in vivo study of human mandibular distraction osteogenesis ... · acta of bioengineering and...

Acta of Bioengineering and Biomechanics Original paperVol. 14, No. 4, 2012 DOI: 10.5277/abb120401

In vivo study of human mandibular distraction osteogenesis.Part I: Bone transport force determination

ANNE-SOPHIE BONNET1*, GUILLAUME DUBOIS2, PAUL LIPINSKI1, THOMAS SCHOUMAN3

1 Laboratory of Mechanics, Biomechanics, Polymers and Structures, Ecole Nationale d’Ingénieurs de Metz, Metz, France.2 Orthopédie Biomécanique Locomotion, Chatillon, France.

3 AP-HP, Department of Maxillofacial Surgery, Pitié-Salpêtrière University Hospital, Faculty of Medicine, Paris, France.

Distraction Osteogenesis (DO) is a surgical technique used to reconstruct bone defects. The evolution of forces acting during DO isknown to be strongly influencing the clinical issue of the treatment. The aim of this study was to determine experimentally the time-dependent forces imposed on bone regenerate by a distraction device in the case of a mandibular DO consecutive to a gunshot wound. Toevaluate the bone transport forces, some fixing pins of the distraction device were equipped with strain gauges. Measurements were doneduring the first weeks of the treatment. An equilibrium analysis was achieved to determine the forces acting in bone regenerate fromstrains in the pins. Those quantities evolved during the records approximately from 5 N to 3–4 N and 2 N to 0–0.3 N for the tension andshear forces, respectively, depending on the record duration. For the longest record, the callus lengthening reached 0.17 mm during75 minutes. This decrease of force and simultaneous callus extension can be attributed to the viscosity of regenerate and the elastic en-ergy release of the device. Essential data were obtained concerning forces, extension and their evolution during mandibular DO. The lowforce level obtained was attributed to the absence of resistance of the soft tissues in the case of ballistic trauma restoration.

Key words: experimental method, mandibular distraction osteogenesis, transport force

1. Introduction

Distraction osteogenesis (DO) consists in thelengthening of a bone by means of a mechanical de-vice. Bone is split into two parts and one is thenmoved away from the other at an adapted rate. If thisrate is too high, DO will fail to create new bone in thegap and if it is too low, exaggerated resistances inlengthening can be encountered due to early consoli-dation (ILIZAROV [1]). Bases for DO have been set upin the fifties of the 20th century (ILIZAROV [2]). Sincethe end of the 1980’s, DO has been widely and suc-cessfully used in craniofacial surgery for differentpurposes (KARP et al. [3], MCCARTHY et al. [4]). Theindications of this technique are mainly congenital

malformations and craniofacial traumas. For the firstcategory of indications, internal devices are generallyused to achieve the distraction osteogenesis. Con-cerning traumas with significant bone defects, exter-nal bone transport devices are often indicated (LABBÉet al. [5]).

Bone distraction forces in maxillofacial surgeryhave already been studied in the past in human man-dible (ROBINSON et al. [6]). Unidirectional screw-bolttype devices were used and the distraction forces weredetermined from the activation torque. Patients wereundergoing unilateral or bilateral DO following con-genital diseases. This study was aimed at measuringthe maximal distraction forces at each increment withno consideration of the bone regenerate time-dependentbehavior.

______________________________

* Corresponding author: Paul Lipinski, Laboratory of Mechanics, Biomechanics, Polymers and Structures, Ecole Nationaled’Ingénieurs de Metz, 1 route d’Ars Laquenexy CS 65820, 57078 Metz Cedex, France. Tel.: 33 3 87 34 42 63, 33 3 87 34 69 35, e-mail:[email protected]

Received: March 26th, 2012Accepted for publication: September 28th, 2012

A.-S. BONNET et al.4

LOBOA et al. [7] measured, by ex vivo experi-ments on a rat model, the tension forces and dis-placements and quantified the stresses and strainsoccurring throughout mandibular distraction. In ad-dition, several studies explored bone transport forcesin lower limb DO. Most of them attempted to deter-mine the link existing between forces and clinicalissue (ARONSON [8]; ARONSON and HARP [9];BRUNNER et al. [10]). Others tried to determine theorigin of these forces (GARDNER et al. [11];GARDNER et al. [12]; HOLLIS et al. [13]; WOLFSONet al. [14]; YOUNGER et al. [15]).

The purpose of this work was to determine theevolving transport forces of bone regenerate duringthe early phases of human mandibular DO.

2. Materials and Methods

2.1. Patient and treatment planning

One 37-year-old male with multi-tissular facial de-fect from a gunshot trauma was considered in thisstudy. The patient was undergoing DO reconstructionfor a 60 mm defect of the mandible and attached softtissues, including skin cover. An external custom-made bone transport device (DEOS, OBL, France, seefigure 1a) was used. After surgery, a latency period ofsix days was observed. The distraction system wasactivated twice a day at a mean daily rate of 1 mm.

Fig. 1. Views of the whole custom-made bone transport device (a)and pin equipped with strain gauges (b)

The patient gave full specific consent and the re-search protocol received approval by a local ethic

committee. The measurement procedure did not inter-fere with the treatment planning.

The apparatus includes two fixed carriages usedto anchor the device on bone and at least one mobileone for bone segment transport. Pins are clamped toeach carriage to fix the device on bone using plateimplants. The mobile carriage is displaced thanksto an endless screw – nut set. The nuts are incorpo-rated into the carriages. The endless screw that canbe referred to as the rail has for particularity to becustom-bent for each patient to optimize the bonepath. In the case presented here, the treatment con-sisted in a bifocal DO with one mobile carriage(LABBÉ et al. [5]).

2.2. Measurement equipment

To determine distraction forces during the acti-vation process, two pins of the distraction devicewere equipped with strain gauges (Vishay USAEA-06-015LA-120), namely the posterior pin of themobile carriage and the anterior pin of the fixed one.As distraction forces are relatively moderate com-pared to the device strength, it was necessary to iden-tify a part of the device undergoing significant defor-mations to ensure precision of the measurements. Theflexible pins appeared to be good candidates. They aresubmitted to significant bending stresses and play therole of reservoirs of elastic energy enabling continu-ous bone distraction. They were chosen for the place-ment of the gauges. Besides, the distraction devicehad to be sterilized before surgery and thus enduredexposition during 20 minutes to steam at a tempera-ture of 134 °C under the absolute pressure of 200 kPa.The measuring gear was chosen and set up in conse-quence.

Since it is impossible to glue the strain gaugesafter the DEOS set up, the mounting position wasforeseen with the surgeons’ help using CT-scan filesexploited in a CAD software package (3-matic, Mate-rialise Software, Belgium). An example of a distrac-tion pin equipped with strain gauges is shown infigure 1b.

2.3. Modus operandi

Strain values were recorded by means of a data ac-quisition system (SCXI, National Instruments, USA)connected to a PC computer. The acquisition wasdone using a LabView program (National Instru-ments, USA). Records started about one minute be-

a) b)

In vivo study of human mandibular distraction osteogenesis. Part I: Bone transport force determination 5

fore the activation of the bone transport device andended about one hour later.

The activation in the first week was conductedduring hospital stay for clinical surveillance and pa-tient’s education. The recording procedure describedabove was repeated for each distraction incrementduring this week. The patient was discharged home atthe end of the week and was asked to keep activatingthe device himself twice a day. Two additional meas-urements were done during check-ups of the patient atpost-operative weeks 4 and 6.

2.4. Data post processing

Figure 2 illustrates a typical geometric configura-tion of the DEOS during DO. To develop the formulaeused in data post processing, a local orthogonal coor-dinate system is attached to each carriage. Its x-axis isoriented towards the regenerate and is contained in therail plane. The y-axis is parallel to the pins clamped tothe carriage considered and points the bone. The far-thest pin from the regenerate is named “pin 1”. Thefollowing geometrical parameters are used to describesuch a configuration:

• l1 and l2 are the working lengths of pins,• a indicates the distance between the two pin

axes of the same carriage,• ω is the angle orienting the instantaneous dis-

traction direction with respect to the coordinate sys-tem,

• g1 and g2 locate the pin section comprisinggauges with respect to the holder,

• 2e defines a mean value of the bone segmentthickness,

• b is the distance between pin 2 extremity andthe regenerate section,

• w is the distance between pins’ extremities.Let δ be the displacement increment imposed to the

mobile carriage (typically 0.5 mm) tangent to the rail.This displacement causes deformation of the pins andgenerates the following distraction forces acting oncallus: ,,, ccc CSF i.e., the tension force, shear forceand bending moment, respectively. The system issketched in figure 2 in initial and deformed configura-tions.

By the action–reaction principle, the forcesapplied to the transported bone segment are definedby:

ccc CCSSFF −=−=−= ,, . (1)

To determine these actions, the equilibrium of thesystem composed of two pins and the transportedbone segment was analyzed. The details of this analy-sis are provided in Appendix A.

In the general case sketched in figure 2, the solu-tion of the above problem is given by:

where:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

11

11

22

221AElT

AElT

aθ (3)

and• T1 and T2 are the internal tension forces,• E1 and E2 are the pins’ Young’s moduli,

⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+Δ+

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+Δ+

−

−+

+−

++⎟

⎠⎞

⎜⎝⎛ Δ

−+−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

+−+

Δ+Δ

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+−

+−

+−

−+Δ

=

θθ

θθ

θθ

θθ

2

222

22

2

1

111

11

2

222

222222

111

111121

211

21222

2222

111

111122

21222

22

111

11

22

2

11

122

)2()(2

)2()(2

)2(2

)2(2)(

)()2(

)(2)2(

)(2)(

)()2()2(22

2)(

lIEB

glwealbwl

lIEB

glwealbwl

gllIEgBl

gllIEgBl

wle

wabTTaTC

TTagllIElB

gllIElBl

alwS

TTlgll

IEgll

IEgl

Bgl

Baal

wF

(2)


• A1 and A2 are the pin sections,• B1 and B2 are the internal bending moments,• I1 and I2 are the moments of inertia,

for pins 1 and 2, respectively.In the case of “perfectly mounted” device defined by

the following relationships, with i taking values 1 and 2:

,;;

;;;

EEIIAA

ggawll

iii

ii

===

===(4)

the above solution of equilibrium problem takes muchmore compact form:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−+−−−−

−+−+

−=

⎥⎦⎤

⎢⎣⎡ −+

−−=

+=

bTbaTglAaelgITT

glelBC

TTAaIB

glF

TTSi

21122

122

21

)()2()(4)(

2)2(2

)(2

4

(5)

Moreover, it is assumed in the present work thatthe moment C can be neglected. This hypothesisis well verified here, as the distance between thepins is large enough compared to their diameter,i.e., a p d. In that case, the measurement of internalactions of one pin per carriage is sufficient to de-termine the distraction forces F and S. System (5)becomes:

)(4))(2()(8)2(2)2( 222

2

gelIbaglaATgelIBleaATglAaS

−+−+−−+−+−−

=

)(4))(2()2(4)(48 222

gelIbaglaATbaIBbaAaIBF

−+−+−+−+−

=

The regenerate interface displacement can be cal-culated using:

yx ebeefCd )v()()( 22 θθ +=−= (7)

with

⎪⎪⎭

⎪⎪⎬

⎫

−−

−−−

=

=

)2(63)2(

23

v

32

22

2

22

glEIlb

al

EAST

glglf

EAlT

(8)

corresponding respectively to the elongation and de-flection of the second pin, see figure 2.

The rotation angle θ is given by:

.)2( 2

EAaSTl −

=θ (9)

The internal actions Ti and Bi produce strains ε1,ε2 and ε3, at gauges G1, G2, and G3, respectively. Bymeasuring these strains, one can find the internalactions as well as the unknown angle ϕ defining therelative orientation of gauge G1 with respect to the

Fig. 2. Diagram of static analysis parameters: (a) initial configuration, (b) deformed configuration

(a) (b)

Transported bonesegment

(6)


bending plane. As illustrated in figure 3, the orienta-tions of gauges G2 and G3 with respect to gauge G1

were defined by the angles α and β, respectively.The following relations detailed in Appendix B arevalid:

⎪⎭

⎪⎬

⎫

=

=

max2

Bii

i

Tiii

dIEB

EAT

ε

ε

(10)

where the strain components due to the tension forceand bending moment are given by:

)sin(sinsinsinsin)sin( 321

αββααεβεαβεε

−+−+−−

=T , (11)

,sin

)1(tan

)sin(sinsinsin)(sin)(

2

1312max

ΔΔ

+×

−+−−−−

±=

ϕ

αββααεεβεεεB

(12)

with .sin)(sin)(

cos)(cos)(tan2113

211332

βεεαεεβεεαεεεεϕ

−+−−+−+−

=

Δ is called the gauge angle and is defined by thegauge width p and pin diameter d:

.dp

=Δ (13)

The gauge length is denoted by λ. This procedurewas automated and applied to all records througha Fortran program. The values of various parameterscorresponding to the DEOS and geometric character-istics of the gauges are summarized in table 1. Thistable also specifies the initial length 0

cl and average

thickness of bone regenerate e as well as its section Ac

and quadratic moment Ic estimated thanks to the pre-distraction CT scan exam of the mandible.

Table 1. Values of the different parameters used

Parameter Unit Posteriormobile pin

Anteriorfixed pin

a mm 10b mm 5 20d mm 2.5g mm 23 19l mm 72 66

DEO

S

E MPa 135,000λ mm 0.51p mm 0.51α ° 90 90G

auge

s

β ° 225 225e mm 50cl mm 1.5

Ac mm2 285Bon

ere

gene

rate

Ic mm4 3780

The forces acting on each side of the distractedbone regenerate were determined considering sepa-rately the strain measurements on the fixed anteriorand mobile posterior pins. This enabled us to plot thecurves of bone regenerate loads versus time and toestimate the quality of our results by comparing theactions computed from the fixed and mobile sides.

3. Results

3.1. Strain measurements

The above-mentioned procedure enabled us to cre-ate a database containing nine strain records. Al-though the device was manipulated by the patient withcare, some parts of the measuring system were pro-gressively damaged making the last two records(weeks 4 and 6) incomplete or mistrustful. For thesereasons they are not reported here. Finally, only six ofthem permitted the calculation of internal actions ofthe pins, callus force and callus extension as the dataconcerning activation 6 were unusable. Besides, forvarious reasons, during the first and second activations,the strains of only one pin were exploitable, namely onmobile carriage for the first activation and on fixedcarriage for the second one.

The duration of each record (tend), depending onthe patient’s convenience, is summarized in table 2 forthe six successfully performed records. All data that

Fig. 3. Gauges positioning parameters


could be treated led to similar curves. It must also benoted that all the values of the recorded strains had

returned to zero before the beginning of a new dis-traction, indicating a complete unloading of the dis-traction device after a twelve-hour period. For allthese reasons, records are only presented for a typicalactivation, i.e., the one done on the fourth day morning(seventh activation).

Table 2. Record duration

Activationnumber 1 2 3 4 5 6 7

tend [s] 93 400 5400 1130 1100 1090 4400

The strains arising in mobile posterior and fixedanterior pins are illustrated in figure 4. It can be notedthat records are quite perturbed due to patient’s ac-tivities. The mean measured strain magnitude isaround 100 µm/m. Comparing with the precision ofthe gauges (1 µm), these values are significant andconfirm the good choice of their positioning. The rec-ords are fairly dissimilar for both pins as their angularorientations defined by angle ϕ are different. The pro-cedure presented above enabled us to estimate theseangles. The values obtained are quite the same for allsix records and equal –40° and 80° for the posteriormobile and anterior fixed pins, respectively. Theseresults are not illustrated in this paper.

3.2. Tension and sheardistraction forces

Figure 5 shows the evolution of the tension dis-traction force F obtained from the posterior mobilepin and the anterior fixed one. The results show thesame tendency for both curves confirming the va-lidity of our approach. Two distinct phases can beclearly identified: the activation phase where theforce rapidly grows up from zero to its maximalvalue and the relaxation one where a slow decreaseof the force is observed. Ignoring noise, one canestimate the maximal value to be approximately4.6 N for both pins. After 75 minutes the distractionforce has been reduced to about 2.8 N for the mo-bile pin and 3.6 N for the fixed one, correspondingrespectively to a loss of 38% and 20% of their ini-tial values.

The shear forces S are not illustrated in this workas they do not exceed 2 N during the whole distractionprocess. The maximum value is obtained at the end ofeach activation.

a)

-300

-200

-100

0

100

200

00:00:00 00:20:00 00:40:00 01:00:00 01:20:00

time (hh:mm:ss)

stra

in (µ

m/m

)

G1 G2 G3

b)

-300

-200

-100

0

100

200

00:00:00 00:20:00 00:40:00 01:00:00 01:20:00

time (hh:mm:ss)

stra

in (µ

m/m

)

G1 G2 G3

Fig. 4. Longitudinal strains versus time for the 7th activation:(a) posterior mobile pin, (b) anterior fixed pin

Fig. 5. Distraction forces versus time for the 7th activationobtained from the posterior mobile and anterior fixed pins


3.3. Deflection of the pinsand callus elongation

The deflection of both pins and the consequentialcallus extension are depicted in figure 6. The deflectioncurves are very similar during the whole distraction pro-cess. At the end of activation phase, i.e., after around 10seconds, the sum of these deflections is close to 0.5 mm.This value corresponds to the increment of displacementimposed on the mobile carriage. During the remainingmeasurement time, the deflections decrease monotoni-cally to reach a value of approximately 0.18 mm.

Fig. 6. Deflection of posterior mobile pin, anterior fixed oneand callus lengthening versus time for the 7th activation

The regenerate extension is obtained using thefollowing expression:

21 ffu −−= δ .

This expression is valid only after the end of acti-vation. The evolution of u is monotonically increasingduring distraction. After 75 minutes of distraction,a regenerate lengthening of 0.17 mm is obtained.

3.4. Evolution of distraction forceversus regenerate extension

The evolution of the distraction force as a functionof the callus lengthening is illustrated in figure 7 forall valid activations. The number of points depends onthe record duration (see table 2). The time instant forwhich the maximal value of distraction force isreached is called peak time and denoted tpeak. For allmeasurements, the forces and corresponding extensionvalues at tpeak and tend have been estimated. Besides, if

available, these values were also estimated at tpeak + 60,tpeak + 180, tpeak + 600, tpeak + 1800. It clearly appears infigure 7 that all data are consistent. The slope of thefitting curve provides the stiffness of the distraction de-vice. This stiffness is estimated to be 10.03 N/mm. Theaverage peak distraction force remains roughly constantfor all activations and can be evaluated to 4.69 N.

0

1

2

3

4

5

6

7

0 0.1 0.2 0.3 0.4

Regenerate extension - u (mm)D

istra

ctio

n fo

rce

- F(N

)

activation 1 activation 2 activation 3activation 4 activation 5 activation 7linear fitting

F = -10.03u + 4.69

Fig. 7. Distraction force versus callus lengthening

4. Discussion

4.1. Strain measurements

The slightly noisy character of the signals of thegauges (cf. figure 4) could not be avoided regardingthe measurement conditions. As the strain magnitudeis quite small (around 100 µm/m), high sensitivity wasrequired. Besides, the device inertia is important dueto the mass of the assembly suspended to long pins.Consequently, small motions of the patient’s headengendered perturbations of the recorded strain val-ues. Even if the patient was asked to stay immobile,some motions such as those due to deglutition couldnot be avoided leading to a noisy response. It can alsobe noted that the records obtained were particularlyperturbed during the activation phase when the dis-traction device was handled by the surgeon or patient.This testifies to its high sensitivity.

4.2. Distraction force

All curves of distraction force versus time exhibitthree distinct phases illustrated by figure 5. The firstphase corresponds to the application of the desiredmobile carriage displacement of 0.5 mm. During this


phase, as the bone regenerate opposes to the motion ofthe mobile bone segment, the distraction force in-creases rapidly to reach its maximal value at the endof activation, typically after 10 to 60 seconds. It canbe seen in figure 7 that this maximal value is quasiindependent of the activation number, i.e., of the cur-rent callus length which expanded about 0.5 mm peractivation. This indicates a viscous behavior of theregenerate. Indeed, the viscosity impedes instantane-ous callus lengthening during relatively rapid activa-tion phases. During this phase almost the whole acti-vation energy is stored in the DO device mainlyinducing a pure bending of flexible pins. As the re-sulting deflection of these pins is roughly constant forall activations, a relatively stable value of the maximaldistraction force is obtained. The scatter of maximaldistraction force observed in figure 7 can partially beattributed to the various activation time-spans.

The second phase is characterized by an exponen-tial decrease of the distraction force typically during300 to 500 seconds. This fast force drop is due to thecallus viscoelastic relaxation process. This phenome-non was accompanied by a regenerate lengthening ofabout 0.1 mm, see figure 6.

The third phase is distinguished by a continuousdecrease of the distraction force in a nearly linearmanner. On average, this force diminished from 3.7 to3.2 N during a period of 4000 s and was accompaniedby an additional callus extension of about 0.05 mm. Itclearly appears that such an evolution of the force isdue to another viscous mechanism of regeneratelengthening characterized by a relaxation time muchlonger than that of the second phase.

The maximal or peak value of distraction forcesobtained in this work is surprisingly feeble (≈5 N)compared to the one obtained in the unique in vivostudy interested in early distraction force measure-ments of human mandible (ROBINSON et al. [6]). Thesame rate and rhythm as in our analyses were appliedfor 8 patients and led to an average peak distractionforce value of 35.6 ± 13.4 N.

At least two reasons allow explaining these differ-ences. First, for all patients treated by Robinson et al.,relatively stiff distraction devices were used. There-fore, the whole imposed displacement was practicallyinstantaneously applied to the regenerate. So, thelengthened tissue underwent a “relaxation test” withimposed strain step. This type of loading generateshigh reaction of the viscous tissues as was observedby many authors. The DEOS used in our analysis ischaracterized by low stiffness (≈10 N/mm) and highstored energy. Immediately after a relatively slowactivation period, the callus extension for all records

did not exceed 0.05 mm for the displacement incre-ment of 0.5 mm leading to a much smaller strain stepand consequently to the feeble value of peak distrac-tion force. Moreover, in our case, during the rest ofthe distraction cycle, the tissue underwent a creep likeloading with progressively diminishing applied force.Finally, in contrast to the study of ROBINSON et al. [6]concerning the treatment of indications such as man-dibular hypoplasia, the soft tissues of our patient weregreatly destroyed by a bullet and did not participate inresistance to the lengthening.

In their experimental analysis with a rat model ofmandible distraction in ex vivo conditions, LOBOA etal. [7] determined the average distraction force peaksranging from 0.3 N for an end latency specimento 4.1 N for post-operative day 13. For the purposeof comparison with our study, two additional differ-ences have to be pointed. The rat’s callus cross-section area was approximately 10 times smallerthan the human one and the rate of distraction was0.25 mm twice daily. The important stiffness of theex vivo testing apparatus illustrated in figure 1C of [7]as well as the short loading time of 0.5 s led to a re-laxation type loading. The mean value of peak forceon 2 and 5 distraction days, well corresponding to ourtests, is about 0.65 N. Multiplying this value by theratio of the cross-section areas, a force of 6.5 N isobtained which is 30% higher than the value of 4.6 Ndetermined in our study. This difference can probablybe attributed to the much higher load rate applied inthe tests by LOBOA et al. [7]. The soft tissues wereeliminated in Loboa et al.’s experiments.

These two comparisons clearly suggest that the softtissues bear a large part of lengthening load. In the lit-erature concerning lower limb distraction, only 25%of this load is generally attributed to the soft tissue re-sistance; ARONSON [8]; ARONSON and HARP [9];BRUNNER et al. [10]; Younger et al. [15], Hollis et al.[13]. This percentage seems to be highly underestimated.Our results support the WOLFSON’S et al. [14] andGARDNER’S et al. [11] thesis of a crucial role of the softtissues in the limb distraction resistance phenomenon.

References

[1] ILIZAROV G.A., The Tension-Stress Effect on the Genesisand Growth of Tissues. Part II. The Influence of the Rate andFrequency of Distraction, Clin. Orthop. Relat. Res., 1989,239, 263–285.

[2] ILIZAROV G.A., The principles of the Ilizarov method, Bull.Hosp. Jt Dis. Orthop. Inst., 1988, 48, 1–11.

[3] KARP N.S., THORNE C.H., MCCARTHY J.G., SISSONS H.A.,Bone lengthening in the craniofacial skeleton, Ann. Plas.Surg., 1990, 24, 231–237.


[4] MCCARTHY J.G., SCHREIBER J., KARP N., THORNE C.H.,GRAYSON B.H., Lengthening the human mandible by gradualdistraction, Plast. Reconstr. Surg., 1992, 89, 1–8.

[5] LABBÉ D., NICOLAS J., KALUZINSKI E., SOUBEYRAND E.,SABIN P., COMPÈRE J.F., BÉNATEAU H., Gunshot Wounds:Reconstruction of the lower face by osteogenic distraction,Plast. Reconstr. Surg., 2005, 116, 1596–1603.

[6] ROBINSON R.C., O’NEAL B.J., ROBINSON G.H., Mandibulardistraction force: laboratory data and clinical correlation,J. Oral. Maxil. Surg., 2001, 59, 539–544.

[7] LOBOA E.G., FANG T.D., WARREN S.M., LINDSEY D.P., FONGK.D., LONGAKER M.T., CARTER D.R., Mechanobiology ofmandibular distraction osteogenesis: experimental analyseswith a rat model, Bone, 2004, 34, 336–343.

[8] ARONSON J., Experimental and clinical experience withdistraction osteogenesis, Cleft Palate-Cran. J., 1994, 31,473–481.

[9] ARONSON J., HARP J.H., Mechanical forces as predictors ofhealing during tibial lengthening by distraction osteogenesis,Clin. Orthop. Relat. Res., 1994, 301, 73–79.

[10] BRUNNER U.H., CORDEY J., SCHWEIBERER L., PERREN S.M.,Force required for bone segment transport in the treatmentof large bone defects using medullary nail fixation, Clin.Orthop. Relat. Res., 1994, 301, 147–155.

[11] GARDNER T.N., EVANS M., SIMPSON A.H., KENWRIGHT J.,A method of examining the magnitude and origin of soft andhard tissue forces resisting limb-lengthening, J. Biomed.Eng., 1997, 19, 405–411.

[12] GARDNER T.N., EVANS M., SIMPSON H., KENWRIGHT J.,Force-displacement behaviour of biological tissue duringdistraction osteogenesis, Med. Eng. Phys., 1998, 20, 708–715.

[13] HOLLIS J.M., ARONSON J., HOFMANN O.E., Differential loadsin tissues during limb lengthening, Trans. Orthop. Res. Soc.,1992, 14, 388–389.

[14] WOLFSON N., HEARN T.C., THOMASON J.J., ARMSTRONG P.F.,Force and stiffness changes during Ilizarov leg lengthening,Clin. Orthop. Relat. Res., 1990, 250, 58–60.

[15] YOUNGER A.S.E., MACKENZIE W.G., MORRISON J.B., Femo-ral forces during limb lengthening in children, Clin. Orthop.Relat. Res., 1994, 301, 55–63.

Appendix A

The obvious geometrical relations are considered:

wl

wll Δ=

−= 12sinω , (A1)

.coswa

=ω (A2)

Let X1, Y1, M1 and X2, Y2, M2 be the reactionsappearing at holders (see figure 3). The problembeing plane, three equilibrium equations may bewritten. They correspond to the projection of forces

on x and y-axes and the projection of moments onz-axis:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−+−−

++++++

=+++

=−++

0)sincos)((

)cossin)((

0cossin

0sincos

211

22121

21

21

ωω

ωω

ωω

ωω

ebYYaY

eblXXMMC

SFYY

SFXX

(A3)

Referring to equations (A1) and (A2), system (A3)can be solved to express the unknown forces acting ontransported bone segment as a function of reactionsapplied to pins:

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ Δ

−+++

⎟⎠⎞

⎜⎝⎛ +

Δ++−−=

+−+Δ+Δ

−=

+Δ+++Δ

−=

wle

wabYYaY

wae

wlblXXMMC

YYaXXlal

wS

YYlXXaal

wF

)(

)(

)]()([)(

)]()([)(

211

22121

212122

212122

(A4)

Each pin is considered as a beam clamped by theholder. Thus, the well-known solution of the canti-lever beam problem may be exploited to determinethese reactions experimentally. Let T1( y) and T2( y)be the internal tension forces and B1( y) and B2( y) theresultant internal bending moments acting in a givenpin section defined by the y coordinate. These inter-nal actions can be expressed as a function of theproblem reactions:

.)(

,)(

iii

ii

MyXyB

YyT

−−=

−=

(A5)

On the other hand, they can be deduced from thedeformation of the pins. This point is addressed lateron. For the moment, it is supposed that these internalactions are known at the locations y = gi, so the fol-lowing relations hold:

,

,

iiii

ii

BXgM

TY

−−=

−=

(A6)

where Bi = Bi(gi). These four equations have tobe completed by two kinematic relations. The


transported bone segment can be supposed perfectlyrigid in comparison to the pins and callus stiffness.So, the distance between any two points of the seg-ment is preserved and its rotation angle, around aninstantaneous rotation axis ,k is unique during mo-tion. If this rotation angle is small (θ ^ 1), the fol-lowing relation holds for any two points of a rigidbody:

ABkAuBu ∧+= θ)()( . (A7)

Let fi, vi and θi be the deflection, elongation androtation of the i-th pin extremity fixed to the bonesegment (points A and B in figure 2). It follows fromthe above hypothesis and expression (A7) specifiedfor zek = that:

)(vv 1122

21

yxzyxyx eleaeeefeef Δ+∧++=+

==

θ

θθθ

which leads to:

⎪⎪⎭

⎪⎪⎬

⎫

=+

Δ−=

==

21

12

21

vv a

lff

θ

θ

θθθ

(A8)

On the other hand, the following results hold for thecantilever beam problem:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛ +=

−=

⎟⎠⎞

⎜⎝⎛ +=

ii

iii

ii

ii

iii

iii

ii

ii

lXMIE

l

AElY

lXMIE

lf

2

v

62

2

θ

(A9)

where Ei, Ai and Ii are Young’s modulus, cross-sectionarea and inertia of pin (i), respectively. Using (A9),systems (A8) and (A6) can be solved with respect tothe unknown reactions:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

−=

θ

θ

i

iiiii

iii

i

iii

iii

ii

lIEgBl

glM

lIEB

glX

TY

22

1

22 (A10)

where

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

11

11

22

221AElT

AElT

aθ . (A11)

Substitution of (A10) into (A4) leads to the prob-lem solution:

⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+Δ+

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+Δ+

−

−+

+−

++⎟

⎠⎞

⎜⎝⎛ Δ

−+−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

+−+

Δ+Δ

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+Δ+

−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+−

−+Δ

=

θθ

θθ

θθ

θθ

2

222

22

2

1

111

11

2

222

222222

111

111121

211

21222

2222

111

111122

21222

22

111

11

22

2

11

122

)2()(2

)2()(2

)2(2

)2(2)(

)()2(

(2)2(

(2)(

)()2()2(22

2)(

lIEB

glwealbwl

lIEB

glwealbwl

gllIEgBl

gllIEgBl

wle

wabTTaTC

TTagll

IElBgll

IElBlal

wS

TTlgll

IEgll

IEgl

Bgl

Baal

wF

(A12)


Appendix B

Two coordinate systems are considered, see fig-ure 3. The first one (O; ),, zyx eee is attached to thepin and has been introduced before. The second one(O; ),, **

zyx eee corresponds to the bending plane

)( *zii eBB = and is defined by angle ϕ with respect to

the first gauge G1. Assuming that the strain field ina pin section considered is mainly due to the tensionforce Ti and bending moment Bi, the correspondingstrain components εT and εB exerted at point P aregiven by:

ii

iT EA

T=ε , (B1)

**)( xEI

Bxii

iB =ε .

This last strain can be expressed using the angle ψpositioning the current point P with respect to *

xe .

ψεψψε coscos2

)( maxBi

ii

iB

dEI

B== (B2)

where

2maxi

ii

iB

dEI

B=ε . (B3)

The total strain is obtained by addition of (B1) and(B2):

ψεεψε cos)( maxBT += . (B4)

Let p be the width of any gauge glued on a pin ofdiameter di. When p is comparable with di, the strainmeasured by the gauge can be significantly differentfrom that corresponding to the position of its centre.Let Δ be the gauge angle such that:

dp

=Δ . (B5)

The average strains recorded by three gauges aregiven by:

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

Δ+=

+Δ

=

Δ+=

+Δ

=

Δ+=

+Δ

=

∫

∫

∫

∫

∫

∫

Δ++

Δ−+

Δ++

Δ−+

Δ++

Δ−+

Δ++

Δ−+

Δ+

Δ−

Δ+

Δ−

ψψεε

ψψεεε

ψψεε

ψψεεε

ψψεε

ψψεεε

βϕ

βϕ

βϕ

βϕ

αϕ

αϕ

αϕ

αϕ

ϕ

ϕ

ϕ

ϕ

d

d

d

d

d

d

BT

BT

BT

BT

BT

BT

cos2

)cos(21

cos2

)cos(21

cos2

)cos(21

max

max3

max

max2

max

max1

(B6)

where angles α and β locate gauges G2 and G3 withrespect to gauge G1.

After integration (B6) becomes:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

+ΔΔ

+=

+ΔΔ

+=

ΔΔ

+=

)cos(sin

)cos(sin

cossin

max3

max2

max1

βϕεεε

αϕεεε

ϕεεε

BT

BT

BT

(B7)

This system has to be solved with respect toϕ, εT and εBmax. After some transformations onegets:

)sin(sinsinsinsin)sin( 321

αββααεβεαβεε

−+−+−−

=T . (B8)

This result enables determination of angle ϕ:

βεεαεεβεεαεεεεϕ

sin)(sin)(cos)(cos)(tan

2113

211332

−+−−+−+−

= .

(B9)

A special case of pure tension leading to the divi-sion by zero has to be considered separately.

Finally, the bending strain can be calculated usingthe expression:


ϕεεεcossin

)( 1max Δ

Δ−= T

B .

Using (B8) and the well-known relation

)1(tan

1cos2 +

=ϕ

ϕ leads to: .sin

)1(tan

)sin(sinsinsin)(sin)(

2

1312max

ΔΔ

+×

−+−−−−

=

ϕ

αββααεεβεεε B

(B10)

in vivo study of human mandibular distraction osteogenesis ... · acta of bioengineering and...

Documents