Volume of a Prolate Ellipsoid

Date: 07/07/99 at 10:32:06
From: William Hansen
Subject: Calculation of Volume using areas and length

I have researched the equation for a prolate ellipsoid, which 
resembles closely the left atrial chamber of the heart. Could you 
explain how the equation for a prolate ellipse:

     V = 4/3*pi*a^2*c

where a = minor diameter of the ellipsoid and c = "length" or major 
diameter of the ellipsoid, indicates a DIRECT relationship between 
length of the ellipsoid and its volume; while another equation to 
derive volume:

     V = (0.85*A1*A2)/L

where V = volume, A1 = machine derived 2D area of one plane of a 
chamber (left atrium), A2 = machine derived 2D area of a plane 90 
degrees to A1 of a chamber, L = length of chamber (averaged from the 
two planes), indicates an INVERSE relationship of length to volume?  
I.e., in the latter equation, the GREATER the length, the LESS the 
volume yielded?

I understand the mathematical yielding of correct units (cubed cm) in 
the latter formula, but I can't understand why (how) the length of a 
chamber (ellipsoid) could be INVERSELY related to its volume. Am I 
thinking about this correctly?

Date: 07/07/99 at 12:23:41
From: Doctor Peterson
Subject: Re: Calculation of Volume using areas and length

Fascinating question!

What you are comparing are a precise formula for an ellipsoid, and an 
approximation for near-ellipsoids. If the shape were an actual prolate 
spheroid, the two areas you are measuring, which I understand to be 
both lengthwise, would be identical. So I'll work with the more 
general formula for the ellipse,

     V = 4/3 pi a b c = 4.19 a b c

where we're assuming a = b and c = length.

Your other formula is based on two cross-sectional areas, which, 
assuming we have an actual ellipsoid and the areas are in the 
direction of the minor axes a and b, will be ellipses sharing major 
axis c, which is half the length L:

     A1 = pi a c

     A2 = pi b c

      V = .85 A1 A2 / L
        = .85 (pi a c) (pi b c) / (2c)
        = .85 pi^2 a b c^2 / (2c)
        = 4.19 a b c

This agrees with the exact formula, in the case of an exact ellipsoid. 
Your factor .85 is actually 4/3 pi divided by pi^2/2, or 8/(3 pi) = 
0.8488.

You can see why we have to divide by L: in multiplying two areas that 
share that dimension, we have multiplied by L twice (my c^2) and have 
to remove one to get the right dependency on L.

The volume is not really inversely proportional to L, because the 
areas also depend on L. If you keep the areas the same while 
increasing L, you have to narrow the ellipsoid so much that the volume 
will decrease. If you just increase L, the areas will both increase 
proportionally, and the volume will remain proportional to L.

I hope that restores your confidence in your formula and in the 
consistency of math.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/