Mathematical Modelling of Biological Soft Tissues

Mathematical Modelling of Biological Soft Tissues 1., William J. Parnell 1., Barbara Lynch 2., Hazel R.C. Screen 3. and I. David Abrahams 4. 1. University of Manchester 2. Ecole Polytechnique 3. Queen Mary University of London 4. University of Cambridge

Skin Biological soft tissues Anterior cruciate ligament rupture Energy storing and positional tendons Arteries and muscles Heart Tendon

Biological soft tissues Anterior cruciate ligament rupture Energy storing and positional tendons Anterior cruciate ligament rupture

Biological soft tissues Anterior cruciate ligament rupture Energy storing and positional tendons Different tendons can have radically different stiffnesses Positional tendons (dashed - equine CDET) are stiffer than energy storing tendons (solid - equine SDFT) How do these differences occur despite these tendons being made of the same fundamental materials?

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons Diagram of tendon microstructure, from (Kastelic et al., 1978)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons Simplified hierarchical structure: Fibril Fascicle Tendon Crimp Extracollagenous matrix

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 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. 1978 Kastelic et al. (1978) observed that the fibril crimp angle varies through the radius of a fascicle Variation of crimp angle

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons Sequential straightening and loading model, from (Kastelic et al., 1980)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ(ρ) is a function that satisfies θ(0) = 0, θ(1) = θ o

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ(ρ) is a function that satisfies θ(0) = 0, 2. Calculate the strain in a fibril at radius ρ: l(ρ) l(ρ)+ l(ρ) θ(1) = θ o θ(ρ) L L+ L

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ(ρ) is a function that satisfies θ(0) = 0, 2. Calculate the strain in a fibril at radius ρ: l(ρ) l(ρ)+ l(ρ) θ(1) = θ o θ(ρ) L ǫ = L L, L+ L ǫ f(ρ) = l(ρ) l(ρ), l(ρ) = L cos θ(ρ)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons To calculate the stress-strain response of the fascicle: 1. Assume a form for the fibril crimp angle distribution: θ(ρ) is a function that satisfies θ(0) = 0, 2. Calculate the strain in a fibril at radius ρ: l(ρ) l(ρ)+ l(ρ) θ(1) = θ o θ(ρ) L ǫ = L L, L+ L ǫ f(ρ) = l(ρ) l(ρ), l(ρ) = L cos θ(ρ) L+ L = l(ρ)+ l(ρ) ǫ f (ρ) = cosθ(ρ)(ǫ+1) 1

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 3. Calculate the stress in a fibril at radius ρ (we assume the fibrils obey Hooke s law): σ f (ρ) = Eǫ f (ρ)

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

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 3. Calculate the stress in a fibril at radius ρ (we assume the fibrils obey Hooke s law): σ f (ρ) = Eǫ f (ρ) 4. By integrating, we can calculate the total fascicle stress: τ = 2 R 0 σ f (ρ)ρdρ R p is the radius within which all fibrils are taut for a given fascicle strain. 5. If we choose θ(ρ) carefully, we can calculate this integral analytically: E τ = 3sin 2 θ o ( 2ǫ 1+ ) 1 (ǫ+1) 2

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons We note that the previous relationship only holds for 0 ǫ ǫ, where ǫ = 1/cosθ o +1, and for ǫ > ǫ, we have τ E = 2 1 0 σ f (ρ)ρdρ = β(ǫ+1) 1, β = 2(1 cos3 θ o ) 3sin 2 θ o

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons We note that the previous relationship only holds for 0 ǫ ǫ, where ǫ = 1/cosθ o +1, and for ǫ > ǫ, we have τ E = 2 1 0 σ f (ρ)ρdρ = β(ǫ+1) 1, β = 2(1 cos3 θ o ) 3sin 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 τ = (2λ 3+ 1λ ) 3sin 2 θ 2, 1 λ 1 o cosθ o τ = E(βλ 1), λ > 1 cosθ o

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons We also need to account for the deformation of the extra-collagenous matrix, which results in a model containing four parameters: Constitutive parameters Structural parameters µ matrix shear modulus φ collagen volume fraction E fibril Young s modulus θ o crimp angle of outer fibrils (1 φ)µ ( λ 2 1 ) λ σ = (1 φ)µ ( λ 2 1 ) λ + φe λ < 1 ( ) 3sin 2 θ 2λ 3+ 1 o λ 1 λ 1 2 cosθ o (1 φ)µ ( λ 2 1 λ) +φe(βλ 1) λ > 1 cosθ o

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons The outer crimp angle θ o can be measured by scanning electron microscopy SEM image of fibril crimp (Franchi et al., 2010)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons The outer crimp angle θ o can be measured by scanning electron microscopy SEM image of fibril crimp (Franchi et al., 2010) The matrix shear modulus µ can be measured via mechanical tests on isolated matrix Tension test (Thorpe et al., 2012)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons The outer crimp angle θ o can be measured by scanning electron microscopy The fibrilyoung s modulus E canbe measuredby atomic force microscopy SEM image of fibril crimp (Franchi et al., 2010) Isolated fibril (Svensson et al., 2011) The matrix shear modulus µ can be measured via mechanical tests on isolated matrix Tension test (Thorpe et al., 2012)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons The outer crimp angle θ o can be measured by scanning electron microscopy The fibrilyoung s modulus E canbe measuredby atomic force microscopy SEM image of fibril crimp (Franchi et al., 2010) Isolated fibril (Svensson et al., 2011) The collagen volume fraction φ and fibre alignment vector M can be measured by X-ray computed tomography The matrix shear modulus µ can be measured via mechanical tests on isolated matrix Tension test (Thorpe et al., 2012) XCT image of patellar tendon (Shearer et al., 2014)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons The Holzapfel-Gasser-Ogden (HGO) model is one of the most commonly used anisotropic, non-linear elastic models of biological soft tissue. It gives: { ( ) c λ σ HGO 1 = λ λ < 1 c ( λ λ) 1 +4k1 λ 2 (λ 2 1)e k 2(λ 2 1) 2 λ 1 The parameters c, k 1 and k 2 are chosen to match a given set of experimental data.

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons The Holzapfel-Gasser-Ogden (HGO) model is one of the most commonly used anisotropic, non-linear elastic models of biological soft tissue. It gives: { ( ) c λ σ HGO 1 = λ λ < 1 c ( λ λ) 1 +4k1 λ 2 (λ 2 1)e k 2(λ 2 1) 2 λ 1 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 slide.

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 12 10 8 6 4 2 stress (MPa) 0.005 0.010 0.015 0.020 0.025 0.030 strain 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 (Holzapfel et al., 2000). 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

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons Tendons whose fascicles have a helical fibril arrangement σ = σ(µ,e,φ,θ o,α) Right: SEM image of a fascicle from (Yahia and Drouin, 1989)

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons This model was used to fit stress-strain data on energy storing and positional tendons (equine SDFT and CDET) using θ o and α as fitting parameters to predict their values. 200 stress (MPa) 150 100 50 0.05 0.10 0.15 0.20 0.25 strain

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 36 data sets were fitted with a minimum coefficient of determination (R 2 value) of 0.979 across all 36 data sets.

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 36 data sets were fitted with a minimum coefficient of determination (R 2 value) of 0.979 across all 36 data sets. Results: Tendon Crimp angle (θ o ) Helix angle (α) CDET 15.1 ±2.3 7.9 ±9.3 SDFT 15.8 ±4.1 29.1 ±10.3

Tendon hierarchical structure A typical stress-strain curve and fibril crimp The sequential straightening and loading (SSL) model Positional and energy storing tendons 36 data sets were fitted with a minimum coefficient of determination (R 2 value) of 0.979 across all 36 data sets. Results: Tendon Crimp angle (θ o ) Helix angle (α) CDET 15.1 ±2.3 7.9 ±9.3 SDFT 15.8 ±4.1 29.1 ±10.3 This shows that the differences in the stress-strain response of these two tendon types could be entirely due to differences in the alignment of their fibrils. This agrees with qualitative observations in the literature.

Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues Langer s lines are lines of greatest tension in skin Collagen fibres thought to be co-aligned with them Is all the collagen aligned this way?

Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues Unixial test data on rabbit skin along (blue) and perpendicular to (red) Langer s lines, from (Lanir and Fung, 1974)

One fibre family Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues Two fibre families Data fitted using HGO model

Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues

Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues f(λ c ) = 0, λ c < a, 2(λ c a) (b a)(c a), a λ c c, 2(b λ c) (b a)(b c), c λ c b, 0, λ c > b,

Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues

Langer s lines Transverse isotropy vs orthotropy A more general model for soft tissues

Viscoelastic properties of ligaments and tendons The model Experiments Results Hysteresis

Viscoelastic properties of ligaments and tendons The model Experiments Results Hysteresis Strain-rate dependence

Viscoelastic properties of ligaments and tendons The model Experiments Results Hysteresis Strain-dependent relaxation Strain-rate dependence

Viscoelastic properties of ligaments and tendons The model Experiments Results Hysteresis Strain-dependent relaxation Strain-rate dependence We aim to develop a model that captures all of these behaviours but is still mathematically tractable.

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibrils and matrix modelled as linear viscoleastic: σ f (t) = E(0)e f (t)+ t 0 E (t s)e f (s)ds,

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibrils and matrix modelled as linear viscoleastic: σ f (t) = E(0)e f (t)+ t 0 E (t s)e f (s)ds, where E(t) = E 0 +E 1 e t/τ 1 +E 2 e t/τ 2

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibrils and matrix modelled as linear viscoleastic: σ f (t) = E(0)e f (t)+ t 0 E (t s)e f (s)ds, where E(t) = E 0 +E 1 e t/τ 1 +E 2 e t/τ 2 Matrix strain equals fascicle strain: e m (t) = e(t)

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril behaviour is more complex: Loading: e f (t) = e(t) e c 1+e c

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril behaviour is more complex: Loading: e f (t) = e(t) e c 1+e c Unloading: Fibril slackens when σ f = 0 e f (t) = t 0 J (t s)σ f (s)ds J(t) = J 0 +J 1 e t/t 1 +J 2 e t/t 2

Viscoelastic properties of ligaments and tendons The model Experiments Results e(t) Relaxation tests e f (t) t σ f (t) t t A t

Viscoelastic properties of ligaments and tendons The model Experiments Results e(t) Relaxation tests e(t) Cycle test e f (t) t e f (t) t σ f (t) t σ f (t) t t A t t A t B t C t t Dt B A t C t D t

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril parameters determined from tests on individual fibrils: (Yang et al., 2012) E = 3.0 GPa, E 1 = 950 MPa, E 2 = 810 MPa, τ 1 = 1.9 s, τ 2 = 52 s.

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril parameters determined from tests on individual fibrils: (Yang et al., 2012) E = 3.0 GPa, E 1 = 950 MPa, E 2 = 810 MPa, τ 1 = 1.9 s, τ 2 = 52 s. Collagen volume fraction set to 80% (Screen et al., 2005)

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril parameters determined from tests on individual fibrils: (Yang et al., 2012) E = 3.0 GPa, E 1 = 950 MPa, E 2 = 810 MPa, τ 1 = 1.9 s, τ 2 = 52 s. Collagen volume fraction set to 80% (Screen et al., 2005) Matrix neglected to reduce number of parameters

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril parameters determined from tests on individual fibrils: (Yang et al., 2012) E = 3.0 GPa, E 1 = 950 MPa, E 2 = 810 MPa, τ 1 = 1.9 s, τ 2 = 52 s. Collagen volume fraction set to 80% (Screen et al., 2005) Matrix neglected to reduce number of parameters Crimp distribution (truncated) normal distribution

Viscoelastic properties of ligaments and tendons The model Experiments Results Fibril parameters determined from tests on individual fibrils: (Yang et al., 2012) E = 3.0 GPa, E 1 = 950 MPa, E 2 = 810 MPa, τ 1 = 1.9 s, τ 2 = 52 s. Collagen volume fraction set to 80% (Screen et al., 2005) Matrix neglected to reduce number of parameters Crimp distribution (truncated) normal distribution Distribution parameters unknown, so fitted to data

70 σ(t) (MPa) 70 Viscoelastic properties of ligaments and tendons The model Experiments Results σ(t) (MPa) 70 60 60 60 50 50 50 40 40 40 30 30 30 20 20 t (s) 10 20 t (s) 10 0 80 200 400 600 800 σ(t) (MPa) σ(t) (MPa) 0 80 200 400 600 800 σ(t) (MPa) 0 80 60 60 60 40 40 40 20 20 0.5-20 1.0 1.5 2.0 t (s) t (s) 10 200 400 600 800 σ(t) (MPa) 20 0.5-20 1.0 1.5 2.0 t (s) 0.5 1.0 1.5 2.0 t (s) -20 Relaxation tests are fitted, cycle tests are predicted.

Viscoelastic properties of ligaments and tendons The model Experiments Results The model exhibits hysteresis and strain-dependent relaxation σ (MPa) 40 30 20 10 1 2 3 4 5 6 e (%)

Viscoelastic properties of ligaments and tendons The model Experiments Results The model exhibits hysteresis and strain-dependent relaxation σ (MPa) σ(t)/σ(0.01) 1.0 40 0.8 30 0.6 0.4 20 0.2 50 100 150 200 250 300 t (s) 10 1 2 3 4 5 6 Need to include matrix phase for initial-strain dependent relaxation e (%)

Tissue engineering Conclusions References Lab-grown meat Imagine we want a tendon with a given stress-strain behaviour We could use mathematical modelling to determine the required crimp distribution parameters You could then produce a lab-grown tendon with this crimp distribution This could then be used in surgery instead of autograft or allograft Lab-grown tendon

Tissue engineering Conclusions References Mathematical modelling gives us insights into the structure-function relationships of biological soft tissues

Tissue engineering Conclusions References Mathematical modelling gives us insights into the structure-function relationships of biological soft tissues The differences between positional and energy storing tendons may be due to the helical fibril arrangement in energy storing tendons

Tissue engineering Conclusions References Mathematical modelling gives us insights into the structure-function relationships of biological soft tissues The differences between positional and energy storing tendons may be due to the helical fibril arrangement in energy storing tendons Skin must be at least orthotropic, but is likely more complex

Tissue engineering Conclusions References Mathematical modelling gives us insights into the structure-function relationships of biological soft tissues The differences between positional and energy storing tendons may be due to the helical fibril arrangement in energy storing tendons Skin must be at least orthotropic, but is likely more complex Many complex viscoelastic phenomena arise naturally from the geometrical arrangement of the fibrils in a soft tissue

Tissue engineering Conclusions References Mathematical modelling gives us insights into the structure-function relationships of biological soft tissues The differences between positional and energy storing tendons may be due to the helical fibril arrangement in energy storing tendons Skin must be at least orthotropic, but is likely more complex Many complex viscoelastic phenomena arise naturally from the geometrical arrangement of the fibrils in a soft tissue In the future we may be able to use mathematical modelling to lab-grow soft tissues with custom mechanical properties

Tissue engineering Conclusions References Shearer, T. 2015 A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure. J. Biomech. 48, 290 297 Shearer, T. 2015 A new strain energy function for modelling liagments and tendons whose fascicles have a helical arrangement of fibrils. J. Biomech. 48, 3017 3025 Shearer, T., Thorpe, C.T., Screen, H.R.C. 2017 The relative compliance of energy-storing tendons may be due to the helical fibril arrangements of their fascicles. J. Roy. Soc. Inteface, 14, 20170261. Shearer, T., Parnell, W.J. Lynch, H.R.C. Screen, J.-M. Abrahams, I.D. 2017 Sequential straightening and loading viscoelasticity: A model of the timedependent behaviour of tendon fascicles incorporating fibril recruitment. In preparation. Shearer, T. Rawson, S. Castro, S.J. Ballint, R. Bradley, R.S. Lowe, T. Vila-Comamala, J. Lee, P.D. Cartmell, S.H. 2014 X-ray computed tomography of the anterior cruciate ligament and patellar tendon. Muscles Ligaments Tendons J. 4, 238 244 Shearer, T., Bradley, R.S., Hidalgo-Bastida, L.A., Sherratt, M.J., Cartmell, S.H. 2016 3D visualisation of soft biological structures by microct. J. Cell Sci. 129, 2483 2492 Balint, R., Lowe, T., Shearer, T. 2016 Optimal Contrast Agent Staining of Ligaments and Tendons for X-ray Computed Tomography. PLoS ONE 11, e0153552.