San José State University

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Animal Longevity and Scale

A useful line of analysis is to consider the effect of scale changes for creatures which are similar in shape and only differ in scale. As the scale of an animal increases the body weight and 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: The surface area of the lungs for absorbing oxygen and evacuating carbon dioxide

The surface area of the intestines for absorbing food nutrients

The surface area of the skin for dispersing heat generated within the body

The cross sectional area of the arteries, capillaries and the veins for carrying the blood required to transmit the oxygen, carbon dioxide, food nutrients and excess heat.

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 Creature Blood

Pressure

(mm Hg) Height of

Head Above

Heart

(mm) Weight (grams) Human 120 500 90000 Cow 157 500 800000 Duck 162 100 2000 Cat 129 100 2000 Guinea Pig 60 25 100 Goat 98 400 30000 Pig 128 200 150000 Monkey 140 200 5000 Dog 120 200 5000 Turkey 193 300 15000 Frog 24 25 50 Giraffe 300 3000 900000 Snake 55 25 100

The linear regression of the logarithm of pressure on the logarithm of height yields the following result:



*log(Pressure) = 1.203 + 0.377*log(Height)

R2 = 0.675



The linear regression of the logarithm of pressure on the logarithm of weight yields:



*log(Pressure) = 1.45 + 0.154*log(Weight)

R2 = 0.619



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

(beats per

minute) Weight

(grams) Human 60 90000 Cat 150 2000 Small dog 100 2000 Medium dog 90 5000 Large dogs: 75 8000 Hamster 450 60 Chick 400 50 Chicken 275 1500 Monkey 192 5000 Horse 44 1200000 Cow 65 800000 Pig 70 150000 Rabbit 205 1000 elephant 30 5000000 giraffe 65 900000 large whales 20 120000000

A regression of the logarithm of heart rate on the logarithm of weight yields the following equation:

*log(heart rate) = 2.89 - 0.202*log(Weight)

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



heart rate = A(Scale)b

where b= -0.6

:

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 Creature (grams) (/minute) (years) (billions) Human 90000 60 70 4200 2.21 Cat 2000 150 15 2250 1.18 Small dog 2000 100 10 1000 0.53 Medium dog 5000 90 15 1350 0.71 Large dogs 8000 75 17 1275 0.67 Hamster 60 450 3 1350 0.71 Chicken 1500 275 15 4125 2.17 Monkey 5000 190 15 2850 1.50 Horse 1200000 44 40 1760 0.93 Cow 800000 65 22 1430 0.75 Pig 150000 70 25 1750 0.92 Rabbit 1000 205 9 1845 0.97 elephant 5000000 30 70 2100 1.1 giraffe 900000 65 20 1300 0.68 large whale 120000000 20 80 1600 0.84

Although the lack of dependence is clear visually the confirmation in terms of regression analysis is:



*log(Lifetime Heartbeats) = 9.006 - 0.0046*log(Wt)

R² = 0.0018



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.

Lifespan = B*(Scale)0.6

which in term of animal weight would be

Lifespan = C*(Weight)0.2

The data for testing this deduction are:

Lifespan versus Weight for Various Creatures Creature Weight (grams) Life Span

(years) Human 90000 70 Cow 800000 22 Duck 2000 10 Cat 2000 15 Guinea Pig 100 5 Goat 30000 15 Pig 150000 25 Monkey 5000 25 Dog 5000 15 Turkey 15000 5 Frog 50 3 Giraffe 900000 25 Snake 100 10

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



*log(Lifespan) = 0.970 + 0.191*log(Weight)

R² = 0.527



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



*log(Lifespan) = 0.970 + 0.573*log(Scale)



This says that if an animal is built on a 10 percent larger scale it will have a 6 percent longer lifespan.