mechanical modeling of soft tissue

MECHANICAL MODELLINGMECHANICAL MODELLINGMECHANICAL MODELLINGMECHANICAL MODELLING OFOFOFOF

SOFT TISSUESOFT TISSUESOFT TISSUESOFT TISSUE A Literary ReviewA Literary ReviewA Literary ReviewA Literary Review

January 2002

Christopher M. AnthonyChristopher M. AnthonyChristopher M. AnthonyChristopher M. Anthony

Department of Mechanical EngineeringDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringDepartment of Mechanical Engineering Imperial College of Science, Technology and MedicineImperial College of Science, Technology and MedicineImperial College of Science, Technology and MedicineImperial College of Science, Technology and Medicine

Exhibition Road Exhibition Road Exhibition Road Exhibition Road London SW7 2BX.London SW7 2BX.London SW7 2BX.London SW7 2BX.

Abstract This report is a literature review of the mechanics of biological soft tissue. It provides a brief introduction into the necessity for research in the field, particularly for the purposes of trauma assessment due to impact and engineering for safer environments and practices. Theory is given on the biology of the tissue and how this affects its mechanical properties as a result. From this there is presentation of a number of methods for describing these complex mechanical properties mathematically, particularly in the form of constitutive equations representing stress-strain or similar relationships. Where appropriate, older methods are presented as the foundations for the newer developments in the field in order to illustrate how specific cases can be derived from fundamental principals both biologically and mathematically. It was found that most biological tissues are quasi-incompressible, non-homogeneous, anisotropic, non-linear viscoelastic materials with large deformation. Tissues discussed in the report include that of the lung, heart, tendons, ligaments, skeletal muscle, smooth muscle, brain, skin, liver and kidney. The report also gives some insight into the approaches towards the finite element solutions of the governing equations. Finally an overview is given of methods in the study of high rate loading on various biological tissues such as that of the brain and lung, with respect to high velocity impact.

Contents 1. Introduction 3. 2. Soft Tissue Modelling 4.

Physiology of Soft Tissue Non-Linear Elasticity and Bioviscoelasticity 5. Constitutive Modelling of Bioviscoelasticity 6. Uniaxial Models Viscoelastic Models Stress Response in Loading and Unloading 7. Quasi-Linear Viscoelastic Theory 8. The Elastic Response 10. The Reduced Relaxation Function Pseudo-Elasticity 11. Generalised Viscoelastic Relations and Multidimensionality 12. 3D Quasi-Linear Viscoelasticity 15.

3. Skeletal Muscles 15. Sliding Element Theory 17. The Hill Model 4. Tendons and Ligaments 19. 5. The Lung 20. 6. The Heart 21. 7. The Brain 22. 8. Smooth Muscles 23. 9. Continuum Mechanics Solution Overview 24.

Data Acquisition 25. Muscle Action Modelling 26. Finite Element Modelling 27. Finite Element Implementation 28. Tendons and Ligaments 29. Skin 30. Passive Skeletal Muscle 31. Muscle Contraction 32. Myocardium Tissue 38. Lung Tissue 39.

10. Impact and Trauma 46. 11. Discussion 51. 12. References 53.

1. Introduction Throughout the medical field, biomechanical engineers have strived to develop systems representative of human tissue for the purpose of simulation. Experimentation has been performed for centuries on humans and animals alike, in order to increase our understanding of the human biology and, as importantly, increase the safety of certain processes. Car manufacturers regularly place high-tech crash test dummies into new cars and drive them into walls at great expense both financially and in terms of set-up/analysis time. It is only recently that the biomechanical understanding of the body has been coupled with new technologies to produce useful results from such tests, but as with all processes even they have their limitations. As the field of finite element and finite volume methods has developed in the 20th and into the 21st century, it has become far cheaper, faster and easier to create and fully test models and prototypes virtually before a single part has been manufactured. But whilst constitutive models for common construction materials such as plastics and metals are well described and we can usefully predict the effect of large force upon them (such as in simulated crash testing of a vehicle), those of biological tissue and organs are still very much in development. Ability to model tissue accurately will significantly help designers produce safer products as we’ll be able to directly see the effects propagate through a virtual human. The process would be clean and perfectly repeatable. Stress, strain, velocity, vibration, force (and so on) information could be extracted from any number of places with no need for worry about whether the measuring system will fail. The information can then be used from a medical perspective to illustrate potential injuries and aid in the development of safer practices. This report is a literature review of past and current methods for establishing the mechanical properties of different types of soft tissue. It discusses different types of organs and how basic biomechanical models can be extended to more specific cases, such as skeletal muscle, cardiac muscle, tendons and ligaments, brain tissue, lung tissue and skin. It is important to bear in mind that the intention is to simulate a real biological process and so in each case the physiology of each organ or tissue type is discussed from a purely biological perspective. Discussion is then given for the derivation of these models from their associated physiology. A discussion is included on the finite element formulations of the proposed constitutive equations as well as some insight into considerations from a practical point of view. No information is given on the numerical solutions of the finite element process itself as it is assumed that the reader is either familiar with the concept, or as in most cases, such methods are hidden within FE software. Detailed explanation of the finite element method can be found in any good continuum mechanics book and are beyond the scope of this report. Brief discussion is also given to indicate how the surface mesh data can be acquired for the internal organs and how the input data leads to a solved tissue deformation. Where appropriate, advantages and disadvantages of the methods presented have been included, along with any potential limitations. Whilst it is not possible to review every method available, an attempt has been made to present the most significant work in the field a long with a history of previous work that has led up to it. It has been shown that a lot of research in this field is based on modifications to old methods and it is hoped that the models included here will at least provide an indication of how future research can be developed.

2. Soft Tissue Modelling [MAUREL98b] describes soft tissues as quasi-incompressible, non-homogeneous, anisotropic, non-linear viscoelastic materials in large deformation. The general procedure for describing the temporal and spatial “evolution” of the medium, involves treatment of it as a continuum in conjunction with their associated stress-strain constitutive relationships. The application and subsequent numerical solution of these equations along with the conservation equations through discretisation of the geometry into finite elements, and time will lead to such an “evolution”. It is assumed that the reader of this report has some knowledge of continuum mechanics and so the nature of the incremental/iterative methods for solving these equations is beyond the scope of this report. However such information can be found in most continuum mechanics books. This discrete approach is essential in this area of study due to the geometric and physical non-linearities involved in mathematical modelling of soft tissues. As such, analytical solutions are not only impractical, but also often impossible to find. Hence a discrete numerical approach will give results at nodes on the system as an approximation to the continuum. As we are interested in the deformation of the soft tissue, a Lagrangian description is utilised over an Eulerian one. Following a rigid body simulation of the desired skeletal motion in a system, the resulting skeletal muscle forces can then be used as the basis for simulation of the dynamic deformation of muscle and other soft tissue by direct application of continuum mechanics methods. Similarly, deformation and motion of other, non-skeletal (i.e. smooth) muscular organs such as the lung and heart can be simulated from the mathematical equations that govern them. A number of models have been suggested for such constitutive relations some of which are reviewed in this text. Physiology of Soft Tissue Prior to formulation of the governing constitutive equations of soft tissues, or indeed any material, it is essential that its mechanical properties are observed carefully and thoroughly understood. [FUNG72] suggests that in general, all soft tissues in the body are composed of collagen, elastin, reticulin and ground substance (a hydrophilic gel). Soft tissues do, however, show a large variety of mechanical properties depending on their function. It is believed that their structure varies greatly depending on the organ. For example, the microstructure of skeletal muscles is constructed from parallel fibres. Clearly this type of muscle exhibits a different mechanical behaviour based on whether the stresses occur parallel or perpendicular to the direction of the fibres.

Non-Linear Elasticity and Bioviscoelasticity Non-linear elasticity is an important property of all soft tissues. Parallel-fibered collagenous tissues such as skeletal muscle, show a non-linear stress-strain relationship when stressed uniaxially. [WOO89] describes this stress relationship as having an initial low modulus region, caused by the unfolding of the undulating fibrils, which exist in a relaxed tissue. This is followed by an intermediate region of gradually increasing modulus from the stress directly causing strain on the fibrils. During this direct loading, the modulus of the tissue increases. Once the fibrils become taught, the modulus reaches a peak and the tensile stress begins to increase linearly with increasing strain until finally the fibrils begin to reach their ultimate tensile stress and the modulus decreases representing the rupture of the tissue itself. Fig. 1 shows a graphical example of this behaviour. Whilst this may be observed in the slow moving, unidirectional case, [FUNG72] suggests that this is not strictly true for the dynamic situation. Since muscles are particularly dynamic, clearly something more appropriate is required. It is suggested in this text that for any given strain, the stress level is higher in the dynamic case than in the static or slow moving one i.e. the system has a strong dependence on the rate of strain. [FUNG81] suggests that three main temporal properties of (bio)viscoelastic materials are particularly important -

• Stress-relaxation [Fig. 2(a)] Upon suddenly applying a constant strain to a material, the stress in the material will reduce over time. This is known as stress-relaxation. .

• Creep [Fig. 2(b)] Creep involves applying a constant force to a material. The material will subsequently undergo some extension and the strain rate of lengthening decreases with time.

• Hysterisis [Fig. 2(c)]

The cyclic loading of a material is often path dependent. Typically the loading curve is higher than the unloading curve, the difference in area representing the energy lost in the process.

Fig. 1 Load-Extension Curve

a) Stress-Relaxation [VIIDIK80] b)Creep [VIIDIK80] c)Hysterisis [FUNG72] Fig. 2 Bioviscoelastic Behaviours (images taken from [MAUREL98b]) Constitutive Modelling of Bioviscoelasticity Before forming the constitutive equations for soft tissue, or indeed any material, it is essential to classify it according to the most appropriate method for modelling it. It has been discussed that soft tissues have a wide range of dynamic properties. [MAUREL98b] suggests that it would be suitable to classify it according to whether it is uniaxial or multi-dimensional, elastic or viscoelastic, phenomenological or structural. It is also suggested that the phenomenological approach (fitting mathematical equations to experimental curves) is convenient for generalisation and predicting behaviour. Structural models however, are constructed with respect to the assumed behaviour of the structural components of the tissue. These are more useful for trying to understand the reasoning behind the observed response by direct analogy of the material microstructure to mechanical components such as springs and dashpots. Uniaxial Models The earliest models of soft tissue were based on uniaxial constitutive relations which related the Lagrange stress, T=F/S0, to strain, ε=(L-L0)/L0, or the extension ratio, λ=1+ε. A number of similar relationships were derived from curve fitting from experimental data or continuum mechanics, such as ε2 = aT2 + bT for tendons [WERTHEIM47] ε = a(1-ebT) for elastic fibres [CARTON62] T = aeb for skin [KENEDI64] Viscoelastic Models The general approach to developing viscoelastic models is through the “discrete element combination” approach. Both empirical observation of the microstructure of the tissue and the experimental data can lead directly to estimations of useful combinations of commonly used elements such as linear and non-linear springs and dashpots in series and parallel. Three commonly used viscoelastic models are the

Maxwell, Voigt and Kelvin Bodies. Fig. 3 shows each of these physical combinations along with their associated response curves.

a) Maxwell Body b) Voigt Body c) Kelvin Body

Fig. 3 Common Viscoelastic Models (taken from [MAUREL98b]) But these models can also be used through series and parallel combinations to directly form other models and general solutions for the given models. For example Fig. 4 shows a model and response for generalised Maxwell model as an infinite number of individual Maxwell elements in series. [VIIDIK68] proposed the idea of a model composed of Hooke, Newton, Coulomb and Maxwell bodies in order to describe the non-linear viscoelastic behaviour of soft tissues. In [CAPELO81] we see an attempt to model passive cardiac muscle with two exponential springs and a hyperbolic sinusoidal dashpot.

Fig. 4 Generalised Maxwell Model (from [FUNG81]) The constitutive relations formed are often stated in the differential form and then integrated with the given boundary conditions. Stress Response in Loading and Unloading If we consider a one-dimensional specimen loaded in tension, we can define the tensile stress, T, as the force divided by the reference area, the stretch ratio, λ, as the length divided by the reference length, and the tensile strain, ε, as (λ2-1)/2. This reference state is arbitrary, but traditionally it is taken as the value in the “natural” state (i.e. the fully relaxed, unstressed state). The Eulerian stress, σ, referring to the cross-section of the deformed specimen (i.e. true stress) is trivially given by,

λλσ TAP

AP

ref

=== - Eq. 1

[FUNG72] and [FUNG81] offer a differential form of the stress-strain relationship of tissues as,

)( βαλ

+= TddT - Eq. 2

which can be directly integrated to give the general form,

ββ λλα −+= − *)(* )( eTT - Eq. 3 Here, the integration constant can be found from experimentally finding a point on the curve where T=T* and λ=λ*. An important observation made by Fung is that if λ is referred to the natural state, by definition, T=0 when λ=1, and this is only possible when,

)1*(

)1*(*

1 −−

−−

−= λα

λα

βeeT - Eq. 4

A number of expressions have been proposed, but those presented here have been used extensively to approximate a number of different tissues. They are also observed to be applicable for strains of up to approximately 30%. Quasi-Linear Viscoelastic Theory [FUNG72] and [FUNG81] describe an important mathematical description based on continuum mechanics, which can be used to characterise many soft tissues. Fung’s belief is that for oscillations of small amplitude about an equilibrium state, linear viscoelastic theory should be applicable. But for systems of finite deformation it is necessary to take into account the non-linear stress-strain characteristics. The proposal of quasi-linear viscoelasticity involves considering the relaxation function K(λ,t), as being composed of a reduced relaxation function, G(t) and an elastic response, T(e)(λ), a function of λ (i.e. the stretch) only. The relaxation function (i.e. the history of the stress response) can be assumed to be of the form, K t G t T e( , ) ( ) ( ),( )λ λ= G( )0 1= - Eq. 5 It can then be assumed that the stress response to an infinitesimal change in stretch, δλ(τ), is superimposed on a specimen in the stretched state, as a function of λ at an instant of time, τ, is (for t > τ),

[ ]G t

T e

( )( )

( )( )

−τ∂ λ τ

∂λδλ τ - Eq. 6

and, assuming that the superposition principle applies, such that,

[ ]T t G t

Td

et

( ) ( )( )

( )( )

= −−∞∫ τ

∂ λ τ∂λ

δλ τ τ - Eq. 7

Here, the tensile stress at time, t, is the sum of the contributions of the previous changes, each governed by the same reduced relaxation function. This can be written in the form,

T t G t T det( ) ( ) ! ( )( )= −

−∞∫ τ τ τ - Eq. 8

This shows us that the stress response is a described by a linear relationship between the stress, T, and the elastic response, T(e). The function T(e)(λ) is analogous to the strain, ε, in conventional viscoelasticity. This equation can be inverted as,

[ ] ∫ ∞−−=

t ee dTtJtT τττλ )()()( )()( ! - Eq. 9

which defines the reduced creep function, J(t). [FUNG81] suggests that if we say that T(e)(λ)=F(λ) and λ=F-1(T(e)) is the inverse of F(λ), where the stretch ratio corresponds to the tensile stress, then for unit step change of tensile stress at t=0, the stretch ratio becomes,

[ ])()( 1 tJFt −=λ - Eq. 10 Now, if the motion starts at time, t=0, and σij=eij=0 for t<0, we get

ττ

τλτ dTtGtGTtTt e

∫ ∂∂−+=

0

)( )]([)()()0()( - Eq. 11

and also given that tT e ∂∂ /)( and tG ∂∂ / are continuous in the range 0 ≤ t < ∞, it can be shown from the above that

∫ ∂∂−+=

t ee dGtTtTGtT0

)()( )()()()0()( ττττ - Eq. 12

or

∫ −∂∂=

t e dGtTt

tT0

)( )()()( τττ - Eq. 13

With G(0) = 1, as stated above, we have

[ ] ∫ ∂∂−+=

t ee dGtTtTtT0

)()( )()()()( ττττλ - Eq. 14

Fung observes that,

“the tensile stress at any time, t, is equal to the instantaneous stress response, T(e)[λ(t)] decreased by an amount depending on the past history, because ∂G(τ)/∂τ is generally of negative value.” The Elastic Response “By definition, T(e)(λ) is the instantaneous tensile stress generated in the tissue when a step function of stretching λ is imposed on the specimen.” [FUNG81]. From an experimental point of view, measurement of the elastic response function is difficult due to the generation of transient stress waves upon the sudden (i.e. step) loading of the specimen. Fung suggests that if we assume that the relaxation function, G(t) is a continuous function, then the elastic response may be approximated by the tensile stress response in a loading experiment with a suitable high rate of loading – T(λ) can be obtained as T(e). He justifies this by assuming that the continuous function, G(t), is a decreasing function. If by some monotonic process λ is increased from 0 to λ in time, ε, then at time t=ε,

∫ ∂∂−+=

ετ

τττλλε

0

)()( )()]([)()( dGtTTT ee - Eq. 15

As τ increases from 0 to ε, the integrand does not change sign and so

∂∂+= )(1)()( )( cGTT e

τελε - Eq. 16

with 0 ≤ c ≤ ε. Now, since ∂G/∂τ is finite, and small ε, such that ε|∂G/∂τ|<<1, then we can clearly see that T(e)(λ) ≈ T(ε), as required. The Reduced Relaxation Function Fung suggests that the reduced relaxation function, G(t), is usually analysed in terms of a sum of exponential functions, identifying each exponent with the rate constant of a relaxation mechanism.

∑∑ −

=i

tvi

CeC

tGi

)( - Eq. 17

From an experimental point of view, it should be noted that if the experiment is terminated too early, it is possible to achieve an incorrect limiting value, G(∞), corresponding to v0=0 in the above equation. It is also important to note that the representation of empirical data as a sum of exponentials is a non-unique process, so vi should not be interpreted literally. To place this in perspective, relaxation times in soft tissue could very realistically reach well above 1000 minutes. [FUNG72] presents a relaxation function of the form,

ττ

ττ τ

dS

deStG

t

∫∫

∞

∞ −

+

+=

0

0

/

)(1

)(1)( - Eq. 18

where, S(τ) = (τσ/τ)-1, or more specifically,

><=

≤≤==21

21

,0)(

ττττ

τττττfor

forcS - Eq. 19

Pseudo-Elasticity On the whole, soft tissues do not exhibit the criteria required for elastic behaviour. There is no single valued relationship between stress and strain – they exhibit hysterisis under cyclic loading. Under constant strain, they exhibit relaxation and under constant stress they exhibit creep. They are anisotropic and have highly non-linear stress-strain relationships. These properties also vary significantly under different physiological conditions. Clearly this makes the mechanical modelling of these tissues particularly difficult. The incremental law can be used to approach such problems. This method linearises the relationship between incremental stresses and strains by perturbing the material by a small amount about the equilibrium point. The elastic constants are only meaningful if the initial state is known, and are then only applicable to perturbations about that state. This incremental modulii depend strongly on the initial stress state and, in some tissues, the strain history also. It is common for the incremental law to be derived for a well defined physiological state or some other meaningful equilibrium state. [FUNG67] warns that the intermittent loading and unloading cycles are not parallel to each other, nor are they tangent to the encompassing curves. As such, “the incremental law varies with the level of stress and is not equal to the tangent of the loading and unloading curve of finite strain.” This is an important observation as in the past, derivations have been made relating the stress and strain in a finite strain situation, claiming that the derivative is the modulus of the incremental law where as really they should be determined by incremental experiments. The linearity between the incremental stresses and infinitesimal incremental strains, thus allows for the application of classical linearised theory of elasticity. As the stress and strain are uniquely related in both the loading and unloading states of a cyclic process, it is possible to treat the material two different elastic materials in each direction. And so the theory of elasticity can be used to describe and potentially solve for fundamentally inelastic materials. This process of using elastic theory to describe inelastic materials is known as pseudo-elasticity and is thus not an intrinsic property of the material, but rather a convenient way of describing the stress-strain response of a material in cyclic loading. A useful property of pseudo-elasticity is that it is relatively insensitive to strain rate. Indeed, an increase in strain rate of 103 is usually coupled by an increase in change of stress for a given strain of a factor of only 1 or 2. Similarly, for the most part, energy dissipation does not change by more than a

factor of 2 or 3 with a frequency change from 1KHz to 10MHz in ultrasound experimentation. Hence for most tissues, attenuation per cycle is practically independent of frequency. However, [MCELHANEY67] discovered that the stress increases by 2.5 times at a given strain upon increasing the strain rate from 0.001 to 1000s-1. As useful as pseudo-elasticity is for soft tissues, it must be noted that it is still only an approximation. Essentially though, it allows for a useful description of the stress-strain relationship in both directions of cyclic loading to be analysed and described by elastic rather than viscoelastic laws. Generalised Viscoelastic Relations and Multidimensionality The description of the relevant constitutive equations of bioviscoelastic materials is presented in [FUNG81] as an extension of the uniaxial stress state described by quasi-linear viscoelasticity theory to higher order (i.e. 2 and 3 dimensionally). Eq. 11 is generalised into a tensor equation,

∫ ∂∂

−+=t e

klijklijkl

eklij d

EStGtGStS

0

)()( )]([

)()()0()( ττ

ττ - Eq. 20.

where, Sij is the Kirchhoff stress tensor, E is Green’s strain tensor (with components Eij) and Gijkl is reduced relaxation function tensor, with Gijkl =1 for t=0 and )(e

ijS as the elastic stress tensor corresponding to E, representing the instantaneous stress upon sudden increase in strain from 0. The elastic response, )(e

ijS , can be approximated by a pseudo-elastic theory. It can be observed that Gijkl will only have two independent components in an isotropic material, and is likely to have a reduced relaxation spectrum, G(t), as described above. [LANIR83] presents a method for describing fibrous connective tissue in three dimensions based on the microstructure and the energy of the system. It works by assuming that the tissue is composed of a series of different types of fibres embedded in a fluid matrix and then forming a strain energy function based on their anisotropic properties. Considering a unit volume of tissue, the distortional energy, W1 is expressed as the sum of the strain energies of each type of fibre, thus,

∑=k

kWW1 where ∫Ω ΩdwuRSW kkkk )()( λ - Eq. 21

where, Sk is the volume fraction of unstrained k-type fibres, Rk(u) is the density of the fibres in the unit direction, u, wk(λ) is the uniaxial strain energy function of the k-type fibres, and λ and Ω are the individual fibre stretch ratios and the range of fibre orientations.

A fibre strain-energy function, )(* λkw was then introduced to account for the non-uniform undulation of each k-type fibre along it’s direction, u, by means of an undulation density distribution function, Dk,u(X). Lanir developed this for skin tissue, but later extended it to lung tissue [LANIR83b] as will be discussed later. [HOROWITZ88] used this technique to model the properties of the myocardial muscle by assuming a composition of muscle fibres and collagen fibres in a gelatinous ground matrix substance. Incompressibility was assumed and the following stress tensor was defined,

S = EIL

EW

∂∂

+∂∂ 31 - Eq. 22

where, W1 is the fibre distortional energy function from Lanir, L is the Lagrange multiplier for the matrix hydrostatic pressure and I3 is the 3rd invariant of the Cauchy-Green right dilation tensor, C. Now, by assuming a fibre waviness distribution function, Dk,u, and a straight fibre constant stiffness, Ck, the stress-strain relationship of the wavy fibres was presented as,

∫='

0 ,* )()()'(

εεε dxfxDf kukk - Eq. 23

where, fk(ε) = Ckε is the stress-strain relationship for straight fibres,

xx

21'+−= εε true wavy fibre strain,

rs

sr

E'1

'1

'ξξ

ξξε

∂∂

∂∂= total fibre strain, and

Ers is the global tissue strain component.

Fig. 5 The orientations of the fibre coordinate system. Horowitz considers that the wavy fibre stress, fk

* derives from the strain energy wk*(ε)

thus finally leading to the global tissue stress tensor,

S = EILd

EfuRS kkk ∂

∂+Ω

∂∂

∫Ω 3* ')'()( εε - Eq. 24

where,

')(

)'(*

*

εεε

∂∂

= kk

wf

[SHOEMAKER86] used the work of Lanir and Horowitz et al. to develop methods for modelling skin. Here it is assumed that the material was composed of a compliant component (from a gelatinous matrix) and a component directly from the fibres. This was then used to derive an expression for the second Piola-Kirchoff stress tensor, S, as S = SF + SC

SF = ∫− ∂∂2/

2/)(

π

πθθ d

EE

SD ff Fibrous stress

SC = ∫ ∞− ∂∂−Λ

tdEtg τ

τττ )()( Compliant Stress - Eq. 25

where, D(θ) is the fibre orientation distribution function, θ is the angle of fibre with respect to the x axis, Sf is the fibre stress measure, g(t) is the reduced relaxation function, Ef is the fibre strain measure and Λ is a constant tensor. The fibre stress measure, Sf is also assumed to be a viscoelastic function of Ef, thus,

∫ ∞− ∂∂

−=t f

f dE

tGS ττ

ττ

)()( - Eq. 26

with, G(t) = G0g(t),

∫∫

∞

∞ −

+

+=

0

0

/

)(1

)(1)(

ττ

ττ τ

dfa

defatg

t

,

f(t) = e-pt and a = constant also,

Ef = ])1[(102

0

Σ−Σ+Σ− −bbλ

- Eq. 27

where, Σ and λ are the fibre strain and stretch ratio respectively, Σ0 and λ0 are the effective straightening fibre strain and stretch ratio respectively, Σ- the perpendicular line strain, and, G0 and b constants, with 0≤ b ≤1. 3D Quasi-Linear Viscoelasticity The natural extension of Fung’s one-dimensional quasi-linear viscoelasticity is to three dimensions. In order to do this, it needs to be written in the form,

∫ ∞− ∂∂

−=t e

klijklij d

EStGtS τ

τττ )]([

)()()(

- Eq. 28

with, Gijkl(t) = G(t)(1)ijkl As with the CHARM implementations described above, this was then approximated with a Prony series to form, ∆Sij = (1-Dv)∆Sij

(e) – (Sv)ij - Eq. 29 where,

∑

−

∆−= ∆− )1(1 / ατα

ατ

et

gDv and

∑=

∆− −−=N

ije

ijt

ij tAtSe1

)(/v ))()()(1()(S

α

ατα

3. Skeletal Muscles Skeletal muscles form the large part of the human body’s muscle system. They are controlled by voluntary nerves and provide the motion of the skeletal system. The tetanized state is defined as the state in which the muscle is stimulated at a sufficiently high frequency to generate and maintain a constant maximal tension [FUNG81]. A resting muscle is a viscoelastic material. It is most often studied in the tetanized state, as this is most relevant to its primary function. The stress in the tetanized muscle is significantly higher than that in the fully relaxed muscle. As such, the resting stress plays little role in skeletal muscle mechanics other than to set the resting length of the muscle. Fig. 6 shows the composite structure of skeletal muscle. The main units of skeletal muscle are the fibres. The fibres are arranged in bundles of varying size, called fasciculi. Within a bundle, the space between muscle fibres is filled with connective collagenous tissue. Similarly, a stronger sheath of connective tissue surrounds each bundle. The whole muscle itself is further surrounded by a stronger sheath of tissue.

The skeletal muscle fibre has a diameter of 10-60µm, with a length ranging from several millimetres up to 30 centimetres. Some fibres can stretch who whole length of the muscle, whilst others span only a fraction of the total muscle length, terminating in a tendon at the edge of the muscle, or further connective tissue attached to other fibres. The cytoplasm is itself divided into long threads called myofibrils, each of which has a diameter of approximately 1µm. The myofibrils contain different regions within them. These regions, or bands include the isotropic, or I-bands, which are bisected transversely by the zwischenscheibe, or Z-bands, and the anisotropic A-bands which have a highly ordered sub-structure and are bisected by H-bands. Muscle contraction involves the I- and H-bands narrowing, whilst the A-bands remain unaltered. Each myofybril is constructed from a number of myofilaments. They are divided crosswise by the Z-bands into a series of repeating units called sarcomeres, approximately 2.5µm in length (depending on the force acting on the muscle, and it’s state of excitation). There are two types of myofilament in each sarcomere – fine ones, which are actin molecules, approximately

5nm in diameter, and thicker, myosin ones which are 12nm diameter. The actin filaments are each attached to a Z-band at one end, and unattached at the other, where they overlap with the myosin filaments. Referring back to the banding structure, the A-band consists of the myosin filaments, and the I-band consists of the actin filaments which do not overlap with the myosin ones. The H-bands are in the centre of the A-bands where there are no actin filaments. A final M-band lies across the middle of the H-bands. This band consists of fine

Fig. 6 Skeletal Muscle Structure [GRAY91]

strands of interconnecting adjacent myosin filaments in a hexagonal pattern [FUNG81][GRAY91]. Sliding Element Theory It is known that when a muscle contracts, the actin and myosin filaments slide relative to each other with relatively little change in their own lengths. The myosin filament consists of two moieties – a light meromyosin, which is the tail of the filament, and a heavy meromyosin, which represents the head. Each myofilament is formed by spiralled sets of moieties as shown in Fig. 6. The heads of which are clearly visible in the diagram, are at 120o to each other and equally spaced at 14.3nm intervals. In the relaxed state, these heads lie flat and when excited, they point towards the actin filaments. Collectively these heads are called cross-bridges. Cross-bridge theory has been investigated, and could provide an excellent predictor of steady-state muscle force and energetics but generally it is computationally prohibitive to consider the activity of each cross-bridge in each muscle. The constitutive equations that are used to describe skeletal muscle motion are essentially attempting to describe the properties and behaviour of populations of cross-bridges [LIEBER00]. The ability for a muscle to respond to a stimulus is related to the size of a motor unit. The motor unit is defined as a group of the muscle fibres that are connected to and controlled by a single motor nerve fibre. Overall, small muscles that react quickly and have precise control, have small motor units and many nerve fibres going into each muscle. Large muscles, on the other hand, do not require the same control and response as small ones, and so have many (up to a thousand) muscle fibres in each motor unit. The fibres in adjacent motor units often overlap by ten to fifteen fibres, and this is responsible for some communication and accordance between them. Motor units can vary significantly in strength and size. Smaller motor units are usually more easily excited than larger ones as they are activated by smaller nerve fibres in the spinal cord with higher levels of excitability. This leads to “steps” in weak muscle contraction, which increase with the magnitude of contracting force. This is usually accounted for by offsets in the firing of different motor units such that as one unit is contracting, another is relaxing and so on. Overall, the biomechanics research into the activation of muscles is often considered on the muscle fibre or sarcomere level upon which the mechanics of muscle bundles can be derived by addition of other constituents [FUNG81]. The Hill Model Archibald Vivian Hill is most noted for deriving the Hill equation and, consequently, his proposed three-element muscle model. Both of these have been referred to extensively in modelling of muscle mechanics and have largely been extended. Hill’s equation is given as, ( )( ) ( )v b P a b P a+ + = +0 - Eq. 30 where, P = tension in muscle, v = velocity of contraction and a, b, P0 are constants,

It is has been referred to as the most well known equation in muscle mechanics and more or less represents an equivalence between the rate of work done and the chemical conversion of energy which provides it. Essentially though, the equation refers to the ability of a tetanized skeletal muscle to contract and was discovered empirically from experimentation into the electrical stimulation of frog satorius muscle. The equation shows that there is a hyperbolic relationship between the tension in the muscle, P, and the velocity of contraction, v. Larger loading is coupled with slower contraction velocity and higher tension in the muscle. This is a direct contrast to the viscoelastic behaviour of passive materials, where higher velocity of deformation requires larger forces to cause the deformation. Further experimentation by Gordon et al. [GORDON66] on frog muscle cells with an unloaded sarcomere length of 2.1µm revealed a number of properties of muscle contraction. When the sarcomere length is between 2.0 and 2.2µm, the maximum tension is not dependent on the length of the muscle. However, outside this range, the maximum developed tension is smaller. This happens because of variations in the number of cross-bridges between the myosin and actin fibres. In the case that the muscle is too long, the filaments are too far apart and the number of cross bridges decreases coupled with a decrease in tension. If the muscle is below this range, the actin filaments obstruct each other, hindering the ability for cross bridges to form.

Whilst this equation is a good indication of the tensions within a tetanized muscle, it cannot say anything about single muscle twitches or wave summations from multiple twitches, as it has no temporal components to do so. Similarly it cannot describe the force-velocity relationship of a tetanized muscle in slow release and there is no provision for the strain rate at any particular time. As a more complete model, Hill proposed his three-element model as shown in Fig. 7. Since Hill first proposed this model, a number of modifications have been made to it to

take into account new ideas and discoveries in the field. Examples of this will be discussed later. Although new models look quite different, they basis is almost always the original Hill model. The three-element model is constructed from a contractile element which is freely extensible at rest, and capable of shortening upon activation, an elastic element in series with the contractile element and another elastic element in parallel with these, representing the elasticity of the muscle at rest. The contractile element is representative of the sliding actin-myosin molecules and the generation of tension from active cross-bridges. The series element is implemented to account for the

Fig. 7 Hill’s Three-Element Model

elasticity of the actin and myosin molecules and cross bridges, Z-band and connective tissue. Now, if it is assumed that there is no tension in the resting muscle, its stress-strain relationship determines the constitutive equation of the parallel element. Thus the difference between the muscle properties and that of the parallel element determines the effect of the contractile and series elastic elements together. However it is very difficult to distinguish between the strains in these series elements. Once this has been achieved though, three velocities can be considered with respect to the overall speed of muscle contraction – the rate of change of muscle length, contractile element and series element along with two tensions – series (S) and a parallel (P). Hill’s equation specifically describes the tension in the contractile element (as the tension in the parallel element was negligible in his experimentation). The series tension is related to d∆/dt or dη/dt which are equal and opposite, where ∆ is the overlap length between actin and myosin filaments, and η is the extension of the series elastic element in the sarcomere. Hill’s equation can then be written for the tetanized muscle, in the form,

)/(1)/(11

0

0

0 SScSS

dtd

v +−

=∆ - Eq. 31

where, c is a constant, S0 is the tension in a tetanized isometric contraction and v0 is the velocity of d∆/dt when S = 0. A temporal extension of this is presented in [FUNG81], which derives the following for the time for the stress to change from σ1 to σ,

∫ −++==

θ

θθγ

αγ 1 )1)((1|

01 dx

xkxx

vt - Eq. 32

Hill’s model exhibits a number of problems. The contractile element is assumed to be stress free and freely extensible in the resting state, which is not possible. The series and parallel elements are not perfectly elastic, and the whole muscle cannot realistically be modelled by such a simple arrangement of parallel and series elements. The contractile element itself is really just a feature which is designed empirically by relating an input to an output but never really taking into account the mechanics behind it [BOGERT98]. Natural extensions to the model include making the elastic elements viscoelastic (or more precisely, bio-viscoelastic) and having larger, more complex formations of them. Further modifications to the basic Hill approach are discussed later in this report. 4. Tendons and Ligaments Tendons and ligaments are connecting tissues in the body. Tendons connect muscle with bone whilst ligaments connect bone to bone. As with most biological structures, tendons and ligaments have highly non-linear stress-strain properties. The tissue is composed of a network of collagen fibrils in a hydrated matrix of elastin and proteoglycans. At rest, the collagen fibrils are significantly crimped or wavy so that

initial loading acts primarily to straighten them. At higher strains, the straightened collagen fibrils must be further lengthened, which requires a greater stress. As a result, tendons and ligaments are more compliant at low loads, and less compliant at higher loads. The highly non-linear low-load region is often referred to as the toe region and occurs at up to approximately 3% strain and 5MPa stress [ZAJAC89].

Fig. 8 Tendon Hierachy [KASTELIC78] (image taken from [FUNG81])

Typically, tendons and ligaments have linear properties from approximately 3% until ultimate strain, which occurs at roughly 10%. The tangent modulus in this region is approximately 1.5GPa. Typical physiological operation of tendons is in the region of 5 to 10MPa and they are usually considered in uniaxial operation. Further description of their properties is given later in the report. 5. The Lung As with most active muscles, the mechanical properties of the lung are of great importance to its normal and pathological function in the body. The stress distribution in the lung will directly influence the impedance and subsequent energy consumption during breathing and can directly influence the localisation of some disease. The tissue itself is very unique in the body and this leads to particularly complex mechanical properties. The lungs are composed of a thin external serous coat, a subserous areolar tissue and the pulmonary substance or parenchyma. The subserous areolar tissue contains most of the elastic tissue, and covers

Fig. 9 Cross section of Lung [GRAY91]

the surface of the lung. The parenchyma forms the bulk of the lung tissue. Diaphragm movement reduces pressure in the lungs allowing air to enter a large number of closely packed air ducts and air sacks in the parenchyma called alveoli. Each of these alveoli can be considered as an irregular polyhedron in shape bounded by the alveolar wall membrane. The air in the lung is always at atmospheric pressure as it can flow freely between the alveoli and the environment. The membrane tissue forming the alveolar wall is a delicate layer of simple squamous epithelium. The cells of this are connected at their edges by cement substance. Between the squames are random smaller, polygonal, nucleated cells. Outside this lining is a delicate connective tissue which contains a large number of elastic fibers and a tight network of blood capillaries, which form a common wall to adjacent alveoli. The fibre direction is distributed randomly in the tissue. 6. The Heart The heart muscle has a structure somewhat similar to that of skeletal muscle in that it too is striated with a definite fibre direction. Thus, skeletal muscle solutions are probably a reasonable place to start when attempting to derive models for the heart. The main difference is that it is never tetanized, but rather functions on single “twitches” from nerve (i.e. electrical) stimulation. Each palpitation must be complete before another pulse can stimulate the heart further. The resting heart muscle is also significantly stiffer than the resting skeletal muscle. This indicates that whilst we can potentially ignore the resting stress in the skeletal muscle, in the heart, this stress is not negligible compared to the contractile element. Perhaps the most important difference between these muscles is in the existence of a syncytium with branching interconnecting fibres in the myocardium tissue (see Fig. 9). Each myocardium cell is separated laterally from adjacent cells by cell membranes called sarcolemmas. Length-wise each cell is separated from adjacent cells by dense structures called intercalculated discs, joined to the sarcolemma. The workings of this syncytium is to provided a constant response throughout the heart, unlike in skeletal muscle which can have graded contraction from activation of different numbers of cells. The heart’s cells are all active or all relaxed.

Fig. 10 Cardiac Muscle Fibres [BERNE72] (image from [FUNG81])

Fig. 11 Cardiac Muscle Fibres [BERNE72] (image from [FUNG81])

Cardiac muscle also contains significantly more sarcosomes than in skeletal muscle. The energy produced by the body is used to synthesize adenosine tri-phosphate (ATP), which is then transported out of the sarcosomes and into the myosin-actin matrix, allowing them to contract and relax. This is a regular action that occurs at a relatively fixed rhythm for a lifetime, unlike skeletal muscle, which is usually activated for short periods. The cardiac muscle also requires a constant supply of oxygen whilst the skeletal muscle can compensate by undergoing an anaerobic process for energy conversion. Hence there are fewer sarcosomes required in skeletal muscle than in myocardium. For similar reasons, the myocardium also has a larger density of capillaries in it (approximately one per fibre), unlike the skeletal muscle. A series of transverse T tubules, deep within the sarcolemma help with the exchange of substances between the capillaries and the myocardial cells. Longitudinal branches in rectangular formation connect these to each other. The resting tension in the heart is important in determining the end-dialostic (i.e. relaxed state) volume and thus its stroke volume. 7. The Brain The brain is divided into a number of sections each controlling different aspects in the body. Of these, the cerebrum is the largest and usually, but not exclusively, the subject of a majority of the research in the mechanical modelling of the brain. This is divided into two distinct but connected hemispheres on the left and right sides of the skull. Each hemisphere is composed of three surfaces. The surfaces of the hemispheres are moulded into a number of irregular eminences, named gyri or convolutions, and separated by furrows termed fissures and sulci. Within the hemispheres, there are large amounts of white substance surrounded by thin layers of grey substance interspersed with a large number of blood vessels.

Fig. 12 Sections of the Cerebrum [GRAY91] The grey substance is predominantly composed of nerve cells, while the white substance contains only the nerve fibres. In the grey substance, signals are received, stored, and transformed into efferent impulses. In the white substance, these signals are conducted through the spinal cord to the desired organ. The nerve cells vary in both size and shape dependent on their function. The body of the nerve cell is known as the cyton, consists of a finely fibrillated protoplasmic material. Each nerve cell also has delicate neurofibrils running through its substance. The medullated fibres form the white matter in the brain and can be considered to be homogenous in structure. The medullary sheath is considered to be a fatty matter in a fluid state, which insulates and protects the axis-cylinder part of the nerve.

Fig. 13 Brain Cell Structure [GRAY91] 8. Smooth Muscles Muscles with no obvious fibre direction or striations are called smooth muscles. As with all other muscles in the body, they contain actin and myosin and work on the

conversion of ADP to ATP for energy. However, there are a number of differences between them. Smooth muscle cells are smaller than skeletal and cardiac cells. Really it is difficult to class smooth muscle properties together as each organ containing smooth muscle is assembled differently. Membrane material and collagen filaments fill the narrow spaces between muscle cells within bundles. In multi-unit smooth muscles such as the vascular smooth muscles of large arteries, each fibre operates independently, often controlled by individual nerve endings as with skeletal muscles.

Fig. 14 Single- and Multi-Unit Smooth Vessel The contractile process in smooth muscle is essentially the same as with skeletal muscles. This is assumed from the fact that they contain the same contractile proteins (actin and myosin). Cross-bridges can bee seen in smooth muscle tissue under electro microscopes. This is a further indication that the contraction mechanisms are the same. The tension generated per filament in a smooth muscle is larger than in striated muscle by approximately 40% due to the thick filament width also being bigger. The thick filaments in vascular smooth muscle placed in groups of three to five. The thin filaments are randomly attached to either dense bodies or plasma membrane in the sarcoplasm. The sarcomeres are not periodically spaced, and so the smooth muscle action is slower than skeletal muscle. On average there are fifteen times as many thin filaments than thick in smooth muscle. The Hill equation and three-element model have been used to model in vitro smooth muscles. However, the validity of this is really partly dependent on the organ that is being modelled. One argument against the use of the Hill models is that the parallel element is inseparable from the contractile. This is because when the electric stimulus stops, the contractile mechanism is still not entirely relaxed. The stress relaxation under constant strain shows that the stress relaxes to zero in the steady state. This suggests that smooth muscles have an almost plastic behaviour [FUNG81]. 9. Continuum Mechanics Solution Overview Whilst a complete description of the numerical solution to the continuum mechanics equations is inappropriate here and is usually achieved through use of FEA software, a general solution pipeline may aid the reader in the appreciation of the workflow for such problems. A number of continuum mechanics resources such as [MASE70] propose a similar set of steps.

The problem can initially be broken down into four general stages. Initially, knowledge about the real world problem is required and description of the desired material properties, solution domain and initial/boundary conditions are established. The next step involves the understanding of the physics of the situation, which leads to formulation of the constitutive relationships and governing equations. Following this, a numerical method is constructed by setting up a system of discretised equations is formed in both time and space. This set of equation itself requires a solution algorithm, formed by linearising the equations and solving with a linear equation solver. The results are then passed to stress analysis software for interpretation and presentation in numerical and graphical form. Dynamic properties are established through repeated application of the numerical method to the previous solution for each subsequent time step. Data Acquisition The data required for the finite element formulation needs to be modelled initially as a polygonal surface mesh in a CAD package or similar. Data from CT scans can be used to define the dimensions of these meshes. Indeed, the Visible Human Data Project was set up in 1995 by the U.S. National Library of Medicine to provide data

for such tasks. The subject of this project was a deceased male human who was frozen in a block of ice and successive 1mm thick slices were sliced off and scanned. A pre-processing stage was implemented to convert the scans into the labelling software.

The different tissue types could then be traced and isolated in this software [Fig. 16] in a semi-automatic process. Following this, the reconstruction process involved taking these slices and placing them in a stack such that they could be lofted together to form the 3D mesh of each organ. Finally the meshes were post processed to ensure that they were suitable for the FE simulation (e.g. ensuring closed ends, no anomalies, correction for misalignments etc.). The post processing could also optimise the resolution of the organs to account for the very high demands that the simulation would have in terms of computing time and memory.

Fig. 16 Visible Human Data Cryosection

Fig. 15 Brachialis Mesh [MARTINS98]

Muscle Action Modelling We are often concerned with the actions of soft tissue under their natural actions such as the rhythmic beating of a heart or skeletal muscle contraction or some other force such as the effects of crash testing, when the motion of the body would be a consequence of the motion of the vehicle. The mechanics of each are different, and both are covered in this report where appropriate. If we are concerned with the latter case, then we may be attempting to describe the gross deformation of skeletal muscle in a particular driven action. This might be due the intrinsic contraction of the skeletal muscle in an intended action (any controlled movement) or from external forces. It might, for example, be interesting to describe the propagation of force from a local impact to the internal organs. Some method is required to direct this motion. When the motion is intended, the skeletal muscles internally contract as described above to perform this operation. But many muscles move in synergy with continuous feedback to the brain and from an experimental perspective it would be far more useful to control the gross skeletal movement and from this inversely calculate the forces and deformations which must have occurred to achieve this. Similarly if we are concerned with a driven action such as the crash test example, it would be useful to control the gross body motion by a variant of rigid body dynamics simulations and again inversely derive the deformation and stresses that must have achieved it. Essentially this gross movement would need to be animated for each time-step. This could be achieved procedurally (e.g. the rigid body dynamics mentioned above), hand animated by the setting down of keyframes or poses in space and time, or optical/magnetic motion capture from an athlete or actor performing the action and having it recorded in three dimensions. For most scientific research, motion capture would be the most obvious choice. Obviously for high rate loading such as in a crash test, it would not be suitable as the person would have to undergo the testing themselves thus defeating the purpose of increasing the safety of the test. However this system is often used in fields such as athletics where a problem in the way an athlete repeats a particular movement can be observed. Keyframed animation would only be practical for scientific experimentation in very simple situations, usually analogous to the experimentation used to derive the constitutive equations (such as simple uniaxial or biaxial loading). As in many CAD packages, the bones of the skeleton can be considered as a series of links. The joints can be hinge, universal or sliding to represent the different types of joints in the body. These bones may have volume and constitutive properties themselves, but for the purpose of this kinematic approach, it must be considered a rigid link with a single line of action. A lot of work has been done in the field of rigid body dynamics, taking into account the material properties of the links such as mass and centre of gravity. Detail on this kinematic theory could take up an entire report itself, hence it shall not be discussed further.

The virtual skeletal system can then be linked with the muscles through series of attachment points. Essentially the attachment points are analogous to tendons. They represent the points at which the skeletal muscle is connected to the bone and a practical option would be to include a tendon model between them. In early models, each muscle was considered to have a single uniaxial action line. This implies that the muscle force direction is solely along this action line and the muscle has no moment around this line. Clearly this is in appropriate for a lot of larger muscle, which have broad attachment sites. Moment vectors can be significantly altered by small changes in the direction of such forces. [HELM91] proposed a method for modelling the effect of large skeletal muscles by means of a mathematical distribution of the muscle activation sites and fibre distribution. This was then used to produce a large series of force vectors representing the effect of the muscle. This procedure was then optimised to reduce the

number of force vectors whilst maintaining minimal effect on the error in the mechanics of the system. Given the motion of the skeletal system and subsequent force derived to induce it, it is then up to the finite element analysis of the systems to analyse the governing equations of the muscles and produce the deformation. Finite Element Modelling The EU ESPRIT CHARM project is has been set up to develop to develop human biomechanical research in the form of a comprehensive set of tool allowing for the human musculoskeletal system to be constructed and modelled. Following this, a set of tools is being developed to deal with the finite element simulation of the deformation of the soft tissue and muscle contraction. As this is currently one of the most comprehensive and successful projects that implement the methods described in this review, it is appropriate to specifically discuss their finite element implementations. It should be noted that this is not the only implementation. A number of finite element methods have been developed for the simulation of soft tissue dynamics, some of which are discussed here. But before considering the governing equations, it is important to choose the type of finite element to represent the soft tissue. Finite element software, such as ABAQUS, offers a choice of finite element shape, each with a specific range of uses. The choice of meshing procedure is usually determined by the shape of the object to be represented, and the type of constitutive equation likely to be applied to it (e.g. uniaxial, biaxial etc).

Fig. 17 Example Joint System

Volume Meshing is mainly used for large, arbitrarily shaped thick objects. In this case, it is most appropriate for skeletal muscle and other organs such as the liver. Predominantly hexahedron and tetrahedral shapes are available, the latter of which is more suited to complex shapes such as human organs. Line Meshing is particularly applicable to materials which are mainly orientated in a single direction. Here, it is particularly applicable to the deformation of tendons, which have fibres orientated uniaxially and is described here by a one-dimensional constitutive equation in this direction. The volume deformation can assume incompressibility by means of a constricting coefficient such as the Poisson ratio. Surface Meshing is essential for layered materials, and thus is particularly useful when describing the skin. Many constitutive equations for the skin only refer to the epidermis and dermis, with the hypodermis ignored due to low resistance to applied forces. The constitutive equations could be applied to the three layers as a whole, but in reality all of the layers, even the hypodermis, contribute to the total deformation of the skin. Thus it is much more appropriate to consider the individual layers. [MAUREL98b] proposes the idea of considering the upper and subcuteous layers independently, in the form of a two layer composite shell with individual constitutive relationships for each. Finite Element Interfaces between surfaces need to be considered in depth before the governing equations are solved. We know, for example, that muscles work in groups in order to perform a specific task, and so they need to interact with each other and all be calculated at the same instant. There are two common interface elements provided in finite element software, namely static contacts which involves a matching of nodes such that they do not move relative to each other, and sliding contacts in which the bodies are in contact but allowed to move relative to each other given friction and relative displacement of the surfaces. For the most part, human tissue groups tend to be connected to each other through networks of collagenous fibres. It is therefore somewhat appropriate to use static contacts. A simplification that [MAUREL98b] proposes is to analyse the skin independently of the muscle as it has relatively little effect, and to save computational time, could easily be relaxed onto the muscle surface and then deformed accordingly. Of course, this is dependent on the required analysis, and assumes that the skin, for most purposes (e.g. impact) really does have little effect. Finite Element Implementation The CHARM project relies on the use of finite element software, ABAQUS to achieve its deformation analysis. This was chosen because of its ability to analyse solids under large deformation by direct input of the constitutive equations. The following is largely but not exclusively taken from [CHARM96], [MAUREL98a] and [MAUREL98b]. It is usual for the material tangent stiffness matrix of the incremental constitutive relationship, Kn

M to be acquired from the second-order derivation of a strain energy function, W with respect to the strain tensor, E, thus, ∆S = Kn

M ∆E + SnR - Eq. 33

with,

KnM 2

2

nEW

∂∂=

Reverting back to the work of [HOROWITZ88] into three-dimensional constitutive laws for passive myocardium, the stress tensor, S, was defined as,

EIL

EWS

∂∂

+∂= 31 - Eq. 34

with, W = W1(I1,I2) + W2(I3), W1 = ∑ ∫∑ Ω

Ω=k

nkkkk

k dwuRSW )()( '* γ and

W2 = L(I3-1) The stress Sn and stress increment, ∆S at n were then derived as,

nn

nn E

IL

EWS

∂∂

+∂

= 31 - Eq. 35

with,

LEIE

EILddJd

ERfS

LEIE

EIL

EWL

LSE

ESS

nnn

n

ncncc

nnn

nn

n

n

n

∆∂∂

+∆

∂∂

+∂∂

=

∆∂∂

+∆

∂∂

+∂∂

=∆∂∂

+∆∂∂

=∆

∫ ∫ ∫−

33/

3/

2

0 0 23

2

2

'2'*

23

23

2

21

2

),,()(π

π

π παθφεφθαε

This led to an elastic stiffness matrix of the form,

KnM= =

∂∂

+∂∂

=∂∂

23

2

21

2

2

2

nn

nn EI

LEW

EW

∫ ∫ ∫ ∫− ∂∂

+∂∂

+3/

3/

2

0 0 23

2

2

'2

0),,(

21)(π

π

π π εαθφεφθα

nn

n

nc

cc E

ILddJd

ERdx

xxD

S c - Eq. 36

This could then solved by means of a finite element incremental iterative modified Newton-Raphson scheme, where the tangent stiffness matrix was updated after each load increment. Tendons and Ligaments The parallel orientation of the fibre, leading to a uniaxial behaviour implied that there was no three-dimensional constitutive equation for tendons and ligaments. A physiological consideration is that tendons always work in extension and the

transverse constriction is always small compared to this. Thus they were modelled with a one-dimensional, incompressible, non-linear viscoelastic relationship as described above. Fung’s quasi-linear viscoelastic was implemented as follows,

∫ ∞−−=

t e

dd

dTtGtT ττ

τ)(

)()( - Eq. 37

with,

)/ln(1)/ln(1

)(12

2

ττττγ

ccctG

+−−

≈ as the relaxation function, as proposed in [FUNG72]

and a modified elastic response,

)1()1( )1()( −−−= − λλ ABeAT Be - Eq. 38 with A and B as constants. The relaxation function was then expressed in the form of a Prony series [CHARM96] to,

∑=

−−−=N

tegtG1

/ )1(1)(α

τα

α - Eq. 39

with, gα as the relative modulii and, τα as the relaxation times. The constitutive Jacobian for elastic response was defined as,

)]1()1)(()()[( )1)(( −−∆++∆+∆+=∆∆ −∆+ ttB

v eABDttttTttdd λλλ

εσ - Eq. 40

with, σ = λt and, ε = ln λ Skin Two models are employed for the CHARM implementation of skin modelling. The first is based on the isotropic hyperelastic constitutive relationship presented by [VERDONA70], in the form of a strain energy function in terms of the first two strain invariants, thus,

)()3(]1[ 22)3(

11 JgIcecW I +−+−= −β - Eq. 41

with c1, c2 and β as constants.

This was deemed to be the simplest relationship for modelling isotropic hyperelastic behaviour as it allows direct modification of the different energy components. It was also particularly applicable to the ABAQUS system, which has predefined routines for handling hyperelasticity. This required assumption of incompressibility such that J = 1 and g(1) = 0, leading to the principal partial derivatives of W as,

)3(1

1

1 −=∂∂ Iec

IW ββ 2

2

cIW =

∂∂ )3(2

121

21−=

∂∂ Iec

IW ββ - Eq. 42

with the other components taken as 0. The other skin implementation used in the CHARM project was proposed in [TONG76] as,

22211522

2114

3222

311122114

2213

2123

2222

2111 2

21

221142213

2333

2222

21112

1 )2(EEEEEEEEaEaEaEaEace

EEEEEEWγγγγ

ααααα+++++++++

++++= - Eq. 43

The ABAQUS implementation required further development of the constitutive Jacobian in the form,

jlsnrmklmn

ikirjssjirijrsrs

ij FFFEWF

J

∂∂+++−=

∆∂∆∂

2

21σδσδσδεσ

- Eq. 44

Passive Skeletal Muscle For passive skeletal muscle, the most appropriate method would be a strain energy formulation to account for the large three-dimensional non-linear deformation of the system. Maurel’s implementation [MAUREL98b] relies on the work of Horowitz [HOROWITZ88], as described above. This is mainly due to the proved application in finite element modelling, and the structural properties it describes. The CHARM implementation is based upon the work of [HUMPHREY87]and [HUMPHREY90] into the incompressible, transversely isotropic, hyperelastic model for passive cardiac tissue. It has been implemented in ABAQUS to describe both the passive and active skeletal muscle tissue. This model is given in the quasi-incompressible form as, W = WI(I1) + Wf(λf) + WJ(J) - Eq. 45 where, WI = )1( )3( 1 −−Ibec is the strain energy of the embedding isotropic matrix,

Wf = )1(2)1( −−faeA λ is the strain energy of each muscle fibre group,

WJ = 2)1(1 −JD

is the strain energy associated with the volume change, and,

A, a, b, c, D are constants.

An active contractile strain, ζCE was then introduced to represent Zajac’s musculotendon model [ZAJAC86]. Thus the fibre strain energy became [MARTINS98], Wf = Wf(λf,ζCE) = WPE(λf) + WSE(λf, ζCE) - Eq. 46 with, WPE(λf) = ∫

fdfT PE

M

λλλ )(0 and

WSE(λf,ζCE) = ∫f

dfT CESE

M

λλζλ ),(0

The complex material tangent operator was then described as,

H = ES

EW

∂∂=

∂∂

2

2

- Eq. 47

Fig. 18 Zajac’s Musculotendon Model Muscle Contraction Whilst passive systems of muscle movement (i.e. by assuming external forces at the tendon) have been discussed, the effect of internal contraction mechanism in the sarcomeres should also be considered. Here, the stresses within the muscle are calculated with respect to the length and shortening velocity, taking into account the anisotropic properties due to fibre direction. Ultimately though, both methods are really reaching the same results based on different input criteria.. The former method is inversely derived from a defined skeletal motion, whilst the latter considers the muscle dynamics and deformation responsible for driving the skeleton. [MAUREL98a] considers an interesting approach to muscle contraction. It initially assumes that the muscle fibres are organised parallel to each other in a manner analogous to fluid streamlines. The muscle is considered as above, as a continuous, deformable three-dimensional object. The fibrous structure in each individual muscle is described by a vector field, d, with a unit tangent vector, f. There are a number of different muscle formations, as shown below, and in his work, Maurel considers the fusiform, triangular and spiral variety.

Fig. 19 Muscle Pennations At any time, the muscle fibres may have different strengths and activation rates. Indeed, it is believed that the smaller units are recruited before the larger ones and the larger are released before the smaller [WINTER90]. [MAUREL98a] defines a contraction force vector at a point, P, as, FC(P) = -f C(P)δδδδ(P) - Eq. 48 With, f C(P) = a(P)s(P)fM(lM,vM), s(P) as the strength scale distribution function, a(P) as the activation scale distribution function, δδδδ(P) as the unit vector along the local contraction force direction, fM as the uniform contraction force at maximum activation, l as the muscle length and v as the shortening velocity. The uniform contraction force is considered to be uniform across the muscle and, as can be seen, only a function of the length and shortening velocity. It can be approximated from the following force-length and force-velocity diagrams [PANDY90],

with, P0 as the peak isometric active contraction force = )( 0

MCEFA lP

Ml0 as the muscle length at which P0 can be developed Tl0 as the tendon slack length T0ε as the tendon strain for PT=P0

α0 as the pennation angle for which MM ll 0= Fig. 20 Force-Length and Force-Velocity Relationship that,

M

MMMM

VvlPf

0

),(= - Eq. 49

where, PM(lM,vM) is derived from the force-length and force-velocity diagrams and, V0

M is the initial muscle volume. Now, the fibres of the fusiform muscle can most appropriately be represented by parabolic curves, in the cylindrical co-ordinate form, r(z)= r0(1+afz2) - Eq. 50 for, (θ,z)∈ [0,2π]x[-h,h] with, (r0, θ0)∈ [0,r2]x[0,2π],

22

12

hrrraF

−−= and,

20 1 zarr

F+=

Fig. 21 Proposed Fusiform Muscle Co-ordinate System (edited from [MAUREL98b]) now, the fibre direction at a point, P(r,θ,z) can be defined as,

d(P)=

==+

=

10

22

z

F

Fr

dd

zaarzad

θ - Eq. 51

The fibres of the triangular muscle are represented by straight lines in a Cartesian co-ordinate space as, y(z) = y0(1+aTz) - Eq. 52 for,

],0[2

,2

),( heezx ×

−∈ ,

],0[2

,2

),( 100 reeyx ×

−∈ ,

hrrraT

1

12 −= and,

zayy

T+=

10

Fig. 22 Proposed Triangular Muscle Co-ordinate System (taken from [MAUREL98b]) Here, the fibre direction can be described as,

d(P)=

=+

=

=

11

0

z

T

Ty

x

dza

yad

d

- Eq. 53

The proposed spiral muscle model consists of straight lines in a cylindrical polar co-ordinate space. Here,

θcos0rr = and

)( 0θθ −= Saz - Eq. 54 with,

20αθ −= for

−∈

2,

2ααθ

20απθ −= for

+−∈

2,

2απαπθ ,

αhaS = and,

r0 = rcosθ

Fig. 23 Proposed Spiral Muscle Co-ordinate System (edited from [MAUREL98b]) such that,

d(P)=

==

=

Sz

y

x

adrd

rd θtan - Eq. 55

Some observations that [MAUREL98a] makes of both the mathematical descriptions and the physiological structure of these models indicate that each of the fibres can be considered to have a constant strength along it’s length, have a constant activation along it’s length and undergo a constant tension along it’s length. The contraction force vector is always tangent to the fibres and both the strength and activation gradient vectors are normal.

The active strength scale, ω(P), a combination of the strength and activation scales distribution functions, s(P) and a(P), can now be considered thus, fC(P) = ω(P)fM(lM,vM)δδδδ(P) - Eq. 56 with, ω(P) = a(P)s(P) and, ∇ω (P)δδδδ(P) = 0 Now, for the fusiform muscle, ω(P) must be a positive, decreasing function of the radial distance, r. Maurel proposes the form,

0)()(1

22 =

∂∂+

∂∂

+ zP

rP

zarza

F

F ωω - Eq. 57

leading to,

F

F

F

zar

FFF ezr

ηϖ

ωθω

+−

Ω+==210),,( - Eq. 58

where,

FFFF ηϖω ,,,0 Ω are constants Similarly, the active strength scale for the triangular muscle must be a positive, decreasing function of y. The following is proposed,

0)()(1

=∂

∂+∂

∂+ z

PyP

zaya

T

T ωω - Eq. 59

leading to,

T

T

T

zay

TTT ezyx

ηϖ

ωω

+

−

Ω+== 10),,( - Eq. 60 where,

TTTT ηϖω ,,,0 Ω are constants. Finally, ω(P) for the spiral muscle must be a positive, decreasing function of radial distance, r, and independent on z, due to z always being proportional to θ at any point.

0)()()(tan =∂

∂+∂

∂+∂

∂zPaP

rPr S

ωθ

ωωθ - Eq. 61

leading to,

( ) SS r

SSS ezrηθϖωθω cos0),,( −Ω+== - Eq. 62

where,

SSSS ηϖω ,,,0 Ω are constants.

Similarly, temporal dependency can be accounted for in any muscle type by considering factor functions as functions of time. Suggestions for finite element implementation of this type of contraction simulation can be found in [MAUREL98b] and are beyond the scope of this report. However, it is appropriate to discuss the limitations of this model. The first, and perhaps most obvious limitation in this model is that, as with the other skeletal muscle models, the composite structure is not independently accounted for. Whilst the model takes into account the fibre directions, it assumes that the properties throughout the cross-section of the muscle are considered for the same type of material. The state of internal stress is likely to be different for different tissue components within the muscle, but this model assumes a certain homogeneity only really governed by the active strength scale. Indeed, some tissue components are likely to respond completely differently to the state. For example, during an active contraction, the stress inside the contractile fibres is likely to increase in tension, whilst the stress in the surrounding passive tissues will increase in compression. This is analogous to Hill’s three-element model in which during an active contraction, the series element will be extended by the shortening of the contractile element, whilst the parallel element shortens. This model assumes that the muscle is continuous and that the state of stress in a point in a medium is unique and cannot correspond to a state of tension and compression at the same time. The methods described above do not truly describe the phenomenon of muscle contraction, but rather represent the contraction as a result of external compressive forces applied on a fully passive material. To simulate this would require a greater consideration of the constitutive equations of each component within the tissue, with appropriate finite element interactions between them. In theory, this is very possible, but computationally demanding, and the requirement for setting up the conditions for each of the components in each muscle would be very difficult by hand. It may be possible to automate the set up by a scripting method for distributing and assigning conditions (element type, interface type, constitutive equations et al.). Maurel proposes the idea that, “rather than taking the form of an external contractile force applied to a fully passive tissue, the contraction would result from an internal change in the mechanical state of the contractile fibres, induced by a change of activation. This change of internal state of stress of the contractile fibres would affect the equilibrium of the medium, which would then evolve and stabilize by compressing the surrounding passive tissues.” [MAUREL98b] Myocardium Tissue A resting heart muscle is an inhomogeneous, anisotropic, incompressible, temperature and chemically dependent material [FUNG81]. It exhibits relaxation under a fixed strain, creep under a fixed stress and hysterisis under a cyclic loading. This hysterisis curve is independent of temperature in the range 5o-37o and is relatively independent of strain rate. Hence it can comfortably be described as (bio)viscoelastic and the pseudo-elasticity theory is valid. In the both loading and unloading direction, the stress exists as an exponential function of strain with different constants for each. In creep tests, this muscle also exhibits significant creep strain under constant load.

In the heart, the reduced relaxation function, G(t), is essentially independent of temperature, particularly in periods of less than 1 second (comparable to the length of a heartbeat) and in the temperature range 5o-37o. Similarly, under constant strain, the reduced relaxation function is independent of the stretch ratio for strains up to 30%. Thus the theory of quasi-linear viscoelasticity can be applied to the heart in the end-diastolic condition. The dynamic behaviour of active myocardium relies on knowledge of the constitutive equation of the heart muscle in the systolic (contracted) heart, the diastolic (relaxed) heart and the states in between. It had been thought that Hill’s three-element model coupled with a known contractile element could be used to describe the heart’s constitutive equation. The model needs to be modified to introduce an active state factor to relate the maximum force and maximum velocity with time after stimulation. It was found that this was applicable if the muscle was tested at a length small enough that the resting tension in the parallel element was negligible in comparison with the active tension. Experimentally it was found that the resting tension could not be ignored and two modifications were proposed to solve this. The parallel resting tension model involved allowing the parallel element in the three-element model to bear a resting tension, whilst the series resting tension model involved rearranging the components such that the parallel element was only in parallel with the contractile element and the series element was in series with them. Finally, this series element was also allowed to bear a resting tension. Unfortunately it was also found that the Hill equation did not provide the desired results and the force-velocity relationship was too complex and so this was largely abandoned in favour of newer methods. Lung Tissue The first methods for describing the mechanics of the lung tissue included modelling the alveoli structure with two-dimensional hexagonal sets of stochastic non-linear springs and the natural extension of this to three-dimensional dodecahedrons. Most of the models were derived on the basis that the primary mechanical resistance in the tissue came from the alveoli membranes. [LANIR83b] applied the fibrous approach described here for modelling skeletal muscle, to model the parenchymal microstructure of lung tissue. It too considers the stochastic approach to the alveoli structure. Another consideration in this model is that there is a thin layer of liquid on the surface of the alveoli, on the tissue-air interface, leading to a significant interfacial tension. Fig. 24 shows a number of proposed approaches for modelling the alveoli in the lungs. The uppermost images in the figure have been shown to be physically incorrect. The lung models presented there will collapse due to pressure differences between any alveoli sharing the same branch. This pressure difference would cause the air from the higher pressure alveoli to flow into those with lower pressure unless the surface tension is zero or the elastic force is much greater than the surface force [FUNG90].

Fig. 24 Some Proposed Alveolar Models In reality, the alveoli volumetrically fill the lung in a manner such that the alveolar wall is an alveolar septum. The inner surface of one alveolus is the outer surface of another. The pressure difference across the septum is negligible and they must be minimal surfaces (with zero mean curvature). Thus it is believed that space filling shapes such as tetrahedral, cubes, icosahedra and so on, are most appropriate for modelling of the alveoli. Fung proposes the use of a second order 14-hedron surrounded by 14 identical polyhedra. Air duct are then created by removal of central faces where connected polyhedra meet. This wall removal causes the equilibrium to be disturbed and this is compensated for by means of curving the wall. It is believed that this is truly representative of why the alveoli walls are always observed to be rounded structures. Tree methods can then be employed to consider the lengths of ducts formed by branching. Considering each branch, the ducts get shorter with increasing generations. It is believed that the square root of the width and the cube root of the curvature of the collagen and elastin fibres in the interalveolar septa are normally distributed along with the fourth root of width in the alveolar mouths.

This suggests that whilst the fibres show significant orientation in individual septum, there is no obvious direction over all of the septa as a whole. The wider fibres tend to be straighter and the mean value of the curvature in collagen is larger than in the elastin fibres. Fung proposes a simplified constitutive equation of lung by treating the

alveoli as a series of connected cubes. The principal stress formulation derived in the text is,

Fig. 25 14-Hedra and Tree Structures

)1()](2[)](2[ 3)(

31312)(

212111 hLpNN Aee −−∆++∆+= λγλγσ

)1()](2[)](2[ 1)(

12123)(

323222 hLpNN Aee −−∆++∆+= λγλγσ

)1()](2[)](2[ 1)(

23231)(

131333 hLpNN Aee −−∆++∆+= λγλγσ Eq. 63

where, 2γ is the surface tension for both inner and outer membranes, h is the thickness of interalveolar septa, L is the length of intercepts of the septa per unit cross sectional area of lung, ∆ is the spacing between the septa and, N(e) is the elastic tension. Consideration of a pseudo-strain energy function of the lung parenchyma leads to,

)]()()2()([0)(0

231

223

212113333222211

233

222

211 EEEEEEEEEEEEe eCW ++−++×++++

∆= βαβα

ρρρ Eq. 64

where, α, β, and C are a material constant, ρ is the density of parenchyma in the deformed state, ρ0 is the density of parenchyma in the zero stress state. This method has a number of disadvantages, whilst at the same time being able to represent major features of the stress-strain relationship in the lung. Modelling the alveoli as cubes is clearly too simple, and more suitable polyhedra would be far more useful [LANIR83b]. The physical meaning of the constants is unclear and they cannot be obtained from knowledge about the microstructure of the tissue. They can be obtained from triaxial loading experimentation, but it can be particularly difficult to work with this tissue type and obtain good values. This model also assumes that the macrostrain does not differ from the microstrain as the alveolar ducts and mouths do not provide extra degrees of freedom to the system. This model also does not take into consideration the effect of the surrounding pleura layers, which can contribute up to 25% of the bulk modulus of the lung.

Fig. 26 The Bronchi and Lobules of the Lung [MARTINI98] The problem of considering the surface tension at the air-interalveolar septa interface can be tackled by consideration of the equilibrium of the surface. For a curved

surface, the effect of surface tension on the balance of pressures normal to each side of the septum is governed by Laplace’s formula,

oi pprr

T −=

+

21

11 Eq. 65

where, T is the surface tension, r1 and r2 are the principal radii and pi and po are the internal and external pressures respectively.

Fig. 27 Stresses on Curved Membrane The surface tension-area relationship in cyclic changes of area can be described by a number of different methods. [FUNG90] offers,

+−+= ∑

∞

=1minmaxmin sin)(

nn nc πξξγγγγ Eq. 66

where, γ, γmin and γmax are the surface tensions at current, minimum and maximum areas, ξ = (A-Amin)/(Amax -Amin) and cn are constants. Other models include that of [FLICKER74], γ = c1(A/Amin)c2 Eq. 67 that of [VAWTER82],

[ ]min'3 /'

2'1 1 AAcecc −−=γ Eq. 68

and [WILSON82], who proposed the idea of considering the total energy of an air filled lung, E, as being the sum of the tissue elastic energy which consists of the energy of a saline filled (i.e. surface tension independent) lung and a strain energy

function representing the distortion of the alveolar wall caused by surface tension. This is added to a separate surface energy function. This is presented in the form,

∫+∆+=S

S dSSVUVUE0

),()( γ Eq. 69

where, US(V) is the energy of the saline filled lung as a function of lung volume, V, and ∆US(V,S) is the alveolar wall distortion energy as a function of volume and total alveolar wall surface area, S. Fung provides a diagram for describing the relevant steps for deriving a macroscopic stress-strain relationship for the lung. Lung Fig. 28 Development of the Stress-Strain Relationship in the Lung [FUNG90] It is suggested by Fung that a simple and useful solution to the problem as a whole is by means of a minimising the potential energy of the system. Intially the strain distribution for both the inner and outer surfaces in each interalveolar septum and alveolar mouth would be established in tensor form. Then the stress-strain relationships of interalveolar septa and alveolar mouths would be established. Based on the mechanical properties of the collagen and elastin fibres in the ground substance. Following this, the strain of each interalveolar septum and duct edge is derived and each energy function is summed together. Minimisation of the energy given specified boundary conditions would then yield the solution. Perhaps the biggest problem here is that throughout the lung there are a number of different structures and types of collagen, each of which ideally needs to be modelled. The work of [LANIR83b] involves consideration of a density distribution function for the orientation of the membranes in the unstrained state, R(n), was given such that the

Macro Zero-Stress State

Macrostrain Micro Zero-Stress State

Microstrain Interalveolar Septa

Microstrain Alveolar Mouth

Equilibrium Compatability Constitution

Zero Stress-State Equilibrium Stress Constitutive Eqs.

Ducts and Alveoli

Collagen and Elastin Fibres in Septa and Mouth

Stress

Macro Stress

number of membranes, n, occupying a spatial angle, ∆Ω, is given by R(n)∆Ω. The assumption is made that the membranes are thin, planar, perfectly flexible, transversely isotropic and subject to in-plane loading with atmospheric air filling the voids between them. Plane strain is applicable considering the thickness of the membranes. The elasticity of the alveolar wall is considered in terms of a strain energy function with two components,

it WWW += - Eq. 70 where, Wt is the tissue strain energy function and Wi is the contribution of the fluid interfacial tension. The planar contravariant stress components, can be found from the strain energy function for the tissue by,

αβ

αβτeW

J tmt ∂

∂= −1 - Eq. 71

where, Jm is the ratio of volume of strained to unstrained membrane and,

ij

ii

xxe γθθ

βααβ

∂∂

∂∂= for α,β = 2, 3

is the plane strain as a function of the general strain. Here, the strain energy function can be represented as a function of three strain invariants,

αβαβλ AaI += 2

1 ,

aAJ /21 αβλ= ,

2/)1( 21 −= λK - Eq. 72

The contravariant stress components are then given by,

αβαβαβ ψφτ Aat += - Eq. 73 where,

1

5.01 )/2(

IW

J t

∂∂

=φ and,

1

5.012

JW

J t

∂∂

=ψ

Another model (this time structural rather than continuum) for the tissue strain energy function is proposed in [LANIR83]. This considers the wall elastic potential as a sum of the elastic potential of its components. The fibrous constitution is fundamentally of importance here, whilst the effect of the ground substance is negligible in comparison. The stress is given as,

∑ +∆••=v

mt pAvqefJ αβαβ βαπτ ),()()/1( - Eq. 74

where, q(2,2) = cos2v; q(2,3) = q(3,2) = sinv.cosv; q(3,3) = sin2v, f(e) is the axial stress in a single fibre as a function of strain and, p is the hydrostatic pressure. The strain, e, is given by,

vvevevee cossin2sincos 232

332

22 ++•= - Eq. 75 The interfacial tension comes from the resistance of the fluid layer to changes in area. It shows considerable hysterisis and there is evidence of time dependent properties. It is suggested in [FUNG88] that this could be considered as a pseudo elastic material with separate consideration for inflation and deflation phases of the respiration. Considering that the surface tensions is an interfacial pseudo strain energy function, the stress is given as,

mmi Jh

JAeWiJ

0

2)/1(αβ

αβ

αβ ητ •=∂∂= - Eq. 76

with,

dJdWi=η2 is the surface tension coefficient per unit area.

The total stress for the alveolar wall is then the direct combination of the above, so,

∑ ++∆••=v

mmt AJhJpvqefJ αβαβ ηβαπτ )/2(),()()/1( 0 - Eq. 77

The elasticity of the parenchyma in the lung is modelled by consideration of the elastic potential in a unit volume of tissue. For adiabatic, isothermal conditions, the elastic potential is function of the strain alone and the stress or equivalent strain energy function can be directly derived from it. The stress function is given as,

∑ ∆Ω•∂∂

•=n ij

mij

ee

JnRJS )()()/( αβαβττ - Eq. 78

where, S is the fraction of material’s total volume occupied by the membranes. This is then combined with the stress function of the alveoli wall to produce,

∑∑ ∆••=vn

t vqefnRJS ),()()/1)[(()/( βαπτ αβ

∆Ω•∂∂

•+•+ij

m ee

AhJJp αβαβη ])/2( 0 - Eq. 79

The non-homogeneous viscoelastic form of this is offered as,

∑ ∆Ω•ℑ∂∂

=n

tm

ij

ij tetJe

nRtJSt ]),([)()(|))(|)(()( 0τγ

θτ αβαβαβ

θ - Eq. 80

where, ∑ •ℑ+∆•ℑ=ℑ

v

tp

tfm

t AtevqteJt αβαβ τβατπ ]),([),(]),([)/1()( 00 and,

pf ℑℑ , are functions for the behaviour of an equivalent fibre and matrix. 10. Impact and Trauma Biomechanical engineers are often concerned with high rates of loading/high strain rates. In such conditions, human safety is questionable. It is therefore important to be able to predict the forces that the human body will experience in a particular situation, in order to be able to analyse the level of safety and consequently make sufficient adjustments. Many sports involve high impact actions. Even non-contact sports that involve actions as trivial as running and jumping are subject to impact analysis to aid in the improvement of technique. Through any sort of impact or high loading rate situation, the stresses will propagate through the body, initiated at the point of impact, or as a consequence of inertia. Waves will also propagate through the body based on the particular material and the attachment/boundary conditions. In theory, a complete set of constitutive equations should be able to describe everything dynamic in the system (the FE analysis provides a deformation as a function of time, and from this the vibration/wave response can be found, if desired). The non-linearity of stress-strain history in biological tissue has already been discussed. An increase in stress induces an increase in the elastic modulus and this “strain-stiffening” property can lead to shock waves throughout the organ. The shock waves can propagate faster through particles with high stresses. At certain speeds, the wave front causes a jump condition in the form of the shock wave. Curved features can also cause wave focussing by concentrating the energy into small regions of the material. This leads to stress concentrations in these areas and if either stress or strains reach past certain critical values, undesired effects may occur. In the case of biological tissue, if the ultimate tensile stress of the material has been reached from such processes, the tissue will rupture leading to haemorrhaging (internal bleeding) or other injury. [SEGERS00] discusses the role of tapering in wave reflection from aortic valves. Systolic pressure is a very important cardiac risk factor and so such measurements are clearly of great importance. Here there were at least two interacting phenomena in the form of a continuous reflection from the tapered geometry and local reflections from the branches at near the diaphragm. But this problem is not only local to the heart as most human arteries will show similar reflection patterns. [FUNG90] gives an example of trauma of the lung to impact in the form of oedema and haemorrhaging. In plane progressive waves, the stress is given by, σ = ±ρcv - Eq. 81 where, ρ is the mass density of the tissue, c is the velocity of sound (i.e. longitudinal waves) in the tissue and,

v is the tissue velocity. The ± is given to represent the waves travelling in each direction. If the rate of loading is high, then so will the induced velocity in the tissue. If the Mach number of the impact in the lung, where, Ma = v/c - Eq. 82 exceeds 1, the waves will propagate at supersonic speeds, leading to a shock wave. The velocity of sound in the lung is relatively low compared to other tissues in the body and so it is much more susceptible to damage from shock waves as the tissue velocity doesn’t have to be as high for the Mach number to exceed 1. High speed impact is generally considered to be a local effect. The focusing effect caused by curvature of the wave front will increase the damage caused by the shockwave. Indeed, it has been observed that the haemorrhaging in the lung is at worst near the spine, heart and ribs for this reason. [YEN88] describes experimentation on rabbit lung tissue to study lung oedema - a swelling caused by water retention due to pressure changes across the cell membrane. Impact was controlled by two methods. A series of 2cm2 area, lightweight (1.49g) pellets were fired into the tissue at different initial speeds. A shock tube was also used in a separate set of tests to send air shock waves of varying Mach numbers into the tissue. It was found that for impact velocities of less than 11ms-1, there was no oedema in the tissue. At 11.5ms-1, after one hour, the oedema had reached 20% of the weight of the lung. Above 13.5ms-1, the oedema was massive and developed quickly. This experimentation suffers in that there are numerous other factors which should be considered, and it is rare for an equivalent impact to occur. However, it is still fairly indicative of the criteria for injury in the lung. It was similarly found that above 15ms-

1, there was damage to the alveolar regions in the lung. Due to the complex structure of the body, the wave formations are more complex and stress concentrations can be higher for lower impact velocities for the whole body than the isolated lung. The properties of soft tissue exhibit good strength in compression, but since impact is a compressive action, superficially this doesn’t seem logical. [FUNG88] attempts to explain this by suggesting that the tensile and shear stresses are induced by reflection from the alveolar wall. This can cause certain airways to collapse, locking air in the alveoli at a critical strain. When they open again, the sudden expansion can produce extra forces on the alveolar wall. As all organs in the body contain fluids or gasses, this introduces a further complication in the prediction of the trauma a tissue. The straining of tissue that holds fluids or gasses increases its permeability. This allows for the fluids to pass between tissues – a very dangerous process medically. Starling’s law of membrane filtration is given as,

)(0 outinoutin ppKJ ∏+∏−−= - Eq. 83 where, J0 is the volumetric flow rate of fluid per unit area of membrane, K is the permeability constant of the membrane,

pin and pout are the fluid pressures on each side of the membrane and, Πin and Πout are the concentrations of the fluids on each side of the membrane. Oedema is essentially governed by this equation as the change in pressure of the fluids causes the fluid to move from the interstitium space to the alveolar. This change in permeability is thus governed by the stresses in the tissue from the impact loading and the injury is not directly from the tensile stress. A finite element approach to analyse stress waves within the lung showed that the pressure waves in the lung from impact are very non-uniform and it is very reasonable to expect tensile strains in some areas, of comparable magnitude to initial compressive strain from impact. Ideally these considerations should be taken into account in the simulation of the effects of impact on the lung. [COLLINS72] describes a method for imposing a high strain rate on aortic tissue. This was done by means of a shock tube. The proposed stress function is given as,

)1)(18.028.0( 12 −+= εεσ e! for 15.3 −< sε! - Eq. 84 [HONGYAN99] describes a model for the tensile failure of ligaments. It assumes, as described by [LANIR83] for tendons, that the mechanics of the ligament involve uncrimping and then stretching of the collagen fibres. It assumes a Gaussian distribution of fibre length but does not take into account the strain rate effects (viscous effects) such as fibre movement through ground substance and interfibre gliding. The model also accounts for the mechanical losses due to rupture of fibres, and presents the following stress function,

∫−= −−λ µ λ

πλσ

1

2/)( 22

21)( dx

xxe

sE sx - Eq. 85

where, λ ≥ 1+α, E is the Young’s Modulus, λ is the stretch, x is the intermediate specimen stretch, s is the standard deviation of the normalised initial lengths of the fibres, µ is the standard deviation of the normalised initial lengths of the fibres and, α is the stretch limit.

Fig. 29 Model Fit of Ligament Tensile Response to Abrupt and Prolonged Failure

This model was then compared with experimental tensile tests on rabbit ligaments. It was found (Fig. 24) that the model was useful in describing the material when the failure was abrupt, but less suited to modelling slower failures. The collagen modulus was found to be from 300 to 680MPa and the failure strains from 6 to 22% and this was partly dependent on the age of the subject that the tissue came from. Recent work into the constitutive modelling of brain tissue can be found in [MILLER97] (in vitro experimentation) and [MILLER00] (in vivo experimentation). In the in vitro methods, it was observed that the relationship between the Lagrange stress and the true strain contained no meaningful linear portion, and there was an increase in stiffness with increase in loading speed, indicative of a strong stress-strain relationship for stretch rates in the region of 0.6-0.7s-1. The proposed constitutive model was presented in the preliminary research by treatment of the brain as a non-linear, viscoelastic, single-phase medium. The strain energy function of the form,

∑=

−−=N

ji

jiij IICW

1,21 )3()3( - Eq. 86

Where, ][1 CTrI = ,

3

21

2 2][

ICTrII −

= ,

1det3 == CI are the strain invariants and, C = is the left Cauchy-Green strain tensor. The non-zero Lagrange stress components were given by,

z

WTλ∂

∂= - Eq. 87

for a uniaxial stretch parameter. For modelling the velocity dependent behaviour, the coefficients of the energy function were presented as the exponential series,

∑=

−∞ +=

n

k

tijkijij

keCCC1

/ τ - Eq. 88

leading to,

∫ ∑

−−

∂∂−=

=

t N

ji

ji

zijzz dJJ

ddtCT

01,

21 ))3()3(()( τλτ

τ - Eq. 89

which was considered to the N=2 term. It was believed that this method was applicable for compressions of up to 30% and loading over 5 orders of magnitude. Further work on the constitutive modelling of the brain under low loading rates is

discussed in [MILLER00]. Here, a hyper-viscoelastic model is proposed on the basis of a strain energy function in the form of the convolution integral,

∫ ∑ ∑

−−×

−−=

= =

−t N

ji

jin

k

tkij dJJ

dtdegCW k

01,

211

/0 ])3()3[())1(1( ττ - Eq. 90

The work of [MILLER00b] describes the three-dimensional constitutive modelling of abdominal internal organs, focusing in particular the kidney and liver tissue. This was done for the purpose of investigating high rate loading as in high impact situations. Experimental data confirmed that the tissues were non-linear with a strong dependence between stress and strain rates. The proposed models are in the form of a single-phase, hyper-viscoelastic material based on the strain energy function. It was noted that the tissue of the liver and kidney were in many ways similar to that of the brain. As such, these equations were based on their earlier work as described above. The tissue was assumed to be isotropic in the fully relaxed state, and incompressible. The stress function was given as,

∫ −−+−−= −−t

zzzzz ddtC

ddtCT

0

301

210 )2()()22()( λ

ττλλ

ττ

)22()32)(( 21220

−− −−+−+ zzzz ddtC λλτ

λτ

τλτ

λλτ dddtC zzz

−−+−+ −− )2()32)(( 32

02 - Eq. 91

It was found the theoretical models proposed in the paper compared well for compression of up to 35% and a wide range (two orders of magnitude) of loading velocity. The strain rates for which this model gave good results indicted that the model could be useful for simulation of high speed vehicle impact testing. However the model suffers from being rather poor at modelling low strain rate loading. It also does not take into account the methods of attachment of the organs in the body. It is stated that these would be essential in determining the boundary conditions required for a good finite element simulation. Essentially, when considering the effects of high rates of loading such as impact onto organs in the body requires more than just consideration of the ultimate tensile stresses and strains tissue in the tissues. Whilst this might be suitable for a construction material (such analysing when a steel frame will deform plastically or fracture totally), many biological tissues (particularly organs, which deal with gas and fluids) are far more complicated in the situations that can lead to trauma. Of course, this is not true of all tissue, for example tendons and ligaments can almost certainly be represented purely by their tensile properties to fracture. A hybrid finite element and finite volume analysis may even be required to deal with the fluids contained in the body. However, if it is known experimentally at what stress or strain levels at which a particular injury will occur, an FE method which only taking into account the stresses and strains and, the wave dynamics (including the ability for a wave to travel at supersonic speeds) may be suitably indicative. The precise nature of the trauma and the mechanics that describe it may not be important for most safety applications, as

long as there is sufficient indication of the relative forces and suitable safety factors can be taken into account. 11. Discussion This report has described the physiology and subsequent mechanical models of a number of biological tissue structures in the body. Clearly the research into methods in this field is vast and yet still very much in its infancy. Computational methods have only really been practical since the 1960’s and so it can be expected that there is a lot of work still to be done in both stable and efficient methods of solution. As computing power increases, more complicated models can be solved. Unfortunately engineers are producing the newer, more complex models faster than the rate at which computing power is increasing. This was the problem when finite element/volume methods were introduced – the computing power was not advanced enough for the application. It is clear that this area of research suffers from the sheer complexity of the tissues’ microstructure and dynamics, as well as variety of different tissues and layers that organs are constructed from. This makes the modelling of individual organs difficult enough, whilst the demand is for whole virtual humans to be modelled. Unfortunately isolating individual organs for the purpose of complex simulations such as crash test is perhaps not as useful as one might hope. In-vivo simulations are highly dependent on accurate consideration of all internal features. Organs are tightly packed into the body and thus the mechanical responses of each are dependent on the contact forces with those surrounding them. High and low rate loading occurs at the skin level and the subsequent force has to travel in a wave-like form, through numerous layers and organs, so solutions must ideally take this into account. Some very positive work in the field has been produced with respect to the modelling of muscle tissue, particularly by the CHARM project which is aiming to unify the knowledge base and bring biomechanics closer to being able to simulate the whole body. However it would seem that constitutive modelling of this tissue benefits well from the intuitive approach proposed by Lanir for handling structures with well-defined fibre direction such as skeletal muscle. Similarly a number of useful methods have been proposed for the modelling of tendons and ligaments, including the quasi-linear viscoelastic method. Sadly, it seems, there are large gaps in the research for a number of organs. It is assumed that this is because of the relative complexity of the microstructure and function of biological soft tissue. Clearly methods intended to provide solutions for an organ by applying a single set constitutive equations uniformly through it, will not provide truly successful results. There is simply not enough research into the mechanics of numerous different tissues within individual organs to be able to model them accurately. As would be expected, research currently still focuses on whichever tissue forms the bulk of the organ. However they may still be indicative, within a (large but possibly acceptable) margin, of the stresses involved in a particular situation and may still be useful in the prevention of injury. Similarly it is apparent that the result may also be dangerously misleading as effects such as focussing of impact waves caused by changes in material structure and shape are very sensitive to the particular model used. This would happen locally but not globally. A particular area where there appears to be relatively little research is in brain tissue modelling. Experimentally it is a difficult material to investigate,

responding somewhere between a gel and a paste. However there is a distinct difference between approaching the modelling by means of purely experimental methods or investigation of the microstructure. The latter method may be more useful for tackling such problems, but ideally a combination of these would be preferable. Another problem, it would seem, is that continuum approaches are applied to model structures where perhaps it isn’t strictly applicable. The earlier work on the lung, for example, made attempts to consider the alveoli properties as part of the constitutive equations when their relative size could be considered to be too large for a continuum approach to apply. The work on treating the alveoli as a branched collection of open polyhedra seems to be a very useful and popular approach. Some work still needs to be done into the distribution of these features in the organ as a whole as the global mechanical properties are very dependent on local regions in the lung. Similarly, there needs to be more work into the distribution of collagen types in the different parts of the organ. It is most encouraging to notice that many of the newer methods in the field of tissue modelling have a very firm basis in earlier research. This may be indicative of the relative infancy of this area of study but also that the early work provides a very good foundation. This is fundamentally due to the vast amount of medical knowledge on the structure and behaviour of the tissue that provides a strong base on which to develop the mathematical models. The long term future of this research lies in the development of better models and constitutive equations for all of the organs, culminating in simulation of the whole virtual human where every organ in the body reacts in response to contact forces with every other one which in turn respond realistically to the gross initial conditions such as rigid body simulation of the skeleton. Fluid interactions abundant in a majority of the organs in the body will be more practical considerations which may be handled by finite element/finite volume methods. It has been observed that even the thinnest of fluid layers on the surface of the alveoli in the lung can have huge influence on its mechanical properties. Better methods will need to be developed to model the effect of features that are too large to be considered part of the continuum and too complicated or numerous to be accurately created in the original polygonal mesh (such as muscle fibres, alveoli and so on). As the mathematical models improve, covering wider ranges of inputs (say high and low rates of loading) and generally increasing the accuracy and applicability of the solutions, the constitutive equations will inevitably become longer and more complicated. Whether new, more efficient algorithms will be developed to solve such complex simulations or raw computing power will be increased to compensate, is yet to be determined. Biological soft tissue is by no means trivial to model and each different type considered, produces many new problems to be solved experimentally or theoretically. Certainly the push to create accurate virtual humans has generated a lot of interest in a number of related fields and hopefully as much motivation to develop the methods required to attain that target. From this there will be new levels of safety analysis which will affect everyday life greatly. When we will actually get there remains to be seen.

12. References [BERNE72] R. M. Berne, M. N. Levy (1972) Cardiovascular Physiology, 2nd Edition. C. V. Mosby, St. Louis (images taken from [FUNG81]) [BOGERT98] A. J. van der Bogart, K. G. M. Gerritsen, G. K. Cole (1998), “Human Muscle Modelling From a User’s Perspective,” Journal of Electromyography and Kinesiology [CAPELO81] A. Capelo, V. Comincioli, R. Minelli, C. Poggesi, C. Reggiani, L. Ricciardi (1981), "Study and Parameters Identification of a Rheological Model for Excised Quiescent Cardiac Muscle," Journal of Biomechanics [CARTON62] Carton (1962), "Elastic Properties of Single Elastic Fibers,” Journal of Applied Physiology. Taken from [MAUREL98b] [CHARM96] E.B. Pires, J.A.C. Martins, J.A.Carvalho, L.R. Salvado, G. Engel (1996), "3-D Finite Element Prototype," Esprit Project 9036 CHARM, IST Lisbon. [COLLINS72] R. Collins, W. C. Hu, (1972) “Dynamic Deformation Experiments on Aortic Tissue,” Journal of Biomechanics [FLICKER74] E. Flicker, J. S. Lee, (1974) “Equilibrium of Force of Subpleural Alveoli: Implications to Lung Mechanics,” Journal of Applied Physiology [FUNG67] Y. C. Fung (1967), "Elasticity of Soft Tissues in Simple Elongation," American Journal of Physiology (extract taken from [Fung90]) [FUNG72] Y. C. Fung, N. Perrone, M. Aliker (1972), "Stress-Strain History Relations of Soft Tissues in Simple Elongation," in Biomechanics: Its Foundations and Objectives, Englewood Cliffs, N. J. Prentice-Hall [FUNG81] Y. C. Fung (1981), Biomechanics: Mechanical Properties of Living Tissues. Berlin: Springer-Verlag [FUNG88] Y. C. Fung, R. T. Yen, Z. L. Tao, S. Q. Liu (1988) “A Hypothesis on the Mechanism of Trauma of Lung Tissue Subjected to Impact Load,” Journal of Biomechanical Engineering [FUNG90] Y.C. Fung (1990), Biomechanics: Motion, Flow, Stress and Growth. Berlin: Springer-Verlag [GORDON66] A. M. Gordon, A. F. Huxley, F. J. Julian (1966), “The Variation in Isometric Tension with Sacomere Length in Vertebrate Muscle Fibres”, Journal of Applied Physiology [GRAY91] H. Gray, T. Pickering, R. Howden (1991), Gray’s Anatomy 15th Rev. Edition, Random House

[HELM91] F.C.T. van der Helm, R. Veenbaas (1991), "Modelling the Mechanical Effect of Muscles with Large Attachment Sites: Application to the Shoulder Mechanism," Journal of Biomechanics [HONGYAN99] L. Hongyan, S. Belkoff (1999), "A Failure Model for Ligaments,” Journal of Biomechanics [HOROWITZ88] A. Horowitz, I. Sheinman, Y. Lanir, M. Perl, R.K. Strumpf (1988), “Structural Three-Dimensional Constitutive Law for the Passive Myocardium,” Journal of Biomechanical Engineering [HUMPHREY87] J.D. Humphrey, F.C.P. Yin (1987), "On Constitutive Relations and Finite Deformations of Passive Cardiac Tissue – Part I: A Pseudo-Strain Energy Function," Journal of Biomechanical Engineering [HUMPHREY90] J.D. Humphrey, RK Strumpf, F.C.P. Yin (1990), "Determination of a Constitutive Relation for Passive Myocardium – Parts I and II," Journal of Biomechanical Engineering [KASTELIC80] J. Kastelic, I. Palley, E. Baer (1980), "A Structural Mechanical Model for Tendon Crimping," Journal of Biomechanics [KENEDI64] R. M. Kenedi, T. Gibson, C.H. Daly (1964), "Bioengineering Studies of the Human Skin," in Biomechanics and Related Bioengineering Topics. Oxford: Pergamon Press. Taken from [MAUREL98b] [LANIR83] Y. Lanir (1983), "Constitutive Equations for Fibrous Connective Tissues,” Journal of Biomechanics [LANIR83b] Y. Lanir (1983), "Constitutive Equations for the Lung Tissue," Journal of Biomechanical Engineering. [LIEBER00] R. Lieber et al (2000), "Musculoskeletal Soft Tissue Mechanics," in The Biomechanical Engineering Handbook, CRC Press LLC [MARTINS98] J.A.C. Martins, E.B. Pires, L.R. Salvado, P.B. Dinis (1998), "A numerical model of the passive and active behavior of skeletal muscles," Computer Methods in Applied Mechanics and Engineering [MARTINI98] R. H. Martini (1998), Fundamentals of Anatomy and Physiology, Fourth Edition, Prentice-Hall [MASE70] G.E.Mase (1970), “Theory and Problems of Continuum Mechanics,” Schaum’s Outline Series, McGraw-Hill, Inc. [MAUREL98a] W. Maurel, Y. Wu, N. Magnenat Thalmann, D. Thalmann (1998), “Biomechanical Models for Soft Tissue Simulation,” (Springer-Verlag Berlin/Heidelberg)

[MAUREL98b] W. Maurel, (1998), “3D Modeling of the Human Upper Limb Including The Biomechanics of Joints, Muscles And Soft Tissues,” PhD Thesis. Ecole Polytechnique Federale De Lausanne [MCELHANEY66] J. H. McElhaney (1966), “Dynamic Response of Bone and Muscle Tissue,” Journal of Applied Physiology [MILLER97] K. Miller, K. Chinzei, (2000), “Constitutive Modelling of Brain Tissue: Experiment and Theory,” Journal of Biomechanics [MILLER00] K. Miller, K. Chinzei, G. Orssengo, P. Bednarz (2000), “Mechanical Properties of Brain Tissue In-Vivo: Experiment and Computer Simulation,” Journal of Biomechanics [MILLER00b] K. Miller (2000), “Constitutive Modelling of Abdominal Organs,” Journal of Biomechanics [PANDY90] M.G. Pandy, F.E. Zajac, E. Sim, W.S. Levine (1990), "An Optimal Control Model for Maximum-Height Human Jumping," Journal of Biomechanics [SEGERS00] P. Segers, P.Verdonck (2000), “Role of Tapering in Aortic Wave Reflection: Hydraulic and Mathematical Model Study,” Journal of Biomechanics [SHOEMAKER86] P.A. Shoemaker, D. Schneider, M.C. Lee, Y.C. Fung (1986), "A Constitutive Model for Two-Dimensional Soft Tissues and its Application to Experimental Data," Journal of Biomechanics [TONG76] P. Tong, Y.C. Fung (1976), "The Stress-Strain Relationship for the Skin," Journal of Biomechanics [VAWTER82] D. L. Vawter, W. H. Shields (1982), “Deformation of the Lung: the Role of Interfacial Forces,” in Finite Elements in Biomechanics, edited by R. H. Gallagher. Wiley, New York. [VERDONA70] D.R. Veronda, R.A. Westmann (1970), "Mechanical Characterization of Skin-Finite Deformations," Journal of Biomechanics [VIIDIK68] A. Viidik (1968), "A Rheological Model For Uncalcified Parallel-Fibered Collagenous Tissue," Journal of Biomechanics [VIIDIK80] A. Viidik, J. Vuust (1980), Biology of Collagen: Proceedings of a Symposium, Aarhus, London: Academic Press [VIIDIK87] A. Viidik (1987), "Properties of Tendons and Ligaments", in Handbook of Bioengineering, edited by R. Skalak, S. Chien. New York: McGraw-Hill [WERTHEIM47] M.G. Wertheim (1847), "Mémoire Sur L'élasticité et la Cohésion des Principaux Tissus du Corps humain," Annls. Chim. Phys. Taken from [MAUREL98b]

[WINTER90] D.A. Winter (1990), "Muscle Mechanics", in Biomechanics and Motor Control of the Human Movements, New York: Wiley [WOO89] S L-Y. Woo, M.K. Kwan (1989), "A Structural Model to Describe the Non-Linear Stress-Strain Behaviour for Parallel-Fibered Collagenous Tissues," Journal of Biomechanical Engineering [YEN88] R. T. Yen, H. H. Ho, Z. L. Tao, Y. C. Fung (1988), “Trauma of the Lung Due to Impact Load,” Journal of Biomechanics [ZAJAC86] F.E. Zajac, E.L. Topp, P.J. Stevenson (1986), "A Dimensionless Musculotendon Model," Proceedings of 8th Annual Conference IEEE Engineering Med. Biol. Soc [ZAJAC89] F.E. Zajac (1989), "Muscle and Tendon: Properties, Models Scaling and Application to Biomechanics and Motor Control,” CRC Critical Review in Biomedical Engineering

mechanical modeling of soft tissue

Documents