2.1.

Simulation Method of Stationary Light Fields Inside and Outside Multilayered Tooth Tissue

Tooth tissue is assumed to be a three-layered medium composed of enamel (top layer), dentin (intermediate layer), and dental pulp (bottom layer). All these layers are highly turbid, so one can represent them as uniform infinite slabs in the direction perpendicular to the light incidence. We will use this assumption in further simulations. Optical and geometrical characteristics of the multilayered medium to describe light fields inside the tissue and backscattered by it must be specified. Unfortunately, there are no so much published data on the scattering and absorption coefficients, ${\mu }_s$ and ${\mu }_a$, and on the phase function (or its integral parameters) of tooth tissue as there are data concerning soft biotissues. The phase function of each layer is usually represented²¹ as the sum of the totally diffuse component (with relative weight $f_d$) and the Henyey–Greenstein function with relative weight ($1-f_d$) and asymmetry parameter $g$ (or the mean cosine of the scattering angle).Several works^21–28 have been devoted to experimental or model estimations of the said optical characteristics at several wavelengths $\lambda$.

We used the compilation of data^21,24,26,28 on ${\mu }_s$, ${\mu }_a$, $f_d$, and $g$ for enamel and dentin to simulate spectral light fields inside and outside tooth tissues. For intermediate wavelengths where published results were lacking, the literature data were extrapolated and interpolated as needed. Note that the employed optical model is rough. In particular, it does not take into account the anisotropy of tissue scattering properties caused by the orientations of enamel prisms and dentinal tubules.^29–32 We ignored this effect for our estimations as many investigators do while studying light propagation through tooth tissue.

The dental pulp is the connective tissue. Its main optically essential chromophores are blood, interstitial fluid, collagen bloodless tissue, and some other minor components. The optical properties of the pulp were assumed below to be the same as those of soft biotissues. According to their various models,^33–35 scattering and especially the absorption properties of the soft tissue depend on the blood volume fraction $C_V$ (blood volume per tissue volume). To estimate the value of $C_V$, let the pulp volume be³⁶ $V_p=0.02{\mathrm{cm}}^3$, pulp density (pulp consists of 75% to 80% water³⁷) ${\rho }_p=1\mathrm{g}/{\mathrm{cm}}^3$, and specific blood flow through pulp³⁸ $F_0=40$ to $50\mathrm{mL}/\min$ per 100 g of pulp. Then the blood volume $V_b=v_b·\mathrm{\Delta }t·S·C_V$ passes through the pulp section area $S\approx {(V_p)}^{2/3}$ per time $\mathrm{\Delta }t$, where $v_b$ is the blood velocity in pulp. On the other hand, $V_b=F_0·{\rho }_p·V_p·\mathrm{\Delta }t$. From these two equations, one can estimate $C_V=0.025/v_b$ for the specific $F_0$ value, where $v_b$ is in $\mathrm{mm}/\mathrm{s}$. The velocity $v_b$ depends on the blood vessel type and diameter. It approximately equals³⁹ 0.08 to 0.36, 0.3 to 2.5, and 0.5 to $1\mathrm{mm}/\mathrm{s}$ for capillaries, arterioles, and venules, respectively. Therefore, concentration $C_V$ varies from 0.01 to 0.3. This range agrees rather well with measurements⁴⁰ and will be used in the model calculations below.

We simulated the diffuse reflectance (usually measured by an integrating sphere) and depth distributions of the fluence rate over three-layered tooth tissue. The calculation procedure to do so as it applied to skin tissue, was published^18,41,42 earlier. This method is based on the known analytical solutions to the radiative transfer equation,⁴³ accounting for multiple re-reflections between tissue layers and the surface. The goals of the simulations are to evaluate whether blood conditions will be seen in the reflected light and to estimate the light penetration depth in tooth tissue.

2.2.

Experimental Setup

The schematic and general views of the device for monitoring blood microcirculation are shown in Fig. 1. The setup was tested and certified by the Belarus State Institute of Metrology. The setup and study protocols were approved for use on human subjects (Certificate No. IM-7.2263).

Fig. 1

Schematic (upper part) and general (lower) views of the setup: 1 – laser, 2 – transmitting fiber, 3 – receiving fiber, 4 – recording module, 5 – sensor assembly, 6 – biological object. Distances $S_1$ and $S_2$ are selected to form the required sizes of the illuminating and received light beams and to provide the maximal possible contrast of speckle images.

The setup⁴⁴ includes a laser illuminating module, a recording module, and a processing unit. The illuminating module contains a single-mode semiconductor laser diode HL6501MG conjugated with a single-mode transmitting fiber SM600, a connector FC, a module for monitoring laser power, and a low-voltage power supply. The wavelength of the diode is $\lambda =660\mathrm{nm}$ and the light power is $P_{\text{out}}\le 7.5\mathrm{mW}$ at the output of the fiber.

The recording module includes a sensor assembly, a receiving fiber, a photo-electronic multiplier (PEM) (FEU-114), an amplifier, a band-pass filter, an analog-to-digital converter (ADC), and a high-voltage power supply. The sensor assembly contains a special mouth piece providing a nondisturbing contact between the assembly and the tissue surface. The mouth piece [shown in the lower part of Fig. 1(b) as applied to skin tissue] also serves as a screen for light scattered by adjacent areas of the object studied. The PEM is supplied by high voltage to provide the required spectral sensitivity (about ${10}^{-10}\mathrm{lm}/\mathrm{Hz}$) of the sensor. The PEM outputs an electric signal to the amplifier with a controllable gain that properly amplifies it for the ADC. The filter rejects the DC component of the PEM signals and restricts their frequency band within a controllable range. The ADC digitizes the analog signal from the filter and delivers the output to the processing unit.

The latter includes a PC and operating software. It controls the amplifier’s gain and the filter band and performs the pre- and final processing operations. These operations include gathering the experimental data, their Fourier transform, and statistical processing of the results obtained. The filter band is usually set to transmit frequencies in the range of 10 to 2000 Hz.

2.3.

Patients

Dental care was provided for patients with a diagnosis of deep or middle occlusal caries. The patients were referred by themselves to a dental establishment. They volunteered for the study and the proper consent was obtained. The x-ray studies of the teeth have shown that the distances between the bottoms of carious cavities and the pulp cameras were 1.5 to 2 mm. All the in vivo teeth used for the measurements were separated into the following two groups (the number of the teeth in each group is indicated in brackets):

2.4.

Parameters of Speckle Patterns

The temporal dependences of a light signal backscattered by a multilayered in vivo tooth tissue were measured at different stages of caries treatment. Then the speckle patterns were preprocessed (filtered, time- or space-averaged, etc.). Two kinds of parameters were investigated. They are spectral (items a to d below) and spatial characteristics (e). Note that assuming the ergodicity hypothesis in respect to the temporal and spatial distributions of the speckles, one can replace the space averaging with the time averaging. Then after making the Fourier-transform $W(f)$ ($f$ is the frequency) of the preprocessed temporal signal, the following integral quantities of the transform were calculated as

In Eqs. (1) to (4), $f_{\text{min}}$ and $f_{\text{max}}$ are the minimal and maximal frequencies used in the calculations, $f_1$, $f_2$, $f_{\text{low}}$, and $f_3$, $f_4$, $f_{\text{high}}$ are the frequencies taken, respectively, in the low- and high-frequency regions, and $\mathrm{\Delta }f=10\mathrm{Hz}$ is the fixed increment.

We also calculated (after the time averaging of the signals) (e) contrast $C$ of a speckle pattern, which is defined as the ratio of the standard deviation $\sigma$ of the illuminance $I$ in the observation plane to the background or mean illuminance $\langle I\rangle$ (here brackets $\langle \dots \rangle$ denote space averaging)

Additionally, other characteristics, such as autocorrelation function, asymmetry coefficient of the spectrum with respect to the mean frequency, $A={\int }_{f_{\mathrm{low}}}^{\langle f\rangle }W(f)\mathrm{d}f/{\int }_{f_{\mathrm{low}}}^{f_{\mathrm{high}}}W(f)\mathrm{d}f$, and the ratio of the asymmetry coefficient to the mean frequency, were tested. However, these parameters showed large dispersions and did not enable one to clarify the general relations between them and blood flow rate. For this reason, only the quantities of items (a) to (e) will be used.

3.1.

Simulated Stationary Reflectance/Fluence Rate

Figure 2 shows the simulations of spectral diffuse reflectance [Fig. 2(a)] and fluence rate distributions over depth $z$ [Fig. 2(b)]. Figure 2(a) gives the calculations for varying enamel thickness $d_e$ and a specific dentin thickness $d_d=4\mathrm{mm}$. The similar results obtained for other $d_d$ values show that dentin with $d_d>2\mathrm{mm}$ can be practically regarded as an infinitely thick layer with respect to light reflection. In other words, the diffuse reflectance in the visible to the near IR is essentially independent of $d_d$ in this case. This is apparently due to the rather large optical thickness of dentin. For the same reason, the diffuse reflectance is independent of blood volume content $C_V$. This tells us that it is impossible to monitor blood conditions using stationary backscattered light or habitual spectral photometric measurements “by reflection.” The situation is understood to be opposite when one observes transmitted light. In such a case, this light can be promoted to monitor various blood parameters.^39,45 Figure 2(a) also gives the experimental data²³ on diffuse reflectance (symbols). One can see that the experimental results show a behavior similar to our theoretical simulations. This is surprising at the first glance, if one recalls that the base for the calculations is a rather rough optical model.

Fig. 2

(a) Spectral dependence of tooth diffuse reflectance, calculations (curves 1 to 5) and experiment²³ (symbols), $d_e=0.2$ (1), 0.4 (2), 0.8 (3), 1.6 (4), and 3.2 mm (5), $d_d=4\mathrm{mm}$; and (b) depth dependence of normalized fluence rate inside tooth tissue at $\lambda =450$ (1), 632 (2), and 800 nm (3), $d_e=0.2\mathrm{mm}$, $d_d=4\mathrm{mm}$, $C_V=0.15$, blood oxygen saturation 0.75.

Depth dependences of the fluence rate are shown in Fig. 2(b). Here, the ordinate data are dimensionless so as to be normalized by the incident power density. One can see that fluence rate near the tooth surface up to some fractions of mm is greater than unity. This is due to the large amount of backscattered light at the topical tooth region. Note two points with respect to Fig. 2(b). First, the fluence rate in the pulp at the blue—violet wavelengths attenuates quickly with depth because of high blood absorption there. In the red to near IR, blood absorption is lower, and the shown dependences, for all practical purposes, do not change their slope. Second, one can roughly estimate the light penetration depth $z_0$ in tooth tissue, which is on the order of 2 mm in the visible to near IR. Here, $z_0$ values are assumed to be the depths where the fluence rate decreases by 10 times as compared with that incident to the tooth surface.

3.2.

Experimental Characterization of Dental Pulp Hemodynamics

For each tooth the parameters of Eqs. (1) to (5) were obtained. Twelve ranges were checked for the spectral power and the band coefficient, 15 for the $\mu$ coefficient, and 10 for the mean frequency to select the integration limits of Eqs. (1) to (4). The most informative limits will be given below. The measurements for each tooth were repeated 10 times. Data dispersion was peculiar to each observation so that particular measurements were excluded from the experimental dataset if they were characterized by large deviations of all parameters from the mean results in all considered frequency ranges. On average, 2 to 4 observations were excluded from 10 repetitions. Data for one tooth were excluded for Groups 1 and 2 because the corresponding results did not agree with the others.

Then for each Group of tested teeth and for each stage of caries treatment, the mean values of the said parameters and their standard deviations were calculated. They are shown below as, respectively, the ordinates of the bar graphs and the error bars in Figs. 3Fig. 4 to 5. The statistical significance of the differences in the parameters at each stage of caries treatment was evaluated according to the well-known Student’s $t$-test. The significance level ($p$-value) for all the parameters and for all stages was less than 0.01. For specific parameters and specific stages, the $p$-values were less than 0.001.

Fig. 3

Changes in spectral power, relative units (a), band coefficient (b), coefficient $\mu$ (c), and mean frequency, Hz (d) for an “average” tooth from Group 1: (a) $f_{\text{min}}=10$ and $f_{\text{max}}=600$ (or 700) Hz; (b) $f_1=50$, $f_2=100$ (or 150), $f_3=900$, and $f_4=1000\mathrm{Hz}$; (c) $f_{\text{low}}=100$ (or 50) and $f_{\text{high}}=900\mathrm{Hz}$; and (d) $f_{\text{min}}=10$ and $f_{\text{max}}=800\mathrm{Hz}$ before (1), immediately after anesthesia (2), and in 2 h after anesthesia (3).

Fig. 4

Changes in speckle contrast (a) before (1), immediately after anesthesia (2), and in 2 h after anesthesia (3) for teeth from Group 1; and (b) before (1) and after the preparation (2), after etching (3), after tooth restoration (4), after irradiation (5), and after polishing (6) for teeth from Group 2.

Fig. 5

Changes in spectral power, relative units (a), band coefficient (b), coefficient $\mu$ (c), and mean frequency, Hz (d) before (1) and after the preparation (2), after etching (3), after tooth restoration (4), after irradiation (5), and after polishing (6) of a tooth from Group 2. (a) $f_{\text{min}}=10$ and $f_{\text{max}}=600$ (or 700) Hz, (b) $f_1=50$, $f_2=100$ (or 150), $f_3=900$, and $f_4=1000\mathrm{Hz}$, (c) $f_{\text{low}}=100$ (or 50) and $f_{\text{high}}=900\mathrm{Hz}$, (d) $f_{\text{min}}=10$ and $f_{\text{max}}=800\mathrm{Hz}$. The legend is shown in Fig. 5(a).

Figures 3 and 4(a) show the diagrams of changes in the above parameters observed in experiments with the teeth of Group 1, namely before anesthesia (1), immediately after the anesthesia (2), and 2 h after anesthesia (3). Anesthesia will obviously give rise to a reduced blood flow rate. Two h after anesthesia, the spectral power slowly recovers. A similar conclusion can be made from the comparison of the data for $f_{\text{low}}=100$ and 50 Hz in Fig. 3(c).

Changes in speckle contrast [Fig. 4(a)] also testify to the above qualitative estimations of the speckle image contrast at varying blood flows. Immediately after the anesthesia, the contrast increases due to the obvious reduction in blood flow. However, the contrast becomes even lower than its initial value 2 h after the anesthesia. The latter qualitatively tells us that the blood flow rate is higher in this case compared with its values before treatment. This fact disagrees somewhat with the data of Fig. 3 and requires further investigations.

Figures 4(b) and 5 illustrate the changes in the considered integral parameters of the speckle structure at various stages of caries treatment, namely before (1) and after the preparation (2), after etching (3), after tooth restoration (4), after irradiation (5), and after polishing (6). These stages are correspondingly shown from the left to right. The data on the changes in the speckle contrast, Fig. 4(b), highly correlate (synchronously decrease and increase at the illustrated stages) with those on $S$ and $\mu$ values, Figs. 5(a) and 5(c). On the other hand, these results anticorrelate with the data of Figs. 5(b) and 5(d), i.e., when the contrast increases, $K_b$ and $\langle f\rangle$ decrease, and vice versa. A similar correlation and anticorrelation can be mentioned with respect to the corresponding data shown in Fig. 3.