7.5 Behavior of Soft Tissues under uniaxial Loading

1. 7.5 Behavior of Soft Tissues under uniaxial Loading • Pure biological materials: actin, elastin, collagen. • Tissues: several aforementioned materials & ground substance. • Experimental approach to constitutive equation. • Single axial tension test on cylindrical specimen, load & elongation are recorded, stress-strain relationship. • Wertheim (1847): non-Hookean, tissues is under stressed in physiological state, artery shrunk from cut, broken tendon retravts

2. Preconditioning Cyclic response of dog’s carotid artery l1: stretch ratio referred to zero-stress length of segment, 37 deg C, 0.21 cycles/min

3. Hysteresis of rabbit papillary (乳頭) muscle Increasing strain rate

4. Relaxation of rabbit mesentery(腸繫膜)

5. Long-term relaxation G(t) Log 10 t

6. Creep of papillary muscle of rabbit Log 10 t

7. summary • Hysteresis, relaxation, creep at lower stress ranges are common for mesentery of rabbit, cat & dog, ureter of animals, papillary muscles at resting • Difference: degree of distensibility • Mesentery 100%-200% from relaxed length • Ureter 60% • Heart muscle 15% • Arteries & veins 60% • Skin 40% • Tendon 2%-5%

8. 7.5.1 Stress Response in loading and unloading Lagrange stress

9. Notes: • For Hookean materials d T/d l = const • Piece-wise linear model (practical)

10. 7.5.2 Other expressions • For finite deformation of elastic body, strain energy (or elastic potential), W, is often used • For elastic, isotropic material W is function of strain invariants. • Examples: Mooney(1940), Rivlin(1947), Rivlin & Saunders(1951), Green & Adkins (1960)*

13. Mechanical properties of Cornea

14. 7.6 Quasi-Linear Viscoelasticity of Soft Tissues • Biological materials not elastic, history of strain affects stress, loading unloading difference. • Linear theory of viscoelasticity, continuous relaxation spectrum (sec. 2.11), combination of an infinite no. of Voigt & Maxwell elements. • Nonlinear theory, a sequence of springs of different natural length with no. of springs increases with increasing strain. • Linear viscoelasticity for small oscillation; for finite deformation, nonlinear stress-strain characteristics.

15. hypothesis • Consider a cylindrical specimen subjected to tensile load, a step increase in elongation imposed, stress ~ function of time t & stretch ratio l G(t) 1 0 t

17. Tensile stress = Instantaneous response + decrease due to past history Experimental determination of T(e)[l] and G(t)

18. 7.6.1 Elastic Response (experimental) • By definition T (e)(l) is instantaneous tensile stress generated by a step stretch; transient stress waves due to sudden loading will be added. • Assumptions: • G(t) is continuous • T (e)(l) can be approximated by T(l) with high loading rate

19. Justification • G(0)=1, if l is increased from 0 to l in time interval e, at t= e we have

20. 7.6.2 Reduced Relaxation Function • Assume Relaxation function = sum of exponential functions, and identify exponents & coefficients • Experiment cut off too early can induce error • Non-uniqueness of the fitting • Notes • A law based on G(t) as t goes to infinite is unreliable • Other experiments should be utilized to determine the relaxation function.

21. 7.6.3 Special Characteristics of Hysteresis of Living Tissues • Hysteresis loop is almost independent of strain rate within several decades of rate variation. • Incompatible with viscoelastic model with finite no. of springs & dashpots. (discrete relaxation rate constants) • A continuous distribution of exponents ni should be considered Discrete spectrum Continuous spectrum

22. 7.6.4 G(t) related to Hysteresis ts: time constant for creep at const stress te: time constant for relaxation at const strain ER: residual of elastic response • Standard linear solid

23. Frequency response function Lead compensator in control engineering Note: tand is a measure of internal damping, if it is not frequency dependent, the peak must spread out – superposing a large no. of Kelvin models

24. 7.6.5 Continuous Spectrum of Relaxation • Relaxation function

25. How to find S(t) • Idea: find S that will make G(t), J(t) and M(w) to match with the experimental data. M(w) to be nearly constant for a wide range of frequency.

26. Constant damping for t1<1/w<t2 Maximum damping at w=1/√t1t2

27. Maximum damping

28. Reduced relaxation function Note: continuous relaxation spectrum

30. Reduced Creep Function • Laplace transform is used to solve for creep function from reduced relaxation function

32. Notes • Relaxation spectrum Eqs.(30)(31) & relaxation function Eq.(34) & complex modulus Eq. (32) & damping (33) work very well for many living tissues. • Creep functions Eqs. (45)(46) does not work so well; good for papillary muscle not for blood vessels & lung tissues • Creep is a more nonlinear process & does not obey the quasi-linear hypothesis. Microstructure movement in creep process is different from that of relaxation or oscillation, analog to metal at high temperature.

33. Maxwell Voigt Kelvin Hysteresis vs. frequency Biological soft tissues 7.6.6 A graphic Summary • Generalization • Large no. of Kelvin units in series • Nonlinear elastic springs, same type • Size distribution of springs & dampers

34. 7.6.7 History Remarks • Above theory (Fung, 1972) • Hysteresis insensitive to frequency (Becker & Foppl 1928) • Structure damping (Garrick 1940) • Earth’s crust internal friction (Routbart & Sack 1966) • Kink in dislocation line & kink energy barrier (Manson 1969) • Special plasticity theory (Bodner 1968) • Eq (47) by Wagner (1913)

35. 7.6.8 Oscillatory Stretch If the amplitude is small then linear viscoelasticity theory can be used. arteries: amplitude of strain < 4% on top of l = 1.6 Reproducible up to 16h following removal of tissue nonlinear elasticity of the tissue

36. 7.6.9 Example: Collagen Fibers in Uniaxial Extension • Experimental results in 7.3; regimes • Small strain toe region, Lagrange stress is nonlinear function of stretch ratio • Linear regime • Non-physiological, overly extended, & failing regime • Stress-strain relationship in toe • Both toe and linear region (Mooney-Rivlin material) T: Lagrange stress, l: stretch ratio, m=finite in toe region, m=0 in linear region

37. Note: Mooney-Rivlin is isotropic, collagen fibers are transverse orthotropic; let x3 axis be axis of fiber then strain energy function • If stress T in Eq.(50) is the elastic response T(e) then from quasi-linear theory, stress at time t due to strain history l(t) is:

38. Limitations & Extensions • QLV model work reasonable for skin, arteries, veins, tendons, ligaments, lung parenchyma, pericardium, muscle & ureter in relaxed state. • In reality, specific tissue may have a spectrum with a localized peaks & valleys not considered in QLV • No experimental identification of a relaxation function which is dependent on invariants of stress, strain & strain rate. • General theory of nonlinear viscoelastic materials (Green & Rivlin 1957), tensorial power series expansion.

39. 7.7 Incremental Laws • Mechanical properties of soft tissues such as arteries, muscle, skin, lung, ureter, mesentery are inelastic (hysteresis, anisotropic, nonlinear stress-strain relationship) • Incremental laws: linearized relationship between incremental stresses & strains by small perturbation about an equilibrium condition.

40. l between 1 & 1.24 Rabbit Mesentery Note: Small loops are not parallel to each other, neither are tangent to loading unloading curves Incremental moduli should be determined by incremental experiments Pseudo-elasticity laws is simpler for full range of deformation

41. 7.8 Pseudo-Elasticity • Simplification of QLV to pseudo-elastic equation (for preconditioned tissue) • For loading & unloading branches, the stress-strain relationship is unique • Treat the material as one elastic material in loading and another in unloading • Hysteresis independent of strain rate • 1000-fold change of strain rate vs 1 to 2 fold change of stress of a given strain. Ultrasound experiments suggest lower limit of relaxation time 10-8, • To describe stress-strain relationship in loading & unloading by a law of elasticity and it can be further simplified if assuming a strain energy function exists.

42. 7.9 Biaxial Loading Experiments on Soft Tissues Rectangular specimen of uniform thickness in biaxial loading

43. Typical display of specimen on VDA monitor Applications: Testing of skin Lung tissue, with thickness measurement Digital computer control of stretching in two directions • Key Issues: • Need to control boundary conditions, edges must be allowed to expand freely • In target region stress and strain should be uniform, away from outer edges • Strain is measured optically to avoid mechanical disturbance

44. 7.9.1 Whole organ experiments • Alternative: test whole organ • For lung, whole lobe in vivo or in vitro • For artery, deformation when internal/external pressure are changed or longitudinal tension is imposed. • In whole organ test, tissues not subjected to traumatic excision, close to in vivo condition • Difficulty in analyzing whole organ data • Complement to excised experiments

45. 7.10 Three-dimensional Stress and Strain States • Consider a rectangular plate of uniform thickness, orthotropic material • Two pairs of forces F11, F22 act on the edges; no shear stress and x, y are principal axes • s– Cauchy stress (~equilibrium Eq. ) • T- Lagrange stress (~lab) • S- Kirchhoff stress (~strain energy) • r0– density at zero stress

46. For large deformation

47. 7.11 Strain-energy Function • Strain potential or Strain-energy function • W: strain energy per unit mass of tissue (J/kg) • r0 density in zero-stress state (kg/m3) • r0W: strain energy per unit volume (J/m3) • Let W be expressed in terms of strain components E11, E22, E33, E12, E21, E23, E32, E31, E13, and using symmetric properties.

49. General relationship between Cauchy, Lagrange & Kirchhoff Stresses • s– Cauchy stress (~equilibrium Eq. ) • T- Lagrange stress (~lab) • S- Kirchhoff stress (~strain energy)

50. Pseudoelasticity & pseudo-strain energy function • If a material is perfectly elastic then a strain-energy function exists (thermodynamics) • Living tissues are not perfectly elastic a strain energy function not exist. • Fact: after preconditioning, cyclic loading & unloading stress-strain relationship are strain-rate independent. • Loading & unloading curves can be treated as two uniquely defined stress-strain relationships, each associated with a strain-energy function • Pesudo-elasticity curve, pseudo-strain energy function • Must be justified by experiments