ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 908–915.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
Yu.V. Vassilevski, V.Yu. Salamatova, A.V. Lozovskiy, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 935–942.
Concise Formulas for Strain Analysis
of Soft Biological Tissues
Yu. V. Vassilevski
, V. Yu. Salamatova
, and A. V. Lozovskiy
Institute of Numerical Mathematics of the Russian Academy of Sciences,
Moscow, 119333 Russia
Moscow Institute of Physics and Technology (State University),
Dolgoprudnyi, 141701 Russia
Received February 2, 2017
Abstract— We describe a method for the approximate solution of nonlinear elasticity prob-
lems in the framework of ﬁnite deformation for the case of hyperelastic isotropic materials.
This method enables one to write the resulting equations from the ﬁnite element method in
analytical form, which reduces the amount of computations and simpliﬁes the implementation.
This approach is implemented for several types of hyperelastic materials used to describe the
mechanical behavior of soft biological tissues.
In the recent years, the role of mathematical modeling in the solution of various biomedical
problems has been steadily increasing. Predictive modeling of various types of surgery procedures
and the development of telesurgery, where surgery is carried out by robots, may serve as examples.
An adequate description of the mechanical behavior of soft biological tissues by mathematical
modeling methods is of key importance for successful progress in this ﬁeld of medicine.
The development of minimally invasive surgery was the ﬁrst impetus to the development of
methods for modeling soft tissue deformations [1–3]. In particular, this was motivated by the
development of surgical simulators for surgeon training . Since the methods to be used were re-
quired to produce results online, simpliﬁed models were chosen such as linear models or mass-spring
models. Although these models were fairly easy to implement, they failed to produce an adequate
description of the mechanical behavior of soft tissues.
Experimental data show that the mechanical behavior of soft tissues is extremely nonlinear,
which necessitates solving nonlinear elasticity problems with ﬁnite (large) strains taken into ac-
count. The paper  suggests an approach in which the strain of nonlinear membranes made of
a Saint Venant–Kirchhoﬀ material (which is one of the simplest nonlinear models) is modeled by
a set of nonlinear springs, which is more eﬃcient than the conventional approach from the view-
point of implementation and the amount of computations. The paper  also suggests to use the
interpolation properties of barycentric coordinates and the principle of minimum potential energy;
in the case of triangular ﬁnite elements for a Saint Venant–Kirchhoﬀ material, this provides all
required formulas in concise analytical form.
The concept suggested in  can be applied to the whole class of isotropic hyperelastic materials
that can be used to describe the nonlinear behavior of soft biological tissues. The present paper
develops an algorithm for the approximate solution of nonlinear elasticity problems for the case
of ﬁnite strains of hyperelastic isotropic materials. By analogy with the Saint Venant–Kirchhoﬀ
material, we obtain a concise analytical representation of all required equations, which makes it
fairly easy to implement arbitrary constitutive equations for a hyperelastic isotropic material. This