Yaş kemikler için elektromekanik bir model

Abstract

Bu çalışmada yaş kemiğin, birisi piezoelektrik özellikli katı matris, diğeri de içerisinde çeşitli tuz ve iyonların bulunduğu polar bir akışkandan oluşan bir karışım olduğu kabul edilerek aralarında kütle ve elektrik yükü taşınımı olan bu karışıma ait denklik yasaları ve bünye ilişkileri elde edilmeye çalışılmıştır. Bu çalışma beş ana bolümden oluşmaktadır. Birinci Bölümde, konunun tarihsel gelişimi kısaca anlatılmış, konu ile ilgili önceden yapılmış olan çalışmalar özetlenmiş ve şimdiki çalışmanın gereği ve so nuçta neler yapıldığı anlatılmıştır. ikinci Bölümde N tane farklı bileşenden oluşan bir karışım için genel denklik denklemleri, elektromanyetik alan denklemleri ve entropi eşitsizliği elde edilmiştir. Üçüncü Bölümde ikili bir karışım için bağımsız bünye değişkenleri saptanarak genel bünye denklemleri elde edilmiş, daha sonra viskozitenin ihmal edildiği halde non-lineer bünye denklemleri bulunmuştur. Üçüncü Bölümün ikinci kısmında li neer bünye denklemleri elde edilmiştir. Bu denklemlerin (Cg) hexagonal simetri halinde aldıkları şekiller yine bu bölüm içerisinde ele alınmıştır. Çalışmanın Dördüncü Bölümünde bir önceki bölümde elde edilmiş olan lineer bünye denklemleri kullanılarak yaş kemikler içerisinde harmonik dalga yayılımı problemi incelenmiş, problemin basitliği açısından tek boyutta dalga yayılması problemi ele alınmıştır. Bu problemde genel dispersiyon bağıntısı elde edilmiş ve büyüklükler boyutsuzlaştırılarak nümerik olarak çözülmüştür. Sonuçta ortamda iki dalga olduğu görülmüştür. Bunlardan akışkan içerisinde yayılan dalganın akustik dalga olduğu katı faz içerisinde yayılan dalganın ise dispersif bir dalga olduğu dalga hızının dalga sayısı ile arttığı görülmektedir. Bu bölümün sonunda dışarıdan kemiğe verilen bir elektrik dalgası olması halinde kütle taşınım hesabı yapılmıştır. Son olarak Beşinci Bölümde, yapılan çalışmanın sonuçlan verilmiş ve bu konuda daha sonra yapılabilecek çalışmalardan söz edilmiştir.

Until recently, the scientists who have been dealing with bones were only in terested in the mechanical properties of bones and the materials that they used in connecting the broken bones to each other. Their primary problems were the stress concentrations around the holes in which the certain types of nails are inserted in holding the broken bones, and the medical complications caused by plates and nails used during orthopeadic surgery. In addition, a group of scien tists were working on the transmission of forces and moments from one part of the body to the other parts through the joints. These were the only problems that the orthopeadic surgeons expecting from bioengineers. In the last four decades or so, as other biological tissues, the bony tissues have received a great deal of interest from the scientists working in the area of biomechanics. The scientists were interested in not only the mechanical prop erties of bones but also the mechanism of regeneration and remodelling of bony tissues. These efforts brought many scientists from different disciplines such as chemists, physicists, engineers and medical peoples together to investigate this multydisciplinary subject. The majority of scientists working on the remodelling and regeneration of bones stated that the adaptation of bones to external or physiological forces is pos sible only through a biological control and/or a feedback mechanism. For this pur pose, although some researchers (eg,KUMMER[5], BASSETT[6], BECKER[7]) tried to construct models about the feedback mechanism, it is not yet sufficiently clear how the bone remodels and adapts itself to external mechanical forces with out changing the totality of materials. However, the majority of scientists have agreed on the point that the mechanical forces create some bioelectrical signals in bones and that they are assumed to be the causes of this feedback mechanism. Although there are different views on the origin of these bioelectrical signals the majority of researchers have agreed on that this is due to the piezoelectric prop erties of bones. When working on the dynamic properties of bones, YASUDA and his co workers^] observed some piezoelectric behaviour in bones. Later FUKADA and his co-workers [9] studied the direct and converse piezoelectric effects in bones and found a linear relation between the polarization and the pressure. BASSETT[4], and MARINO and his group[15] studied the electrical properties of bones and their connections to the remodelling processes. But all these and sim ilar researchers had tried to measure the piezoelectrical coefficients for a classical linear piezoelectric material as applied to bony tissues. However, the remodelling is a dynamic process and requires the mass and electrical charge transfer between various components of bones and regions. Considering that the bone consists of a piezoelectric matrix and charged conducting fluid phase, some kind of mixture model that takes the mass and charge transfer into account should be presented. In this context, the first attempt for electromechanical modelling of wet bones had been made by PETROV[40,41], in which he used the mixture theory for elec trically charged mixtures, proposed by DEMIRAY[42,43,44]. Later DEMİRAY[46] and DEMIRAY et al[45] presented models for electromechanical behaviour of wet bones, where they used all the thermodynamic restrictions. GUZELSU and SAHA [47,48] suggested an electromechanical model for living bones and investi gated the remodelling of bones by studying the wave propagation in long bones. But this derivation is quite complicated for the analysis of some practical prob lems. Furthermore, they have only taken the polarization of the solid matrix into consideration and neglected the polarization of the fluid component. In the present work, a mixture model that takes the piezoelectric properties, mass and charge transfers into account is presented. The bone is assumed to be composed of an elastic piezoelectric solid bone matrix and a polar and electri cally conducting fluid phase. It is also assumed that there is a mass and charge transfers in the course of remodelling processes of bone under the effects of the mechanical forces. In chapter 2, the kinematics and balance laws for an electrically charged mix ture consisting of N different components are given for each species as well as the whole of the mixture. These subjects are summarized below: Kinematics: We consider a mixture consisting of N species that occupies a material volume V at time t=0. Upon application of the external forces the particles move along their paths. We assume that the space point x at time t is occupied at least by one particle from each species. Therefore the motion is described by x = x(o)(X(a),i), (a = 1,2,...AT) (1) where X^ are the material coordinates of ath particle. The transformation (1) is one-to-one and onto, thus the inverse motion is given by X«=X«(xM,t), with J = det(^Ş^)#0 (2) The velocity and the acceleration vectors of the ath component are defined by (a) _ fofc dt «W - Tk X(«) * dt Dt + "fit*>«£r,f> (3) Here the operator *j%- is used to denote the time derivative along the path of ath componenet. VI Balance Laws: In this subsection we briefly give the balance laws for each species of mixtures as well as the whole mixture, (i) Conservation of Mass: This law is stated in local form as ^ + (P(.)4a)),fc = c(a), £c(«) = ° (4) and ^ + (Pvk),k = 0 (5) where p^ is the mass density of the a component, C(a) is the rate of mass transfer from other species into ath species p is the total density and vk is the average velocity of the mixture and they are defined by ^ a Here a comma after a subscript denotes the partial derivatives with respect to that space coordinate. In order to save the space, the jump conditions associated with these balance laws are not given here, they are listed in the main text. (ii) Balance of Linear Momentum: This law is stated in differential form as till + p^f^+g^-pn^ + ^ + cavi-v^^o, JX^o (7) a and tki,k + pfı + gi~pvi = 0 (8) where vjjj' is the partial stress tensor, /j is the external body force density, g\a ' is the electromagnetic force density, vi is a reference velocity (which is not defined yet) and Rf' is the rate of linear momentum density from other species into ath component. By summing up equation (7) over the species number and making use of the following definitions Vll the balance of linear momentum equation (8) can be obtained. Here «j. = vk' - vk is the diffusion velocity and prime denotes the material derivative along the trajectory of ath component of the mixture. The expression of electromag netic forces is given in the Appendix A of the main work. (iii) Balance of Angular Momentum This law is stated in local form as İl + İMmP + ta)) = 0 (10) f>îa) = o (11) £»=1 where l\a' is the electromagnetic moment density and mf' is the rate of an gular momentum transfer from the other species to the ath component. t)%l stands for the anti-symmetric part of the partial stress tensor of ath component and defined by #-§ (*'-*') (12) The expression of the partial electromagnetic moment density is given in the Ap pendix A. (iv)Balance of Energy: The equation of energy balance in local form is stated as follows P(a)S(a) + c(a)e(a) = t$v$ - ç$ + p{a)h{a) + -c(a)Jka)Jka) + Rka)u{ka) + e{a) + E{a) (13) and pi, = tklvlfk - qk,k + ph + E (14) where £(a) is the partial internal energy density, h(a) heat source, q£*' heat flux, E(a) the electromagnetic energy density and e(aj is the rate of energy transfer from the other species into atk species. The energy balance for the mixture is obtained from (13) by summing this equation over the species number and making use of the following definition /* = £ /»<«)«<«) + 2'<«>w!ft)un ' 4a) = ^-4a) ft = £ [«P + «A («w + 5 W) «Î* - W] pfc = £[«.)*(.) + «°)//°ty*)], £ = £[£,") t^S»'] (15) (v) Electromagnetic Field Equations: Since the mechanical fields are interacting with the electromagnetic fields, we need the governing equations of the electromagnetic field which are given by VxE = 0, V-D = ?, m dt + J = 0 (16) where E is the electric field vector, D is the dielectric displacement vector, q is the free charge density and J is the current density. Here we have neglected the magnetic effects and defined D in terms of polarization vector P(a) as N D = e0E + ]£P(a (17) where e0 is the dielectric constant of the vacuum, (vi) Entropy Inequality: The second law of thermodynamics is stated in local form as N E P(a)Vi«) + C(«)l7(a) + (~J-h ~ q >0 (18) where 9 is the common temperature and 7/(a) is the entropy density of ath component. Introducing the free energy density $(a) as £(a) = $(a) + 0V(<*) + EiPl («) PW (19) and eliminating the h(a) between equation (13) and (18) we obtain Wo, E (20) Here we have neglected the effects of polarization transfer term. Inequality (20) should be valid for all admissible thermodynamic processes. Constitutive Equations: In this subsection we have derived a set of constitutive relations suitable for living bones. Considering that a living bone consists of a piezoelectric matrix and IX an electrically conducting polar viscous fluid we have selected the independent state variables as Ek, FkK, 4? i wf2) ¦» P(a), 9(c), 6 (21) where Ek is the electric field vector, FkK deformation gradient of solid body, <4i is the rate deformation tensor for fluid, u>\ ' = v\ '- vj ' the relative velocity, /)(a) mass density, Ç(a) electrical charge density and 9 the common temperature. Here the solid matrix and the fluid are labelled, respectively, by subscript (or superscript) (1) and (2). Thus a typical constitutive dependent variable may be given by $(a) = $(a)(Ek, Fuc, 4?, "f2). P(a), (!), 0(1), FiK, 0, Ei), $(2) = $(2)(/>(2), Ç(2), 0, Ei). 4?-*)gg**-,W- (a = 1,2), Pİa) = -p{a) Ö$(ı) Ö$ («) 0*(1) ^â^ + ^^ö^J^ B*Ü - 2 g$(2), ^(2)",(2) (a = 1,2), q' (2) *«,$' = *$' + **!; *(*) w = o (23) The remaining part of the entropy inequality becomes j(2) (2) Ö$(2) _ #$ $(2) + ^^r2 - *(i) - P(l) (1) 9/>(i). (24) where dV-Ş is the dissipative part of the fluid stress. These constitutive equations are further restricted by the axioms of objectiv ity and material symmetry. By using these axioms the most general nonlinear constitutive relations are obtained for the mixture under consideration. Unfortu nately the result is so complicated that one cannot solve any practical problem by use of these nonlinear equations. To simplify the problem further, the depen dence of constitutive relations on deformation rate tensor and the charge density of the piezoelectric body is neglected, and a considerably simple set of nonlinear constitutive relations are obtained. Even this simplified constitutive relations are being nonlinear, the solution of any physiologically important problem cannot be given by use of these equations. Linearized Constitutive Equations: In order to see the novelty of the present derivation one must solve some prob lems of practical importance. Unfortunately the solution of such problems by use of the nonlinear constitutive theory is not too easy. Therefore, to simplify the problem further we have investigated the linearized constitutive equations which may be given by t\V = - (ap(1) + 2akıhı + ckEk) Sij + 2o-,iWew + SijkEk + -£-p{\) (25) P(l) *{? = - (biPV) + hq) Sij (26) PV = Skuhi + XijEj - -Ş-p(1) (27) p?) = -/3Ei (28) #!2) = //%) + /P%> + fU + $Ei + flP4] + fTZmn (29) C(2) = cxp(x) + c2p(2) + c3q + diEi + giU}j2) + gij&ij (30) where a, ck, ak\, Sijk, ... -9ij are some scalars, vectors and tensor coefficients characterizing the electro-mechanical properties of the material under consideration. Other quantities are: p(a) increments in mass densities, q is the increment in charge density of fluid phase, uT = relative velocity of fluid phase with respect to solid velocity and e*/ is the infinitesimal strain tensor defined by hi = ğ (uk,i + ui,k) (31) Here uk axe the displacement components of the piezoelectric body. The constitutive equations are further restricted by the axiom of material symmetry. The studies on electro-mechanical properties of bones(see FUKADA and YASUDA[9]) show that it has the polar hexzagonal ( C& ) symmetry. This axiom puts further restrictions on the material coefficients (see (3.2.61)) but they will not be given here. In order to obtain the governing equations for fo), />(2), q, uk, vk2 and Ek we need the linearized field equations which could be obtained from (4) - (13). If this is done and the linearized constitutive equations (25) - (30) are introduced into these equations the following differential equations are obtained -gp- + cip(1) + c2p{2) + c3q + diEi + giuW + g^ = 0 (32) ?M j. ". "« _,. a... -,-^ _..« -A.I?. (2) _ "..s.. _ n (VK\ XI - aP(l),l - aij (ui,jl + Uj,il) + Ciipji + CTklmn(umtnk + UnıTnk) - SklmJ 5 - J333 c4 = d3, cs = 533, ce - g3 (39) ö^m A dip du n,.nS -ğ^- + CıP(ı) + c2^(2) + c3q - c4 -t^ + cs ^j + ceu = 0 (40) d/>(2) " dv _ dip du n.,,. dp(l) d2u d2cp 0 d2u a°~bT + 0lâ? " °2ö? ~ pvW " /oV "" Ml) -M2)-/3? + /4^-/5^ = 0 (42), 9p(2), dq o dv -hfc ~blo~z~ p(*)di + foV + Md + Mî) +A* -/«£ + /.£ = » (43) -*S?-*3?-*£+*£+*&-« <«> -e°aldl ~ Xo~dT ~ Xldlbl + *2dlbl + ^ " q°v (45) We shall seek a harmonic wave type of solution to these field equations. For this we set tp = Eexp[i(ut - kz)], where w is the angular frequency, k is the wave number and E is the complex amplitude (the same are being valid for other field variables) and introduce the result into (39) - (45), and obtain six algebraic homogenous equations in six unknown coefficients. In order to have a non-zero solution for these unknowns, the determinant of the coefficient matrix must van ish. If this is done we get the following dispersion relation xm p\x)(ju^ - ipfac4X0u)4 - (aın + a2Xx + pfaftb^kPu3.\-i{a\CiX0 + p°^b\c4X0 - aic^,X0 - a0C5fi - a0C4Xi)k2u}2 +6ı(aı/i + a2X\)kAuj + 161(0205^0 + a0csfi + a0c\X\ - a\c4X0)kA = O (46) In obtaining this we have neglected the effects of linear momentum term. Al though the dispersion equation is cubic in u>2 there are only two longitudinal waves propagating in the medium. The third root corresponds to a disturbance which decades in time. The dispersion relation is discussed for several special cases numerically and analytically, whenever it is possible, and the results are depicted on some graphs. The numerical results reveal that the wave associated with fluid phase is less dispersive as compared to the wave associated with piezoelectric solid matrix. One of the most important implication of wave propagation in such a medium is to observe the effect of the electrical wave on the remodeling and the healing of the bones. This is characterized by the transfer term among the components of the mixture. The effects of the external wave to the mass transfer is calculated and the results is given by C(2) = uk{Xxc4 + pes) X0c$ - iu)X\ Eexp[i(ut - kz)] (47) If these electromechanical material coefficients are measured through exper imental measurements, the amount of mass transfer resulting from an external electrical wave can be calculated by use of the equation (47).