Modelling heat transfer in a bone–cement–prosthesis system

Journal of Biomechanics 36 (2003) 787–795

Modelling heat transfer in a bone–cement–prosthesis system Eskil Hansen* Numerical Analysis, Centre for Mathematical Sciences, Lund University, Box 118, Lund S-221 00, Sweden Accepted 19 December 2002

Abstract The heat transfer in a general bone–cement–prosthesis system was modelled. A quantitative understanding of the heat transfer and the polymerization kinetics in the system is necessary because injury of the bone tissue and the mechanical properties of the cement have been suggested to be effected by the thermal and chemical history of the system. The mathematical model of the heat transfer was based on first principles from polymerization kinetics and heat transfer, rather than certain in vitro observed properties, which has been the common approach. Our model was valid for general three-dimensional geometries and an arbitrary bone cement consisting of an initiator and monomer. The model was simulated for a cross-section of a hip with a potential femoral stem prosthesis and for a cement similar to Palacos R. The simulations were conducted by using the finite element method. These simulations showed that this general model described an auto accelerating heat production and a residual monomer concentration, which are two phenomena suggested to cause bone tissue damage and effect the mechanical properties of the cement. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Acrylic bone cement; Heat transfer; Polymerization kinetics; Mathematical model; Finite element analysis

1. Introduction When performing a total hip-joint replacement surgery it is common to fix the prosthetic component with an acrylic bone cement. The aseptic loosening in this surgery has been suggested to be effected by e.g. the mechanical properties of the cement (Lewis, 1997) and thermal and chemical injuries of the bone tissue in the bone–cement interface (Albrektsson and Linder, 1984; . Mjoberg et al., 1984; Toksvig-Larsen et al., 1991). Thermal injuries can be the result of the heat production from the polymerization process in the cement and chemical injuries could be inflicted by a long-term leakage of toxic compounds from the cement. The chemical processes also effects the mechanical properties, e.g. residual monomer in the hardened cement acts as a plasticizer (Hasenwinkel et al., 2002; Vallo, 2002). Thus, in order to fully understand the complex process behind the loosening of the prosthesis, it is necessary to have a quantitative understanding of the heat transfer and the polymerization kinetics in the bone–cement–

*Corresponding author. Tel.: +46-46-222-36-64; fax: +46-46-22205-95. E-mail address: [email protected] (E. Hansen).

prosthesis system. Modelling such systems has been the subject of previous studies e.g. Borzacchiello et al. (1998), Huiskes (1980), Swenson et al. (1981) and Vallo (2002). In the models presented by Huiskes (1980) and Swenson et al. (1981) the polymerization kinetics and the heat source term, respectively, are modelled by preset time-dependent functions. Both these models neglect the coupling between the heat production, chemical reactions and the temperature distribution, preventing them from describing temperature- and space-dependent properties like the location of the auto accelerating heat production. In Borzacchiello et al. (1998) and Vallo (2002) the polymerization kinetics are modelled by a single polymerization equation. Borzacchiello et al. (1998) uses @x ¼ k1 ðTÞðk2 ðTÞ  xÞn xm ; @t while Vallo (2002) presents the equation ( k3 ðTÞxð1  xÞ when xox0 ; @x ¼ 2 @t k4 ðT; xÞxð1  xÞ when xXx0 ;

ð1Þ

ð2Þ

where x is the degree of reaction, t is the time, T is the absolute temperature, ki with i ¼ 1; y; 4; are empirically determined functions and n; m and x0 are

0021-9290/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0021-9290(03)00012-5

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E. Hansen / Journal of Biomechanics 36 (2003) 787–795

the bone–tissue boundary was assumed to be governed by Newton’s law of cooling. 2.2. Chemistry of the cement Expressions of the form d½X X i ¼ hX dt i

Fig. 1. Cross-sections of the femur with a potential cemented stem prosthesis. The gray areas represent the cemented regions. Crosssection like the one presented to the right, will be used in the finite element simulations of the model.

empirically determined constants. These models agree well with the specific in vitro measurements of the temperature, which the models are based on. However, these models are not necessarily valid in the general in vivo case, because the empirically determined parameters may be dependent of e.g. the composition of the cement and the system geometry. The aim of this article was therefore to suggest an alternative approach, which would produce a more general model. This was done by constructing a model based on first principles from polymerization kinetics and heat transfer. The model was simulated for a crosssection of a hip with a potential femoral stem prosthesis (Fig. 1) and for a cement similar to Palacos R. The simulations were conducted by using the finite element software Femlab.

2. Methods 2.1. General setting The model was constructed in the context of bone– cement–prosthesis system with a general system geometry and with an arbitrary bone cement consisting of an initiator and monomer. The heat capacity and the thermal conductivity of the materials in the system, the volume of the cement and the temperature of the tissue surrounding the system were all considered to be constant. Furthermore, the heat transfer across the inner boundaries (cement–prosthesis and bone–cement) was assumed to be continuous while the transfer across

ð3Þ

are frequently derived, where [X] denotes the concentration of the substance X in mol=m3 ; and hiX denotes the ith term in the sum which is equal to the time derivative of [X]. The dominant process in the bone–cement is a polymerization, which is achieved in three steps: initiation, propagation and termination. In the initiation step, free radicals ðR Þ must be introduced to start the reaction. This can be accomplished by adding an initiator (I), which decomposes as I! 2R : k

ð4Þ

d

Reaction (4) is a first-order reaction with the rate constant kd : Due to the order of the reaction h1I ¼ kd ½I and h1R ¼ 2kd ½I: Some of the free radicals (with an assumed constant fraction given by f ) are connected to the monomers (M) creating monomer-ended radicals ðM1 Þ; R þ M-M1 :

ð5Þ

Reaction (5) is almost instantaneous, thus h1M ¼ fh1R and h1M ¼ fh1R : 1 In the propagation step, monomers are successively added to the monomer-ended radicals in a second-order reaction Mj þ M ! Mjþ1 kj

with j ¼ 1; y; N;

ð6Þ

p

where Mj represents the radical R  Mj1  M : Due to the small size of the monomer, in comparison to the radicals, the mobility of the monomer will only be marginal effected by the presence of radicals. Thus, one can assume that the rate constant is independent of the chain length ðkpj ¼ kp Þ: Then h2M ¼ kp ½M½M ; where ½M  denotes the concentration of monomer-ended radicals of any length. Note that reactions (6) do not change ½M : The only change of ½M  (considered so far) is the production of ½M1  in reaction (5), implying that h1M ¼ h1M : The adding of a monomer to a radical 1 in reactions (5) and (6) is an exothermal process and this is the cause of the heat production in the cement. In the termination step, the sequence of monomer additions is ended by mutual annihilation of two monomer-ended radicals and the dead polymer is produced. There are two such reactions Mn þ Mm ! Mnþm ; kn;m td

ð7aÞ

E. Hansen / Journal of Biomechanics 36 (2003) 787–795

Mn þ Mm : Mn þ Mm ! kn;m

ð7bÞ

tc

789

kt

Both reactions are of second order and if the average rate constants (ktc and ktd ) are considered, then h2M ¼ 2kt ½M 2 with kt ¼ ktc þ ktd : The chemistry could therefore be described as d½I ¼ h1I ¼ kd ½I; dt

ð8aÞ

d½M ¼ fh1 þ h2 gM ¼ 2fkd ½I  kp ½M½M ; dt

ð8bÞ

d½M  ¼ fh1 þ h2 gM ¼ 2fkd ½I  2kt ½M 2 : dt

ð8cÞ

The temperature dependence of the rate constants is described by an Arrhenius-type expression kX ¼ AX eEX =RT ;

ð9Þ

where AX is the pre-exponential factor, EX is the activation energy, R is the gas constant and T is the absolute temperature. Note that the activation energy of the (almost) instantaneous reaction (5) was modelled as zero. Finally, the chain length dependance of kt needs to be considered. The average chain length n of the reactants in the termination reaction is approximated as the ratio of the number of monomer units consumed per active center ðM1 Þ produced, which is equivalent to    h2  k ½M½M   M p n¼ 1 ¼ : ð10Þ  hM   2fkd ½I

ν

e

ν

Fig. 2. The principle appearance of the termination rate kt as a function of n: The triangled and squared curves correspond to the instantaneous and gradual transition models of kt : Note how the two models generated the same decreasing behaviour (except in a small region around ne ).

Using Eqs. (8), (9) and (12) the dynamics of the chemistry could finally be described as @½I ¼ kd ðTÞ½I; @t

ð13aÞ

@½M ¼ 2fkd ðTÞ½I  kp ðTÞ½M½M ; @t

ð13bÞ

  @½M  2n0 kt0 ðTÞ ½M  ¼ 2fkd ðTÞ½I 1  : @t kp ðTÞ ½M

ð13cÞ

1

The rate constant kt is then often modelled as ( n1=2 when none ; kt p 3=2 when nXne ; n

ð11Þ

where ne is a critical chain length where the polymer solution changes from a non-entangled to an entangled regime (Buback et al., 2001). The transition can macroscopically be observed as a very large and rapid increase of the cement’s viscosity. In order to limit the complexity of the model, the instantaneous transition in relation (11) was approximated by kt pn1 for all n; which could be interpreted as a more gradual transition between the non-entangled and the entangled regime. This approximation had the same properties as relation (11) for all values of n; except in a small interval around ne (Fig. 2). Hence kt ¼

n0 0 2n0 fkd ½I 0 kt ¼ k ; kp ½M½M  t n

ð12Þ

where n0 is an unknown constant and kt0 is the Arrhenius-type expression presented in Wunderlich (1989), where no consideration has been taken to the chain length dependence of kt :

Note that the derivatives in (13) are partial because T is a function of both time and space. 2.3. Heat transfer The heat transfer in the system can be described as s

@T ¼ r ðlrTÞ þ Q; @t

ð14Þ

where s is the density times the heat capacity, l is the thermal conductivity and Q is the heat source term. It was assumed that the density, heat capacity and thermal conductivity were constant for all materials. In the bone–cement–prosthesis system the only heat production is in the cement (due to the exothermal reactions (5) and (6)), hence Qbone ¼ Qprosthesis ¼ 0: The heat production in the cement is proportional to the rate of

E. Hansen / Journal of Biomechanics 36 (2003) 787–795

790

disappearance of monomers: Qcement ¼ Qc

The right-hand side was given by

@½M ; @t

ð15Þ

where Qc is a positive constant. The heat transfer across the inner boundaries (cement–prosthesis and bone–cement) was assumed to be continuous while the transfer across the bone–tissue boundary was modelled by using Newton’s law of cooling: n ðlrTÞ ¼ aDT;

ð16Þ

where n is the outward unit normal, DT is the temperature drop over the boundary and a is a positive constant. Furthermore the tissue was assumed to have a constant temperature T0 : 2.4. The complete model If the variables T; [I], [M] and ½M  were normalized,   T ½I ½M ½M  T T u¼ and v ¼ ½v1 ; v2 ; v3  ¼ ; ; ; T0 I0 M0 M0 Eqs. (9), (13)–(16) were rewritten and the system % in Fig. 3 was considered, geometry (closed set O) the model could be stated as follows: The polymerization kinetics was modelled as an initial value problem in O2 @v ¼ fðu; v; n0 Þ; t > 0; @t   ½Ijt¼0 ½Mjt¼0 ½M jt¼0 T vjt¼0 ¼ ; ; : I0 M0 M0

ð17aÞ

ð17bÞ

NΩ

Ω2

Ω1

3

Ad eEd =RT0 u v1

7 6 2fAd I0 Ed =RT0 u 7 6 e v1  Ap M0 eEp =RT0 u v2 v3 7 6  7ð: 18Þ 6 M0 6  7  4 2fAd I0 2n0 At M0 Ep Et =RT0 u v3 5 Ed =RT0 u v1 1  e  e M0 A p M0 v2 The heat transfer in O was governed by the diffusion equation s

@u Q c M0 ¼ r ðlruÞ  XO2 f2 ðu; vÞ; @t T0

ð19Þ

t>0

with the boundary and initial conditions a ð1  uÞ on @O for t > 0; n ðruÞ ¼ lð@OÞ ujt¼0 ¼ u0

ð20aÞ ð20bÞ

in O:

Note the nonlinear coupling of the dynamics via the source terms in Eqs. (17a) and (19). Furthermore, XO2 is the characteristic function of the set O2 and s; l and u0 are functions, constant over the open sets O1 ; O2 ; O3 and the closed set @O: Observe that the model is also valid for a three-dimensional system geometry, but a twodimensional geometry was sufficient in order to show the properties of the model. 2.5. Simulations

y

Ω3

fðu; v; n0 Þ ¼ 2

x

Fig. 3. The principal geometry of the system. The sets O1 ; O2 and O3 corresponds to the prosthesis, cement and bone region, respectively.

The heat transfer was simulated by using the finite element program Femlab. The advantage of this software was that one could simulate very general systems of equations with a quite short computing time. In order to simulate the heat transfer and the polymerization, an explicit test case was needed. For this purpose the commercial bone cement Palacos R was considered. The compounds in Palacos R that effects the model were benzoyl peroxide (initiator) and methyl methacrylate (monomer). This combination of compounds gave the values of the rate constants in Table 1 (Masson, 1989; Rudin, 1998; Wunderlich, 1989). The initial concentrations in the vacuum mixed cement were 20 mol=m3

Table 1 The pre-exponential factors ðAÞ and activation energies ðEÞ of the rate constants obtained in a polymerization based on benzoyl peroxide and methyl methacrylate Rate constant

A

E ðkJ=molÞ

kd kp kt0 f

5:7  1017 =s 4:8  104 m3 =mol s 6:6  107 m3 =mol s 0.5

130 31 21 0

E. Hansen / Journal of Biomechanics 36 (2003) 787–795 Table 2 Heat transfer related constants used in the simulations Constant 3

s ðMJ=m KÞ l ðJ=m s KÞ Qc ðkJ=molÞ a ðkJ=m2 s KÞ

Prosthesis

Cement

Bone

3.6 19

1.7 0.21 54

3.1 0.29 0.50

benzoyl peroxide and 3800 mol=m3 methyl methacrylate. It was also assumed that the cement initially did not contain any monomer-ended radicals. Hence, ½Ijt¼0 ¼ I0 ¼ 20 mol=m3 ; ½Mjt¼0 ¼ M0 ¼ 3800 mol=m3 and ½M jt¼0 ¼ 0: The temperature of the surrounding tissue T0 was set to 311 K (body temperature) and the initial temperatures of the system were chosen to be 304 K in the bone (Toksvig-Larsen et al., 1991) and 296 K (room

0.02

y (m)

y (m)

0.013

0

-0.013 -0.013

0

0.013

x (m)

(a)

0

0.013

x (m)

(b) 0.02

y (m)

y (m) (c)

0

-0.02 -0.013

0.013

0

-0.013 -0.013

791

0 x (m)

0.013

0

-0.02 -0.013 (d)

0

0.013

x (m)

Fig. 4. Geometries and meshes used. The geometry in (a) was used when investigating the impact of the parameter n0 upon the system and the geometry in (b) was used in the final simulation.

E. Hansen / Journal of Biomechanics 36 (2003) 787–795

792

temperature) in the cement and prosthesis. The values of the density times the heat capacity, thermal conductivity and the constants Qc and a; introduced in Eqs. (14) and (16), are presented in Table 2 (Huiskes, 1980; Swenson et al., 1981). What remained unspecified was the parameter n0 ; introduced in Eq. (12). The simulations were therefore conducted in two steps: First the impact of the parameter n0 was investigated. As this demanded that many simulations were done the computation time was minimized by using a simplified system geometry (Fig. 4a) with a 3 mm thick cement region. Secondly, the simulations were made for a cross-section of a hip with a potential femoral stem prosthesis and a cement region with thicknesses in the range 2–5 mm (Fig. 4b). This simulation was conducted with a value of n0 which gave temperatures and concentrations in the earlier simulations that agreed with results from other studies. The meshes (Fig. 4c and d) for the simulations were generated by Femlab. The mesh in Fig. 4c contained of 1600 nodes and the mesh in Fig. 4d contained of 2400 nodes. It was observed that using twice as many nodes as in Fig. 4c and d generated results that deviated less than 5%, for any simulated quantity in any common node at any time, from the results obtained with the coarse mesh. Furthermore, the computation time increased drastically when using the finer mesh. Hence, the coarse meshes were used.

3. Results Fig. 5 shows the simulated temperature and monomer concentrations for some different n0 values in a point located in the corner of the bone–cement interface (see geometry in Fig. 4a). From the temperature simulation shown in Fig. 5a, it was observed that the maximum temperature decreased as n0 increased. When varying n0 from 0.2 to 1 the maximum temperature decreased from 45 C to 41 C: From the simulated monomer concentrations in Fig. 5b it was observed that the speed of the monomer decomposition decreased when n0 increased. When varying n0 from 0.2 to 1 the monomer concentration after 2000 s increased from 50 to 220 mol=m3 : These results could be summarized as: The smaller n0 ; the more extreme is the solution. In the simulation of the test case with the prosthesis geometry (Fig. 4b) the parameter choice n0 ¼ 0:5 was made. The simulated temperature distributions for some different times in the prosthesis geometry are shown in Fig. 6. The temperature initially rose in the system but after approximately 120 s it started to decreased towards the temperature of the surrounding tissue. The maximum temperature was 53 C and located in the center of the broadest part in the cement region. The maximum temperature on the bone–cement interface was 48 C: Fig. 7 shows the simulated monomer distribution for some different times in the prosthesis geometry. The monomer was unevenly distributed and the highest concentrations were found in the thinnest regions of the 4000

46 44 42

3000

[M] (mol/m3)

Temperature (C)

40 38 36

2000

34 1000

32 30

0

28 0 (a)

500

1000 Time (s)

1500

2000 (b)

0

500

1000

1500

2000

Time (s)

Fig. 5. The simulated temperature (a) and monomer concentration (b) for some different n0 values in a point located in the corner of the bone– cement interface of the simplified geometry. The squared, circled and triangled curves in both figures correspond to simulations with n0 ¼ 0:2; 0.5 and 1, respectively.

E. Hansen / Journal of Biomechanics 36 (2003) 787–795

t=70s

0.02

t=120s

36

0.02

0.01

53

-0.01

Temperature (c)

Temperature (c)

0.01

0

0

-0.01

-0.02 (a)

793

-0.02

27 -0.01

-0.005

0

0.005

0.01

(b)

34 -0.01

-0.005

0

0.005

0.01

t=2000s 37.5

0.02

Temperature (c)

0.01

0

-0.01

37

-0.02 (c)

-0.01

-0.005

0

0.005

0.01

Fig. 6. The temperature as a function of space for different times. Note how the largest heat production was obtained in the broadest cement regions.

cement. The residual (after 2000 s) monomer concentration was between 2% and 3%.

4. Discussion In vivo studies have found maximum temperatures ranging from 40 C to 48 C in the bone–cement interface

of a 3–5 mm thick cement layer (Biehl et al., 1974; Reckling and Dillon, 1977; Toksvig-Larsen et al., 1991). In vitro studies found the maximum temperatures ranging from 36 C to 53 C; depending on the mixing technique, in a setup simulating the bone–cement interface (Dunne and Orr, 2002). Finally, empirically based models predicted the maximum temperature in the bone–cement interface of 3 mm thick cement region

E. Hansen / Journal of Biomechanics 36 (2003) 787–795

794

t=120s

t=70s

0.02

3550

0.02 2700

0.01

0

-0.01

0

-0.01

-0.02 (a)

[M] (mol/m3)

[M] (mol/m3)

0.01

3150 -0.01

-0.005

0

0.005

0.01

-0.02 (b)

1600 -0.01

-0.005

0

0.005

0.01

t=2000s 150

0.02

[M] (mol/m3)

0.01

0

-0.01

50

-0.02 (c)

-0.01

-0.005

0

0.005

0.01

Fig. 7. The monomer concentration as a function of space for different times. Note how the highest monomer concentrations are found in the thinnest cement regions.

to lie between 35 C and 43 C (Borzacchiello et al., 1998; Vallo, 2002). Hence, choosing n0 ¼ 0:5 gave a maximum interface temperature of 43 C (Fig. 5a), which agreed well with earlier studies. This choice also resulted in a residual monomer concentration of 3% of the initial concentration (Fig. 5b), which coincided with previous studies that found the residual monomer to be in the

range of 2–6% (Huiskes, 1980; Kuhn, . 2000). Thus, setting n0 ¼ 0:5 was justifiable. The finding of the highest temperatures and the lowest monomer concentrations in the broadest parts of the cement in the simulations with the prosthesis geometry was due to auto accelerating heat transfer. The auto acceleration occurred when the polymerization

E. Hansen / Journal of Biomechanics 36 (2003) 787–795

produced heat that increased the local temperature. The local temperature rise increased the reaction rates, resulting in a locally increased heat production. This rapidly increased the temperature in the broad part of the cement, as the largest heat production was obtained in the regions with the largest amount of reactants (the local heat production is proportional to the local initiator and monomer concentrations). The acceleration ceased when the polymerization had consumed most of the monomer and initiator, resulting in low monomer concentrations in the broad parts of the cement. Contrary to the broad regions, the thin parts of the cement had initially a small local amount of reactants. This only generated an insignificant initial temperature raise, which prevented the subsequent monomer consumption. Similar observations were made by Vallo (2002) for a cylindric system. Furthermore, a maximum temperature of 48 C in the bone–cement interface and a residual monomer concentration in the range 2–3% corresponded well to the ranges discussed above. Finally a comment regarding the test case. Most commercial bone cements use a two-component initiator system I1 þ I2 !0 2R ; ð21Þ kd

in order to obtain a cement that is initiated at room temperature when the two compounds are mixed. In the case of Palacos R, the system consists of benzoyl peroxide and dimethyl-p-toluidine. Data concerning the kinetics of this particular reaction is scarce. Hence, reaction (4) and the values of Ad and Ed obtained from a thermally induced initiation was used. The fact that the results of the simulations agreed with previous studies suggests that the kinetics of the two-component initiator system was quite similar to the one presented here. In conclusion, a model derived from first principles was developed in order to investigate the heat transfer in a bone–cement–prosthesis system. The model was valid for an arbitrary system geometry and with an arbitrary bone cement consisting of an initiator and monomer. The model was simulated for a cross-section of a hip with a potential femoral stem prosthesis and for a cement similar to Palacos R. The numerical results agree well with in vivo studies and earlier models based on an empirical description of the polymerization process.

Acknowledgements . The author would like to thank Gustaf Soderlind and Leif Ryd for the opportunity to write this article and the department of Numerical Analysis, Centre for Mathematical Sciences, Lund University and the Department

795

of Orthopaedics, Lund University Hospital for all their support.

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