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Bone emodeling is a biological p ocess ha de elops in bone issue h oughou i s whole li e ime. I deno es he
p ocess o new bone o ma ion and old bone eso p ion ha con inuously modi ies he in e nal mic os uc u e
and composi ion o bone. The main esul s o bone emodeling a e: (i) o epai he in e nal damage gene a ed
by small-ampli ude loads; (ii) o adap he bone s i ness and s eng h o he speci ic mechanical demand; and
(iii) o con ol he calcium equilib ium in he skele on1,2. Du ing he i s s age o bone emodeling, old bone
is emo ed (wi h i s in e nal c acks) by he os eoclas s. These cells a e ac i a ed by he os eocy es, as he cells
esponsible o de ec ing he en i onmen al signals (e.g., s ains, luid low, change in concen a ion o g adien s
ͷ ǡ ǡ ǡǡ Ǥ
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ȋȌǡǢ×ȋǦ×ȌǢ
× ± Àǡ ȋǦȌǡ Ƭ ǡ
ͻǡͷƪǡǡȀǡͻͶͶͷ;ǡǤ *ǣǤ
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o biochemical subs ances, e c.). This is ollowed by he second s age o new bone p oduc ion pe o med by he
os eoblas s ha ill he a eas p e iously eso bed by he os eoclas s1.
Unde s anding his p ocess is impo an in many applica ions, such as op imizing he ea men o diseases
like os eopo osis, main aining bone densi y in ex eme si ua ions like mic og a i y o long- e m pe iods o dis-
use, o assessing he long- e m e olu ion o he bone su ounding p os heses a e implan a ion. In pa icula ,
os eopo osis in he elde ly (men and women) and especially in pos -menopausal women, is highly p e alen 3.
A educ ion in physical ac i i y o he use o d ugs such as s e oids may p omo e excessi e bone eso p ion,
accele a ing os eopo osis. This disease migh accele a e he educ ion in bone quali y a e implan a ion o os eo-
syn hesis de ices, join p os heses o den al implan s, inc easing he p obabili y o bone ac u e4,5. Finally, i may
cause a educ ion in calcium concen a ion below i s physiological le el, which may p omo e o he diseases6.
Consequen ly, he ea men o os eopo osis wi h an i eso p i e d ugs such as bisphosphona es o denosumab
is widely used7,8.
A pa icula case s udy o he e ec o hese d ugs on he os eopo o ic bone is he one o den al implan a ion.
The inc easing li e expec ancy in de eloped coun ies has boos ed he demand o den al implan s in pa ien s
wi h os eopo osis9. Se e al e iew s udies4,10,11 conclude ha bone healing ime inc eases in os eopo o ic pa ien s
which may endange he success o den al implan a ion12. His omo phological s udies on bone e olu ion a ound
Ti anium implan s in ibia13, in animal models wi h induced os eopo osis14,15, indica ed ha his disease leads o
slowe bone u no e and poo e bone-implan adhesion, which p omo es educ ions in he s i ness and s eng h
o he bone-implan in e ace, which may d i e o low abecula bone densi y. Also, i has been epea edly dem-
ons a ed ha he ailu e a e o den al p os heses and implan s, as well as he associa ed o hopedic equipmen ,
inc eases when ea ing os eopo o ic o low-quali y bone16. Despi e all his, he e is no enough e idence o ban
den al implan s in os eopo o ic pa ien s, al hough a deepe s udy and addi ional imp o emen s a e equi ed.
Se e al me hods ha e been p oposed o imp o e he s abili y o den al implan s in pa ien s wi h os eopo o-
sis, including modi ica ions in he implan design17, in he implan su ace18,19, less in asi e su gical echniques
and complemen a y medical ea men . D ugs like bisphosphona es and denosumab a e usually used o ea
os eopo osis20, despi e ha hei long- e m use o a high dose may cause os eonec osis21–23. These an i eso p i e
agen s dec ease os eoclas ac i i y, hus educing bone eso p ion, bu simul aneously, hey also educe he bone
emodeling a e, which ul ima ely may cause slow mic oc ack epai and a mo e b i le bone24. Al hough bo h
bisphosphona es and denosumab educe he os eoclas ac i i y, hei ac ion mechanism is di e en . Bisphos-
phona es binds o he bone mine al, p e en ing he inhibi o y e ec o ma u e os eoclas s, while denosumab
p ecludes he binding o RANK-L o i s ecep o RANK25.
The disco e y o he RANK/RANK-L/OPG pa hway has been animpo an p og ess in he unde s anding o
bone emodeling26–28. RANK is a p o ein sec e ed by he os eoblas s ha ac s as a ecep o a he memb ane o
p ecu so os eoclas s27, wi h an impo an e ec in he o ma ion, unc ion, and su i al o os eoclas s. Binding
o RANK o i s ligand (RANK-L) causes he di e en ia ion o p ecu so os eoclas s o ma u e os eoclas s28, as
well as he biochemical signalling be ween os eoblas s and os eoclas s, con olling bone emodeling. OPG is a
decoy ecep o o RANK-L26 wi h a highe a ini y han RANK. When OPG a aches o he ecep o si es in he
p ecu so os eoclas memb ane, i p ecludes RANK/RANK-L binding, educing bone eso p ion27. In addi ion
o his main pa hway, o he g ow h ac o s, cy okines, and ho mones, such as
T
GFβ and PTH, a e also in ol ed
in bone homeos asis27. A comple e unde s anding o his RANK/RANK-L/OPG pa hway and i s in e ac ion wi h
he mechanical s ain and wi h ex e nal d ugs such as hose men ioned would help o iden i ying he op imal
dose o pa ien s wi h bone diso de s.
Pe e e al.29,30 analyzed he e ec o an i eso p i e d ugs on he bone emodeling p ocess in pa ien s wi h
os eopo osis u ilizing a ini e elemen model a ound a hip implan a e applica ion o alend ona e. They used
a phenomenological bone emodeling model o in es iga e he e ec o such d ug on he os eoclas ac i i y
ying o es ablish a ela ion be ween he d ug dose and he pa ame e s o he eso p ion pa o he densi y
a e-s imulus cu e. Hambli e al.31 in es iga ed he denosumab e ec on bone emodeling, conside ing a couple
PK/PD and FE model. Al hough hei wo k ga e ise o good p edic ions on he mean bone mine al densi y,
hey did no conside he e ec o di e en loads, no di e en doses on di e en bone ypes. Also, hei damage
model did no conside damage epai . Finally, he e ec on long- e m mine aliza ion and he damage inc ease
induced by he highe mine aliza ion-induced b i leness we e no conside ed ei he . Ma inez e al.32 s udied
he e ec o denosumab on he bone mine al densi y, bu wi hou aking in o accoun he damage e ec , nei he
di e en ypes o bones unde a ious loads.
The e o e, he de elopmen o a pha macokine ic-dynamic (PK/PD) model o bone emodeling ha also
akes in o accoun he mechano-chemical coupling can consequen ly help in p edic ing he bone e olu ion and
beha io a e implan a ion in pa ien s wi h os eopo osis and in op imizing he ea men wi h di e en ypes o
d ugs. In his wo k, we in es iga e he e ec o di e en doses o d ugs on bone emodeling wi h he help o he
PK/PD model p o ided by Ma a he e al.8,33. This model is complemen ed he e wi h a sub-model ha couples
he mechanical signal wi h he RANK/RANK-L/OPG pa hway34. Finally, he esul ing mechano-PK/PD model
is used o analyze he e olu ion o bone in no mal and os eopo o ic mandibles a e den al implan a ion wi h
di e en d ug dosages.
Fi s o all, we ied o alida e he biochemical model desc ibed abo e. Wi h such pu pose, we calcula ed he
e olu ion in ime o wo bioma ke s o bone u no e , Se um N- e minal elopep ide (sNTX), a e applica ion
o di e en doses o denosumab, and u ine C- e minal elopep ide (uCTX) a e applica ion o di e en doses
o Iband ona e. Figu e1a,b show he e olu ion du ing 90 days o sNTX and plasma concen a ions a e admin-
is a ion o a single dose o denosumab. In he i s days, a signi ican dec ease in sNTX was obse ed o any
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dosage. This educ ion is slowly eco e ed om 55 o 95
%
o he ini ial baseline, in 80 days, depending on he
dose. Tha ini ial dec ease is highe o highe doses, al hough he di e ence be ween doses o 0.3 and 1 mg/kg
is small. On he con a y, he plasma concen a ion inc eases by o de s o magni ude in he i s days a e d ug
adminis a ion, wi h subsequen eco e y owa ds he ini ial baseline. Figu e1c shows he e olu ion o uCTX
concen a ion a e in a enous adminis a ion o Iband ona e o 180 days and an in e al be ween successi e
doses o 90 days. The concen a ion o uCTX shows a s ong educ ion in he i s days a e d ug adminis a ion,
up o alues o 80
%
o educ ion o a dose o 2mg o Iband ona e. Then he concen a ion s a s o ise owa ds
i s ini ial baseline, which is eob ained a abou 90 days a e d ug adminis a ion. All hese esul s a e in good
ag eemen wi h hose p esen ed in o he s udies8,33,35,36.
The e olu ion o he bone olume ac ion (Eq.(2)) o di e en bone ypes (os eopo o ic, ρ=0.5 g/cm3 ,
abecula ,
1
.0 g/cm
3
, and co ical, 2.05 g/cm
3
) unde di e en mechanical s imuli o disuse (
ξ=0 ), equilib ium
(
ξ=ξ∗ ), o e load (
ξ=5ξ∗ ) and high-o e load (
ξ=7ξ∗ ), wi h ξ
∗
deno ing he e e ence s imulus (Eq.(7)) a e
depic ed in Fig.2. The ini ial alues a e ob ained by sol ing he s a iona y s a e o Eq. (1), i.e wi hou conside ing
he d ug e ec and unde equilib ium s imulus (
ξ=ξ∗ ). When inc easing he d ug dose, he olume ac ion
inc eases o all ypes o bone. Fo ρ=1.0 g/cm3
,
he maximum inc ease o olume ac ion in he equilib ium
condi ion wi h espec o he con ol case o 0.1, 0.3, 1.0 and 3.0 mg/kg o denosumab was abou 16%, 32%,
53% and 107% espec i ely, while o 0.25, 0.5, 1.0 and 2.0 mg o Iband ona e hese inc eases we e 15%, 27%,
46% and 90% espec i ely.
Figu e1. (a) E olu ion o he se um NTX concen a ion a e adminis a ion o a single dose o denosumab
and compa ison wi h Ma a he’s wo k33 and wi h expe imen s36; (b) e olu ion o he plasma concen a ion a e
adminis a ion o a single dose o denosumab and compa ison wi h Ma a he’s wo k33 and wi h expe imen s36;
(c) changes in he concen a ion o u ine CTX om he baseline a e adminis a ion o a single dose o
Iband ona e and compa ison wi h Ma a he’s wo k8 and wi h expe imen s35.
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The e olu ions o ash ac ion, bone olume ac ion and damage a e shown in Fig.3 when he bone is sub-
jec ed o cons an s ess alues o σ=1.0 MP
a
o ρ=0.5 g/cm3 , σ=7.0 MP
a
o ρ=1.0 cm3 and σ=34.
0
and 54 MP
a
o ρ=2.05 g/cm3 , ha co espond o simila s imuli, and a e adminis a ion o di e en doses
Figu e2. E olu ion o he bone olume ac ion (Eq.(2)) o he con ol case wi hou d ugs and o di e en
doses o denosumab (0.1, 0.3, 1.0 and 3.0 mg/kg) and Iband ona e (0.25, 0.5, 1.0 and 2.0 mg) when applying
di e en s imuli o disuse (
ξ=
0
), equilib ium (
ξ=ξ∗ ), o e load (
ξ=5ξ∗ ) and high-o e load (
ξ=7ξ∗ ) o
di e en bone ypes (
ρ=0.5, 1.
0
and 2.05 g/cm
3
).
ͻ
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o denosumab and Iband ona e. As shown in34 and inFig.3, he ash ac ion dec eases du ing he i s s age o
emodeling, because o he ac i i y o os eoclas s ends o inc ease. When applying one o hose wo d ugs, he
ash ac ion inc eases o all ypes o bones. In he cases o os eopo o ic bone unde σ=1.0 MP
a
, and abecula
bone, he ash ac ion i s inc eases, and hen dec eases o doses o 1.0 and 3.0 mg/kg o denosumab. A e
his pe iod, he ash ac ion begins o inc ease again. The bone olume ac ion inc eases o all ypes o bones,
as shown in Fig.3. This same igu e also shows ha he bone olume ac ion ends o dec ease wi h he s ess
le el in abecula and os eopo o ic bones. The same end can be obse ed when p esc ibing bisphosphona es.
The bone olume ac ion in os eopo o ic bone inc eases in ha la e case up o he maximum densi y allowed.
Also, he inc ease a e o he bone olume ac ion is highe when using denosumab han when applying bis-
phosphona es (Fig.3). Finally, he damage le el in co ical bone (
ρ=2.05 g/cm3 ) shows a g ea e inc ease han
o he wo o he bone ypes. Using denosumab wi h doses o 0.1 and 0.3 mg/kg does no inc ease he damage
le el in os eopo o ic and abecula bones, while, on he con a y, o co ical bone, o hose doses, he damage
le el inc eases. In he case o bisphosphona es, doses o 0.25 and 0.5 mg do no inc ease he damage le el o
os eopo o ic bone. The change in ash ac ion o ρ=1.0 g/cm3 subjec ed o σ=7.0 MP
a
o o 0.25, 0.5, 1.0
and 2.0 mg o denosumab wi h espec o he con ol case a ime =3000 days was abou – 2%, 14%, – 4% and
20% espec i ely, while hese changes o 0.25, 0.5, 1.0 and 2.0 mg o Iband ona e we e abou – 0.7%, – 0.9%,
– 1.5% and 5% espec i ely.
The e olu ions o bone olume ac ion, ash ac ion, and damage o di e en doses o d ugs and di e en
dosage in e als o he os eopo o ic bone, in pa icula , a e depic ed in Figs.4 and 5. Fo a denosumab dose
0.1 mg/kg, inc easing he dosage in e al dec eases he bone olume and ash ac ions. In con as , o a dose
o 0.3 mg/kg and a ime in e al o 60 days, he damage eaches he maximum le el allowed. A e eaching
ha maximum damage, he ash ac ion shows a highe educ ion a e han o o he doses. Simul aneously,
he olume ac ion o his dose is sligh ly educed and hen eaches a new s eady alue a e he maximum
damage is eached. Fo o he ime in e als and he same dose, he bone olume ac ion inc eases wi h ime,
depending on he ime in e al, since damage does no each i s maximum le el. In con as , he ash ac ion
dec eases when inc easing he ime in e al. Inc easing he d ug dose inc eases damage, while sho e ime
in e als d i e o a as e inc ease in damage up o i s maximum le el. Mo ing now o Iband ona e, a dose o
0.25 mg inc eases he damage up o i s maximum alue a ime in e als o 60 and 90 days. The ash and bone
olume ac ions, unlike denosumab, do no s op inc easing a e eaching maximum damage. The same esul
is obse ed o o he doses and dosage ime in e als. I is possible o obse e as e inc eases in he ash and
bone olume ac ions when dec easing he dosage ime in e al o all doses. Reduc ions in olume ac ion
o ρ=0.5 g/cm3 a e 3000 days om he case o 60 days o 90, 120, 150 and 180 days o adminis a ion o 0.1
mg/kg o denosumab we e abou 13%, 31%, 37% and 37% espec i ely, while o Iband ona e hese alues we e
45%, 45%, 56% and 57% espec i ely.
Figu e6 shows he inal densi y dis ibu ion in he whole mandible a e applica ion o he phenomenological
bone emodeling model wi h addi ional iews o se e al c oss-sec ions o compa e such esul s wi h co espond-
ing CT images. Figu e6 shows a cu iew o he densi y dis ibu ion o he educed compu a ional model, while
Fig.6g depic s he assumed densi y dis ibu ion a e applying he densi y educ ion due o os eopo osis.
Finally, he applica ion o he coupled PK/PD and emodeling models he e desc ibed o he mandible model
a e den al implan a ion, physiological mas ica ion loads, and adminis a ion o di e en d ug doses d i es o
he esul s shown in Figs.7, 8, 9 and 10. The bone olume ac ion (Fig.7), damage le el (Fig.8) and ash ac-
ion (Fig.9) a e compa ed wi h he base case wi h he applica ion o no d ug. Fo he wo d ugs analyzed he e,
any dose inc eases he bone olume and ash ac ions and co esponding densi y. Doses o 1.0 and 3.0 mg/kg o
denosumab and 1.0 and 2.0 mg o bisphosphona e p oduce highe damage. As consequence, he elas ic modulus
dec eases in hose egions (Fig.10). The ash ac ion (Fig.9), like damage, inc eases when inc easing he d ug
dose, especially in he abecula bone a ound he implan h eads.
One o he main objec i es o his s udy was o in es iga e he e ec o di e en doses o an i eso p i e d ugs
on bone beha io . An i eso p i e d ugs a eused o ea diseases such as os eopo osis in which he balance
be ween he ac i i y o os eoclas s and os eoblas s is dis u bed37. Denosumab and Iband ona e a e an i eso p-
i e os eoclas - a ge ing d ugs used in he ea men o os eopo osis25,38. These d ugs a ec he bone emodeling
p ocess by di e en ac ion mechanisms. Denosumab binds o RANK-L, educing he binding be ween RANK and
RANK-L, he eby he concen a ion o ac i e os eoclas s on he bone su ace. Bisphosphona es as Iband ona e,
on he con a y, in e e e wi h he os eoclas ac i i y by binding o he bone mine al su ace25. As a consequence,
hese d ugs educe he eso p ion s age in bone emodeling, inc easing, he e o e, he bone olume ac ion
and densi y, and, wi h ha , imp o ing he long- e m bone quali y in os eopo o ic pa ien s. Besides, he mine al
con en o bone inc eases, which causes bone o become mo e b i le and damaged. This may p o oke local
ac u es despi e he highe s i ness and s eng h o he ea ed bone.
The e o e, his ambi alen e ec o an i eso p i e d ugs makes i di icul o p edic hei ne e ec on os eo-
po o ic bone. This p ocess is especially complex when ea ing wi h such d ugs a e implan a ion since in hose
case, i is no only he e ec o d ugs bu also he c i ical change in he mechanical condi ions o he su ound-
ingbone which con ibu es o modi ying he long- e m bone in e nal mic os uc u e. Implan a ion in pa ien s
wi h os eopo osis is, he e o e, challenging39 and i s clinical ea men u ilizing hese d ugs may nega i ely a ec
he success o he implan wi h inc easing os eonec osis24,38.
This happens, o example, a e den al implan a ion, a p ac ice ha has inc eased in ecen yea s in he elde ly,
who ha e an inc eased isk o os eopo osis in he mandible bone, which jus i ies why his p oblem has a ac ed
he in e es o se e al au ho s14,15,40,41. In pa icula , ma hema ical models a e use ul in analyzing hese complex
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p oblems. In p inciple, an ideal ma hema ical model o bone emodeling should ake in o accoun he di e en
bone cells in ol ed and hei main ac i i ies such as p oli e a ion, di e en ia ion, mig a ion, dea h, biochemical
signals p oduc ion, changes in hei exp ession due o biochemical o mechanical signals, and issue eso p ion o
Figu e3. E olu ion o he ash ac ion, olume ac ion and damage o he con ol case, and di e en doses o
denosumab (0.1, 0.3, 3.0 mg/kg) and Iband ona e (0.25, 0.5, 1.0 and 2.0 mg) when applying cons an s ess o 1.0
MPa o an ini ial densi y o ρ=0.5 g/cm3 , 7.0 MPa o an ini ial densi y o ρ=1.0 g/cm3 and 34 and 54 MPa
o an ini ial densi y o ρ=2.05 g/cm3.
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p oduc ion. Also, i should conside he di usion, decay and p oduc ion o g ow h ac o s, ho mones, p o eins,
and o he biochemical subs ances ha con ol he cell beha io . Finally, he pa icula mechanical mic oen i on-
men and i s in e ac ions wi h he chemical eac ions and cell p o ein exp ession should be aken in o accoun .
To implemen all hese p ocesses equi es e y complex mechano-chemo-biological models wi h se e al coupled
mechanisms no ye ully unde s ood, a lo o pa ame e s, many imes unmeasu ed, and a di icul alida ion
due o lack o expe imen al esul s in a su icien numbe and a ie y o si ua ions. E en wi h hese limi a ions,
ma hema ical modeling is a powe ul ool o s udying complex biological sys ems since hey allow us o ind
ou impo an ends, and o quan i y, o a ce ain ex en , he ela ionships be ween causes and e ec s, o es
heo e ical hypo heses, quan i y he e ec s o he di e en pa ame e s indi idually on he beha io o he bio-
logical sys em and o do i ual expe imen s in “wha i ” si ua ions42.
In his pape , we p esen a combina ion o a PK/PD model and a ully-coupled chemo-mechano-biological
bone emodeling app oach ha inco po a es hes imulus e ec on he signaling pa hway be ween os eoclas s
and os eoblas s. The e ec o damage on he signaling pa hway and he local ma e ial p ope ies ha e also been
conside ed. Finally, he mine aliza ion le el is moni o ed along he whole bone li e ime. We ha e p o en ha
hese ypes o models can be used o p edic ing he bone beha io in he mandible a e den al implan a ion
when using an i eso p i e d ugs o imp o ing he long- e m quali y o os eopo o ic bone. In o de o s udy he
applicabili y o his model, he long- e m e ec s o di e en doses o denosumab (0.1, 0.3, 1.0 and 3.0 mg/kg)
and Iband ona e (0.25, 0.5, 1.0 and 2.0 mg) on he mandibula bone su ounding a den al implan we e s udied.
We i s examined he e ec o di e en d ugs and o he mechanical s imulus on he bone beha io . Deno-
sumab does no bind o he bone mine al su ace, unlike Iband ona e, so Iband ona e e ec s las longe a e
s opping i s adminis a ion24,43,44. Conside ing his ac and compa ing he inc ease in bone olume ac ion
a e adminis a ion o hese wo d ugs, we can specula e ha he inc ease in olume ac ion and in mine al
con en in bone induced by Iband ona e will be highe han ha o denosumab. As commen ed, he d ugs he e
analyzed inhibi he ac i i y o os eoclas s, he eby educing bone eso p ion and he e o e, inc easing he bone
olume ac ion and he mine aliza ion. On he o he hand, due o he e ec o calcium, he bone becomes mo e
b i le, and damage inc eases. By compa ing Figs.4 and 5, a low doses o 0.1 mg/kg o denosumab and 0.25 mg
o Iband ona e, we ound a mo e signi ican inc ease in he bone olume ac ion when applying Iband ona e,
while, con a ily, his d ug p oduces highe inc eases in bone ash ac ion and damage han denosumab, lead-
ing o a mo e b i le bone. Also, hese d ugs inc ease he ash ac ionini ially (Fig.3), while he mine alized
po ion o he bone is also ini ially educed. This s age is hen ollowed by he illing o he eso bed bone by he
Figu e4. E olu ion o he olume ac ion, ash ac ion and damage o di e en doses o denosumab (0.1, 0.3,
1.0 and 3.0 mg/kg) when applying a s ess o 1.0 MPa o a bone ini ial densi y o ρ=0.5 g/cm3 , and di e en
dosage ime in e als o 60, 90, 120, 150 and 180 days.
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Figu e5. E olu ion o he olume ac ion, ash ac ion and damage o di e en doses o Iband ona e (0.25,
0.5, 1.0 and 2.0 mg) when applying a s ess o 1.0 MPa o a bone ini ial densi y o ρ=0.5 g/cm3 , and di e en
dosage ime in e als o 60, 90, 120, 150 and 180 days.
Figu e6. Bone densi y dis ibu ion (
g
/cm3 ) in he mandible and se e al c oss-sec ions. (a) Mandible, (b) c oss
sec ion con aining he inciso , (c) CT image o he co esponding inciso c oss sec ion, (d) c oss sec ion o
second igh mola , (e) CT image o he co esponding second igh mola sec ion, ( ) lingual-labial cu iew o
he isola ed model and (g) densi y o he os eopo o ic s a e in he same cu iew.
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os eoblas s, o ming new bone wi h he co esponding nex mine aliza ion. Inc easing he d ug dose in any o
wo cases conside ed also inc eases he bone olume ac ion and damage o all ypes o bone. Fo co ical bone,
o ins ance, which does no need d ug ea men , e en low d ug doses cause bone o become highly b i le,
eaching he maximum le el o damage as e . In his bone, no dec ease in he mine alized ac ion was de ec ed
a e d ug ea men , so mic odamage p og esses in ime, and a s ess ac u e may occu .
The s ess le el also in luences his beha io by p omo ing o delaying he damage a e. Fo abecula o
os eopo o ic bone, low d ug doses inc ease he bone olume ac ion wi hou subs an ial damage inc ease.
This is mo e e iden in os eopo o ic bone. Consequen ly, d ug applica ion is bene icial in low-densi y bone,
al hough his ea men always inc eases b i leness, which may comp omise he success o den al implan a-
ion. This sub le con ol be ween hese wo opposi e e ec s is essen ial when de ining he ea men p o ocol
Figu e7. Bone olume ac ion dis ibu ion a e 540 days o simula ion o di e en doses o (a) 0.1, (b) 0.3,
(c) 1.0, (d) 3.0 mg/kg o denosumab and doses o (e) 0.25, ( ) 0.5, (g) 1.0, (h) 2.0 mg o bisphosphona es and (i)
con ol implan .
Figu e8. Damage dis ibu ion a e 540 days o simula ion o di e en doses o (a) 0.1, (b) 0.3, (c) 1.0, (d)
3.0 mg/kg o denosumab and doses o (e) 0.25, ( ) 0.5, (g) 1.0, (h) 2.0 mg o bisphosphona es and (i) con ol
implan .
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Ǥ Damage is he e associa ed wi h he densi y o mic oc acks and s ongly a ec s he mechanical
p ope ies o bone as well as he signaling p ocess among cells. He e, he mic oc ack densi y, h, is assumed o
ha e a linea ela ion wi h he damage le el d, such as h=kd , wi h k=0.0003
4
55.
Bone, as a li ing issue, is able o epai hose mic o-c acks, so damage inc eases when ha ing high s esses/
s ains,
˙
dacc (damage accumula ion a e), while, a he same ime, mic oc acks a e emo ed in egions whe e
bone is eso bed,
˙
d e
p
(damage epai a e)2. We can hen w i e:
As s a ed in34 he damage accumula ion o a ce ain numbe o cycles is a unc ion o he load ampli ude and
he ype o s ess s a e ( ension, dacc
,
, o comp ession, dacc
,
c ) can be w i en as:
which δ1
,
2 ,
γ
1
,
2 and
C
1
,
2
,
3 a e pa ame e s. N is he numbe o load cycles and ε=√2u/E is he equi alen s ain
in each o hose cycles, which desc ibed be o e, wi h u he s ain ene gy densi y and E he elas ic modulus. E∗
is he he e e ence elas ic modulus which o undamaged (
d=
0
) co ical bone he a io E
/
E∗ is equal o one.
Fa igue li e (
N ) in comp ession and ension calcula ed as:
(15)
˙
d=˙
dacc −˙
d e
p
(16)
dacc,c=−
1
γ1
ln1−C1εδ1N,
dacc, =1−γ2
1
C3
lneC3−C2εδ2N,
δ1=10.3, γ1=−5.238E/E∗ε−6100+710−3,C1=
1−e−γ1
9.333 ×1040 ,
δ2=14.1, γ2=−0.018E/E∗ε−4100+12, C2=
eC3−1
1.445 ×1053 ,C3=−20
,
Table 2. Values and desc ip ion o he mechanical pa ame e s.
Desc ip ion Uni Value Re e ences
Denosumab
Dose D ug dose mg/kg 0.1, 0.3, 1.0, 3.0 33
ka
Abso p ion a e 1/day 0.167 33
k
in
D ug–ligand complex in e naliza ion 1/day 2.67 ×10−
2
33
kel
Elimina ion a e o d ug om cen al compa men 1/day 2.12 ×10−
2
33
K
D
KD=
k
o
ff
/
k
on M3.0 ×10−1
2
33
VC
/
FCen al compa men olume l/kg 0.114 33
R
ss S eady-s a e ee ligand concen a ion nM 1.07 33
I
ma
x
Maximal ac ional ex en o inhibi ion – 0.331 33
I
C5
0
Concen a ion p oducing 50
%
o maximal inhibi ion nM 2.64 33
k
ou Ra e o loss o esponse 1/day 0.572 33
Iband ona e
Dose D ug dose mg 0.25, 0.5, 1.0, 2.0 8
Vpl Plasma compa men olume l 4.30 35
Vp1Pe iphe al-1 compa men olume l 2.80 35
Vp2Pe iphe al-2 compa men olume l 8.70 35
V
b
Bone compa men olume l 609.00 35
Q
p
1
Plasma-pe iphe al-1 compa men al clea ances l/day 69.43 35
Vp2Plasma-pe iphe al-2 compa men al clea ances l/day 18.57 35
V
b
Plasma-bone compa men al clea ances l/day 51.71 35
CL Renal clea ance l/day 57.00 35
KS uCTX o ma ion a e μg mmol CR−1day−
1
231.43 35
KD uCTX deg ada ion a e 1/day 0.68 35
R
a Limi ing alue o uCTX o ma ion a e μg mmol CR−1day−
1
194.29 35
k
q
q
Ra e cons an by which
R
a ob ained l/day 0.0024 35
I
C5
0
Iband ona e concen a ion p oducing 50% o maximum esponse μgl−
1
0.37 35
nHill coe icien – 1.92 35
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Finally, a Mine ule59 was used he de e mine he inc ease in damage o a ce ain numbe o cycles, using
he bone a igue li e (
N ) o each s ain le el and o a pa icula bone calcium con en as s a ed by Ma inez
e al.57, being his la e di ec ly ela ed wi h he ash ac ion as
[
Ca]=
2
5
9
.
2
0.69
α
60. The ela ionship be ween K
and
he amoun o calcium (
[
Ca
]
) in he bone is exp essed by he ollowing equa ions:
whe e δ
i
and
β
a e cons an s. ε
u
is ul ima e s ain which has ela ion wi h calcium con en :
Finally damage epai e olu ion is calcula ed as ollows:
whe e ˙ R=k
esC
is he a e o bone olume ac ion due o os eoclas s ac i i y.
Ǥ Finally, as a i s app oach, and despi e he well-known local o ho opy o bone
issue61, we assumed bone issue as he e ogeneous and iso opic wi h i s mechanical p ope ies de ined by he
ollowing co ela ion be ween he olume ac ion (
b
), ash ac ion (
α
) and damage (d), wi h he bone elas ic
modulus55,62:
A summa y o he mechanical pa ame e s used in hese s udy a e p esen ed in Table3.
Finally, a scheme o he chemo-mechano-biological bone emodeling model, coupled wi h he PK/PD models
o he wo d ug con en s, is illus a ed in Fig.11.
Ǥ Compu ed omog aphy images o a heal hy adul woman we e used o con-
s uc he 3D geome ic model o he mandible. A e segmen ing he mandible and ee h in MIMICS 10 (Ma e-
ialise, Leu en, Belgium) and ob aining he STL iles, CATIA (CATIA V5, Dassaul Sys èmes, Vèlizy-Villacou-
blay, F ance) was used o c ea e he inal h ee-dimensional geome y o he mandible, ee h, and PDLs (see
Fig.12). The gaps be ween he mandible and he ee h we e used o ob ain he geome y o he PDLs. The
implan selec ed o his analysis is based on he INTRI design wi hou in e nal esilien pa s68. The heigh o
he implan was 11mm and i s diame e s a he op and bo om we e 5.1mm and 4.5mm wi h wo h eaded
s eps o 1 and 0.5 mm espec i ely. Finally, he c own was designed o he i s igh mola , aking in o accoun
he implan neck.
The model he e de eloped is based on physiological mechanisms and p ope ies, so, con a y o o he phe-
nomenological bone emodeling models, i leads o w ong esul s when he ini ial densi y dis ibu ion is no
physiological and ela ed o he ini ial alues o he cell concen a ions. The e o e, he geome y in Fig.12 was
used i s o ob ain he ini ial densi y dis ibu ion o he nex simula ions. A phenomenological bone emodeling
model56,69 was used o his pu pose, conside ing he mas ica ion muscles’ eac ion o ces, while he bounda y
condi ions we e applied o each ee h in ol ed in he mas ica ion p ocess based on p e ious s udies63,70. A e
(17)
N
=Ki
εδi
=⎧
⎨
⎩
9.333×1040
E
E∗ε10.3 in comp ession
1.445×1053
E
E∗ε14.1 in ension ,i=ccomp ession, ( ension
)
(18)
K
([Ca])=107
εu([Ca])
β
δ
(19)
logεu=25.425 −11.341log[Ca]
(20)
˙
d ep =˙ R
d
b
,
(21)
E
=84370
2
.5
8
bα
2
.74(1−d)
Table 3. Values and desc ip ion o he mechanical pa ame e s.
Desc ip ion Uni Value Re e ences
NNumbe o cycles – 10000 (500 o mandible) 55,63,64
mWeighing exponen – 4 55,56,65
ξ
∗
0
Re e ence equilib ium s imulus 0.0025 55
cS imulus ac i a ion cu e pa ame e – 0.0025 55
aDamage ac i a ion cu e pa ame e – 20 –
d0
Ini ial damage – 0 55
α
ini Ini ial ash ac ion – 0.6 55,65,66
α0
Minimal ash ac ion – 0.45 55,66,67
α
ma
x
Maximum ash ac ion – 0.7 55,66,67
κ
Seconda y mine aliza ion pe iod yea s 6 55
β
Fa igue limi coe icien – 5 –
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simula ion o he comple e mandible (540 simula ion s eps), we co ela ed such ini ial densi y dis ibu ion wi h
he elas ic modulus poin -wise, using he ollowing co ela ions: E=1736ρ3.2 and E=2014ρ2.5 o co ical and
abecula bone, espec i ely63. Finally, hose alues we e modi ied o ake in o accoun os eopo osis. Reduc ions
o 33% 66% ha e been epo ed o he modulus o elas ici y o co ical (
ρ>1.2g/c
c
) and abecula bones
(
ρ<1.2g/c
c
), espec i ely, in os eopo o ic pa ien s17. The e o e, we modi ied he ini ial dis ibu ion o he elas ic
moduli in each o he bone ypes conside ing such alues.
To educe he compu e ime, a cu o bone, which includes he p emola oo h and i s PDL, he second
mola and i s PDL and he implan , was isola ed o pe o m he nex simula ions. The whole geome ic model,
oge he wi h he densi y dis ibu ion (and co esponding ma e ial p ope ies), was hen expo ed o ABAQUS
(ABAQUS 6.11, Dassaul Sys èmes, Vèlizy-Villacoublay, F ance) o pe o m he ini e elemen simula ions. A
ou -noded solid e ahed al mesh was buil in ABAQUS-CAE. The inal numbe o elemen s was ob ained
a e ensu ing su icien accu acy in a p e ious con e gence analysis. The inal esul ing numbe o elemen s o
he whole mandible, ee h, PDLs, implan , and c own in he inal model was 860064, 233083, 30369, 46939 and
5325, espec i ely, while he numbe o elemen s in he model o he sec ion cu o bone, ee h, and PDLs was
448265, 12896 and 4729 espec i ely.
Comple e osseoin eg a ion was assumed o he bone-implan , bone-PDL and PDL- oo h in e aces. Dis-
placemen a he nodes o he mesial and dis al su aces o he cu model was imposed wi h alues de i ed om
he esul s o he comple e mandible model (Fig.12). The Ti anium implan and c own ma e ials we e assumed
as linea ly elas ic wi h E=118
G
Pa
,
ν=0.35 and E=82.8
G
Pa
,
ν=0.33 espec i ely71,72. Finally, he bone
mechanical p ope ies change du ing he bone emodeling p ocess, so a use ma e ial (UMAT) sub ou ine
o ABAQUS was implemen ed o compu e such p ope ies along he loading p ocess acco ding o he model
desc ibed abo e. To simula e he e ec o d ug ea men in he bone su ounding he den al implan , di e en
d ug doses we e used and he co esponding esul s compa ed wi h he con ol model wi hou he d ug.
Recei ed: 29 July 2020; Accep ed: 12 Janua y 2021
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