|San José State University|
& Tornado Alley
A useful line of analysis is to consider the effect of scale changes for creatures
similar in shape and only differ in scale. As the scale of an animal increases the body
volume increase with the cube
of scale. The volume of blood flow required to feed that bulk also increases with the cube
of scale. The cross sectional area of the arteries and the veins required to carry that
blood flow only increases with the square of scale. There are other area-volume relationships
which impose limitations on creatures. Some of those area-volume constraints, including the above one, are:
Thus to compensate for the body needs which increase with the cube of scale but the areas increase with only the square of scale the average blood flow velocity must increase linearly with scale. Blood flow velocity is driven by pressure differences. The pressure difference must be great enough to carrying the blood flow to the top of the creature and great enough to overcome the resistance in the arteries and veins to the flow. The pressure required to pump blood from the heart to the top of the creature is proportional to scale. The pressure difference required to overcome the resistance to flow through the arteries into the capillaries and back again through the veins is more difficult to characterize in terms of scale. The greater cross sectional area reduces the resistance but the long length increases resistance. The net result of these two scale influences seems to be that the pressure difference required to drive the blood through the bulk of the creature is inversely proportional to scale. The pressure difference imposed would be the maximum of the two required pressure differences.
Shown below are the typical blood pressures for creatures of different scales.
|Blood Pressure versus Height and Weight for Various Creatures|
The linear regression of the logarithm of pressure on the logarithm of height yields the following result:
The linear regression of the logarithm of pressure on the logarithm of weight yields:
If blood pressure were proportional to scale then the coefficient for *log(Height) would be 1.0 and for *log(Weight) would be 0.333 since weight to proportional to the cube of scale. The regression coefficients are not close to the theoretical values but they are of the proper order of magnitude for accepting blood pressure as being proportional to scale.
The volume of the heart of a creature is proportional to the cube of scale. The volume of the blood to be moved is also proportional to the cube of scale. From the previous analysis the flow velocity is proportional to scale. Therefore the time required to evacuate the heart's volume is proportional to scale. This means that the heartbeat rate is inversely proportional to scale. The following table gives the heart rates for a number of creatures.
|Heartbeat Rates of Animals|
|Creature||Average Heart Rate
A regression of the logarithm of heart rate on the logarithm of weight yields the following equation:
If heart rate were exactly inversely proportion to scale the coefficient for *log(weight) would be -0.333. This is because scale is proportional to the cube root of weight. The coefficient of -0.2 indicates that the heart rate is given an equation of the form
One salient hypothesis is that the animal heart is good for a fixed number of beats. This hypothesis can be tested by comparing the product of average heart rate and longevity for different animals. Because the heart rate is in beats per minute and longevity is in years the number of heart beats per lifetime is about 526 thousand times the value of the product. The data for a selection of animals are:
|Lifetime Heartbeats and Animal Size|
|Weight||Heart Rate||Longevity||Product||Lifetime Heartbeats|
Although the lack of dependence is clear visually the confirmation in terms of regression analysis is:
The t-ratio for the slope coefficient is an insignificant 0.15, confirming that there is no dependence of lifetime heartbeats on the scale of animal size.
If a heart is good for just a fixed number of beats, say one billion, then heart longevity is this fixed quota of beats divided by the heart rate. From the above equation for heart rate, lifespan (limited by heart function) would be proportional to scale raised to the 0.6 power.
The data for testing this deduction are:
|Lifespan versus Weight for Various Creatures|
|Creature||Weight (grams)||Life Span|
For the data in the above table, admittedly very rough and sparse, the regression of the logarithm of the lifespan on the logarithm of weight gives
Thus the net effect of scale on animal longevity is positive. Taking into account that weight is proportional to the cube of the linear scale of an animal the above equation in terms of scale would be
This says that if an animal is built on a 10 percent larger scale it will have a 6 percent longer lifespan.
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