# Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals

## Abstract

**:**

## 1. Introduction

## 2. The Cardiovascular System

_{I}(with units of pressure) is described from engineering theory by the equation

_{R}denotes blood-flow acceleration (in units of volume per time per time). Also, the coefficient β* denotes a factor that takes into account branching of vessels from the vessels under consideration. For all but the (non-branching) capillaries, its value may be estimated as ½, and for the capillaries it value is unity. The viscous resistance (with units of pressure) is similarly described by the Poiseuille equation with branching factor, that is

^{2}and the associated average flow is about 275 mL/s, as noted in earlier work of the writer [17]. The ratio Q

_{R}/Q in Equation (3) can accordingly be estimated as 11,000/275 or 40/s. Thus, with typical values of blood density ρ of 1.05 g/cm

^{3}and viscosity µ of 0.04 dynes-s/cm

^{2}[18] and with a radius r of 0.5 cm used for a representative artery, the equation gives the ratio f

_{I}/f

_{v}of 33, thus indicating that inertial forces can be expected to dominate in the arterial system. In contrast, with a radius r of 0.0005 cm chosen for a representative vessel from the capillary system, the equation gives 3.2 × 10

^{−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.

## 3. Theoretical Scaling Laws for Mammals—Review and Discussion

#### 3.1. Scaling of Heart Dimensions

#### 3.2. Scaling of Blood Vessels

_{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

_{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:

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

_{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

^{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

_{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

## 4.Comment on Theory and Variation of Solution

## 5. Some Comparisons with Measurements

#### 5.1. Arterial and Venous Vessels

^{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}= 0.175 M

^{3/8}(r

^{2}= 0.99) and L

_{a}= 20.6 M

^{1/4}(r

^{2}= 0.99)

_{v}= 0.379 M

^{7/24}(r

^{2}= 0.95) and L

_{v}= 8.30 M

^{5/12}(r

^{2}= 0.96)

#### 5.2. Capillary System

_{c}of the capillaries. These measurements involve the number of nephrons in the kidneys of mammals of various sizes. The nephron is the basic unit in the kidney and consists of a collection of capillary vessels from which fluid is extracted from the blood and wastewater, or urine, is produced. Assuming similar function, the number of capillaries per nephron will be the same for any mammal, and hence a count of nephrons in a kidney can be considered to be proportional to a count of capillaries in the kidney. The measurements of Kunkel (mouse to ox range) were analyzed by Adolph [20] and shown to obey a power law relation, with mammal mass raised to the power 0.62, which may also be taken as 5/8 and thus is consistent with the third of Equation (13b).

_{c}given by Equation (13b), since the diameter of the renal capsules may be considered proportional to the diameter of the contained capillaries within them.

_{c}varies with mammal mass to the power 0.07 and that net capillary length n

_{c}L

_{c}varies with mammal mass to the power 0.86. These exponents are in good agreement with the theoretical values from Equation (13b), that is, 1/12 (or 0.083) and 5/6 (or 0.83), respectively.

_{c}and net capillary length n

_{c}L

_{c}of individual mammals, as described by Equation (13b). This matter has been considered earlier by the author [12,13]. Best-fit calculations for the coefficients have been calculated from the data as

_{c}= 0.0027M

^{1/12}(r

^{2}= 0.95)

_{c}L

_{c}= 160M

^{5/6}(r

^{2}= 0.99)

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

#### 5.3. Heart Rate and Cardiac Output

**Figure 2.**Data illustrating variation of resting heart rate with mammal mass to the negative ¼-th power, as required by present scaling theory.

**Figure 3.**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.

## 6. Effects of Vascular Size on Function

#### 6.1. Oxygen Consumption Rate

_{c}(2π r

_{c}) L

_{c}. It must also be inversely proportional to capillary wall thickness h

_{c}. Assuming tentatively that the ratio of oxygen pressures inside a capillary and (resistive) pressure immediately outside is relatively invariant under change of scale and that the ratio of capillary wall-thickness to capillary radius is also invariant, the resulting similarity relation is found expressible as

_{0}denotes oxygen pressure in the blood

_{.}

_{c}L

_{c}can be seen from Equation (13b) to be proportional to mammal mass to the power 5/6. The product of the two in Equation (18) is therefore such that oxygen transfer and consumption rate are predicted to vary as mammal mass to the power 3/4, in agreement with observation when a wide range of mammal sizes is considered.

#### 6.2. Blood Pressures

**Figure 4.**Measurements illustrating relative invariance of blood pressure with mammal size. Data from literature survey by Seymour and Blaylock [9].

#### 6.3. Circulation Time

^{−3/4}n L r

^{2}, with n, L, and r denoting variables in any one of the sets of three variables given by Equation (13).

#### 6.4. Fluid Flow across Capillary Walls

_{f}across capillary walls can accordingly be expressed as

## 7. Validity of Scaling Relations for Children

**Figure 5.**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 [28]. Reference for circulation data is Seckel [29] and that for filtration-rate data is Schwartz and Work [30].

## 8. Application to Pediatric Drug Dosage

_{B}for a cycle of blood circulation, (b) a circulation number N to account for capillary exchange processes having a time scale different from that set by the blood circulation, (c) the net outward drug flow Q across the capillary walls, (d) the dose M

_{D}associated with body mass M, and (e) the blood volume V

_{B}. With relative dose M

_{D}/M denoted by D, the general relation is that the ratio of drug mass CV

_{B}in the blood to the initial drug mass DM must vary directly with the ratio of drug transport across capillary walls Qt to blood volume V

_{B}and with a general function f

_{1}of the ratio of time t to net circulation time NT

_{B}, that is,

_{B}and body mass M are proportional for adults and children and assuming proportionality between C and Q, the expression for the free concentration can thus be written (with Qt = QNT

_{B}× t/NT

_{B}) as

_{2}= f

_{1}× t/NT

_{B}. Scaling theory for physiologic processes, discussed above, requires further that the flow Q across capillary walls must vary with body mass M to the power 5/6 and that the time for circulation T

_{B}must vary as M to the power 1/4. Thus, the term QT

_{B}/M in Equation (22) can be seen to be proportional to M to the power 1/12.

_{B}is fixed, the ratio on the left hand side of this relation will be proportional to the first ratio on the right. This provides the scaling law for the relative concentration C/D as a function of time. The factor U is generally the same for children as adults. The desired scaling relations for concentration at time may thus be written, for the child relative to the adult, as

^{1/12}= K*, where K* denotes a constant which may be evaluated in terms of adult values to provide the scaling relation

_{D}/M written for D.

_{A}= 70 kg) and child (M

_{C}= 11 kg) are available for adult dose M

_{D}/M of 0.714 mg/kg and for child dose of 2.43 mg/kg. For the adult dose, Equation (25) requires a value for the child of 0.833 mg/kg. Measurements were for a dose of 2.43 mg/kg so that, for application here, concentration values for the child need to be reduced by the factor 0.833/2.43.

_{B}. Results confirming this variation are shown in Figure 6, thus providing support for the validity of the scaling relations considered here.

**Figure 6.**Illustration of validity of Equation (25) in predicting the dose level required to match concentrations of adult and child for scaled times. Basic data source: Walsh et al. [32].

## 9. Effects of Strenuous Exercise

## 10. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

- Clark, A.J. Comparative Physiology of the Heart; Cambridge University Press: Cambridge, UK, 1927. [Google Scholar]
- Brody, S. Bioenergetics and Growth; Reinhold Publishing: New York, NY, USA, 1945. [Google Scholar]
- Holt, J.P.; Rhode, E.A.; Kines, H. Ventricular volumes and body weight in mammals. Am. J. Physiol.
**1968**, 215, 704–715. [Google Scholar] - Holt, J.P.; Rhode, E.A.; Holt, W.W.; Kines, H. Geometric similarity of aorta, venae cavae, and certain of their branches in mammals. Am. J. Physiol.
**1981**, 241, 100–104. [Google Scholar] - Schmidt-Nielsen, K.; Pennycuik, P. Capillary density in mammals in relation to body size and oxygen consumption. Am. J. Physiol.
**1961**, 200, 746–750. [Google Scholar] - Gehr, P.; Mwangi, D.K.; Ammann, A.; Malooig, G.M.D.; Taylor, C.R.; Weibel, E.R. Design of the mammalian respiratory system. V. Scaling morphometric pulmonary diffusing capacity to body mass: wild and domestic mammals. Respir. Physiol.
**1981**, 44, 61–86. [Google Scholar] [CrossRef] - Hoppeler, H.; Mathieu, O.; Weibel, E.R.; Krauer, R.; Lindstedt, S.L.; Taylor, C.L. Design of the mammalian respiratory system, VIII. Capillaries in skeletal muscles. Respir. Physiol.
**1981**, 44, 129–150. [Google Scholar] [CrossRef] - Dlugosz, E.M.; Chappel, M.A.; Meek, T.H.; Szafransks, P.A.; Zub, K.; Konarzewski, M.; Jones, J.H.; Bicudo, J.E.P.W.; Nespolo, R.F.; Careau, V.; Garland, T. Phylogenetic analysis of mammalian maximal oxygen consumption during exercise. J. Exp. Biog.
**2013**, 216, 4712–4721. [Google Scholar] [CrossRef] - Seymour, R.S.; Blaylock, A.J. The principle of Laplace and scaling of ventricular wall stresss and blood pressure in mammals and bird. Physiol. Biochem. Zool.
**2000**, 73, 389–405. [Google Scholar] [CrossRef] - White, C.R.; Blackburn, T.M.; Seymour, R.S. Phylogenetically informed analysis of the allometry of mammalian basal metabolic rate supports neither geometric nor quarter power scaling. Evolution
**2009**, 63, 2658–2667. [Google Scholar] [CrossRef] - White, C.R.; Seymour, R.S. The role of gravity in the evolution on mammalian blood pressure. Evolution
**2014**, 68, 901–908. [Google Scholar] [CrossRef] - Dawson, T.H. Engineering Design of the Cardiovascular System of Mammals; Prentice Hall: Englewood Cliffs, NJ, USA, 1991. [Google Scholar]
- Dawson, T.H. Similitude in the cardiovascular system of mammals. J. Exp. Biog.
**2001**, 204, 395–407. [Google Scholar] - Dawson, T.H. Scaling laws for capillary vessels of mammals at rest and in exercise. Proc. R. Soc. Lond. B
**2003**, 270, 755–763. [Google Scholar] [CrossRef] - Dawson, T.H. Allometric scaling in biology. Science
**1998**, 281. [Google Scholar] [CrossRef] - Selkurt, E.E.; Bullard, R.W. The Heart as a Pump: Mechanical Correlates of Cardiac Activity. In Physiology; Selkurt, E.E., Ed.; Little Brown: Boston, MA, USA, 1971; pp. 275–295. [Google Scholar]
- Dawson, T.H. Modeling the Vascular System and its Capillary Networks. In Vascular Hemodynamics; Yim, P.J., Ed.; Wiley-Blackwell: Hoboken, NJ, USA, 2008; pp. 1–35. [Google Scholar]
- Elad, D.; Einav, S. Physical and Flow Properties of Blood, Chapter 3. Standard Handbook of Biomedical Engineering and Design; McGraw-Hill: New York, NY, USA, 2004. Available online: http://www.digital engineering library.com (accessed on 11 February 2014).
- Kunkel, P.A., Jr. The number and size of the glomeruli in the kidney of several mammals. Bull. Johns Hopkiins Hosp.
**1930**, 47, 285–291. [Google Scholar] - Adolph, E.F. Quantitative relations in the physiological constitution of mammals. Science
**1949**, 109, 579–585. [Google Scholar] - Kleiber, M. Body size and metabolism. Hilgardia
**1931**, 6, 315–353. [Google Scholar] - Brody, S.; Procter, R.C. Relation between basal metabolism and mature body weight in different species of mammals and birds. Univ. Missouri Agr. Exp. Station Bull.
**1932**, 166, 89–102. [Google Scholar] - Schmidt-Nielsen, K.; Larimer, J.L. Oxygen dissociation curves of mammalian blood in relation to body size. Am. J. Physiol.
**1958**, 195, 424–428. [Google Scholar] - Woodbury, R.A.; Hamilton, W.F. Blood pressure studies in small animals. Am. J. Physiol.
**1937**, 119, 663–674. [Google Scholar] - Gregg, D.E.; Eckstein, R.W.; Fineberg, M.H. Pressure pulses and blood pressure values in unanesthetized dogs. Am. J. Physiol.
**1937**, 118, 399–410. [Google Scholar] - Prosser, C.L.; Brown, F.A., Jr. Comparative Animal Physiology; W. B. Saunders: Philadelphia, PA, USA, 1961. [Google Scholar]
- Dawson, T.H. Scaling adult doses of antifungal and antibacterial agents to children. Antimicrob. Agents Chemother.
**2012**, 56, 2948–2958. [Google Scholar] [CrossRef] - Dawson, T.H. Scaling adult dose and schedule of anticancer agents to children. J. Cancer Res. Clin. Oncol.
**2013**, 139, 2035–2045. [Google Scholar] [CrossRef] - Seckel, H. Blood volume and circulation time in children. Arch. Dis. Child
**1936**, 11, 21–30. [Google Scholar] [CrossRef] - Schwartz, J.G.; Work, D.F. Measurement and estimation of GFR in children and adolescents. Clin. J. Am. Soc. Nephrol.
**2009**, 4, 1832–1843. [Google Scholar] - Graham, G.R. Blood volume in children. Ann. R. Coll. Surg. Engl.
**1963**, 33, 149–158. [Google Scholar] - Walsh, T.J.; Adamson, P.C.; Seibel, N.L.; Flynn, P.M.; Neely, M.N.; Schwartz, C.; Shad, A.; Kaplan, S.L.; Roden, M.M.; Stone, J.A.; et al. Pharmacokinetics, safety, and tolerability of caspofungin in children and adolescents. Antimicrob. Agents Chemother.
**2005**, 49, 4536–4545. [Google Scholar] [CrossRef] - Dawson, T.H. Scaling laws for plasma concentrations and tolerable doses of anticancer drugs. Canc. Res.
**2010**, 70, 4801–4808. [Google Scholar] [CrossRef] - Clark, D.L.; Andrews, P.A.; Smith, D.D.; DeGeorge, J.J.; Justice, R.L.; Beitz, G.J. Predictive values of preclinical toxicology studies for platinum anticancer agents. Clin. Cancer Res.
**1999**, 3, 11161–11167. [Google Scholar] - Baudinette, R.V. Scaling of heart rate during locomotion in mammals. J. Comp. Physiol.
**1978**, 127, 337–342. [Google Scholar] [CrossRef] - Taylor, C.R.; Wiebel, E.R. Design of the mammalian respiratory system. Respir. Physiol.
**1981**, 44, 1–10. [Google Scholar] [CrossRef] - Wiebel, E.R.; Hoppeler, H. Modeling design and functional integration in the oxygen and fuel pathways to working muscle. Cardiovasc. Eng.
**2004**, 4, 5–18. [Google Scholar] [CrossRef] - Bishop, C.M. Heart mass and the maximum cardiac output of birds and mammals: Implications for estimating the maximum aerobic power input of flying mammals. Phil. Trans. R. Soc. Lond. B
**1997**, 352, 447–456. [Google Scholar] [CrossRef] - Bishop, C.M. The maximum oxygen consumption and aerobic scope of birds and mammals: Getting to the heart of the matter. Proc. R. Soc. Lond. B
**1999**, 266, 2275–2281. [Google Scholar] [CrossRef]

© 2014 by the authors; licensee MDPI, Basel, Switzerland. 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/).

## Share and Cite

**MDPI and ACS Style**

Dawson, T.H.
Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals. *Systems* **2014**, *2*, 168-185.
https://doi.org/10.3390/systems2020168

**AMA Style**

Dawson TH.
Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals. *Systems*. 2014; 2(2):168-185.
https://doi.org/10.3390/systems2020168

**Chicago/Turabian Style**

Dawson, Thomas H.
2014. "Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals" *Systems* 2, no. 2: 168-185.
https://doi.org/10.3390/systems2020168