Tooth Canal Working Length Electric Meter

1Abstract—Electrical apical constriction locators are not harmful for humans and allow to measure exactly the physiological apex of tooth root canal. The original apex locator measurement circuit is presented. It has the power supply sources of low and high frequencies. Each measurement cycle consists of three measurement intervals. The dependence of measurement accuracy on capacitance between the electrodes and other canal electric properties is eliminated. The analysis of measurement circuit is performed. The results of experimental investigation are presented. The apical constriction is found reliably and exactly.


I. INTRODUCTION
One of more frequently performed endodontic procedures is processing of tooth root canal.The root canal ends with apical constriction (Fig. 1).It is very important that endodontic instrument could not pass beyond the apical constriction (anatomical apex) and could not break soft tissues of the patient.Therefore the position of apical constriction must be found exactly.In endodontology the measurement of apical constriction position is called tooth root canal working length measurement.
Initially, radiographs for canal working length were used.Later alternative measurement instruments -Apex locatorswere developed.In Apex locators the electric resistance between two electrodes is measured.The possible arrangement of Apex locator electrodes is shown in Fig. 1(d) is distance to apical constriction tip -physiological apex (P).A is anatomical apex.One electrode (E2) is placed on the patient's lip or is kept in hand while the other electrode (E1) which can be superposed with endodontic instrument is inserted in canal.
Using Apex locators for tooth root working length measurement allows decreasing the quantity of radiographs and patients are not affected by harmful X-rays.Apex locators have been used for more than 40 years [1]- [4], but diversity of human teeth and anatomy requires permanent development of these devices to guarantee reliable measurement regardless patient age, canal geometry, used medication, etc. Original Apex locator has been developed in Lithuania.It is protected by UK patent [5].It was improved additionally to increase its accuracy and reliability.The improved Apex locator is presented in this paper.

II. THE ELECTRIC CIRCUIT OF BODY BETWEEN ELECTRODES
The electric properties of the human body tissue between electrodes can be presented by electric circuit shown in Fig. 2. Different electric signals of biological nature act in human body.Galvanic internal voltage is generated in the junction of two materials with different electrochemical potentials.There are several junctions in the body part between electrodes: electrode -mucosa, electrodemoisturizing fluid, soft tissues -moisturizing fluid.In Fig. 2 voltages of these junctions as equivalent voltage source en(t) are presented.The voltage of this source can be presented as direct voltage corrupted by low frequency noises.The main part of these noises is concentrated in the range of 0 Hz-100 Hz.The capacitance Cs presents the phenomenon of electric charge accumulation in the junctions of materials with different electrochemical potentials.Rs is resistance of soft tissues between electrode E2 and tooth canal junction with soft tissues, Rp and Cp are correspondingly, the resistance and capacitance between electrode E1 and abovementioned junction.When the distance d between the E1 and apical constriction varies (see Fig. 1), Rp varies, too.For Rp measurement the stable voltage source must be connected to electrodes E1 and E2.To suppress the influence of voltage en(t) on measurement result it is sufficient to use the sinusoidal voltage ul0 with frequency f > 100 Hz and narrow band filter.The resistance Rp depends on electric conductance of liquids used for canal processing and may vary by the order of six.The capacitance Cp also depends on the liquid type too.Theoretical and experimental investigations [1]- [4] show, that for accurate resistance Rp measurement two voltage sources of different frequency are needed.One of these sources benefits to form the signal which compensates the resistance variation because the canal processing conditions variation.But realization of such compensation can be different.

III. BLOCK DIAGRAM OF MEASUREMENT
The electric circuit of measurement canal of Apex locator [5] is presented in Fig. 3.Here u10 and uh0 are sources of low and high frequency sinusoidal voltages, R0 -constant resistor, Zx = Zx(ω, Rs, Cs, Rp, Cp) complex (phasor) impedance between electrodes, D1 -resistor divider with controlling division coefficient, composed of resistors Rv1, Rv2, D2 -resistor divider with constant division coefficient, composed of resistors Rk1, Rk2, K1 and K2 -three-pole switches.Frequency of the source u10 is greater than 100 Hz, frequency of the source uh0 is at least ten times greater than the frequency of the source ul0.Therefore the influence of parasitic source ep (Fig. 2) on measurement signal is insignificant.The impedance Zx has capacitive character.It is presented in Fig. 3 by parallel RxCx circuit.Rx and Cx may be expressed by the elements of equivalent circuit Fig. 2 and angular frequency ω as follows: In second measurement cycle switch K1 is commutated into position II.The high frequency source uh0 is connected to measurement circuit.The divider's D1 division coefficient remains unchanged.The output voltage phasor of D1 is UxII.
In the third cycle the source uh0 remains connected to measurement circuit, but measurement circuit output is connected to divider D2 with output signal phasor Uk.Signals UxII and Uk are fed into processor P via narrow band amplifier-detector SSD2 (Fig. 4).

IV. ANALYSIS OF MEASUREMENT CIRCUIT
Voltage in measurement circuit is formed by sources of sinusoidal voltage of low frequency ul0  Uml0sinlt and high frequency uh0  Umh0sinht.Suppose that non-equalities Zxh << Rv1 + Rv2, Zxh << Rk1 + Rk2 and Zxl << Rv1 + Rv2 are valid.Here Zxh and Zxl are the moduli of complex (phasor) impedances Zxh and Zxl for high and low frequency, correspondingly.Let represent the voltages ul0 and uh0 by phasors Ul0 and Uh0, correspondingly.The current phasors Ih and Il for high and low frequency, correspondingly, are: The phasors of divider output voltages Uxh and Uxl for high and low frequency, correspondingly, may be written: Division coefficient of divider D1 by ( 3) and ( 7) is Substituting this expression into (6) produces Taking into account that Rx and Cx are connected in parallel the following may be expressed: where Rxh and Cxh are expressed by ( 1) and ( 2), when ω = ωh, Rxl and Cxl are expressed by ( 1) and ( 2), when ω = ωh.Substituting (10) and ( 11) into (9) the following may be written Modulus Uxh of phasor Uxh is (1 ) , ( 1) where Uh0 and Ul0 are effective values of voltages uh0 and ul0, correspondingly.The frequencies must be chosen such that the following non-equalities could be valid: The (13) expression may be simplified as follows In third cycle the measurement circuit output signal phasor Uk can be expressed by (6), substituting Rk1/(Rk1 + Rk2 instead of Rv1/(Rv1 + Rv2).Taking into account (10), ( 15) and ( 16) Uk effective value Uk can be expressed as We obtain measurement signal Nx by dividing Uk by Uxh where 19) is independent on Cx if it does not change during the second and third measurement cycles.
Taking into account ( 14) and ( 16) from ( 1) it follows that Rxl ≈ Rs + Rp.When working instrument coupled with electrode E1 (see Fig. 1) approaches the apical constriction, Rp → 0 and Rxl → Rs.Setting Rs during Apex locator calibration apical constriction point may be determined exactly.

V. TECHNIQUE OF EXPERIMENTAL INVESTIGATION
The experimental investigation was performed in vitro.The block diagram of investigation is shown in Fig. 5.The working length of tooth root canal was measured in canals of four different sizes with endodontic instruments of four different diameters and using six different solutions for canal filling.
A plexiglas model of the tooth was used that had the cavity of the same shape as the tooth root canal.The apex in the model was formed by a segment of wire inserted into cavity 0,5 mm short of its end.The conditions of the oral mucosa were simulated using imitator -RC circuit.One simulator output was connected to the wire segment that imitated the tooth root tip, the other output was connected to the passive electrode of the apex locator E2.Different Protaper endodontic instruments were used.They are shown in Fig. 6.The maximal diameters of instruments are F1 -0,2 mm, F2 -0,25 mm, F3 -0,3 mm and F4 -0,4 mm.At first canals in the Plexiglas tooth models were formed by different Protaper instruments.These models were numbered analogically as instruments: F1 (maximal diameter dmax = 0,2 mm), F2 (dmax = 0,25 mm), F3 (dmax = 0,3 mm) and F4 (dmax = 0,4 mm).
The investigated tooth model was connected into the circuit shown in Fig. 5.At first, the experiment was performed with the instrument that had the same number as the imitator.Next, instruments with smaller numbers were used.To measure the real working length instrument was inserted into dry canal to contact with wire segment that imitated the tip.The length of inserted instrument part was measured by electronic calliper.When the real working length was measured, the testing solutions were one by one injected into tooth imitator cavity and the working length was measured using Apex locator.For chosen instrument and chosen solution the measurements repeated 50 times.

VI. EXPERIMENT RESULTS AND DISCUSSION
The main aim of Apex locator is to indicate reliably that endodontic instrument approaches the apical constriction.If it indicates this point after the moment when the instrument touches the apical constriction can be too late due to moving inertia (overinstrumentation).It is important, that Apex locator should operate when distance d of instrument tip to apical constriction (physiological apex) is approximately 0 ÷ 2 mm.
The number of 1500 measurements was performed and in all cases apex locator was fixed such that apical constriction is reached though working instrument without touching the tip imitating wire.The distance d was in interval [0,08 ÷ 1,92] mm for all 1500 cases.Therefore, it may be claimed that the investigated Apex locator functioned very reliably.
The summary of results obtained in canal of 0,4 mm using different instruments is presented in Table I.It can be seen that using solutions of smaller conductivity the distance to the apical constriction was measured more precisely.But difference among results is not big considering difference among the conductances.For example, conductance of NaOCl is 10 6 times greater than conductance of distilled water.Significantly greater scattering of results using instrument F1 with minimal diameter can be explained by the fact that the tip of this instrument may be in different positions in canal with respect to tooth root tip.But in all cases deviation from the mean value was not greater than ±0,3 mm.Distribution of mean distances at apical constriction for 0,4 mm canal and different instruments is shown in Fig. 7. Similar distribution is for 0,3 mm canal (see Fig. 8).The results for canals 0,2 mm and 0,1 mm are closed to results presented in Fig. 8.Typical distribution of 48 measurement results obtained with the same instrument in the same liquid is presented in Fig. 9.
Figure 9 shows that the measurement result distribution law is very close to normal.
VII. CONCLUSIONS 1. Measurement accuracy dependence not only on canal electrical properties, but on capacitance between the electrodes as well may be eliminated by using three different measurement intervals and power supply sources of low and high frequencies.2. Experimental investigation in vitro showed that Apex locator with three measurement intervals allows reliably and exactly to find apical constriction point using instruments of different diameter and solutions which electric conductance that may vary in a wide range.

Fig. 3 .Fig. 4 .
Fig. 3. Electric circuit of measurement cycles.Measurement consists of three cycles.In the first cycle the switches K1 and K2 are in the position I. Measurement circuit input is connected to low frequency source ul0, circuit output is connected to divider D1 (resistors Rv1, Rv2).The output voltage Ux of divider D1 is connected to input of narrow band amplifier -detector SSD1 which is part of control circuit (see Fig.4).In SSD1 signal Ux is filtered, K times amplified and rectified.In comparator CM rectified signal Ul = KUx1 is compared with reference voltage Ur.In the output unit OU control signal CS (output signal of control circuit CC) is formed.This signal is fed to control input of divider D1 part Rv1 (typically, field transistor, ) where it changes its resistance and divider output voltage Ux.Divider D1, amplifier -detector SSD1, comparator CM and output unit OU contain the circuit of negative feedback.This circuit changes Ux such that error signal U  KUxl-Ur which is input signal of comparator CM could be infinitely small.This way the effective value UxI of phasor UxI is set in the first cycle as follows / .xlr U U K (3)

Fig. 7 .
Fig. 7.The mean distance d of apex locator action point to apical constriction using instruments F1 -F4 and different solutions for 0,4 mm canal.

Fig. 8 .
Fig. 8.The mean distance d of Apex locator action point to apical constriction using instruments F1 -F3 and different solutions for 0,3 mm canal.

TABLE I .
THE MEAN DISTANCE OF APEX LOCATOR ACTION POINT AT APICAL CONSTRICTION AND ITS DEVIATION.