This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The modeling of the cardiovascular system of mammals is discussed within the framework of governing allometric relations and related scaling laws for mammals. An earlier theory of the writer for resting-state cardiovascular function is reviewed and standard solutions discussed for reciprocal quarter-power relations for heart rate and cardiac output per unit body mass. Variation in the basic cardiac process controlling heart beat is considered and shown to allow alternate governing relations. Results have potential application in explaining deviations from the noted quarter-power relations. The work thus indicates that the cardiovascular systems of all mammals are designed according to the same general theory and, accordingly, that it provides a quantitative means to extrapolate measurements of cardiovascular form and function from small mammals to the human. Various illustrations are included. Work described here also indicates that the basic scaling laws from the theory apply to children and adults, with important applications such as the extrapolation of therapeutic drug dosage requirements from adults to children.

The general subject of this paper is the modeling of the cardiovascular systems of mammals. The modeling to be dealt with is that associated with the derivation and discussion of allometric relations and scaling laws for the system. The present paper also involves discussion of the significance of these laws in understanding physiological processes of mammals. Earlier experimental measurements on various aspects of the cardiovascular system of mammals have been reported by Clark [

Scaling laws and the attendant similarity they indicate, when confirmed by measurements, are important in the general understanding of the cardiovascular system of humans. They can serve to illustrate that all mammals are “designed” according to the same general theory. Hence, the study of the system on the basis of measurements from other mammals can reveal important information for the human regarding physiological processes and proposed explanations. In a broader sense, the study of the system can provide guidance in dealing with complex modeling issues where different scaling laws apply to different parts of the same system.

Overall design of the cardiovascular system of mammals is well known. It consists of left and right sides of the heart (the pumps) and two major parts of the circulatory system (the pipes). The

Relative estimates of the inertial resistance (associated with blood acceleration) and viscous resistances (associated with blood velocity) in the various groups of vessels may be formed from representative values of vessel size. The average inertial resistance _{I}_{R}

Now, measurements of initial outflow from the left side of heart of the human show a relatively uniform rise to 550 mL/s in about 0.05 s [^{2} and the associated average flow is about 275 mL/s, as noted in earlier work of the writer [_{R}^{3} and viscosity ^{2} [_{I}_{v}^{−5} which indicates that viscous forces can be expected to dominatein the capillary system. Similarly, for the venous system, viscous forces can be expected to dominate because of the resulting near- steady flow of the blood after passing through the capillaries.

Directly associated with scaling laws for the vascular system are those associated with the heart itself. The author has considered the matter earlier [

Now, the two terms in the first of the above relations must both be proportional to body mass if their sum is proportional. These two relations, together with Equation (4b), therefore require that all three dimensions of the heart ventricles must scale with mammal mass according to the relations:

The author has derived earlier the scaling laws for the arterial, venous and capillary vessels of mammals [_{R}^{2}^{2}^{2}^{2}_{0} denoting the amplitude of ventricular wall force associated with periodic contractions and _{b}_{a}_{a}_{c}_{c}_{c}_{v}_{v}_{0}/^{2}l

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 _{a}_{a}_{v}_{v}_{c}_{c}_{c}_{c}_{c}

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

^{β}

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

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 [_{a}_{v}

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 [

It is worthwhile to note the differences that arise if values of

The value of 2/3 for the exponent

Holt ^{2} (for the coefficients) from the regression analysis also indicate excellent agreement between theory and measurement. Results are summarized below in Equations (16a) and (16b):
_{a}^{3/8} (r^{2} = 0.99) and _{a}^{1/4} (r^{2} = 0.99)
_{v}^{7/24} (r^{2} = 0.95) and _{v}^{5/12} (r^{2} = 0.96)

Considering now the capillary system, it may first be noted that direct experimental measurements are limited regarding the systemic side of the circulation, but the data available are consistent with the theory described here. In particular, there are the counting measurements of Kunkel [_{c}

There are also measurements of Kunkel [_{c}

Fortunately, for the pulmonary side of the circulation, there are more direct measurements of capillary volume and surface area of the pulmonary capillaries in mammals of widely different size [_{c}_{c}L_{c}

These data may also be examined for the capillary radius _{c}_{c}L_{c}_{c}^{1/12} (r^{2} = 0.95)
_{c}L_{c}^{5/6} (r^{2} = 0.99)

Values for the capillary radius, so determined in connection with Equation (17a), are shown for illustration purposes in

Data (circles) for capillary radius from pulmonary side of circulation compared with 1/12-th relation of Equation (17a).

Two remaining variables need discussion in connection with the basic Equation (6) and these refer to heart rate and cardiac output as associated with Equations (14) and (15). According to the theory, the former should vary with body mass to the negative ¼-th power and the latter with body mass to the positive ¾-th power. Data have been collected from works of Seymour and Blaylock [

Data illustrating variation of resting heart rate with mammal mass to the negative ¼-th power, as required by present scaling theory.

Graphical display of averaged resting cardiac output per unit of body mass. Open circles denote measurements from the right ventricle and closed circles denote those from the left ventricle.

The preceding discussion has indicated that strong similarity exists in the design of mammals of vastly different size. Regarding the effect of vascular size on physiological function, two examples have already been cited, namely heart rate and cardiac output. A few more will be mentioned here.

Resting oxygen consumption rate has long been a subject of interest in physiology because of its role as fuel supply in bodily function. Measurements by Kleiber [

The matter may also be examined from basic considerations, recognizing the equivalence of resting oxygen consumption rate and resting oxygen transfer rate from the capillaries. Total oxygen transfer from the capillaries is governed by the well-known diffusion equation for gases and is directly proportional to the product of the difference in oxygen pressure inside and immediately outside the capillaries and the capillary surface area _{c} (2π r_{c}) L_{c}. It must also be inversely proportional to capillary wall thickness _{c}_{0} denotes oxygen pressure in the blood_{.}

Earlier work by the author [_{c}L_{c}

Systolic and diastolic blood pressures in the arterial system of mammals have been known for many years to be essentially independent of mammal size, at least in the resting state [

In addition to the earlier works of Woodbury and Hamilton [

Measurements illustrating relative invariance of blood pressure with mammal size. Data from literature survey by Seymour and Blaylock [

The time for complete circulation of a small volume of blood around the cardiovascular system is known to be dependent on mammal size [^{−3/4} n L r^{2}

Application of this latter relation to the arterial connecting vessels, using Equation (13a), show that the time for travel through the arterial vessels varies as mammal mass to the power 1/4. Similarly, application of Equations (13b) and (13c) for the capillary vessels and venous connecting vessels will give the same result. The same may be expected for the pulmonary system. Thus, since the total time

Prosser and Brown [

Fluid flow across capillary walls is described by a relation similar to Equation (18) for oxygen flow except that the driving force is the (essentially) scale-invariant blood pressure rather than the oxygen pressure in the blood. The equation for fluid flow _{f}

The validity of this last relation can be demonstrated using measurements of glomerular flow in the kidneys of resting mammals as presented by Adolph [

A matter of importance with regard to the allometric equations and scaling laws discussed here is their validity when applied to children, particularly in connection with pediatric drug therapy. This matter has been discussed in recent work of the writer [

Data showing variation of resting circulation time and filtration rate with body mass of children and theory-based descriptions of 1/4-th and 5/6-th relations, as reported earlier by the writer [

An important practical application of the scaling theory of the present work concerns the scaling of known therapeutic drug dose and schedule for adult humans to children. This matter is of obvious interest in biomedical engineering and has been dealt with in recent works of the writer [_{B}_{D}_{B}_{D}_{B}_{B}_{1} of the ratio of time _{B}

To limit attention to the effective part of the drug dose associated with the free concentration, the fraction _{B}_{B}_{B}_{2} = _{1} × _{B}_{B}_{B}

Now, if the ratio _{B}

The dosing condition then follows from Equation (22) as ^{1/12} = _{D}

A simple illustration of the adequacy of the above relations may be constructed using data associated with the antifungal agent caspofungin, as reported by Walsh _{A}_{C}_{D}

These adjustments have been made and, according to the present work, the resulting concentrations should coincide with the adult data when both are considered as a function of relative time _{B}

With regard to the physical significance of Equation (25) for dose calculation, reference may be made to earlier work of the writer [

Illustration of validity of Equation (25) in predicting the dose level required to match concentrations of adult and child for

An interesting aspect of the work described in the present paper is that the cardiovascular system of mammals appears to be designed on the basis of resting conditions. This aspect has been noted earlier by the writer [

In particular, oxygen consumption rates of mammals in strenuous exercise have been found to vary with mammal mass raised to a power of about 7/8 and heart rate has been found to vary with mammal mass to a power of about −1/8, in contrast with the corresponding resting values of 3/4 and −1/4, respectively. These results for strenuous exercise are not in question, but they do, in fact, indicate that similarity, and the accompanying general scaling laws described here for the resting state, can no longer apply for the intense exercise state.

Modeling of the cardiovascular system of mammals has been discussed here, with the goal of describing relevant scaling laws for mammals of vastly different size. A main conclusion to be drawn from this work is that similarity exists for the cardiovascular system of mammals, as well as for related physiological processes, and that this similarity exists for the resting state. Associated with this similarity are scaling laws that can provide predictions of measurements from small mammals to humans for increased understanding of the cardiovascular system. The scaling laws can also provide tools to eliminate unnecessary experiments because of answers already provided by theory. Finally, as a specific application, the scaling laws can provide a means for extrapolation of therapeutic drug dose and schedule from adult to child.

I am grateful to Guest Editors Paul Agutter, Lloyd Demetrius, and Jack Tuszynski for including me in this special issue of

The author declares no conflict of interest.