Areas of Interest

My academic interests fall under the following relatively broad categories:

Current Research — The Mathematics and Mechanics of Growth in Tissue

Abstract

Growth in biological tissue depends upon cascades of complex biochemical reactions involving several species, as well as their transport through the extra cellular matrix and diffusion across cell membranes. In this work, a theoretical and numerical framework for the macroscopic treatment of growth is formulated within the context of open system continuum thermodynamics. Assumptions central to classical mechanics being too restrictive to capture such detail, this treatment involves the introduction of additional quantities (including mass sources/sinks, mass fluxes, terms for energy and momentum transfer between species) and deduces implications for balance laws. The framework, consistent with classical mixture theory, accounts for the multiple inter-converting and interacting species present in the tissue. Systematic adherence to fundamental physical principles, such as frame indifference of the response in the reference configuration and non-violation of the dissipation inequality, result in constitutive laws whose forms are clearly motivated and arise naturally from the treatment. Notably, the transport of the extra cellular fluid relative to the matrix is shown to be driven by the gradients of stress, concentration and chemical potential—a coupling of mass transport and mechanics that emerges directly.

A finite element formulation employing a staggered scheme is implemented to solve the coupled partial differential equations that arise from the theory. Nonlinear projection methods are utilised to handle incompressibility, mixed methods for stress-gradient driven fluxes and energy-momentum conserving algorithms are used for dynamics. Our current tissue of interest being engineered in vitro tendon, the numerical examples are in this context. The examples serve to demonstrate aspects of the coupled phenomena as the tissue grows. The classes of initial and boundary conditions imposed, and model geometry match parallel experiments (which form another integral part of a larger project studying growth and remodelling of engineered tissues). Representatively, concentration or flux boundary conditions (tissue exposed to fluid in a bath, fluid injected in at the boundary) are imposed for the balance of mass governing the fluid phase. Conditions in an experiment being consistent with specification of quantities in the current configuration, noteworthy differences that arise when solving these equations in terms of quantities in this configuration, as opposed to the reference configuration, where the equations are initially developed, are accentuated.

During the course of applying the theory to our system of interest, various meaningful modelling choices are made to tailor it to be representative of tendon constructs. While this tends to hint at a loss of generality, it is important to recognise that the fundamental theory and physical principles employed are still applicable to a larger class of problems, both from biology (injury mechanisms, wound healing, scarring and surgical repair, …) as well as from other diverse fields (from porous soil mechanics, to diffusion of air through anisotropic rubber materials in automobile tyres). With further refinements to the theory and concurrent maturation of the resulting computational framework, it is of future interest to extend the application of the theory to other classes of problems.

1. Introduction

1.1. Specific goals

The fundamental objectives of this research project are as follows:

1.2. Background and motivation

The processes involved in the development of biological tissue, though numerous and involving several cascades of complex interactions, are generally broken down into the distinct processes of growth, remodelling, and morphogenesis in biomechanics literature [nn]. This present work treats these processes as mathematically independent and its focus, growth, is defined to be an addition (or depletion, if one is dealing with the converse process of resorption) of mass through the processes of mass transport and biochemical reactions. Additionally, this is a continuum treatment at a macroscopic scale, rather than at a cellular or sub-cellular level.

Our Constructs
Figure 1.1: Different stages in the development of engineered tendon constructs [nn].

Recognising the complexity of the system presented (that it is open with respect to mass and energy, and contains numerous species which are capable of interacting and inter-converting) and the limitations of classical mechanics, additional terms (scalar mass sources/sinks, vectorial mass fluxes and terms for momentum and energy transfer between species) are introduced enhancing classical balance laws. The complex cascades of biochemical reactions are treated in an elementary fashion, using source-sink terms to govern inter-conversion and mass fluxes that supply nutrient and remove byproducts.

In the context of biological growth, the notion of a mass source was first introduced in [nn]. The notion of a mass flux is a more recent introduction [nn], but this work regarded fluxes purely as irreversible fluxes of momentum and entropy. In [nn], configurational forces motivate mass flux where the transported species is the same material as the tissue itself. These few cited examples of previous work are just a subset of a large body of theoretical and computational literature in this area. But, while the details vary, the body of literature represented by these works is largely based upon a single species undergoing transport and being produced/annihilated.

In addition to the possibility of multiple species undergoing cascades of reactions, the full range of driving forces for mass transport, such as chemical potential gradients, stress gradients, external body forces such as gravity, has not been systematically treated previously. Most previous coupling between transport and mechanics has been through growth-induced residual stress, as described in Section 2.1.4. As indicated at the outset, this work is aimed at a complete treatment mass transport, coupled with mechanics, for the growth problem.

Though the formulation is applicable to a large class of open systems with multiple species potentially participating in reactions, here it is used to model and predict the response and evolution of one specific tissue of interest to us, our engineered tendon constructs. These are functionally immature tendons formed by the self assembly of tendon fibroblasts in vitro [nn]. Figure 1.1 shows the development these constructs in time. During the course of development of this tissue, it undergoes numerous complex processes, but for the purposes of this growth model, we are focused on the evolution of concentrations of substances such as collagen (see Figure 1.2), as well as their dependence on mechanics.

[ Appropriate figure goes here ]

Figure 1.2: The evolution of collagen concentration with age [nn].

There are compelling clinical reasons to study tendons. Tendons consist mostly of collagen and perhaps provide the simplest physiologically relevant setting to understand the chemical and mechanical factors affecting proper collagen deposition. Collagen is the most important structural component of soft biotissue. Errors in collagen deposition cause important types of tissue dysfunction ranging from cardiomyopathy (hypertrophy of collagen makes the heart too stiff to undergo proper volumetric change) to hypertrophic scarring in burns (where the morphology of collagen is whirled rather than aligned in overly stiff scars).

Additionally, there are a large number of musculoskelital injuries each year which result in damage to soft tissues, including tendon. For tendons damaged beyond repair, replacement is necessary. This replacement must incorporate most native properties of tendon to restore function. However, such transplantation is limited by the availability of viable autograft, resulting in the use of synthetic materials which are unable to restore long term function due to incompatibility. Thus, a need exists for replacements which incorporate as many native properties as possible, necessitating a systematic study of engineered tendon.

2. Preliminary Work

2.1. Mathematical formulation of growth

In the following sections, the basic dynamical equations of the continuum formulation developed from physical principles governing the behaviour of growing tissue are summarised (detailed derivations of the same can be obtained from [nn]). Section 2.1.1 helps define the system and introduces fundamental quantities characterising it. Sections 2.1.2 and 2.1.3 present the balance of mass and balance of momenta respectively. Section 2.1.4 describes the treatment of growth kinematics. Key concepts from thermodynamics, the conservation of energy and the entropy inequality, are the subject of Section 2.1.5. Finally, the functional forms of the constitutive relations derived from the Clausius-Duhem inequality are highlighted in Section 2.1.6.

2.1.1. Defining the system

The tissue of interest is an open subset of with.

With that brief background out of the way, we now see how classical balance laws are enhanced by addition of these sources/sink and flux terms. Without really getting into the details, I am going to present relevant equations here. The terms that show up in the equations and figures have been described in slides 49 - 52 of this presentation. They have been written out for a species, in the reference configuration. As a final detour before I delve into the mathematics, it is beneficial to consider the continuum potato description of the system.

Continuum potato
The continuum potato picture of things.

It is easiest to comprehend the equation for the balance of mass by considering the two terms on the right hand side individually. The first arises due to the existence of a source term for the species under consideration, and the second arises due to the existence of a flux. (The negative sign arises from the convention that an outward normal is positive. And an outward flux will tend to reduce the concentration of that species.)

Balance of Mass
The balance of mass.

Similarly, we go on to see how linear momentum balance is affected by terms involving mass transport. In order to see the physical relevance, a material velocity, analogous to the flux in the balance of mass, relative to the solid is introduced. We then get the balance of linear momentum to be of the form,

Balance of Linear Momentum
The balance of linear momentum.

Again, it is easiest to see how this relates to classical results when each of the terms on the right hand side are analysed separately. The first two, corresponding to the body force and partial stress respectively, are classical terms. The final term is an addition in momentum to the system due to the flux of the species.

Before we proceed, we realize a subtlety with respect to the kinematics of growth. We thus apply a suitable multiplicative decomposition of the deformation gradient into a growth tensor and an elastic deformation gradient. The split will be apparent in the figure below.

Kinematics Continuum potato
The continuum potato picture describing growth kinematics.

Proceeding in a similar manner to what was followed for balance of mass and momentum, the (first law of thermodynamics) balance of energy is written out.

Energy
The balance of energy.

We recognize the terms on the right hand side to be the stress power terms, energy loss due to heat flux outwards, energy gain due to pointwise heat generation, energy gain due to internal energy inflow, and energy loss due to flux of species outwards. Finally, we write out the (second law of thermodynamics) entropy inequality, and combine it with the energy balance equation to get the dissipation inequality below.

Entropy
The dissipation inequality.

Assuming a suitable constitutive hypothesis for the stored energy (that it is a function of the elastic deformation gradient, the species density, and entropy) results in the following constitutive relations in accordance with the dissipation inequality.

Consitutive relations
Constitutive relations - 1.

The relations above are fairly standard and expected. The most interesting is the final one below.

Consitutive relations
Constitutive relation - 2.

This relation describes how the flux of a given species relates to the different driving forces. It can be thought of as a product of a mobility and the driving forces due to various phenomena (inside the parenthesis). It is easy to imagine this in terms of a wet sponge. If the sponge is moved with an acceleration in a certain direction, the fluid it holds will be accelerated in the opposite direction. This is the first term, the driving force due to inertia. The second is plain to see. If the sponge is under the influence of a body force, say gravity, the driving force on the fluid will be along that direction. The next term is the stress gradient term (Darcy's law). If you squeeze the sponge to a certain stress state, the fluid it contains will be driven in a direction as indicated by that term. The final term can be shown to be the driving force arising from chemical potential.

Features of the Computational Formulation

In order to solve the coupled system of partial differential equations arising from the theory detailed above, a finite element formulation employing a staggered scheme (Armero [1999], Garikipati et al. [2001]) was implemented in FEAP, a general purpose finite element program. This code base is continually being refined and enhanced as our understanding of the system grows. Some of the specifics of our computational formulation include:

Selected Numerical Examples

For the following numerical examples, we consider systems similar in geometry, material properties and boundary conditions to our biological models — engineered tendons in fluid baths. The tendon itself is modelled as a right circular cylinder, 12 mm in length and 1 mm2 in cross sectional area. The following is a representative spatial mesh used to represent it in the computations.

Relatively coarse mesh used in computations

In order to visualize the results of the computations, I have also linked to videos that show evolution of various fields with time. Clicking on specially marked images below should link you to the corresponding video. The plots are generated in FEAP, and the videos from these plots were rendered using Imagemagick's ccnvert.

1. Tendon Growing in a Fluid Rich Bath

In this example, a biphasic model — consisting of a solid and a fluid phase — was used to represent the tendon. The solid phase is representative of the collagen and proteoglycan network, and is modelled using a nonlinear material model based on the worm like chain model. The fluid phase, representing the extracellular fluid and dissolved solutes, is modelled as an ideal, nearly incompressible fluid with the bulk compressibility of water. An artificial source term, following a simple first order rate reaction law, is used to model the creation of collagen. The fluid mobilites were extracted from literature (Han et al. [2000]).

The boundary conditions, consistent with experiments, are immersing this tendon in a bath rich in fluid, i.e. the fluid concentration is maintained a constant (high) on the boundary and it is allowed to flux in through and out of all surfaces. The mechanical conditions are traction free. As we intuitively expect, we see the nutrient rich fluid diffusing into the tendon, facilitating production of collagen (growth of the tendon). The following are some results of the computations.

Collagen concentrations before and after growth

Videos of evolution of related quantities

Changes in volume of cylinder clearly showing diffusion and growth dominated regimes

Stress=extension curves showing a stronger tendon post growth, imitating experimental observations

Numerical Example
An image describing the numerical example.

Source term
Evolution of the source term.

Total Flux
Total flux driven by all effects.

Stress
Stress in the 33 direction.

(Will be populated when I have the time.)

Material forces and remodelling in biological tissue

Diffusion studies in Silicon

Emission studies in engines modified to LPG