Modeling the lateral movement of teeth

J. Bmnechanrcr,

1976. Vol. 9. pp. 723-734.

MODELING

Pergamon

Press.

Printed

in Great Bntain

THE LATERAL

MOVEMENT

OF TEETH*

GEORGE G. Ross Department of Computer Science, City College of New York, NY 10033. U.S.A. CLEMENTS. LEAR Department of Orthodontics,

Faculty of Dentistry. University of British Columbia, Vancouver, BC. Canada and

RICHARD DECOU Faculty of Dentistry, University of British Columbia, Vancouver. BC. Canada Abstract-Experimental results on displacement patterns of human left and right maxillary central incisor teeth in response to laterally directed forces of low magnitude are reported. Three mechanical models involving one dimensional motion described by deterministic differential equations are introduced to conceptualize the experiments. Analytic and numerical solutions to these equations are presented and the models are comparatively evaluated with respect to the data.

INTRODUCTION Human teeth are subjected to fluctuating laterally-dir-

ected forces derived from routine function of the faciolingual muscles. A previous study of the magnitude, duration and wave-form characteristics of such muscle forces showed that “resting” forces could be as low as 1 g, but peak forces during mastication, deglutition and speech sometimes exceeded 20 g. However, in the majority of these oral activities the forces fluctuated between 5 and log (Lear and Moorrees, 1969). The tooth oscillation patterns that result from these small faciolingual forces have received little attention. The rationale for investigating the details of such force/displacement phenomena relates to the finding that low forces (< 5 g) are capable of causing transient displacement of human teeth (Lear, Mackay and Lowe, 1972; Lear and DeCou, 1972) and such repetitive, low-magnitude forces have the potential for affecting the location of a tooth, and thereby for influencing the alignment of teeth in the dental arch (Weinstein, 1967). This paper reports on the displacement patterns of human left and right maxillary central incisor teeth in response to lingually directed forces of varying duration, in the 5-10 g range. These applied forces simulated those occurring from physiological faciolingual muscle force patterns (Lear and Moorrees, 1969). Moreover, it is the intent of this paper to present a mathematical model mimicing the observed experimental phenomena. Modeling can be defined formally as the setting up of a many-to-one correspondence between the parts (state variables) of a real system with their attendant state transformations and the parts and transformations of a conceptual representation called the model (Ashby, 1956). The degree to which the representation can be made faithful depends on the total knowledge of the system. Never can man hope to * Received 20 January 1916.

know all the parts and transformations of a real system, but he may hope to establish a model which mimics it, e.g. in the case of a dynamical system simulations of the model should satisfactorily mimic observed real time behavior. Modeling should be carried out by an inductivedeductive process. Initially the investigator inductively forms conjectures concerning the mechanics of a phenomenon for which he has a quantity of observed data. He then examines the conjectured mechanism deductively through the medium of mathematical analysis or, in the case of a dynamical system, simulation, to determine the properties of the system resulting if the conjectured mechanism is postulated. The next step involves a comparison of these properties against experimental data, testing whether the two are compatible. The model is accepted, rejected. or returned for modification on the basis of the comparison. Technically a model, once conceived, provides some consequences from its defining axioms which are interpreted as propositions about the real system. The many-to-one character of the correspondence of course sacrifices some of the behavior alternatives of the real system but it is an indication of successful modeling if the salient features of real behavior are preserved in the correspondence. The propositions of the real system derived from the model must be related to the total available experimental knowledge bearing on the system and the model rejected or modified if found wanting. The repeated modify-test procedure suggested is called system identification in engineering terminology and lies at the heart of what is commonly known as the scientific method. The choice of a model should depend on the quality of data available, the tractability of the mathematics proposed to deduce properties of the model, and the testability of propositions, e.g. existence of additional data independent of that used

723

GEORGEG. Ross, CLEMENTS. LEAR and RICHARDDECOU

724

to evaluate the model parameters which can be called upon to identify its predictions. There is no point in creating elaborate models which are too difficult to analyze and so produce no hypotheses or those for which there is no convincing experimental evidence to confirm the attainable hypotheses. Such a model could never be validated. The authors have restricted themselves to one dimensional tooth displacement, and have coordinated their theoretical and experimental investigations, subject to the modify-test criterion. The approach is a comparative analysis of three models, beginning with the simplest linear case, proceeding to a time varying spring constant and finally a nonlinear spring. Model parameters are evaluated by sensitivity analysis, model predictions are compared with the actual experimental data, and the results are discussed in relation to one another. METHODS AND MATERIALS The apparatus used to reproduce faciolingual muscle wave forms on isolated individual teeth, as well as the displacement-sensing apparatus and associated graphic recorder have already been described (Lear and Mackay, 1972). The system used in the present study was modified, however, to record tooth displacement data on F.M. tape (Ampex recorder type SP-700) simultaneously with an oscillographic trace, the latter being used for monitoring during the course of each test, Subsequent to the test, the F.M. tape was replayed and the displacement waveform data displayed and analyzed on a storage oscilloscope (Tektronix type 5013N). Although tape-to-oscilloscope systems have been used extensively in biomedical research, previous human and animal tooth-oscillation studies have not reported on the use of such techniques. It thus seems appropriate to briefly review the advantages of F.M. tape-oscilloscope systems for toothdisplacement studies. First, tape/oscilloscope techniques can readily cover frequency ranges from d.c. to beyond 2500 cps. Thus for myological studies no data are lost from inadequate response times. Moreover, the write-out is made without pen or mirror lag or overshoot, thus obviating waveform distortion from mechanical inertia.

Second, since the signals are stored in analog form, they may be reexamined any number of times with oscilloscopes or computers, or the signals may be duplicated, with appropriate filter networks, to provide specific information without any change to the original taped data. Third, signal-to-noise ratios of better than 2O:l are achievable for the entire system, even with amplifications high enough to display a 5 p tooth displacement as a 5 cm pattern on the oscilloscope screen. Accordingly, the above tape recording techniques were utilized for studying oscillation patterns of maxillary central incisor teeth. One female and four male subjects were chosen. Their ages ranged from 24 to 32 years. All had regular dental alignment of the test teeth, which were also judged to be periodontally “sound” on the basis of: (a) minimal depth of the gingival crevice (~2 mm); (b) alveolar bony crests which were less than 1.5 mm from the cementoenamel junction as seen in periapical radiographs; (c) no “thickening” of the periodontal “space”; and (d) soft tissues which appeared clinically healthy. Some of the test teeth had “tight” mesial and distal contact judged by resistance to the passage of fine dental floss. Other test teeth did not have tight contacts (Table 1). Since human faciolingual force fluctuations include various waveforms ranging from brief thrust patterns (< 250 msec) to relatively prolonged plateaux (Lear and Moorrees, 1969) (> 1 set), a program was devised to evaluate the relative contributions to force duration and force magnitude in making a tooth deflect at 5.0, 8.0 and lO.Ogm force levels. The thrust durations were 100, 250, 500, 1000, 1500 and 2500 msec at each force level, and there was a relaxation period of 20 set between each thrust. Tests were conducted on 3 separate sessions for the left central incisors and on 3 separate sessions for the right. The splint which held force-producing and displacement measuring devices against the test tooth was similar to that used in previous studies (Lear, Mackay and Lowe, 1972). This device applied lingually directed forces to the center of the facial enamel of the tooth, at right angles to its long axis. In vitro tests were carried out to determine the limits of accuracy of the mechanical and electronic components of the entire system. It was found that maximum baseline drift was less than the equivalent of 0.25 pm over 2 min. Variations between calculated

Table 1.

Subject

Maximum distance between alveolar crest and cemento-enamel junction .

1 2 3 4 5

1.5 mm 1.0 mm l.Omm l.Omm l.Omm

Interproximal contact Left Distal + -

Right Mesial + +

Mesial + +

Distal -

125

Modeling the lateral movement of teeth

(i.e. response of tooth 11 compared to tooth 21 for any given subject) were negligible (J < 0.1). Accordingly, appropriate data for each subject were pooled. Composite data are represented as isolated points in Figs. l-5 for applied forces of 5, 8 and log. There was always an initial rapid movement of the tooth away from the force, followed by a significantly reduced rate of movement. Termination of force application immediately resulted in an inversion of the ascending waveform.

and actual forces delivered by the solenoid to the tooth did not differ by more than +O.l g. Application of a 1OOmsec electrical square wave (90” rise and fall) to the solenoid produced a mechanical square wave to the tooth with less than 2% waveform distortion. DATA

REDUCTION

Displacements of teeth in response to “square wave” forces were measured 10, 50 and 100 msec after application of forces 1OOmsec in duration. For force applications longer than 1OOmsec the ascending displacement slope was also measured, as appropriate, at 250, 500, 1000, 1500 and 25OOmsec intervals. The descending slope (i.e. return of the tooth towards datum position after removal’ of the force), was measured at intervals corresponding to the rising slope measurement points. Within-subject variation in displacement patterns

MODELS

Simple linear spring

The initial attempt to model the displacement behavior of the teeth is to pose the simple second order ordinary differential equation with constant coefficients : d”X

m~+ed~+kx=F[u(t)-u(t-TI1.

15r-

Time,

Fig. l(a).

set

dx/dt + 0.3 x = 7 u (t, 2.5). (B) dx/dt + 5 (1 - 0.4 e-2.9f)x = 20~ (t, 2.5). (C) dxldr + (0.64x + 0.35)x = 25~ (t, 2.5). u (t, y) = 1, 0 5 t I y = 0, t > y.

%. (4

15r

(b)

i

Time,

Fig. l(b).

8g.

set

(A) dx/dt + 0.3 x = 11 u (t, 2.5). (B) dxidt + 5 (1 - 0.4e-2.9’)x = 32~ (t, dx/dt + (0.64x f 0.35)x = 40~ (t, 2.5). u (t. y) = 1, 0 < t 5 y = 0, t > y.

726

GEORGEG.

Ross. CLJWENT S. LEARand RICHARDDECOU

Time,

set

Fig. l(c). log. (A) dx/dt + 0.3 x = 14~ (t, 2.5). (B) dx/dt + 5 (1 - 0.4 e-‘.‘*))x = 40~ (t, 2.5). (C) dx/dt + (0.64.x + 0.35)x = 50~ (t, 2.5). u (t, y) = 1, 0 I t I y = 0, t > y. The mechanical configuration corresponding to the equation is a mass m moving in one dimension under the action of a driving force F, subject to the restraint of a spring with elastic constant k and in a medium with linear damping constant c. x(t) measures the displacement of the mass from a standard datum. Note F is constant from time 0 to time T and then drops to 0. A routine application of the Laplace transform technique yields for the transform of the displacement

where u(Y) =

0 if y
Thus for times between 0 and T while the external force is in effect 2” a

&f

- kl(k2 - k,) + k2(kz - k,)

1’

and when t > T in the decay portion of the solution:

L(x(tN=

F k21k,el” (1 - eekzTl

x(t) = m where k = -c/m+

J(mrn

1

i

kl)

-

kl(kf’1 11- emklTl]. k

-I

)

The constants k,, k, are to be chosen so that the solution best approximates the experimental data D(ti). Note that F -=k,k,m

1

F k

s(s - k,)(s - kz) _

W,kz + - llkl(kz - U + l/h (kz- kJ s - k,

s

s-k,

’

into partial fractions and evaluating the residues of the transform’s numerator at the simple poles, the following expression for the inverse is obtained: F

x(r) = ; -

+

[I

-- 1 k,k2

ekz’

&’

k#z - k,) + kz(kz - M @t-T)

1 -klkz k&z - k,) ,+t - T)

kz(kz - W

1 1 u(t-T)

,

u(t)

is the asymptotic value the solution approaches if the external force is not removed and time grows large. Standard optimization methods were employed to minimize

where i ranges over all data points for a given subject. Results obtained indicated wide disparity between the values of k, and k2, for each subject, typically (subject lb kl = -0.022 setk2 = -5Osec-‘.

’

Modeling

the lateral

movement

TV%!,

721

of teeth

set

Fig.

2(a). 5g. (A) dx/dt + 0.27~ = 5u (t, 2.5). (B) dx/dt + 4.5 (1 - 0.3e-Z.*‘)~ = 15 u (t, 2.5). (C) dt + (0.59x + 0.37)x = 21 u (t, 2.5). u (t, y) = 1, 0 < t I JJ= 0, t > y.

Fig.

2(b). 8g. (A) dx/dt + 0.27x = 7.9 u (t, 2.5). (B) dx/dt + 4.5 (1 - 0.3 ew2.1r)x = 24~ dx/dt + (0.59 x + 0.37)x = 35 u (t, 2.5). u (t, y) = 1, 0 5 t < y, t > y.

Time,

Time,

Fig.

set

(t, 2.5). (c)

set

2(c). (A) dx/dt + 0.27 x = 9.6~ (1, 2.5). (B) dx/dt + 4.5 (1 - 0.3 e-‘.“)x = 31 u (t, log. dx/dt + (0.59 x + 0.37)x = 45 u (t, 2.5). u (t, y) = 1. 0 5 t I y = 0, t > y.

2.5).

(C)

GEORGEG. Ross. CLEMENTS. LEAR and RICHARDDECOU

128

Time,

set

Fig. 3(a). 5g. (A) dx/dt + 0.26x = 3.7~ (t, 2.5). (B) dx/dt + 4 (1 - 0.3 e-2.01)x = 8 u (t, 2.5). (C) dx/dt + (0.58 x + 0.33)x = 11 u (t, 2.5). u (r, y) = 1, 0 s t I y = 0, t > y.

0.5

IO

1.5

2.0 Time,

3.0

25

3.5

4.0

4.5

set

Fig. 3(b). 8g. (A) dx/dt + 0.26x = 6.2 u (t, 2.5). (B) dx/dt + 4 (1 - 0.3 em2.01)x= 12 u (t, 2.5). (0 dx/dt + (0.58 x + 0.33)x = 17 u (t, 2.5). u (t, y) = 1, 0 I t I y = 0, t > y. 15

E ‘O =l_ 5 E B .o

n5

0.5

1.0

1.5

2.0

Time,

2.5

30

3.5

4.0

45

set

Fig. 3(c). (A) dx/dt + 0.26x = 7.3 u (t, 2.5). (B) dx/dt + 4 (1 - 0.3e-‘.“)x = 16.3~ (t, 2.5). (C) log. dx/dt + (0.58 x + 0.33)x = 20.9 u (t, 2.5). u (t, y) = 1, 0 I t I y = 0, t > y.

Modeling

05

10

15

the lateral

movement

20

2.5

Time,

30

729

of teeth

35

40

45

SW

Fig. 4(a). 5g. (A) dxldt + 0.27 x = 4.1 u (t, 2.5). (B) dx/dt + 4.1 (1 - 0.3 e-*,oqx = 10 u ([, 2.51. (c) dxldt + (0.58x + 0.33)x = 15~ (t. 2.5). u (t. J) = I, 0 5 t < J = 0, t > !‘, 15 (b) F

05

IO

I.5

20

Time, Fig.

2.5

30

35

40

45

set

4(h). 8g. (A) dxldt + 0.27x = 6.5~ (t, 2.5). (B) dqdt + 4.1 (1 - 0.3~-~.~‘)~ = 16~ (f, 2.5). cc) dx/dt + (0.58 x + 0.33)x = 23.6 u (t, 2.5). u (t, y) = 1, o 2 t 5 y = 0, t > J’. '5 (c)

05

1.0

15

20

Time,

25

30

35

40

45

set

Fig. 4(c). log. (A) dxidt + 0.27 x = 8.0 u (t, 2.5). (B) dxldt + 4.1 (1 - 0.3 e--'."y~ = 19.5 II (l. 2.5). (c) dxldt + (0.58x + 0.33)x = 29.3~ (t, 2.5). IA(r, y) = 1. 0 I t $ y = 0, t > y.

730

GEORGEG. Ross, CLEMENTS. LEAR and RICHARDDECOU

05

IO

1.5

21)

25

Time, Fig.

5(a). 5g.

3.0

35

4.0

4.5

set

(A) dx/dt + 0.27x = 4.5 u (t, 2.5). (B) dx/dt + 4.4 (1 - 0.3 e-‘.‘*)x = 13 u (t, 2.5). (C) dx/dt + (0.59x + 0.33)x = 17~ (t, 2.5) u (t, y) = 1, 0 I t 2 y = 0, t > y.

Time,

set

Fig. 5(b). 8g. (A) dx/dt + 0.27x = 7 u (t, 2.5). (B) dx/dt + 4.4 (1 - 0.3 e-*.“)x = 21 u (t, 2.5). (( dx/dt + (0.59 x + 0.33)x = 27 u (t. 2.5). u (t, y) = 1, 0 I t I y = 0, t > y. 15

(C)

r

Time, Fig.

set

5(c). log. (A) dx/dt + 0.27x - 9 u (t, 2.5). (B) dx/dt +.4.4 (1 - 0.3 e-*.lt)x dx/dt + (0.59x + 0.33)x = 35u (t, 2.5). u (t, y) = 1, 0 5 t < y = 0,

=

25 u (t, 2.5). (C)

t > y.

731

Modeling the lateral movement of teeth The solution to the mechanical analog, since it presumes an initial velocity of zero, has positive curvature near t = 0. The data do not seem to reflect this characteristic, indicating the inertial effect is very small relative to the damping and elastic effects. This appears natural in light of the small mass of the tooth relative to a 1Ogm force, and is confirmed by the disparity in the values found for k, and k2, i.e. if kL is close to zero. the term c/m must be very large to dominate 4k and hence m must be small. If k, is considered large in comparison with k2. the solution, for times when the external force is in effect, reduces to x(t) = ;(I

- hl’).

a simple exponential approach to the asymptotic Flk. However the above solution for the numerically “optimal” choice of k, and k2 (k = klkz = 1.1) does not represent the data faithfully for any of the five subjects under the action of the applied forces of 5, 8 and 10 gm. The curves labelled A in “Figure i (jgm)” represent the best simple linear spring model solution for subject i under the action of a force of jgm. It is evident that the initial slope cannot be made steep enough to satisfy the tendency of the observations if we are to maintain a reasonable asymptotic level, and not have the tooth leave the head. From consideration of this model two features emerge : (i) that the second order derivative representing inertial forces in the model can be omitted. This can be supported by noting that the reduced solution (F/k)(l - eklt) is precisely the solution to cd.x/dt f kx = F. (ii) that the representation of the retention force by a spring obeying Hooke’s law is an oversimplifying assumption and we should look for a model involving either a time varying linear elastic property or a nonlinear retentive behavior.

on the system the acceleration term is neglected. and taking the variable spring constant with exponential time variation between initial value of cfl - c’) and final value of c: dx

+ c(l - c’e-“‘)x = F.

Z

where x(t) represents the tooth displacement as a function of time. c(1 - c’e-“I) represents the timedependent ratio between elastic and damping effects and F is the ratio between force applied and the damping effect. The constants c, c’ and a are to be evaluated by sensitivity analysis directly from the experimental data. The solution to the equation can be obtained by introducing a change in the dependent variable x which results in two ordinary first order equations. the first homogeneous and separable. and the second having constant coefficients. Setting x(t) =.f(t)v(tX If@) # O), substituting in the differential equation and dividing through by f(t):

z+ __+c(1 6

dfldt .f

c’e-“) 4’ = Fif. I

after setting

dfldt __ + c(1 - c’e-a’) = c. f and simplifying to obtain the homogeneous separable equation i In f = cc’ eCa’, integration yields

Info = $1 _ f(O)

e-“)

a

’

and, exponentiating both sides, f(t) = f(0) ecc’/ue-Icc’!l)e I’,

The

variant linear spring

The results of the previous section indicate that modeling the elastic properties of the tooth-supporting fibers by a mechanical spring whose Hooke’s constant is independent of both time and displacement, does not provide a solution adequately representing the observations. Accordingly the assumption is made that the resistance of the tooth supporting system stiffens as the time it is under stress progresses, perhaps due to a hydraulic effect in the capillaries or lymphatics or the tissue fluids. The model implementing the assumption requires the introduction of a time-dependent coefficient of the displacement term in the differential equation. In consideration of the remarks at the end of the previous section concerning the smaI1 effect of inertial forces

With this choice of f(t) the inhomogeneous for y(t) has constant coefficients

dy z+

equation

F CY =fT,’

and it has complementary solution e-“. Using variation of parameters the solution to the inhomogeneous equation for y(t) is: y(t) =

* s

F c foe”‘-“dz,

y(t) = Fe-”

732

GEORGEG.

Ross, CLEMENT S. LEARand

To evaluate this last integral set

and after changing the integration variable to n, substituting the Taylor series for the exponential e-“, integrating term by term, and some algebraic manipulation: e(cc’/a)e-”

ec* &

=

! ;

_

c!

j=o

Substituting in the expression for y(t):

RICHARD DECOU

recorded data point D(i) (the constant F in the solution x(i) is set to the force magnitude used in observing data point D(i)). The problem reduces to the minimization of a highly nonlinear function of three real variables CI,c, c’. To carry out the optimization appeal is made to numerical procedures recently developed by Davidon (1959) and Fletcher and Powell (1963). These procedures are sophisticated extensions of Newton’s method which rely on variable metric considerations and avoid the calculation of the Hessian matrix. Functional and first partial derivative evaluation for a trial value of CI,c, c’ have been carried out using a FORTRAN IV program running on the Control Data Corporation 6600 at the Courant Institute of Mathematical Sciences at New York University. A local minimum for the deviation function (1) was found at LX= 2.9X-’ c = 5 gmjsec”

and finally, since x(t) = f(t) y(t),

c’ = 0.4 (dimensionless),

,

or

Further light can be shed on the solution by expanding the right-hand side of the above as a power series in t around t = 0. The mathematical details, although straight forward, are cumbersome and have been omitted. It is sufficient to present here the following simply demonstrated characteristics of the solution: x(0) = 0

(9 (ii) (iii) (iv)

dx(O) __ = dt

lim x(t) = F/c

lim dxO = 0 f-Lm dt ’

1 Co(i) - xW, is conducted

and is easily seen to be x(t) = x(T)*e- C(t-T).e[-Cc’/.)(e-“~-e-‘T)l ,

NONLINEAR SPRING

t-cc

where the i summation

dx X + c(1 - c’e-“‘)x = 0.

where T is the time at which the force F is removed and x(T) is the maximum observed displacement of the tooth.

F

These elementary properties conform to our intuitive concept of tooth displacement by exhibiting asymptotic approaches to a constant value under the action of the external force. The objective is to use the experimental data to select values for the parameters of this model so that the model solution best agrees with the totality of available data. The parameters involved are c, c’ and GLand the model solution is “fitted” by asking for a minimum of (1)

for the totality of data for subject 1 under a 5 g force. The model solution corresponding to these parameters is displayed as curve B in Fig. 1 (5 g). Similar results for the time varying spring constant model for subject i under a jg force are labelled B in Figure i (j g). The optimal parameter values tl, C, C' also appear in the Figs. For all subjects under all forces there is a marked improvement in the model approximation of the experimental data. The decay part of the solution to this model solves

over each

The second attempt to overcome the deficiencies of the solution of the model with simple linear spring constant involves taking the spring stiffness to be a function of the displacement, instead of a function of time, as in the previous section. We assume the stiffness increases as the displacement increases. A physiological interpretation is that the matrix supporting the tooth develops its elastic characteristics as the tooth is forced out of position. Mathematically our model equation takes the form dx

z+

:x+-

k

x=--,

F

C> C ( where 01/cx represents the nonlinear stiffening behavior considered in this section. For this equation it is possible to separate the variables, expand the x

733

Modeling the lateral movement of teeth dependencies in partial fractions, and integrate to arrive at the following implicit relation defining the solution:

where

kl = -

k-,/m y

2a

special case of subject 1 under the action of a 5 g force. The corresponding model solution is displayed as curve C in Fig. 1 (5 g). Similar results for the nonlinear spring model for subject i under a j g force are identified as C in Figure i (jg). Optimal parameters values a/c, k/c appear on the Figs. The decay part of the solution to the nonlinear model can be solved explicitly for the dependent variable after a routine integration. The result after taking as initial condition the displacement at r = X the time when the force was removed from the tooth, is: x(T)*(l - ax(T)e-BT)-e-P(*-T’ x(t) = ___

1 - crx(T)e-O’

and k =_k+dL2z 2

2a ’ Except in special cases the relation cannot be solved explicitly for x as a function of t. However, at those values of t where observations were taken, the relation can be solved numerically for the displacement predicted by the model and compared with the experimental data as in the examination of the two previous models. Below is presented a schema of the computer program written to calculate the values of the parameters k,, k2 and c so that the deviation from actual data is minimized. (1) Choose initial trial values for k,, k,, c. (2) Calculate for each time of observation and magnitude of applied force F the displacement x(t) predicted by the model, i.e. solve the implicit relation using successive approximation techniques. (3) Calculate the sum of deviations of model predictions from actual data for each experiment and then sum over all experiments. (4) Calculate the partial derivatives with respect to kl, k2, c of the function defined by the totality of deviations above by a centered difference formula. (5) Enter the variable metric optimization procedure of Davidon et al. noted in the previous Section. (6) Receive as output from 5 new trial values for the parameters k,. k,, c. (7) If no reduction in functional value (total deviation) has been realized since the last trial then stop; else transfer control to step (2). Implementation of the above schema as a FORTRAN IV program yielded for subject 1 under the action of a five gram force: k, = -6.54cm/sec2 k, = 3.50cm/sec2

c = 0.20sec-‘. The above generate the optimal values LZ/C= 0.64 mg/(sec-cm) k/c = 0.35 gm/sec

describing the elasticity of the model spring for the

Figures 1-5, curve C, show the correspondence between the optimal model solutions and the experimental data. It is evident that this model gives the best results of those tried. We have finally succeeded with the nonlinear spring model in generating the steep initial slope together with the moderate level asymptote. DISCUSSION

Fellow modellers of tooth support systems have observed that single linear mechanical models do not satisfactorily mimic experimental data (Picton, Wills and Davis, 1972). A supporting conclusion was reached here in the first modeling effort where the neglecting of tooth acceleration was justified. The choice of the mechanical constants producing simulated displacement best approximating the available data did not yield an initial slope as steep nor an asymptote as low as those indicated by the experimental results. Picton, Wills and Davis (1972) have attacked the problem by constructing a complex of linear spring and damping mechanisms arranged in series and parallel. Having more model parameters to adjust, they were naturagy able to make the complex system more accurately simulate the real system. This approach appears equivalent to subdividing the time scale into various intervals and then piecewise choosing an optimal linear model peculiar to the displacement data in each interval. If such modeling philosophy is extended to the extreme, the data might be arbitrarily well approximated by piecewise exponentials (or straight lines for logarithmicaliy reduced data) in arbitrarily small intervals. A model complex consisting of a large number of linear mechanisms could then be postulated which would simulate the data well. It could not, however, be expected to relate in a general way to other data on tooth displacement, nor to provide insight as to the structure of tooth support. An analog in curve fitting arises when one allows m free parameters to fit m data points and gets a perfect but not significant “fit”. The authors contend that the desirability of a model is inversely related to the number of free parameters and that

734 an investigator

GEORGEG. Ross, CLEMENTS. LEAR and RICHARDDCou

must be alert to minimize the free parameters and to base his model on some simple theoretical conceptualization to be confirmed or rejected by the experimental results. One point of departure from the simple linear system is to hypothesize an elastic behavior of the tooth’s supporting fibers which is influenced by either time (model 2) or displacement (model 3). It is to be stressed that the model parameters were chosen to optimally simulate the experimental data taken while the external force was still applied to the tooth. The data taken after the force was removed served as a check of the models’ predictions of tooth behavior when no external influence is exerted but there was an initial displacement from equilibrium. A comparative examination of Figs. l-5 reveals that curve C for all subjects most closely conformed to the data both in the rising (force applied) and falling (force released) portions. It is concluded that the three parameter nonlinear spring model is superior to the single linear spring and time-variant elastic models and in fact provides a remarkably consistent simulation of the experimental results. The physiological interpretation that the tooth changes its elastic properties as the tooth is forced out of position is supported by the above results. Investigation of the structural details of this developing elasticity must consider the microstructure of the periodontial ligament and the identification and separation of its elas-

tic and hydraulic components. Additional experiments will be necessary in which fluid pressures and fiber extensions in the matrix supporting the tooth are monitored.

REFERENCES

Ashby,

W. R. (1956) An Introduction

to Cybernetics,

Wiley,

NY.

Davidon, W. C. (1959) Variable metric method for minimization, ANL-5990, Atomic Energy Commission Res. Dev. Rpt. Fletcher, R. and Powell, M. J. D. (1963) A radially convergent descent method for minimization. Comput. J. 6, 163-168.

Lear, C. S. C. and Moorrees, C. F. A. (1969) Buccolingual muscle force and dental arch form. Am. J. Orthod. 56, 379-393.

Lear, C. S. C., Mackay, J. S. and Lowe, A. A. (1972) Threshold levels for displacement of human teeth in response to laterally directed forces. J. dent. Res. 51, 1478-1482. Lear, C. S. C. and De Cou, R. E. (1974) Threshold levels for displacement of human maxillary central incisors in response to lingually directed forces. J. dent. Rex 53, 946.

Lear, C. S. C. and Mackay, J. S. (1972) Force waveform generation and recording lateral movement of teeth. Med. Biol. Engng 10, X6-521. Picton, D. C. A., Wills, D. J. and Davies, W. I. R. (1972) An investigation of the viscoelastic properties of the periodontium in monkeys. J. periodont. Res. 7, 42-51. Weinstein, S. (1967) Minimal forces in tooth movement. Am. J. Orthod. 53, 881-903.