Bone Regeneration During Distraction Osteogenesis: Mechano-Regulation by Shear Strain and ﬂuid Velocity

ARTICLE IN PRESS

Journal of Biomechanics ] (]]]]) ]]]–]]] www.elsevier.com/locate/jbiomech www.JBiomech.com

Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and ﬂuid velocity

Hanna Isakssona,b, Olivier Comasb, Corrinus C. van Donkelaarb, Jesus Mediavillab, Wouter Wilsonb, Rik Huiskesb,c, Keita Itoa,b,Ã

aAO Research Institute, Clavadelerstrasse 8, 7270 Davos Platz, Switzerland bDepartment of Biomedical Engineering, Eindhoven University of Technology, The Netherlands cDepartment of Orthopaedics, University of Maastricht, The Netherlands

Accepted 18 September 2006

Abstract

Corroboration of mechano-regulation algorithms is difficult, partly because repeatable experimental outcomes under a controlled mechanical environment are necessary, but rarely available. In distraction osteogenesis (DO), a controlled displacement is used to regenerate large volumes of new bone, with predictable and reproducible outcomes, allowing to computationally study the potential mechanisms that stimulate bone formation. We hypothesized that mechano-regulation by octahedral shear strain and fluid velocity can predict the spatial and temporal tissue distributions seen during experimental DO. Variations in predicted tissue distributions due to alterations in distraction rate and frequency could then also be studied. An in vivo ovine tibia experiment evaluating bone-segment transport (distraction, 1 mm/day) over an intramedullary nail was used for comparison. A 2D axisymmetric finite element model, with a geometry originating from the experimental data, was created and included into a previously developed model of tissue differentiation. Cells migrated and proliferated into the callus, differentiating into fibroblasts, chondrocytes or osteoblasts, dependent on the biophysical stimuli. Matrix production was modelled with an osmotic swelling model to allow tissues to grow at individual rates. The temporal and spatial tissue distributions predicted by the computational model agreed well with those seen experimentally. In addition, it was observed that decreased distraction rate (0.5 mm/d vs. 0.25 mm/d) increased the overall time needed for complete bone regeneration, whereas increased distraction frequency (0.5 mm/12 h vs. 0.25 mm/6 h) stimulated faster bone regeneration, as found in experimental findings by others. Thus, the algorithm regulated by octahedral shear strain and fluid velocity was able to predict the bone regeneration patterns dependent on distraction rate and frequency during DO. r 2006 Elsevier Ltd. All rights reserved.

Keywords: Tissue differentiation; Mechanobiology; Finite-element analysis; Tissue growth; Bone-segment transport

1. Introduction mostly because repeatable experimental outcomes under controlled mechanical environments are required, but Osteogenesis has frequently been studied experimentally rarely available in experimental or clinical studies. and computationally. In osteogenesis, associated differen- Distraction osteogenesis (DO) is a procedure by which tiation of precursor cells is sensitive to the local mechanical controlled displacement of a bone fragment is used to environment. There have been several propositions of how generate large volumes of new bone that have been lost due this is mechano-regulated. However, corroboration of to trauma, infection or tumour resection (Ilizarov, 1989a; these algorithms are difﬁcult (Isaksson et al., 2006a), Richards et al., 1998). It can also be used to correct a variety of orthopaedic deformities and malformations. The Ã outcome is predictable and reproducible. Therefore, it is a Corresponding author. AO Research Institute, Clavadelerstrasse 8, suitable model for studying the potential mechanisms that 7270 Davos Platz, Switzerland. Tel.: +41 81 414 24 50; fax: +41 81 414 22 95. stimulate bone formation and for examining the role of E-mail address: [email protected] (K. Ito). mechanical forces. DO is usually separated into three

Please cite this article as: Isaksson et al., Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity. Journal of Biomechanics (2006), doi:10.1016/j.jbiomech.2006.09.028 ARTICLE IN PRESS 2 H. Isaksson et al. / Journal of Biomechanics ] (]]]]) ]]]–]]] phases. The first is the latency phase, immediately 2. Methods following osteotomy before distraction. The second is the distraction phase in which there is active distraction of the 2.1. Experimental model bony segments for a certain time at specific rates (total distance/day) and frequencies (number of distractions/ Data from an ovine in vivo experiment for evaluation of bone segment transport over an intramedullary nail, previously conducted in our day). During this period, tissue differentiation is initiated, institution, was used for comparison (Brunner et al., 1993, 1994). A distal with some sparse bone formation. The third is the diaphyseal defect, either 20 or 45 mm, was created in the left tibia. The consolidation phase, during which there is no distraction, tibia was then stabilized with an unreamed static interlocking nail. After which finally leads to bony union. The rate of bone corticotomy, bone segments were transported (distracted) using subcuta- formation during DO is directly related to the distraction neous traction wires over the nail (Fig. 1(a)). Distraction started on post- operative day 1 with a rate of 1 mm/d until the defect was closed, followed rate (Ilizarov, 1989b; Li et al., 1999, 2000), frequency by consolidation. Animals were sacrificed after 12 weeks for the short (Aarnes et al., 2002; Ilizarov, 1989b; Mizuta et al., 2003) defects and 16 weeks for the long defects. Daily distraction forces were and the strain/stress generated in the distraction gap (Li measured before (resting force), during (peak force) and 5 min after et al., 1997, 1999). Meyer et al. (2001a) showed that the distraction. The resting force represented the tension between the magnitude of distraction-generated mechanical tension distracted segment and the fixator before distraction. The peak force was the sum of all forces between fixator and distracted segment after directly influenced the phenotypic differentiation of the 1 mm distraction. The relaxation behaviour was calculated as the cells within the distraction gap. difference between resting force and peak force, divided by the peak Although DO provides an attractive setting for the study force, and was used as a measure of the viscoelasticity of the tissue. of mechanical effects on bone regeneration, very little Weekly standardized radiographs and undecalcified histology at the time computational evaluation has been performed. Morgan of completed transport were available for comparison. et al. (2006) investigated the local physical environment within an osteotomy gap during long bone DO and 2.2. FE model correlated tissue dilatation (volumetric strain) with differentiation of mesenchymal tissue. They evaluated distrac- A 2D axisymmetric FE mesh was created based on the geometry of the tion and tissue relaxation during one single day of the tibia, the nail and the callus from the experimental data (Brunner et al., 1994)(Fig. 1(b)). The initial corticotomy gap was set to 1 mm. Boundary distraction period. Loboa et al. (2005) used finite-element conditions were applied according to the experimental model. The ends of (FE) analysis to correlate bone formation with magnitudes bone and marrow and the external callus boundary were assumed of tensile strain and hydrostatic pressure (Carter et al., impermeable. Distraction (1 mm/d for 20 or 45 d) was applied to the top of 1998) during mandibular DO at four time points. So far no the cortical bone and started on post-operative day 1. Distraction was studies have described the process of tissue differentiation followed by consolidation, where no active mechanical stimulation was applied, according to the experimental protocol. Each iteration simulated during DO both spatially and temporally during the 1 d, where distraction was performed over 1 s followed by 24 h of complete distraction process. This type of computational relaxation, during which reaction forces were monitored. All tissues were evaluation of DO will provide useful information about the assumed to follow linear poroelasticity theory with properties taken from local stress and strain magnitudes that lead to the highest literature (Table 1). The intramedullary nail was assumed to be rigid amount of bone regeneration, and enable optimization of compared to the biological tissues and the interfaces between the nail and the tissues were modelled using finite sliding and zero friction (ABAQUS, treatments. v 6.5, ABAQUS Inc. Pawtucket, RI, USA). The mechano-regulation algorithm based on octahedral shear strain and fluid velocity was proposed by Prendergast et al. (1997) as a general tissue differentiation scheme. The threshold values for this algorithm were initially determined to predict bone formation around implants (Huiskes et al., 1997). Thereafter it has been shown to predict tissue differentiation during secondary fracture healing (Lacroix and Prendergast, 2002; Lacroix et al., 2002; Isaksson et al., 2006b), in bone chambers (Geris et al., 2003, 2004) and during osteochondral defect repair (Kelly and Prendergast, 2005). Recently, we demonstrated that it was more consistent with bone healing under both shear and compressive deformations (Isaksson et al., 2006a) than other algorithms (Carter et al., 1998; Claes and Heigele, 1999; Isaksson et al., 2006b). During DO the tissue is subjected to tension, a mechanical mode for which this algorithm has never been Fig. 1. Experimental and computational model. (a) The experimental tested. For this study, we hypothesized that mechano- model from Brunner et al. (1994) including initial defect and corticotomy, regulation by octahedral shear strain and fluid velocity can followed by a distraction phase (bone segment transport) and final consolidation period. (b) The initial two-dimensional axisymmetric finite also predict spatial and temporal tissue distributions seen element model was created from the experimental measurements. The experimentally during DO, including variations due to initial gap size was 1 mm. The nail diameter was 7 mm, the cortical bone’s alteration in distraction rate and frequency. inner diameter 14 mm, and outer diameter 20 mm.

Table 1 Material properties used to describe the tissues in this study

Cortical bone Marrow Granulation tissue Fibrous tissue Cartilage Immature bone Mature bone

Young’s modulus (MPa) 15750a 21 2b 10c 1000 6000d Permeability (m4/Ns) 1E-17e 1E-14 1E-14 1E-14b 5E-15f 1E-13 3.7E-13g Poisson’s ratio 0.325h 0.167 0.167 0.167 0.167i 0.325 0.325 Solid bulk modulus (MPa) 17 660a 2300j 2300j 2300j 3400k 17 660a 17 660a Fluid bulk modulus (MPa) 2300 2300 2300 2300 2300 2300 2300 Porosity 0.04l 0.8 0.8 0.8 0.8m 0.8 0.8

a(Smit et al., 2002). b(Hori and Lewis, 1982). c(Lacroix and Prendergast, 2002). d(Claes and Heigele, 1999). e(Johnson et al., 1982). f(Armstrong and Mow, 1982). g(Ochoa and Hillberry, 1992). h(Cowin, 1999). i(Jurvelin et al., 1997). j(Anderson, 1967). k(Tepic et al., 1983). l(Schafﬂer and Burr, 1988). m(Mow et al., 1980).

without any precursor cells. The precursor cells could then migrate into the callus from the boundaries between callus, marrow and periosteum, at an unlimited supply. This was simulated as a diffusive process to incorporate migration and proliferation of cells (Lacroix et al., 2002; Isaksson et al., 2006b). The distraction was applied and the biophysical stimuli were calculated in the FE analysis at maximal distraction. The new tissue phenotypes were predicted according to the local magnitudes of octahedral shear strain and fluid velocity (Prendergast et al., 1997). The cells within an element of callus tissue were able to differentiate into fibroblasts, chondrocytes or osteoblasts and to produce their respective matrices. Cell differentiation and the type of matrix produced by the present cells were only restricted by the mechanical environment. The differentiation of the cells between one phenotype and another was not explicitly modelled, but by having the type of matrix modulated by the mechanical environment, tissue transformation over time and space was modelled. There was one additional requisite that bone was only allowed to form on already calcified surfaces (Claes and Heigele, 1999). Matrix production of different tissue types was modelled to occur separately, at individual rates, depending on cell type and cell density. Matrix production and growth were simulated by applying a swelling pressure to the growing element and considering the subsequent volume Fig. 2. Bone regeneration as simulated in a mechano-regulated adaptive expansion as being an increase in matrix. The biphasic swelling model of model in MATLAB. The iterative procedure starts with a masstransport Wilson et al. (2005) was adopted for this growth simulation. In this model analysis to determine cell concentrations followed by a stress analysis swelling pressure is given by where the biophysical stimuli, i.e. octahedral shear strain and fluid velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 are calculated. The tissue phenotype is determined for each element Dp ¼ RTð cF þ 4cextÞ2RTcext, (1) followed by matrix production simulated with a biphasic swelling model. The callus geometry are re-meshed and the properties re-mapped, before where R is the gas constant, T the absolute temperature, cext the external the tissue properties and cell concentrations are updated and next iteration salt concentration and cF the fixed charged density which can be expressed begins. as a function of the tissue deformation as nf;0 cF ¼ cF;0 , (2) nf;0 1 þ J 2.3. Adaptive tissue differentiation model where nf,0 is the initial fluid fraction of the tissue, cF,0 the initial amount of negative charges in the tissue and J the determinant of the deformation The adaptive tissue differentiation process was accomplished through tensor. Before a simulation, all negative charges were set to zero and the custom-written subroutines (MATLAB, The Mathworks Inc. v 7.1) displaced geometry after the previous distraction served as input. Identical (Fig. 2) and based on an earlier adaptive model (Isaksson et al., 2006b). geometrical boundary conditions were applied. Growth was induced by The meshing of the callus was performed by an automatic meshing introducing a non-zero amount of fixed charges in the growing element, algorithm into triangular elements, which were transformed into quad- dependent on the cell type stimulated in the element (Table 2). The fixed- rilateral elements with a maximum area of 0.1 mm2 for the FE analysis charge density changes were chosen such that they resulted in growth/ (Brokken, 1999). The initial corticotomy consisted of granulation tissue, volume changes within the range of those found experimentally for each

Table 2 Material properties and constants used in the osmotic swelling model to predict tissue growth

Model parameters Resulting growth

3 nf,0 cF,0 (meq/mm ) Volume growth Bone growth rate

Fibrous tissue 0.8 7 102 15–20% Cartilage 0.8 3.5 102 5–7% Appositional bone growth 0.8 3.5 102 5 mm/da Endochondral bone growth 0.8 5.25 102 25 mm/db R ¼ 8.3145 Nmm/mmolK T ¼ 298 K 4 3 cext ¼ 1.5 10 mmol/mm

The assumed fluid fraction nf,0 and the fixed charged density cF,0 are inputs and the volume growth and bone growth rate are the calculated growth of the various tissue types. a(Eriksen et al., 1984; Vedi et al., 2005). b(Wilsman et al., 1996a, b). tissue type (Table 2). For bone, the amount of added fixed charges was Additionally, to explore the models’ potential to explain biological also dependent on whether bone formation was through intramembranous phenomena during DO, simulations with altered distraction rate (0.5 and or endochondral bone formation, i.e. which tissue type was previously 0.25 mm/d) and frequency (0.5 mm/12 h, and 0.25 mm/6 h) were con- located in that element. The tissue was allowed to swell for 24 h until ducted. When the distraction frequency was altered, the matrix produc- equilibrium. The final tissue shape was then assumed stress free and used tion/increment was adapted accordingly to ensure equal production rates. as input for the next increment. Hence, all stresses induced by growth were Thus, each iteration simulated 12 or 6 h when the frequencies were 2 or 4 assumed to fully relax within 1 d. The new tissue material properties were distractions/d, respectively. then calculated as the result of matrix production and degradation over the past 5 iterations, using a rule of mixtures: P 3. Results n Pi¼n4Ei vgi Enþ1 ¼ n , (3) i¼n4vgi The experimental results, published in detail elsewhere, where Ei was the elastic modulus at iteration i and vgi was the volume showed good reproducibility (Brunner et al., 1993, 1994). growth fraction, calculated as the elemental volume after swelling divided Post-operatively there was a narrow corticotomy gap. by the elemental volume before swelling pressure was induced. E was nþ1 During the first week of distraction ‘graining’ appeared, i.e. the temporary new elastic modulus before considering cell distribution. The cell concentrations were adjusted to the new tissue volumes (Eq. (4)), small slightly radio-opaque areas appeared throughout the such that the total cells remained the same after matrix production. The distraction gap without any organized pattern. From the final new modulus was calculated assuming a linear relation between the second week of distraction, strips of increased radio modulus of the tissue and the number of cells with the corresponding opacity were running from the two cortical bone ends phenotype (Lacroix et al., 2002; Isaksson et al., 2006a, b) (Eq. (5)): and growing towards each other. A small overlapping

½cellsn callus on the periosteal side was observed. Furthermore the ½cellsnþ1 ¼ , (4) vgn observed bone growth was more substantial on the periosteal side compared to the endosteal side and the nail ½cellsnþ1 ½cellsmax ½cellsnþ1 Enþ1 ¼ E þ EGran, (5) interface. During distraction of the segment, bone forma- ½cells nþ1 ½cells max max tion was clearly observed in the longitudinal direction of where [cells]n and [cells]n+1 were the cell densities in the elements before distraction, particularly with the larger defect size, with and after considering growth, respectively, [cells] was the maximal cell max increasing density over time, and with highest density density (assumed to be 100%), En+1 the final new elastic modulus and Egran the elastic modulus for granulation tissue. closer to the cortical ends, where the initial bone formation was seen. Throughout continued distraction of the bone 2.4. Model implementation fragment an area of soft tissue was located in the middle of the regenerate. During consolidation, reorganization and Marrow and cortical bone were not allowed to change material maturation of regenerated bone occurred. Over time, the properties or to produce matrix. Furthermore, the nail–callus interface did soft tissue gap diminished and bony bridging occurred. In not influence cell processes or matrix production. To avoid highly general the same patterns were observed in the short and deformed elements the callus was remeshed prior to every new increment (Brokken, 1999). After remeshing all tissue properties were mapped from long regenerates, but in the long ones, the different stages the integration points of the previous mesh onto the integration points of of healing were more clearly distinguished. the new mesh, using interpolation (Peric et al., 1996; Brokken, 1999; Overall, the predicted tissue distributions agreed well Mediavilla, 2005). The reaction forces of the transported bone segment with those seen experimentally (Fig. 3). During the first were measured with the computational model at the top of the cortical week, the mesenchymal stem cells that migrated and bone during distraction and were monitored during the 24 h of matrix production to calculate relaxation behaviour. proliferated into the callus mainly differentiated into Temporal and spatial tissue distributions, reaction forces and force fibroblasts. Thus, the predicted tissue distributions were relaxation data were evaluated and compared to experimental results. primarily fibrous tissue. After 7 d, differentiation into

Fig. 3. Tissue differentiation during distraction of the long defects followed by consolidation. Distraction rate and frequency are identical to the experimental study, i.e. 1 mm/d distracted once. (a) Predicted bone regeneration pattern. The tissue type was based on the average element moduli as determined by the mixture theory (Eqs. (3)–(5)). (b) Stimulated cell types.

osteoblasts was first observed along the periosteum and in 70% in the experiment, compared to 65% computation- the gap area (Fig. 3(b), iteration (it) 10). After 15 d bone ally. During distraction the relaxation increased to about tissue could be distinguished close to the periosteum. 80% for both experimental results and the computational During distraction of the segment, bone continued to prediction (Fig. 4(b)). develop. Slow creeping substitution of bone was seen in the When the distraction rate was decreased to 0.5 mm/d or longitudinal direction of distraction, with a higher density 0.25 mm/d the total time for bone regeneration increased, at the periosteal side (Fig. 3(a), it 35). Throughout even though the amount of bone formation at the same distraction of the bone segment, there was a gap of soft magnitude of total distraction increased with decreasing tissue in-between the bony ends, which reached a steady rate (Fig. 5). When the distraction frequency increased to size between day 30 and 40. The predicted areas of bone 0.5 mm distraction two times/day, or 0.25 mm distraction mainly remained immature until distraction was finalized. four times/day, the overall rate of bone formation During consolidation, maturation of the bone occurred increased (Fig. 6). During the first week of distraction the followed by final bony bridging. Similar distributions and tissue distributions were similar and mainly fibrous for all stages of tissue differentiation were seen for both defect three frequencies, but as distraction proceeded into the sizes (both in simulations and experiments), but the second and third week the amount of bone formation continued bone growth during distraction was mainly seen increased with the frequency. Also the consolidation period in the longer defects. necessary to achieve complete bridging became shorter Reaction forces and relaxation behaviour were also with increased distraction frequency. compared. Reaction forces increased almost linearly during the first weeks of distraction in the experiment with a 4. Discussion temporary drop in the rate of increase during the third week in four out of five sheep (Fig. 4(a))(Brunner et al., A mechano-regulation algorithm based on octahedral 1994). Computationally, the peak force was initially higher shear strain and fluid flow was able to predict the bone than experimentally found and over time it decreased formation pattern observed experimentally during DO slightly due to the increased soft tissue regenerate from initial corticotomy to final consolidation. The (Fig. 4(a)). Predicted relaxation forces compared well with comparison of spatial and temporal patterns of bone the experiments (Fig. 4(b)). The stress relaxation curves of regeneration was successful. The first bone formation was the tissues during transport were initially between 60% and seen in the cortical gap at the end of week two of

Fig. 5. Bone regeneration patterns with various distraction rates. The distraction rates simulated were (a) 1, (b) 0.5 and (c) 0.25 mm/d, all with a frequency of 1. The tissue type was based on the average element moduli as determined by the mixture theory (Eqs. (3)–(5)).

Fig. 4. (a) Peak reaction forces after 1 mm distraction and (b) relaxation behaviour calculated by the computational model and measured experimentally (Brunner et al., 1994). distraction in both experiments and predictions. These events were followed by progressive bone growth in the direction of distraction, with increased bone density at the periosteal side. Areas of soft tissue remaining in the gap throughout distraction of the segment, and bone maturation seen during consolidation, were similar in both experiment and computational predictions. The mechano- regulation algorithm has previously been shown suitable to predict fracture healing, as well as other bone regenerative processes. Now it has been taken one step further, by successfully predicting the bone formation patterns during DO. During DO the tissue is subjected to tension, in contrast to fracture healing where compression is predominant. With tensile loads, for example, the fluid velocity is directionally reversed, when compared to compressive loading. The mechano-regulation algorithm only considers the magnitude of the fluid velocity, and not the direction. Hence, in terms of the mechano-regulation algorithm, the loading conditions are not very different. The magnitudes of the two stimuli after distraction are displayed in Fig. 7. Fig. 6. Bone regeneration patterns with various distraction frequencies. The total distraction rate was 1 mm/d divided into (a) 1 (1 mm/24 h), 2 Furthermore, the relaxation behaviour over 24 h is shown. (0.5 mm/12 h), or 4 (0.25 mm/6 h) distractions/day. The tissue type was This confirms that the peak values of the stimuli indeed based on the average element moduli as determined by the mixture theory occur around the time of maximal distraction. The (Eqs. (3)–(5)).

Fig. 7. Spatial distributions and relaxation behaviour of fluid velocity and deviatoric strain at day 5. (a) Model geometry at the beginning of day 5. Two elements are highlighted and further relaxation behaviour of these elements are displayed in (d) and (e). Spatial distributions of (b) fluid velocity and (c) deviatoric strain after 1 mm distraction at day 5. (d), (e) The relaxation behaviour over 1 d of the fluid velocity and the deviatoric strain are displayedin two elements in the callus. Note that the time scale (x-axis) is logarithmic.

magnitudes of fluid velocity decreased rapidly as soon as Still, most likely there were some contributions from the distraction was completed (Fig. 7(c)), while the deviatoric adjacent soft tissue (including muscles) on the measured strain remained high during the beginning of the relaxation forces. Hence, that is one source of discrepancy. Another period (Fig. 7(d)). Depending on the location in the callus, probable cause for the disagreement in forces is that unlike the strain values even slightly increased initially during simple DO, the segment transport model required the relaxation. In those cases, the increases were minimal and creation of a large gap distal to the transported segment, did not affect the predicted phenotype. which would have been filled with soft tissues. With The relaxation behaviour of the tissue corresponded well distraction of the segment, these tissues would have been with what was measured experimentally. This occurs compressed, eventually completely, and the compression because relaxation is dominated by the modulus/perme- would result in an additional force component. Finally, ability ratio of the callus tissue, which did not change much throughout the distraction period, with the longitudinal during the distraction period. In contrast, the reaction alignment of collagen fibers under contract traction (Meyer forces from the model did not agree with the experiment. et al., 2001b), the modulus of the soft tissue in the gap This is probably not due to the mechano-regulation would have increased, similar to other collagen-oriented algorithm itself and the predicted pattern of tissue soft tissues, e.g. fascia, vessels, etc. (Hudetz et al., 1981; differentiation, but to additional factors in the experi- Birk and Silver, 1984; Billiar and Sacks, 2000a, b). None of mental model that were not included in the computational these effects were included in the computational model model. In limb lengthening, or simple DO, there are and, in combination, may be the source of discrepancy in progressively higher and higher tensions on adjacent fascia, reaction forces when compared to experimentally measured tendons and muscles (Simpson et al., 1995; Williams et al., magnitudes. 1999). This is because muscles are often attached across the Even though DO is mechanically well defined, some distraction gap. It can increase reaction forces substantially assumptions were necessary. The peak magnitudes of the (Aronson and Harp, 1994) and also cause considerable mechanical parameters immediately after distraction were pain for patients undergoing DO. However, in the current considered for the mechano-regulation, and the subsequent experimental model of bone segment transport (Brunner et relaxation was assumed to have minimal mechano-biolo- al., 1994), these effects were reduced because the total gical effects. Loading during consolidation was neglected, length of the tibia was kept constant, and most muscles are because the performance of the algorithm during distrac- only attached to the proximal and distal main fragments. tion (tensile displacements) was the main focus of this

Please cite this article as: Isaksson et al., Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity. Journal of Biomechanics (2006), doi:10.1016/j.jbiomech.2006.09.028 ARTICLE IN PRESS 8 H. Isaksson et al. / Journal of Biomechanics ] (]]]]) ]]]–]]] study. Thus, the resorption criteria initially suggested for compute matrix production. Hence, the concentrations of the mechano-regulation algorithm had to be excluded in fixed charges included during matrix production are only this study. The nail was modelled with finite sliding and used to achieve a new geometrical shape of the callus. The was assumed to have no influence on cell processes. This fixed charge densities and their effects on solid/fluid assumption was chosen since experimentally no bone content in the tissue have no physical meaning and are formation on the nail was seen during transport. Addi- not used in subsequent iterations. The assumption about a tionally, to overcome computational difficulties with high stress free geometrical shape after swelling/growth was relative strains, the initial experimental corticotomy made to avoid incremental stress increases in the tissue and (0.5 mm) was modelled as a gap of 1 mm, and the to allow us to use the same set of parameters to achieve the distraction rate was initially chosen 0.5 mm/d, increasing to same amount of volume increase throughout the simula- 1 mm/d at day 3. This did not influence the tissue tion. Furthermore, the relaxation times for the tissues are differentiation process since even with the lower distraction on the order of hours (Weiss et al., 2002; Bonifasi-Lista magnitudes, fibroblasts were stimulated during this period et al., 2005; Park and Ateshian, 2006; Huang et al., 2003). and matrix production of fibrous tissue occurred. Hence, by modelling 24 h relaxation and matrix produc- Tissue growth and matrix production were modelled tion, we believe we are well on the side where the using a new approach. The effect of local matrix assumption can be used. The matrix constitution in each production on tissue morphology was simulated by iteration is based only on the differentiation algorithm and inducing local tissue swelling in response to osmotic the rule of mixtures (Eq. (4)). pressure. The parameters were chosen such that fibrous Experimental findings by others have shown that the rate tissue would grow faster than cartilage and bone. More of bone formation is directly related to the local strain/ specifically, volume increases up to 20% occurred in the stress generated in the distraction gap (Li et al., 1997, regions where fibroblasts saturated the tissue producing 1999), and that the amount of mechanical tension directly fibrous matrix, while the growth rate of cartilage was lower influenced the phenotypic differentiation of the cells within (5–10%). These relative growth rates are compatible with the distraction gap (Meyer et al., 2001a). Our simulations experimental findings. The growth during bone formation with altered distraction rates agreed with those findings. corresponded to a bone apposition rate of 5–10 mm/d (Vedi When the tension in the gap was lowered by a reduction in et al., 2005) and when calcification of cartilage occurred the distraction rate, the bone formation/day increased. Still, volume growth was higher due to hypertrophy prior to the most favourable distraction rate was 1 mm/d, because mineralization (Wilsman et al., 1996a, b) and the corre- the total time needed to regenerate the bone in the defect sponding bone formation rate was about 20 mm/d (Wils- was shorter than for lower rates. This also agrees with the man et al., 1996a, b). Fig. 8 displays an example of this findings of Ilizarov which showed 1 mm/d to be the most model, where the stimulated cell types after distraction are favourable rate. Additionally, experimental studies have compared with the resulting matrix production and tissue shown that further increases in distraction rate can be growth generated with the biphasic swelling model after detrimental to healing (Ilizarov, 1989b; Choi et al., 2004) 10 d of distraction. The osmotic swelling model, originally and lead to a distraction gap filled with mostly fibrous developed to describe cartilaginous tissues, was applied to tissue (Choi et al., 2004). In the current study, distraction rates above 1 mm/d were not examined. Hence, we cannot compare those experimental observations with our computational model. Ilizarov’s studies further showed that the greater the distraction frequency, the better the outcome (Ilizarov, 1989b). Our predictions demonstrated the same pattern, where the rate of bone regeneration increased with distraction frequency (Fig. 6). In our model, the best possible bone regeneration was achieved with a total distraction of 1 mm/d divided into 4 sub-distractions of 0.25 mm/6 h. Experimental studies have also suggested that the division in endochondral and intramembranous bone formation during DO is related to the distraction rate (Li et al., 1999; Mizuta et al., 2003; Kessler et al., 2005). With this model the stimulated cell phenotypes and tissue types produced were altered similarly. With a lower distraction rate the proportion of the cells that differentiated into osteoblasts without first going through a cartilage inter- mediate was increased. Fig. 8. The osmotic swelling model used simulates matrix production in an element specific manner. (a) The stimulated cell phenotypes and (b) the Tissue differentiation during DO, by a mechano-regula- resulting volume growth after 10 iterations with distraction rate of 1 mm/d tion algorithm based on octahedral shear strain and fluid and frequency of 1 distraction/day. velocity, was successfully simulated from distraction to

Please cite this article as: Isaksson et al., Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity. Journal of Biomechanics (2006), doi:10.1016/j.jbiomech.2006.09.028 ARTICLE IN PRESS H. Isaksson et al. / Journal of Biomechanics ] (]]]]) ]]]–]]] 9 consolidation and was confirmed by experimental observa- normal individuals employing a kinetic model for matrix and mineral tions in a model of bone segment transport. The rate of apposition. Metabolic Bone Disease and Related Research 5, 243–252. bone formation depended on distraction rate and fre- Geris, L., Van Oosterwyck, H., Vander, S.J., Duyck, J., Naert, I., 2003. quency, similar to experimental observations. These Assessment of mechanobiological models for the numerical simulation of tissue differentiation around immediately loaded implants. Com- relationships can now be further investigated with this puter Methods in Biomechanics and Biomedical Enginering 6, algorithm, which could potentially help adapt and optimize 277–288. DO treatment protocols. Geris, L., Andreykiv, A., Oosterwyck, H.V., Sloten, J.V., Keulen Fv, F.F., Duyck, J., Naert, I., 2004. Numerical simulation of tissue differentiation around loaded titanium implants in a bone chamber. Journal of Acknowledgements Biomechanics 37, 763–769. Hori, R.Y., Lewis, J.L., 1982. Mechanical properties of the fibrous tissue The authors would like to thank Prof Ulrich Brunner for found at the bone–cement interface following total joint replacement. providing experimental results and the AO Foundation, Journal of Biomedical Materials Research 16, 911–927. Huang, C.Y., Soltz, M.A., Kopacz, M., Mow, V.C., Ateshian, G.A., 2003. Switzerland, for financial support. 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