(Bone) Fracture Healing

(Bone) Fracture Healing Part 2/2 Computational Biomechanics Summer Term 2017 Martin Pietsch

Fracture Healing Biology Healing Phases: Inflammation Blood coagulates blood clot Peak within 24 h, completed after ~ 7 days (rats) Repair Intramembranous ossification (~2 weeks) Revascularizaion of the hematoma commences Endochondral ossification (chondrocytes) Remodeling 5 8 weeks (rats; humans: years) Woven bone -> Lamellar bone Bailón-Plaza & van der Meulen 2001, Geris et al. 2009

Fracture Healing Biology Direct Fracture Healing Requires very stable fixation Tiny gaps, no inflammation, no callus formation Contact healing Gap < 0.01 mm BMUs directly remodel lamellar bone cross-fracture Bony union and restoration of Haversian system Gap healing Gap < 0.8 mm Gap filled with woven bone Gradually replaced by oriented revascularized osteons Claes et al. 2012

Fracture Healing Biology 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 dicult to predict intuitively the behaviour of the system When the time/length scales of various events under investigation are signicantly dierent When the system exhibits clearly nonlinear (nonobvious) behavior Experiment in vivo in vitro model in silicio static dynamic

Fracture Healing Biology 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 dierential equations 3. Organ-scale models Primary focus on mechanical stimuli Strong coupling to mechanoregulatory models

Mechanoregulatory Tissue Differentiation Hypotheses Roux & Krompecher Roux (1881): specific stimulus specific tissue type Proposed that cells within tissues engage in a competition for the functional stimulus (Weinans & Prendergast 1996) Differenzirende u. gestaltende Wirkungen der function. Reize. Selbstgestaltung (self-organization) Compressive bone Tensile fibrous connective tissue Compressive/tensile + high shear stress cartilage Krompecher (1937) Agrees with Roux, but Hydrostatic pressure cartilage Wilhelm Roux (1850-1924) Martin-Luther Universität Halle-Wittenberg

Mechanoregulatory Tissue Differentiation Hypotheses Pauwels Eine neue Theorie über den Einfluss mechanischer Reize auf die Differenzierung der Stützgewebe (1960) Challenges Roux s hypothesis Tensile stimuli also stimulate bone formation Long bones: bending loads Refutes Roux s specific stimulus for cartilage formation New hypothesis Bone deposit on an existing framework protecting it from non-physiological deformations Cell-level combinations of pure distortional strain & pure volumetric strain determine differentiation Niemeyer 2013 (after Pauwels 1973)

Mechanoregulatory Tissue Differentiation Hypotheses Carter et al. Proposes osteogenic index as a function of peak cyclic shear and peak cyclic hydrostatic stress I = i n i S i + kd i Influence of vascularity Carter et al. 1998 Carter 1987

Mechanoregulatory Tissue Differentiation Hypotheses Claes & Heigele Reinterpretation of Pauwels (Heigele 1998) Assumptions Local hydrostatic stress and local strain state as determining stimuli Bone formation on existing bony surfaces if both hydrostatic stress and shape changing strains stay below certain thresholds Thresholds determined based on combined in vivo & FE investigation Vaguely defined strains Probably normal strain of max. absolute value along x/y axes Claes & Heigele 1999

Mechanoregulatory Tissue Differentiation Hypotheses Claes & Heigele, Why -> Experimental output: After week 0; 4; 8 Visual tissue distribution Interfragmentary movement (IFM) Claes & Heigele 1999

Mechanoregulatory Tissue Differentiation Hypotheses Claes & Heigele, Why -> Claes & Heigele 1999

Mechanoregulatory Tissue Differentiation Hypotheses Claes & Heigele, Why -> Strain x-direction Strain y-direction Hydrostatic pressure Claes & Heigele 1999

Mechanoregulatory Tissue Differentiation Hypotheses Prendergast et al. Biological tissue as biphasic material (poroelastic) Solid phase (matrix) Fluid phase (interstitial fluid) Tissue differentiation guided by Octahedral shear strain γ Fluid flow (flow velocity) v Combined stimulus S = γ a + v b Niemeyer 2013 (after Lacroix et al. 2002)

The Ulm Bone Healing Model Biological Processes Endochondral ossification Bone maturation Cartilage Woven Bone Lamellar Bone Cartilage destruction Chondrogenesis Intramembranous ossification Bone resorption Connective tissue Bone resorption Avascular tissue Angiogenesis Vessel destruction Optimal perfusion

The Ulm Bone Healing Model Representing Biological State 1.0 1.0 c soft c lamellar c woven c cartilage c vascularity 0.0 0.0 FE mesh Tissue composition Vascularity c: Ω 0, + 0, 1 5 with Ω R 3 c: x, t c woven, c lamellar, c cartilage, c soft, c vascularity where c soft = 1 c woven c lamellar c cartilage i T c i x, t = 1.0 with T soft, cartilage, woven, lamellar

The Ulm Bone Healing Model Predicting Tissue Concentrations c x, t 1 = c 0 + t 0 t 1 t c x, t dt unknown t c x, t f m x, t, b x, t Mechanical stimuli m x, t = M x, t, c(, t 0 t), u BC, F BC Biological stimuli b x, t = B(x, t, c, t, c BC ) Depends on history Depends on finite neighborhood of x

The Ulm Bone Healing Model Numerical Implementation Initialization (geometry, meshing, tissue distribution) Rule of mixture Analyze mechanical response (FEA) Compute effective stimuli from strain history Time integration loop Compute average concentrations Tissue differentiation (fuzzy logic) Remeshing and state mapping t n t n+1 Niemeyer 2013

The Ulm Bone Healing Model Mechanics Mechanical Stimuli Pure volume change Pure shape change ε = 1 3 ε 1 + ε 2 + ε 3 γ = 1 2 ε 1 ε 2 2 + ε 1 ε 3 2 + ε 2 ε 3 2 where ε = ε 1 0 0 0 ε 2 0 R 3 R 3 3 0 0 ε 3 Simon et al. 2002, 2010, Shefelbine et al. 2005

Cortex Haematoma The Ulm Bone Healing Model Mechanics Rule of Mixture & Structural Analysis (FEA) Composite linear-elastic material properties (Carter & Hayes 1977, Shefelbine et al. 2005): E x, t = i T E i c i 3 x, t Muscle Medullary cavity ν x, t = ν i c i x, t i T Fracture gap Skin γ ε Niemeyer 2013

The Ulm Bone Healing Model Mechanics Mechano-regulated Tissue Differentiation Distortional strain (%) Tissue destruction Cartilage 16 Connective tissue Bone 0.07 0.045 Bone resorption Simon et al. 2002, 2010, Shefelbine et al. 2005, Niemeyer 2013-5.0-0.85-0.02 0 0.02 0.85 5.0 Dilatational strain (%)

The Ulm Bone Healing Model Biology Biological Stimuli b = c woven, c lamellar, c cartilage, c soft, c vascularity, s b, s v Non-local influence http://www.snipview.com/q/paracrine_signaling http://web.mit.edu/smart/research/biosym/biosym-sub-projects- Thrust%203.html

The Ulm Bone Healing Model Biology Appositional Growth

The Ulm Bone Healing Model Biology Biological Stimuli b = c woven, c lamellar, c cartilage, c soft, c vascularity, s b, s v s b x, t = c bone, t G σ in 2D = + + x Non-local influence c bone ξ, υ; t G σ x ξ, y υ dξ dυ s v x, t = c vasc, t G σ x e.g. in two spatial dimensions G σ x, y 1 2πσ 2 exp x2 y 2 2σ 2

Fuzzification Rule evaluation Defuzzification The Ulm Bone Healing Model Biology Evaluating f with Fuzzy Logic Local bone concentration t c x, t f m x, t, b x, t Weighted-average bone tissue concentration Fuzzy logic controller Cartilage concentration Change in bone concentration Local vascularity Change in cartilage concentration Weighted-average vascularity Change in vascularity Distortional strain (immediate and effective) Dilatational strain (immediate and effective) if vascularity is medium and not bone is low and dist_strain_eff is low then delta_bone is positive

The Ulm Bone Healing Model Biology Fuzzy Logic Primer Example (1/3): Fuzzification & Premise Eval. if c_vasc is sufficient and not c_bone is low then delta_c_bone is positive c_vasc is sufficient and not c_bone is low = 0.45 and not 0.25 = 0.45 and 1-0.25 = min(0.45, 0.75) = 0.45

The Ulm Bone Healing Model Biology Fuzzy Logic Primer Example (2/3): Implication c_vasc is sufficient and not c_bone is low = 0.45 and not 0.25 = 0.45 and 1-0.25 = min(0.45, 0.75) = 0.45 if 0.45 then delta_c_bone is positive positive

The Ulm Bone Healing Model Biology Fuzzy Logic Primer Example (3/3)

The Ulm Bone Healing Model Numerical Implementation Initialization (geometry, meshing, tissue distribution) Rule of mixture Analyze mechanical response (FEA) Compute effective stimuli from strain history Time integration loop Compute average concentrations Tissue differentiation (fuzzy logic) Remeshing and state mapping t n t n+1 Niemeyer 2013

Haematoma Cortex The Ulm Bone Healing Model Applications Callus Healing (Sheep, External Fixator) Axis of symmetry F Muscle Medullary cavity Fracture gap Non-linear spring Skin Simon & Niemeyer, 2015

Relative bone concentration (%) The Ulm Bone Healing Model Applications Simulation Results 7 % initial IFS Time (days) 7 14 21 28 35 42 49 56 360 30 % initial IFS Simon & Niemeyer, 2015

The Ulm Bone Healing Model Applications Simulation Results 1.50 7 % initial IFS 30 % initial IFS IFM (mm) 0.75 0.00 7 14 21 28 35 42 49 56 Time (days) Simon & Niemeyer, 2015

New Bone Healing Model Motivation Interface Capturing Motivation

Rot. axis New Bone Healing Model New <-> Old Interface Capturing Motivation Simulation area Ω Subdomains i N Ω i = Ω Ω j Force Ω Interfaces Ω i Ω j = Γ ij Rotational symmetry (3D problem) Γ ij Ω i Old New Value of interest c i R + 0 Ω [0,1] Γ ij t Ω Tissue growth c i R + 0 Ω R v ij Γ ij t R 2

New Bone Healing Model Growth description Tissue growth description Phase volume fraction α i α i Ω Ι 0,1 with α i = 1, on Ω i 0, on Ω j Ω i Advection equation for α i α i t Velocity field j tt v ij α j = 0 v ij = θ ij n ij with n ij the vector normal to the interface Γ ij θ ij M i ε, γ R 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.20 0.07 0.00 0.00 0.75 1.00 0.65 0.00 0.00 0.40 0.98 0.43 0.00

New Bone Healing Model Growth description How to get n ij Level-Set function φ i Ω R with Γ i = x Ω φ i x = 0 1. φ i 0 = 1.5 x (α i 0.5) 2. φ i τ sign φ i 0 1 φ i τ = 0 3. φ i τ equals signed distance function near Γ i after few iterations 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.20 0.07 0.00 0.00 0.75 1.00 0.65 0.00 n ij = φ φ κ ij = n ij 0.00 0.40 0.98 0.43 0.00

New Bone Healing Model Solutions Solutions Small initial strain Day 35 Day 28 Day 14 Day 7 Day 0

New Bone Healing Model Solutions Solutions Big initial strain Day 60 Day 58 Day 44 Day 24 Day 0