2.1.

Effects of Light Wave External to the Tooth Propagating Directly to Gingiva on TLP Signal

The effects of the light wave external to the tooth traveling through the gingiva on the TLP signal were examined in the human volunteers. After the protocol for human study was approved by the Institutional Review Board of the Tokyo Medical and Dental University, the optical measurements in the upper central incisor of human volunteers were conducted using the multiwavelength TLP system (Fig. 2 ).¹⁶ Preceding the study, a resin adaptor (as shown in Fig. 3 ) for each subject was fabricated to fix the optical fibers against a tooth. As a light source, we used the $522\mathrm{nm}$ wavelength LED pulsed at $560\mathrm{Hz}$ . The transmitted light through the tooth at $560\mathrm{Hz}$ was sampled to minimize the background light. At first, the dc component (absolute level of the transmitted light signal) and the ac component (periodically varying signal amplitude synchronized to heart beat) of the TLP signals at $522\mathrm{nm}$ wavelength were recorded in each subject for $\sim 2\min$ . Then, the silicone impression material, Exafine Regular Type (GC Dental Products Corp., Tokyo Japan) was injected into the gingival sulcus around the resin cap and proximal surface of the tooth to block the light propagation between the optical fiber and the gingiva, followed by the recording of the dc and ac signals again. The significance in the difference between the two measurements with and without the silicone material was evaluated with $p<0.01$ .

Fig. 2

Multiwavelength optical plethysmograph.

Fig. 3

Resin cap placed on the upper central incisor with optical fibers.

Figure 4 showed the changes in dc and ac/dc signals of the TLP in the upper central incisors before and after injection of the silicone impression material $(n=12$ , $25–31\text{years}$ old with mean $27.3\text{years}$ old). With the silicone injected in the gingival sulcus, the average value of the optical density (OD) changed from $6.55\pm 0.07\text{to}6.60\pm 41$ , and ac/dc ratio changed from $0.069\pm 0.031\text{to}0.064\pm 0.025$ . For both OD and ac/dc values, there were no significant differences with and without silicone impression material. This study demonstrated that the light wave traveling on the tooth surface directly from the light source to the gingiva had little affect on TLP measurements.

Fig. 4

OD and ac/dc ratio obtained in human volunteers showing the effects of the silicone material injected around the gingival sulcus.

2.2.

Monte Carlo Simulation of Light Propagation Inside the Tooth

Figure 5 showed a two dimensional Monte Carlo simulation model of the tooth Fig. 1 showing layers of enamel, dentin, pulp, and gingiva. The dimensions of the model reflect those of the average size tooth as measured using the X-ray machine (refer to Table 1 data). In the microscopic domain of dentin, each dentin layer consisted of a collagen fiber and mineral substance having a $30\text{-}\mu \mathrm{m}$ thickness, each with the differential optical properties between them affecting the light propagation. Table 2 shows the optical properties of the dentin material for the wavelengths of 522 and $810\mathrm{nm}$ .²⁰ The Monte Carlo simulation of the light propagation for the model tooth of Fig. 5 was derived by an iterative calculation using Eqs. 1, 2, 3, 4.¹⁷ The photon step size and its angle of scattering at one scattering were determined by Eqs. 1, 2, 3, yielding the maximum step size of photon in the mineral substance to be $\sim 37\mu \mathrm{m}$ for 522 and $810\mathrm{nm}$ . We thus employed the mesh size of $30\mu \mathrm{m}$ in the Monte Carlo simulation. The iterative equations used for calculation were

Fig. 5

Two-dimensional tooth model vertically cut in the plane of the optical fibers for the Monte Carlo simulation.

Table 1

Dimensions of the selected teeth.

Table 2

Optical properties of dentin and gingiva.

In order to emphasize the photon paths in the model, the result was shown by the level of absorption intensity. The numbers of photons that had scattered off the gingiva and then traveled through the tooth and those that did not reach gingiva were tracked separately to show the effects of gingiva on the transmitted light intensity. Figure 6 showed the absorbed light intensity for 522 and $810\mathrm{nm}$ wavelengths in the macroscopic scale. In Fig. 7 absorption intensities in the $500\times 500\mu \mathrm{m}$ microscopic scale were shown, where the photons scattered in the enamel layer became guided along the dentin tubule toward pulp chamber, the effect similar to that reported by Kiele and Hibst²¹ Table 3 summarizes the statistical results showing how many photons have traveled to hit the gingiva and scattered off toward the detector. Out of 20 million photons injected inside the model tooth, only 5 photons (0.000025%) traveled to the gingiva, while 497 photons (0.0025%) at $522\mathrm{nm}$ and 876 photons (0.0044%) at $810\mathrm{nm}$ , over 100 times those that hit the gingiva, traveled through pulp to the detector without routing the gingiva. The Monte Carlo simulation has thus indicated the minimal effects of the gingival circulation on the optical transmission measurements in an upper central incisor model.

Fig. 6

Absorption intensity for $522\mathrm{nm}$ and $810\mathrm{nm}$ obtained by the Monte Carlo simulation. Left side are two-dimensional plots and right side three-dimensional plots.

Fig. 7

Microscopic absorption intensity for $522\mathrm{nm}$ (top) and $810\mathrm{nm}$ (bottom) in the left panel without enamel layer. In the right panel, for $522\mathrm{nm}$ , with the enamel layer, the scattered light in the enamel layer was directed along the dentin tubules with the highest intensity at the center.

Table 3

Results of the Monte Carlo simulation.

3.1.

Dentin-Pulp Chamber-Dentin Three-Layer Model

The tooth structure was simplified to a three-layer model comprising of dentin-pulp chamber-dentin layers (Fig. 8 ). Although there are paths around the pulp chamber for the photons to propagate, the tubules in the dentin layer are organized in such a way to guide the light toward the pulp chamber.²¹ The Monte Carlo simulation presented above supported the light-guiding phenomenon, as illustrated in the microscopic absorption intensity in the dentin model. The scattering property of the dentin layer is so much greater than that of the enamel layer that the tooth appearance is determined mainly by the scattering property of the dentin tubules.^{22, 23} The exclusion of the enamel layer from the model thus does not affect the relative changes of the model performances in terms of the red blood cell concentration inside the pulp chamber.

Fig. 8

Simplified TLM consisting of dentin, pulp, and dentin layers.

3.2.

Three-Layer Model (TLM) Equation

In deriving a mathematical equation for the transmitted light intensity through a three-layer model (TLM), we assumed that each layer was homogeneous and also the diffuse radiation was incident upon the medium so as to justify the application of the two-flux model introduced by Kubelka with no reflection at the layer boundary (Fig. 9 ).¹⁸ The differential equations for the two fluxes, $I$ in the positive direction and $J$ in the negative direction are written as

Fig. 9

Two flux model for diffuse radiation incident on a homogeneous medium.

Now we consider when three homogeneous layers with different optical properties were next to each other as shown in Fig. 10. The solution can be given in a closed form as a function of the transmittance and reflectance of each layer as follows:¹⁹

Fig. 10

TLM with the transmitted and reflected light fluxes.

3.3.

Optical Constants of Dentin and Pulp Chamber

The scattering and absorption properties of the medium are wavelength dependent, particularly of hemoglobin contained inside the red blood cells. In this study, the transmission through three layers $T_{123}$ was computed for the isosbestic wavelengths of the oxy- and deoxy-hemoglobin, in the green region at $522\mathrm{nm}$ and the near-infrared region at $810\mathrm{nm}$ [Fig. 11].²⁴

For the absorption and scattering constants of the dentin layer, $K_{\mathrm{d}}$ and $S_{\mathrm{d}}$ , though its properties are dependent on the density of tubules, in this study we adopted $K_{\mathrm{d}}=0.40{\mathrm{mm}}^{-1}$ , $S_{\mathrm{d}}=28.09{\mathrm{mm}}^{-1}$ for $522\mathrm{nm}$ , and $K_{\mathrm{d}}=0.344{\mathrm{mm}}^{-1}$ , $S_{\mathrm{d}}=26.97{\mathrm{mm}}^{-1}$ for $810\mathrm{nm}$ , as reported by Fried ²⁵

As for the optical constants of the pulp chamber, we assumed that the chamber was a homogeneous medium comprising of the whole blood. The scattering and absorption constants of the pulp chamber, $S_{\mathrm{b}}$ , $K_{\mathrm{b}}$ , were thus derived from the optical constants of the whole blood. According to Twersky,²⁶ $S_{\mathrm{b}}$ can be derived from the scattering and absorption cross sections of a red blood cell, ${\sigma }_{\mathrm{s}}$ and the red blood cell concentration given by ${\mathrm{Hct}}_{\mathrm{p}}∕{\mathrm{V}}_{\mathrm{rbc}}$ , where ${\mathrm{Hct}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{rbc}}$ the hematocrit of the blood in the pulp chamber and the mean corpuscular volume of red blood cells, respectively, as follows:

As for $K_{\mathrm{b}}$ value, ${\sigma }_{\mathrm{a}}$ is usually expressed as a linear sum of the absorption cross section of the oxygenated and deoxygenated red blood cell, ${\sigma }_{\mathrm{ao}}$ , ${\sigma }_{\mathrm{ar}}$ , and oxygen saturation $\mathrm{S}{\mathrm{O}}_2$ and red blood cell concentration ${\mathrm{Hct}}_{\mathrm{p}}∕{\mathrm{V}}_{\mathrm{rbc}}$ as follows;²⁸

Fig. 11

Absorption coefficients of the human oxygenated and deoxygenated hemoglobin in the visible and near-infrared regions.

3.4.

Experimental Verification of the Three-Layer Model

The experiments consisted of (i) selection of the human upper central incisors for optical measurements, (ii) transmitted optical intensity measurements in the selected sample tooth to obtain relationship between the tooth’s OD versus pulp chamber hematocrit ${\mathrm{Hct}}_{\mathrm{p}}$ , and to verify the accuracy of the TLM for estimation of ${\mathrm{Hct}}_{\mathrm{p}}$ , (iii) microscope examination of the light propagation through a tooth to provide the basis for OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ relationships obtained in (ii), and (iv) optical measurements in human volunteers to quantify the hematocrit of the tooth using the TLM.

3.4.1.

Selection of the sample teeth

In selecting the tooth for optical measurements, the mesiodistal (MD) width and buccolingual (BL) thickness of the 25 extracted human upper central incisors free of caries, restoration, attrition, and discoloration were measured at $3\mathrm{mm}$ from the cervical line (Fig. 1) using X-ray equipment, Compuray (Yoshida Dental Mfg., Co. Ltd., Tokyo, Japan). Figure 12 showed the scatter plots of the BL thickness of the pulp chamber $\left({\mathrm{BL}}_{\mathrm{p}}\right)$ vs. BL thickness of the tooth $\left({\mathrm{BL}}_{\mathrm{T}}\right)$ with a linear regression line ${\mathrm{BL}}_{\mathrm{p}}\left(\mathrm{mm}\right)=0.223\times {\mathrm{BL}}_T\left(\mathrm{mm}\right)-0.996(r=0.568,p<0.01)$ . Out of 25 samples, five teeth (Nos. 1–5) were selected for optical measurements. Table 1 summarized ${\mathrm{BL}}_{\mathrm{T}}$ and ${\mathrm{BL}}_{\mathrm{p}}$ of the selected five teeth. As for tooth 4, the pulp chamber was enlarged twice during the optical measurements denoted as Nos.4-1, 4-2, and 4-3, each having different pulp chamber size to see the effects of the pulp chamber size on the OD.

Fig. 12

Scatter plot between the BL thickness of the pulp chamber $\left({\mathrm{BL}}_{\mathrm{p}}\right)$ versus BL thickness of the whole tooth ${\mathrm{BL}}_{\mathrm{T}}$ .

In order to introduce the blood into the pulp chamber, the root of all the sample teeth was cut off approximately $3.0\mathrm{mm}$ from the root apex. The root canal was enlarged to $1.5\mathrm{mm}$ diam, and the remnant pulp tissues were removed.

3.4.2.

Optical measurements in the selected teeth

A resin cap was prepared for each tooth to support the tooth as well as to fix the optical fibers as shown in Fig. 13 . Two plastic optical fibers ( $2.0\mathrm{mm}$ diam and $1.0\mathrm{m}$ length each), one for transmitting the light signal from the LED to the tooth and the other for carrying the transmitted light signal from the tooth to a photodiode located in the multiwavelength optical plethysmograph,¹⁶ were aligned perpendicular to the longitudinal axis of the tooth illuminating from the palatal surface and collecting the transmitted light from the labial surface.

Fig. 13

Experimental setup for the optical transmission measurements in a extracted model tooth.

The pulp chamber was filled with saline first to obtain the reference signal, followed by the measurement with the human blood having different ${\mathrm{Hct}}_{\mathrm{p}}$ . After obtaining the informed consent, $\sim 10\mathrm{ml}$ of fresh human blood was withdrawn from one of the volunteers, anticoagulated with heparin ( $5000\text{units}∕5\mathrm{ml}$ , Mochida Pharmaceutical Co., Ltd., Tokyo, Japan), centrifuged, and rinsed by phosphate buffer solution (PBS). The hematocrit of the blood was adjusted by adding PBS from 0.35, 0.30, 0.20, 0.10, 0.05, 0.03 to as low as 0.01. The optical measurements were made at 522 and $810\mathrm{nm}$ wavelengths. For each wavelength, the incident light intensity $V_0$ in voltage was derived from the LED current level, while the transmitted light intensity $V_{\mathrm{t}}$ in voltage was recorded to derive the optical density defined as $\mathrm{OD}=\ln ({\mathrm{v}}_{\mathrm{o}}∕{\mathrm{v}}_{\mathrm{c}})$ . The OD measurements were repeated seven times for each wavelength with different LED current levels and the data were expressed as mean ± SD.

3.4.3.

Examination of the optical propagation through a tooth

Following the OD measurement, the optical propagation through a tooth was examined under a microscope to verify the experimental results obtained in Sec. 3.4.2. One of the extracted human upper central incisors was cut horizontally at the plane of the incident light $I_{\mathrm{o}}$ , placed under a surgical microscope with the cut surface upward and illuminated from the palatal side with the $522\mathrm{nm}$ wavelength light using an optical fiber, as shown in Fig. 14 to observe the optical propagation process in the tooth close to the cut surface. The microscope pictures of the pulp chamber being filled with (a) saline, whole blood having hematocrit of (b) 0.03, or (c) 0.30 were obtained to elucidate the causes for the OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curves. In addition, the blood was hemolyzed to reduce scattering and the hemoglobin solution equivalent to the cases of (b) or (c) were introduced into the pulp chamber to examine the effect of red blood cell scattering on the OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curves obtained in Sec. 3.4.2.

Fig. 14

Experimental setup for examining optical propagation patterns in an extracted model tooth under a microscope.

3.4.4.

${\mathrm{Hct}}_{\mathrm{p}}$ estimation in the extracted model tooth

Calibration procedure

From the preliminary study we confirmed qualitative agreement between the TLM and the experimental results, but there was a discrepancy in the absolute levels. To utilize the TLM for quantification of the ${\mathrm{Hct}}_{\mathrm{p}}$ , the discrepancy in the absolute signal levels between the TLM and the experimental data was compensated for by using the experimental data. The OD vesus ${\mathrm{Hct}}_{\mathrm{p}}$ data of the reference tooth 4-2 ( $6.0\mathrm{mm}$ ${\mathrm{BL}}_{\mathrm{T}}$ and $0.40\mathrm{mm}$ ${\mathrm{BL}}_{\mathrm{p}}$ ), which was closest to the regression line of Fig. 12 was used to adjust the absolute OD level derived by the TLM. The gain constants for the saline and blood samples, denoted as $G\left(0\right)$ and $G\left({\mathrm{Hct}}_{\mathrm{p}}\right)$ , were both defined as follows:

Eq. 16

$$G\left(0\right)=\frac{{\mathrm{OD}}_{\mathrm{EX}-4\text{-}2}({\mathrm{Hct}}_{\mathrm{p}}=0)}{{\mathrm{OD}}_{\mathrm{TLM}-4\text{-}2}({\mathrm{Hct}}_{\mathrm{p}}=0)}$$

Eq. 17

$$G\left({\mathrm{Hct}}_{\mathrm{p}}\right)=\frac{{\mathrm{OD}}_{\mathrm{EX}-4\text{-}2}\left({\mathrm{Hct}}_{\mathrm{p}}\right)-{\mathrm{OD}}_{\mathrm{EX}-4\text{-}2}({\mathrm{Hct}}_{\mathrm{p}}=0)}{{\mathrm{OD}}_{\mathrm{TLM}-4\text{-}2}\left({\mathrm{Hct}}_{\mathrm{p}}\right)-{\mathrm{OD}}_{\mathrm{TLM}-4\text{-}2}({\mathrm{Hct}}_{\mathrm{p}}=0)}.$$

Here, ${\mathrm{OD}}_{\mathrm{EX}\text{-}4\text{-}2}\left({\mathrm{Hct}}_{\mathrm{p}}\right)$ , ${\mathrm{OD}}_{\mathrm{EX}\text{-}4\text{-}2}({\mathrm{Hct}}_{\mathrm{p}}=0)$ corresponded to the OD measurements in tooth 4-2 as a function of ${\mathrm{Hct}}_{\mathrm{p}}$ and saline, respectively, while ${\mathrm{OD}}_{\mathrm{TLM}\text{-}4\text{-}2}\left({\mathrm{Hct}}_{\mathrm{p}}\right)$ , ${\mathrm{OD}}_{\mathrm{TLM}\text{-}4\text{-}2}({\mathrm{Hct}}_{\mathrm{p}}=0)$ the OD calculated by the TLM for reference tooth 4-2 as a function of ${\mathrm{Hct}}_{\mathrm{p}}$ and saline, respectively.

${\mathrm{Hct}}_{\mathrm{p}}$ Estimation.

In obtaining ${\mathrm{Hct}}_{\mathrm{p}}$ estimate from the OD data measured in the selected teeth (1, 2, 3, 4-1, 4-3, and 5), the ${\mathrm{OD}}_{\mathrm{TLM}}(T,{\mathrm{Hct}}_{\mathrm{p}})$ was first calculated for each tooth (where $T$ is the tooth number) as a function of ${\mathrm{Hct}}_{\mathrm{p}}$ , then its absolute level was adjusted using the gain constants as follows;

Eq. 18

$${\mathrm{OD}}_{\mathrm{TLM}}^{\prime }(T,{\mathrm{Hct}}_{\mathrm{p}})=[{\mathrm{OD}}_{\mathrm{TLM}}(T,{\mathrm{Hct}}_{\mathrm{p}})-{\mathrm{OD}}_{\mathrm{TLM}}(T,{\mathrm{Hct}}_{\mathrm{p}}=0)]G\left({\mathrm{Hct}}_{\mathrm{p}}\right)+{\mathrm{OD}}_{\mathrm{TLM}}(T,{\mathrm{Hct}}_{\mathrm{p}}=0)G\left(0\right).$$

In Eq. 18, ${\mathrm{OD}}_{\mathrm{TLM}}^{’}(T,{\mathrm{Hct}}_{\mathrm{p}})$ is the adjusted absolute OD level by the TLM, $G\left({\mathrm{Hct}}_{\mathrm{p}}\right)$ , $G\left(0\right)$ the gain constants defined by Eqs. 16, 17. The estimation of the ${\mathrm{Hct}}_{\mathrm{p}}$ was carried out graphically as shown by Fig. 15 to determine the ${\mathrm{Hct}}_{\mathrm{p}}$ value, where the adjusted OD curve by the TLM crossed the measured OD line. The accuracy of the method was quantified for both 522 and $810\mathrm{nm}$ wavelengths for the ${\mathrm{Hct}}_{\mathrm{p}}$ values below 0.10.

Fig. 15

Graphical representation of how to determine ${\mathrm{Hct}}_{\mathrm{p}}$ estimate from the TLM calculation and experimental OD measured in a tooth.

4.1.

Experimental Verification of the Three-Layer Model

4.1.1.

OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ in the extracted model tooth

The OD changes of tooth 4 versus ${\mathrm{Hct}}_{\mathrm{p}}$ were shown in Fig. 16 when the ${\mathrm{BL}}_{\mathrm{p}}$ was varied from 0.28, 0.40, to $0.51\mathrm{mm}$ . The OD at $522\mathrm{nm}$ increased greatly as the ${\mathrm{Hct}}_{\mathrm{p}}$ increased from 0.0 to around 7.0–8.0% with a small increase beyond that. The OD change of $810\mathrm{nm}$ , on the other hand, was much milder than that of $522\mathrm{nm}$ wavelength for the entire range of ${\mathrm{Hct}}_{\mathrm{p}}$ . As for the effects of the $\mathrm{B}{\mathrm{L}}_{\mathrm{p}}$ size on the OD level, the OD increased at both 522 and $810\mathrm{nm}$ wavelengths due to increased light loss when the pulp chamber was enlarged.

Fig. 16

OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ in tooth 4 for 522 and $810\mathrm{nm}$ wavelengths showing the effects of the pulp canal enlargement on the measured OD.

Figure 17 showed the OD changes versus ${\mathrm{Hct}}_{\mathrm{p}}$ for two different tooth samples, tooth 1 with smaller pulp chamber, while tooth 5 with a larger chamber from Fig. 12. Similar patterns were observed at both 522 and $810\mathrm{nm}$ wavelengths, although the OD level of saline did not agree between the two samples due to differences in the optical characteristics of the tooth material. For a smaller pulp chamber like in tooth 1, the absolute level of the OD was smaller with less light attenuation in the pulp chamber.

Fig. 17

OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ of two different extracted teeth showing the discrepancy in the OD level with saline in the pulp chamber.

4.1.2.

Optical propagation in a tooth

Figure 18 showed the microscope pictures of the light propagation through the tooth seen from the top as the pulp chamber was filled with (a) saline, and the blood having the ${\mathrm{Hct}}_{\mathrm{p}}$ of (b) 0.03 and (c) 0.30, or hemoglobin solution with hemoglobin content of (d) $1.2\mathrm{g}∕\mathrm{dL}$ or (e) $10.0\mathrm{g}∕\mathrm{dL}$ . The corresponding OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curves was shown in Fig. 19 to connect the external measurement to the internal optical process.

Fig. 18

Microscope pictures showing (a) optical propagation patterns with saline in the pulp chamber, (b) with 3.0% ${\mathrm{Hct}}_{\mathrm{p}}$ blood in the pulp chamber, and (c) with 30.0% blood in the pulp chamber. (d) and (e) show the propagation patterns when the blood of (b) and (c) was hemolyzed to reduce scattering effects.

Fig. 19

OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ or hemoglobin content showing the effects of scattering in the extracted model tooth when the pulp chamber was filled with whole blood or hemoglobin solution.

4.2.

Accuracy of the Three-Layer Model (TLM)

4.2.1.

OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ by the TLM

In Fig. 20 , OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ obtained by the TLM for the wavelengths of 522 and $810\mathrm{nm}$ were shown to simulate the results of tooth 4 when the $\mathrm{B}{\mathrm{L}}_{\mathrm{p}}$ varied from 0.28, 0.40 to $0.51\mathrm{mm}$ . The TLM results agreed qualitatively with the experimental results shown in Fig. 16.

Fig. 20

OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ of the tooth 4 for varying pulp chamber size generated by the TLM.

4.2.2.

${\mathrm{Hct}}_{\mathrm{p}}$ estimation in the extracted model tooth

Figures 21a and 21b showed the scatter plot between the ${\mathrm{Hct}}_{\mathrm{p}}$ estimates by the TLM versus the actual ${\mathrm{Hct}}_{\mathrm{p}}$ measured in the extracted model tooth at $522\mathrm{nm}$ [Fig. 21a] and at $810\mathrm{nm}$ [Fig. 21b]. Although the error bars increased with the increase in ${\mathrm{Hct}}_{\mathrm{p}}$ , the mean of the ${\mathrm{Hct}}_{\mathrm{p}}$ estimate versus actual ${\mathrm{Hct}}_{\mathrm{p}}$ showed an excellent agreement with the correlation coefficient of 0.957 $(p<0.0001)$ for $522\mathrm{nm}$ , while $r=0.849$ $(p<0.0001)$ for $810\mathrm{nm}$ wavelength. Figures 22a and 22b showed the mean and SD of the errors between the ${\mathrm{Hct}}_{\mathrm{p}}$ estimate and the actual ${\mathrm{Hct}}_{\mathrm{p}}$ for 522 [Fig. 22a] and $810\mathrm{nm}$ [Fig. 22b] wavelengths. The mean error for $522\mathrm{nm}$ was $-0.00115$ with SD of 0.00733, while for $810\mathrm{nm}$ they were $+0.09157$ and 0.02493.

Fig. 21

Scatter plots between the ${\mathrm{Hct}}_{\mathrm{p}}$ estimate by the TLM and the ${\mathrm{Hct}}_{\mathrm{p}}$ actual in the extracted model tooth for (a) $522\mathrm{nm}$ and (b) $810\mathrm{nm}$ .

Fig. 22

Bland and Altman plots showing distribution of errors between the ${\mathrm{Hct}}_{\mathrm{p}}$ estimate and the ${\mathrm{Hct}}_{\mathrm{p}}$ actual in the extracted model teeth for (a) $522\mathrm{nm}$ and (b) $810\mathrm{nm}$ light-source wavelength.

5.1.

Optical Propagation in an Extracted Model Tooth

The optical process in the extracted model tooth depended on the red blood cell concentration or ${\mathrm{Hct}}_{\mathrm{p}}$ , $\mathrm{B}{\mathrm{L}}_{\mathrm{p}}$ , and the light source wavelength. The wavelength-dependent absorption and scattering by hemoglobin and red blood cells in the pulp chamber and those of dentin layer determined the relation between the tooth’s OD and ${\mathrm{Hct}}_{\mathrm{p}}$ . Although the OD for $810\mathrm{nm}$ increased monotonously with its change of $<1.0$ for ${\mathrm{Hct}}_{\mathrm{p}}$ from 0.0 to 0.35 (Figs. 16 and 17), OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curve at $522\mathrm{nm}$ comprised of two regions (Fig. 19), region I with ${\mathrm{Hct}}_{\mathrm{p}}$ $<0.10$ and region II with ${\mathrm{Hct}}_{\mathrm{p}}$ $>0.10$ . In the region I, OD increased by 1.0–3.0 as ${\mathrm{Hct}}_{\mathrm{p}}$ increased from 0.0 to 0.10 with its rate and final level depending on the pulp chamber size, while $>0.10$ it was similar to that of $810\mathrm{nm}$ .

In explaining the OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curve of $522\mathrm{nm}$ , microscope pictures of Fig. 18 and summary graph of Fig. 23 were helpful. When the pulp chamber had non-scattering fluid like saline [Figs. 18a and 23a], the light reaching the dentin-pulp chamber boundary propagated mainly in the forward direction with some photons scattered off along the boundary. The dentintubules guided the light particles toward pulp chamber as well as from pulp chamber toward labial side of the tooth. When the pulp chamber had a tenuous medium with ${\mathrm{Hct}}_{\mathrm{p}}$ $<0.10$ , photons entering the chamber were multiply scattered to propagate in all directions or absorbed by hemoglobin [Figs. 18b and 23b]. The OD level in the region I thus increased to reach a plateau as the ${\mathrm{Hct}}_{\mathrm{p}}$ level increased to 0.10 because the photons penetrating the pulp chamber were reduced due to scattering and absorption. The OD level at plateau depends on the amount of photons routing around the pulp chamber to reach labial side. There was a large difference in OD level between the 522 and $810\mathrm{nm}$ wavelengths because of higher absorption at $522\mathrm{nm}$ , $\sim 50$ times the $810\mathrm{nm}$ , and also larger scattering at $522\mathrm{nm}$ than $810\mathrm{nm}$ .

Fig. 23

Summary of optical propagation patterns in the extracted model tooth, (a) with saline in the pulp chamber, (b) with a tenuous medium in the pulp chamber, and (c) with dense medium in the pulp chamber.

In region II with ${\mathrm{Hct}}_{\mathrm{p}}$ $>0.10$ , as shown in Figs. 18c and 23c, the light hitting the pulp chamber was completely blocked from advancing into the second dentin layer due to increased absorption as well as scattering in the pulp chamber. Only those photons that scattered off at the dentin-pulp chamber boundary and that routed around the pulp chamber probably reached the labial side. The light level at $522\mathrm{nm}$ wavelength beyond ${\mathrm{Hct}}_{\mathrm{p}}$ of 0.10 is probably equal to the amount of photons routing around the pulp chamber. When the pulp chamber size was increased both in width and thickness, the amount of the light that was blocked by the pulp chamber increased simultaneously reducing the photons routing around the pulp chamber to consequently increase the OD level. When the pulp chamber size was narrowed to decrease the microvessel density, the OD level decreased with less attenuation of the light in the pulp chamber.

Comparison of the OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ characteristics of the $522\mathrm{nm}$ wavelength against those of $810\mathrm{nm}$ , where the pulp chamber was filled with whole blood or hemoglobin solution, suggested that in the region I extremely high absorption by hemoglobin determined the characteristics of the OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curve at $522\mathrm{nm}$ wavelength although scattering had some effects, while at $810\mathrm{nm}$ wavelength the scattering was dominant in both regions, but its effect was much smaller in comparison to the absorption effect at $522\mathrm{nm}$ (Fig. 19).

5.2.

TLM

In modeling the optical process in the extracted model tooth, particularly the effect of the blood volume change in the pulp chamber on the tooth’s OD, the tooth structure was simplified to a TLM comprising dentin, pulp chamber, and dentin. The enamel layer had much lower scattering properties in comparison to the dentin tubules, and thus, it was excluded from the analysis.^{22, 23} The guided scattering by the dentin tubules as suggested by Kiele and Hibst and demonstrated by the Monte Carlo simulation in the preliminary study²¹ was helpful in simplifying the model to a TLM. Although there were paths around the pulp chamber for the light to propagate, because of the tubule organization the light entering the dentin layer was mainly guided toward the pulp chamber. The microscope examination of optical process in the tooth revealed that some photons propagated along the dentin-pulp chamber boundary to exit in the second dentin layer. The light propagating in the forward direction out of the pulp chamber was then guided along the dentin tubules spreading radially toward labial side.

The two-flux model proposed by Kubelka^{18, 19} was utilized to obtain an analytical equation relating the OD to tooth’s dimensions, optical constants of each layer, and light-source wavelength. The optical constants of each layer were expressed by the bulk scattering and absorption constant, $S$ and $K$ , by assuming that each medium was optically homogeneous. For the dentin layer, the experimentally derived values were used,²⁵ while for the pulp chamber they were derived from the red blood cell properties and ${\mathrm{Hct}}_{\mathrm{p}}$ . Because the scattering properties of the dentin layer depend on the dentin-tubule density, both scattering and absorption constants require adjustment from person to person.^{29, 30} In addition, optical properties of the model components including enamel, dentin, and pulp tissue vary with time due to growth, aging, pathological conditions, and discoloration due to surface characteristics, which, in turn, all affect the light transport through a tooth and absolute OD level.³¹

Concerning the boundary conditions for light transport in a complex medium, such as tooth, boundary conditions at the dentin-pulp chamber boundary need careful attention in improving the accuracy of the model, for the photons might be scattered off at the boundary depending on the ratio of the refractive indices between the dentin and pulp chamber. This effect was shown by the microscope examination where the light level at the dentin-pulp chamber boundary varied depending on the ${\mathrm{Hct}}_{\mathrm{p}}$ level. If we assumed the index of refraction of the dentin to be 1.50 at $522\mathrm{nm}$ ,³² because that for the pulp chamber takes a value from 1.33 to 1.45 for the ${\mathrm{Hct}}_{\mathrm{p}}$ level varying from 0.0 to 0.30, the ratio of the dentin to pulp chamber refractive indices changes from 1.13 to 1.03, the mismatch in index of refraction becoming lesser at higher ${\mathrm{Hct}}_{\mathrm{p}}$ level. The ratio becomes closer to 1.0 at higher ${\mathrm{Hct}}_{\mathrm{p}}$ , and hence, the absorption becomes a dominant factor for attenuating the light propagation at green wavelength, while for red and near-infrared wavelengths scattering is dominant to mask absorption. Hence there is a steady but small increase in the OD level. The OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ curves of 522 and $810\mathrm{nm}$ obtained in the extracted model tooth with its pulp chamber filled with whole blood or hemoglobin solution verified wavelength-dependent effects of scattering depending on the red blood cell concentration (Fig. 19).

Although the model yielded the qualitative description of the changes in the OD as a function of ${\mathrm{Hct}}_{\mathrm{p}}$ and tooth dimensions, it required adjustment in the absolute level to quantify ${\mathrm{Hct}}_{\mathrm{p}}$ of the dental pulp. In addition to improving the optical constants of the dentin and pulp chamber, the inclusion of the boundary condition, for example, such as $p_1∕p_2={(n_1∕n_2)}^2$ at the boundary between layer 1 and 2 where $p_1$ , $p_2$ the photon densities and $n_1$ , $n_2$ , the index of refraction of the layer 1 and 2, respectively, as reported by Takatani ³³ for the two-layer model, could improve the quantitative accuracy of the model.

5.3.

Quantification of ${\mathrm{Hct}}_{\mathrm{p}}$ in the Extracted Model Tooth

In connecting the experimental and theoretical studies, a method was developed to quantify ${\mathrm{Hct}}_{\mathrm{p}}$ of the extracted model tooth by utilizing the TLM equation. The difference in the absolute level between the experimental and theoretical OD was adjusted by calibrating the theoretical OD by the TLM against the experimental data of a selected reference tooth. After measuring the physical dimensions of the tooth, the tooth 4 with the ${\mathrm{BL}}_{\mathrm{T}}$ and ${\mathrm{BL}}_{\mathrm{p}}$ dimensions closest to the linear regression line of Fig. 12 was picked to obtain the gain constants required to normalize the theory to the experimental data. Although other teeth having dimensions not close to the regression line of Fig. 12 were tried, they did not yield results as good as tooth 4. For the ${\mathrm{Hct}}_{\mathrm{p}}$ level of $<0.10$ , where OD sensitivity to ${\mathrm{Hct}}_{\mathrm{p}}$ change was much higher at $522\mathrm{nm}$ , it was possible to achieve a higher accuracy with mean error of $-0.00115$ , while the mean error increased to $+0.09157$ at $810\mathrm{nm}$ because of low sensitivity of OD with respect to ${\mathrm{Hct}}_{\mathrm{p}}$ . As evident from the increasing SD values at higher ${\mathrm{Hct}}_{\mathrm{p}}$ for both 522 and $810\mathrm{nm}$ wavelengths in Fig. 21, errors in ${\mathrm{Hct}}_{\mathrm{p}}$ estimation based on the method described here increased because of the decreased sensitivity in OD versus ${\mathrm{Hct}}_{\mathrm{p}}$ at higher ${\mathrm{Hct}}_{\mathrm{p}}$ values.

5.4.

Estimation of ${\mathrm{Hct}}_{\mathrm{p}}$ in Human Volunteers

In translating from the theory, to ex vivo study and finally to in vivo application, the TLM as calibrated by the ex vivo data was utilized to estimate ${\mathrm{Hct}}_{\mathrm{p}}$ in 10 young volunteers. As summarized in Table 4, the mean ${\mathrm{Hct}}_{\mathrm{p}}$ estimate in human volunteers based on the TLM was found to be 0.032 with a SD of 0.017. In our previous study,¹⁶ we speculated the ${\mathrm{Hct}}_{\mathrm{p}}$ in vivo from the following approximation:

5.5.

Assessment of Pulp Vitality by ${\mathrm{Hct}}_{\mathrm{p}}$

The mean ${\mathrm{Hct}}_{\mathrm{p}}$ of the upper central incisor of human as estimated by the TLM agreed fairly well with the theoretical speculation assuming the systemic to microvascular hematocrit ratio of 0.50 as reported in the literature.³⁴ In addition, the effective fractional volume of the microvascular system ${\mathrm{BV}}_{\mathrm{F}}$ was estimated to be 0.144 and again showed excellent agreement with the value measured from the cross section of the cat’s canine tooth.³⁵ We thus here propose a new quantity “ ${\mathrm{Hct}}_{\mathrm{p}}$ or ${\mathrm{BV}}_{\mathrm{F}}$ ” of the pulp chamber to assess microcirculatory changes related to dental pulp vitality.

On the basis of the TLP, the dental pulp vitality may be assessed using the light source wavelength in the green region most likely at the isosbestic wavelength of $522\mathrm{nm}$ . The measured signal can be processed in two ways, one in the pulse mode or the other absolute mode as reported in this study. The pulse mode can diagnose rapidly changing characteristics of the vascular system or compliance of the microvascular bed synchronized with the heartbeat, while the absolute mode can uncover accumulation or depletion of the blood in the dental pulp. The clinical diagnosis of circulatory status of dental pulp based on the absolute TLP should be established through accumulation of clinical data. In addition to the measurement of the absolute and time-varying blood volume at the isosbestic wavelength of $522\mathrm{nm}$ , the nonisosbestic wavelength of $\sim 467\mathrm{nm}$ , where the optical absorption is dependent on the oxygenation of the blood, can provide the average tissue oxygen level from which oxygen demand of the dental pulp may be assessed.¹⁶ The determination of the blood volume shift in the pulp chamber by the isosbestic wavelength of $522\mathrm{nm}$ together with tissue oxygenation possibly determined from the nonisosbestic wavelength of $467\mathrm{nm}$ can provide the circulatory as well as metabolic dynamics reflecting vitality of the dental pulp.