Time-Dependent Collagen Fibered Structure in the Early Distraction Callus: Imaging Characterization and Mathematical Modeling

O iginal A icle
Time-Dependen Collagen Fibe ed S uc u e in he Ea ly Dis ac ion
Callus: Imaging Cha ac e iza ion and Ma hema ical Modeling
PABLO BLA
´ZQUEZ-CARMONA ,
1
JOSE
´A. SANZ-HERRERA,
1
JUAN MORA-MACI
´AS,
2
JUAN MORGAZ,
3
JAIME DOMI
´NGUEZ,
1
and ESTHER REINA-ROMO
1
1
Escuela Te
´cnica Supe io de Ingenie ı´a, Uni e sidad de Se illa, A enida Camino de los Descub imien os s/n, 41092 Se ille,
Spain;
2
Escuela Te
´cnica Supe io de Ingenie ı´a, Uni e sidad de Huel a, 21007 Huel a, Spain; and
3
Hospi al Clı´nico Ve e ina io,
Uni e sidad de Co
´ doba, C a. Nacional IV-A, Km 396, 14014 Co
´ doba, Spain
(Recei ed 28 Feb ua y 2022; accep ed 9 June 2022; published online 22 June 2022)
Associa e Edi o Umbe o Mo biducci o e saw he e iew o his a icle.
Abs ac —Collagen is a ubiqui ous p o ein p esen in egen-
e a ing bone issues ha expe iences mul iple biological
phenomena du ing dis ac ion os eogenesis un il he deposi-
ion o phospha e c ys als. This wo k combines fluo escence
echniques and ma hema ical modeling o shed ligh on he
mechano-s uc u al p ocesses behind he ma u a ion and
accommoda ion- o-mine aliza ion o he callus issue. O ine
me a a sal bone calluses we e analyzed h ough con ocal
images a di e en s ages o he ea ly dis ac ion os eogenesis
p ocess, quan i ying he fibe o ien a ion dis ibu ion and
mean in ensi y as fibe densi y measu e. Likewise, a ma h-
ema ical model based on he expe imen al da a was defined
o mic omechanically cha ac e ize he appa en s i ening o
he issue wi hin he dis ac ed callus. A eo ganiza ion o he
fibe s a ound he dis ac ion axis and inc eased fibe densi y
we e ound as he bone agmen s we e g adually sepa a ed.
Gi en he deg ee o significance be ween he ma hema ical
model and p e ious in i o da a, eo ganiza ion, densifica-
ion, and bundle ma u a ion phenomena seem o explain he
appa en mechanical ma u a ion obse ed in he issue
heo e ically.
Keywo ds—Collagen, Con ocal mic oscopy, Dis ac ion
os eogenesis, O ien a ion, Ma hema ical modeling, Mine al-
iza ion.
INTRODUCTION
Dis ac ion os eogenesis is a widely-known and ma-
u e clinical field ha is now being spun ou in o o ho-
pedic applica ions, including he ea men o ex emi ies
leng hening, bone de o mi ies, o de ec s.
41,47
The bases
o his p ocedu e a e es ablished on a su gical induc ion
o an os eo omy s abilized by an ex e nal fixa o , a la-
ency pe iod o an ini ial issue o ma ion, and a dis-
ac ion phase when bony agmen s a e g adually
sepa a ed.
19
The e a e no fixed alues o he la ency
phase du a ion, a e, and equency o dis ac ion o all
bone models. Howe e , i s p ope choice is c i ical o
a oid non-unions, bone weakness, o p ema u e consol-
ida ion.
17,19
A 1 mm/day dis ac ion a e and a la ency
pe iod be ween 5 and 7 days a e p edominan alues in
long bone expe imen s.
5,6,20,35,37
Once he equi ed
leng h is eached, a consolida ion phase begins, including
subsequen bone emodeling.
Nume ous s udies ha e a emp ed o un a el he
complex biological p ocess behind dis ac ion os eo-
genesis.
1,17,39,53,55
Following an inflamma o y
esponse and a hema oma o ma ion a e os eo omy,
a syn hesis o a bone callus ma ix occu s in he
neighbo hood o he os eo omized issue. The
sequen ial biopsy analysis o Vauhkonen e al.
55
e-
ealed ha an o ganic ma ix apidly fills he dis ac-
ion gap. Al hough his ma ix is composed o
collagenous and non-collagenous p o eins, ea ly col-
lagen fibe s sec e ed p ima ily by os eoblas s become
i s majo componen , especially he e opolyme s ype
I.
17,30,52,55
The eby, hey assume he ole o collagen
Add ess co espondence o Pablo Bla
´zquez-Ca mona, Escuela
Te
´cnica Supe io de Ingenie ı
´a, Uni e sidad de Se illa, A enida
Camino de los Descub imien os s/n, 41092 Se ille, Spain. Elec onic
mail: [email p o ec ed]
Annals o Biomedical Enginee ing, Vol. 50, No. 12, Decembe 2022 (2022) pp. 1798–1809
h ps://doi.o g/10.1007/s10439-022-02992-3
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0090-6964/22/1200-1798/0 2022 The Au ho (s)
1798
fibe s as he p ima y s uc u al elemen o he ea ly
callus issue.
26
This is one o he phenomena ac i a ed
by he cascade o molecula signals igge ed by he
mechanical o ces in ol ed in dis ac ion.
39
No only
do he collagen fibe s inc ease in densi y du ing he
load-induced dis ac ion phase, bu hey a e also
g adually aligned in he di ec ion o elonga ion
17
and
ma u e s uc u ally. This ma u a ion is unde s ood as
he p ocess in which he fibe s a e c osslinked and
packaged o accommoda e mine aliza ion.
26
Thus, a
cen al fib ous in e zone is o med in he bone callus.
Os eoblas s a his fib ous in e zone a e also espon-
sible o deposi ing os eoid on he collagen bundles o
u he c ys alliza ion.
1
Acco ding o Tomoaia and
Pasca,
53
he eal mechanism o mine aliza ion is s ill
unknown since collagen canno induce he o ma ion
o he ini ial amo phous phase o he calcium phos-
pha e.
All his being said, he influence o he dis ac ion
loads and he mechanical en i onmen on he callus
p ope ies seems undeniable.
27
This ela ionship is no
pa icula ly new and has been in es iga ed o many
yea s in he field o callus issue mechanobiology. Fo
ins ance, he in e es in quan i ying in a di ec way he
mechanical p ope ies o he callus led o he appea -
ance o se e al ex i o s udies based on mechanical
es s a mac o-scale,
40
o nanoinden a ion.
2,28,34
Ne e heless, hei ou comes a e limi ed by di e en
bounda y condi ions o he es s and he numbe o
samples and ime-poin s analyzed. Ins umen ed
fixa o s we e also widely used o con inuously moni o
he dis ac ion o ces in i o and assess he co ec
e olu ion o he callus ossifica ion p ocess indi-
ec ly.
3,8,13,36
None heless, hese s udies a e gene ally
no able o dis inguish be ween he mechanical
beha io o he di e en issues in ol ed in he dis-
ac ion p ocess, including he bone callus, endons,
muscles, o skin. Mo e ecen ly, Bla
´zquez-Ca mona
e al.
6
decoupled he loss o he su ounding so issue
iscoelas ici y and he mechanical e olu ion o he
elas ic fibe s and he callus ex acellula ma ix om
expe imen al da a. A s i ening o he callus issue,
unde s ood as an inc ease in i s elas ic modulus and a
highe esis ance o de o ma ion, was also epo ed
du ing he dis ac ion phase.
6
Acco ding o he biological p ocesses p e iously
desc ibed, he main ing edien s which a e sugges ed o
con ol his mechanical e olu ion a e he syn hesis o
new fibe s men ioned abo e, hei ea angemen in he
di ec ion o ac ion,
50,55
and hei ma u a ion. F om
he au ho ’s poin o iew, he absence o quan ifiable
assessmen s o hese pa ame e s is s ill a majo bo -
leneck in de eloping mo e ad anced nume ical models
o dis ac ion han he cu en ones.
7,21,44
Highly-
de eloped ma hema ical models could po en ially
p edic he mechanobiological e olu ion o he bone
callus issue and expand knowledge o he phenom-
ena esponsible o issue s i ening and i s accom-
moda ion o mine aliza ion du ing dis ac ion
os eogenesis.
S anda d his ological analyses o so issue do no
allow p ope cha ac e iza ion o he o ganiza ion o
fib ils due o i s plana isualiza ion.
30,48
P io s udies
ha e also explo ed a collagenous analysis h ough
scanning elec on mic oscopy. Howe e , i s applica-
bili y is limi ed o connec i e s uc u es made up
mos ly o collagen (e.g., endons, join ca ilage, o
hyd ogels) o isola ed fibe s.
15,38,42
Fo mo e
he e ogeneous so issues, mos ea ly app oaches
succeeded in combining specific dyes, p ope s aining
p o ocols, image pos -p ocessing so wa e, and fluo-
escence mic oscopy, including linea pola ized ligh ,
59
con ocal,
10
o mul ipho on mic oscopy.
4,60
In he
same ein, he objec i e o he p esen s udy is o
implemen imaging echniques on bone callus issue o
a comple e unde s anding o he geome ical and
s uc u al ene s ha could a ec he e olu ion o he
dis ac ion o ces and issue s i ening. The fibe o i-
en a ion and densi y a e quan ified om image s acks
o di e en in e zones o he callus a a ious ime-
poin s o egene a ion. These quan i a i e da a a e
combined in his wo k wi h ma hema ical modeling o
u he in es iga e issue s i ening h ough he bio-
logical phenomena de i ed om dis ac ion os eogen-
esis.
MATERIALS AND METHODS
Tissue P epa a ion
The samples used in his s udy come om in i o
expe imen s o dis ac ion os eogenesis in he igh
me a a sus o six skele ally-ma u e (3–5 yea s old) e-
male Me ino sheep. Animal use and su ge ies we e
app o ed by he Animal E hics o he Uni e si y o
Co
´ doba (Re e ence 2021PI/21) ollowing he Eu o-
pean (2010/63/UE) and na ional (RD 1201/2005) eg-
ula ions. All he specimens ollowed he same su gical
and bone egene a ion p o ocol used in p e ious
s udies
5,6
: os eo omy size o 0.5 mm (blade hickness),
a la ency pe iod o one week, a dis ac ion a e o 1
mm/day du ing 15 days, and an Iliza o - ype ex e nal
fixa ion.
6
The fixa o was composed o ames a ached
o he bony agmen s using six d illed Schanz-pins
and in e connec ed using ex endable ba s wi h a sc ew-
nu mechanism o apply dis ac ion in a con olled
manne .
6
In his s udy, he animals we e slaugh e ed a
di e en ime-poin s o he bone egene a ion p ocess,
as specified in Table 1. Gi en ha his s udy aims o
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Time-Dependen Collagen Fibe ed S uc u e in he Ea ly Dis ac ion Callus 1799
quan i y he e ec s o dis ac ion on collagen fibe s
be o e mine al p ecipi a ion occu s, all he samples a e
so issue, mainly belonging o he dis ac ion s age.
As a e e ence o a u he egene a ion s age, a mos ly
non-mine alized sample o he consolida ion phase was
also included (day 29 a e su ge y o day 22 a e la-
ency). A e sac ifice, ea ed limbs we e ozen a
280 C. A longi udinal sec ion, abou 3 mm hick,
was cu in he sagi al plane o each bony callus
(Fig. 1). This plane was selec ed because no significan
mechanical di e ences we e ound in he on al planes
o dis ac ion samples om p e ious s udies.
34
Cu s
we e pe o med in he limb esh ou o he eeze
using a Femi FM-785XLband saw (Femi, Cas el
Guel o, Bologna, I aly) o ensu e he in eg i y and
conse a ion o he callus so issue.
As shown in Fig. 1a, h ee di e en callus in e zones
we e iden ified o compa e hei collagenous s uc u e:
callus an e io (CA), callus middle (CM), and callus
pos e io (CP). Each in e zone was manually dissec ed.
Samples we e immedia ely fixed in a bu e ed 4%
pa a o maldehyde solu ion a 4 C o 3 h and washed
h ee imes in cacodyla e bu e . A e wa d, hey we e
ozen a 220 C o cu 100 lm sec ions using an
au oma ic c yos a Leica RM 2165(Leica
Mic osys em, We zla , Ge many). A his ological s ain
was used o iden i y collagen om he es o he so
issue. Lo
´pez-Pliego e al.
30
d o e a p e ious his o-
logical s udy in he same o ine me a a sus model
subjec ed o he same biomechanical ac o s.
In amemb anous was epo ed as he p edominan
ype o ossifica ion in he whole se o calluses a di -
e en ime-poin s o he consolida ion phase, days 17
o 98 a e su ge y. The a e age ype I/ ype II a io was
quan ified a 2.48 in hese o ine calluses.
30
Thus, he
p esence o ype II collagen is significan ly educed in
an ea ly s age o mine aliza ion and assumed negligible
in his wo k. Consequen ly, his s udy ocuses on he
mo phological and quan i a i e cha ac e iza ion o he
ype I collagen s uc u es. Pic osi ius ed (PSR) was
selec ed as an ex ensi ely p o en his ological dye o
fluo escence imaging o ype I and III o collagen
fibe s.
4,56,59,60
All issue sec ions we e incuba ed in a
one- ound PSR s aining o 1 h (iden ical solu ion and
du a ion) a oom empe a u e and washed wice wi h
dis illed wa e . The non-aqueous dibu yl ph hala e
polys y ene xylene (DPX) moun ing medium was
applied be o e co e slipping. The sample in he con-
solida ion phase (22 days a e la ency) had a pa ially
ossified CA in e zone. The e o e, only CM and CP
samples we e ex ac ed and analyzed h ough con ocal
mic oscopy.
Fluo escence Image Acquisi ion and Da a Analysis
Fluo escen mic oscopy was pe o med using a lase
scanning in e ed con ocal mic oscope Zeiss LSM7
DUO(Ca l Zeiss AG, Obe kochen, Ge many) wi h
he ollowing objec i es: EC Plan-Neoflua 10x/0.30
Plan-Apoch oma and Plan-Apoch oma 40x/0.95
Co . The 561 nm DPSS @ 40 mW was fixed as he
exci a ion lase line. The esolu ions ob ained wi h
hese objec i es we e 1.731 and 6.926 pixels pe lm,
espec i ely. The absence o au ofluo escence in he
issue was p e iously e ified h ough uns ained sec-
ions o he 7-day sample. Fo each sec ion, z-s acks
we e cap u ed up o he maximum dep h a which
signal was acqui ed (Fig. 1c), ha a ied due o he
biological ma u a ion o he issue.
The fluo escence da a analysis was pe o med using
he open-sou ce image p ocessing package Fiji.
49
The
di ec ional analysis o he collagen fibe s was ca ied
ou on he sum-p ojec ion o 40x s acks o a be e
fibe s’ dis inc ion. The O ien a ionJ plugin was used o
e alua e fibe o ien a ion dis ibu ions based on he
g adien s uc u e enso in a local neighbo hood o
each pixel.
45
A Gaussian-shaped window was fixed o
compu e he s uc u e enso s by sliding he Gaussian
analysis window o e he comple e p ojec ion.
10,12
Gi en ha he me a a sal callus is mos ly subjec ed o
axial o ces due o dis ac ion and limb-loads,
5,6
i
seems easonable o find a g adual longi udinal eo i-
en a ion o he fibe s as epo ed in p e ious wo ks.
50,55
In his way, he p e e ed o ien a ion was assumed as
he di ec ion o dis ac ion owing o he complexi y in
ecognizing he o ien a ion o he specimen du ing he
fixa ion and moun ing p ocesses. Besides, he fluo es-
cence in ensi y was aken as an indi ec measu e o he
fibe densi y e olu ion h ough he egene a ion p o-
cess. In his case, he maximum p ojec ions o he 10 9
z-s acks (8-bi images) we e used o compa e a highe
signal olume. Gi en he di e ence in dep h measu ed
be ween samples, he same numbe o in e media e ocal
planes (4.89 lm z-spacing be ween planes) was used o
his analysis, co esponding o a o al dep h o 39.13
lm. Indeed, he Mean G ay Value was calcula ed,
which compu es he sum o he g ay alues o all he
TABLE 1. Expe imen al in o ma ion o each bone callus
issue analyzed: days a e he la ency pe iod, dis ac ed
leng h un il he day o sac i ice, and phase o he bone
egene a ion p ocess.
Animal
Days a e
la ency
Dis ac ed
leng h [mm] Phase
1 0 0 Dis ac ion
2 3 3 Dis ac ion
3 5 5 Dis ac ion
4 10 10 Dis ac ion
5 15 15 Dis ac ion
6 22 15 Consolida ion
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BLA
´ZQUEZ-CARMONA e al.1800
pixels in he z-p ojec ion di ided by he numbe o
pixels. In no case he backg ound signal was sub ac ed
om he z-s acks because o he low in ensi y compu ed
in hese image in e zones.
Ma hema ical Model o Tissue S iffening
A ma hema ical model was designed o explain and
unde s and he appa en elas ic s iffening o he callus
issue du ing he dis ac ion phase. Once men ioned
hei influence in he in oduc ion sec ion, we
hypo hesized ha h ee key playe s could go e n his
p oblem: he o ien a ion o he fibe s a ound he axial
dis ac ion axis (gh), he collagen and elas ic fibe
concen a ion (gd), and hei bundle ma u i y (gm). In
ligh o he independence o hese h ee s uc u al
agen s, he axial elas ic s i ening was modeled using
Eq. 1.
K ðÞ¼gh ðÞgd ðÞgm ðÞK1ð1Þ
whe e is he dis ac ion ime a e la ency ( 2[0, 15]
days), and K1is a cons an ep esen ing he appa en
s i ness o he collagen fibe s induced a he end o he
la ency phase, wi h he 3D o ganiza ion co esponding
o ha ime-poin ( = 0 days) and wi h a comple e
ma u a ion (gm= 1). The modeling o each o he
biologically conside ed phenomena is de ailed below.
The o ien a ion effec s on fibe ed ma e ials a e
in ica e p oblems in ol ing many ac o s, including
he geome y o he fibe s
23,29
o he ic ion con ac
be ween he fibe s and hei su ounding ma ix. By
in eg a ion o e he uni sphe e, he mac oscopic (o
appa en ) issue s ess in a fibe ed olume becomes
33
:
U ðÞ¼ 1
4p
2p
0
p
0
RTh;/ðÞ Rh;/ðÞph;/ðÞ
sin hðÞdhd/ð2Þ
whe e Rh;/ðÞis he o a ion enso o he fibe o i-
en a ion, Uis he s ess enso a he global configu-
a ion (Fig. 1b, xyz), is he fibe s ess enso a he
local configu a ion x0y0z0
ðÞ,ph;/ðÞis he p obabili y
dis ibu ion unc ion o he fibe s o ien a ion, /ð Þ
ep esen s he azimu hal angle o each fibe in he a-
dial plane (xy-plane), and hð Þis he ele a ion angle o
he fibe s o he longi udinal di ec ion o he me a a -
sus (z-axis), a a gi en ime o analysis. Figu e 1b
shows bo h angles defined in Eq. 2. The eby, Rh;/ðÞis
defined as
33
:
FIGURE 1. (a) Scheme o he cu s pe o med o he issue o each specimen. The h ee in e zones o he bone callus analyzed by
luo escence imaging a e also indica ed: callus an e io (CA), callus middle (CM), and callus pos e io (CP); (b) angle o de ine a
collagen ibe o ien a ion in bo h global (xyz) and local x0y0z0
ðÞcon igu a ions: he azimu hal angle /and he ele a ion angle h; (c)
image p ocessing o o ien a ion calcula ions: acquisi ion o x-s acks, sum-p ojec ion, and pos -p ocessing h ough he
O ien a ionJ pluggin
45
.
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ENGINEERING
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Time-Dependen Collagen Fibe ed S uc u e in he Ea ly Dis ac ion Callus 1801
Rh;/ðÞ¼
cos hðÞcos /ðÞ sin /ðÞ sin hðÞcos /ðÞ
cos hðÞsin /ðÞ cos /ðÞ sin hðÞsin /ðÞ
sin hðÞ 0 cos hðÞ
0
@1
A
ð3Þ
Conside ing his axis configu a ion, he dis ac ion
di ec ion would co espond o he angle h= 0 ad (o
p ad). Conce ning Eqs. 2and 3, he elas ici y in he
fibe di ec ion (z¢) was assumed o con ol he
mechanical p ope ies a he local configu a ion.
23
Thus, he no malized fibe o ien a ion coe ficien gh,
which accoun s o he mechanical p ope ies in he
global z-di ec ion, can be defined as:
gh¼1
4pgh;0
2p
0
p
0
cos2hðÞph;/ðÞsin hðÞdhd/ð4Þ
whe e gh;0is he fibe o ien a ion coe ficien a he end
o he la ency phase ( = 0 days). The p obabili y
dis ibu ion unc ion in he uni sphe e is no malized
as:
2p
0
p
0
ph;/ðÞsin hðÞdhd/¼1ð5Þ
Fo each dis ibu ion ob ained om he expe imen al
image analysis, he s anda d de ia ion was calcula ed
as he squa e oo o he a iance. A linea co ela ion
(as a fi s app oach) o hese s anda d de ia ions was
pe o med o ex apola e he expe imen al da a o he
es o he dis ac ion days. A homogeneous mechan-
ical beha io was assumed in he whole ange o /
h ough a uni o m dis ibu ion. Con e sely, ano mal
p obabili y dis ibu ion unc ion o he angula a i-
able hbe ween [0, p] was buil o each s anda d
de ia ion ex apola ed om he linea fi ing using
Ma lab(Ma hwo ks, Na ick, MA, US). No e ha
he p obabili y densi y unc ion o he ol es wi h ime
in he p ocess o fibe o ien a ion along he dis ac ion
axis. Thus, he in eg al in Eq. 4was nume ically
e alua ed (N= 10e3) o he e ol ing empo al hð Þ
p obabili y densi y unc ions using Eq. 6:
gh ðÞ¼ 1
2gh;0X
N
i¼1
cos2hi
ðÞphi
ðÞwið6Þ
whe e wiis he nume ical weigh , p=N. I should be
ema ked ha ghgh;0has he alue o 1/12pin a
andom fibe a angemen and 1/4pin a ully aligned
fibe configu a ion in he di ec ion o dis ac ion. In
line wi h Kang and Kim,
23
his ac implies a
mechanical con ibu ion h ee imes g ea e in he case
o an alignmen be ween he ac ion di ec ion and he
fibe s compa ed o a andom scena io.
On he o he hand, he mechanics o bio- o na u al-
based ma e ials wi h cellula la ice s uc u es (e.g.,
collagen-based scaffolds, open-cell oam, balsa wood,
o abecula bone) is gene ally well documen ed. In
he li e a u e, hey a e modeled based on he dimen-
sional analysis o de o ma ion mechanisms in hei
o ganized po ous s uc u es.
14
Specifically, assuming
linea mechanics, he elas ic modulus was ound o
depend on he solid p ope ies and he squa e o he
ela i e densi y. In his way, he impac o he collagen
densi y on he callus s i ness h ough he dis ac ion
phase (gd) was calcula ed om he daily fibe densi y
qð Þindi ec ly es ima ed om a empo al co ela ion
o he mean in ensi y da a (linea fi ing as a fi s
app oach), and he ini ial densi y measu ed immedi-
a ely a e he la ency phase (qo):
gd ðÞ¼C1q ðÞ
qo

2
ð7Þ
being he cons an C115.
14,54,58
The naı¨ e and induced fibe s unde go ma u a ion
pa hways o consis en ly inc ease he s abili y o he
ex acellula ma ix and acclima e o he mine aliza-
ion p ocess. This complex p ocess in ol es wide-
anging mechanisms o collagen c osslinking, packag-
ing, and he clea age o C- and N- e minal p opep ides
om he collagen molecules.
25
Koma o a e al.
26
de-
sc ibed he composi e mine aliza ion p oblem h ough
o dina y di e en ial equa ions, including he collagen
ma u a ion, he e ec s o inhibi o s, o he mine al
nuclea ion and g ow h. In pa icula , he o e all ma -
u a ion e ec was modeled wi h a cha ac e is ic con-
s an a e o K2= 0.1 day
21
o di e en clinical
scena ios, including bone de o mi ies and ac u es.
This cons an was es ima ed om he long ime
equi ed o he assembly o he collagen bundles.
9,26
Assuming ha all fibe s a e ini ially naı¨ e a he
induc ion ime, he collagen ma u a ion is exp essed by
he ollowing di e en ial equa ion:
dmc þ 0
ðÞ
d ¼K2nc þ 0
ðÞ ð8Þ
whe e ncand mca e he no malized concen a ion o
naı¨ e and ma u e collagen, espec i ely, and ¢is he
accommoda ion- o-mine aliza ion ime expe ienced by
he collagen fibe s quan ified be o e he dis ac ion
phase (du ing he la ency pe iod, ¢2[0,7] days). Gi en
he o eseeable con inuous induc ion o collagen due o
he mechanical s imulus om dis ac ion, he global
ma u a ion coe ficien o he bone callus (gm) was
calcula ed conside ing he ma u a ion lag be ween he
fibe s om la ency. An in e p e a ion o accumula i e
law, simila o Mine ’s a igue damage accumula ion
ule, was used
32
:
BIOMEDICAL
ENGINEERING
SOCIETY
BLA
´ZQUEZ-CARMONA e al.1802

gm ðÞ¼P
i¼ 0qiðÞmciðÞ
P
i¼ 0qiðÞ ð9Þ
So hen, his ma hema ical model depends unda-
men ally on h ee fi ing pa ame e s: C1, ¢, and K1.
They we e adjus ed using he appa en in i o elas ic
s i ness measu ed du ing dis ac ion in p e ious
s udies
5,6,35
using Ma lab. In he ci ed essays, he
sheep we e subjec ed o he comple e phase o dis-
ac ion using ins umen ed fixa o ba s wi h load cells
Bu s e 8431-6001 (Bu s e , Ge nsbach, Ge many) in
o de o quan i y he eac ion o ce o ha d and so
issues o dis ac ion. The ope a ed limb was aised o
a oid al e ing he dis ac ion o ce wi h in e nal me-
a a sal loads. Dis ac ion o ces we e moni o ed o
20 minu es a e applying he 1 mm bony agmen
sepa a ion. Bla
´zquez-Ca mona e al.
6
also applied he
gene alized Maxwell heological model o he aw da a
so as o disc imina e he mechanical beha io o he
majo componen s: collagen fibe s, ex acellula ma-
ix, and su ounding so issues (e.g., endons, mus-
cles, o skin). Hence, he e olu ion o s a ic callus
esponse a e elaxa ion was associa ed wi h he col-
lagen fibe s s i ening. These collagen fibe s’ s i ening
was he basis o he adjus men o he ee pa ame e s
o he p edic i e model p esen ed abo e. Mo e de ailed
in o ma ion can be ound a Bla
´zquez-Ca mona e al.
5,
6
Bound cons ain s we e imposed on fi ing he
pa ame e ¢in he specified la ency ime ange (0–7
days). Coe ficien s o de e mina ion (R
2
and p- alue)
we e calcula ed o e alua e he significance o e e y
co ela ion and fi ing pe o med.
RESULTS
Figu e 2shows he empo al e olu ion o he col-
lagen s uc u e h ough sum p ojec ions o ep esen-
a i e s acks aken wi h bo h 910 and 940 objec i es
om one callus in e zone, and an image o he ea ed
bone slice cu p io o issue ex ac ion. Quali a i ely,
he fibe s seem o o ien in a p e e ed di ec ion and
inc ease densi y as co ical bony agmen s sepa a e.
Du ing consolida ion, a b oad fluo escence signal can
be seen h oughou he comple e p ojec ion.
The empo al e olu ion o he o ien a ion o he
collagen fibe s is shown in Fig. 3. The no malized
equency o ien a ion dis ibu ion o each callus
in e zone ( ed cu es) and i s a e age (blue cu e) o e
he comple e angula ange, om 2p=2 op=2 ad, a e
ep esen ed o each ime-poin . In his angula e e -
ence, he axial di ec ion o dis ac ion is assumed o be
0 ad. The fi s dis ac ion sample epo s a p ac ically
iden ical beha io in he h ee in e zones wi h a an-
dom fibe o ien a ion acco ding o he dis ibu ions.
Like he fluo escence images, he fibe o ganiza ion
ends o a p e e ed di ec ion as he bony agmen s
a e dis ac ed. In addi ion, he e a e sligh ly mo e
spa ial in e -di e ences wi hin he same bone callus,
especially in he CA in e zone o he end-o -dis ac ion
sample (day 22). In he consolida ion issue, homo-
genei y in o ien a ion seems o be eco e ed wi h a
simila a e age dis ibu ion o he p e ious one. Fig-
u e 4a shows he s anda d de ia ion o he dis ibu-
ions o e he days a e la ency. Despi e he limi ed
analyzed da a due o he limi ed numbe o specimens
and in e zones analyzed, his de ia ion seems o co -
ela e wi h he egene a ion ime significan ly, R
2
=
0.8699 and p- alue <0.01. Conce ning he fibe den-
si y, he ime e olu ion o he mean in ensi y is shown
in Fig. 4b(R
2
= 0.7290 and p- alue <0.01). This
quan i y unde goes a s ong inc ease a e he fi s days
o dis ac ion. Howe e , s abiliza ion is obse ed in
la e s ages o dis ac ion p io o new g ow h du ing
ea ly consolida ion.
A e fi ing o model pa ame e s (which yielded
alues shown in Table 2), e olu ion in he no malized
concen a ion o naı¨ e and ma u e collagen fibe s,
ollowing he model in Koma o a e al.,
26
a e ep e-
sen ed in Fig. 5a. Despi e being a long biological
p ocess, hal o he fibe s a e p one o mine aliza ion
a e a week o ma u a ion (Fig. 5a). Also, changes in
he dis ac ion phase in he ma hema ically modeled
s i ening coe ficien s o he callus issue a e shown in
Fig. 5b. Inc easing h ough di e en beha io s, he
o ien a ion, densi y, and ma u a ion coe ficien a y
be ween 1–1.16, 0.91–2.28, and 0–0.77, espec i ely. In
his espec , Fig. 5c compa es he e olu ion o he
p oposed collagen s i ening ma hema ical model
h ough he dis ac ion phase and he in i o s i ness
o he fibe ed componen o he callus measu ed by
Bla
´zquez-Ca mona e al.
6
and Mo a-Macı´as e al.
36
The fi ing pa ame e s o achie e his end a e shown
in Table 2. Bo h models and da a epo a g adual
s i ening, eaching a ound 50 N/mm a e 15 days o
dis ac ion. Addi ionally, he coe ficien o de e mina-
ion e eals a significan ela ionship be ween he
expe imen al da a and he defined model: R
2
= 0.9795
and p- alue <0.01.
DISCUSSION
This wo k p o ides aluable quan i a i e da a and
ma hema ical models o u u e in silico and in i o
esea ch o del e in o he ole o he mechanics in he
mine al p ecipi a ion and callus ossifica ion o egen-
e a ion p ocesses.
Nume ous wo ks had al eady p o ed a eo ien a-
ion o he fibe s in he di ec ion o dis ac ion o ces
BIOMEDICAL
ENGINEERING
SOCIETY
Time-Dependen Collagen Fibe ed S uc u e in he Ea ly Dis ac ion Callus 1803
h ough his ochemical analysis,
52
demine alized his-
ology and pola ized ligh mic oscopy,
22
o elec on
mic og aph.
43
Ne e heless, as a as he au ho s a e
conce ned, no p io s udy has quan i a i ely cha ac-
e ized his mo phological ea u e e olu ion in bone
dis ac ion. Fluo escence echniques ha e been used o
e alua e fibe s in o he so issue pa hologies,
including ocal old ex acellula ma ix,
4
uninju ed
de mis and sca issue,
10
o in e e eb al disk degen-
e a ion.
60
The closes esea ch ound in es iga ed he
dis ac ion o he small in es ine.
16
They epo ed a
mo e sha ply aligne eo ien a ion in he longi udinal
FIGURE 2. Sum p ojec ion o he luo escence images aken wi h bo h 310 and 340 objec i es and bone callus slices be o e
dissec ion a di e en ime-poin s a e la ency.
BIOMEDICAL
ENGINEERING
SOCIETY
BLA
´ZQUEZ-CARMONA e al.1804
di ec ion o he submucosal collagen fibe s h ough he
pos -ope a i e ime. Ne e heless, he changes
epo ed we e no as no able as hose o he p esen
s udy, p obably due o an ini ial p e e ed di ec ion in
ha issue. Mo e suppo ing s udies can be ound
ega ding fibe densi y da a. Focusing on he dis ac-
ion phase, ou esul s show an inc ease in he p o-
po ion o collagen (Fig. 4b). Vauhkonen e al.
55
his ologically also epo ed a g ow h in he collagen
p opo ion in he o al p o eins om 53 o 88% du ing
4 weeks o o ine adius elonga ion a a lowe dis ac-
ion a e, 0.3–0.5 mm/day. In he same bone model as
he p esen wo k, Lo
´pez-Pliego e al.
30
desc ibed a
his ological inc ease om 1.17 o 3.35% in he collagen
pe cen age o he egene a ed issue be ween he las 5
days o dis ac ion. A simila fibe densi y g ow h a e
is measu ed in he cu en s udy acco ding o he
coe ficien nc(Fig. 5b), being 2.48 du ing he comple e
dis ac ion phase.
The ma hema ical modeling o he elas ic callus
s iffening (Fig. 5) also p o ides in e es ing insigh s.
The achie ed le el o fi ing significance (R
2
= 0.9795
and p- alue <0.01) be ween he model and he
expe imen al da a om Bla
´zquez-Ca mona e al.
6
and
Mo a-Macı´as e al.
35
shows ha , in di e en deg ees,
he h ee modeled coe ficien s (o ien a ion, densi y,
and ma u a ion) play a undamen al ole in he col-
lagenous ne wo k mechanical esponse. Collec i ely, a
simila pa e n o esul s was ob ained in o he wo ks
FIGURE 3. O ien a ion dis ibu ion o collagen ibe s a he
di e en ime-poin s analyzed a e la ency: dis ibu ion pe
callus in e zones ( ed cu es) and a e age be ween all
in e zones (blue cu e). Dis ac ion axis is h= 0 ad, plo ed
in he cen e (2p=2 op=2) o a clea e iew on he wid h o
he dis ibu ion.
FIGURE 4. Time e olu ion o he pa ame e s measu ed by con ocal mic oscopy (mean and s anda d de ia ion pe in e zones) in
he bone callus issue h ough he p ojec ions o he z-s ack luo escence images: (a) s anda d de ia ion o he o ien a ion
dis ibu ion; (b) mean in ensi y.
TABLE 2. Fi ing pa ame e s in he ma hema ical model o bone callus issue s i ening.
Pa ame e Desc ip ion Value
C1Cons an in Eq. 70.92
¢Ma u a ion ime be o e he beginning o dis ac ion 0.15 days
K1Appa en s i ness o ma u e ibe s a e la ency 23.87 N/mm
BIOMEDICAL
ENGINEERING
SOCIETY
Time-Dependen Collagen Fibe ed S uc u e in he Ea ly Dis ac ion Callus 1805
in he b oade li e a u e. As a composi e, many exis -
ing s udies ha e examined a clea influence o fibe
o ien a ion on he mechanics o se e al biological
ma e ials, including co ical bone
46
o collagen gels.
51
Kanungo and Gibson
24
also epo ed an in i o ela-
ionship be ween densi y and mechanical p ope ies in
collagen–glycosaminoglycan sca olds. Fo example,
hei comp essi e elas ic modulus and s eng h in a d y
s a e shi ed wi h he densi y om 32 o 127 kPa and
om 5 o 19 kPa, espec i ely. Rega ding he ma u-
a ion coe ficien , Depalle e al.
11
desc ibed an in silico
ein o cemen o he elas ic modulus, maximum s ess,
and oughness as he densi y o imma u e di alen o
ma u e i alen c osslinks g ows. O he wise, he fi -
ing alues o he pa ame e s specified in Table 2ap-
pea o ha e consis ency when compa ed wi h o he
fi ings in he li e a u e. The cons an o he densi y
coe ficien C1app oaches 1, as ound by Wa en and
K aynik
58
in a e akaidecahe al uni cell model o by
Vajjhala e al.
54
in h ee-dimensional Vo onoi oams.
Be o e he fi s dis ac ion es , he ma u a ion ime
was ela i ely low, ¢= 0.15 days. This sugges s ha
he ma u a ion p ocess o he fibe s occu s mainly
du ing he dis ac ion phase, p obably because la ency
biology ocuses p ima ily on hema oma o ma ion
su ounded by fib oblas s.
17
Finally, he ini ial issue
s i ness (K1= 23.87 N/mm) is in line wi h he p o-
ided in i o da a in he fi s days o dis ac ion.
6,36
Compa ing he ma hema ical model agains o he
expe imen al da a is no an immedia e issue due o he
wide a ie y o bone models and dis ac ion p o ocols
applied in he li e a u e.
18,31,57
Figu e 5d compa es
FIGURE 5. Model ou comes: (a) changes in ime in he no malized concen a ion o naı
¨ e (nc, g een cu e) and ma u e collagen
(mc, blue do ed cu e) du ing a mine aliza ion p ocess
26
; (b) e olu ion o he mul iplica i e coe icien s o he appa en s i ness o
he bone callus h ough he dis ac ion phase: o ien a ion (gh, blue do ed cu e), densi y (gd, ed dashed cu e), and ma u a ion
(gm, yellow cu e); (c) compa ison be ween p oposed ma hema ical model o he elas ic collagen s i ening du ing dis ac ion wi h
expe imen al da a om p e ious in i o wo ks
6,36
; (d) compa ison wi h o he epo ed da a in he li e a u e om Meye s e al.
31
,
Hyodo e al.
18
, and Waande s e al.
57
BIOMEDICAL
ENGINEERING
SOCIETY
BLA
´ZQUEZ-CARMONA e al.1806