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Non-equilibrium Thermodynamics of Heterogeneous Growing Biosys-
tems
Natalya Kizilova1
1 Kharkov National University (Ukraine)
Abstract
Biological bodies are formed from heterogeneous tissues which are mostly composite materials with complex
mechanical, thermal, electrical, chemical properties, central and local regulation. Biomaterials have complicated
internal structures adapted for external loads and exhibit optimal mechanical properties like maximal perfor-
mance at total lightweight design [1]. Optimal structure of the tissues, organs and bodies is formed during their
growth and development, and knowledge of principles of optimal design in nature is important for technical and
biomedical applications.
Biosystems are open thermodynamic systems which are in permanent mass and energy exchange with environ-
ment. Biological growth as irreversible variations in body mass/volume/size is provided by new matter uptake
from environment and it delivery and distribution to the growing cell through special conducting pathways
by delivering liquids (blood, plant sap, trophic uids and tissue uids). Optimal performance of biosystems
is connected with minimum energy loss and entropy production which can be computed for dierent systems,
states and processes [1,2]. New matter delivered to the tissues is accumulated on the surfaces of the extra-
cellular structures and consumed by cells. In that way growth is connected with liquid-solid phase transitions
and can be described by formalism of non-equilibrium thermodynamic. Since physical and chemical reactions
at the surfaces determine the growth rate and direction, the surface phenomena must be taken into account
[3]. Collagen-based biotissues possess piezoelectric properties, and a wide range of electrokinetic phenomena is
proper to the growing tissues. Mass, heat and charge transfer at the membranes and solid surfaces are coupled,
and the relations between the forces and uxes must be generalized for an active free energy driven medium
with central and local electrical and chemical regulation.
Continual models are most relevant for mathematical modeling of growing biological bodies as multi-phase and
multi-component systems [4-7]. In this paper a thermodynamic model of the heterogeneous growing material
with internal variables is proposed. The material consists of several solid (extracellular matrix, connective tis-
sues, vessel walls) and liquid phases (intracellular, extracellular and delivering liquids). The internal parameters
describe the microstructure formation in solid phases. The mass, momentum and energy balance equations are
formulated and the relationships between uxes and forces are obtained from the entropy production term. The
model is applied for the problem of the heterogeneous tissue growth in a rigid scaold from a biodegradable
material as slow ltration of a multicomponent liquid in a porous medium with gradually increasing porosity.
The optimal regimes allowing a balance between the tissue growth and scaold degradation are proposed for
tissue engineering. Introduction of the vessel wall as separate solid phase gives an opportunity to model and
control the vascularized tissue growth in scaolds and bioreactors.
References
[1] Bejan A. Shape and Structure, from Engineering to Nature. Cambridge Univ. Press, 2000.
[2] Kjelstrup S., Bedeaux D., Johannessen E., Gross J. Non-equilibrium thermodynamics for engineers. World
Scienti c Pub., 2010.
[3] Kjelstrup S., Bedeaux D. Non-equilibrium thermodynamics of heterogeneous systems. World Scienti c Pub.,
Ser. Advances in statistical mech., Vol.16, 2008.
[4] Kizilova N.N. Constitutive equations and inverse problem solution in mechanics of growing biocompos-
ites. In: "Advanced Methods in Validation and Identi cation of Nonlinear Constitutive Equations in Solid
Mechanics". Moscow University Press. 2004.
[5] Kizilova N.N. Identi cation of rheological parameters of the models of growing biological continuous media.
J. Biomech. 2006. vol.39,N1.
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[6] Kizilova N. Thermodynamic modelling of biological growth. Proc. "Advanced Problems in Mechanics. Re-
pino. 2009.
[7] Kizilova N.N., S.A. Logvenkov, Stein A.A. Mathematical modeling of transport-growth processes in multi-
phase biological continua Fluid Dyn. 2012. v.47,N1.
E-mail: n.kizilova@gmail.com