#### 3.2. Scaling of Blood Vessels

The author has derived earlier the scaling laws for the arterial, venous and capillary vessels of mammals [

12]. Details are summarized here. First, the numbers of arterial and venous vessels in the body are assumed to be constant, independent of size. Next, it is noted that the flow rate

Q_{R} and flow

Q in Equations (1) and (2) may, for scaling purposes, be expressed in terms of heart rate

ω and ventricular volume

πa^{2}l as

ω ^{2}a^{2}l and

ω a^{2}l, respectively. Thus, with

F_{0} denoting the amplitude of ventricular wall force associated with periodic contractions and

E denoting the elastic modulus (with units of pressure) associated with subsequent relaxations, the equation relating cardiac output

Q_{b} (average outflow) to vascular resistance may be written in dimensionless (non-unit) form as

where

f denotes a general unspecified function,

r_{a} and

L_{a} denote radius and length of arterial vessels, respectively;

r_{c},

L_{c} and

n_{c} denote radius, length and number of capillary vessels; and

r_{v} and

L_{v} denote radius and length of venous vessels. Scaling laws follow from this relation by noticing that the left side will be fixed, independent of size, if the four ratios on the right side are fixed. Assuming constant values for the blood density

ρ, the blood viscosity

µ, the elastic modulus

E and contractile stress amplitude

F_{0}/

hl, the following relations may be written:

where

M has been substituted for the product

a^{2}l, as indicated by Equation (5).

In addition to these relations, three others may be written associated with the fact that the total blood volume in mammals varies directly with mammal mass. The blood volume in the connecting vessels and the capillary system can therefore similarly be assumed to vary in this manner. The following relations thus apply:

Equations (7) and (8) provide six relations between the eight variables

r_{a},

L_{a},

r_{v},

L_{v},

r_{c},

L_{c},

n_{c}, and

ω. Two additional relations are thus needed for determining their variation with mammal mass. The idea behind the development of the needed additional relations is that the variables associated directly with the “characteristic” capillary system described above can be expected to apply also to the capillaries of the ventricles, since their mass is proportional to body mass. Thus, the number of capillaries in the ventricles can be considered proportional to the number of capillaries

n_{c} associated with the Equations (7) and (8). The number of related cardiac cells in the ventricles can also be considered to be proportional to the number of capillaries supplying them. Thus, the volume of a single cardiac cell can be considered proportional to the ratio of heart mass to capillary number; or, since heart mass and body mass are proportional, the volume of a single cardiac cell can be considered proportional to the ratio

M/n_{c} . The characteristic length

d* of a cardiac cell is therefore expressible as

with length, diameter and wall thickness of the cells all scaling in this manner.

Now, cardiac muscle tissue consists mainly of contraction fibers which, when excited, provide the pumping action of the heart. The fibers consist of series connections of cardiac cells. Contraction is initiated in the upper heart by electrical discharge and subsequently spreads over the heart by signal propagation, causing the influx of ions into the cardiac cells making the fibers. There are two matters to be considered: (1) the heart rate as influenced by the diameter of the cardiac fibers, with the latter assumed the same as the linear dimension d* defined above; and (2) the heart rate as influenced by the influx of ions into the fibers.

Heart Rate and Fiber Diameter. It is generally known that the propagation speed

c of an electrical signal in cardiac muscle fiber varies with fiber diameter

d, with propagation faster in the larger diameter fibers. A simple power-law expression with

c proportional to

d ^{β} is typically assumed, with

β denoting a constant. A value of

β equal to 2/3 was determined appropriate by the author in earlier work [

12] and will be used here. Implication of other values will also be considered in later remarks. Regarding heart rate, it is assumed that the period between resting heartbeats is proportional to the ratio of heart length

l to signal speed

c. The heart rate

ω is, of course, equal to the reciprocal of this period, so that, with Equation (9) applying to

d and with

ω = c/

l, the following relation results

Heart Rate and Ion Movement. A second relation follows from consideration of the transfer of ions into and out of the cardiac cells making the fibers so as to cause fiber contraction and recovery. Let

m denote the mass of ionic substance moving into (or out of) a cardiac cell in time

t. The relation for rate of transfer

m/

t may be written (by analogy with Fick’s law for diffusion) for a cardiac cell of surface area

S and wall thickness

h as

where

K denotes a constant (in units of time per area) and Δ

F denotes the driving force. With cell diameter

d and length

l, the cell surface can be represented as the product of circumference π

d and length

l and the time for a heart cycle can be expressed as 2π/

ω. Equation (11a) can thus be written as

The left-hand side of this equation is equal to the ionic mass per cell volume, and this may tentatively be assumed to be independent of mammal size. With the constant

K and driving force Δ

F also tentatively assumed independent of size, it can be seen that the product of heart rate

ω and cell dimensions

h and

d must likewise be constant under change of scale. With

h,

d and

l all scaling the same way, as

d* in Equation (9), it can thus be seen that

heart rate is expressible as

Equations (10) and (12), together with the six expressions of Equations (7) and (8), provide a solution for the scaling laws for the arterial, venous, and capillary vessels and heart rate. The complete solution is expressible as [

12]

where it is noted that the number of arterial and venous connecting vessels

n_{a} and

n_{v} are invariant with scale change. The scaling relation for the heart rate is also determined from Equations (10) or (12) as

Using Equation (5) for heart dimensions, the left-hand side of Equation (6) provides the scaling relation for the cardiac output, namely

These last two relations are in agreement (at least in an average sense) with earlier studies when a wide range of mammals were considered [

12,

13]. This agreement accordingly provides support for the tentative assumptions used in developing Equation (13) through Equation (15).