Journal of Theoretical Biology 450 (2018) 37–42 Contents lists available at ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/jtbi A mathematical model for pressure-based organs behaving as biological pressure vessels Aaron R Casha a,∗ , Liberato Camilleri b , Marilyn Gauci a , Ruben Gatt c , David Sladden d , Stanley Chetcuti e , Joseph N Grima c a Medical School, Faculty of Medicine, University of Malta, Msida, Malta b Department of Statistics and Operational Research, University of Malta, Msida, Malta c Metamaterials Unit, Faculty of Science, University of Malta, Msida, Malta d St. Bartholomew’s Hospital, Barts Health NHS Trust, London, UK e Cardiovascular Center, University of Michigan, Ann Arbor, MI, USA a r t i c l e i n f o Article history: Received 27 November 2017 Revised 26 March 2018 Accepted 25 April 2018 Available online 26 April 2018 Keywords: Allometry, isometry Pressure vessel Biomechanics Physiology Dinosaurs a b s t r a c t We introduce a mathematical model that describes the allometry of physical characteristics of hollow or- gans behaving as pressure vessels based on the physics of ideal pressure vessels. The model was validated by studying parameters such as body and organ mass, systolic and diastolic pressures, internal and exter- nal dimensions, pressurization energy and organ energy output measurements of pressure-based organs in a wide range of mammals and birds. Seven rules were derived that govern amongst others, lack of size efficiency on scaling to larger organ sizes, matching organ size in the same species, equal relative effi- ciency in pressurization energy across species and direct size matching between organ mass and mass of contents. The lung, heart and bladder follow these predicted theoretical relationships with a similar rel- ative efficiency across various mammalian and avian species; an exception is cardiac output in mammals with a mass exceeding 10 kg. This may limit massive body size in mammals, breaking Cope’s rule that populations evolve to increase in body size over time. Such a limit was not found in large flightless birds exceeding 100 kg, leading to speculation about unlimited dinosaur size should dinosaurs carry avian-like cardiac characteristics. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Any vessel that is exposed to a trans-mural pressure gradient can be called a pressure vessel, with even plants being described in pressure vessel terms (Surridge, 2016). In the body there are sev- eral organ systems that are designed to contain fluid under pres- sure (Casha et al., 2015). These organs or organ systems may act as biological pressure vessels. All cells in the body are organized into a hierarchy of tissues, organs and organ systems. Such organ systems are constructed in a similar fashion across various mammalian species in that the basic building block of a pressure-generating organ is muscle tis- sue (Kohn, 2014). The optimization of efficiency of such tissues is important as such pressure-generating functions are by neces- sity energy-depleting since energy, required for pressure genera- tion, must follow physical laws and is energy intensive. The ther- ∗ Correspondence to: Medical School, University of Malta, Mater Dei Hospital, Msida, Malta. E-mail address: aaron.casha@um.edu.mt (A.R. Casha). modynamic energy or work required to compress a gas against a constant external pressure, or pressure-volume work, is fixed but the energy utilized by a biological system is higher due to thermo- dynamic losses, with the efficiency of muscle enthalpy (heat and work) production being around 12%, although higher values have also been quoted (Laughlin, 1999; Smith et al., 2005; Nelson et al., 2011). Organ systems can be divided into solid organs like the liver, pancreas and kidneys, and hollow organs that perform pressure- based work such as the heart, lungs and bladder (Casha et al., 2017). Hollow organs that behave as biological pressure vessels are subject to the same laws of physics as other material pressure ves- sels. It is likely that different pressure-based hollow organs in an- imals will demonstrate similar pressure vessel characteristics due to the similarity in the building blocks of muscle and bone, simi- lar size and genetic relationship (Katz, 1969; Hess, 1970; Ashmore, 1971; Alexander et al., 1979; Aerssens et al., 1998; Fritsch et al., 2009). This paper investigates the physical characteristics and allome- try of ideal pressure vessels to provide a mathematical basis to the scaling in size of pressure-based organs; and calculates the rela- https://doi.org/10.1016/j.jtbi.2018.04.034 0022-5193/© 2018 Elsevier Ltd. All rights reserved.