A MICRO-MACRO EVALUATION OF THE VERTEBRAL BONY ENDPLATE PERMEABILITY BASED ON
COMPUTATIONAL FLUID DYNAMICS
1And ea Maland ino, 2Ch is ian Hellmich, 1Jé ôme Noailly and 1Damien Lac oix
1Ins i u e o Bioenginee ing o Ca alonia, Ba celona, Spain; email: amaland ino@ibecba celona.eu, web: www.biomechanics.es,
2Ins i u e o Mechanics o Ma e ials and S uc u es. Vienna, Aus ia,
SUMMARY
The in insic pe meabili y is an impo an pa ame e ha
desc ibes he esis ance o a po ous s uc u e o luid low. I
has a key ole in po oelas ic ini e elemen models o spinal
segmen s, especially a he e eb al endpla e, i.e. he in e ace
be ween in e e eb al disc and e eb a. In he unde s anding
o he p ope ies o he complex endpla e sys em, an explici
e alua ion o pe meabili y o subchond al bone is missing.
Thus, a new me hod was p oposed o e alua e he in insic
pe meabili y o he bony endpla e. CT-based econs uc ions
o he bony endpla e om a lumba e eb a we e analyzed
using compu a ional luid dynamics, and he in insic
pe meabili y and po osi y o he s uc u e we e calcula ed.
Resul s showed ha he pe meabili y did no depend on he
luid low di ec ion, and was s a is ically simila o bo h he
supe io and in e io endpla es. Pe meabili y alues a ied
wi hin he ange o abecula bone, while po osi y alues
we e lowe han abecula bone cha ac e is ic alues. Finally,
in insic pe meabili y co ela ed well wi h po osi y h ough he
Kozeny-Ka man model, which o e s pe spec i es o
pa ame ic s udies in ol ing degene a i e o age- ela ed
changes a he disc-bone in e ace.
INTRODUCTION
Resis ance o luid low is a key p ope y o he in e ace
be ween in e e eb al disc and e eb ae, which s ongly
in luences spinal segmen biomechanics and anspo o
nu ien s [1]. A pa ame e ha desc ibes such a esis ance is
he pe meabili y, which is o cen al impo ance in po oelas ic
modeling o he in e e eb al disc [2]. Due o ex emely
educed hickness o he subchond al bone a he disc
in e ace, expe imen al echniques we e no able o explici ly
e alua e pe meabili y o he bony endpla e po ion a he
mac oscopic le el. As such, pe meabili y alues ex ac ed
om pe mea ion expe imen s as well as he low di ec ion-
dependen di e ence o he esul s [1,3] we e likely mo e
ep esen a i e o he adjacen ca ilaginous laye ha
consolida es mo e and is less pe meable han bone. Howe e
wi h aging, degene a ion and/o endpla e calci ica ion, he
ca ilage endpla e becomes hinne [3], inc easing he
impo ance o he subchond al bone pe meabili y. A di ec
e alua ion o subchond al bone pe meabili y and po osi y
would he e o e p o ide inpu po oelas ic pa ame e s o
modeling degene a ion/calci ica ion a he endpla es.
Mic o-compu e omog aphy ( CT ) o e s a bone s uc u al
mapping om which i is possible o simula e he pe mea ion
o luid and hus e alua e he esis ance o luid low. Thus, by
using CT-based ini e elemen models, he aim o his s udy
was o calcula e he pe meabili y o he bony endpla e h ough
compu a ional luid dynamics (CFD) analyses.
METHODS
CT scans (X emeCT, Scanco Medical AG, Swi ze land) o a
L1 human e eb a we e used. The oxel esolu ion was 12
m. 3D pa allelepiped (2.5 x 2.5 x 3 mm3) models we e
gene a ed o CFD analyses using mixed hexahed al and
e ahed al ini e olumes wi h ScanIP© (Simplewa e L d.).
The models conside ed a squa ed c oss sec ion (2.5x2.5 mm2)
o he bony endpla e. The hi d dimension was aligned wi h
he supe io -in e io di ec ion and coincided wi h he luid
lux di ec ion. T abecula bone was emo ed in o de o lea e
he comple e endpla e pa in he model. Thus, he hickness o
he po ous zone was a iable om sample o sample,
depending on he pa icula mo phology obse ed du ing he
image segmen a ion (Fig. 1). Endpla e pe mea ion was hen
simula ed. A mass lux bounda y inle o 10-5 kg/s was applied
a a dis ance o app oxima ely 2 mm om he endpla e. The
p essu e a he ou le c oss sec ion was null. Flow-in ( om
e eb a o disc) analyses we e pe o med on 11
pa allelepipeds om he in e io pa o he e eb a and 11
om he supe io one using FLUENT© CFD so wa e. Flow-
ou ( om disc o e eb a) analyses we e also pe o med on
ou pa allelepipeds om he supe io pa and ou om he
in e io one. The mac oscopic pe meabili y was e alua ed
wi h he Da cy ela ion
)( ou in
inin
PPA
LQ
PA
LQ
K
whe e A is he c oss sec ional a ea, K is he hyd aulic
pe meabili y, Qin is he mass lux applied a he inle , is he
luid densi y and L is he dis ance be ween he wo sec ions
whe e he p essu es Pin and Pou we e compu ed. The in insic
pe meabili y, k, independen o he pa icula luid simula ed
was calcula ed om K and he dynamic iscosi y, :
k = K
Po osi y was e alua ed o each sample by di iding he
segmen ed luid olume wi hin he endpla e by he o al
olume o luid- illed s uc u e. Resul s we e s a is ically
analyzed wi h Mini ab® (Mini ab Inc.) o sc een ou he e ec
o loca ion (supe io s. in e io ) and low di ec ion (in s.
ou ).
Figu e 1: Ske ch o he 3D model used o CFD analyses
( low-in case).
RESULTS AND DISCUSSION
No s a is ically signi ican di e ences we e ound be ween
in insic pe meabili y alues om he supe io and in e io
endpla es (p=0.19). The mean alue o he supe io endpla e
was 10.7×10-10 m2 while o he in e io endpla e, he alue
was 6.74×10-10 m2 (Table 1). The a e age in insic
pe meabili y alue o bo h supe io and in e io
pa allelepipeds was 8.73×10-10 m2. No s a is ically signi ican
di e ences we e ound be ween low-in and low-ou
pe meabili y (Pai ed T- es , p=0.443).
The po osi y was also no s a is ically di e en be ween
in e io and supe io endpla es (p=0.291), he mean global
alue being 64%. A posi i e co ela ion was ound be ween
he pe meabili ies and po osi ies pooled o supe io and
in e io endpla e by using a Kozeny-Ca man model (R2=0.79)
o he ype
2
V
S
c
k
, whe e SV is he speci ic su ace, i.e. he
in e nal su ace o he po es pe uni olume o he solid
ma ix [4].
Pe meabili y
[×10-10 m2]
Po osi y [%]
Supe io
10.7 7.10
65.1 7.4
In e io
6.74 6.77
62.9 7.1
Table 1: Pe meabili y and po osi y alues ound and ela i e
s anda d de ia ions.
The ange o in insic pe meabili ies, k, ound in his s udy
(1.16×10-10 o 2.73×10-9 m2) can be compa ed wi h p e ious
measu emen s on abecula bone (2.68×10-11 o 2.00×10-8 m2)
[5], whe eas i is ou o he ange o alues ound o co ical
bone (5×10-15 o 6.35×10-13 m2) [4]. Howe e , he mean
po osi y ound o he endpla e was lowe han he
cha ac e is ic alue usually aken o abecula bone and
highe han ha o co ical bone, i.e. 80% and 0.05%,
espec i ely [6]. This suppo s he idea ha he ascula
openings p esen in he endpla e subchond al bone a e
specialized o bo h he exchange o nu ien s be ween
e eb ae and in e e eb al discs, and he emo al o disc
was e p oduc s [7]. The e o e, endpla e pe meabili y could be
an impo an pa ame e o moni o o he unde s anding o
in e e eb al disc degene a ion in case o se e e subchond al
bone calci ica ion.
The non-co ela ion be ween luid low-in and low-ou in he
p esen s udy con i ms ha he obse ed dependency in
expe imen s [1,3] should be a ibu ed only o ca ilaginous
endpla e and no o subchond al bone [1]. The co ela ion
ound be ween po osi y and pe meabili y, oge he wi h u u e
s udies using mic o- o mac o homogeniza ion app oaches,
could be use ul o pa ame ically explo e he disc anspo
and biomechanics in spinal segmen s as a unc ion o he
subchond al bone calci ica ion s a e.
To ou knowledge, his is he i s s udy ha a emp s an
explici e alua ion o he subchond al bone pe meabili y
wi hou associa ing i o he co ical o abecula bone. Also,
he p oposed me hod o compu e he mac oscopic pe meabili y
excludes a possible biasing om he ca ilaginous pa o he
in e ace disc- e eb a.
CONCLUSIONS
The esul s om his CFD-based e alua ion o he
mac oscopic pe meabili y o he e eb al endpla e bone
showed ha bony endpla e pe meabili y is compa able o ha
o abecula bone. No in luences o he e ical loca ion and
luid low di ec ion we e ound. The alues o pe meabili y
and po osi y ob ained in his s udy a e ele an o modeling
pu poses and may be use ul in he u u e o imp o e ou
unde s anding o he degene a i e changes occu ing a he
disc-bone in e ace.
ACKNOWLEDGEMENTS
Financial unding is acknowledged om he Eu opean
Commission (NMP3-LA-2008-213904) and he Spanish
Minis y o Science and Inno a ion (Acción In eg ada
AT2009-002 and FPU p e-doc o al ellowship AP2008-
03317).
The au ho s a e also g a e ul o Heinz Redl and Ka l K opik
om he Aus ian Cen e o Tissue Enginee ing and Ludwig
Bol zmann Ins i u e o Expe imen al and Clinical
T auma ology, o p o iding a CT da a se o a human
e eb a.
REFERENCES
1. Ayo e D C, e al., Jou nal o O hopaedic Resea ch.
19:1073-1077, 2001.
2. Maland ino A, e al., Jou nal o Biomechanics. 42:2780-
2788, 2009.
3. Accadbled F, e al., Spine. 33:312-619, 2008.
4. Cowin. Bone Mechanics Handbook, CRC p ess, Boca
Ra on, Flo ida, 2001.
5. Nauman E A, e al., Annals o Biomedical Enginee ing.
27:517-524, 1999.
6. Recke , R R, Bone His omo phome y: Techniques and
In e p e a ion, CRC P ess, Boca Ra on, Flo ida, 1983.
7. Benneke L M, e al., Spine. 30:167-173, 2005.
