PourJa Tavakkolj Avval Department ot Mectianical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada e-mail: [email protected]

Vaclav Klika Department ot Mattiematics, FNSPE, Czech Tectinicai University in Pragje, Trojanova 13, Prague 120 00, Czecti Repubiic e-mail: [email protected]

Habiba Bougherara^ Department ot Mectianicai and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada e-mail: [email protected]

Predicting Bone Remodeling in Response to Total Hip Arthroplasty: Computational Study Using Mechanobiochemical Model Periprosthetic bone loss following total hip arthroplasty (THA) is a serious concern leading to the premature failure of prosthetic implant. Therefore, investigating bone remodeling in response to hip arthroplasty is of paramount for the purpose of designing long lasting prostheses. In this study, a thermodynamic-based theory, which considers the coupling between the mechanical loading and biochemical affinity as stimulus for bone formation and résorption, was used to simulate the femoral density change in response to THA. The results of the numerical simulations using 3D finite element analysis revealed that in Gruen zone 7, after remarkable postoperative bone loss, the bone density started recovering and got stabilized after 9% increase. The most significant periprosthetic bone loss was found in Gruen zone 7 (—17.93%) followed by zone 1 (-13.77%). Conversely, in zone 4, bone densification was observed (+4.63%). The results have also shown that the bone density loss in the posterior region of the proximal metaphysis was greater than that in the anterior side. This study provided a quantitative figure for monitoring the distribution variation of density throughout the femoral bone. The predicted bone density distribution before and after THA agree well with the hone morphology and previous results from the literature. [DOI: 10.1115/1.4026642] Keywords: bone remodeling, mechanobiochemical model, irreversible thermodynamics, total hip arthroplasty, finite element method

1

Introduction

Periprosthetic bone loss following THA is one of the most serious concerns compromising patients' life quality and is believed to be the result of stress shielding and osteolysis [1,2]. Stress shielding caused by load sharing between bone and implant is associated with bone résorption and occurs through bone remodeling [3-7]. The mechanism of bone remodeling was first postulated by Wolff's law stating that the reduction of mechanical stress causes bone to adapt itself by reducing its mass, either by getting thinner (external remodeling) or by becoming more porous (internal remodeling) [8-11]. In the case of hip arthroplasty, severe mass reduction may cause painful loosening of the implant which leads to the failure of hip replacement. Eventually, this situation requires a revision surgery which compared to primary THA, is more complicated with less satisfactory outcomes [12,13]. Periprosthetic bone loss is attributed to implant design [14,15] and pre-operative bone quality [16,17]. To investigate bone loss, many researchers have measured the bone mineral density (BMD) after THA using dual energy X-ray absorptiometry [18-21]. Since it is not always feasible to follow-up the long-term behavior of the bone in response to implantation, it is important to develop realistic models to predict the bone evolution and thus monitor bone remodeling. To date, several bone remodeling models have been developed [4,5,9,22-30]. Mechanical-based models predict bone adaptation based on mechanical stimuli such as stress, strain, strain energy, or mechanical Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 27, 2013; final manuscript received January 15, 2014; accepted manuscript posted February 6. 2014; published online April 10,2014. Assoc. Editor: Carlijn Bouten.

Journal of Biomechanical Engineering

damage [4,9,27,29]. In this category, the strain energy .model [4] is probably the most popular one. It proposes that bone remodeling mechanism is regulated by strain energy. In other words, a relation between strain energy and change of bone density (with respect to the time) defines bone remodeling governing rule. Another example of such models, developed by Levenston and Carter [27], is based on cyclic energy dissipation as a measure of bone damage. In this model, the stimulus which initiates bone remodeling is formulated as proportional to the damage energy dissipation summed over all daily loading cycles. Although these models have been to some extent successful in predicting normal bone architecture, they suffer from some major drawbacks: These models use only mechanical signals to stimulate bone remodeling and fail to consider the underlying biological mechanisms that control the bone remodeling process [4,31]. Mechanobiological models aim at taking into effect biological as well as mechanical factors responsible for bone remodeling. These models were initially introduced via Frosts' mechanostat theory [23]. This theory states that bone adapts its strength to keep the strain caused by physiological loads close to a set point. If strain level is below a set point, bone is resorbed, and if it exceeds this set point, new bone is formed. The mechanostat theory is a qualitative theory because the set-point is not specified [31]. Nevertheless, several mechanobiological models have been presented so far [25,32-34]. A further improved model of this type, developed by Huiskes et al. [25], is based upon the separation of osteoblastic and osteoclastic activities. According to this model, osteoclasts are recruited and activated either where microcracks occur or in disused areas of the bone. The dynamic forces of daily living are known to produce microcracks. Mechanosensitive cells, called osteocytes, sense mechanical signals due to the external

Copyright © 2014 by ASME

MAY2014, Vol. 136 / 051002-1

load transfer through the architecture and locally recruit osteoblasts to do the bone formation. The mechanical signal which is sensed by osteocytes is assumed to be the strain energy density. Biochemical models are based on the activities of osteoblasts and osteoclasts to obtain an insight into the bone remodeling process at a cellular level. The first model of this kind describes the differential activity of the parathyroid hormone which acts as a regulator for bone résorption and formation [35,36]. Another biochemical model, developed by Komarova et al. [37], studies the role of autocrine and paracrine interactions in the regulation of bone remodeling. In addition, Lemaire et al. [38] proposed a signaling pathway known as RANK/RANKL/OPG to regulate bone cell activities. The most noticeable defect of these cell-based models is that they do not take into account the mechanical stimulus in the bone remodeling process [31]. In order to improve the understanding of bone remodeling, it is crucial to include all factors (i.e., mechanical, biological,"and biochemical) concurrently. To do so, some relatively new models (mechanobiochemical models) were proposed in which the mechanical factors are linked to the chemical and biological one [31,39,40]. The model of Rouhi et al. [40] is based on the mixture theory combined with chemical reactions. It is intended for bone résorption and could be extended for modeling the growth and adaptation of bone according to the authors. Another model of this kind, proposed by Bougherara et al. [31] and Klika and Marsik [39], is based on irreversible thermodynamics in which bone is considered as an open self-organizing system capable of exchanging matter, energy and entropy with its surroundings. The main contribution of the current study is the mechanobiochemical approach used to predict the long-term behavior of the femoral bone in response to THA. In addition, another key feature of thermodynamic-based model [31,39] is its ability to consider the coupling between the biochemical reactions (representing bone biochemistry) and the mechanical loading (represented by strain). Our model is more comprehensive compared to the existing ones; some of them [4,5] leads to complete bone résorption in the absence of mechanical loading (which is not realistic). However, by using thermodynamic-based model, our study predicted that 52% of bone is resorbed due to the presence of chemical stimulus only (see Appendix). This was validated by an in vitro animal study suggesting that when limbs are placed in casts (i.e., the bone loading is minimal), bone is rapidly lost and reaches a new steady state after about 50% bone loss [41]. It should be mentioned that there exist some improved models [24,32,33,42] which, in the absence of mechanical stimulus, predict partial bone résorption. For instance, in the model proposed by Schriefer et al., bone remodels based on the principle of cellular accommodation; strain history is integrated into cellular memory so that the reference state for adaptation (set-point) constantly changes [42]. In spite of successful outcome, this model does not take into account the actual biology of bone remodeling process (e.g., osteoblasts-osteoclasts activities and biochemical reactions). Also, the models of [32,33], are not complete for full understanding of the coupling between mechanical stimulus (damage accumulation) and bone biological response. However, in our model, for considering the coupling phenomenon in bone remodeling mechanism, a unifying language/theory (nonequilibrium thermodynamics) is introduced which provides a suitable framework for analysis of such an interconnected phenomena (i.e., the effect of mechanical loading on bone biochemistry).

2

Materials and Methods 2.1

Thermodynamic-Based Theory for Bone Remodeling

2.1.1 Biochemical Reactions Involved in Bone Remodeling. In the thermodynamic-based theory [31,39], bone résorption and formation phases, performed by the bone cells known as osteoclasts and osteoblasts, were only considered as the steps of bone remodeling process. Bone was hypothesized as an open thermodynamic system exchanging matter, energy and entropy with its surroundings (Fig. 1). All biochemical reactions (see Ref. [31]) describing the mechanism of bone remodeling have the general form of Menten-Michaelis enzyme reaction [43], i.e.. E+P

(1)

Enzyme (E) retroactively binds to substratum (S) to form the complex of SE. Afterward, this complex breaks down into enzyme and product (P). Osteoclasts are the only cells that are able to resorb bone tissues. These cells initially exist as mononuclear cells, MCELL. In order to be activated, mononuclear cells have to be coupled in multinucleated complex, MNOC, whose formation can be characterized by the following reaction: (a=l)

• MNOC + ÍV4

(2)

A^i is the mixture of substances initiating the reaction with the mononuclear cells. ^4 is the remaining product from the first reaction (a = 1). In the next step (a = 2), which is so-called bone decomposition, multinucleated osteoclasts act on the bone to break it down (a = 2)

MNOC + Old-B ;

(3)

where Old_B is the abbreviation for old bone. N¿ and ^7 are the products resulting from the degradation of the old bone. ^7 is then used in the production of osteoblasts activator as described in the following reaction: (a = 3)

Old_B ;

• Activ.OB

(4)

The activators act on the osteoblasts, OB, which causes them to secrete collagen and produce Osteoid which is the unmineralized bone (a = 4)

Activ.OB + OB ;

Dynamic load

Osteoid-I-Wi 2

(5)

Nutriiioii

1 Metabolic Mechanism

Thermodynamic-based model, combined with finite element analysis, is capable of predicting the effect of stress shielding around the joint prosthesis which can potentially be used in the design and optimization of implants. Moreover, since this model deals with the bone remodeling mechanism at the cellular level, the authors believe that it is capable of investigating bone résorption related diseases such as osteoporosis. This can be achieved by implementing the effect of disease on the concentration of substances participating in bone remodeling process. However, to the best knowledge of the authors, such a relationship (between disease and the concentration of substances) is still unknown. 051002-2 / Vol. 136, MAY 2014

Bone résorption

old bone

New booe (output)

Bone fonnatíon Bone growth factor

Hormones

Possible medical treatment

Fig. 1 Schematic representation of bone as an open thermodynamic system

Transactions of the ASME

Ng and Nn are the remaining product of the third (a = 3) and forth (a = 4) reactions, respectively. The last and longest reaction (a = 5) in bone remodeling process is the mineralization of Osteoid. A'i3, the substratum initiating the bone calcification, is embedded into the bone matrix to form the new bone (New_B) according to (a = 5)

+ Osteoid •

(6)

Nis is the remaining material in the bone formation reaction. The above reactions (Eqs. (2)-(6)) contain 15 substances {N\, MCELL, MNOC, ^4, Old J , . . . , Wij) whose concentration will be denoted by [N,], [MCELL], [MNOC], [A'4], [Old_B],..., [N,¡], respectively. 2.1.2 Mechanobiochemical Coupling. In the thermodynamicbased model, it is assumed that the coupling between the mechanical and biochemical fluxes or forces drives bone remodeling process. Mechanobiochemical coupling is in its early stages of recognition. However, there are experimental findings confirming this phenomenon, e.g., in the well controlled chemical environment of Belousov-Zhabotinski reactions in gels [44]. Including the effect of mechanical factors on biochemical ones is a crucial component of modeling the bone adaptation process. For studying these coupling phenomena in such an interdisciplinary problem (i.e., bone remodeling), it is very useful to utilize some unifying language/theory; such a tool is nonequilibrium thermodynamics which provides a suitable framework for analysis of such an interconnected phenomena. Linear and nonlinear coupling has been successfully developed within classical irreversible

[Old_B]

[Activ_OB] = -

thermodynamics (used for the current study) [45,46] and GENERIC framework [47], respectively. In order to include the mechanical effects on biochemical reactions, the standard law of mass action was replaced by a modified version, represented by Eq. (7), which takes into account the coupling between the applied mechanical loads and the biochemical affinity of the reactions (or concentration of substances)

=^\-

'' - k^, f[ [ (7) where r^, and A^ are the reaction rate and affinity of the considered reaction (ath reaction), respectively. Phenomenological and reaction rate coefficients are denoted by /y and k±j, respectively, ¿/(j) stands for the first invariant of the strain rate tensor representing the rate of volume change. The concentration of the different substances involved is denoted by [Nj] and finally, u«,- and u^ are stoichiometric coefficients of mixture of substance N¡ entering and leaving the ath reaction, respectively. Using the modified version of the law of mass, Bougherara et al. [31] derived a system of differential equations governing the bone remodeling process. Solving this system yielded the time evolution of concentration (Eqs. (8)-(12)) of substances participating in the biochemical reactions of bone remodeling [31] 1 [MCELL] = 2 ( -Al

(8)

- 2[MCELL])' + 4

(9)

(10)

-\ß

[Osteoid] = -

(11)

[New_B] = -[Osteoid] + ß^Q

— O4

= (54[Activ_OB]

(12)

where [A^,] is the normalized concentration of the ¡th substance, /?, is the sum of the initial molar concentration of relevant substances, ¿J, denotes the ratio of the rate of ath reaction to that of second reaction, /, is the flux of the /th substance. And, D^ represents the effect of the mechanical loading (an approximation of d(i)) on the ath biochemical reaction which is expressed as follows [31]:

A?5(ref)

"^

(13)

where Ai is the time of loading, £(i) is the trace of strain tensor, ¿•(ref) and flc((ref) are constants describing the reference strain

Journal of Biomechanical Engineering

rate value and the influence of the reference strain rate on the ath reaction, respectively. The values of model parameters which characterize the biochemical reactions have to be measured experimentally or adjusted through comparison with real data from clinical observation [31]. The values of model parameters which were used for both pre- and postoperative bone remodeling simulation were tabulated in Table 1. 2.2 Density and Elastic Modulus of Bone. The bone density (p) is related to the initial bone density (po), normalized concentration of the new bone ([New_B]) and old bone ([OldJB]) according to the following law of mass and mixture [31]: [Old_B] + [NewJB]")

(14)

MAY2014, Vol. 136 / 051002-3

Table 1 Values of the parameters of thermodynamic-based model Mode] Parameters (dimension-less)

Value

Pi

5.23 30.0 6.0 2.28 3.0 2.38 5.07 20.29 10.03 5.75 3.08 2.44 1.26 5.85 1.30 4.68

ß3 ßl

ßw ßn h Ju ¡4 öi(ref) Ö2(ref)

In the literature, different empirical and experimental relationships between the density and modulus of elasticity of the bone are provided [48]. Using the empirical power law relationship provided by Carter and Hayes [49], the modulus of elasticity of the bone was calculated by [Old_B] E =

£oid_B

[Old_B] + [New_B] /

[New_B]

\[01dJ3] + [New_B]

(15)

where ÊOM.B and £New_B are the modulus of elasticity of old and new bone, respectively. A sensitivity analyses for two different density-modulus relationships have been carded out by Weinans et al. [50]; the first and second relationship were taken from Refs. [49,51], respectively. Results showed that the finite element (FE) models provide consistent stress-shielding patterns in the bone, independent of the choice of the bone density-modulus relationship used in the computer model. 2.3 Development of the CAD Model. A large left forth generation composite femur (model 3406, sawbones, Vashon, WA [52]) was fully scanned by computed tomography (CT). A series of two-dimensional cross-sectional images was taken at every 0.5 mm along the length of the femtir across the coronal, sagittal, and axial planes of the body. MIMICS® Medical Imaging Software (The Materialise Group, Leuven, Belgium) was used to produce a 3D model from the CT results. The 3D model included geometry data for both cortical and cancellous bones. The 3D model already saved as an initial graphic exchange specification file was imported into ANSYS Workbench 12.1 (ANSYS, Inc., Canonsburg, Pennsylvania) to separate cortical from cancellous bone. Next, the model was imported into SolidWorks 2011 (Corp., Dassault Systems, Concord, Massachusetts) using which the hip implant was inserted into the proximal end of the femur. It should be noted that the hip stem was implanted into the femur where the location of hip stem was already removed. Then, the head (and neck) of the femur was split from the rest of the femur to create three distinct parts: femoral shaft, femoral head (and neck) and hip implant (Fig. 2). Therefore, we created the computer-aided design (CAD) model of femur whose femoral shaft was used for both pre- and postoperative bone remodeling simulations. This enabled us to make a precise comparison between the density distribution of the intact (pre-operative) and postoperative femur. 2.4 Finite Element Modeling. As shown in the bone remodeling algorithm (Fig. 3), the prepared CAD model was imported

051002-4 / Vol. 136, MAY 2014

Fig. 2 CAD model.: (a) Femoral shaft, (b) femoral head and neck, and (c) hip implant. into ANSYS, then discretized and forces and constraints were applied. In the first step of numerical simulation, outer layer (cortical) and inner layer (cancellous) were modeled as homogenous structures with the initial density of 0.98 gr/cm^ (po) which is the average value of cancellous bone with 0.32gr/cm'' and cortical bone with 1.64gr/cm^. Unlike several bone remodeling algorithms, themiodynamic-based model was not sensitive to the initial density. In other words, the initial density did not have any significant effect on the final bone density distribution similar to Refs. [26,53]. To model the intact femur, the cancellous bone material properties [52] were assigned to the hip implant as well as to the inner layer of the femur. Also, the outer layer was considered to consist of cortical bone whose material properties obtained from Ref [52]. As illustrated in the bone remodeling algorithm, for the next steps (iterations), the new properties of each element were calculated based on the concentration of Old and New bones remained/ produced inside the element. Therefore, the femur became an inhomogeneous structure. However, similar to several other studies [5,7,26,54,55], each bone element was considered as an isotropic material. In other words, one single Young's modulus, which is calculated by Eq. (15), was sufficient to define the mechanical properties of each element. The contacts between all surfaces were considered to be perfectly bonded in finite element model (FEM). Using surface-tosurface contact elements (CONTA174 and TARGE170), all the interfaces were meshed. The CAD model was discretized by tennode higher order 3D tetrahedral solid elements (SOLID 187) with three degrees of freedom at each node. Mesh sensitivity was investigated and the appropriate mesh size was obtained. To simulate physiological loading, hip and muscle forces regarding 45% of gait cycle were applied to the models. The force components were taken from Refs. [56,57]. To avoid stress Transactions of the ASME

CAD model

Loads and constraints

Structurai analysis Thennodynamic based mode} (with calibrated parameters) Calculation of old and new bone concentration

Calculation of bone densit\-

Calculation of modulus of elasticity for each element (for nest iteration)

Fig. 3 Iterative process of the thermodynamic-based model for bone remodeling simulation concentration, the hip joint and muscle forces were distributed over several nodes of the femoral head and greater trochanter, respectively. All the degrees of freedom of all nodes at the distal epiphysis were fixed to avoid rigid body motion (Fig. 4). The thermodynamic-based model, defined by Eqs. (8)-(12), was implemented into finite element analysis through a userdefined macro in the ANSYS parametric design language (APDL). The finite element analysis yielded the trace of strain Ê(1) for each element which in turn gave the values of D^, which represents the effect of mechanical loading on ath reaction, through Eq. (13). D-, were then entered into Eqs. (8)-(12) to calculate the values of [Old_B] and [New_B] for each element. Using Eqs. (14) and (15), the density and modulus of elasticity of each element were obtained, respectively. These new material properties were then assigned for the next iteration of finite element analysis, and the process was repeated until no significant change in the density of the elements was observed. It was assumed that the convergence and consequently the final density distribution throughout the intact femur are obtained when ConvCri < 0.0001 (see Eq. (16)) 1 ConvCri ^ -

(16)

n and t are the number of elements and iteration number, respectively. To simulate bone remodeling in response to THA, the elements of the head (and neck) of the intact femur were removed (unselected). The material properties of CoCrMo (Young's modulus = 210GPa, Poisson's ratio = 0.31, density = 8.28gr/cm' [58]) were then assigned to the hip implant. Moreover, the initial material properties of each element of immediate postoperative femur were chosen from the same element in the after convergence-intact femur (inhomogeneous femur). The joint Journal of Biomechanical Engineering

(a)

(b)

Fig. 4 Graph showing the location of constraints and loadings on the models, (a) Intact femur, and (b) post-operative femur.

reaction force was also transferred to the femoral ball of the hip implant (Fig. 4(0)). Finally, the thermodynamic-based model was again applied to simulate bone response to THA. Equation (16) was again used to control the convergence. 3

Results

3.1 Bone Density Convergence. The iterative process of bone remodeling simulations was considered to be converged when no significant change in the density of the elements was observed. As mentioned before, it was controlled by Eq. (16). The convergence of simulations for both intact (pre-operative femur) and postoperative femur is shown in Fig. 5. As seen, the average of change in the density of all elements has approached to zero after approximately 50 iterations. 3.2 Pre-Operative Bone Density Distribution. Figure 6 illustrates the bone density distribution obtained from the thermodynamic-based model throughout the intact femur. The range of density along the intact femur was between 0.33 and 1.63gr/cm^. The maximum value of density was observed in the outer cortex (cortical bone) in the midshaft. MAY2014, Vol. 136 / 051002-5

0.6 1

Dense cancelious bone Pre-operative femur Post-operative femur

0.4 •

0.3 •

(a)

20

30

40

(b)

Fig. 7 Bone density distribution in the coronal section of intact femur. Obtained from (a) thermodynamic-based model, and (d) X-ray (Reproduced with permission from Ref. [59]).

Number of Iteration

Fig. 5 Convergence of the bone remodeling simulations for pre- and post operative femur

(1.70gr/cm^) was observed in the oblique surface of the proximal femur in a point adjacent to the implant. To examine the density distribution in details, the femur was divided into seven zones (Fig. 8(c)) known as the Gruen zones [60]. The density evolution of zone 7 in response to THA is plotted in Fig. 9. It can be seen that after a significant decrease in density, BMD started recovering and eventually reached and remained at 0.9 gr/cm^. 3.4 Percent Change in the Bone Density. The final density of each element of intact and postoperative femur was exported from APDL to calculate the percent density change of each element. Then; in APDL, a table containing the percent density change of each element was defined to present a quantitative 3D figure for monitoring the distribution variation of bone density. Figure 10 demonstrates the percentage of bone loss/formation in every point of the postoperative femur. As seen in the figure, the maximum bone loss and formation of —46.25% and -1-128.12% were observed, respectively. Moreover, it was revealed that the bone density loss in the posterior region (-11.32%) of the proximal metaphysis was greater than that in the anterior side (—3.75%), as is qualitatively visible in A-A segment in Fig. 10(0). The change (pre- versus postoperative) in the average density over all elements located at each zone is shown in Fig. 11. Among the zones, the maximum bone loss was seen in zones 7 (—17.93%) and 1 (-13.77%). For the zones of 2, 3, 5, and 6, the bone loss of -4.55%, -3.48%, -1.11% and -5.45% were observed, respectively. Conversely, only in zone 4, a moderate densification with the percentage of -1-4.63% happened. This is due to the fact that at the distal tip of the stem, forces were transferred to the bone and resulted in a high stress concentration which consequently induced a prominent deposition of bone mass.

Fig. 6 Bone density distribution of the intact femur obtained from the thermodynamic-based model (gr/cm^). (a) Anterior view, and (b) posterior view.

The density distribution in the coronal section of proximal intact femur obtained from the model and CT (CT reproduced with permission from Ref. [59]) is presented in Fig. 7. Our results showed that a dense cancelious bone between the calcar and the location of hip joint reaction force was formed, in addition to a dense cortical bone around the medullary canal. 3.3 Postoperative Bone Density Distribution. Postoperative bone density distribution is shown in Fig. 8, in anterior and posterior view as well as in coronal section of the femur. The range of density throughout the postoperative femur was obtained between 0.33 and 1.70gr/cm''. The maximum value of density 051002-6 / Vol. 136, MAY 2014

4

Discussion

The bone density throughout the intact femur was predicated to be in range of 0.33 and 1.63 gr/cm'' which agrees with the clinical observations [61,62]. Furthermore, the density distribution in the intact femur is consistent with the CT results obtained by Truong et al. [59]. In both cases, in addition to the formation of dense cortical bone around the medullary canal, a dense cancelious bone was observed between the calcar and the location of hip joint reaction force. Similarly, such a dense trabeculae carrying the stress from the superior contact surface to the calcar region of the medial cortex was seen in the computational study of Kuhl and Balle [26]. Moreover, a dense cortical bone in both proximal and distal region of the femoral neck was seen, alike to the CT results. In the postoperative bone, the location of maximum density (1.70gr/cm^) shifted up to a point adjacent to the implant in the oblique surface of the proximal femur. This can be explained by Transactions of the ASME

Fig. 8 Bone density distribution of post-operative femur obtained from the thermodynamicbased model (gr/cwr). (a) Anterior view, (b) posterior view, and (c) coronal section of posterior view.

the presence of stress concentration due to the proximal load transfer to the femur from the implant. It was also observed that BMD in zone 7 decreased dramatically until iteration 11, then started to increase and eventually 1.15

10

20

30

40

50

60

Number of Iteration

Fig. 9 Graph showing the post-operative bone density versus, iteration for zone 7

Journal of Biomechanical Engineering

after 9% recovery, it reached a stable condition with the density of 0.90gr/cm^. This pattern agrees well with StukenborgColsman's observation reporting that after a progressive bone loss in zone 7 during the first 6 months postoperation, BMD increased and recovered slightly by 7% at the end of the first year [21]. Therefore, this study suggests that the periprosthetic bone loss is not necessarily progressive and even restoration of bone density can occur, analogous to the observations of Trevisan et al. and Wixson et al. [63,64]. In addition, Kroger et al. [2] reported that after a rapid bone loss in response to THA, BMD got stabilized which agrees with the present findings. The maximum periprosthetic bone loss predicted by the model was —46.25% observed in the calcar in zone 7, which is consistent with Niinimaki and Jalovaara s' observation reporting the maximum bone loss of —40% in the calcar area [65]. The 6% discrepancy between ours and Niinimaki and Jalovaara s' results can be justified by the difference between the implants stiffness (CoCrMo versus Steel); the more flexible the implant is, the less the effect of stress shielding will be. According to our study, the maximum bone formation of -1-128.12% was seen in a point located in the oblique surface of the proximal femur adjacent to the implant. The high level of stress transferred to the proximal femur from the implant could result in such a bone densification which in turn may increase the risk of fracture in the femoral neck. MAY2014, Vol. 136 / 051002-7

128.12 108.74 89.37 69.99

"1

50.62 31.24 11.86 -7.50 -26.88 -46.25 (b)

Fig. 10 Percent bone loss/formation in response to THA. (a) Posterior view, (b) three transverse segments, and (c) medial view.

Gruen Zones Fig. 11 Grapli showing changes in the average bone density in Gruen zones after convergence (%)

A severe bone résorption was observed around the stem in the proximal area (A-A segment in Eig. 10). Moreover, some cortical résorption as well as some trabecular densification was found at the medial edge of the implant similar to the simulation results of Huiskes et al. [5]. The results showed that the periprosthetic bone loss decreased as the segment plane moved distally from the proximal metaphysis to mid-diaphysis such that the dramatic range of 051002-8 / Vol. 136, MAY 2014

bone loss (-46.25% to -26.88%) vanished in B-B and C-C segments, and even some bone densification occurred in the area adjacent to the tip of the stem. Our findings also revealed that the bone density loss in the posterior region of the proximal metaphysis was greater than that in the anterior side, analogous to Shim's observation reporting the posterior region of the metaphysis gets more unloaded compared to the anterior side after THA [66]. Among Gruen zones, the maximum periprosthetic bone loss occurred in zone 7 followed by zone 1 with the percentage of -17.93% and -13.77%, respectively (Eig. 11). Similarly, in Stukenborg-Colsman's clinical examination, using a Ti alloy implant, the strongest decreases in BMD were observed in zone 7 (--12%) and zone 1 (-11%) [21]. In Turner's work simulating bone remodeling process after THA with a CoCr implant (Epoch), the maximum bone loss of —10% was seen in zone 7 after 2yr postOperation [7]. In terms of the location (zone) of maximum bone loss, our findings are consistent with Turner's work. However, there is an 8% difference quantitatively. With that in mind, our result (-17.93%) was closer to the clinical value (-16%) used as a touchstone in their study. Therefore, it may be concluded that the thermodynamic-based model can more realistically predict the maximum bone loss in response to THA. Moreover, Li et al. [18] reported that 2yr after THA with a CoCr implant, the maximum bone loss of -19.7% in zone 7 was observed which is consistent with our result (—17.93%). In zones 2 and 6, our model underestimated the bone loss (while it still agreed with Glassman's Transactions of the ASME

Black (Simple): Li e! al C200") Red(IaUe): Thermoi^namicbased mßdet

-9.2% 1

^ ^ Ê^

^ ^ ^

^K'

-8.3% -4.Si%

-3.2%

/

-19.7%

K

-17.93%

là

B HH

/

-9.8%

S

^^M H

M

-SJSft

•

••

•

-0.9%

I

•

-J.i/K

Fig. 12 Graph comparing the periprosthetic bone loss/formation observed by Li et al. [18] (Simple font) with those obtained by thermodynamic-based model (Italic font). Re-drawn from Ref. [18] with permission.

study reporting the density change of —3.24% and -7.72% for zones 2 and 6, respectively [67]) compared to the Li's observation. However, for the rest of regions, our results were close to the clinical one (Fig. 12: -13.77% versus -9.2% for zone 1; -3.48% versus —3.2% for zone 3; -1-4.63% versus +2.4% for zone 4; -1.11% versus -0.9% for zone 5; -17.93% versus -19.7% for zone 7). The present study also showed that the bone loss in the Gruen zones progressively declined as the region of interest moves distally from the proximal metaphysis to mid-diaphysis (Fig. 11) which agrees with Li's observations [18]. Furthermore, Li's study confirmed our results that among the Gruen zones, zone 4 is the only region showing increase in BMD. As mentioned before, a bone loss of —17.93% in zone 7 was reported as the highest change in the average density among Gruen zones. However, the maximum bone loss of —46.25% was observed in the most proximal/medial area of the femur after THA which is even more than twice the average bone loss in zone 7 (—17.93%). That being said, thermodynamic-based model warns that the average bone loss in the Gruen zones which is mainly used as a touchstone to report the periprosthetic bone loss might be an unsuitable since some locations in the bone with a huge bone loss may exist that may result in unpredictable local failure.

Limitations of the Study Although the results agreed very well with the bone morphology' and literature, the model has some limitations and simplifications that a user has to keep in mind. First, the concept of coupling is based on linear nonequilibrium thermodynamics and thus is phenomenological. Therefore, we cannot relate it directly to actual mechanosensing or mechanotransduction processes in cells. Second, the mechanical stimuli in the simulations do not include any viscous effects. We have also assumed that the trace of strains regulates bone cells response; however, other mechanical stimuli such as fiuid now and transport of nutrients could be

Journal of Biomechanical Engineering

considered. Thirdly, we disregarded kinetics of some of the known control mechanisms in bone remodeling such as the RANKL-RANK-OPG chain which may play a role in regulating the bone remodeling process. Fourthly, in the proposed model the computational steps are utirelated to real time. However, the convergence of the iterative process was assumed when no significant change in the density of the elements was observed which, from the clinical point of view, is considered as the long-term response when the femoral density gets to the stable condition. Fifthly, in Fig. 12, the FE results were compared with those corresponds to uncemented THA. A direct comparison should be done with precaution. We have conducted this comparison based on several research papers comparing the outcome of cemented versus uncemented arthroplasty. A detailed literature review, conducted by Ni et al. [68], compared cemented and uncemented THA in terms of bone remodeling, pain, thigh pain, quality of life, micromotions, etc. In terms of bone remodeling, for the midterm period (2-1 Oyr after operation), they concluded that cemented THA is identical to uncemented one. Another study, conducted by Dan et al., showed no differences in BMD loss between patients with cemented and uncemented implants in Gruen zones 2-7 [69]. Sixthly, in the current work, we tried to bring insight into bone loss due to stress-shielding phenomena, and the density decrease caused by osteolysis which is a purely chemical disease was not considered. It should be mentioned that the proposed model is capable of considering the effect of chemical environments on bone remodeling through the biochemical reaction rates and the concentration of substances participating in the reactions. However, to the best knowledge of the authors, the effect of osteolysis on the concentration of substances is still unknown. Further research is indeed needed to quantify the effect of osteolysis on chemical reactions. Seventhly, similar to several studies [5,7,26,55], only bonded contacts are considered in the current study. This bonded contact was meant to mimic complete interdigitation of bone around the hip stem. It should be noted here, that the bonded contact could underestimate the effect of stress shielding. Eighthly, in the present work, anisotropic behavior of the bone was ignored as each element was assumed to be an isotropic material.

5

Conclusions

Thermodynamic-based theory which considers the coupling between the mechanical loading and biochemical affinity as stimulus for bone remodeling, was implemented in this study to simulate the long-term behavior of the femur in response to THA. This study provided a quantitative figure for monitoring bone density changes throughout the femoral bone. It was revealed that in Gruen zone 7, after remarkable postoperative bone loss, the bone density started recovering and got stabilized after 9% increase. The most significant periprosthetic bone loss was found in Gruen zone 7 and 1 with the percentage of —17.93% and —13.77%, respectively. Conversely, the bone densification was observed in zone 4 (-1-4.63%). The results have also shown that the bone density loss in the posterior region of the proximal metaphysis was greater than that in the anterior side. The predicted bone density distribution before and after THA agree well with the bone morphology and previous results from the literature. The proposed model could provide valuable information on stress-shielding effect and bone résorption which should be taken into account for the design of any new implant. Conflict of Interest The authors declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgment This research was supported by the Natural Sciences and Engineering Research Council of Canada-Discovery Grant (NSERC).

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Nomenclature Aoi = affinity of octh reaction d(i) = first invariant of strain rate tensor Dai = effect of the mechanical loading (an approximation of i/(i)) on the ath reaction Do,(ref) = influence of the reference strain rate on the ath reaction £^oid_B> £-New_B = modulus of elasticity of old and new bone, respectively /, = flux of the rth substance k±j = reaction rate coefficients /y = phenomenological coefficients [Ni] = concentration of N, [Ni] = normalized concentration of N¡ í'c = rate of ath reaction 5(ref ) = reference strain rate ßi = sum of the initial molar concentration of relevant substances ¿5, = ratio of the rate of ath reaction to that of second reaction Ai = time of loading £(1) = trace of strain tensor p = bone density Po = initial bone density «oci, ü'^ = stoichiometric coefficients of substance TV; entering and leaving ath, reaction, respectively

.48 ,33

Appendix A.I Presence of Chemical Stimulus in Bone Remodeling Mechanism Figure 13 illustrates that in the absence of mechanical loading, thermodynamic-based model predicts partial bone résorption due to the presence of chemical stimulus. A.2 Effect of High and Low Level of Uniaxial Stress. Figure 14 illustrates how the high (±80MPa) and low level (±10MPa) of uniaxial stress affect the density distribution of a 1 cm^ cortical bone. The cube was fixed from the rear plane and loaded on the front plane (shown in the figure). Applying high level stress (±80MPa) induced the average bone density of 1.51 gr/cm"', which is in a good agreement with the clinical results [62]. However, the reduction of mechanical stress to ±10MPa caused the average bone density of the cube to decrease by 41%.

(b) Fig. 13 Distribution of bone density (gr/cm^) in post-operative femur, (a) In the presence of mechanical loading (d) in the absence of mechanical loading.

A.3 Sensitivity Analysis of Model Parameters. Sensitivity analysis has been performed for all 16 model parameters (in the range of ±15%). According to the following results, the average density of the intact femur was most sensitive to ßj. In other words, among all parameters, ßj was the one with the highest effect on the density of the intact femur, especially for the lower band (ßj==5.\) (see Figs. 15-17). Therefore, this critical

Fig. 14 Graph showing the bone density distribution (gr/cm^) in 1 cm^ cortical bone resulting from (a) high level stress, and (b) low level stress. 051002-10 / Vol. 136, MAY 2014

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Average Density (gr/cm^ 128.12 108.74 89.37 69.99 50.62 31.24 11.86 -7.50 -26.88 -46.25 change

Fig. 15 Average density of intact femur (gr/cm^) for different values of j!y

(b)

p¡g » 1

Percent bone loss/formation for (a) ^7 = 6, and (b)

parameter was chosen to investigate its effect on the postoperative density. Bone remodeling simulation on postoperative femur with new value of ^7 (/?7 = 5.1) revealed that the pattern of bone loss and formation approximately remained same as the base pattern. In addition, the maximum bone loss and formation only changed by 5% (Fig. 18). References

% change

Fig. 16 Average density of intact femur (gr/cm^) for different values of J/and d.

Average Density (gr/cm^

.-Dl (ref) -D2(ref) -D3(ref) -D4(ref) - 0 5 (ref)

-

0,66—

Fig. 17 Average density of intact femur (gr/cm^) for different values of D, (ref)

Journal of Biomechanical Engineering

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