A new strain energy function for the hyperelastic modelling of ligaments and tendons

Transcription

1 A new strain energy function for the hyperelastic modelling of ligaments and tendons University of Manchester BMC-BAMC 2015

2 Anterior cruciate ligament reconstruction surgery Ligament and tendon hierarchical structure Talk outline The anterior cruciate ligament (ACL) is the most frequently injured knee ligament, but does not heal naturally when ruptured In surgery, it is commonly replaced by the patellar tendon (PT) or hamstring tendons (HT) Open question: which replacement tendon is more similar to the ACL mechanically, and why? Diagram of right knee, from Wikimedia Commons

3 Anterior cruciate ligament reconstruction surgery Ligament and tendon hierarchical structure Talk outline Diagram of tendon microstructure, from Kastelic et al. 1978

4 Anterior cruciate ligament reconstruction surgery Ligament and tendon hierarchical structure Talk outline We shall derive a model for the elastic behaviour of a ligament or tendon based on a simpler hierarchical structure: Fibril Fascicle Tendon Crimp Fascicular membrane

5 Anterior cruciate ligament reconstruction surgery Ligament and tendon hierarchical structure Talk outline We shall derive a model for the elastic behaviour of a ligament or tendon based on a simpler hierarchical structure: Fibril Fascicle Tendon Outline Crimp Fascicular membrane Derivation of the stress-strain response of a single fascicle

6 Anterior cruciate ligament reconstruction surgery Ligament and tendon hierarchical structure Talk outline We shall derive a model for the elastic behaviour of a ligament or tendon based on a simpler hierarchical structure: Fibril Fascicle Tendon Outline Crimp Fascicular membrane Derivation of the stress-strain response of a single fascicle A new strain energy function for ligaments and tendons

7 Anterior cruciate ligament reconstruction surgery Ligament and tendon hierarchical structure Talk outline We shall derive a model for the elastic behaviour of a ligament or tendon based on a simpler hierarchical structure: Fibril Fascicle Tendon Outline Crimp Fascicular membrane Derivation of the stress-strain response of a single fascicle A new strain energy function for ligaments and tendons Comparison with the most commonly used existing model

8 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model

9 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model The crimp waveform of the fibrils within a fascicle can be viewed with an optical microscope θ o Increasing radius Fascicle micrograph, from Kastelic et al Kastelic et al. (1978) observed that the fibril crimp angle varies through the radius of a fascicle Variation of crimp angle

10 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model Sequential straightening and loading model, from Kastelic et al. 1980

11 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ p (ρ) = θ o ρ p, or ˆθ p (ρ) = sin 1 (sin(θ o )ρ p )

12 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ p (ρ) = θ o ρ p, or ˆθ p (ρ) = sin 1 (sin(θ o )ρ p ) 2. Calculate the strain in a fibril at radius ρ: l p (ρ) l p (ρ) + l p (ρ) θ p (ρ) L L + L

13 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ p (ρ) = θ o ρ p, or ˆθ p (ρ) = sin 1 (sin(θ o )ρ p ) 2. Calculate the strain in a fibril at radius ρ: l p (ρ) l p (ρ) + l p (ρ) θ p (ρ) ǫ = L L, L ǫf p(ρ) = l p(ρ) l p (ρ), l p(ρ) = L + L L cos θ p (ρ)

14 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ p (ρ) = θ o ρ p, or ˆθ p (ρ) = sin 1 (sin(θ o )ρ p ) 2. Calculate the strain in a fibril at radius ρ: l p (ρ) l p (ρ) + l p (ρ) θ p (ρ) ǫ = L L, L ǫf p(ρ) = l p(ρ) l p (ρ), l p(ρ) = L + L L cos θ p (ρ) L + L = l p (ρ) + l p (ρ) ǫ f p(ρ) = cos θ p (ρ)(ǫ + 1) 1

15 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model 3. Calculate the stress in a fibril at radius ρ (we assume the fibrils obey Hooke s law): σ f p(ρ) = Eǫ f p(ρ)

16 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model 3. Calculate the stress in a fibril at radius ρ (we assume the fibrils obey Hooke s law): σ f p(ρ) = Eǫ f p(ρ) 4. By integrating, we can calculate the total fascicle stress: τ p = 2E Rp 0 σ f p(ρ)ρdρ R p is the radius within which all fibrils are taut for a given fascicle strain.

17 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model 3. Calculate the stress in a fibril at radius ρ (we assume the fibrils obey Hooke s law): σ f p(ρ) = Eǫ f p(ρ) 4. By integrating, we can calculate the total fascicle stress: τ p = 2E Rp 0 σ f p(ρ)ρdρ R p is the radius within which all fibrils are taut for a given fascicle strain. 5. To evaluate this expression explicitly, we must choose a specific value for p and whether to use θ p or ˆθ p. Using ˆθ p gives simpler expressions, as we shall see on the next slide.

18 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model Stress obtained when using ˆθ 1 (ρ) (compared with that when using θ 1 (ρ): ( ) E 1 ˆτ 1 = 3 sin 2 2ǫ 1 + θ o (ǫ + 1) 2 ( τ 1 = E 2 ) ǫ(ǫ + 2) θ o cos 1 (ǫ + 1) 1 (cos 1 (ǫ + 1)) 2 2ǫ 0.10 τ p E or ˆτp E Non-dimensionalised stress as a function of strain. Solid: ˆτ p/e, dashed: τ p/e. Black: p = 1, blue: p = 2. θ o = π/6. ǫ

19 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model We note that the previous relationship only holds for 0 ǫ ǫ, where ǫ = 1/ cos θ o + 1, and for ǫ > ǫ, we have ˆτ 1 1 E = ˆσ 1(ρ)ρdρ f = β(ǫ + 1) 1, 0 β = 2(1 cos3 θ o ) 3 sin 2 θ o

20 A typical stress-strain curve Fibril crimp The sequential straightening and loading (SSL) model We note that the previous relationship only holds for 0 ǫ ǫ, where ǫ = 1/ cos θ o + 1, and for ǫ > ǫ, we have ˆτ 1 1 E = ˆσ 1(ρ)ρdρ f = β(ǫ + 1) 1, 0 β = 2(1 cos3 θ o ) 3 sin 2 θ o By using the relationship λ = ǫ + 1, where λ is the stretch in the direction of the fascicle, we can write E ˆτ 1 = (2λ 3 sin λ ) θ 2, 1 λ 1 o cos θ o ˆτ 1 = E(βλ 1), λ > 1 cos θ o

21 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model Body B X 3 Body b x 3 X R x r X 2 x 2 X 1 x 1 The deformation gradient tensor is given by F = Gradx, F ij = x 1 X 1 x 2 x 1 X 2 x 2 x 1 X 3 x 2 X 1 x 3 X 2 x 3 X 3 x 3 X 1 X 2 X 3 (in Cartesian coordinates)

22 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The left and right Cauchy-Green deformation tensors are given by B = FF T, C = F T F Using the above, we can define the isotropic strain invariants I 1 = trc, I 2 = 1 2 (I 2 1 tr(c 2 )), I 3 = detc

23 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The left and right Cauchy-Green deformation tensors are given by B = FF T, C = F T F Using the above, we can define the isotropic strain invariants I 1 = trc, I 2 = 1 2 (I 2 1 tr(c 2 )), I 3 = detc For a transversely isotropic material, there is a preferred direction defined by the vector M, which leads to the existence of two additional invariants: I 4 = M (CM), I 5 = M (C 2 M)

24 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The left and right Cauchy-Green deformation tensors are given by B = FF T, C = F T F Using the above, we can define the isotropic strain invariants I 1 = trc, I 2 = 1 2 (I 2 1 tr(c 2 )), I 3 = detc For a transversely isotropic material, there is a preferred direction defined by the vector M, which leads to the existence of two additional invariants: I 4 = M (CM), I 5 = M (C 2 M) Any constitutive law for a transversely isotropic nonlinear elastic material must depend only on these five strain invariants.

25 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The constitutive behaviour of an elastic material is governed by a strain energy function, in our case: W = W(I 1, I 2, I 3, I 4, I 5 ), Incompressibility W = W(I 1, I 2, I 4, I 5 )

26 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The constitutive behaviour of an elastic material is governed by a strain energy function, in our case: W = W(I 1, I 2, I 3, I 4, I 5 ), Incompressibility W = W(I 1, I 2, I 4, I 5 ) The Cauchy stress tensor for an incompressible transversely isotropic non-linear elastic material is given by T = QI + 2 W I 1 B + 2 W I 2 (I 1 B B 2 ) + 2 W I 4 m m where Q is a Lagrange multiplier, and m = FM. + 2 W I 5 (m Bm + Bm m)

27 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The constitutive behaviour of an elastic material is governed by a strain energy function, in our case: W = W(I 1, I 2, I 3, I 4, I 5 ), Incompressibility W = W(I 1, I 2, I 4, I 5 ) The Cauchy stress tensor for an incompressible transversely isotropic non-linear elastic material is given by T = QI + 2 W I 1 B + 2 W I 2 (I 1 B B 2 ) + 2 W I 4 m m where Q is a Lagrange multiplier, and m = FM. Note that we can now write I 4 = m W I 5 (m Bm + Bm m)

28 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model We now assume that our strain energy function takes the form W = W(I 1, I 4 ) = (1 φ)w m (I 1 ) + φw f (I 4 ) where φ is the collagen volume fraction in order to obtain a simpler expression for the Cauchy stress: T = QI + 2(1 φ)w m(i 1 )B + 2φW f (I 4)m m

29 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model We now assume that our strain energy function takes the form W = W(I 1, I 4 ) = (1 φ)w m (I 1 ) + φw f (I 4 ) where φ is the collagen volume fraction in order to obtain a simpler expression for the Cauchy stress: T = QI + 2(1 φ)w m(i 1 )B + 2φW f (I 4)m m The stress associated with the fascicles is T f = 2W f (I 4)m m, and the corresponding traction is t f = T f ( ) m m = 2 W f I 4 (m m) ( ) m m = 2 W f m m m = 2 W f m m I 4 m I 4

30 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The component of this traction in the direction of the fascicles is given by ( t f m = 2 W ) ( ) f m m m = 2 W f m 2 dw f = 2I 4 m I 4 m I 4 di 4

31 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The component of this traction in the direction of the fascicles is given by ( t f m = 2 W ) ( ) f m m m = 2 W f m 2 dw f = 2I 4 m I 4 m I 4 di 4 We now equate the general expression for the traction acting on the fascicles with the expression derived for the stress-strain response of a single fascicle: 2I 4 dw f di 4 = E 3 sin 2 θ o (2λ 3 + 1λ 2 ), 1 λ 1 cos θ o 2I 4 dw f di 4 = E(βλ 1), λ > 1 cos θ o

32 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model Since I 4 is the square of the stretch in the fibre direction, λ = I 4, so that dw f E 2I 4 = (2 ) di 4 3 sin 2 I I4 1, 0 I 4 θ o cos 2 θ o 2I 4 dw f di 4 = E(β I 4 1), I 4 > 1 cos 2 θ o

33 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model Since I 4 is the square of the stretch in the fibre direction, λ = I 4, so that dw f E 2I 4 = (2 ) di 4 3 sin 2 I I4 1, 0 I 4 θ o cos 2 θ o E W f = 6 sin 2 θ o 2I 4 dw f di 4 = E(β I 4 1), I 4 > cos 2 θ o We can then solve the above equations for W f : ( 4 I 4 3 log(i 4 ) 1 + γ I 4 W f = E γ = 3, ( β I 4 1 ) 2 log(i 4) + η, I 4 > 1 ), 1 I 4 η = 1 ( 2 cos2 θ o 1 sin 2 log θ o cos θ o 1 cos 2 θ o ) 1 cos 2 θ o

34 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The strain energy function We will assume that the extra collageneous matrix behaves like a neo-hookean material, so that the final form for our strain energy function is W = (1 φ) µ 2 (I 1 3), I 4 < 1 W = (1 φ) µ 2 (I 1 3)+ ( φe 6 sin 2 4 I 4 3 log(i 4 ) 1 ) 1 3, 1 I 4 θ o I 4 cos 2 θ o W = (1 φ) µ ( 2 (I 1 3)+φE β I 4 1 ) 2 log(i 1 4) + η, I 4 > cos 2 θ o

35 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The outer crimp angle θ o can be measured by scanning electron microscopy SEM image of fibril crimp, from Franchi et al. 2010

36 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The outer crimp angle θ o can be measured by scanning electron microscopy SEM image of fibril crimp, from Franchi et al The matrix shear modulus µ can be measured via mechanical tests on isolated elastin Biaxial test, from Gundiah et al. 2007

37 The outer crimp angle θ o can be measured by scanning electron microscopy A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The fibril Young s modulus E can be measured by atomic force microscopy SEM image of fibril crimp, from Franchi et al The matrix shear modulus µ can be measured via mechanical tests on isolated elastin Isolated fibril, from Svensson et al Biaxial test, from Gundiah et al. 2007

38 The outer crimp angle θ o can be measured by scanning electron microscopy A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The fibril Young s modulus E can be measured by atomic force microscopy SEM image of fibril crimp, from Franchi et al The matrix shear modulus µ can be measured via mechanical tests on isolated elastin Isolated fibril, from Svensson et al The collagen volume fraction φ and fibre alignment vector M can be measured by X-ray computed tomography Biaxial test, from Gundiah et al XCT image of patellar tendon, from Shearer et al., 2014

39 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The Holzapfel-Gasser-Ogden (HGO) strain energy function is the most commonly used anisotropic, non-linear elastic model of biological soft tissue. It is given by: W = c 2 (I 1 3), I 4 < 1 = c 2 (I 1 3) + k 1 k 2 (e k 2(I 4 1) 2 1), I 4 1 It is a phenomenological model and the parameters c, k 1 and k 2 are chosen to match a given set of experimental data.

40 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The Holzapfel-Gasser-Ogden (HGO) strain energy function is the most commonly used anisotropic, non-linear elastic model of biological soft tissue. It is given by: W = c 2 (I 1 3), I 4 < 1 = c 2 (I 1 3) + k 1 k 2 (e k 2(I 4 1) 2 1), I 4 1 It is a phenomenological model and the parameters c, k 1 and k 2 are chosen to match a given set of experimental data. To compare the new model with the HGO model we will analyse their ability to match experimental tension test data on human patellar tendon on the following slides.

41 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model We model the patellar tendon as a cylinder and consider a deformation of the form A a r = R ζ, θ = Θ, and z = ζz

42 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The resulting Cauchy stresses are given by T zz = (1 φ)µ(ζ 2 ζ 1 ) ζ < 1 (1 φ)µ(ζ 2 ζ 1 ) + φe 3 sin 2 (2ζ 3 + ζ 2 ), 1 ζ 1 θ o cos θ o (1 φ)µ(ζ 2 ζ 1 ) + φe(βζ 1), ζ > 1 cos θ o { c(ζ Tzz HGO 2 ζ 1 ) ζ < 1 = c(ζ 2 ζ 1 ) + 4k 1 ζ 2 (ζ 2 1)e k 2(ζ 2 1) 2, ζ 1

43 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model The resulting Cauchy stresses are given by T zz = (1 φ)µ(ζ 2 ζ 1 ) ζ < 1 (1 φ)µ(ζ 2 ζ 1 ) + φe 3 sin 2 (2ζ 3 + ζ 2 ), 1 ζ 1 θ o cos θ o (1 φ)µ(ζ 2 ζ 1 ) + φe(βζ 1), ζ > 1 cos θ o { c(ζ Tzz HGO 2 ζ 1 ) ζ < 1 = c(ζ 2 ζ 1 ) + 4k 1 ζ 2 (ζ 2 1)e k 2(ζ 2 1) 2, ζ 1 On the following slide, the nominal stress S zz = T zz /ζ is plotted against the longitudinal strain e = ζ 1. c = (1 φ)µ was chosen to be small (= 0.01MPa); φe, θ o, k 1 and k 2 were fitted to the experimental data.

44 A very brief overview of anisotropic nonlinear elasticity Derivation of the new strain energy function How to experimentally determine the model s parameters Comparison with the Holzapfel-Gasser-Ogden model stress (MPa) Stress-strain curves comparing the ability of the new model to reproduce experimental stress-strain data for human patellar tendon taken from (Johnson et al., 1994) with that of the HGO model. Solid black: new model, dashed blue: HGO model, red circles: experimental data. Parameter values: c = (1 φ)µ = 0.01MPa, θ o = 0.19rad = 10.7, k 1 = 25MPa, k 2 = 183, φe = 552MPa e

45 References We have derived a new strain energy function for modelling ligaments and tendons based on their microstructure

46 References We have derived a new strain energy function for modelling ligaments and tendons based on their microstructure All the parameters in the new strain energy function can be directly measured via experiments

47 References We have derived a new strain energy function for modelling ligaments and tendons based on their microstructure All the parameters in the new strain energy function can be directly measured via experiments The new model exhibits better agreement with experimental data than the HGO model Average relative errors: δ = 0.053, δ HGO = 0.57 Average absolute errors: = 0.12MPa, HGO = 0.31MPa

48 References We have derived a new strain energy function for modelling ligaments and tendons based on their microstructure All the parameters in the new strain energy function can be directly measured via experiments The new model exhibits better agreement with experimental data than the HGO model Average relative errors: δ = 0.053, δ HGO = 0.57 Average absolute errors: = 0.12MPa, HGO = 0.31MPa The new model shows better agreement because it is non-linear even in the small-strain limit

49 References References Shearer, T., A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fibre microstructure. Journal of Biomechanics 48, Shearer, T. Rawson, S. Castro, S.J. Ballint, R. Bradley, R.S. Lowe, T. Vila-Comamala, J. Lee, P.D. Cartmell, S.H., X-ray computed tomography of the anterior cruciate ligament and patellar tendon. Muscles, Ligaments and Tendons Journal 4, Kastelic, J. Galeski, A. Baer, E., The multicomposite structure of tendon. Connective Tissue Research 6, Kastelic, J. Paller, I. Baer, E., A structural model for tendon crimping. Journal of Biomechanics 13, Franchi, M., Ottani, V., Stagni, R., Ruggeri, A., Tendon and ligament fibrillar crimps give rise to left-handed helices of collagen fibrils in both planar and helical crimps. Journal of Anatomy 216, Gundiah, N. Ratcliffe M.B. Pruitt, L.A., Determination of strain energy function for arterial elastin: experiments using histology and mechanical tests. Journal of Biomechanics 40, Svensson, R.B., Hansen, P., Hassenkam, T., Haraldsson, B.T., Aagaard, P., Kovanen, V., Krogsgaard, M., Magnusson, S.P., Mechanical properties of human patellar tendon at the hierarchical levels of tendon and fibril. Journal of Applied Physiology 112, Johnson, G.A. Tramaglini, D.M. Levine, R.E. Ohno, K. Choi N-Y. Woo, S.L-Y., Tensile and viscoelastic properties of human patellar tendon. Journal of Orthopaedic Research 12, Yahia, L-H. Drouin, G., Microscopial investigation of canine anterior cruciate ligament and patellar tendon: Collagen fascicle morphology and architecture. Journal of Orthopaedic Research 7,