2 Curvature of the Spine: Hydrostatic Pressure as an Indicator of Scoliosis

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1 Report on problem studied t the UK Mthemtics-in-Medicine Study Group Nottinghm 2001 < > MMSG Curvture of the Spine: Hydrosttic Pressure s n Indictor of Scoliosis John Billinghm, Birminghm; Chris Brewrd & Peter Howell, Oxford. 2.1 Introduction Scoliosis refers to n bnorml lterl curvture of the spine. It ffects between 0.5 nd 2% of the popultion nd is more common in femles, usully becoming pprent during the dolescent growth spurt. The exct cuses of scoliosis re unkown t present. The purpose of the present study is to investigte currently populr hypothesis tht it rises from systemtic symmetric loding of the spine nd thus of the intervertebrl discs. This my led, it is thought, to permnent remodelling of the discs so tht they ssume wedge shpe. This symmetric reshping plys n importnt role in the progression nd permnence of the disorder. The intervertbrl discs form the joints tht llow bending of the spine. They lso ct s shock bsorbers nd help to support the lods pplied by muscles nd ligments s well s weight of the hed. Ech disc consists of two distinct regions. The nnulus fibrosus is composed of fibrous lmelle, mde up of bundles of collgen fibres, which form onionlike rings round the outside of the disc, The nucleus pulposus is highly hydrted gel contined inside these rings. The whole disc is composed mostly of wter which, long with proteoglycns nd collgen, comprises 90 95% of the tissue. The proteoglycns role is to generte n osmotic pressure tht drws wter into the disc nd holds it there. A pressure trnsducer hs been used to mesure the pressures inside intervertebrl discs. In fct, by rotting the trnsducer, it is possible to mesure the norml stresses in two perpendiculr directions. In helthy disc, the pressure is found to be roughly uniform cross the disc, except in nrrow regions ner the edge where it drops rpidly to zero, s indicted in figure 1. The solid nd dotted lines in this figure indicte tht the pressure mesured is generlly independent of the orienttion of the trnsducer. This uniformity of the pressure is found to hold even when n symmetric lod is pplied. In dmged disc, however, the pressure is found to be less isotropic nd to vry more cross the disc, s shown schemticlly in figure 1b. p b p x x Figure 1: Typicl mesurements of pressure p versus distnce x cross n intervertebrl disc; helthy disc, b dmged disc. The solid nd dotted lines indicte the pressures mesured in two perpendiculr directions.

2 MMSG Semipermeble membrne Rigid plte Poroelstic disc c b Figure 2: Schemtic of poroelstic disc. The uniform, isotropic stress sketched in figure 1 suggests tht ny pplied force is idelly supported minly by the fluid held inside the disc. A nonuniform, nisotropic stress like tht shown in figure 1b is more consistent with n elstic response, nd probbly mens tht the solid mtrix of the disc is experiencing significnt portion of the lod. Presumbly, remodelling nd/or dmge result if the stress pplied to the tissue, rther thn to the liquid trpped inside it, becomes too gret. We propose to develop mthemticl model for n intervertebrl disc by treting it s poroelstic mtrix tht is sturted with viscous liquid. The osmotic pressure due to the presence of proteoglycns is introduced s pressure difference between the solid nd liquid phses. Our model will determine the motion of the porous elstic mtrix nd the trnsport of fluid through it s the disc is deformed under n pplied lod. It will lso predict the distribution of stress between the two phses, nd how this depends on the mteril properties of ech. Our eventul im is to build in the grdul evolution of those properties s the tissue remodels under the stresses pplied to it. Of prticulr interest is the possibility of feedbck loop, whereby remodelling cts to enhnce ny initil symmetry in the system. In this report, we limit out ttention to uniform, isotropic deformble porous medium, nd only consider smll deformtions, so tht the solid phse my be described using the clssicl theory of liner elsticity. We begin, however, by presenting the equtions in some generlity, so tht nonuniformity, nisotropy nd nonlinerity my ll be introduced in further refinements. 2.2 Model for poroelstic disc Model description We model the intervertebrl disc s consisting of poroelstic mteril tht is stturted with fluid. The vertebre re modelled by rigid pltes ttched to the top nd bottom of the disc, s shown in figure 2, while stress-free boundry conditions re pplied t the curved edges. The disc is ssumed to be enclosed by membrnes tht re semi-permeble to the fluid. Since fluid my lso lek into the porous vertebre, the rigid pltes re likewise treted s semi-permeble. Now, if n xil force e.g. the weight of the hed is pplied to the disc, we cn

3 MMSG expect it to deform s shown in figure 2b. The lod is supported both by the pressure in the trpped fluid nd by elstic stress in the poroelstic mtrix. Over time, the liquid is grdully squeezed out through the semi-permeble boundries; this is mnifested in noticeble decrese in person s height over the course of ech dy. The elstic component of stress must therefore grdully increse. Our im is to quntify this distribution of stress between the fluid nd elstic components, the ide being tht excessive elstic stress my led to dmge nd consequent wekening of the mtrix. Of prticulr interest is the possibility of bending moment being pplied to the spine, resulting in deformtion like tht shown in figure 2c. In this cse, we wish to determine the symmetry in the stress field cused in the disc. Our hypothesis is tht this might led to grdul wekening of one side of the disc nd thus mke it still more susceptible to bending under everydy lods Governing equtions Both the solid mteril comprising the porous mtrix nd the fluid occupying the pores re ssumed to be incompressible. Conservtion of mss for ech phse therefore implies the two equtions φ t + φq l = 0, φ t + {1 φq s } = 0, 2.2 where φ is the fluid volume frction or porosity, while q l nd q s re the velocities of the fluid nd solid phses respectively. Next we impose conservtion of momentum for ech phse, ssuming tht, over the timescles of interest typiclly few hours, inerti my be neglected. The only liquid stress considered is n isotropic pressure p l. Thus mcroscopic viscous effects re ignored, lthough viscosity plys role in the pore-scle inter-phse drg D. The porous mtrix is chrcterised by n elstic stress stress tensor τ s well s n isotropic pressure p s ssocited with the ssumed incompressibility of the solid phse. These ssumptions give rise to the equtions φ p l = D, φ p s = τ D. 2.4 Note tht the stress τ is mesured per unit re occupied by both phses; in other words it is the stress tensor tht would be mesured in the mtrix s whole if no liquid were present. To close the equtions, we need to impose some constitutive lws. For the interphse drg we ssume tht the fluid flow reltive to the solid mtrix stisfies Drcy s lw: Kφ D = µ l φ q s q l, 2.5 where µ l is the fluid viscosity nd Kφ is the permebility tensor. In relting the two pressures p l nd p s, we incorporte the osmotic pressure π, due to the presence of proteoglycns, s n interphse pressure difference. Since the proteoglycns re ssumed to be ttched to the incompressible solid mtrix, their concentrtion, nd hence the osmotic pressure, is decresing function of the porosity φ: p s p l = πφ, π φ <

4 MMSG Finlly, we need consititutive reltion between the stress tensor τ nd the displcement field u, given in terms of the velocity q s by the kinemtic condition u t + q s u = q s. 2.7 In prctice, the solid mtrix is certinly nisotropic nd probbly experiences strins of order unity s the disc is compressed by significnt frction of its initil thickness. A fully nonliner nisotropic elsticity formultion would clerly led to n extremely complicted model, which would require difficult nd time-consuming numericl solution. Furthermore, we do not t present hve enough detiled informtion bout the rheology of the disc constituents to be ble to identify the mny physicl prmeters in such model. In this report we use isotropic infinitessiml elsticity theory to describe the solid mtrix. This both leds to reltively trctble elstic problem nd llows us to linerise the other governing equtions , s described below in While not prticulrly relistic, these simplifictions result in compct model from which we cn quickly drw useful qulittive conclusions. Another pproch tht incorportes nisotropy in prgmtic wy nd my be relevnt is to ssume tht the solid deformtion is unidirectionl, s in [1]. The governing equtions re further simplified s follows. The liquid velocity q l nd solid pressure p s my be eliminted using 2.5 nd 2.6 respectively. Then, by dding 2.1 to 2.2 nd 2.3 to 2.4, we obtin two equtions representing net conservtion of mss nd momentum for the two-phse continuum s whole, nmely q s φµl Kφ p l = 0, 2.8 nd Along with 2.2, or τ = p l + 1 φ πφ. 2.9 φ t = {1 φq s }, 2.10 this comprises closed system for φ, q s nd p l, once τ hs been constituted The linerised isotropic problem Now we simplify the equtions derived heretofore by treting the solid phse s isotropic nd ssuming tht the displcement is smll enough for infinitessiml elsticity theory to pply. Thus the solid stress tensor tkes the form τ ij = λ uδ ij + µ ui x j + u j x i, 2.11 where λ nd µ re the Lmé constnts for the whole mtrix, recll, not just the solid phse. It is consistent with 2.11 to ssume tht φ vries only slightly from its initil vlue φ 0 ssumed uniform: φ = φ 0 + φ, φ

5 MMSG If the solid mtrix is isotropic, then the permebility tensor tkes the form K = kφi, where I is the identity, i.e. to leding order, K kφ 0 I In the infinitessiml theory, the kinemtic condition 2.7 reduces to while the osmotic pressure is pproximted by u t q s, 2.14 π πφ 0 + π φ 0 φ The linerised version of 2.10 my be integrted with respect to t, ssuming tht u nd φ re both zero initilly, to obtin φ = 1 φ 0 u The other two governing equtions 2.8 nd 2.9 reduce to φ0 1 φ 0 kφ 0 φ t = 2 p l, 2.17 µ l λ + µ u + µ 2 u = p l + 1 φ 0 π φ 0 φ It is strightforwrd to eliminte p l nd u from to obtin diffusion eqution for φ: φ t = D 2 φ, D = φ 0 kφ 0 µ l { λ + 2µ 1 φ0 2 π φ 0 } Although this is not prticulrly helpful in solving the problem, since the boundry conditions see below re not esily stted in terms only of φ, it does give the useful insight tht chnges in the porosity due to displcement of the boundry diffuse through the mtrix, with n effective diffusivity D. Notice tht we expect π to be negtive, so there is no potentil difficulty ssocited with D chnging sign Boundry conditions Before ny forces re pplied, the poroelstic continuum is ssumed to occupy disc of rdius L nd thickness 2. We dopt the coordinte system x, y, z shown in figure 3, with the x- nd y-xes in the mid-plne of the disc nd the z-xis norml to tht plne, so tht the initil configurtion is given by x 2 + y 2 L 2, z. The sitution is complicted by the fct tht the curved edge of the disc is free boundry. Since we re using infinitessiml elsticity theory, however, the displcement field is ssumed to be smll. It is symptoticlly consistent with this pproch to pply the boundry conditions on the undeformed boundry to the order of pproximtion used in deriving , Eulerin nd Lgrngin vribles re equivlent. As described in 2.2.1, the upper nd lower surfces of the disc re ttched to rigid pltes, whose displcements re ssumed to be specified. We further suppose, s indicted

6 MMSG z L y x Figure 3: Definition sketch for the coordinte system x, y, z. in figure 2c, tht the force/couple systems pplied to the upper nd lower surfces re equl nd opposite, so tht symmetry in the plne z = 0 is preserved. We therefore hve u = 0, 0, ±W on z = ±, 2.20 where W is given function of x, y nd t, typiclly liner in x nd y. Recll tht the end pltes re supposed to be semi-permeble, to model the possible lekge of fluid into the porous vertebre. We therefore ssume tht the flux of liquid through ech plte is proportionl to the difference between the liquid pressure p l nd the externl mbient pressure p. It follows tht n q l q s = hp l p, 2.21 where n is the outwrd norml nd h is sclr prmeter representing the permebility; h 0 implies impermebility while h if fluid my pss freely through the plte. In generl h should be function of x nd y, since the vertebre re known to be more permeble ner the centre thn t the edges. If the velocities re eliminted, 2.21 becomes Robin boundry condition for the pressure: kφ 0 p l z = µ lhp l p on z = ± The outer surfce, initilly given by x 2 + y 2 = L 2, z, is denoted by S. It is ssumed to be open to fluid t uniform pressure p, so tht 1 φp s p n + τ n = 1 φp l + π p n + τ n = 0, x S, 2.23 where the outwrd-pointing norml is here given by n = x/ x 2 + y 2, y/ x 2 + y 2, 0 T. As for the upper nd lower pltes, S is ssumed to be semi-permeble to fluid, with permebility η, nd this gives rise to Robin condition of the form kφ 0 n p l = µ l ηp l p, x S In principle, provide enough boundry conditions to solve for φ, p l nd u. The first-order motion of the free boundry is then found by evluting the displcement u t S. We cn lso use 2.5, 2.11, 2.6, etc. to clculte the elstic stress, fluid velocity nd ll other quntities of interest.

7 MMSG Initilly, we ssume tht the porosity tkes its undisturbed vlue nd therefore φ = 0 when t = We re not t liberty to specify initil conditions for p l or u, both of which stisfy elliptic boundry-vlue problems. It might be nticipted tht the disc should strt t equilibrium with p s = p l = 0. Becuse of the dditionl osmotic pressure π in 2.23, however, the displcement field u will not in generl be zero initilly. For the moment, we void this difficulty by ssuming tht the osmotic pressure is zero when the disc is in equilibrium, i.e. tht πφ 0 = 0. Otherwise, it is nontrivil to determine the elstic stress in the disc even if no displcement is pplied to the top nd bottom, problem tht wrrnts further investigtion. 2.3 Lubriction solution in two dimensions Lubriction scling The simplifictions we hve employed thus fr hve enbled us to reduce the model to liner boundry-vlue problem on known, fixed domin. Further progress with the system would require numericl scheme to be devised, lbeit of firly stndrd type. Insted, we reduce the model further by mking use of the fct tht the spect rtio of disc is typiclly rther lrge, round 5:1, which enbles us to dopt lubriction pproximtion. To simplify things s much s possible, we lso restrict our ttention to two dimensions, which llows us to investigte more esily wedging of the disc by twisting of the vertebrl bodies. Our im here is to gin some insight into the qulittive behviour to be expected; in ny cse, there is little hope of our gretly simplified model producing ny trustworthy quntittive informtion. The disc now occupies the region L x L, z in terms of the twodimensionl Crtesin coordintes x, z. The inverse spect rtio, or slenderness prmeter, is defined to be ǫ = L, nd the lubriction pproximtion is obtined by tking the limit ǫ 0. We use W 0 s representtive vlue of the pplied displcement W nd nondimensionlise the equtions nd boundry conditions s follows: u = W 0 ǫ û, x = L x, z = ẑ, W = W 0 Ŵ, w = W 0 ŵ, t = 2 D t, φ = 1 φ0 W 0 θ, p l = p + µ ldw 0 L 2 φ 0 kφ 0 3 p, where u = u, w T is the two-dimensionl displcement field. With the hts now dropped for ese of nottion, 2.17 nd 2.18 become ǫ 2 u t x + w = ǫ 2 2 p z x + 2 p 2 z2, 2.26 p x = ǫ2 A u x x + w + 1 A ǫ 2 2 u z x + 2 u, z 2

8 MMSG where p z = ǫ2 A u z x + w + ǫ 2 1 A ǫ 2 2 w z x + 2 w, z 2 A = λ + µ 1 φ 0 2 π φ 0 0, 1. λ + 2µ 1 φ 0 2 π φ 0 The porosity perturbtion is determined posteriori from The boundry conditions re θ = u x + w z u = 0, w = ±Wx, t, u x + 2µ w λ + 2µ z = 0, where Hx = p z = ǫ2 Hxp t z = ±1 for 1 x 1, 2.30 u z + ǫ2 w x = 0, p x = Y zp t x = ±1 for 1 z 1, L 2 µ l Lµl hx, Y z = ηz. kφ 0 kφ Recll tht h nd η re the dimensionl permebilities of the horizontl nd verticl surfces of the disc, respectively, nd therefore need not be equl. As we shll see, the richest symptotic limit is obtined by ssuming they re such tht H nd Y re both O1 s ǫ Leding-order equtions We expnd the dependent vribles s follows: p p 0 + ǫ 2 p , u u 0 + ǫ 2 u , w w 0 + ǫ 2 w At leding order, 2.28 shows tht p 0 = p 0 x, t, so tht 2.26 is stisfied utomticlly t leding order. Eqution 2.27 then gives us nd hence 1 A 2 u 0 z = p 0 2 x u 0 = z2 1 p 0 21 A x is the solution tht stisfies u 0 = 0 t z = ±1. At Oǫ 2, 2.28 shows tht 2.32 p 1 z = A 2 u 0 x z + 2 w 0 = Az 2 p 0 z 2 1 A x + 2 w 0 2 z,

9 MMSG while 2.26 gives nd hence, from 2.33, t u0 t x + w 0 = 2 p 0 z x + 2 p 1 2 z 2 { z p 0 21 A x + w 0 2 z } = 1 2 p 0 1 A x + 3 w 0 2 z 3 { = 2 z p 0 z 2 21 A x + w } 0. 2 z This is simply the leding-order version of the diffusion eqution 2.19 for φ. If we integrte with respect to z nd use the fct tht w 0 is n odd function of z to eliminte the constnt of integrtion, we rrive t where ψ = At leding order, the boundry condition 2.30 becomes or, in terms of ψ, p 1 z = ψ t = 2 ψ z 2, z A 3 z p 0 x + w A 2 p 0 1 A x + 2 w 0 = Hp 2 z 2 0 t z = 1, 2 p 0 x Hp 2 0 = ψ t z=1 The boundry condition t z = 1 is identicl, by symmetry. We cn lso write the boundry condition on the verticl displcement, w 0 = Wx, t t z = 1, in terms of ψ, to obtin 2 p 0 x = 31 A W ψ 2 z= Agin, symmetry requires tht n identicl boundry condition be imposed on z = 1; indeed, we cn restrict our ttention to the hlf-disc z > 0 by setting ψ = 0 t z = To conclude, we hve to solve the diffusion eqution 2.34 for ψ with the boundry conditions 2.36, 2.37 on z = 1 nd 2.38 on z = 0. The initil condition for ψ is simply ψ = 0 t t = Notice tht ψ stisfies boundry-vlue problem in z nd t, with x merely prmeter. On the other hnd, p 0 effectively stisfies n ordinry differentil eqution in x, depending only prmetriclly on t. The coupling of these through the boundry conditions 2.36 nd 2.37 leds to mthemticlly interesting, lthough rther tricky problem.

10 MMSG To complete the system, we lso need the leding-order boundry conditions on p 0 t the edges of the disc, nmely p 0 x = Y p 0 t x = ± Note tht this requires us to ssume tht Y is independent of z, which seems physiologiclly resonble. Otherwise, there must be boundry lyers ner x = ±1, of thickness Oǫ, over which z-vritions in p 0 decy. The stress-free boundry conditions 2.31,b must likewise be stisfied cross edge lyers; this structure is consistent with stress profiles like those shown in figure 1. Once we hve solved for ψ nd the dimensionless fluid pressure p, the leding-order devitoric stress components my be determined from 2.11: τ xx = W 0 λ + 2µ u x + λ w z W 0 2µ z2 1 2 p 0 21 A x + λ ψ z τ xz = µw 0 u z + ǫ2 w x µw 0 z p ǫ 1 A x τ zz = W 0 W 0 λ u + λ + 2µ w x z 2µ z A 2 p 0 + λ + 2µ ψ x2 z We my thus quntify the stress experienced by the elstic mtrix of the disc, with the eventul im of estimting the possible consequent dmge nd/or remodelling Simple solutions The initil/boundry vlue problem given by 2.34 to 2.40 is difficult to solve in generl. It my be reduced to n ordinry differentil eqution using Lplce trnsforms, but this isn t relly much help. For the purposes of this report, we consider few simple scenrios for which exct solutions exist. To begin with, suppose tht H is constnt nd we seek time-hrmonic solution with W = e iωt. It is strightforwrd, without solving the diffusion eqution for ψ, to show tht where p 0 = iω cosh kx k sinh k + cosh k 1 Y k 2 = 31 AH 31 A + iω. e iωt, It is then strightforwrd to find the long-time solution for ψ in the form Y k 2 H coshkx sinh z iω ψ = Hk sinh k + cosh k + H sinh iω eiωt. 2.44

11 MMSG Similrly, if the upper nd lower surfces re impermeble, with H = 0, we hve ψ t + 31 A ψ y=1 = 31 AW, y=1 which cn be solved independently of the diffusion problem. Suppose we condider the cse where the disc is uniformly squeezed, so tht W = 1 sy. Agin, it is esy to obtin the pressure, which reds p 0 = Ae 31 At Y x2. The pressure obtined in this cse is kin to tht shown in figure 1, lbeit without the flt centre. We solve for ψ to find tht ψ = sin 31 Az sin 31 A e 31 At z + [ ] 4 nπ sin nπ e n2 π 2t sin nπ1 z, 2.45 nπ 2nπ sin 2nπ provided A m 2 π 2 /3 which cn never hppen since 0 A 1. We cn then solve for θ using θ = ψ 31 z = Ae 31 At cos 31 Az sin 31 A 1 4 [ ] nπ sin nπ e n2 π 2t cosnπ1 z, 2nπ sin 2nπ nd we cn redily see tht θ 1 s t, indicting tht the liquid volume frction decreses s the disc is compressed. We cn lso use the solutions for p 0 nd ψ to find the stress components in the disc; τ xx W 0 3µz 2 1e 31 At + λ 1 4 [ nπ sin nπ 2nπ sin 2nπ 31 Ae 31 At cos 31 Az sin 31 A ] e n2 π 2t cosnπ1 z, 2.46 τ xz 3 µw 0 ǫ xze 31 At, 2.47 τ zz W µz 2 1e 31 At + λ + 2µ Ae 31 At cos 31 Az sin 31 A 1 4 [ ] nπ sin nπ e n2 π 2t cosnπ1 z nπ sin 2nπ Note tht the long time behviour is tht the stress components re constnt; τ xx W 0 λ/, τ xz 0 nd τ zz W 0 λ + 2µ/ s t. The nticipted qulittive behviour, whereby the fluid pressure decys t lrge times nd the elstic stress tkes up the lod, is thus reproduced. Finlly, we consider the sitution in which the discs re deformed, so tht they converge s x increses. We set w = x, nd find tht the pressure is given by p 0 = 1 A e 31 At x 2 { Y Y x2 }.

12 MMSG The solution for ψ in this cse is given by x times the solution given in 2.45 nd, therefore θ x s t. Thus the volume frction of the liquid phse decreses linerly from the centre to the compressed edge of the disc. We clculte tht the components of the stress in the disc in this cse re τ xx W µz 2 1xe 31 At + λx Ae 31 At cos 31 Az sin 31 A 1 4 [ ] nπ sin nπ e n2 π 2t cos nπ1 z, nπ sin 2nπ τ xz µw Y 2ǫ ze 31 At 1 + Y 3x2, 2.50 τ zz W µz 2 1xe 31 At + λ + 2µx Ae 31 At cos 31 Az sin 31 A 1 4 [ ] nπ sin nπ e n2 π 2t cos nπ1 z nπ sin 2nπ Note tht the long-time behviour of the stress components in this cse re τ xx λw 0 x/, τ xz 0 nd τ zz W 0 λ + 2µx/ s t. At lrge times, the stress is nonuniform cross the disc, in contrst with uniform hydrosttic pressure. 2.4 Conclusions In this report we hve developed mthemticl model describing the deformtion of n intervertebrl disc under n pplied lod. We hve modelled the disc s poroelstic mtrix tht is sturted with viscous liquid. We model the presence of the proteoglycns s pressure difference between the two phses. We imposed conservtion of mss nd momentum to the two phses nmely mtrix nd fluid. We reduced the stted model to three coupled equtions for the mtrix velocity, the volume frction of the fluid, nd the pressure in the fluid phse. We simplified the model by ssuming tht the deformtions of the mtrix re governed by liner-elsticity theory. We then utilised the slenderness of the disc to further reduce the model. We solved the simplified model for severl prescribed displcements of the free surfce. The model is first step towrds describing the behviour of n intervertebrl disc. There re numerous extensions to this preliminry investigtion tht could be undertken. Firstly, the coupled problem for ψ nd p 0 eqution 2.34 nd the ssocited boundry conditions could be solved for physiclly relistic sptilly vrying permebility. Secondly, the thin-disc nlysis should be extended to the nturlly more relistic circulr geometry. Thirdly, the non-slender version of the model could be solved numericlly. Finlly, we should enhnce the model to include the effects of chnging the prmeters over time, so tht we cn ssess whether or not remodelling cts to enhnce ny initil symmetry in the system.

13 MMSG References [1] Fitt, A.D., Howell, P.D., King, J.R., Plese, C.P. & Schwendemn, D.W Poroelstic multiphse flow modelling for pper squeezing. Euro. J. Appl. Mths, to pper.