Fracture Healing 2. Implementation overview and level-set approach. Martin Pietsch. Summer Term Computational Biomechanics

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1 Seite 1 Modeling background Fracture Healing Fracture Healing 2 Implementation overview and level-set approach Martin Pietsch Computational Biomechanics Summer Term 2016

2 Seite 2 Modeling background Fracture Healing When do we use mathematical models? P. Pivonka and Colin R Dunstan, 2012, Role of mathematical modeling in bone fracture healing When simultaneous multiple events make it difficult to predict intuitively the behaviour of the system When the time/length scales of various events under investigation are significantly different When the system exhibits clearly nonlinear (nonobvious) behavior Experiment in vivo in vitro model in silicio static dynamic

3 Seite 3 Modeling background Fracture Healing Modelling types of bone fracture healing P. Pivonka and Colin R Dunstan, 2012, Role of mathematical modeling in bone fracture healing 1 Cellular-scale models Cell population and temporal evolution Concentration of regulatory factors 2 Tissue-scale models Mostly continuous spatio-temporal models Based on partial differential equations 3 Organ-scale models Primary focus on mechanical stimuli Strong coupling to mechanoregulatory models

4 Seite 4 Modeling background Fracture Healing Modelling types of bone fracture healing P. Pivonka and Colin R Dunstan, 2012, Role of mathematical modeling in bone fracture healing

5 Seite 5 Modeling background Fracture Healing Tissue-scale model (Geris et al., 2008) Density of 12 variables: mesenchymal stem cells (c m ) endothelial cell (c ν ) chondrocytes (c c ) osteoblasts (c b ) fibroblasts (c f ) vascular matrix (m ν ) fibrous extracellular matrix (m f ) cartilaginous extracellular matrix (m c ) bone extracellular matrix (m b ) generic angiogenic growth factor (g ν ) generic osteogenic growth factor (g b ) chondrogenic growth factor (g c )

6 Seite 6 Modeling background Fracture Healing Tissue-scale model (Geris et al., 2008) Evolution description with 12 (partial) differential equations cell population (example): c m t = [D m c m C mct c m (g b + g ν )... ] F 1 c m... differentiation (example): m b t = P bs (1 κ b m b )c b growth factor (example): g b t = [D gb g b ] + E gb c b d gb g b

7 Seite 7 Modeling background Fracture Healing Tissue-scale model (Geris et al., 2008) Mouse model

8 Seite 8 Modeling background Fracture Healing Tissue-scale model (Geris et al., 2008)

9 Seite 9 Modeling background Fracture Healing Biophysical influence of fracture healing Niemeyer 2013 (after Pauwels 1973)

10 Seite 10 Modeling background Fracture Healing Biophysical models of fracture healing (not complete) Frost Determining parameter: bone deformation = L L Distinction between three modes: disuse, conservative, overuse Carter et al. Osteogenic index: I = i n i(s i + kd i ) Consider history of mechanical stimuli Claes & Heigele Based on experimental findings Mechanical stimulation at the existing calcified surfaces Prendergast et al. Poroelastic material (solid filled with fluid phase) Flow velocity as mechanical stimulator

11 Seite 11 Modeling background Fracture Healing Biophysical model Frost H. M. Frost, J Bone Miner Metab (2000) 18: DWindow disuse AWindow adapted MOWindow mild overload POWindow pathological overload MES = microdamage strain thresholds

12 Seite 12 Modeling background Fracture Healing Biophysical model Carter et al. Carter et al. 1998

13 Seite 13 Modeling background Fracture Healing Biophysical model Claes & Heigele, 1999

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15 Seite 14 Modeling background Fracture Healing Biophysical model Claes & Heigele, 1999

16 Seite 15 Modeling background Fracture Healing Biophysical model Claes & Heigele, 1999 Experimental output: after week 0; 4; 8 visual tissue distribution interfragmentary movement (IFM)

17 Seite 16 Modeling background Fracture Healing Biophysical model Claes & Heigele, 1999

18 Seite 17 Modeling background Fracture Healing Biophysical model Claes & Heigele, weeks Initial connective tissue (ICT) E = 3 MPa, ν = 0.4 Fascie (F) E = 250 MPa, ν = 0.4 Soft callus (SOC) E = 1000 MPa, ν = 0.3 Intermediate stiffness c. (MSC) E = 3000 MPa, ν = 0.3 Stiff callus (SC) E = 6000 MPa, ν = 0.3 Chondroid oss. zone (COZ) E = MPa, ν = 0.3 Cortex (C) E = MPa, ν = 0.3

19 Seite 18 Modeling background Fracture Healing Biophysical model Claes & Heigele, weeks 4 weeks

20 Seite 19 Modeling background Fracture Healing Biophysical model Claes & Heigele, 1999 a) strain in x-direction b) strain in y-direction c) hydrostatic pressure [MPa]

21 Seite 20 Modeling background Fracture Healing Biophysical model Claes & Heigele, 1999

22 Seite 21 Modeling background Fracture Healing Biophysical model Prendergast et al. Niemeyer 2013 (after Lacroix et al. 2002)

23 Seite 22 Different implementation Fracture Healing Comparisson of the models (Isaksson et al., 2006) External callus: h = 14mm Ø = 28mm Load cases: a) y = 0.6mm (0.5Hz) F max = 360N b) φ = 7.2 (0.5Hz) M max = 1670Nmm Cort. Marrow Gran. Fib. Cart. Immat. Mature E ν in MPa

24 Seite 23 Different implementation Fracture Healing Model realization (Isaksson et al., 2006) Calculation circle: 1 Cell population dynamics d dt n = D 2 n, where D is chosen to reach a steady state after 16 weeks 2 Material parameter calculation E = nmax n n max E gran. + n n max E tissue 3 FEM analysis of the biophysical stimuli

25 Seite 24 Different implementation Fracture Healing Axial load situation (Isaksson et al., 2006)

26 Seite 25 Different implementation Fracture Healing Comparisson of the models (Isaksson et al., 2006) Axial 4 weeks Axial 8 weeks a) No, external bridging too early Reasonable, but inhibited final healing due to high pressure b) No, external bridging too early, Reasonable, but inhibited final unstable tissue predictions healing due to high pressure c) Yes, bony external callus prior No, fails to bridge due to high to bridging fluid velocity d) No, external bridging too early Reasonable, but too quick a) Carter et al., b) Claes & Heigele, c) Lacroix & Prendergast, d) Dev. strain Discussion: The algorithms 2 [...] correctly simulated some features of early or intermediate healing, but none of them predicted final healing. Although complete healing was not seen in vivo either, it is believed that once a fracture has bridged with bone, it becomes stable enough for complete healing. 2 Carter et al., Claes and Heigele

27 Seite 26 Different implementation Fracture Healing Torsional load situation (Isaksson et al., 2006)

28 Seite 27 Different implementation Fracture Healing Comparisson of the models (Isaksson et al., 2006) Torsion 4 weeks Torsion 8 weeks a) No, strain limits too low No, failed to bridge b) No, strain limits too low No, failed to bridge c) Yes, mature bone formation in Yes, including bone formation gap prior to bridging in gap area d) No, threshold values not No, extreme strain magnitudes transferable to torsion This event 3 was not assessed in the in vivo study, due to the experimental time line, but would likely have occured at a later stage. 3 [...] resorption of the internal callus [...]

29 Seite 28 Different implementation Fracture Healing Examples of different implementations (not complete!) Repp et al., 2015 The connection between cellular mechanoregulation and tissue patterns during bone healing Checa & Prendergast, 2008 A mechanobiological model for tissue differentiation that includes angiogenesis: A lattice-based modeling approach Niemeyer (et al.), 2013 Fuzzy-logic implementation of the Ulm healing model Pietsch (et al.), 201x Level-Set approach of the fracture healing process

30 Seite 29 Different implementation Fracture Healing Repp et al., 2015 Diffusion process of a biological potential: c t = D 2 c Evolution of the young s modulus: E(r, t) = c(r, t) k tt (ms) t ms := V /V

31 Seite 30 Different implementation Fracture Healing Checa & Prendergast, 2008 Describe the cell migration as random walk Model the growth of blood vessels Near O 2 level approximation Biased random walk Cell differentiation: Prendergast mechanical response model (shear strain and fluid/solid velocity) Oxigen level at the surrounding area

32 Seite 31 Different implementation Fracture Healing Ulm model of the biological background (Niemeyer, 2013) Endochondral ossification Bone maturation Cartilage Woven bone Lamellar bone Bone resorption Intramembranous ossification Chondrogenesis / Cartilage destruction Soft tissue Bone resorption Avascular tissue Angiogenisis Vessel destruction Optimal perfusion

33 Seite 32 Different implementation Fracture Healing The Ulm healing model (Simon et. al, 2011) tissue c : R + 0 Ω [0, 1]4 with Ω R 3 and perf. p : R + 0 Ω [0, 1] n c i = 1 i=1 c soft t. c cart c woven c lamellar c avasc c vasc FEM mesh Time evolution: Tissue composition Vascularity d c = F (c, Mechanics, Biology) dt

34 Seite 33 Different implementation Fracture Healing 2 Level-Set motivation Figure: Baron & Kneissel, Nature Medicine 19, (2013) !"#$%&'()*+,-./ :;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{ }~ Figure: Claes & Heigele, 1998

35 Seite 34 Different implementation Fracture Healing Difference of Ulm healing model and Level-Set approach Simulation area Ω Subdomains Ω s Ω b = Ω Ω s Ω b Γ b Bone boundary Ω s Ω b = Γ b Rotational symmetry (3D problem) Ω Old New Value of interest: c b : R + 0 Ω [0, 1] ċ b : R + 0 Ω R Value of interest: Γ b (t) Ω v b : Γ b (t) R 2

36 Seite 35 Different implementation Fracture Healing Resolving the bone boundary Γ(t) Level-Set function φ : Ω I R Request : Γ(t) = {x Ω φ(x, t) = 0} φ(x, t) > 0 x Ω b, t I φ(x, t) < 0 x Ω s, t I Motion of the interface φ t + φ x t = 0 with the velocity v := x t

37 Seite 36 Different implementation Fracture Healing Getting the velocity(-field) v Motion of the interface φ t + φ x t = 0 with the velocity v := x t Normal to the interf. (ν : Ω R) consequently: v = ν φ φ φ t + ν φ = 0 Figure: Baron & Kneissel, 2013

38 Seite 37 Different implementation Fracture Healing Getting the growth velocity ν velocity ν typically is defined by the physical problem (pressure difference in a bubble system) Level-Set needs a velocity-field on Ω to update φ γ 0 η 0 ν in not constant on Ω, not even on Γ Extension methods are needed to project ν on Γ into Ω in a way that no new 0-Level-Sets are created

39 Seite 38 Different implementation Fracture Healing Two possible definitions of φ Signed distance function φ(x) := γ min x Γ Γ ( x x Γ ) with γ = { 1, if x Ω s 1, if x Ω b Smoothed heavyside function 0, ( ) if φ sd < ɛ 1 φ(x) := 2 + φ sd 2ɛ + 1 2π sin πφsd ɛ, if ɛ < φ sd < ɛ 1, if φ sd > ɛ

40 Seite 39 Different implementation Fracture Healing The signed distance function φ Signed distance function φ(x) := γ min x Γ Γ ( x x Γ ) with γ = Advantage (Euclidean norm) { 1, if x Ω s 1, if x Ω b φ = γ min (x x Γ ) 2 + (y y Γ ) 2 φ = 1 φ t = ν φ φ 1 φ 2 ν x Problem: If ν is not constant in normal direction from φ = 0, one have to handle a lot of things!

41 Seite 40 Different implementation Fracture Healing Reinitialization Direct distance calculation (need of exact boarder position) Reinitialization equation with pseudo-time stepping 5 φ Re τ ( sign(φ Re 0 ) 1 φ Re 0 = φ(t i ) ) φ Re =

42 Seite 41 Different implementation Fracture Healing Conservative formulation (after E. Pllson and G. Kreiss 2005) Smoothed heavyside function 0, ( ) if φ sd < ɛ 1 φ(x) := 2 + φ sd 2ɛ + 1 2π sin πφsd ɛ, if ɛ < φ sd < ɛ 1, if φ sd > ɛ Time evolution: φ t + v φ = γ [ ( ɛ φ φ(1 φ) φ )] φ

43 Seite 42 Different implementation Fracture Healing Height [mm] Height [mm] Height [mm] Radius [mm] Radius [mm] Radius [mm] Height [mm] Height [mm] Height [mm] Radius [mm] Radius [mm] Radius [mm]