Chapter 4 Multifield Surface Bone Remodeling
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1 hr Mulifild Surf on Rmodling In hr, h horil nd numril rul of inrnl on rmodling wr rnd. Exnion o mulifild urf on rmodling i diud in hi hr. horil rdiion of urf on rmodling in h dihyi of h long on undr vriou xrnl lod r md wihin h frmwork of div li hory. h lod inlud xrnl lrl rur, lri nd hrml lod. wo oluion r rnd for nlyzing hrmolroli rolm of urf on rmodling. h nlyil oluion h giv xlii formulion i l of modling homognou on mril, nd h mi-nlyil oluion i uil for nlyzing inhomognou. umril rul r rnd o vrify h rood formulion nd o how h ff of mhnil, hrml nd lri lod on h urf on rmodling ro. In hi hr, h dvlomn in [-] r rifly ummrizd.. Soluion of urf modling for homognou hollow irulr ylindril on In hi ion, nlyil oluion for hrmolroli rolm of urf on rmodling, d on h hory of div liiy [], r rnd o udy h ff of mhnil, hrml, nd lri lod on h urf on rmodling ro. h oluion i ud o invig h urf on rmodling ro on h i of uming homognou on mril [5]... R quion for urf on rmodling h quion for h hory of div liiy. n xndd o inlud izolri ff y dding om nw rm low [,]
2 U n, Q ij Q, Q, n, Q E Q, E Q, ij zz ij zz rz rz i E r r i E z z i. whr zzzz urf rmodling offiin. E E rz rz r r z z, i r.. Diffrnil fild quion for urf rmodling W now onidr gin h hollow irulr ylindr of on hown in Fig... h hollow irulr ylindr i ujd o n xrnl mrur hng, qui-i xil lod P, n xrnl rur, nd n lri onil lod or/nd hown in Fig... h oundry ondiion r dfind y Eq.5 nd.6. h oluion of dilmn u r, u z nd lri onil o h rolm ov wr givn y Eq.9-.6 in hr. Suiuing Eq.-.6 ino Eq. yild U ln ln U ln ln whr..
3 * * F F zz zz. * * F F zz.5 * P F zz.6 r zr 5.7 * * F F zz zz.8 * * F F zz.9 * P F zz.
4 5 zr r... nd h uri or ui nd rfr o riol nd ndol, rivly. For h ron d in Sion.., Eq. n ud hr, h i U d, d d U. d hu, Eq. nd. n wrin d d ln ln.5 d d ln ln.6 whr... Aroximion for Smll hng in Rdii I i rn h Eq.5 nd.6 r non-linr nd nno, in gnrl, olvd nlyilly. owvr, h quion n roximly linrizd whn hy r lid o olv rolm wih
5 mll hng in rdii. In h on urf rmodling ro, Qin l [] umd h h rdii of h innr nd our urf of h on hng vry lil omrd o hir originl vlu, w don in Sion... hi mn h h hng in nd r mll. hi i ronl umion from h viwoin of h hyi of h rolm. o inrodu h roximion, wo nondimnionl rmr [,],.7 r dod in h following lulion. A rul, nd n wrin,,,.8 Sin oh nd r fr mllr hn on, hir qur n ignord from h quion. onqunly, w n hv h following roximion:.9. ln ln...
6 ln. whr,, ln,.5 hu, Eq.5 nd.6 n roximly nd in rm of nd follow:, d d d d.6 whr
7 .. Anlyil Soluion of Surf Rmodling An nlyil oluion of Eq.6 n oind if homognou rory i umd for on mril. In h, h inhomognou linr diffrnil quion ym.6 n onvrd ino h following homognou on d d, d d. y inroduing wo nw vril uh h,., d M d M.5 M.6 d M.7 h oluion of Eq., uj o h iniil ondiion h nd, n xrd in four oil form h fulfill h hyi of h rolm, i.., whn, nd mu limid qunii, nd h oluion mu l. h form of h oluion dnd on h roo of h following qudri quion: whr rm d M.8 rm.9 All h horilly oil oluion r hown follow:
8 A: whn, nd, Eq.8 h wo diffrn roo, nd, oh of whih r rl nd diin. hn h oluion of h quion r,. whih n lo wrin,. h formul for h vriion of h rdii, i.., nd, wih im n oind y uiuing Eq. ino Eq.7. hu,. h finl rdii of h ylindr r hn lim, lim. : whn, nd, Eq.8 h wo qul roo,. h oluion of h quion r
9 . whih n lo wrin,.5 h formul for h vriion of nd wih im n oind y uiuing Eq.5 ino Eq.7,.6 h finl rdii of h ylindr r hn lim, lim.7 : whn nd, h oluion of h quion r
10 whih n lo wrin,.8,.9 h formul for h vriion of nd wih im n oind y uiuing Eq.9 ino Eq.7, follow:,.5 h finl rdii of h ylindr r hn, lim lim.5 D: whn nd, h oluion of h quion r whih n lo wrin,.5,.5 h formul for h vriion of h rdii wih im n oind y uiuing Eq.5 ino Eq.7. hu.5 h finl rdii of h ylindr r hn, lim lim.55
11 All h ov oluion r horilly vlid. owvr, h fir i h mo likly oluion o h rolm, i i hyilly oil whn []. hrfor i n ud o lul h on urf rmodling.. Aliion of Smi-Anlyil Soluion o Surf Rmodling of Inhomognou on h mi-nlyil oluion rnd in Sion. n ud o lul rin nd r ny oin on h on urf. h rul form h i for urf on rmodling nlyi. hi ion rn liion of oluion.58 o h nlyi of urf rmodling hvior in inhomognou on. [ I i nod h urf on rmodling i im-dndn ro. h hng in h rdii or n hrfor luld y uing h rngulr lgorihm of ingrl Fig... h rodur i drid hr. Fir, l h ring im nd h lngh of im o onidrd, nd divid h im domin ino m qul inrvl / m. A h im, lul h rin nd lri fild uing Eq.-.6. h rul r hn uiud ino Eq. o drmin h norml r of h urf on rmodling. Auming h i uffiinly mll, w n rl U wih i mn vlu U h im inrvl [, + ]. h hng in h rdii or im n hu drmind uing h rul of urf vloiy. Aordingly, h rin nd lri fild r udd y onidring h hng in h rdii. h udd rin nd lri fild r in urn ud o lul h norml urf vloiy h nx im inrvl. hi ro i rd u o h l im inrvl m, ]. Fig.. how h rngulr lgorihm of ingrl whn w rl U wih i iniil vlu hn i mn vlu U h im inrvl [, + ]. U rhr
12 Fig.. Illurion of h rngulr lgorihm. Surf Rmodling Equion Modifid y n Inring Mdullr Pin Suiuing Eq.65 ino Eq.5 nd.6 yild M d d ln ln.56 M M d d ln ln.57 whr F M zz.58
13 F M zz.59 M.6 I n n h Eq.56 nd.57 r imilr o Eq.5 nd.6. onidring Eq.7, Eq.56 nd.57 n imlifid, Y Y Y d d Y Y Y d d.6 whr 5 M Y.6 5 M Y.6 M Y M M Y M M Y.66 6 Y M M.67
14 wih.68 F 5 F F.7 F 7 F.7 Eq.6 i imilr o Eq. nd n hu olvd y following h oluion rodur drid in Sion.... umril Exml for hrmoizolri on onidr gin h fmur ud in Sion.7. h gomril nd mril offiin of h fmur r h m ho ud in Sion.6 x h h volum frion hng i now kn o zro hr. In ddiion, h urf rmodling r offiin r umd o
15 9.6 m /, 7. m /, 5. m /, 8. m /,.6 m /,.8 m / zr zz 9.6 m /, zr m /, nd r zz, 9 V m /.87 m/,.58 m/ nd, r umd. In h following, numril rul r rovidd o how h ff of mrur nd xrnl lri lod on h urf on rmodling ro. h rul for h ff of mhnil loding, inrd in, nd mril inhomogniy on h urf rmodling hvior r omid hr; hy n found in []. W diinguih fiv loding. n MP n=.8,.9,,. nd., P=5, nd no ohr y of lod i lid. h xndd rul for hi loding r hown in Fig.. o udy h ff of xrnl rur on h on rmodling ro. Fig.. illur h h rnvr nd xil lod hv h m ff on h on. h innr rdiu of h on dr whil h our rdiu inr h xrnl rur inr. hi rul in n inr in h on ro ionl r nd, onqunly, hikr nd rongr on. Whn h xrnl rur dr, h innr rdiu inr nd h our rdiu dr, whih mn h h on om hinnr nd wkr. On h ohr hnd, g rur n inr h r of on urf rmodling, whih n lr h rovry of injurd on. Fig.. rn n inring hng of rdiu gin im. h our rdiu of h on inr fir h rnvr rur inr. I gin o dr fr fw dy nd finlly onvrg o l vlu h i g hn i iniil vlu. A imilr rul w rnd y owin nd Vn ukirk []. h rlir nlyi w d on modl of urf rmodling nd w riud
16 o h uounding on mril oming mor or l iff rhr hn ing du o urf movmn dy MP.9MP.MP.8MP.MP... MP.MP.MP.9MP.8MP dy Fig.. Vriion of nd wih im P5, ohr. vru ; vru
17 9.5, 9.8,,.,.5, V, MP, P=5 Fig.. how h ff of mrur hng on on urf rmodling. In gnrl, h rdii of h on dr whn h mrur inr nd hy inr whn h mrur dr. I n lo n from Fig.. h nd r lmo h m. Sin, h hng of h our urf rdiu i normlly g hn h of h innr on. h r of h on ro ion dr h mrur inr. hi lo ugg h lowr mrur i likly o indu hikr on ruur, whr wrmr nvironmn my imrov h rmodling ro wih l hik on ruur. hi rul m o oinid wih ul f. hikr nd rongr on rh mk ron living in Rui look rongr hn on living in Vinm. I hould mniond hr h how hi hng my ff h on rmodling ro i ill n on quion. A n iniil invigion, h uro of hi udy i o how how on my rond o hrml lod nd o rovid informion for h oil u of imod xrnl mrur fild in mdil rmn nd in onrolling h hling ro of injurd on dy 9.5 o 9.8 o o. o.5 o
18 dy 9.5 o 9.8 o o. o.5 o Fig.. Vriion of nd wih im for vrl mrur hng. vru ; vru -6V, -V, V, nd 6V, MP, P= 5, nd =. Fig.. how h vriion of nd wih im for vriou vlu of lri onil diffrn. I n n h h ff of h lri onil i xly ooi o h of mrur. A dr in h inniy of h lri fild rul in dr of h innr nd our urf rdii of h on y lmo h m mgniud. horilly, h rul ugg h h rmodling ro migh imrovd y xoing on o n lri fild. lrly, furhr horil nd xrimnl udi r ndd o invig h imliion of hi xour for mdil ri.
19 V 6V V -V -6V dy... V 6V V -V -6V dy Fig.. Vriion of nd wih im for vrl onil diffrn. vru ; vru P=5, nd inrnl rur i rodud y inring rigid in who rdiu * i g hn. h vlu of nd gin for * -=.mm,.mm, nd.5mm r hown in Fig..5. I n n h oh h innr nd h our urf rdii of h on r rdud fr h in h n inrd, whih will inr h ighn of fi. A ighly
20 fid in n lo inr h r of on urf rmodling, whih n lr h rovry of h injurd on. On h ohr hnd, h rdiu of h in inr, h our urf rdiu of h on dr mor ignifinly hn do h innr urf rdiu. hi rul in dr of h on ro-ionl r nd, hn, in hinnr nd wkr on ruur. Furhrmor, if h fi i oo igh, h rur on h inrf will u dmg o h on ruur. hu h rdiu of h in hould k wihin n rori rng = m = m = m 5 5 dy dy =m =m =5m Fig..5 Vriion of nd wih im indud y olid in. vru ; vru 5 A hollow, inhomognou irulr ylindril on ujd o xrnl lod.
21 h xrnl lod r =MP, P=5, =, nd =V. h gomril nd mril rmr of hi rolm r h m ho ud in h ov, x h ll mril onn h ginning of hi ion r now modifid y mulilir [---r/-], whr nd n rng rduion of iffn h innr urf of h on. I i worh mnioning h y uing h mi-nlyil roh, h form of iffn vriion in h rdil dirion n riy. Fig..6 how h rul of nd h ouid urf of h on for =.995,.99 nd.98. In gnrl, h rmodling r dlin h iniil iffn of h innr on urf dr. h inhomogniy of h on h ignifin ff on h on urf rmodling. A highr urf rmodling r i lwy rld o highr lvl of inhomogniy dy..8.6 =.995 =.99 = dy
22 Fig..6 Vriion of nd wih im for inhomognou on ujd o ouling lod. vru ; vru.5 Exnion o hrmomgnolroli Solid In h of hrmomgnolroli olid, h oniuiv quion i dfind y Eq.7 nd h rmodling r quion Eq. mu ugmnd y om ddiionl rm rld o mgni fild follow: U ij n, Q ij Q ij Q i Ei Q Ei Q G i i Q i Q zzzz rzrz r Er zez Gr r Gz z.7 whr G i i=r,z r urf rmodling offiin, nd E E G G.7 zz zz rz rz r r z z r r z z W now onidr gin hollow irulr ylindr of on, ujd o n xrnl mrur hng, qui-i xil lod P, n xrnl rur, n lri onil lod or/nd nd mgni onil lod nd/or. h oundry ondiion r dfind y Eq.6 nd.77. h govrning quion r dfind y Eq.7, hn h oluion of U nd U o h rolm ov hv h m form ho of Eq. nd. x h h offiin nd r rld y 5 G 5 zr r zr r.75 5 G 5 zr r zr r.76
23 hrfor, h oluion o nd hv h m form ho drid in Sion... A numril illurion of h nlyil oluion drid in hi ion, w onidr gin fmur diud in Sion.7 x h h volum frion hng i now kn o zro hr. In ddiion, h urf rmodling r offiin givn in Sion. r ud. h urf rmodling r offiin for mgni fild i G r m / Amr. W diinguih hr loding :.5, -.,,.,.5, V, A, =MP, P dy -.5 o -. o o. o.5 o
24 dy -.5 o -. o o. o Fig..7 Vriion of nd wih im for vrl mrur hng. vru ; vru Similr o h hown in Fig.., Fig.7 how h ff of mrur hng on on urf rmodling undr h ominion loding of mhnil, lri, nd mgni fild. h rul oind i imilr o ho of Fig.. nd h nlyi for Fig. li hr. -6V, -V, V, nd 6V, MP, P 5, A nd.5 o V V V -V -6V dy
25 ... 6V V V -V -6V dy Fig..8 Vriion of nd wih im for vrl lri onil diffrn. vru ; vru Fig.8 how h vriion of nd wih im for vriou vlu of mgni-fild diffrn. Similr o h hown in Fig.., i n lo n h h ff of h mgni-fild i xly ooi o h of mrur. A dr in h inniy of h mgnifild rul in dr of h innr nd our urf rdii of h on y lmo h m mgniud. -A, -A, A, nd A, MP, P 5, V nd..5..5a.a A -.5A -.A dy
26 dy.a.5a A -.5A -.A Fig..9 Vriion of nd wih im for vrl mgni onil diffrn. vru ; vru Fig.9 how h vriion of nd wih im for vriou vlu of mgni onil diffrn. h hng of h our nd innr urf of h on du o h mgni influn r imilr o h in []. Rfrn [] Qin Q, Qu Y, Y JQ: hrmo lroli oluion for urf on rmodling undr xil nd rnvr lod. iomril 6, [] Qu Y, Qin Q, Kng Y: horil rdiion of urf on rmodling undr xil nd rnvr lod, 7-8. In: Pro. of 9h In onf on Inion, Aril, Rir & Minnn of Sruur, Rn WX, Ong KG, nd n JSY d.. I-Prmir PY D, 5 [] Qin Q-: Muli-Fild on Rmodling Undr Axil nd rnvr od. In: oomingon DR d. w Rrh on iomril ov Sin Pulihr, w York 7 [] owin S, Vn ukirk W: Surf on rmodling indud y mdullry in. Journl of iomhni,
27 [5] Qin Q, Y JQ: hrmolroli oluion for inrnl on rmodling undr xil nd rnvr lod. Inrnionl Journl of Solid nd Sruur 9-, 7-6
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Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..