Non-linear modeling of active biohybrid materials

https://doi.org/10.1016/j.ijnonlinmec.2013.03.005Get rights and content

Highlights

  • The non-linear theory of pseudo-elasticity is used to model biohybrid materials.

  • The constitutive theory includes the active-passive transition of muscle fibers.

  • Tensor quantities required for implementing the model in Abaqus are derived.

  • Numerical results to validate and verify the theory are given.

Abstract

Recent advances in engineered muscle tissue attached to a synthetic substrate motivate the development of appropriate constitutive and numerical models. Applications of active materials can be expanded by using robust, non-mammalian muscle cells, such as those of Manduca sexta. In this study, we propose a model to assist in the analysis of biohybrid constructs by generalizing a recently proposed constitutive law for Manduca muscle tissue. The continuum model accounts (i) for the stimulation of muscle fibers by introducing multiple stress-free reference configurations for the active and passive states and (ii) for the hysteretic response by specifying a pseudo-elastic energy function. A simple example representing uniaxial loading–unloading is used to validate and verify the characteristics of the model. Then, based on experimental data of muscular thin films, a more complex case shows the qualitative potential of Manduca muscle tissue in active biohybrid constructs.

Introduction

In nature, muscles perform as efficient, robust actuators capable of large non-linear elastic deformations. The internal structure is hierarchical with fibers embedded in a matrix of connective tissue, which transition from passive to active states and vice versa [13]. Additionally, their ability to generate tension and provide positive mechanical work under an action potential [57] makes muscles unique amongst biological tissue and attractive for use in engineered devices. Developments in tissue engineering have made it possible to fabricate constructs of muscle cells which may be combined with an artificial substrate, such as polydimethylsiloxane (PDMS), to create an active biohybrid material. These innovative materials have been used in various demonstration projects to function as bioactuators. Cultured muscle tissues are advantageous over engineered actuators because of their ability to vary in size from the micro- to macro-scale [17], [24] and achieve life-like movements [22], [34]. Furthermore, muscle tissue may be directly actuated with the application of a simple electrical stimulus [1] and can take advantage of dense energy sources, such as glucose and lipids [54]. Potentially, hindering further development of bioactuators is the time and cost associated with culturing cells for prototype devices. These obstacles may be mitigated by using mathematical models to first evaluate potential designs.

An early application of bioactuation was demonstrated using frog skeletal muscles for a swimming robot [22]. The robot used explanted muscle tissue, which did not allow for flexibility in actuator design. To overcome this limitation, cultured muscle tissue on a polymer substrate have recently been used. This approach has been applied using mammalian cardiomyocytes to create proof-of-concept devices including a force transducer [58], fluidic pump [54], gripper [17], biomimetic propulsion [34] and potential drug delivery device [19]. An advantageous characteristic of mammalian cardiomyocytes’ is their ability to spontaneously contract without external input. However, they require well-controlled environmental conditions, which limit the possible applications [1], [4]. Other studies have focused on insect muscle cells, which can function under a wider range of conditions for a longer period of time. For example, cultured lepidopteran dorsal vessel tissues have demonstrated the ability to displace a micropillar array [1] and could potentially be used as an actuator. The cultured myoblasts of the tobacco hornworm, Manduca sexta, provide yet another alternative, as a recent study has demonstrated [4]. Potential advantages of Manduca tissue include a simpler muscle fiber structure and the ability for oxygen to be supplied to the tissue via diffusion, eliminating the need for a vascular system. These attributes make it an attractive choice for its application in bioactuators and will be addressed in this study.

Concurrent with the success of bioactuator prototyping is the need to develop mathematical models of biohybrid materials, which may assist in the design of devices. Modeling the continuous transition between stimulation states is of particular significance to evaluate actuator performance. The creation of active tension in a muscle is commonly accounted for by introducing an additional term in the stress formulation. Models following this approach have been developed for myocardium and adapted for skeletal muscles [33], [35], [45], [55], [56]. A prominent example of modeling active stress in skeletal muscle is the work by Hill [23], where an active force–length relationship is defined. An alternative approach is based on the use of an augmented energy function, where an additive term is defined to account for the stimulated muscle fibers [15], [28], [37].

The method, known as active strain, utilizes the multiplicative decomposition of the strain tensor [3]. Nardinocchi and Teresi [32] model cardiac muscle as a hyperelastic material with an isotropic fiber distribution where the active deformation is assumed to be a physiologically-based constant parameter. In a more recent study, the model is expanded to include a relation between the active strain and the chemical potential, i.e. the calcium ion concentration in the muscle [8]. Multiplicative decomposition of the deformation gradient is employed by Boel et al. [6] and Shim et al. [50] to model the changing reference configuration of cardiac muscle. The resulting model accounts for the response of both the active and passive fiber contributions while neglecting energy dissipation. For smooth muscles, Stålhand et al. [53] propose a coupled chemo-mechanical free energy function to describe the behavior of active tissue accounting for an anisotropic fiber distribution. The strain-dependent contribution to the energy function is additively decomposed into passive and active parts, with the latter being dependent on the cross-bridge interaction.

In this paper, we expand the formulation used by Paetsch et al. [43] to model the continuous transition of muscle tissue between passive and active states, assume the general case of compressible materials and simulate the complex movement of active biohybrid constructs. We begin in Section 2 with an overview of the basic equations required to model a transversely isotropic hyperelastic material. Next, we define the transient network theory using the notation of Rajagopal and Wineman [49] and provide a brief review of pseudo-elasticity as it applies to muscle tissue. Section 3 contains the increments of kinematic and energy formulations necessary to derive the fourth-order elasticity tensor for a transversely isotropic material, with and without considering the effect of stress-softening. These general forms are specialized, in Section 3.3, to a specific constitutive law for a muscle transitioning between activation states. Uniaxial loading–unloading is used in Section 4 to verify and validate the characteristics of the model. We conclude with the simulation of a biohybrid gripper and show that cyclic contraction of muscle fibers induces coiling and uncoiling of a rectangular strip.

Section snippets

Basic equations

Muscles possess a complex and hierarchical structure, with the active tension of the whole muscle generated by proteins interacting on the molecular scale [30]. Our approach does not aim to model the individual micro-constituents of muscle tissue. Instead, we attempt to capture the mechanical behavior using phenomenological models. In this section, we give an overview of the kinematic tensors and associated invariants, along with a general pseudo-elastic energy function and the related

Numerical solution

Muscles, like many biological systems, have an inherently irregular geometry and often experience complex loads, making analytical solutions difficult. Exact solutions are possible only for a limited number of problems restricting the use of a constitutive model to simple geometries and boundary conditions. In this section we develop the tensor quantities required for implementing a nearly incompressible, transversely isotropic, pseudo-elastic material in the finite element software Abaqus [51]

Numerical results

We present numerical results to validate and verify the development of the previous sections. The numerical simulations are carried out in Abaqus/Standard, where the expressions of the Cauchy stress and the fourth-order pseudo-elasticity tensor are defined in the subroutine UMAT [51].

Experimental data have been used to determine material model parameters by Paetsch et al. [43] and the same values are used for the numerical simulations in this section. The additional parameter κ, used to define

Conclusion

This paper presents a generalization of the constitutive law presented by Paetsch et al. [43] to simulate muscle transitioning between active and passive states. The material considered is pseudo-elastic, nearly incompressible and transversely isotropic with fiber orientation parallel to the longitudinal direction of the muscle. To account for fiber activation we use the multiplicative decomposition of the deformation gradient and define stress-free configurations for both the active and

Acknowledgments

This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The work of C.P. was supported in part by the National Science Foundation IGERT Grant DGE-1144591.

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