The second requirement we set out was that of passive cellular properties, these properties determine how the cell responds to forces applied by other cells or the boundary conditions. Before we discuss how such a relationship is represented within the model framework, we describe how forces act on continuous bodies using stress and how through a constitutive relationship this relates to the strain of the cell. This constitutive relationship then defines the passive properties of cells within the model.
2.3.1
Stress
In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring regions of a continuous material (the cell in this case) exert on each other. A summary of the major results of stress theory that we will make use of are presented in Appendix A.
Forces acting on a continuous body can be divided into two major groups, body forces and surface forces:
• Body Forces are forces that act on the volume of the cell, such as magnetic fields, gravity or inertia. These are assumed to be zero for the purposes of the model.
• Surface Forces These forces result from the physical contact of cells with other objects, such as other cells or substrates. These forces form the core of the model.
For a cell within a tissue, body forces are typically negligible: the effects of gravity, inertia and magnetism are small when considering a small cell in a viscous tissue.
Cells interact with their neighbours (defined using the Voronoi tessellation) through surface forces and these surface forces are divided into three categories: contact forces, viscous forces and active forces.
Contact forces represent normal forces to the interface between two cells such as pressure terms. They are calculated using a contact force model, which uses the information about the packing and stresses of neighbouring cells to determine the contact force between them.
Viscous forces are defined by the viscous model (which we will describe in the next section when considering cell-cell dissipation) and occur whenever two neighbouring cells move relative to one another.
Active forces represent active processes that occur between cells within a tissue such as protrusive activity (lamellipodia etc.). In the framework of the model, the only restrictions on the active forces are that they are balanced, so a cell must pull on other cells to move. We refer to modelling systems without and with active forces as passive and active systems respectively.
These three classes of surface force represent stresses applied between cells, we expect that these stresses affect the shape (strain) of the cell we are interested in. To explicitly capture these mechanical properties of the cell we must use a constitutive relationship that defines how the applied stresses affect the strain.
2.3.2
A Constitutive Equation
The aim of using a model is to provide a method of probing the mechanics under- lying processes within morphogenesis. This requires a relationship between the shape of the cell and its stress. Experiments performed on cells using atomic force microscopy (Alcaraz et al.[2003]), optical tweezers (Wottawah et al.[2005b]) and other techniques yield many different stress-strain relationships depending on the
environment, type of cell and the scale of the measurement. A relationship be- tween stress and strain is known as a constitutive equation and three candidates are:
• Linear Elasticity The strain of the material depends linearly on the ap- plied stress. Response times are instantaneous and there is no loss of energy with repeated loading and unloading.
• Viscoelasticity The strain of the material is related to the current stress and some information about the history and rate of deformation of the object. Many different viscoelastic models exist, but all involve a dissipative term, that means repeated loading and unloading cycles lead to energy being lost as heat. This is the typical model used for red blood cells (Thurston
[1972]).
• Power Law In many cells, the complex viscoelastic responses cannot be captured using only a few viscous and elastic elements. There are a dis- tribution of dissipation times within the cell as seen in other soft glassy materials (Balland et al. [2006]).
For the purposes of our model, we will use a linear elastic relationship. This has the advantage of being simple and due to the cell-cell viscous dissipation model, cells within the model tissue will exhibit viscoelastic behaviour.
This constitutive relationship could be modified to capture either viscoelastic- ity or power law behaviour, but for the purposes of this thesis we will use linear elasticity and observe the complexity of the behaviour that this simple assump- tion can generate, (this will be discussed further at the end of Chapter 4). We will now discuss linear isotropic elasticity in more detail to derive the results that we will use in this thesis.
Linear Isotropic Elasticity
Many materials including metals, concrete and plastics exhibit linear elasticity for small deformations. In the same way that linear transformations between rank 1 tensors (vectors) can be represented by rank 2 tensors (matrices), linear
transformations between the stress and strain rank 2 tensors is represented by a rank 4 tensor known as the fourth-order elasticity tensor, C. Using suffix notation: σij = Cijklkl (2.4)
Some materials such as wood and crystalline materials exhibit different stress strain relationships depending on the direction in which they are measured, this behaviour is known as anisotropy. This model assumes that the mechanical prop- erties of cells are isotropic, i.e. the same in all directions. This assumption will clearly break down for cells which are highly anisotropic behaviour, such as my- ocytes and neurons.
Isotropy further reduces the stress-strain relationship to the following:
σij = λkkδij + 2µij (2.5)
Here, δij is the identity tensor or the Kronecker delta. λ and µ are known as the
Lam´e constants. The above relationship is known as generalised Hooke’s Law. The three dimensional form constitutive equation that is useful in the model is:
σ = Kvol() + 2Gshear()
which decomposes the stress of the cell into two separate contributions: one related to the variation in the volume of the cell (vol()) and another related to the deformation of the cell at a constant volume (shear()).
The two elastic parameters are the bulk (K) and shear (G) moduli and • Bulk Modulus K describes how difficult it is to change the volume/area
of the object.
• Shear Modulus G describes how difficult it is to change the shape (or equivalently shear) of the object at constant volume/area.
The other elastic moduli that are typically used are the Young’s Modulus E and the Poisson Ratio ν, and these are represented in terms of K and G as E = 3K+G9KG and ν = 2(3K+G)3K−2G .
Linear, isotropic elasticity and the other results derived in this subsection provide the basis for defining the constitutive relationship relating the shape of
the cell to its stress that is used within the model. The model we use is two dimensional, so in the next section we will discuss how to move from the three dimensional results derived above to the two dimensional results used within the model.
Moving to Two Dimensions
As described previously, the modelling approach developed here will initially be used to probe systems that can be reduced to two dimensions. The results derived above for three dimensions are typically reduced to two dimensions in two ways: plane strain and plane stress.
• Plane Stress: The three dimensional object is bounded by two parallel planes separated by a distance that is small in comparison to the other dimensions of the problems. These parallel planes are then assumed to be stress free.
• Plane Strain: The three dimensional object is very large in one dimension compared to the others, so that the strain in this direction is assumed to be zero.
The model described in this thesis uses the plane strain assumption (since we are only interested in strains within the plane) to reduce the problem to two dimensions. Again we get:
σ = K2Darea() + 2G2Dshear()
It can be shown that the 2D elastic parameters (K2D and G2D) can be ex-
pressed in terms of the 3D elastic parameters: G2D = G
K2D = λ + G = K +G/3
For the remainder of this thesis, we will drop the 2D subscripts for clarity, but it should be remembered that the elastic moduli (K and G) relate to the two dimensional parameters rather than three dimensional ones.
This concludes our discussion of the passive cellular properties in the model. In the next chapter, when discussing implementation, we will describe how the different distributions of forces on a cell lead to different shapes and how these relationships can be used to generate dynamics within the model framework.