Author's personal copy A plausible explanation for heart rates in mammals J. Flores a , E. Corvera Poire ´ a,� , J.A. del Rı ´o b,� , M. Lo ´pez de Haro c,�,1 a Departamento de Fı ´sica y Quı ´mica Teo ´rica, Facultad de Quı ´mica, Universidad Nacional Auto ´noma de Me´xico, Ciudad Universitaria, Me´xico D.F. 04510, Mexico b Centro de Investigacio ´n en Energı ´a and Centro de Ciencias de la Complejidad, Universidad Nacional Auto ´noma de Me´xico, A.P. 34, 62580 Temixco, Morelos, Mexico c Departamento de Fı ´sica, Universidad de Extremadura, E-06071 Badajoz, Spain article info Article history: Received 29 September 2009 Received in revised form 8 April 2010 Accepted 1 June 2010 Available online 8 June 2010 Keywords: Heart rate Allometric relations Viscoelastic fluid abstract We consider a simple model to give a plausible mechanical explanation of what are the actual resting heart rates of mammals optimized for. We study what is the optimal frequency for a viscoelastic fluid circulating in a pulsatile way through a network of tubes and conclude that the heart rate is not optimized to transport blood through the whole net. Rather, actual resting heart rates of mammals happen at frequencies that optimize flow in vessels of radii that correspond to large arteries, which bring oxygenated blood rapidly far away from the heart, towards head and limbs. Our results for the optimal frequencies, obtained using observed radii of femoral arteries in mammals, agree best with the heart rates observed. We find a theoretical allometric relation between optimal flow frequency and radius: n � R �1 . This one, agrees with the exponent obtained when plotting observed heart rates versus radii of both, femoral arteries and carotids in mammals of different sizes, from mice to horses. � 2010 Elsevier Ltd. All rights reserved. 1. Introduction The description of blood flow in a cardiovascular system represents a difficult and only partially understood problem. Due to its intrinsic importance, it has received a lot of attention from both, the physiological (Womersley, 1955; McDonald, 1960; Lipowski et al., 1978; Fibich et al., 1993) and the fluid dynamics communities (Lighthill, 1972; Pedley, 1980; Sarkar and Jayara- man, 2001). Traditional studies have considered the pulsatile flow of a newtonian fluid through a net of vessels (Krovetz, 1965; West et al., 1997; Pedrizzetti and Perktold, 2003) in which the elasticity of the vessels has been taken as a key point in the description. Such studies are basically concerned with the description of velocity profiles (Zamir, 2000). Among these studies, it has been found that wave reflection in the cardiovascular system may enhance, rather than impede the flow, and may thus be a reason for the branching nature of the arterial tree (Zamir, 1998). Basic questions like ‘why are arteries the size they are?’ or ‘what is the driving physiological principle determining artery size?’ are important to the physiological community because their answers could provide valuable clinical insight, possibly leading to novel diagnosis approaches and new therapies (Santamore and Bove, 2008). An important question that has not been answered is why mammal hearts beat at the rates they do. Heart rates during exercise and heart rates during recovery are essential in the diagnosis of multiple diseases. A first step towards the under- standing of heart rates in general is their study while mammals are at rest. These rates are known as resting heart rates. A theory should be able to understand what are resting heart rates optimized for, and to explain why heart rates increase during activity. Theoretical and experimental works on viscoelastic fluids flowing in tubes (del Rı ´o et al., 1998, 2001; Castrejo ´n Pita et al., 2003) have found that the dynamic permeability can increase orders of magnitude at certain frequencies. The dynamic perme- ability is an intrinsic property of the system viscoelastic fluid- confining media and determines the system’s response to different signals of the pressure gradient. It can be considered as a measure of the resistance to flow, the larger the dynamic permeability, the lower the resistance to flow. It has also been shown (Collepardo Guevara and Corvera Poire ´, 2007) that a dynamics for the pressure gradient with a properly chosen frequency provides a way of controlling the magnitude of the flow. The field of biological allometry refers to the relation between attributes of animals of different species as a function of their body mass or size. The field was energized by the publication in 1997 of a theoretical model to explain the scaling of metabolic rate with body mass in mammals (West et al., 1997). Traditional methods used to obtain allometric relations have been put on question because estimates for parameters involved in obtaining them may be inaccurate and misleading. For this reason, scaling Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology 0022-5193/$-see front matter � 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2010.06.003 � Corresponding authors. E-mail addresses: eugenia.corvera@gmail.com (E. Corvera Poire ´), arp@cie.unam.mx (J.A. del Rı ´o), malopez@servidor.unam.mx (M. Lo ´pez de Haro). 1 On sabbatical leave from Centro de Investigacio ´n en Energı ´a, Universidad Nacional Auto ´noma de Me ´xico, A.P. 34, 62580 Temixco, Morelos, Mexico. Journal of Theoretical Biology 265 (2010) 599–603