Thermal problems in biomechanics: from soft tissues to Orthopaedics Текст научной статьи по специальности «Медицинские технологии»

Russian Jour nal of Biomechanics

www.biomech.ac.ru

THERMAL PROBLEMS IN BIOMECHANICS: FROM SOFT TISSUES TO ORTHOPAEDICS

M. Stanczyk, J.J. Telega

Polish Academy of Sciences, Institute of Fundamental Technological Research, Swi^tokrzyska 21, 00-049 Warsaw, Poland, e-mail: [email protected]

Abstract: The aim of this paper is to review available results pertaining to various heat transfer problems of biomechanics. It covers experimental data, modelling as well as analytical and numerical solutions. The division into three classes of problems is proposed: soft tissue heat transfer, orthopaedics-specific problems and cryogenics. Various models of heat transfer in perfused tissue are presented along with the results of experimental and numerical investigations of their predictive capability. Existing criteria for heat-induced tissue damage are reviewed. Thermal problems linked with implantation of cemented endoprostheses and PMMA polymerization are examined. The research to date, concerning frictional heating of tissue due to bone drilling/sawing and joint operation is reported. Mechanisms of tissue damage during cooling and at subzero temperatures are described along with physical phenomena important for modelling of the freezing and thawing processes in the tissue.

Key words: biomechanics, thermal problems, bioheat transfer, thermal injury, hyperthermia, vascular models of heat transfer, frictional heating of implants, cryogenics

1. Introduction

Heat transfer problems, the biomechanics has to cope with, range from cryogenic protocols to procedures during which soft and hard tissues are subjected to considerably elevated temperatures. In the present paper we try to make a sort of a comprehensive synthesis of various aspects pertaining to thermal problems of biomechanics.

Particularly in Sec. 2 various available models of heat transport in soft tissues are discussed. In Sec. 3 thermal problems arising during and after joint replacement are examined. Section 4 is concerned with low temperatures. Here one distinguishes problems related to cryotherapy and cryopreservation.

Two important aspects are emphasized: the modelling of the heat transport phenomena and its applicability for the description of possible mechanisms of heat-induced damage to tissue. Although the ranges of problems in different fields of biomechanics, as outlined in Sections 2 - 4, are different, the mathematical models and criteria are often similar.

2. Heat transport in soft tissues

Comprehension of phenomenon of heat transfer due to blood perfusion in the soft tissue is important for all considerations of heat exchange in such tissues for various reasons. It helps to predict the outcome of modern surgical treatments involving local application of high temperature to diseased tissue (heat sources include fiber optics coupled to Nd: YAG

(neodymium-yttrium-aluminium-garnet) lasers, microwave antennas, radio-frequency transducers, ultrasound emitters, Joule-heated probes, saline-filled balloons and others). It is necessary for constructing whole-body or whole-limb heat transfer models that can be used for various practical purposes like predicting heat loss rate in water, wind cooling, thermoregulation efficacy etc. cf. [89]. Modelling of heat transfer in soft tissue is also necessary for accurate assessment of energy dissipation rate in joints and for thermal analysis of first stages of cryosurgical protocols (before freezing, when effects of blood circulation are present). Problems presented in this section are therefore inevitably linked with the issues presented in the remaining part of the paper.

One of the most important purposes of modelling of the heat transfer in living tissues is to be able to predict the level and area of potential damage caused by extreme temperatures in tissue. In this section the issues concerning damage in soft tissue caused by hyperthermia are briefly discussed. For review of damage mechanisms due to low temperatures the reader is referred to Sec. 4.

2.1 Soft tissue damage due to hyperthermia

Thermal injury in living cells has been a topic of research for a long time. Its significance has grown with the invention of weapons relying mainly on heat, such as the flame thrower, incendiary charges or napalm and their use in numerous conflicts. The early experimental and theoretical studies focused therefore mainly on heat effect on epidermal injury, cf. [39], [40]. The development of various therapeutic techniques relying on application of elevated temperatures to degenerate tissues made consideration of thermal damage mechanisms in other kinds of tissues necessary.

The thermal treatment of cells leads to numerous degenerative events but the sequence of events leading to the cellular death is still not well established. Various internal structures of the cell were implicated to be targets of the thermal treatment but none have been proven conclusively to be responsible for cellular death. The widely-accepted model of thermal damage in soft tissue is the first-order rate process (Arrhenius model), cf. [9], [38], [40], [64].

The measure of injury Q is introduced and its rate is postulated to satisfy the equation:

dQ . ( Ea \ ,.

-= A expl--a I, (2.1)

dr \ BT) '

where B is the universal gas constant and A , Ea are the frequency constant and activation energy respectively; r denotes the time. The constants A, Ea are model parameters usually obtained experimentally. One can assume that Q = 1 marks the threshold of irreversible thermal injury that can be detected experimentally. Once this threshold is attained in a given mode of heating after the time rA the measured thermal history of the system is used to derive the values of model parameters in accordance with the equation:

1 ^J

zA f e \

exp

a

dz.

0 V BT(z)j

For an isothermal regime the time zA is simply the reciprocal of the damage rate defined in Eq. (2.1).

Another way of normalization is to define Q in terms of concentrations of the original (native) tissue C0(z) and the damaged tissue Cd(z), cf. [21]

Q(z) = lni C°(0) ], (2.2)

V1 - Cd (z) j K J

where C0(x) + Cd (x) = 1 holds for every x .

There is a wide variety of experimental methods used to identify the time xA at which irreversible cellular damage occurs. They rely on different physiological effects and therefore the damage mechanisms to which they react are different. Some methods involve using fluorescent dye markers like propidium iodide, trypan blue or neutral red, which diffuse through heat-damaged cellular membranes. These methods measure essentially cellular membrane damage level which not necessarily equals the overall damage measure Q. Another approach consists of assessing the colony-forming ability (clonogenics) of the heat-damaged cells after treatment, cf. [9] for details. While this seems to give a good indication of cell viability, the method needs considerable time for post treatment incubation. In investigations of the viability of the collagenous tissues the heat-induced shrinkage is usually regarded to be a good, measurable indicator of collagen denaturation, cf. [19], [20], [21], [106].

As can be inferred from Table 1 the values of model parameters obtained by using different methods vary in a wide range, cf. also [9] or [64] for comparison. This can give a clue as to what damage mechanisms are responsible for the detected effects. This insight can be gained from the consideration of the activation energy Ea .

In general, the activation energy of any physical/chemical process is the critical minimum energy that must be possessed by the constituents involved for the process to take place. Therefore the rate of the process will be proportional to the fraction of these constituents which do possess the energy at least equal to the value of the activation energy. This fraction f is deduced from the Maxwell-Boltzman energy distribution law [40]

f = expi- ^] .

\ BT I

Table 1. First-order rate process model of thermal injury parameters (see Eq. (2.1)), after [9], [40], [64].

Tissue Activation energy, Ea (kJ/mole) Frequency factor, A (1/s)

Skin 628.5 3.1-1098

Prostate tumor (clonogenics measurements) 526.4 1.04-1084

Prostate tumor (propidium iodide uptake measurements) 244.8 2.99-1037

Prostate tumor (calcein leakage measurements) 81.33 5.069-1010

Arterial tissue 430 5.6-1063

Erythrocyte membrane 212 1031

Hemoglobin 455 7.6-1066

Whole blood 448 7.6-1066

Since the constant Ea as used in Eq. (2.1) can be viewed as the mean activation energy of the physical and chemical processes leading to heat-induced cellular damage (according to certain experimental criterion) the measured value of this constant, in comparison with the values of activation energy of various well-known processes provides foundation for speculations about the mechanisms of cellular injury [9], [40]. Henriquez [40] divided the potential damage mechanisms into three categories:

(i) Thermal alterations in proteins. Proteins contribute to the maintenance of cell life in various ways and undoubtedly even minor heat-induced alterations in these molecules can lead to irreversible damage. Studies on the subject indicate that alterations in proteins occurring at the temperature range 0-100?C at measurable rates are not unusual. The activation energy of these processes are often well in excess of 200 kJ/mole and can be strongly dependent on pH (heat denaturation of egg albumin: Ea =553 kJ/mole at pH=5; heat inactivation of invertase Ea =461 kJ/mole at pH=4 and Ea =218 kJ/mole at pH=5,7; of hemoglobin Ea =318 kJ/mole

at pH=5), see [9], [40].

(ii) Other possible alterations in metabolic processes. This class of effects includes the temperature influence on kinetics of metabolic processes that do not involve proteins. These are changes in the rates of diffusion, formation and degradation of chemical reactants, etc. The activation energy of these processes is usually of order of 40-80 kJ/mole. These effects are usually regarded of minor importance to cellular thermal injury as compared to previous group.

(iii) Nonprotein induced alterations in the physical characteristics of cells. The physical phenomena characteristic of protoplasm but not primarily affected by the thermal alterations of proteins, e.g. diffusion of metabolites through an unaltered cell wall.

The model presented by Eq. (2.1) gives a definite connection between time-temperature history and damage accumulation. It can facilitate design of the hyperthermic treatment procedures allowing, in principle, accurate damage prediction provided the temperature field is known, the model constants are chosen appropriately and the temperature range is suitable. However, it has several shortcomings. For instance, it does not take into account the history of thermal insult - larger and smaller thermal loads produce the same result, irrespective of their relative order.

The problem not accounted for by the model (2.1) is the observation that marginally lethal or nonlethal temperatures lead to complete cell destruction, if they are preceded by short preheating at high temperature. Such cells become extremely sensitized to further temperature treatment, which would not otherwise cause death, see [38] for details. This phenomena can be viewed as the considerable lowering of the activation energy Ea by the preheating.

Another issue brought up by experimental investigations is that pH variations may play a similar sensitizing role for some kind of cells, cf. [38] and references therein. It is probable that preheating as well as acid pH may cripple the cell's capacity to accumulate and/or repair sublethal heat damage which is in turn different for different kinds of cells and may depend on mechanical loading cf. [20], [21]. Surprisingly, the influence of the latter is usually neglected. While in vitro tests in thermal baths and on isolated cells yield data usually specific to unloaded specimens, in the clinical in vivo experiments the tissue is often loaded in unknown and uncontrollable manner. The investigations of heat-induced shrinkage of collagenous tissue indicate that the increase in the mechanical loading during heating delays the denaturation, cf. [20], [28]. Introducing the characteristic time of damage process r2 one may write

Fig. 1. Steps involved in modelling heat transfer (e.g. during hyperthermia treatment) in soft tissue.

r2 = exp

/ \

a + pP + — T

where a, p and m are experimental constants. Scaling the time variable of experimental results obtained in different temperature-load regimes with r2 proved to be an effective way to reduce them to a single master-curve, cf. [20], [21], [28], [106].

Another possible disadvantage of Q as a measure of thermal damage, as defined by Eq. (2.1) is that most experimental data suggests that thermal damage tends toward an asymptotic value at large r (under isothermal conditions) or constant T (constant heating rates). The model presented, however, does not display such a behaviour. The following model can be suggested to overcome this difficulty, cf. [21] and references therein,

= m0 i1" f)+ mdf,

where f = 1 - e Q and m0, md denote the considered physical property at the original and

damaged state respectively. However some physical properties (e.g. shrinkage) do not follow this law and display piecewise linear behaviour, cf. [21].

2.2 Modelling of heat exchange in perfused tissue

Heat transport in living soft tissues is supplemented by one important factor not presented in in vitro experiments, namely the vascularity. It needs to be taken into consideration in certain problems, while it can be omitted in others with only a slight penalty in accuracy. The procedure usually applied during modelling of the heat transfer in soft tissue is schematically depicted in Fig. 1. The discussion of numerous heat depositions phenomena (surface convection, irradiation, Joule heat generation, etc.) along with analytical and numerical results for different boundary conditions is discussed at length in [30]. Here, we focus on the proposed forms of the bioheat equation which plays a important role in the modelling process, as shown in Fig. 1.

The theoretical analysis of the heat transport in soft tissues was carried out by a lot of

investigations. The proposed models can be basically classified under two categories (cf. Baish et al. [3]): vascular and continuum models. The former try to reproduce the real vascularity of the tissue and to describe all local variations in the temperature near the individual vessels. They require detailed knowledge of the vascular geometry unlike the continuum models which account for the blood perfusion rate by means of the effective conductivity of the tissue which is dependent on the blood flow rate or by means of other global parameters.

2.2.1 Continuum models 2.2.1.1 Pennes' equation

The first continuum model was introduced by Pennes in 1948 to analyze heat transfer in a resting human forearm [63]. The equation has the following form

DT

ptct = XtV2Tt + cblcbl (Tt - Ta )+ qv, (2.4)

Dt

where X is thermal conductivity (W/m-K), p and c are its density (kg/m3) and specific heat

(J/kg-K) respectively. The subscript bl labels the blood, t - the tissue, qv is the volumetric,

metabolic heat generation rate (W/m3); Ta denotes arterial supply blood temperature. When

modelling of hyperthermia treatment with volumetric energy deposition the right-hand-side needs to be supplemented with appropriate term. The internal heat generation rate is in general dependent on temperature. Studies of metabolism of fevered patients indicate that whole-body heat generation increases by some 0.7% for each 0.5?C increase in temperature, giving approximate relation, cf. [48]

Qv = 85 -1.07 T-310)/ 05 (W). (2.5)

Basically, the Pennes' equation (2.4) is the classical heat transport equation supplemented with a linear heat sink, which arises from the thermal equilibration of the blood in capillaries with the surrounding tissue. The idea behind the Pennes' equation is that the blood is supplied to the tissue at arterial temperature Ta, perfuses the tissue at rate cbl (kg/s-m3) attaining the thermal equilibrium with it and then is collected in the veins. The perfusion rate is thus the key parameter in calculating the heat transfer. Blood perfusion rates in different tissues are estimated in Table 2. The data were given by Valvano (1987) and are cited by Davidson et al. in [26].

As shown in Table 2 the values of maximum perfusion rate are often greater by an order of magnitude than those encountered in regular conditions. Increasing perfusion rate value can occur as a result of vasodilation, which is a temperature-dependent thermoregulatory mechanism, cf. [48], [89] or can be induced pharmacologically. The effects of vasodilation, which can be modelled via single parameter c bl in Pennes' model were more

Table 2. Blood perfusion rates of various human tissues, after [26]. Values in (ml/100-gram-min).

Tissue Normal perfusion Maximum perfusion

Skeletal muscle 2.5 60

Liver 29 176

Heart 70 400

Fat 8 30

Skin 200 497

Kidney 400 466

thoroughly examined by means of vascular models, cf. [85], [86], [112] and also Sec. 2.2.2. In the hyperthermic treatment of tumors the significant nonuniformity of the perfusion throughout the cancerous tissue has to be accounted for in order to obtain the agreement with experiment. The necrotic core of the tumor is perfused much less than the outer boundary, where the tumor is growing. Furthermore, the geometry of the vasculature in the tumor can change rather rapidly, cf. [23]. Modelling of such nonuniformities can be done with several regions of uniform perfusion, cf. [23], [77] or by means of continuous function of depth, for example for lymphosarcoma, cf. [44],

cbl = 120 +1513%2 (ml/100-gram-min), (2.6)

where % = r/R, r is the radial position within the tumor and R is the tumor radius. The formulation, analytical solution and numerical evaluation of the solution for multi-region transient Pennes' equation applicable to cases of simple spatial variation of perfusion rate are given in [32], [33].

Pennes' equation (2.4), also called bioheat equation or heat-sink model, is widely used for prediction of temperature elevation during hyperthermia, cf. e.g. [23], [44], [51], [52], [76], [77] as well as for predictions of temperature response in cryosurgical protocols, cf. e.g. [41], [70], [72]. In the latter case the blood perfusion term is often assumed nonlinear, with respect to temperature, to account for the fact that blood flow ceases at certain low temperature. Common usage of the Pennes' model is not always accompanied by careful examination of its limits of applicability. These limitations originate from neglecting the effects of thermally significant large vessels, cf. [12]. The discussion of the shortcomings of Eq. (2.4) was done by numerous authors, cf. [2], [12], [13], [23], [24], [46], [101], [108], [109] and ranged from putting emphasis on the applicability limitations [23] to complete negation of validity [108]. The main drawbacks of the Pennes' model have been outlined by Wulff in [109]. Most important points are:

(i) The Pennes' model assumes that blood arrives at each point in the tissue at one temperature Ta regardless of the distance, which separates that point from the supply vessel. No transport mechanism has been found to accomplish this. Furthermore the local arterial temperature depends on the temperature gradient in tissue resulting from environmental conditions as shown in [13].

(ii) The blood perfusion term fails to account for the directed character of the blood flow.

(iii) The blood perfusion term has been obtained via the global energy balance for blood and is applied to describe local energy balance for tissue.

(iv) The first-order differentiability condition of numerous physical entities in the equation like heat flux, physical properties and heat generation, is not necessarily met in heterogeneous tissue structure.

The Pennes' equation is also unable to account for local temperature variations due to large vessels - a feature inherent to all continuum models and unacceptable in certain applications like localized hyperthermia. The tumor regrowth, that is usually reported in the immediate vicinity of blood vessels most probably results from underheating of these, blood-cooled locations, cf. [13].

The tissue temperature in Pennes' model is typically defined, cf. [2], [18]

Tt (x) = j- {T(x)dV, (2.7)

SV

where the scale of an averaging volume SV is assumed to be much larger than the size of thermally significant vessels and much smaller than the size of the tissue region itself. Baish [2] pointed out that no such scale exists because the vascular tree consists of vessels of

virtually all sizes.

The thermal equilibration length in the Pennes' model is assumed to be infinite for all vessels except the capillaries and zero for capillaries. This is also a nonphysical assumption. In Table 3 the estimates of this value are given for different kinds of vessels, see also Fig. 6, [3], [4], [101]. We recall, that the thermal equilibration length was defined by Chen and Holmes [18] as the length, over which the temperature difference between tissue and blood is reduced by the factor e.

The Pennes' equation can be supplemented with the correction factor - efficiency coefficient, multiplying the heat-sink term in equation (2.4). It accounts for the thermal equilibration of the returning vein (thus making allowance for the fact that the returning venous blood temperature may differ from tissue temperature). This coefficient is a function of the complex geometry of the system and its derivation is described in detail in [103].

In spite of the serious criticism against the Pennes' model, its predictions are often superior to those of more elaborate formulations (see effective conductivity models description below). This fact gives rise to the need for reconsidering the physical foundations of the approach. Charny et al. [16] suggested that the blood perfusion term in Eq. (2.4) does not represent the isotropic thermal equilibration in capillaries but describes the small vessel bleed-off occurring in the regions of the largest counter-current vessels and supplying the capillary bed in tissue. In a manner the temperature of the blood entering these capillaries is the temperature of the blood in the largest vessels. In the region dominated by smaller vessels this condition is not always satisfied and models other than Eq. (2.4) are preferred [16].

2.21.2 Directed perfusion model

The second continuum model is the directed perfusion model, proposed by Wulff in 1974 [108]. Its derivation relies on the assumption that, unlike in Pennes' model, the blood is completely equilibrated with tissue all the time. Recall that in heat-sink model it is assumed to retain arterial temperature until it reaches the capillaries and then to momentarily equilibrate with tissue. When the energy transport is assumed to be not only due to conduction but also due to convection by the moving blood (in equilibrium with tissue) the energy flux is:

q = -XtVTt + pblhblU , (2.8)

Table 3. Properties of different kinds of blood vessels, after [18].

Vessel Percent of vascular volume Average radius (?m) Average length (mm) Thermal equilibration length (mm)

Aorta 3.30 5000 380 190000

Large artery 6.59 1500 200 4000

Arterial branch 5.49 500 90 300

Terminal art. branch 0.55 300 8 80

Arteriole 2.75 10 2 0.005

Capillary 6.59 4 1.2 0.0002

Venula 12.09 15 1.6 0.002

Terminal vein 3.30 750 10 100

Venous branch 29.67 1200 90 300

Large vein 24.18 3000 200 5000

Vena cava 5.49 6250 380 190000

where hbl is the specific blood enthalpy and U is the Darcy's velocity of the blood flow. The first term at the right hand side in Eq. (2.8) is the usual Fourier-law conductive flux. The second one accounts for the blood directed convection.

Substituting the Eq. (2.8) into the energy balance equation and putting Tbl = Tt one

gets

dT

PC ^ = KV2Tt - PbiCbiU • VTt + qv . (2.9)

It should be noted that this model applies to cases when the blood is in equilibrium (or in near-equilibrium) with tissue and during hyperthermic treatments, it is often not the case (cooling effect of the large vessels is then pronounced). The quick glance at Fig. 6 and data in Table 3 indicates that the assumptions of the model presented in Eq. (2.9) are correct only in a certain range of physical situations, strongly dependent on the kind of vasculature. Creeze and Lagendijk [24] also pointed out that the blood flow is not always unidirectional, vessels are often combined to form counter-current pairs and their orientation is frequently isotropic.

2.2.1.3 Effective conductivity models

The effective conductivity models (also termed keff or Kf models) treat the energy

transport in the tissue in terms of heat conduction only. The effect of vascularity is contained in the thermal conductivity term which is often assumed to be isotropic and dependent on blood perfusion cob. The usually assumed dependence is of the form

Keff = Kt t1 + «1®b )

or

Kef = Kt i1 + «2® 2 I

where K is the intrinsic thermal conductivity of the tissue and «, a2 are parameters depending on vessel size and density. The theoretical foundation of the effective conductivity model has been provided by Weinbaum and Jiji [100]. On the basis of the knowledge of anatomical structure of vasculature and with aid of several simplifying assumptions they developed a compact formulation of the heat transfer equation in the soft tissue. The authors called their model the simplified bioheat equation because it is a simplified form of some of the ideas derived from vascular models that are described in Sec. 2.2.2. The Weinbaum-Jiji effective conductivity model is derived with the following crucial assumptions [100]:

• it is possible to formulate the energy equation in terms of a single local average blood-tissue temperature,

• local average blood temperature can be approximated by local tissue temperature,

• the primary heat transfer mechanism is the incomplete countercurrent exchange in thermally significant vessels (greater than 40 ?m in diameter), which means that the heat loss from the artery is nearly but not quite equal to the heat gained by the vein. Using these assumptions and considering the closely spaced, countercurrent artery-

vein pairs Weinbaum and Jiji derived the expression for the effective conductivity tensor, cf. [100]:

f

Kf = K

nn r

5l3 +-

3 4a

2 „2 (

v K J

Kbl K

Petyj

(2.10)

where n is the vessel number density (geometrical parameter of the vasculature), r is the vessel radius (assumed equal for artery and vein), li are the direction cosines of the vessels relative to the coordinate axes and 83j is the Kronecker delta. Pe is the blood flow Peclet

2

number defined by

Pe = Pr. Re = 2pblCbiru , (2.11)

Xbi

where Pr and Re are Prandtl and Reynolds numbers respectively and u is the average blood flow velocity. a is the shape coefficient describing the thermal coupling between the countercurrent artery and vein obtained from the consideration of two-dimensional conduction in the plane perpendicular to the axes of vessels,

a = J cosh , (2.12)

/ 2r(S)' V J

where l is the distance between the axes of the countercurrent vessels at the point determined by the coordinate S along the axis of the vessels.

From (2.10) one can see that the effective conductivity derived by Weinbaum and Jiji [100] is a tissue intrinsic conductivity with an additional term added, to account for the countercurrent convection.

The Weinbaum-Jiji effective conductivity model energy equation derived in [100] is

— dT d pc— = — dr dx

fy j

+ q— - Pe2 lj f f. (2.13)

4aXt oxi dXj

This equation is the ordinary Fourier-Kirchhoff equation for heat conduction in anisotropic media with effective conductivity and an additional term at the right hand side. This term accounts for the possible variation of vessel radius along its length and for the directed capillary perfusion between artery and vein. It is small and vanishes entirely when the vessels are straight.

From Eq. (2.10) the formula for effective scalar conductivity in one-dimensional case can be obtained.

Xff = X

( 2i,2(-k \2 ^

, nn r

1 + -

4a

lbl Xt j

Pe2

(2.14)

As estimations by Weinbaum and Jiji indicate, for tissue with vessels that have radius of more than 100 ?m, the enhancement in conductivity is noticeable and for larger vessels the effective conductivity is several times larger than the tissue intrinsic property.

The above-mentioned assumptions leading to the development of the Weinbaum-Jiji bioheat equation, mainly that average arterio-venous temperature is equal to tissue temperature, have been criticized in [13] and [105]. Results of computations on elaborated vascular model by Brinck and Werner contradict this assumption, cf. [13].

The existence and importance of countercurrent heat exchange at the certain vessel scale range has been confirmed by numerous experiments both in normothermic and hyperthermic (vasodilated state), cf. [85], [86], [112], [113].

The effective conductivity models do not depend explicitly on arterial temperature. Increase of the blood flow rate results therefore in enhancement in effective conductivity of the tissue, regardless of whether warm blood perfuses cold tissue or cold blood perfuses warm tissue. The consequence is that perfusing cooler tissue with warm blood will result in decrease in temperature owing to enhanced conduction to the surface - the effect opposite to the one predicted by Pennes' equation (2.4). Wissler argues that this discrepancy is in favour of the latter and proposes supplementing the Pennes' equation with additional 'efficiency factor' to account for the incomplete countercurrent heat exchange in a simple way [105].

Weinbaum and Jiji suggested that a hybrid model should be used, consisting of Pennes' model and the effective conductivity approach [102]. They propose that thermal equilibration parameter (thermal equilibration distance normalized by the length of the

representative vessel):

nrPe

e =--(2.15)

2"L

should be used to distinguish whether the Pennes' model (e > 0.3) or the Weinbaum-Jiji bioheat equation (e < 0.3) applies. Similar suggestions were made by Charny et al., who compared the normothermic and hyperthermic response of the Weinbaum-Jiji Keff model, the

Pennes' model and more sophisticated three-equation model [16]. The main conclusion of their study was that the Pennes' model provides a relatively good prediction capabilities and should be therefore incorporated in the proposed hybrid model by means of the criterion (2.15). This means that under normothermic conditions the Eq. (2.4) should be used in regions of tissue containing first generations of supply vessels (larger than 500 ?m in diameter). As the study by Charny et al. have shown, neglecting the countercurrent heat exchange in these regions does not lead to any serious discrepancies [16]. The Weinbaum-Jiji model is more suitable in regions of tissue containing smaller vessels.

Limitations of applicability of Weinbaum-Jiji bioheat equation were also shown by Valvano et al. by means of analysis and thermistor measurement of temperature field in canine cortex [97].

The validity of effective conductivity approach was tested by Baish who viewed the tissue with embedded countercurrent vessels as a composite material consisting of low-conductivity tissue matrix and high conductivity fibers representing paired countercurrent vessels [1]. He obtained the formula for the fiber conductivity:

K =-imc2i cosh-1 f, (2.16)

f 2lrn2K Vr J where 2l is the distance between axes of the countercurrent vessels.

Because of the quadratic dependence on the blood flow rate, the conductivity computed using Eq. (2.16) varies over the very wide range. For example for vessels that have radius of 300 ?m the respective fiber conductivity exceeds 3000 (W/m-K) and for vessel radius of 500 ?m this conductivity is 320 000 (W/m-K). No real material has such a high conductivity. This result is explained by the fact that convection really taking place within the veins is much more efficient heat transfer mechanism than the conduction. The tissue has an intrinsic conductivity of the order of 0.2 - 0.6 (W/m-K) and the fibrous composites containing fibers, that are much more conductive than the matrix, are known to be poorly modelled by the effective conductivity, cf. [1]. If we consider the array of parallel countercurrent vessel pairs embedded in tissue, lying along the z -axis and impose an external temperature gradient parallel to the axes of vessels, the total heat flux can be expressed as, cf. [1]:

dT

q = -Kf-T- + n(Qa + Qv),

dz

where (Qa + Qv) is the heat exchange with tissue by single vessel pair. So the effective conductivity approach is admissible only if (Qa + Qv) is proportional to the tissue temperature

dT

gradient —-. Heat exchange by blood vessel pair is, however, proportional to the gradient of dz

dT dT

mean blood temperature Tm. Therefore, assumption by Weinbaum and Jiji — = —m is

dz dz

necessary in the derivation of effective conductivity model, cf. [1].

The three continuum models of heat exchange in perfused tissue described above can be used to construct a mixed model. For example, Chen and Holmes suggest that thermal contributions from individual large vessels should be calculated separately and for the bioheat

effcond.

heat sink

measurement (least squares fit)

perfusion rate

Fig. 2. Comparison of the effective conductivity and Pennes's (heat sink) model predictions to measurements, after [24]. '+' denotes measurement data point.

equation they propose the following formulation, cf. [18]:

Ptct f^ = V ■ (XeffVTt ) + ®blcbl (t* - Tt) - pblcblU • VTt + qv, (2.17)

where a* and T* are the perfusion rate and arterial temperature respectively, modified to

avoid double-counting of the contribution of large vessels.

Crezee et al. considered a mixed heat sink - effective conductivity approach, formulated in the following way, cf. [25]:

dT

Ptct = Kff V2Tt + fablcbl (Ta - Tt) + qv, (2.18)

where f is a model parameter depending on the local vascularity structure. It can be shown that for closed vessel network 0 < f < 1, cf. [25].

The experimental verification of the presented continuum models was performed in [24], [25]. In [24] measurements of temperature field in bovine kidney cortex were conducted. The comparison between heat-sink and the effective conductivity models were generally in favour of the Xeff theory although the scatter of the measurements was considerable, as can be

seen from Figure 2. In this Figure the reciprocal of the temperature of the vessel wall is depicted against perfusion rate. The increase of the perfusion from 0 to 38 (ml per 100 g min-1) caused significant decrease of the vessel wall temperature suggesting a 6-fold decrease in thermal resistance of perfused tissue. The prediction of Pennes' model was 15% decrease in that case. Other tests confirmed that the Xeff model is superior to heat-sink model, especially

in small scale predictions. In the case of larger vessels the use of discrete (vascular) description is preferred, cf. [25].

Roemer and Dutton argued that the Pennes' perfusion term and the effective conductivity are nonphysiological values that are related to true capillary perfusion in a problem-dependent manner, cf. [78]. They provide detailed derivation of the universal tissue convective energy balance equation in [78].

2.2.2. Vascular models

The idea behind developing vascular heat transfer models for the soft tissue is to use the information about actual placement of blood vessels within the tissue to predict the heat flow. The need for accurate temperature predictions arose in the course of development of modern hyperthermia protocols. The cooling effect of large vessels present at the site of intended tissue temperature elevation escapes the continuum models entirely.

The three basic concepts of vessel placement lead to the unidirectional vessels model,

Fig. 3. Generic tissue cylinder considered in model of unidirectional vessels.

countercurrent vessels model and large-small-large vessels model. These are briefly summarized below. We observe that a comparison of the predictions of these models is given in Baish [3].

2.2.2.1 Unidirectional vessels model

This model relies on the assumption that the blood arriving in the tissue at its artery supply temperature exchanges heat with the tissue along the vessel. Two unknown quantities are modeled here: the tissue temperature Tt and vessel blood temperature Tbl defined along the vessel as the local average quantities:

Tt (S) = — JTdA, (2.19)

Lt A

1 f

Tbl (S) = — \TdA

Abl A,

(2.20)

where At and Abl are domains occupied by the tissue and blood, depicted in Fig. 3 as gray

ring and white circle respectively. The energy conservation in a tissue cylinder surrounding blood vessel is considered.

The equation for the tissue is, cf. [3], [4]

dTt , a2Tt P tct — = ^ t ^ - Mbi + qv

ax

(2.21)

where n is the vessel density and qbl is the rate of heat flow into the vessel in (W/m) derived from the energy balance for vessel, cf. [3], [4]

qbi = *rbi Pbicbiu

— dTbl

dS

(2.22)

Here u is the bulk blood velocity in the vessel and S is the distance measured along the vessel, be it straight or curved. Let us introduce convection boundary condition on the vessel wall, cf. [4],

qbl = 2nrblPbla(Tw - Tbl), (2.23)

where a is the convection film coefficient (W/m K) and Tw is the vessel wall temperature. If

qbl is phrased in terms of the temperature difference (Tt - Ta):

qbl = a (Tt - Tbl), (2.24)

then the shape coefficient can be obtained from the consideration of the heat transfer in the plane perpendicular to vessel axis [4]

a- =-^-, (2.25)

ln r-

rbl

hi Nu

Fig. 4. Generic tissue cylinder considered in the model of counter-current flow vessels.

where Nu is the Nusselt number of the flow inside the vessel:

ah

Nu =

assumed to be equal to 4 as proposed in [17] and used subsequently e.g. in [4]; l0 is the

characteristic dimension of the flow equal to the vessel diameter 2r0. The value given here

corresponds to vessels large enough, whose diameters are much larger than size of the blood cells. This is not the case in the microcirculation, cf. [57].

Eqs. (2.21), (2.22) and (2.24) accompanied by (2.25), constitute the unidirectional vessels model. The model neglects heat conduction in blood along the vessel and heat generation in blood. Furthermore, as experimental analysis of the vasculature indicates, all major vessels start out as a closely juxtaposed countercurrent artery-vein pairs and only a generation prior to terminal vessels they diverge substantially forming a roughly periodic array, cf. [101].

The heat conduction shape coefficient approach, consisting of relating the heat transfer in vessels to the blood-tissue temperature difference by means of the proportionality coefficient, obtained from planar steady state analysis, is successfully used in other formulations, that are presented below.

2.2.2.2 Countercurrent vessel model

In this formulation the vessels are assumed to exist only in the form of countercurrent artery-vein pairs, scattered with the density n. The model assumes that the basic mechanisms of heat transfer are countercurrent heat exchange between arteries and veins and the vessel-tissue heat transfer, which was the only mode considered in models presented above. The difference between arterial and venous blood temperatures is considered to be the driving force of the former, while the difference between average temperature in both vessels and the tissue is assumed to cause the latter mode of heat exchange. Note that for two vessels located symmetrically in the tissue cylinder (see Fig. 4), close to each other, the difference in temperature between the vessels does not influence the net heat transfer to the tissue [4].

The model contains three key variables: tissue, artery blood and vein blood temperatures, denoted Tt, Ta and Tv respectively. The equations are constructed using the

shape coefficient formalism, cf. [3], [4]: (i) tissue

dr

(ii) artery (feeding vessel)

2 -T ^roPbcbu s

(iii) vein (draining vessel)

dT 1 0 2T ,

dS

r T + t ^

T —a—1

V

2

+q*

j

= ä,<Ja(Tv -T) + -£

2

r T + t ^

Tt —a—1

v

2

(2.26)

(2.27)

j

Fig. 5. Generic tissue cylinder considered in the model of large-small-large vessels.

^dTa , _ irr rr \

i T, , rri \

nro Pbcbu

T„ + T

= WT -Ta)--^ Tt. (2.28)

v 2 y

0S ■ u / 2

Here cA is the shape coefficient for the heat exchange between individual vessels in the countercurrent pair while cs is the shape coefficient for the heat exchange between vessel pair and tissue. These coefficients are estimated, similarly as in previous model, from considerations of steady-state temperature distributions in the plane normal to the axes of the vessels using the superposition method. The derivation along with values for several kinds of vessels is given in [4].

2.2.2.3 Large-small-large model

This model is somewhat simplified in comparison to the previous two. It relies on the assumption that vasculature has a hierarchical structure with big arteries feeding smaller and smaller vessels, which in turn feed the capillaries, see Fig. 5. The blood from capillaries drains into larger and larger vessels, which eventually feed the large veins countercurrent with the large arteries. The most important assumption here is that significant heat transfer with tissue occurs only at the level of capillaries. The thermal coupling between large vessels and tissue - cs in Eqs. (2.27), (2.28) is assumed to be small and the equilibration length in the capillaries to be much shorter than their length. This common belief is in contradiction to findings of Chen and Holmes [18], see also Table 3, Fig. 6 and the discussion below.

Table 4. Basic types of heat transfer problems related to joint replacement.

Bone cement polymerization Bone drilling and sawing Frictional heating of the implant

Range of application All cemented implants, hip endoprostheses (acetabular and femoral parts) Joint replacements of any kind; fixation of screws in bone Joint replacements of any kind

Time and duration of exposure to elevated temperature During and immediately after implantation, duration: several minutes Prior to implantation, duration: few minutes During joint operation, temperature cycles repeated many times during the service life of artificial joint

Location Bone tissue-bone cement interface, bone cement domain Cut surface/drilled hole Articulating surfaces interface (acetabular cup and prosthesis stem in hip implants)

Other physical phenomena affected Bone cement polymerization (quantity of toxic monomer leftover), thermal bone necrosis, aseptic loosening of the prosthesis Thermal bone necrosis, surgical trauma Acetabular cup wear and material degradation

Materials Metals, bone cement mixtures (PMMA based), bone and surrounding tissue Bone, metals UHMWPE, ceramics, metal alloys, bone and surrounding tissue

Max. Temperatures Up to approx. 60?C at the bone surface and over 100?C within cement domain (dependent on the type of cement) Over 150?C (sawing without coolant) Around 43?C in the middle of the prosthesis head and less than 40?C on the surface of bone

Fig. 7. Three essential layers of peripheral circulation distinguished by Weinbaum et al., after [101].

In this model, heat sink term is employed to account for vasculature similarly to the Pennes' model, see Sec. 2.2.1. Also similarly, the instantaneous thermal equilibration of the blood with the tissue is assumed to take place in capillaries, so this heat sink strength is proportional to arterial blood - tissue temperature difference, see Eq. (2.4). Unlike the Pennes' model, it is expressed here in terms of decay of the blood flow in artery due to the blood draining to capillaries along the major vessel:

OT O2T O í \

PC^L = + rbl Pbl (T - Ta j+ qv . (2.29)

ot OS OS

This modes reduces to Pennes' equation (2.4) provided that

Mb =-Pb-°(nnr0u). (2.30)

OS

The model presented and the Pennes' model are conceptually different. Equation (2.30) should be understood as a certain idealisation applied at a scale that is not too small. Otherwise, when individual capillary branches are considered, the flow value along the major vessel becomes discontinuous and therefore Eq. (2.30) becomes useless.

Baish et al. [3] performed comparison of the above models. They concluded that in limiting cases, with respect to the vessel thermal equilibration length, the vascular models approach the predictions of continuum models and the conclusions can be made about the conditions, under which such models might apply. When the tissue contains large number of small vessels, that are nearly in thermal equilibrium with surrounding tissue, then the directed perfusion or effective conductivity models would apply.

The linear heat-sink behaviour, such as found in Pennes' equation, exists only in large-small-large vascular model. This model however implies the existence of large vessels that are not equilibrated with tissue and which are the source of the local temperature nonuniformities, that escape the Pennes' model completely. Magnitude of these nonuniformities can be comparable to the temperature elevation predicted by the heat source, see also [108].

The vascular heat transfer models presented above are meant to be used on the complex geometry of the vascular system. This is a formidable task and identification of the thermally significant vessels, in order to simplify it, is fully justified. Since the main drawback of the continuum models is that they can not predict local temperature irregularities caused by the presence of large, relatively sparsely distributed vessels, which are easy to identify, it is reasonable to construct mixed models. Such models account for each large vessel individually and all 'small' vessels are modelled via the heat-sink-type term or via the effective conductivity. The important question is which vessels are small enough.

In Fig. 6 the possible temperature of blood element as it passes through the vascular system is depicted. The two dashed lines denote the temperatures of the two kinds of the solid tissue that the blood element can encounter during its transport - namely cooler and warmer than Ta (arterial temperature). As investigations of Chen and Holmes [18] indicate and as can be inferred from Fig. 6, the equilibration of the blood with the solid tissue takes place between the terminal arterial branches and the precapillary arterioles, not in the capillaries, as it was usually previously assumed.

This hypothesis is fully supported by experimental investigations by Lemons et al. [49] and Weinbaum et al. [101] who measured the temperature field in rabbit thigh in-vivo in order to identify the thermally significant vessels. Their measurements indicate that all arteries of diameter da<100 ?m and all veins dv<400 ?m can be considered fully equilibrated with surrounding tissue in normothermic conditions. This needs not to be true during hyperthermic or cryogenic treatment.

As can be seen from the above considerations large vessels require different modelling

techniques than capillary vasculature. Following this observation and the extensive anatomical study of the surface tissue Weinbaum et al. [101] proposed a three-layer model of microcirculation contributing to the heat transfer in soft tissue, cf. [45]. The model contains layers depicted in Fig. 7:

• Deep tissue layer. The thick region of thermally significant large veins and arteries. For four or more generations they proceed as closely juxtaposed branching countercurrent artery-vein pairs. The cross-sectional surface area of these vessels is large and their characteristic thermal equilibration lengths is large compared to their length (see Table3). In this layer the thermal state is characterized by three different temperatures: of arterial blood, venous blood and of local tissue. The countercurrent exchange between paired vessels is the dominant mode of the heat transfer here. These large vessels during their final generations gradually transform to a periodically arranged array of terminal vessels at the bottom of the next layer.

• Intermediate layer. Blood is assumed to equilibrate almost immediately with the tissue when it enters this layer. The thermal state is therefore characterized by local tissue temperature only, so individual vessels are no longer considered. Horizontal temperature gradients may exist due to temperature nonunuformities in the preceding layer. Terminal vessels are assumed to be regularly spaced.

• Skin. This layer, also termed the cutaneous layer, contains large vessels that act as a volumetric heat source. They are far from the thermal equilibrium with the surrounding tissue. The dominant heat transfer mode is the conduction in the direction parallel to the surface.

This three-layer model of the surface tissue heat transfer is described at length by Jiji et al. [45] and Weinbaum et al. [101].

Another three-dimensional, three-layer model of surface tissue heat transfer is presented by Brinck and Werner in [13]. The examples of vascular geometry served for the numerical calculations in resting, exercise and cold states distinguished by different metabolic heating and environmental conditions. The blood temperatures along the arterial and venous vessels and appropriate tissue temperature fields were calculated. The authors conclude that such vessel-by-vessel approach is limited to small volumes of tissue due to lack of knowledge of the detailed, individual vascular geometry and the large size of resulting computational tasks.

This difficulty has been partly resolved by the model proposed by Baish [2]. The algorithm of vascular growth has been proposed to generate the detailed vascular geometry. The physiologically justified Gottlieb procedure was chosen. This method assumes the following process of growth, cf. [2] and references therein:

1. Begin with sparse tree containing only the supply vessel.

2. Assume grid of cells.

3. Check the distance from each cell to the existing vascular tree.

4. If the cell is further than some threshold distance then new vessel is added between the cell and the nearest point on the existing vascular tree.

5. Decrease the threshold distance, refine the cell grid and repeat the procedure starting from 3 until desired density of the vasculature is reached.

The radius of the blood vessels can be obtained from the modified Murray's law

rn =Z rI, (2.31)

J

where index J denotes daughter vessels of vessel i and the value of n is typically 2.7. The mass flow is usually related to vessel radius by means of the equation:

stem cement bone

Fig. 8. Intramedullary cemented part of femoral prosthesis, the respective domains are labelled with Q and domain boundaries with r supplemented with obvious subscripts.

Fig. 9. Cemented acetabular part of femoral prosthesis.

n

mm r

Mj rj

(2.32)

The above rules allow one to construct the entire vascular tree starting from few supplying vessels. The generation encompasses all sizes of vessels, so no arbitrary differentiation into "thermally significant' and "thermally insignificant' is necessary. Despite the simplicity of the generating algorithm and its numerous flaws the results resemble the real vasculature in many respects. Since they are not any real but 'virtual' vasculature the statistical interpretation of the results of associated model of heat transfer is proposed [2].

3. Thermal problems related to joint replacement

Thermal problems arising during and after joint replacement can be divided into three distinct categories: issues concerning heating up of the limb-implant system by the latent heat of the bone cement polymerization (in cemented implantation), problems of frictional heating, occurring during the normal functioning of the prosthesis and issues concerning rapid heating up of the bone due to sawing and drilling the bone during orthopaedic operations. The characteristics of those three kinds of problems are outlined in Table 4.

3.1 Heat generation during bone cement polymerization - hip endoprosthesis

The physical situation is sketched in Fig. 8 (intramedullary component of cemented hip prosthesis) and in Fig. 9 (cemented acetabular part). The stem of the endoprosthesis is fixated in the medullar cavity of the femur with the aid of polymethylomethacrylate bone cement which is inserted by the surgeon in a "doughy" state and then polymerizes in situ. The

same pertains to the acetabular cup.

When modelling the heat transfer in the cemented bone-implant system the following simplifications are usually imposed:

1. Axial symmetry of the model along the axis of the stem, see [56], [88], [90]. In [43] a two-dimensional, axisymmetric model was considered.

2. Uniform heat generation by the bone cement with constant power, non-zero only in a specified period of time.

3. Uniform heat generation by the bone cement with varying power in a specified period of time, see [43] and [90].

4. Perfect thermal contact between the materials [90]. In [56] several areas of null contact are considered on an otherwise perfect thermal interface.

5. Material properties independent of temperature, see [43], [56], [88], [90].

6. Material isotropy, see [43], [56], [88], [90].

7. Boundary conditions constant and independent of temperature, see [43], [56], [88], [90].

8. Excluding the surrounding muscle tissue from the model, see [43], [56], [88], [90].

These simplifications originate mainly from the lack of knowledge that would be required to deal with more sophisticated models. For example, the thermal conductivity of the bone tissue as reported in numerous papers on the subject (see e.g. [11], [22], [43], [56]) varies from 0.26 to 0.60 (W/m-K) and the specific heat varies from 1150 to 2370 (J/kg-K), etc. Generally, the bone tissue material parameters depend on multiple factors, mainly the bone composition (cortical/spongeous) and bone marrow and water content. These are different along the bone and for different individuals.

One way to deal with this difficulty is to construct mathematically tractable model of heat transfer in bone tissue on the basis of anatomical observation, see [36] for the description of bone structure. While being more realistic this approach presents considerable difficulties, arising from the considerable scatter of measurements of bone properties and the fact, that the very architecture of the bone changes with time [36].

Other possible way of approaching the problem is to use experimental results to obtain the distribution of values of average material properties, needed to solve the problem within the framework of the classical heat conduction model, cf. e.g. [11]. While less prospective, this approach allows immediate construction of simple theoretical models. Such an approach was used in all papers mentioned.

Another issue is the model of cement polymerization. At least three different approaches can be envisaged. The first approach is to refrain from modelling of the process of polymerization and to assume a constant power produced, within the cement domain, during the specified period of time. The advantage of this idealisation is that only the linear equation of heat conduction with constant source term has to be considered. The drawback is that the model is rather oversimplified in comparison with real situation. Furthermore, one can gain no information about the monomer leftover, within the bone, as no calculation of polymerization is performed.

The second approach is to assume uniform heat generation within the cement domain, with rate varying with time. This way also gives no information about the distribution of final monomer leftover but an attempt is made to make the model resemble the real situation more closely. The properties of cement such as retardation time can be modeled here. Still only linear equation of heat conduction needs to be solved. Such an approach was followed by Huiskes in [43] and by Swenson et al. in [90].

The third approach is to model the polymerization process of bone cement by means of an appropriate kinetic equation. The polymerization of monomethylometacrylate is a free-radical polymerization, cf. [34], [66], so it is a first-order reaction with respect to monomer. The rate of polymerization is linearly proportional to the concentration of monomer, cf. [66].

Additional effect, that should be taken into account in constructing the kinetic equation, is the glass transition of the cement. The kinetic equation is coupled with the heat conduction equation via the heat generation term, which makes the problem non-linear. More detailed description of this model will be presented below. Such an approach was proposed by Mazzullo et al. in [56].

The models presented in all mentioned papers excluded surrounding soft tissues from considerations. Extending them to cover the heat exchange in the muscles would be an important step in constructing the mathematical model of heat exchange in human limb or extremity cf. [83], [107], [114].

Below the model of heat transfer during cemented prosthesis fixation is presented and simplifications commonly made are discussed.

3.1.1 Formulation of the polymerization problem

The temperature throughout the considered domain Q is an unknown function. In the models, that exclude surrounding soft tissue, Q is essentially reduced to the sum of geometrical domains of prosthesis stem, cement and bone domains, when intramedullar part is considered, or prosthesis head, acetabular cup, cement and pelvis fragment domains, when acetabular part is studied.

The starting point in construction of a mathematical model is the energy balance equation, cf. [14], [104]

CT

PC^T + Vq = qv, (3.1)

CT

where t denotes time, q is a heat flux vector, qv is a volumetric heat production rate and the

other symbols have been defined in previous sections. Taking into an account the constitutive equation (Fourier's law):

q = -AVT, (3.2)

we obtain the Fourier-Kirchhoff equation:

CT

pc— = V-(AVT) + qv . (3.3)

CT

The parabolic model of heat conduction (Fourier's law) has been chosen here because the considered physical problem is assumed not to justify introduction of more sophisticated models. Hyperbolic models may be considered (e.g. Vernotte law) when the rate of heat flux is very large which is not the case here.

Equation (3.3) is satisfied separately in each stem, cement and bone domain (Q s, Qc

and Qb).

CT

pscs — = V-(XSVT), x e Qs,

CT

CT

PbCb — = V-(AbVT), xeQb, (3.4)

CT

CT

PcCc^ = V-(XCVT), xeQc. CT

Here the subscripts s , b and c denote the prosthesis stem, bone and cement properties respectively. In addition the boundary conditions need to be stated. The boundary of the considered domain is denoted r = c(Qs u Qc u Qb) where bar denotes closure of the domain. Only Fourier boundary condition (also called Newton boundary condition) is considered:

0T

X— = «(Tœ -T), Xer. on

(3.5)

Here T^ and a are functions prescribed on the boundary r representing ambient temperature and convection film coefficient respectively.

To complete the formulation of the problem thermal contact conditions have to be specified. These are the continuity of the heat fluxes across the contact surfaces. The temperature is discontinuous across these surfaces with the jump proportional to the normal heat flux across the interface. The interfaces are denoted by rbc = Qb nflc (the interface

between bone and cement domains), rcs = Qc nfis (the interface between cement and stem domains). The constitutive equation on the bone-cement interface is given by:

0T6 0T

-= = ßbcT> X e rbc .

onb onc

(3.6)

Assuming perfect contact between the bone and the cement means (bc ^ œ in this notation.

In Table 5 the values of the material coefficients collected from the literature ([8], [11], [22], [43], [56], [90] and [111]) are provided. It should be noted that these values correspond to a certain type of prosthesis considered here (metal stem, polyethylene distal plug, cf. [56]). As is well known there is a wide variety of materials used for prosthetic heads, acetabula, stems, etc.

In Table 6 possible values of interface conductivities are provided, cf. [43], [56]. Note that ambient temperatures corresponding to different interfaces may be different (ambient-bone interface means muscle-bone whereas ambient-stem interface is air-stem). The values for thermal contact conductivity between polyethylene acetabular cup and cement and articulating surface of prosthesis head are estimated by Huiskes in [43] to be of the order of 500 (W/m2K). One may suspect higher value at the latter interface due to the presence of the synovial fluid. In [88] different values of the thermal contact conductivity ( were used, namely in the range 102-106 (W/m2K) and the study revealed no significant change in the process, when the values exceeding (=2000 (W/m2K) were used. So, according to the estimations provided in Table 6, the determination of the value of the thermal contact conductivity of the bone-cement interface may prove important.

Table 5. Values of material coefficients for the model

Prosthesis stem (metal) Femur (bone) Bone cement Polyethylene

Thermal conductivity Ä (W/mK) 14 0.26-0.60 0.17-0.21 0.29-0.45

Specific heat c (J/kgK) 460 1260-2370 1460-1700 2220

Density p (kg/m3) 7800 1000-2900 1100 960

Table 6. Values of interface conductivities/convection film coefficients,

the values are in (W/m2-K), after [43].

Cement Ambient

Metal stem 1000 -10000 50 -100

Bone 100 -1000 500 -10000

Fig. 10. Volume-averaged final degree of polymerization dependence on the thermal contact conductivity (W/m2K) taken uniform and the same for all the interfaces. Details of the model are given in [88].

Fig. 11. Polymerization rate model - f (o, T)

dr

Fig. 12. Polymerization rate versus polymerization ratio for different temperatures (isothermal process).

Table 7. Polymerization constants used in Eqs. (3.8) - (3.10), after Mazzullo et al. [56].

Q, (J/kg) a, (1/s) Ea, (J/mol) a Tg, (K B, (J/mol-K)

193-103 2.6397108 62866 9.2 378 8.3143

The values of interface conductivities have a significant influence both on temperature distribution and on the monomer leftover. In Fig. 10 the simulated dependence between the thermal conductivity and the overall volume-averaged final degree of polymerization is given. Details of the numerically simulated model are given in [88].

One lacking parameter in the set of equations (3.4) is the heat generation term in the domain Qc denoted qv. This term is directly proportional to the rate of polymerization, which in turn is modeled via equation of polymerization kinetics, cf. Mazullo et al. [56]:

q, = 8 dr. (37)

da dx

= aexp

dx

( E \

--^ P(T,r),

BT J

(3.8)

where Ea, a are the experimental parameters and the function P(T,a) is a factor that allows to take into account the glass transition of the cement (phase change):

P(T ,r) =

a

-a

a (T) 0, if a > a * (T)

1-1/a (a *(T) - a J+Va, if a < a * (T),

(3.9)

where a is an empirical parameter and a (T) is the equilibrium degree of polymerization at given temperature:

a (T) =

T ,ifT * Tg,

g

(3.10)

1, if T > Tg

g

Here Tg denotes the glass transition temperature of the cement. Mazzullo et al. [56]

completed their model by supplying the required parameters for the commercial cement (Howmedica SIMPLEX P), which are listed in Table 7 (B is the universal gas constant).

The dependence da = f (a,T) is shown in Fig. 11. dx

This model allows one to obtain the polymerization rate as a function of two parameters: the temperature and instantaneous degree of polymerization (monomer conversion). The polymerization rate-ratio curves calculated for various different temperatures are shown in Fig. 12.

These curves can be thought of as rates of isothermal polymerization of cement for different temperatures. It should be remarked, however, that given the same initial conditions the rate of isothermal polymerization is always lower than that, which could occur in any real situation (unless some enormously effective cooling measures are taken). Also the monomer leftover would be the highest in isothermal conditions. These conclusions are a straightforward consequence of the polymerization kinetics equations (3.8)-(3.10).

The other extremal variant with the fastest possible polymerization and the lowest monomer leftover is the model of adiabatic polymerization. Polymerization faster than in adiabatic case would require external heating-up of the cement (which is not considered here). Such a model is constructed by considering the simple point-mass model described by two variables: degree of polymerization a and temperature T. The heat exchange is null. The

model consists of simplified equations for the temperature and the polymerization:

dT _ da

c— = Q —,

dr dr

da dr

( E ^ = a exp( - bT J p(T ,a).

(3.11)

Solution of Eq. (3.11) leads to the linear dependence of temperature on the degree of polymerization a :

T = Q (a- ao ) + To,

c

(3.12)

and the ordinary differential equation for the degree of polymerization, provided that

a < a*(T):

f \

da dr

= aexp

E„

B^ Q (a- ao) + To j

f

acTg

Q(a- ao ) + To

n a-1 (

a

o j

q(a- ao )+ c(to - atg ^

cT

a+1 a

(3.13)

The numerical solution of this equation is depicted in Fig. 13.

Thus, for the adiabatic polymerization to be complete (a = 1) it is sufficient that initial condition satisfies:

Q(1 - ao )-c(Tg - T() )= o. (3.14)

If ao is higher than specified by Eq. (3.14) or To is lower, the polymerization will be incomplete.

In physical situation heat exchange occurs and temperature gradients exist. The polymerization time, the monomer leftover and attained temperatures will be therefore between those characteristic for the adiabatic and isothermal processes.

The complete model of polymerization allows us to find computational solutions of both coupled scalar fields of temperature and polymerization ratio. The specific solution (maximum temperatures at appropriate domains plotted versus time) is depicted in Fig. 14.

It should be noted that two coupled field equations: (3.3) and (3.8) are of different type, namely the Fourier-Kirchhoff equation is a partial differential equation for the unknown temperature T, whereas the polymerization kinetics equation is an ordinary differential equation with respect to the function a . This difference stems from the fact that monomer diffusion phenomena was tacitly omitted.

3.1.2 Exothermic polymerization in situ - possible consequences and remedies

The ultimate goal of calculation of temperature/monomer leftover distribution throughout the implant is to gain knowledge if the chosen prosthesis fixation technique presents considerable danger to the living tissue and to assess the probability of possible loosening of the prosthesis as an effect of resulting damage. For this one needs criteria of thermal/chemical damage of the bone tissue (bone necrosis) that one can compare with computed values of the temperature and the monomer leftover.

The issue of thermal bone necrosis has been investigated by many researchers. The results suggest two basic mechanisms. One is the collagen protein denaturation. According to Swenson et al. [9o] it takes place at temperature range 56-7o?C. The second mechanism is caused by cellular death, which occurs at lower temperatures and is therefore more important.

a

140

130 120 11 0 u 100

CD

3 90

03

aj

80

£

I 70 E

ro 60

S

50 40

0 50 100 150 200 250

Time (s)

Fig. 14. Maximum temperatures in bone and cement domain. Details of the model are given in [88].

Table 8. Experimental constants for thermal bone necrosis criteria appearing in Eq. (3.15), after Mazzullo et al. [56].

M, (s) d, (J/mol) Teff , (K)

1/27.4 1000 310

The results presented in [8], [7], [10], [43], [56], [90] and [92] point out the time-temperature dependence inherent in thermal necrosis criteria. For example, the temperature of 70?C is believed to kill cells instantly, 50?C needs to be maintained for 30 seconds and 45 ?C - for 5 hours. Higher temperatures (of order 70?C) are needed to destroy the regenerative capacity of the bone tissue, cf. [43], [90].

From the available data a straightforward mathematical criteria, based on the 'additivity rule' has been constructed by Mazzullo et al. in [56]. Assume the time necessary to

cause thermal bone necrosis at a given temperature T is

rc = M exp

f \

T > Tf. (3.15)

B(T - Tref )j

In given non-isothermal conditions the local measure of thermal bone tissue damage ] can be constructed as an integral of fractions of exposure time at given temperatures over time

r dt

](X,r) = i (T( t)) . (316)

o rc (T(X, t))

The values of r equal to or in excess of unity indicate local bone necrosis. This criterion is analogous to Palmgren-Miner hypothesis of linear damage accumulation in fatigue mechanics and has similar drawbacks. For example it does not take into account the succesion of the different stages of thermal load - the various intensity heating periods will produce the same 'damage', irrespective of their relative order.

The necessary constants for the model were obtained by Mazzullo et al. in [56] by means of linear regression analysis and are given in Table 8.

While it seems to be a good measure for a single heat shock cellular damage, this criterion may be insufficient when dealing with repeated thermal bone loading as the living tissue can probably adapt to higher temperatures by means of producing 'heat shock' proteins. It is not known whether they also exist in normal human joint, but this is feasible as natural joints heat up by ca 2.5?C during walking and probably more during more intensive activities, cf. [94].

Nowadays researchers tend to believe that heat-induced bone necrosis is the secondary mechanism of tissue damage during implantation of cemented endoprostheses. The toxic leftovers of the polymerization process are considered to be a more challenging problem. The negative influence of residual monomer and free radicals released from the cement dough is substantially prolonged when compared with pure high temperature damage. Willert et al. cit. by Swenson et al. [9o] reports 3 mm necrotic zone in 3 weeks postoperative specimens.

It seems therefore important to model the polymerization process as well as temperature distribution and to develop criteria similar to (3.16) for assessing the tissue damage due to polymerization leftovers.

The eventual effect of bone necrosis is bone resorption at the bone-cement interface. The mantle of fibrous connective tissue is developed and the mechanical load carrying capacity of the interface is seriously compromised, which usually leads to implant loosening and the need for reoperation.

Different measures are taken or considered to remedy this problem. These are:

1. implanting cementless prostheses,

2. development of new low temperature and bioactive bone cements, cf. [79], [8o], [99] and the discussion below,

3. water cooling (acetabular parts of total hip implants, cf. [11o]),

4. shielding layers, cf. [43], [56],

5. lowering thermal contact between cement mantle and the bone which can also have a beneficial effect of lowering monomer leftover, cf. [88],

6. precooling or preheating of the prosthesis stem (possibly the bone and cement mixture),

7. using as little cement as possible.

The most straightforward method is to refrain from using cemented implants at all, in favor of cementless ones. There are, however, extensive clinical data from postoperative follow-ups indicating that percentage of failures, marked by the necessity of revision, is

significantly lower in the case of cemented prostheses, see [65] and the references cited therein.

The most promising method is to develop new cements that polymerize at low temperatures leaving no monomer leftover. Another advantage of lowering the maximum temperature in the system (not necessarily bone temperature) is that some heat-labile cement components (e.g. antibiotics) are not deactivated during the cement setting and also the potential problem of MMA boiling and cement porosity induced in this manner is avoided. The task of lowering polymerization temperature can be in part accomplished by means of adding 'heat sink' additives to the PMMA powder and/or changing initial P/L ratio of cement mixture. Unfavourable outcome of these actions is the lower mechanical strength of the cement since its porosity increases. These issues are discussed in detail in [43].

Precooling of the prosthesis stem is proved to be ineffective, cf. [10], [90], [95]; furthermore it was shown by Bishop et al. [10] that the low stem temperature substantially deteriorates cement-stem interface quality and that preheating should be used instead of precooling. Precooling of the cement mixture also prolongs the setting period of the cement. In that period any motion of the installed prosthesis usually causes fixation failure. Implant has to be extracted from the femur, the cement has to be removed and new implant installed. Water cooling is possible only in acetabular components fixation and is reported to have a positive effect on the bone temperature [110].

The possibility of lowering the thermal contact between cement and bone was also investigated, cf. [43], [56]. Mazzullo et al. [56] showed that introducing thin layer of rubberlike material would have a beneficial effect on both the conversion of the monomer and temperatures of the bone. Such a barrier could also protect the tissue against the diffusion of toxic substances from the cement dough. Unfortunately, in view of the efforts to create the best possible environment for the bone tissue to grow into and 'interlock' with the cement such a solution should be considered impractical.

3.2 Bone drilling and sawing during orthopaedic operations

As outlined in Table 4, temperature increase during orthopaedic operations is of short duration; however considerable temperature values may be reached. Two processes may lead to frictional bone heat-up during operation: sawing (e.g. during bone preparation for endoprosthesis implantation) and drilling (e.g. during preparation for screw fixation). The resulting thermal bone necrosis at the screw site leads to instability and consequently invalidates any benefits from any stabilizing devices fixed to the bone. The correct choice of drilling/sawing parameters is therefore of importance. The parameters under consideration are:

1. rate of rotation of the drill, speed of the saw,

2. force on the drill or saw in the direction of drilling/sawing,

3. the kind of tool, level of wear,

4. predrilling,

5. cooling (irrigation) parameters.

The effectiveness of above measures was measured experimentally, see [47], [54], [96]. The outcome can be summarized as follows:

• The rotational speed of the drill is reported to have no marked influence on the spatial temperature distribution in bone but it affects the duration of the exposure to high temperature - higher drill speeds produced high temperature for shorter periods [54]. Similar measurements made by Krause et al. [47] for high-speed (20,000 rpm and 100,000 rpm) cutting burs confirmed that that there is no general correlation between rotational speed of the bur and bone temperature, tendencies for different kinds of burs being different.

• The increase in force measured in the direction of drilling results in significant decrease of temperature in the vicinity of drill. Matthews and Hirsch [54] report approximately 15?C decrease (from ca 82?C) in location o.5 mm from the drill when force is changed from 2 to 6 kG and further 17?C decrease with force changed to 12 kG, see Fig. 15.

Similar effect was reported by Krause et al. [47] for high-speed cutting burs - higher feed rates, and therefore cutting forces, cause lower temperature elevations. Strength of this tendency varied with kind of bur used.

This effect is much more pronounced when duration of exposure to temperatures over 5o?C is compared. For forces 2, 6 and 12 kG these durations are 35.8 and ca 1 second respectively at the mentioned location. This effect is reported not to occur for dental burs and smooth pins used as drilling tools (as opposed to twist drills). Those have no means of eliminating the bone debris and therefore milling occurs rather than cutting and furthermore the debris gets compacted between the tool and the hole walls thus greatly increasing friction.

• Study of the influence of drill wear conducted in [54] showed that worn drills (used to drill 2oo holes before) could produce temperatures over 2oK higher in the immediate neighborhood than new ones. Also time of exposure was significantly prolonged. As can be inferred from this one and other experimental investigations the shape of the tool has a marked influence on bone temperatures during drilling/cutting [47], [96]. Figure 16 presents a comparison of temperatures measured by Krause et al. [47] for two different saw blades.

• The investigations of the effect of predrilling conducted by Matthews and Hirsch [54] showed temperatures during predrilling (drill diameter 2.2 mm) and subsequent enlarging of the hole (diameter 3.2 mm) to be virtually the same and, at the distance of o.5 mm from the drill, ca 6oK lower than measured in control drilling (3.2 mm) (where temperatures exceeded 1oo?C). This result can be viewed as the proof of beneficial effect of predrilling or as the rough measurement of the influence of drill diameter on maximum temperatures attained, which turns out to be very high.

• Cooling of the drill and surrounding bone by means of irrigation with water is reported to lower the temperature of the bone substantially. Matthews and Hirsch [54] found that for their experimental setup and the diameter of drill they used (32 mm) no significant advantage was gained when raising the coolant flow above 5oo ml/minute. The coolant they used was water at room temperature. Using precooled water would probably allow less coolant to be used. The results of measurements conducted by Krause et al. [47] for different kinds of burs and reciprocating saws confirmed conclusion that cooling may significantly reduce the bone and tool temperature. Studies by Toksvig-Larsen et al. [96] with the prototype oscillating-blade saw showed that with adequate coolant flow the temperature elevations are negligible and well below the values usually associated with bone necrosis.

As can be seen from the above considerations, appropriate choice of tools and cooling methods permits to avoid the danger of thermal bone damage during preparation for orthopaedic operation entirely.

3.2.1 Formulation of the problem

The problem of bone heating-up during sawing can mathematically be stated in the first approximation as finding the temperature distribution in the infinite solid with moving line heat source. For steady conditions the solution is given in [14] as

o

cd

80

70

9- 60

<D

50

40

30

- i i 2 kG force —i— 6 kG force —x — 12 kG force - - * - -

- X \ \ \

\ \ \ \

** - -..

i 1 1 i i

3 0.5 1 1.5 2 mm from drill 2.5 3

Fig. 15. Temperatures recorded in bone for different axial forces during drilling, after [54]. 250n

200-

150-

O

o

cd

s_

03

L_

cd

CL

e

cd

T3 cd

| 100 cd

50-

Blade temperature Bone temperature

0

20 40 60 80 100 Time-sec

Fig. 16. Recorded temperatures for two saw blades, after [47].

T (x, z ) = To +

qi

exp

/ \ Ux

v 2ab J

K

U^x 2 + z 2 2ah

(3.17)

It is assumed here that infinite line source located at y-axis moves with the velocity U (sawing feed rate) in direction of x-axis. K0 is the modified Bessel function of the second

kind of order zero. Xb and ab are the thermal conductivity and thermal diffusivity of the bone

respectively; ql is the rate of heat generation expressed in W/m.

The closed-from solution (3.17) can be obtained thanks to the simplicity of the model. The assumptions not reflecting the physical situation are:

• Treating the bone specimen as an infinite solid. That assumption should result in

underestimating the temperatures as the heat is allowed to escape the real geometrical domain of the bone without any boundary resistance. This is thought to have no significant influence as long as the sawing process is fast.

• Treating the saw as one-dimensional entity. This is reasonable for steady-state situations (deep cut), when the temperature of the saw does not change anymore. The heat flow through blade in its direction and the blade heat capacity are thus neglected.

In such a formulation the crucial parameter qt needs to be obtained experimentally by means of temperature or calorimetric measurements.

Alternatively to this formulation, Rosenthal (1946), cit. by Krause et al. [47] proposed solution for an infinite solid with moving plane heat source instead of line source. The problem of bone drilling could be analogously modeled by infinite solid with moving point heat source. Again, the reported dependence of heat generated on feed rate (velocity U) escapes modelling and needs to be supplied. Furthermore, the diameter of the drill is also not a parameter, whereas experimental data strongly suggest its importance. Such a model should be therefore considered too simplified.

To summarize, we conclude that the problem of bone heat-up during drilling or sawing is nowadays not crucial since modern cutting tools are designed in such a way as to minimize or even completely eliminate the risk of thermal damage to tissue, cf. e.g. [96].

3.3 Frictional heat generation in joints

Friction occurs in all types of joints, both in natural and in artificial. The heat generation and dissipation is therefore a process, that takes place every time the joint is used. In normal human hip joints the measured temperature elevation is of the order of +2.5?C during walking and probably more during running, cf. [94].

The artificial joints are less efficient and one can expect that temperatures attained in such a joint can be higher. Bergmann et al. [6] reported a 3.5?C temperature increase in the case of titanium alloy hip prosthesis after 45 min. of normal walking. Temperature was measured inside the neck of the prosthesis. Value at the bearing surface can be considerably greater, see [31], [5o] for the finite elements estimation of this temperature for different articulating pairs.

Investigation of the problem of temperature distribution in an artificial joint, when the joint is being used is done for two specific reasons:

1. To establish if thermal damage in the tissue can take place. Such a damage may lead to prosthesis, usually hip acetabular cup or knee prosthesis, loosening via the mechanism outlined in Sec. 3. The issue of criteria for thermal damage is more difficult here than in the case of bone cutting or cement polymerization because the thermal loading is now cyclic and this may result in 'thermal bone damage accumulation'. On the other hand it is suggested that bone tissue can develop 'heat shock' proteins and adapt to elevated temperatures, cf. [15], [59]

2. To assess the working temperatures and their influence on wear and creep rate of materials commonly used for articulating surfaces. In the case of polyethelene (PE, UHMWPE) implants the three most important factors contributing to failure are: excessive stress (which may be magnified by the presence of the residual stresses, cf. [6o]), material oxidation due to y -irradiation during sterilization and thermal damage. The latter is vital to long-term prosthesis performance. According to Young et al. [111] the Arrhenius-type relation can be applied to asses the time to failure t of polyethylene at elevated temperature T:

log A = ^CL

tR 2.3B

'1 1 ^

VT TR J

where the subscript R denotes design values, B is a universal gas constant and Eact is the material-dependent activation energy. When the equation is evaluated

for values of constants appropriate for PE it appears that prosthesis service life is shortened by half when the temperature increases by 2K over the design value.

Frictional heat generation has been investigated by various researchers in three distinct

ways:

3. In vivo measurements in patients. In [7] the temperature distribution and forces acting on the head of the implant were measured by means of instrumented endoprosthesis, see also [37] for technical details.

4. In vitro measurements of fictional torque, temperatures and material wear on a laboratory set-up over a prescribed range of joint motions. Such measurements were made by Davidson et al. [26] and [27] and by Lu and McKellop [5o]. Influence of the choice of material for articulating surfaces and a number of other parameters was studied. Some of the results are reviewed below.

In [26] the authors describe so called 'simulated in vivo' test where movement of the prosthesis is replaced by resistance heater embedded in the prosthetic head and the system is assembled in such a way as to resemble the real situation as much as possible (the joint capsule is simulated by pieces of bovine muscle tissue), therefore reproducing occuring in vivo mechanisms of heat transport. However the effect of vascularity and blood flow is not reproduced in such a setup.

5. Computer simulation. Unlike the case of bone cement polymerization the stationary temperature field is usually considered here, see [8]. This corresponds to the thermal equilibrium of the system when heat generation rate by friction equals the heat dissipation rate. This happens approximately after 1 hour of continuous walking (the half of total temperature increase taking place in first 6 min.) according to in vivo measurements by Bergmann et al. [7]. Davidson et al. [27] reported half of this equilibration time for tests in vitro. Lu and McKellop [5o] indicated much longer times (of the order of several hours) which is probably caused by different experimental protocol (much larger quantities of lubricant and lower forces in the joint). The greatest difficulty of computer simulations seems to lay in the fact that one needs to choose the parameters for the model correctly and to do this one needs to resort to some experimental data. The heat generation rate at the surface of contact of articulating components and thermal properties of tissues and synovial fluid must be obtained from experiment and, as was already shown in Sec. 3, experimental results show considerable scatter. Furthermore, if ordinary only Fourier-Kirchhoff equation is used, the effect of vascularity is neglected or can be only roughly approximated.

Below some of the important results reported in the literature are reviewed.

3.3.1 Factors influencing frictional heat generation

3.3.1.1 Materials

Material of articulating surfaces has a marked influence on the heat production and rate of wear. In [26] and [27] the authors provide an extensive study of these issues. Three kinds of articulating pairs are taken into consideration (mentioned here in stem-acetabular cup order):

200 300 400

Cycles

Fig. 18. Frictional torque in lubricated and dry conditions. Peak hip load=5000 N, after [27].

1. Co-Cr-Mo steel on UHMWPE (Ultra High Molecular Weight PolyEthylene),

2. alumina (A2O3) on UHMWPE,

3. alumina on alumina.

Some of the results of in-vitro tests are displayed in Fig. 17, after [26]. Specialized test setup was used to produce rocking motion with variable hip loading. The load-time history was selected to reflect natural hip loading during walking. The results cover experiments conducted with two values of maximum force applied to femoral head, namely 2500 N and 5000 N.

The results presented in [50] show superior performance of alumina-UHMWPE, when compared to metal-UHMWPE pair in vitro.

Bergmann et al. [7] performed experiments which showed the superior performance of alumina-alumina articulation in vivo. The differences in heat production for various pairs come from different friction coefficients and can greatly influence wear performance of artificial joint. More detailed information about the rate of wear for different kinds of surfaces can be found in [27] and [31].

3.3.1.2 Lubrication

Another factor important for frictional heat generation is the lubrication. In natural joints it is accomplished by means of synovial fluid, liquid crystalline, biological substance, see e.g. [91].

The available volume of synovial fluid in the joint capsule is less than 2 ml. It is a yellow, clear and highly viscous liquid. It forms a film layer of thickness varying on the type of joint and location within it, in range from 6 ?m to 1mm. The synovial fluid in a healthy joint consists of 94% of water and 2-3% of the hyaluronic acid by weight. Moreover, the synovial fluid contains some macromolecular components like glycoproteins, phospholipids and low molecular compounds, e.g. liquid crystalline cholesterol esters and small ions [91].

The main purpose of the synovial fluid is the lubrication of the joint. Furthermore, it provides the necessary nutrients for the cartillage and protects it from enzyme activity. Properties of the synovial fluid are affected by pathological processes. The shear viscosity coefficient is smaller for synovial fluid from degenerated joints.

In in vitro tests [27] the lubrication was attained by means of water or hyaluronic acid in different concentrations. The hyaluronic acid was chosen since it is the primary lubricant component of the synovial fluid. Additionally, friction in dry conditions as well as in the presence of 2 mg bone cement powder was investigated. No significant difference between water and hyaluronic acid lubrication was reported, while friction in dry conditions was, as expected, substantially greater. In Fig 18 illustrative experimental relation is depicted between frictional torque in joint and lubrication conditions for steel-UHMWPE articulating pair [27]. To obtain these results hip simulator was used to create a rocking motion over a 46? range. The axial load variation was chosen to reflect the walking loads history, see[27] for details.

The frictional torque rises by an order of magnitude when bone cement is present in the joint, even in a small quantity. As results presented in [27] indicate, there is an almost linear relation between frictional torque and equilibrium temperature rise.

A number of other factors influence the temperature distribution in joint. These are:

4. Stride length and step frequency and consequently the flexion-extension angle and angular velocity, cf. [87].

5. Adaptation. Bergmann et al. [7] reported that two patients with low measured temperatures in implanted joints had high body weight but were very active. This effect is assumed to have been caused by physiological adaptation of vascularity to higher temperatures (higher perfusion rates). It is, however, not always present.

6. Possible head-acetabular cup separation during the joint movement. This effect is not present in vivo, where various supporting structures exist to restrain the femur head (fibrous capsule, acetabular labrum, ligament of the head of the femur and the iliofemoral, ischiofemoral, pubofemoral and transverse acetabular ligaments). During the total hip arthroplasty some of those structures may get surgically removed or resected to facilitate surgical exposure. The kinematics of the artificial joint is therefore different from the natural one. Measurements performed by Dennis et al. [29] prove that articulating surfaces separation occurs in vivo. The influence of this effect on joint temperatures is not known but it is suspected to be beneficial [7]. The gap that opens during the separation would be filled with synovial fluid, which would cool the articulating surfaces and the lubricating film would be renewed.

4. Cryogenics

In all previous sections various conditions leading to temperature rise in the tissue were considered. The problems concerning cryogenics (Greek kryos - cold, freezing) arise from substantial temperature drop within the tissue. It is accompanied by heat transfer, phase transition (freezing of the intracellular and extracellular water) and other phenomena often leading to tissue damage. Taking into account the aim of introducing low temperatures into tissue one can distinguish two classes of problems: cryotherapy and cryopreservation. The former consists mainly of controlled exposure of certain cells (e.g. cancer tumors) to subzero

temperatures (cf. e.g. [67]) while the latter deals with problems of conservation of tissues at low temperature (influence of freezer storage on mechanical properties, water content etc., see e.g. [98]).

A topic of fundamental importance in bone mechanics is how accurately the mechanical properties of bone, measured in some postmortem, ex vivo state, represent bone tissue as it exists in the living body. The central questions here are: 1) how do the mechanical properties of bone tissue change when its cells die and/or it is removed from the body? 2) how do various means of preserving bone tissue against postmortem changes alter its in vivo mechanical properties? These questions are manifest in two broad areas of orthopaedic research. First, they are of obvious concern to those experimentalists who wish to understand structure-function relationships in bone and others, who wish to define changes in mechanical properties caused by the genetic variability, aging, disease, medical treatments and so forth. Second, they are of interest to the surgeon in the context of bone allografts. The storing of bone for these purposes raises questions analogous to these encountered by the experimentalists.

In addition, treatment of allografts to prevent immune responses or the transmission of infections may also affect mechanical properties. Martin and Sharkey [53] reviewed a diverse literature concerning these interrelated and important questions. Particularly, the mechanical effects of preserving bone have been discussed. The available literature covers the range of temperatures from -2o?C down to -196?C and the mechanical tests such as compression, bending, pin pullout, torsion, indentation, etc.

Freeze-drying, also known as lyophilization, is accomplished by deep-freezing the bone (e.g. -8o?C), introducing a high vacuum and gradually raising the temperature at the frozen water sublimates but still keeping the temperature well below the freezing. The end point of the process is usually defined as occurring when the residual water content of the specimen is less than 5. The freeze-dried bone may be stored for 4 to 5 years at room temperature as long as it is sealed against the moisture. Martin and Sharkey [53] have also summarized the effects of various treatments relevant to allograft bone on mechanical properties. For instance, several studies have shown that freeze-drying has a more detrimental effect on the mechanical behaviour of cortical bone than does freezing; the effects of freeze-drying on trabecular bone appear to be less severe.

To obtain general solutions (temperature distribution, heat flow rates, etc.) during the cryotherapy or cryopreservation protocol the tissue has to be treated as a nonideal material, whose properties are temperature-dependent and phase transition occurs over a temperature range [5], [3o]. The effects of blood perfusion and metabolic heat generation in unfrozen regions also have to be accounted for (see Sec.2 for appropriate formulations). In Table 9 some selected material properties of tissue in low temperatures are presented along with water properties, see [69].

Table 9. Thermophysical properties related to freezing of _typical soft tissue, angioma and water, see [69].

Soft tissue Angioma Water

Upper limit of phase transition (-1)-0?C (-1)-0?C 0?C

Lower limit of phase transition (-22)-(-8)?C (-22)-(-8)?C

Thermal conductivity in unfrozen state, W/mK 0.5 0.56 0.6

Thermal conductivity in frozen state, W/mK 2 2.22 2.25

Specific heat in the unfrozen state, MJ/m3K 3.6 3.89 4.19

Specific heat in the frozen state, MJ/m3K 1.8 2.01 1.13

Latent heat, MJ/m3 250 250 331.7

0 -5 -10 -15 -20

Temperature (°C)

Fig. 19. Release of the latent heat as a function of temperature during the equilibrium solidification of

three different NaCl water solutions, after [30].

Values of lower limit of phase transition given in Table 9 refer to quasi-static freezing/thawing process. Rabin et al. [75] reported that they may be as low as -45?C in non-equilibrium conditions and when sources of nucleation are absent.

The volumetric specific heat of soft tissue in the temperature range between lower and upper limits of phase transition is usually modeled in terms of the effective property. Experimental data suggests piecewise linear approximation, with two linear functions starting with intrinsic property values at phase change region boundaries, intersecting at the "peak temperature' (-3?C for phase transition in range (-8) - (-1)?C). The slopes of these functions are chosen in such a manner, that the integral of the function representing the effective specific heat over the phase transition temperature range equals transition enthalpy changes, see [69]. The following theoretical justification for this assumption can be made, see [58],

Let us introduce the heat equation with additional freezing/thawing term (compare with Eq. (3.3)):

c(T) = v(l(T)VT) + Lv f, (4.1)

where Lv is the latent heat of water solidification per unit volume, fs is the frozen state fraction (taking value 0 for pure water and 1 for pure ice) and c(T) is the intrinsic specific heat per unit volume (temperature dependent). Assuming that fs is a function of temperature in phase transition temperature range one may write:

L f = L f^^L,

v dr v dT dr

so then the Eq. 4.1 can be reduced to the standard Fourier-Kirchhoff equation by means of introducing the effective specific heat ceff :

Cf (T) = c(T) - Lv f

For purpose of practical modelling the function fs (T) is often assumed linear. This is

only a crude approximation however - in Fig.19 examples of this function for three different water solutions of NaCl are given after [30].

4.1 Cellular damage mechanisms in cryogenics

In both fields of cryotherapy and cryopreservation the problem of tissue damage caused by freezing and exposure to low temperature has also been investigated. In the

cryotherapy the damage is the desired effect of the treatment (destruction of the undesirable tissues, for instance during treatment of cancer) and the prediction of affected area is vitally important for development of successful surgical procedures. In cryopreservation, the damage of the biological tissue is an altogether undesired effect and one tries to minimize it.

One can distinguish two groups of basic mechanisms of cellular damage occurring in cryogenics, those occurring in phase transition temperature range and those taking place after solidification, cf. [68], [7o], [72], [74], [75]. More complex mechanisms such as the post-thaw vascular injury, characterized by stasis, thrombosis and increased vascular permeability (macromolecular leakage) can be viewed merely as the consequences of the basic mechanisms described below. They, however, contribute to the entire injury response and prolong the period needed for healing [42].

4.1.1 Damage mechanisms occurring during freezing

The first group includes destructive processes taking place in phase transition temperature range and related to the dynamics of the freezing/thawing process. Usually accepted view (see [3o], [62], [84], [93] and the references therein) relates cellular damage to the rate of freezing, when the temperature drops below upper freezing limit. The ice nucleation is most intensive in the larger extracellular spaces of single cell systems or the larger vascular/extracellular space of the whole tissues, so that ice forms first in the extracellular fluid, while cells remain unfrozen. As ice forms extracellularly it rejects solutes. To equilibriate the chemical potential between the supercooled interior of the cells and the partially frozen exterior, the water leaves cells through the partially permeable cell membranes and freezes outside. This process increases the intracellular osmolality and causes chemical damage of the cells due to dehydration. The experimental data (see e.g. [62]) prove that the amount of intracellular ice forming in cells ranges from almost none - for cooling rates < 5 K/min to over 88 percent of the original cellular water for cooling at rate of > 5o K/min. This range of values is typical for the regions from the middle to the periphery of the cryosurgical iceball. Therefore, for accurate modelling and simulation of the damage processes occurring during freezing, the problem of water transport between the intracellular and extracellular/vascular space of the tissue is of vital importance.

To predict dehydration experienced by embedded tissue cells due to extracellular ice formation microcompartment models were developed, see [62] and references cited therein. The change in the single cell volume caused by water loss can be modeled by the Krogh cylinder tissue model, [62]

dV_ LPACBT fln V - Vb AHf f ^

dT Gvm

vtr tj , (43)

V - Vb + (Psnsvw B

where V is the cell volume, Lp is the permeability of the plasma membrane to water, Ac is

the original effective membrane surface area for water transport, B is the gas constant (8.314 J/moleK), G is the cooling rate, vC} is the partial molar volume of water, Vb is the

osmotically inactive cell volume, ns is the total number of moles of solutes in the cell, (ps =2

is the dissociation constant for NaCl in water and AHf is the latent heat of fusion for water.

The cell membrane permeability is temperature-dependent and can be modeled via the exponential law:

lp ~ lpg exp

'ELp ( 1 1 Yl

B

TT

. (4-4)

V B VTR T J J

where ELp is the activation energy, TR denotes reference temperature and Lpg is the cellmembrane permeability at that temperature. Krogh model was originally developed for single

Fig. 20. Relationship between cell survival ratio and cooling rate for different kinds of cells, after [55].

application to freezing of rat liver tissue). It can also be adapted to account for the presence of added cryoprotectant (cryoprotective agent - CPA) such as dimethylsulfoxide (DMSO) see [84]. Such cryoprotectants can substantially decrease tissue damage due to dehydration. Action of such protective additives is reviewed in [55].

With increased cooling rate more ice starts to form intracellularly and the damage due to dehydration is reduced as the water transport is reported to cease at ca -10?C [62], [84]. This "shut-off temperature varies slightly with cooling rate (is lower for higher cooling rates) and depends on the presence of CPA. As experimental data indicate [84] increasing DMSO concentration causes significant drop in the 'shut-off temperature (-15?C for cooling rate 5 K/sec and -25?C for cooling rate 50 K/sec at DMSO concentration 2 M). This effect is offset by the fact, that initial dehydration rate is much lower in the presence of CPA and the final cell volume is greater.

When cooling rate is high enough the intracellular ice formation starts to have mechanical effect on the tissue - the second process resulting in damage during freezing. The volume of intracellular ice that the cell can withstand without damage depends on the cell type. Cracks are often reported during rapid freezing (such as using liquid nitrogen) unless the size of the sample is too small (individual cells). It is postulated, that the direct cause of the damage is the excessive stress resulting from volumetric expansion of water contained in the tissue, cf. [82], [72]. Assuming yield stress of frozen tissue equal to 0.005 Rabin et al. have shown that plastic deformation, induced by thermal dilatation during freezing, is likely to occupy the entire frozen region [70]. However, their model assumed freezing at a single temperature and omitted water-to-ice phase transition volumetric expansion which might have been a dominant effect.

When whole tissues are frozen it is argued that vascular expansion, due to cell dehydration, is another destructive mechanism.

The competition of the dehydration and intracellular ice formation in tissue during freezing determines the final volume of the cell and its survival chances. The least damage is reported for some intermediate cooling rates, when cellular volume changes are minimal. Exact value of this "intermediate rate' depends on the type of the cell, its volume to surface area ratio and permeability to water. It should be lower for larger, spherical cells and for those

less permeable, cf. [55]. In Figure 2o survival ratio is given for various types of cells as a function of the cooling rate, after [55]. The maximum survival peak is clearly visible in all presented cases.

In real situation, the cryosurgery usually involves a metal-tipped cryoprobe introduced into tissue. As the freezing front moves away from the probe the cooling rate changes from very fast (in the immediate vicinity of the probe) to very slow (at the periphery of the iceball). Taking into account all the above considerations of damage mechanisms, it should be observed, that the damage distribution after cryosurgical treatment can be quite complex.

4.1.2 Damage mechanisms in solid state

The second group comprises destructive mechanisms occurring after the phase transition is complete, i.e. in the solid state. Damage is caused by mechanical stress at this stage. It was shown that the stress resulting from a constrained contraction of the frozen tissues (termed thermal stress) can easily reach the yield strength of the frozen tissue resulting in plastic deformation or fracture, cf. [75], [72] and the references therein. The contraction is driven by temperature gradients developed during cooling.

The free water in cells forms ice crystals during freezing, so frozen tissue is often expected to have similar characteristics to pure ice during thermal contraction. The uniaxial compression experiments by Rabin et al. [74] indicate that, although the elastic modulus of frozen rabbit liver, kidney and brain is within the range of values reported for sea ice, the maximum strength appears to be one order of magnitude higher than the yield strength of the ice. These large discrepancies may be explained by the contribution of biological fibers to the frozen solid, making it a composite material [74]. It should also be noted, that the maximum strength of the frozen tissue is usually different than the yield strength in uniaxial compression, so the above comparison can be misleading.

Shi et al. [81] modelled the gradual freezing of the tissue (freezing over the temperature range and the thermal expansion during water-ice phase transition). Their research covered also experiments with potato specimens. The authors concluded that most of the ice formation and fracture occurs in the temperature range o-(-2o)?C while most severe fractures occur at high (2oo [K/sec]) rates of cooling. Faster cooling at lower temperatures seemed to have little influence on tissue damage. The step-freezing cryoprotection protocols with an initial low cooling rate are therefore reported to be superior to constant-freezing-rate protocols.

Thermally induced damage analysis in [68] revealed that the highly inhomogeneous structure of the frozen tissue encourages crack formation during loading. It also plays a quite different role. The cracks that would propagate undisturbed in the homogeneous material are frequently halted or diverted. The highly heterogeneous structure of frozen tissue allows substantial degree of cracking before final failure takes place.

When cryoprotectants are used (e.g. dimethylsulphoxide or glycerol) they tend to vitrify the intracellular fluid. We recall that vitrification is an amorphous solidification of liquid caused by enormous increase of viscosity, cf. [55], [75]. Below the glass transition temperature the solution changes state into glass, becomes more brittle and its thermal expansion coefficient drops significantly, see Fig. 22. In attempts to cryopreserve bulky biological systems (e.g. tissues and organs) coexistence of amorphous and crystalline phases often exists, cf. [75].

The experimental and theoretical analysis of freezing/thawing of water solutions (phase change over a temperature range) reveals that most severe fractures occur at the early stages of thawing, not freezing, cf. [72], [73]. This asymmetry is due to the fact that the deviatoric stress on the freezing front is zero while no such a condition is required for the thawing front [71], [72].

120 -100 -80 Temperature°C

Fig. 21. Thermal expansion coefficients (1/K) of various tissues compared to water and copper in

cryogenic temperature range.

Fig. 22. Effect of cryoprotectant (dimethyl sulphoxide -- DMSO or glycerol) on thermal expansion

coefficients (1/K) of soft tissues.

The property of the tissue describing its mechanical behaviour with changing temperature is the thermal expansion coefficient which is defined in the linear thermoelasticity in the following manner, [35], [61].

The constitutive Duhamel-Neumann law is introduced that relates the stress tensor ), i, j = 1,2,3 to the small strain tensor e = (ej ), i, j = 1,2,3 and temperature:

o

= Cmeu - ^ (T - To). (4.5)

Here T0 is the reference temperature, C is the tensor of elastic moduli and p is the tensor of the thermal moduli. In the general case of anisotropic hyperelastic solid the

following symmetries hold true:

Cijkl - Cklij - C jikl ■ ßu - ß,

'J rji ■

In the isotropic case the elastic and thermal moduli can be described in terms of Young's modulus E, Poisson's ratio v and thermal expansion coefficient a (thermal coefficient of linear expansion):

vE El \

Cjl = (1 - 2v)(l + v) 5lJ5kl + 2(1+^) V'k5Jl + 5ll5Jk ^

J lav5"-

In Fig. 21 polynomial approximations of thermal expansion coefficients of various tissues are compared with that of water and of copper (after [75]). The tissues with high water content (liver and muscle tissues) exhibit characteristics similar to that of pure water while thermal expansion coefficient of hard tissue (bone) is significantly lower.

In Fig 22 the influence of cryoprotectant on the thermal expansion coefficient of two kinds of tissues is shown [75]. The quantitative effect is significantly different for muscle and liver tissue suggesting, that some soft tissues are more susceptible to cryopreservation with the given cryoprotectant than others.

Acknowledgement

The authors were supported by the State Commitee for Scientific Research (KBN, Poland) through the grant No8T11F01718.

References

1. BAISH J.W. Heat transport by countercurrent blood vessels in the presence of an arbitrary temperature gradient. J Biomech Eng, 112: 207-211, 1990.

2. BAISH J.W. Formulation of a statistical model of heat transfer in perfused tissue. J Biomech Eng, 116:521-527, 1994.

3. BAISH J.W., AYYASWAMY P.S., FOSTER K.R. Heat transport mechanisms in vascular tissues: a model comparison. J Biomech Eng, 108: 324-331, 1986.

4. BAISH J.W., AYYASWAMY P.S., FOSTER K.R. Small-scale temperature fluctuations in perfused tissue during local hyperthermia. J Biomech Eng, 108: 246-250, 1986.

5. BARRON R.F. Cryogenic heat transfer. Bristol, PA, Taylor&Francis, 1999.

6. BERGMANN G., GRAICHEN F., ROHLMANN A. In vivo measurement of temperature rise in a hip implant. 37th Annual Meeting Orthopaedic Research Society.

7. BERGMANN G., GRAICHEN F., ROHLMANN A., VERDONSCHOT N., VAN LENTHE G.H. Frictional heating of total hip implants, Part 1: measurements in patients. J Biomech, 34(4): 421428,2001.

8. BERGMANN G., GRAICHEN F., ROHLMANN A., VERDONSCHOT N., VAN LENTHE G.H. Frictional heating of total hip implants, Part 2: finite element study. J Biomech, 34(4): 429-435, 2001.

9. BHOWMICK S., SWANLUND D.J., BISCHOF J.C. Supraphysiological thermal injury in Dunning AT-1 prostrate tumor cells. J Biomech Eng, 122: 51-59, 2000.

10. BISHOP N.E., FERGUSON S., TEPIC S. Porosity reduction in bone cement at the cement-stem interface. J Bone Joint Surg [Br], 78(3): 349-356, 1996.

11. BIYIKLI S., MODEST M.F., TARR R. Measurements of thermal properties for human femora. J Biomed Mat Res, 20: 1335-1345, 1986.

12. BOWMAN H.F. The bioheat transfer equation and discrimination of thermally significant vessels. Annals New York Acad Sci, 335: 155-160, 1980.

13. BRINCK H., WERNER J. Estimation of the thermal effect of blood flow in a branching countercurrent network using a three-dimensional vascular model. J Biomech Eng, 116: 324-330, 1994.

14. CARSLAW H.S., JAEGER J.C. Conduction of heat in solids. Oxford University Press, 1954.

15. CHAMPAGNE M.J., DUMAS P., ORLOV P., BENETT M.R., HAMET P., TREMBLAY J. Protection against necrosis by heat stress proteins in vascular smooth muscle cells: Evidence for distinct modes of cell death. Hypertension, 33: 906-913, 1999.

16. CHARNY C.K., WEINBAUM S., LEVIN R.L. An evaluation of the Weinbaum-Jiji bioheat equation for normal and hyperthermic conditions. J Biomech Eng, 112: 80-87, 1990.

17. CHATO J. Heat transfer to blood vessels. ASME Journal of Biomechanical Engineering, 102: 110-118, 1980.

18. CHEN M.M., HOLMES K.R. Microvascular contributions in tissue heat transfer. Annals New York Acad Sci, 335: 137-154, 1980.

19. CHEN S.S., WRIGHT N.T., HUMPHREY J.D. Heat-induced changes in the mechanics of a collagenous tissue: Isothermal, free shrinkage. J Biomech Eng, 119: 372-378, 1997.

20. CHEN S.S., WRIGHT N.T., HUMPHREY J.D. Heat-induced changes in the mechanics of a collagenous tissue: Isothermal, isotonic shrinkage. J Biomech Eng, 120: 382-388, 1998.

21. CHEN S.S., Wright N.T., Humphrey J.D. Phenomenological evolution equations for heat-induced shrinkage of a collagenous tissue. IEEE Trans. BME, 45(10): 1234-1240, 1998.

22. CLATTENBURG R., COHEN J., CONNER S., COOK N. Thermal properties of cancellous bone. J Biomed Mat Res, 9: 169-182, 1975.

23. CRAVALHO E.G., FOX L.R., KAN J.C. The application of the bioheat equation to the design of thermal protocols for local hyperthermia. Annals New York Acad Sci, 335: 86-97, 1980.

24. CREZEE J., LAGENDIJK J. J.W. Experimental verification of bioheat transfer theories: Measurement of temperature profiles around large artificial vessels in perfused tissue. Phys Med Biol, 35(7): 905-923, 1990.

25. CREZEE J., MOOIBROEK J., LAGENDIJK J. J.W., VAN LENEUVEN G. M.J. The theoretical and experimental evaluation of the heat balance in perfused tissue. Phys Med Biol, 39: 813-832, 1994.

26. DAVIDSON J.A., GIR S., PAUL J. Heat transfer analysis of frictional heat dissipation during articulation of femoral implants. J Biomed Mat Res, 22: 281-309, 1988.

27. DAVIDSON J.A., SCHWARTZ G., LYNCH G., GIR S. Wear, creep, and frictional heating of femoral implant articulating surfaces and the effect on long-term performance-Part II: Friction, heating, and torque. J Biomed Mat Res, 22: 69-91, 1988.

28. DAVIS E.D., DOSS D.J., HUMPHREY J.D., WRIGHT N.T. Effects of heat-induced damage on the radial component of thermal diffusivity of bovine aorta. J Biomech Eng, 122 (3), 283-286, 2000.

29. DENNIS D.A., KOMISTEK R., NORTHCUT E.J., OCHOA J.A., RITCHie A. 'In vivo' determination of hip joint separation and the forces generated due to impact loading conditions. J Biomech, 34(5): 623-629, 2001.

30. DILLER K.R. Modelling of bioheat transfer processes at high and low temperature. Advances in Heat Transfer, 22: 177-357, 1992.

31. DOWSON D., Ed. Advances in medical tribology. Orthopaedic implants and implant materials. London, UK, Mechanical Engineering Publications Limited, 1998.

32. DURKEE JR J.W., ANTICH P.P., LEE C.E. Exact solutions to the multiregion time-dependent bioheat equation. I: Solution development. Phys Med Biol, 35(7): 847-867, 1990.

33. DURKEE JR J.W., ANTICH P.P., LEE C.E. Exact solutions to the multiregion time-dependent bioheat equation. II: Numerical evaluation of the solutions. Phys Med Biol, 35(7): 869-889, 1990.

34. FLORJANCZYK Z., PENCZEK S., Eds. Polymer chemistry, 1997 (in Polish).

35. FUNG Y.C. Foundation of solid mechanics. Prentice Hall, 1965.

36. FYHRIE D.P., KIMURA J.H. Cancellous bone biomechanics. J Biomech, 32(11): 1139-1148, 1999.

37. GRAICHEN F., BERGMANN G., ROHLMANN A. Hip endoprosthesis for in vivo measurement of joint force and temperature. J Biomech, 32(10): 1113-1117, 1999.

38. HENLE K.J., DETHLEFSEN L.A. Time-temperature relationships for heat-induced killing of mammalian cells. Annals New York Acad Sci, 335: 234-253, 1980.

39. HENRIQUEZ F.C., MORITZ A.R. Studies of thermal injury, I. The conduction of heat to and through skin and the temperature attained therein: A theoretical and experimental investigation. Am J Pathology, 23: 531-549, 1947.

40. HENRIQUEZ JR F.C. Studies of thermal injury: V. The predictability and the significance of thermally induced rate processes leading to irreversible epidermal injury. Arch Pathology, 43: 489-502, 1947.

41. HOFFMANN N.E., BISCHOF J.C. Cryosurgery of normal and tumor tissue in the dorsal skin flap chamber: Part I - Thermal response. J Biomech Eng, 123(4): 301-309, 2001.

42. HOFFMANN N.E., BISCHOF J.C. Cryosurgery of normal and tumor tissue in the dorsal skin flap chamber: Part II - Injury response. J Biomech Eng, 123(4): 310-316, 2001.

43. HUISKES R. Some fundamental aspects of human joint replacement. Acta Orth Scan, 185, 1980.

44. JAIN R.K. Temperature distributions in normal and neoplastic tissues during normothermia and hyperthermia. Annals New York Acad Sci, 335: 98-106, 1980.

45. JIJI L.M., WEINBAUM S., LEMONS D.E. Theory and experiment for the effect of vascular microstructure on surface tissue heat transfer - Part II: Model formulation and solution. J Biomech Eng, 106: 331-341, 1984.

46. KLINGER H.G. The description of heat transfer in biological tissue. Annals New York Acad Sci, 335: 133-136, 1980.

47. KRAUSE W.R., BRADBURY D.W., KELLY J.E., LUNCEFORD E.M. Temperature elevations in orthopaedic cutting operations. J Biomech, 15(4): 267-275, 1982.

48. LAW H.T., PETTIGREW R.T. Heat transfer in whole-body hyperthermia. Annals New York Acad Sci, 335: 298-310, 1980.

49. LEMONS D.E., CHIEN S., CRAWSHAW L.I., WEINBAUM S. The significance of vessel size and type in vascular heat transfer. Am J Phys, 253: R128-R135, 1987.

50. LU Z., MCKELLOP H. Frictional heating of bearing materials tested in a hip joint wear simulator. In Dowson D. (Editor) Advances in Medical Tribology. Orthopaedic implants and implant materials.. Mechanical Engineering Publications Ltd, pp.193-200, 1998.

51. MAJCHRZAK E., JASINSKI M. Sensitivity analysis of bioheat transfer in 2D tissue domain subjected to an external heat source. Acta Bioengng Biomech, 3: 329-336, 2001.

52. MAJCHRZAK E., MOCHNACKI B. The analysis of thermal processes proceeding in the tissue with a tumor region using the BEM. J Theor Appl Mech (in press).

53. MARTIN R.B., SHARKEY N.A. Mechanical effects of postmortem change, preservation and allograft bone treatments. In Cowin S.C. (Editor) Bone mechanics handbook. CRC Press, Boca Raton, pp.(20-1)-(20-24), 2001.

54. MATTHEWS L.S., HIRSCH C. Temperatures measured in human cortical bone when drilling. J Bone Joint Surg [Am], 54(2): 297-308, 1972.

55. MAZUR P. Cryobiology-the freezing of biological systems. Science, 168, 939-949, 1970.

56. MAZZULLO S., PAOLINI M., VERDI C. Numerical simulation of thermal bone necrosis during cementation of femoral prostheses. J Math Biol, 29(5): 475-494, 1991.

57. MCHEDLISHVILI G. Blood flow in the microcirculation is a specific scientific field. Rus J Biomech, 4(4): 34-35, 2000.

58. MOCHNACKI B., DZIEWONSKI M. Numerical analysis of cyclic freezing/thawing process of biological tissue. Acta Bioengng Biomech, 3: 361-368, 2001.

59. MOSSER D.D., DUCHAINE J., BUORGET L., MARTIN L.H. Changes in heat shock proteins synthesis and heat sensitivity during mouse thymocyte development. Developmental Genetics, 14: 148-158, 1993.

60. OHTA M., TSUTSUMI S., HYON S.H., KANG Y.B., TANABE H., MIYOSHI Y. Residual stress measurements of ultra-high molecular weight polyethylene for artificial joints. Rus J Biomech, 4(2): 30-38, 2000.

61. OSTROWSKA-MACIEJEWSKA J. Mechanics of deformable bodies. Warszawa, Wydawnictwa Naukowe PWN, 1994 (in Polish).

62. PAZHAYANNUR P.V., BISCHOF J.C. Measurement and simulation of water transport during freezing in mammalian liver tissue. J Biomech Eng, 119(4): 269-277, 1997.

63. PENNES' H.H. Analysis of tissue and arterial blood temperatures in the resting human forearm. J Appl Physiology, 1: 93-122, 1948.

64. PFEFER T.J., CHOI B., VARGAS G., MCNALLY K.M., WELCH A.J. Pulsed laser-induced thermal damage in whole blood. J Biomech Eng, 122: 196-202, 2000.

65. PRENDERGAST P.J. Biomechanical techniques for pre-clinical testing of prostheses and implants. Lecture notes at Centre of Excellence for Advanced Materials and Structures, Warsaw, Institute for Fundamental Technological Research, Polish Academy of Sciences, 2000.

66. RABEK J. Fundamental of polymer chemistry. Wroclaw, Wydawnictwo Politechniki Wroclawskiej, 1977 (in Polish).

67. RABIN Y., JULIAN T.B., OLSON P., TAYLOR M.J., WOLMARK N. Cryosurgery for breast malignancies: apparatus and techniques. Advances in Heat and Mass Transfer in Biotechnology, 65-69, 1999.

68. RABIN Y., OLSON P., TAYLOR M.J., STEIF P.S., JULIAN T.B., WOLMARK N. Gross damage accumulation in frozen rabbit liver due to mechanical stresses at cryogenic temperatures. Cryobiology, 34: 394-405, 1997.

69. RABIN Y., SHITZER A. Numerical solution of the multidimensional freezing problem duing cryosurgery. J Biomech Eng, 120(1): 32-37, 1998.

70. RABIN Y., STEIF P.S. Analysis of thermal stresses around a cryosurgical probe. Cryobiology, 33: 276-290, 1996.

71. RABIN Y., STEIF P.S. Thermal stresses in a freezing sphere and its application to cryobiology. J Appl Mech, 65: 328-333, 1998.

72. RABIN Y., STEIF P.S. Thermal stress modeling of the freezing of biological tissue. Advances in Heat and Mass Transfer in Biotechnology, 183-187, 1999.

73. RABIN Y., STEIF P.S. Thermal stress modeling in cryosurgery. J Solids Structures, 37: 23632375, 2000.

74. RABIN Y., STEIF P.S., TAYLOR M.J., JULIAN T.B., WOLMARK N. An experimental study of the mechanical reponse of frozen biological tissues at cryogenic temperatures. Cryobiology, 33: 472482,1996.

75. RABIN Y., TAYLOR M.J., WOLMARK N. Thermal expansion measurements of frozen biological tissues at cryogenic temperatures. J Biomech Eng, 120(2): 259-266, 1998.

76. RIVOLTA B., INZOLI F., MANTERO S., SEVERINI A. Evaluation of temperature distribution during hyperthermic treatment in biliary tumors: A computational approach. J Biomech Eng, 121: 141-147, 1999.

77. ROEMER R.B., CETAS T.C. Application of bioheat transfer simulations to hyperthermia. Cancer Res, 44: 4788s-4798s, 1984.

78. ROEMER R.B., DUTTON A.W. A generic tissue convective energy balance equation: Part I -theory and derivation. J Biomech Eng, 120: 395-404, 1998.

79. SAITO M., MARUOKA A., MORI T., SUGANO N., HINO K. Experimental studies on a new bioactive bone cement: hydroxyapatite composite resin. Biomaterials, 15(2): 156-160, 1994.

80. SENAHA Y., NAKAMURA T., TAMURA J., KAWANABE K., IIDA H., YAMAMURO T. Intercalary replacement of canine femora using a new bioactive bone cement. J Bone Joint Surg, 78(1): 2631, 1996.

81. SHI X., DATTA A.K., MUKHERJEE S. Thermal fracture in a biomaterial during rapid freezing. J Therm Stress, 22: 275-292, 1999.

82. SHI X., DATTA A.K., MUKHERJEE Y. Thermal stresses from large volumetric expansion during freezing of biomaterials. J Biomech Eng, 120(6): 720-726, 1998.

83. SHITZER A., BELLOMO S., STROSCHEIN L.A., GONZALEZ R.R., PANDOLF K.B. Simulation of a cold-stressed finger including the effects of wind, gloves and cold-induced vasodilatation. J Biomech Eng, 120: 389-394, 1998.

84. SMITH D.J., SCHULTE M., BISCHOF J.C. The effect of dimethylsoxide on the water transport response of ret hepatocytes during freezing. J Biomech Eng, 120(5): 549-558, 1998.

85. SONG J., XU L.X., LEMONS D.E., WEINBAUM S. Enhancements in the effective thermal conductivity in rat spinotrapezius due to vasoregulation. J Biomech Eng, 119: 461 -468, 1997.

86. SONG J., XU L.X., LEMONS D.E., WEINBAUM S. Microvascular thermal equilibration in rat spinotrapezius muscle. Annals Biomed Eng, 27: 56-66, 1999.

87. SOTIN A.V., AKULICH Y.V., PODGAYETS R.M. The calculation of loads ating on the femur during normal human walking. Rus J Biomech, 4(1): 49-61, 2000.

88. STANCZYK M., TELEGA J.J. Thermal problems in artificial joints: influence of bone cement polymerization. Acta Bioengng Biomech, 3: 489-496, 2001.

89. STOLWIJK J. A.J. Mathematical models of thermal regulation. Annals New York Acad Sci, 335: 98-106, 1980.

90. SWENSON JR L.W., SCHURMAN D.J., PIZIALI R. Finite element temperature analysis of a total hip replacement and measurement of PMMA curing temperatures. J Biomed Mat Res, 15(1): 8396,1981.

91. SZWAJCZAK E., KUCABA-PICETAL A., TELEGA J.J. Liquid crystalline properties of synovial fluid. Engng Trans, 49: 315-358, 2001.

92. TAKEGAMI K., SANO T., WAKABAYASHI H., SONODA J., YAMAZAKI T., MORITA S., SHIBUYA T., UCHIDA A. New ferromagnetic bone cement for local hyperthermia. J Biomed Mat Res, 43(2): 210-214, 1998.

93. TATSUTANI K., RUBINSKY B. A method to study intracellular ice nucleation. J Biomech Eng, 120(1): 27-31, 1998.

94. TEPIC S., MACIROWSKI T., MANN R.W. Experimental temperature rise in human hip joint in vitro, in simulated walking. J Orthop Res, 3: 516-520, 1985.

95. TOKSVIG-LARSEN S., FRANZEN H., RYD L. Cement interface temperature in hip arthoplasty. Acta Orth Scan, 62(2): 102-105, 1991.

96. TOKSVIG-LARSEN S., RYD L., LINDSTRAND A. On the problem of heat generation in bone cutting. J Bone Joint Surg [Br], 73(1): 13-15, 1991.

97. VALVANO J.W., NHO S., ANDERSON G.T. Analysis of the Weinbaum-Jiji model of blood flow in the canine kidney cortex for self-heated thermistors. J Biomech Eng, 116: 201-207, 1994.

98. VAN EE C.A., CHASSE A.L., MYERS B.S. Quantifying skeletal muscle properties in cadaveric test specimens: effects of mechanical loading, postmortem time and freezer storage. J Biomech Eng, 122(1): 9-14, 2000.

99. VAZQUEZ B., GINEBRA M.P., GIL F.J., PLANELL J.A., LOPEZ BRAVO A., SAN ROMAN J. Radiopaque acrylic cements prepared with a new acrylic derivative of iodo-quinoline. Biomaterials, 20: 2047-2053, 1999.

100.WEINBAUM S., JIJI L.M. A new simplified bioheat equation for the effect of blood flow on local average tissue temperature. J Biomech Eng, 107: 131-139, 1985.

101.WEINBAUM S., JIJI L.M., LEMONS D.E. Theory and experiment for the effect of vascular microstructure on surface tissue heat transfer - Part I : Anatomical foundation and model conceptualization. J Biomech Eng, 106: 321-330, 1984.

102.WEINBAUM S., JIJI L.M., LEMONS D.E. The bleed-off perfusion term in the Weinbaum-Jiji bioheat equation. J Biomech Eng, 114: 539-544, 1992.

103.WEINBAUM S., XU L.X., ZHU L., EKPENE A. A new fundamental bioheat equation for muscle tissue: Part I - blood perfusion term. J Biomech Eng, 119: 278-288, 1997.

104.WISNIEWSKI S., WISNIEWSKI T. Heat transfer. Warszawa, Wydawnictwa Naukowo Techniczne, 1997 (in Polish).

105.WISSLER E.H. Comments on the new bioheat equation proposed by Weinbaum and Jiji. J Biomech Eng, 109: 226-233, 1987.

106.WRIGHT N.T., CHEN S.S., HUMPHREY J.D. Time-temperature equivalence of heat-induced changes in cells and proteins. J Biomech Eng, 120(1): 22-26, 1998.

107.WU Y.L., WEINBAUM S., JIJI L.M. A new analytic technique for 3-D heat transfer from a cylinder with two or more axially interacting eccentrically embedded vessels with application to countercurrent blood flow. J Heat Mass Transfer, 36(4): 1073-1083, 1993.

108. WULFF W. The energy conservation equation for living tissue. IEEE Trans BME, 21(6): 494-495, 1974.

109. WULFF W. Alternatives to the bioheat transfer equation. Annals New York Acad Sci, 335: 151154, 1980.

110.WYKMAN A. G.M. Acetabular cement temperature in arthroplasty. Effect of water cooling in 19 cases. Acta Orth Scan, 63(5): 543-544, 1992.

111.YOUNG T.H., CHENG C.K., LEE Y.M., CHEN L.Y., HUANG C.H. Analysis of ultrahigh molecular weight polyethylene failure in artificial knee joints: Thermal effect on long-term performance. J Biomed Mat Res, 48(2): 159-164, 1999.

112.ZHU L., LEMONS D.E., WEINBAUM S. A new approach for predicting the enhancements in the effective conductivity of perfused muscle due to hyperthermia. Annals Biomed Eng, 22: 1-12, 1995.

113.ZHU L., LEMONS D.E., WEINBAUM S. Microvascular thermal equilibration in rat cremaster muscle. Annals Biomed Eng, 24: 109-123, 1996.

114.ZHU M., WEINBAUM S., JIJI L.M. Heat exchange between unequal countercurrent vessels asymetrically embedded in a cylinder with surface convection. J Heat Mass Transfer, 33(10): 2275-2284, 1990.

ТЕМПЕРАТУРНЫЕ ПРОБЛЕМЫ В БИОМЕХАНИКЕ: ОТ МЯГКИХ

ТКАНЕЙ ДО ОРТОПЕДИИ

М. Станчик, Й.Й. Телега (Варшава, Польша)

Данная работа является обзорной статьей, посвященной анализу различных температурных проблем в биомеханике. Рассмотрены экспериментальные данные, а также аналитические и численные решения. Объектом рассмотрения являются три класса проблем: теплопередача в мягких тканях, различные задачи ортопедии и лечение при низких температурах. Представлены различные модели теплопереноса в тканях, насыщенных сосудами, а также результаты экспериментального и теоретического анализа их способности к переносу тепла. Дается обзор существующих критериев температурной повреждаемости. В частности, описываются проблемы, связанные с имплантацией цементированных эндопротезов и полимеризацией полиметил-метакрилата. Рассмотрено современное состояние исследований, касающихся функционального нагрева тканей при сверлении и распиливании, а также операциях над суставами. Анализируются механизмы температурной повреждаемости при охлаждении и отрицательных температурах, а также физические явления, важные для моделирования процессов замораживания и оттаивания тканей. Библ. 114.

Ключевые слова: биомеханика, температурные проблемы, уровень теплопроводности в живых тканях, тепловая травма, гипотермия, теплоперенос в сосудах, фрикционный нагрев имплантатов, криогеника

Received 11 November 2001