my opinion. If you believe this is possible, please go back to the top of the article and try again.)
  • The study was done at a time when computing the intercept of a linear regression was considered a mathematical accomplishment (so they did it, whereas stroke volume MUST be 0 for 0 body size).
  • Cardiac output at that time appears to have been estimated using ballistocardiography.

Body Size

While we're at it, let's consider how we determine body surface area in the first place when calculating a cardiac volume index. The usual method in veterinary medicine employs the following formula:

\( A=k W^{2/3}\)

\( A\) is the body surface area in m2, \( W\) is the body mass in kg, and \( k\) is a constant derived from regression, usually about 0.1. So we aren't actually indexing by body surface area per se but rather by some simple function of the body mass. I prefer to make this thoroughly apparent when writing about it by subscripting, e.g. \( A_W\) , or otherwise making it clear that the area determination is derived simply from a body mass measurement.

Where does that exponent (2/3) come from? Well, I would say the correct exponent has been chosen based on the same physical arguments I've been making in this article. 2/3 is a pretty specific number and I doubt that it comes from a regression; I don't actually know where this formula was originally introduced into the literature. What are the physical units of the constant, \( k\) ? Well, it multiplies something that has physical units of kg2/3 and returns something that has units of area; consequently it has units of area/kg2/3 (???). Anybody ever try to raise kilograms to the two thirds power? (Don't try, you're going to hurt yourself!) We CAN'T exponentiation kg2/3! It has units of mass and you can't take the cube root of it. However, we can find the cube root of a volume. Volume has physical units of length cubed (\( L^3\) ); we can take the cube root of a volume, square it, and we have physical units of area \( L^2\) . In that case, physical units of \( k\) are nondimensional; it's a pure number.

So there is an implicit (tacit) step going on that is not showing up. We have to convert the mass to a volume before we can take the cube root. To do that we divide mass by the density,\( \rho\) ; \( \rho\) has units of mass/volume, e.g. kg/m3 and, as noted above is \( O(1)\) . While density certainly varies among individuals, we earthlings are all just a little bit denser than water. You can alter your density, e.g. by going heavy on the starchy foods (which increases your mass but decreases your density) or taking in a big lungfull of air, but your density is not going to stray very far from that of water.

Just so you've given it some thought, there is liable to be a significant amount of error involved in estimating body surface area using this formula. (As an exercise, why don't you take a look at some formulas for body surface area and see which ones adhere to some semblance of the function Order principal, if any.) However, the bottom line about indexing heart size is that we are not really interested in using the actual body surface area, or actual body mass!! Those are quantities that are highly dependent on the body condition of the individual (i.e. ranging from severe cachexia to morbid obesity). This is a significant drawback to using anything derived from the simple measured body weight as a means to indicate the size of the individual. What we really want is some sort of intrinsic body size -- one that does NOT depend on body condition.

There may be more than one way to accomplish this, but the one that occurs to me (and many others) is to use various measurements of the appendicular skeleton to estimate intrinsic body size. The obvious one for people is simply the height and one could attempt to determine whether an individual's weight is appropriate for their height (in people anyway). The body mass index was devised with this intent.

\( BMI=\Large\frac{W}{h^2}\)

Here the weight (mass), \( W\) , is typically measured in kg and height, \( h\) , in m. If people were all the same shape, and if this were a correctly formulated index, then the index value would be similar for individuals of appropriate weight.

I have my usual problem with this formula which is that the numerator and denominator are of different Order; this index can be expected to approach \( 0\) for small body size and \( \infty\) for large. Like the volume indices of the heart, the history of this approach is too extensive to hope for a revision of its calculation and it engenders other shortcomings that would likely take precedence over my own complaint. Let us rather use this example as an exercise to see how we could devise indices that are dimensionally sound and adhere to the function Order principle.

If we were to stick with a single length measurement, e.g. \( h\) alone, then the following suggests itself:

\( BMI_2=\Large\frac{W}{h^3}\)

Here all we've done is fix the disparate Order problem by cubing the height instead of squaring it (and renamed it \( BMI_2\) so it's distinct). The numerical value of this index will, of course, be entirely different than the currently accepted \( BMI\) , but this is irrelevant anyway. The way we actually conduct this determination is to start with a bunch of individuals with normal body composition (Good luck with that! ) and simply divide each individual's weight by their height, cubed; that gives the expected (average) value of the index.

You'll readily find a discussion on the pros and cons of the \( BMI\) ; I don't think I've seen the Order issue brought up in Re the body mass index. But another obvious problem is that one length measurement alone (height) is hardly enough to predict an appropriate weight. Suppose we add a few more well-chosen length measurements for the prediction ( e.g. shoulder-to-shoulder length, wrist-to-shoulder, sternum-to-back, etc.); alternative measurements might be used for a quadruped ( e.g. height at shoulder, crown-rump length, etc.). The bottom line is that we have a number of length measurements (\( L_i\) ) and we need to formulate an index of body mass that is \( O(1)\) . Here's a first go:


\( \Large\frac{W^{1/3}}{\Sigma_{i=1}^n k_i L_i}\) or\( \Large\frac{V^{1/3}}{\Sigma_{i=1}^n k_i L_i}=\Large\frac{V^{1/3}}{k_1 L_1+k_2 L_2+k_3 L_3+ +.. k_nL_n}\)

Looking at the numerator we have \( W^{1/3}\) which is \( O(L)\) (after appropriately dividing by density to get volume, \( V\) ); the denominator comes from a weighted sum of the length measurements (\( L_i\) ). The weights (\( k_i\) ) would be determined from a multiple linear regression. We are basically attempting to predict \( W^{1/3}\) (which is available) using the available lengths.

\( W^{1/3}\) or \( V^{1/3}=k_1L_1+k_2L_2+k_3L_3+ + .. k_nL_n\)

Then the expected value of the index is likely to be pretty close to 1.0 since we're dividing both sides of the above by the right hand side:

\( \Large\frac{V^{1/3}}{\Sigma_{i=1}^n k_iL_i}\approx 1.0\)

This index amounts to statistical prediction of a global length, \( W^{1/3}\) or \( V^{1/3}\) , from a weighted sum of other measured lengths, \( L_i\) . A similar alternative would be as follows:

\( W\) or \( V=k_1L_1^3+k_2L_2^3+k_3L_3^3+ + .. k_nL_n^3\)

\( BMI_3 = \Large\frac{V}{\Sigma_{i=1}^n k_i L_i^3}\)

Here we are predicting a volume ( or mass ) from the weighted sum of the cubed measured lengths. Each of the lengths contributes a volume, but we've strictly maintained the function Order principal.

For a second go around, we could also use our 3 best length measurements to obtain a volume directly:

\( W = k L_1L_2L_3\) (\( k\) from multiple linear regression)

\( BMI_3 = \Large\frac{W}{L_1L_2L_3}\)

The value of \( k\) is irrelevant in this case and has been left out of the index formula which now looks the same as \( BMI_2\) but with more than 1 length. Alternatively, if you want to incorporate more than 3 lengths:

\( BMI_4 = \Large\frac{W}{\left [\Pi_{i=1}^n(L_i)\right]^{3/n}}\)

The notation \( \Pi\) indicates that we multiply all the lengths together which yields a quantity with units \( L^n\) ; we have to exponentiate this product to the 3/n power to get a mass or volume. Once again, the correct functional Order has been maintained.

A feature in common with the above approaches is that the various \( BMI\) s shown are derived using a regression to predict body weight or volume. Take a look at the next couple of examples:

\( \Large\frac{W}{L_0^3}\) or \( \Large\frac{V}{L_0^3} = k_0+k_1\Large\frac{L_1}{L_0}+k_2\Large\frac{L_2}{L_0}+k_3\Large\frac{L_3}{L_0}+ +\) (formula A)

or

\( \Large\frac{W}{L_0^3}\) or \( \Large\frac{V}{L_0^3} = k_0 \left[\Large\frac{L_1}{L_0}\right]^{k_1}\left[\Large\frac{L_2}{L_0}\right]^{k_2}\left[\Large\frac{L_3}{L_0}\right]^{k_3}\) (formula B)

Here we propose 2 regression formulas where we are no longer trying to predict \( W\) or \( V\) , which are dimensional quantities, but \( W/L_0^3\) or \( V/L_0^3\) ; the latter are already body mass indices themselves. \( L_0\) is just a representative length for the individual; go ahead and think of it as the height. The \( L_i\) are other measured lengths ( e.g. shoulder to shoulder length, sternum to backbone, etc. ) and the \( k_i\) are regression parameters that we need to estimate from the data that's been collected. In other words we have all of the indicated lengths and body weights from a sample of normal individuals, so we can simply rewrite these two formulas in terms of ratios:

\( \Large\frac{W}{L_0^3}\) or \( \Large\frac{V}{L_0^3} = k_0+k_1r_1+k_2r_2+k_3r_3+ +\) (formula A)

\( \Large\frac{W}{L_0^3}\) or \( \Large\frac{V}{L_0^3} =k_0\left[r_1\right]^{k_1} \left[r_2\right]^{k_2} \left[r_3\right]^{k_3} ...\) (formula B)

where \( r_i\) has been substituted in place of \( L_i/L_0\) . These ratios are data that we have available so we're all set to perform these regressions. But let's observe a few things about the formulas first:

  • The left-hand side of each of these formulas is already a body mass index (it's \( BMI_2\) ffrom above and I'd be willing to bet you will outperform the standard \( BMI\) right off the bat because it's \( O(1)\) ).
  • The right-hand side of each formula is now written in terms of ratios, each of which is dimensionless and \( O(1)\) . In fact you will note that both formulas are dimensionless with both sides of the equal sign \( O(1)\) . This is a pervasive theme in engineering and physical science analyses and its importance cannot be overstated for understanding how the universe works.
  • The left-hand side of these formulas would be a constant value if everyone were the same shape and body condition. The value will grow with degree of obesity and decrease with degree of cachexia but might also change with body habitus, age, gender, fraction of body fat, etc. The right-hand side of the equation provides the means by which the \( BMI\) can be corrected for various shape factors, the ratios that appear as prediction variables. These ratios are chosen for their degree of influence (correlation) with the \( BMI\) and essentially define aspects of the body habitus; I've referred to them elsewhere as shape factors. Notice that we have no conceptual difficulty exponentiating these ratios to integer or fractional powers ( in formula B) since they are pure numbers. We cannot commit any mathematical faux pas because of the form of the equations; the Order cannot get out of whack. \( k_i\) are all pure numbers also ( no physical units ).

I don't know without real data which of the above 2 formulas (A or B) is likely to be best. Notice however that we would convert formula B to something similar to formula A simply by performing a logarithmic transformation.

\( \ln \left[\Large\frac{V}{L_0^3}\right] = \ln \left[k_0\left[r_1\right]^{k_1} \left[r_2\right]^{k_2} \left[r_3\right]^{k_3} ...\right]\)

\( \ln \left[\Large\frac{V}{L_0^3}\right]=k_{0B}\left[k_1r_1+k_2r_2+k_3r_3++...\right]\)

Either formula could be used to predict the appropriate body weight or volume;

\( W=L_0^3\left[k_0+k_1r_1+k_2r_2+k_3r_3+ +\right]\) (from formula A)

or derive a new body mass index that is corrected for body habitus:

\( BMI_5 = \Large\frac{W}{L_0^3\left[k_0+k_1r_1+k_2r_2+k_3r_3+ +\right]}\) (from formula A)


 

Stenosis Flow Dependence (A Primer on Function Order)

Consider the following 2 equations for the change in pressure associated with flow in a conduit:

\( \Delta p=R q\)

and


\( \Delta p=G q^2\)

The first equation is the simple flow resistance formula with \( \Delta p\) representing the pressure drop ( upstream pressure minus downstream) and \( q\) is the flow rate. For a circular conduit with fully developed laminar flow of a Newtonian fluid, \( R=8 \mu L/(\pi r^4)\) where \( \mu\) is the Newtonian viscosity, \( L\) is the length of the conduit, and \( r\) is the conduit radius. Among other things, you could say that \( \Delta p=O(r^{-4})\) . However we're primarily interested in the effect of flow in this section so \( \Delta p=O(q)\) .

The second equation is a rearranged version of the Gorlin formula where \( G = \rho /(2 A_{e}^{2})\) ; \( \rho\) is the fluid density and \( A_e\) is the effective orifice area. Recall that the simplified Bernoulli equation is \( \Delta p=\rho u^2/2\) where \( u\) is the fluid velocity at the vena contracta. All that's been done to get the Gorlin formula from the sinplified Bernoulli is a substitution of \( q/A_e\) for the velocity and some algebra. ( Stenosis Flow is covered in depth within other articles at this website. )

So first of all, the pressure difference for both of these equations is FLOW DEPENDENT, i.e. they both depend explicitly and completely on the value of \( q\) . However they do so in completely different ways. \( \Delta p\) for the resistance equation is \( O(q)\) (or \( O(q^1)\) ), and it's \( O(q^2)\) for the simplified Bernoulli equation. These two equations are devised to model (describe) entirely different physical processes. They are NOT interchangeable! The resistance equation models a loss of energy due to friction and depends on the viscosity of the fluid and the way in which the fluid lamina slide against each other. \( R=8 \mu L/(\pi r^4)\) is due specifically to the Poiseuille ( parabolic) velocity profile that occurs with fully developed, time-invariant (non-pulsatile) flow in a straight circular tube. In contradistinction, the simplified Bernoulli equation models a change in pressure associated with a change in velocity from one location to another, i.e. convective acceleration of the fluid. This change in pressure has nothing to do with friction (viscosity does not appear in the equation) but is dependent on the fluid density. While the terminology is somewhat imprecise, the simplified Bernoulli equation can be thought of as an exchange of pressure "energy" for kinetic energy as the fluid elements accelerate through an orifice. The fluid density (mass/volume) substitutes for mass in a momentum equation (\( \mathbf{F}=m \mathbf{a}\) ) or in a kinetic energy equation (\( ke = m u^2/2\) ).

Note that the full Bernoulli equation also includes an energy loss term. The latter is very difficult to compute on its own unless you know everything about the flow ( i.e. the entire velocity field) and the relationship between relative fluid motion and energy loss. Consequently the energy loss term is often an "unknown" and we use the full Bernoulli equation to calculate it if we know the value of all the other terms in the equation. For stenosis flow, we know ahead of time (from experience, experimentation, and calculation) that the energy loss on the center streamline between the left ventricle and vena contracta is very small compared to the acceleration (kinetic energy) term. Consequently we can calculate the pressure drop ( so-called "gradient") from the acceleration alone assuming energy loss is 0.

We can rearrange the simplified Bernoulli equation into the Gorlin formula as follows:

\( A_e=q \sqrt{\Large\frac{\rho}{2 \Delta p}}\)


You may not usually think of or use the Gorlin formula this way (where's the 44.3?) but this is the real McCoy with an important goof from the usual clinical formula left out. The Gorlin formula as shown estimates the effective orifice area, not the physical area of the stenosis. The physical stenosis area, \( A_p\) , can be written as \( A_p=A_e/C_D\) where \( C_D\) is what's called the discharge coefficient. A quick rearrangement shows us that \( C_D=A_e/A_p\) , the ratio of the effective orifice area to the physical area. \( A_e\) is necessarily smaller than \( A_p\) and so \( C_D\) has the interpretation of a fraction of the physical area ( the effective area is a fraction of the physical area and that fraction is \( C_D\) ). It is necessarily between 0 and 1 ( and can't actually get to either value); it is \( O(1)\) !! Now we can rewrite the Gorlin formula to compute the physical area as:

\( A_p=\Large\frac{q}{C_D} \sqrt{\Large\frac{\rho}{2 \Delta p}}\)

It is now necessary for you to recognize that \( A_p=O(q^0)\) , i.e. \( A_p=O(1)\) with respect to flow; while the physical area might change somewhat with pressure or flow, it's not going to do anything silly like grow to infinity or shrink to zero depending on the flow rate. This is due to the fact that \( C_D\) may change somewhat with flow rate ( or more specifically, Reynolds number), but we've already established that it is absolutely \( O(1)\) .

Clinical researchers in the 1990s began to recognize that the Gorlin formula exhibits "flow dependence"; the value calculated (effective orifice area) changes systematically with flow rate. The physical phenomena surrounding this finding had previously been fully explored in the bioengineering literature in the 1970s. Clinical researchers proposed a resistance equation to describe the phenomenon. They approximated \( C_D\) as follows:

\( C_D= k \sqrt{\Delta p}\)

Substituted back into the Gorlin formula we have the following:


\( A_p=\Large\frac{q}{k \sqrt{\Delta p}} \sqrt{\Large\frac{\rho}{2 \Delta p}} = \sqrt{\Large\frac{\rho}{2}} \Large\frac{q}{\Delta p}=k_b\Large\frac{q}{\Delta p}\)

In other words, they proposed that the valve area is proportional to \( p/q\) ; it behaves just like the resistance similar to the Poiseuille equation. The problem here is that \( C_D\) is not \( O({\Delta p}^{1/2})\) ! It's \( O(1)\) !
In the following figure, an illustrative representation of \( C_D\) is shown in red and the function \( C_D= k \sqrt{\Delta p}\) is shown in blue. This is only a qualitative figure to serve as a conceptual aid; actual data from various publications will be considered elsewhere.

The discharge coefficient (red) is shown increasing somewhat with "flow rate" to reach a maximum value (flow rate is in quotations here because \( \Delta p\) is written specifically as a function of pressure gradient, not flow). There are numerous publications that confirm this essential finding. The sideways parabola (\( C_D \approx k \sqrt{\Delta p}\) ) has been included to show how clinical researchers might be inclined to choose this simplified function to represent \( C_D\) in the "flow dependent" region. However \( C_D= k \sqrt{\Delta p}\) is only adequate in a qualitative sense; it's a function that increases and is concave downward so that it can suggest the actual relationship, with the right choice for \( k\) , over a limited range of the independent variable. In reality, however, \( C_D\) cannot go to \( 0\) nor can it exceed \( 1.0\) as the blue curve would suggest; \( C_D\) is \( O(1)\) ! The researchers that proposed the resistance formula used regression (curve fitting) methods to represent the "Gorlin constant" over a limited range of geometries and flows and did not apparently appreciate that the discharge coefficient is the specific flow dependent part of the Gorlin formula.

"Flow dependence" of the valve "gradient" is NOT AT ALL the same thing as flow dependence of the Gorlin formula. The gradient is \( O(q^2)\) , i.e. highly flow dependent, increasing without bound with increasing flow. This is an immediate consequence of the Bernoulli equation which correctly models the essential physics of stenosis flow. The area computed by the Gorlin formula is \( O(1)\) with respect to flow. ( Actually it's \( O(L^2)\) ; it's an area for cryin' out loud and should change in proportion to body surface area.) While it varies somewhat with flow, the value is strictly bounded. This kind of "flow dependence" is entirely different from the gradient and should NOT have the same term applied. I keep putting quotes around flow dependence because the term is a misnomer in the first place. We will discuss that elsewhere.

The resistance formula applied to stenosis flow is a clinical index. It does not represent the physical attributes of stenosis flow and is predicated on a mathematical misrepresentation of the discharge coefficient. In so doing, physical impossibilty is implied including an effective orifice that is larger than the physical one.

The resistance formula will be less "flow dependent" than the Gorlin formula for a limited range of flows and stenosis geometry/severity. Overall it is NO LESS FLOW DEPENDENT THAN THE GORLIN EQUATION. In my opinion the resistance formula applied to a stenosis is a big step backwards.

 


 

RocketTheme Joomla Templates