Diffuse optical localization of blood vessels and 3D printing for guiding oral surgery

Abstract

Diffuse optical imaging through centimeters of tissue has emerged as a powerful tool in biomedical research. However, applications in the operating theater have been limited in part due to data set requirements and computational burden. We present an approach that uses a small number of optical source-detector pairs that allows for the fast localization of arteries in the roof of the mouth and has the potential to reduce complications during oral surgery. The arteries are modeled as multiple-point absorbers, allowing localization of their complex shapes. The method is demonstrated using a printed tissue-simulating mouth phantom. Furthermore, we use the extracted position information to fabricate a custom surgical guide using 3D printing that could protect the arteries during surgery.

1. INTRODUCTION

Diffuse optical imaging (DOI) is the extraction of information from measurements of light that has traveled through scattering media such as tissue [1,2]. Instrumentation for DOI is typically inexpensive and safe, and related technology has been used to develop the highly successful pulse oximetry devices [3]. In DOI methods such as diffuse optical tomography (DOT), near-infrared or visible light is used to form volumetric images in deep (>1 cm) tissue by treating photons as diffusing particles [4–6]. However, this is a computationally intensive process, limiting its application in surgery, where information is often needed in real time. For this reason, fast localization methods are an attractive alternative, where information about the location only is extracted using a small measurement data set [7,8]. Here, we present optical localization as a means for finding the greater palatine artery (GPA) in the roof of the mouth, thereby addressing the need for this information in dental surgery. We also describe use of this information to 3D print a surgical guide to aid the surgeon.

The GPA is an important imaging target because it is an anatomical landmark for oral surgery. The depth of the GPA varies, but it is about 5 mm into the roof of the mouth [9], and is not visible with the naked eye. Along with the greater palatine nerve (GPN), it forms the greater palatine neurovascular bundle (GPB), which emerges from the pterygopalatine fossa through the greater palatine foramen and runs anteriorly and anastomoses with branches of the nasopalatine bundle from the nasopalatine canal [10]. The location of the GPB is associated with the height of the palatal vault, ranging from 7 to 17 mm from the cementoenamel junctions of the maxillary premolars and molars [11]. Currently, a surgeon must rely on prior knowledge of the GPB location to avoid damaging it. However, a recent study with human cadavers showed a discrepancy in the location of the GPB of up to 4 mm [12]. For these reasons, localization of the GPA in a patient will assist clinicians in pre-surgical planning and reduce complications during surgeries such as a periodontal and oral surgeries, sinus and ridge augmentations, GPN block, soft tissue biopsies, dental implant, and wisdom teeth removal.

In Section 2, we describe the localization procedure used to find the GPA, as well as the mouth phantom fabrication method and the experimental setup. In Section 3, we present results for the localization of the GPA in the mouth phantom, and we use this information to print a surgical guide for assisting a surgeon.

2. METHODS

A. Optical Localization

The optical parameters of interest in DOI are the absorption coefficient μ_a and the reduced scattering coefficient ${\mu }_s^{\prime }$. These parameters characterize a diffusion equation that can describe the propagation of light within a highly scattering medium [1,5]. For an unmodulated or CW light source, the diffusion equation reduces to

$$\nabla ·[D(\mathbf{r})\nabla \varphi (\mathbf{r})]-{\mu }_a(\mathbf{r})\varphi (\mathbf{r})=-\delta (\mathbf{r}-{\mathbf{r}}_{\mathbf{s}}),$$ (1)

where r denotes the position, ϕ (W/mm²) is the photon flux density, $D=1/[3({\mu }_s^{\prime }+{\mu }_a)](\text{mm})$ is the diffusion coefficient, and a unit amplitude Dirac delta function optical excitation (with implied scaling) at position r_s is assumed.

Localization of the GPA is possible because the μ_a of the blood in the GPA is higher than the background tissue [13]. The problem then becomes an estimation of the position of an absorption inhomogeneity. Previously, Cao et al. [8] described localization of a point absorber, and we extend this treatment to the localization of absorption inhomogeneities with complex geometries. We accomplish this by modeling the absorption inhomogeneity with multiple point absorbers instead of just one.

We assume the inhomogeneity is embedded in a locally homogeneous background having known parameters μ_ao and D_o such that μ_a(r) = μ_ao + Δμ_a(r), where Δμ_a is a change in the absorption due to the inhomogeneity. After substitution into Eq. (1), moving the Δμ_a term to the right-hand side, assuming that $D(\mathbf{r})\approx D_o(\text{since}{\mu }_s^{\prime }\gg {\mu }_a)$, and applying the Born approximation [14], the photon flux becomes

$$\varphi (\mathbf{r};\mathrm{\Delta }{\mu }_a)={\varphi }_i(\mathbf{r};{\mu }_{a_o})+\int \mathrm{d}{{\mathbf{r}}^{\prime }}^{3}g(\mathbf{r},{\mathbf{r}}^{\prime };{\mu }_{a_o}){\varphi }_i({\mathbf{r}}^{\prime };{\mu }_{a_o})\mathrm{\Delta }{\mu }_a({\mathbf{r}}^{\prime }),$$ (2)

where ϕ_i is the photon flux in a homogeneous background and g(r; r′; μ_ao) is the Green’s function for the homogeneous diffusion equation with (Dirac) source location r′. In order to form an analytical solution for fast computation [8], we assume that the inhomogeneity has constant Δμ_a and can be represented by L point absorbers at positions r_al such that

$$\varphi (\mathbf{r};\mathrm{\Delta }{\mu }_a)={\varphi }_i(\mathbf{r};{\mu }_{a_o})+\mathrm{\Delta }{\mu }_a\sum _{l=1}^{L}g(\mathbf{r},{\mathbf{r}}_{a_l};{\mu }_{a_o}){\varphi }_i({\mathbf{r}}_{a_l};{\mu }_{a_o}).$$ (3)

The forward model for a measurement at position r can then be written as

$$f({\mathbf{R}}_a;\mathrm{\Delta }{\mu }_a)=y_o+\mathrm{\Delta }{\mu }_a{f}^{\prime }({\mathbf{R}}_a),$$ (4)

where y_o=ϕ_i(r;μ_ao ), $f^{\prime }({\mathbf{R}}_a)={\sum }_{l=1}^{L}g(\mathbf{r},{\mathbf{r}}_{a_l};{\mu }_{a_o}){\varphi }_i({\mathbf{r}}_{a_l};{\mu }_{a_o})$ and R_a contains the L positions r_al. f ′(R_a) is a compact Fréchet derivative representing sensitivity at the detector to variations in absorption at the discrete points over which the absorbing inhomogeneity is described.

The maximum likelihood (ML) estimate of the positions describing the inhomogeneity, R_a, can be formulated as the minimization of the cost function

$$C({\mathbf{R}}_a)=\underset{\mathrm{\Delta }{\mu }_a}{min}{\Vert \mathbf{y}-{\mathbf{y}}_o-\mathrm{\Delta }{\mu }_a{\mathbf{f}}^{\prime }({\mathbf{R}}_a)\Vert }_{\mathrm{\Lambda }}^2,$$ (5)

where y is a vector of N measurements, y_o is a vector of N expected measurements y_o, f ′(R_a) is a vector of N expected f ′(R_a), Λ⁻¹ = α diag[|y₁|;…; |y_N |] is the noise covariance matrix, for which we assume a shot noise model characterized by α [5], and for an arbitrary vector v, ${\Vert \mathbf{v}\Vert }_{\mathrm{\Lambda }}^2={\mathbf{v}}^H\mathrm{\Lambda }\mathbf{v}$, where H denotes the Hermitian transpose.

A two-step procedure can be used to solve this optimization problem [7,8], where for each combination of positions r_al within a region of interest estimates are formed as

$$\mathrm{\Delta }{\stackrel{\sim }{\mu }}_a({\mathbf{R}}_a)=\frac{Re[{(\mathbf{y}-{\mathbf{y}}_{\mathbf{o}})}^H\mathrm{\Lambda }{\mathbf{f}}^{\prime }({\mathbf{R}}_a)]}{{\Vert {\mathbf{f}}^{\prime }({\mathbf{R}}_a)\Vert }_{\mathrm{\Lambda }}^2},$$ (6)

$$C({\mathbf{R}}_a)={\Vert \mathbf{y}-{\mathbf{y}}_o-\mathrm{\Delta }{\stackrel{\sim }{\mu }}_a({\mathbf{R}}_a){\mathbf{f}}^{\prime }({\mathbf{R}}_a)\Vert }_{\mathrm{\Lambda }}^2.$$ (7)

The ML estimates become

$${\widehat{\mathbf{R}}}_a=arg\underset{{\mathbf{R}}_d}{min}C({\mathbf{R}}_a),$$ (8)

$$\mathrm{\Delta }{\widehat{\mu }}_a=\mathrm{\Delta }{\stackrel{\sim }{\mu }}_a({\widehat{\mathbf{R}}}_a).$$ (9)

In practice, constraints are required in order to reduce the complexity of the minimization. Because R̂_a is the parameter of interest, all linear dependencies in the problem are encapsulated in Δμ̂_a, including the excitation power, Green’s function scalars, source-detector coupling into the medium, and the system parameter α [5,7].

We consider the reflection measurement geometry of Fig. 1, where a detector is at position r. In a typical experiment, there are multiple sources and detectors, resulting in N measurements. For simplicity and fast computation, we use an analytical solution to Eq. (1) in an infinite homogeneous domain as the forward model [7], giving

Fig. 1.

Localization geometry for a single point absorber (●) used to derive the forward model. An isotropic excitation source (X) and detector (O) are placed one scattering length, l* = 3D_o, below the boundary [7].

$$y_o=\frac{exp(-k{r}_{sd})}{r_{sd}},$$ (10)

$$f^{\prime }({\mathbf{R}}_a)=\sum _{l=1}^L\left[\frac{exp(-k{r}_{1_l})}{r_{1_l}}\right]\left[\frac{exp(-k{r}_{2_l})}{r_{2_l}}\right],$$ (11)

where $k=\sqrt{{\mu }_{a_o}/D_o}$, r_1l is the distance from the excitation source to the lth point absorber, and r_2l is the distance from the l th point absorber to the detector.

A more sophisticated analytical forward model could be implemented for improved localization accuracy [15]. For example, an extrapolated zero flux (ϕ = 0) boundary condition (describing the scattering medium interface with free space) can be enforced with the placement of image excitation and absorption sources [16–18]. Numerical methods such as the finite element method would allow for better solutions in complex, heterogeneous scattering media [19]. Additionally, the laser beam profile could be included in the model. However, for the work presented here, we found that Eqs. (10) and (11) were sufficient for demonstrating the localization method.

B. Mouth Phantom Fabrication

To test our approach for the localization of the GPA, we fabricated a mouth phantom using a Form 1+ 3D printer (Formlabs). The phantom was printed using standard white photopolymer resin. By printing a slab of material and fitting experimental time-gated measurements to the analytical transmittance [20], we found the optical parameters to be μ_ao = 0.0076 mm⁻¹ and D_o = 0.166 mm at 633 nm (similar to other tissues, such as those in the breast and brain [21,22]). To our knowledge, these parameters have not been measured for bulk gingiva tissue. However, if they were known, there is a procedure to fabricate the appropriate phantom [23].

The phantom was designed with planar cylindrical cavities of diameter 2.65 mm to simulate the GPA [9,10], as shown in Fig. 2(a). The cavities were filled with a mixture of water and India ink (Higgins, 0.0018 (g/mL) density) [24], simulating the absorption of blood at 633 nm, which is 0.7 mm⁻¹ for a 40% hematocrit level [13]. Figure 2(b) shows the printed mouth phantom. Visual inspection of Fig. 2(b) suggests that μ_ao of the mouth phantom is less than gum tissue because the absorption inhomogeneities are visible under room light. However, the phantom is still useful for development of the localization approach.

Fig. 2.

3D printed mouth phantom. (a) Slice of the printing profile showing the planar cylindrical cavities. The phantom was printed in approximately 6 h. (b) Phantom with absorbing inhomogeneities simulating the GPA.

C. Experimental Setup

We measured scattered light at 633 nm reflected from the mouth phantom using the experimental setup shown in Fig. 3. The laser spot had a diameter of about 1 mm. For high signal levels, quasi-CW data (repetition rate 20 MHz, camera exposure time 25 ms) was detected using a filtered pulsed supercontinuum laser. The average power of the laser was about 15 mW. The CCD camera allows for more measurements (detectors) compared with a fiber-based system. A laser line scanner was used to measure the surface topography and project the camera pixels (detectors) to the phantom’s surface for the reconstruction [20,25]. A motorized mirror enables automation of the data capture and repeated measurements with the same source location.

Fig. 3.

DOI experimental setup. An EXR-20 pulsed supercontinuum fiber laser and VARIA tunable filter (NKT Photonics) emit light at 633 nm with 10 nm bandwidth. The scattered light is detected by a CCD camera (Roper PIMAX). The motorized mirror (Zaber) is used to control the source location.

Referring to Fig. 2, two measurement sets were acquired, one with sources and detectors in the vicinity of the left absorption inhomogeneity and the other with sources and detectors in the vicinity of the right. These two measurement sets were used to localize the left and right arteries, respectively. Source-detector pairs with r_sd < 10l*, where l*= 3D_o, were excluded in order to minimize the contribution of the ballistic light. Using this procedure, the left measurement set contained N_L = 473 measurements using 7 sources and 80 detectors, and the right measurement set contained N_R = 407 measurements using 7 sources and 60 detectors. The number of detectors could be minimized using a sensitivity analysis [8], reducing the computational time.

Both the left and right measurement data sets (referring to Fig. 2) were calibrated in order to account for the source and detector coupling to the scattering medium [20]. Data $y_n^{\text{base}}$, where n is the data index, was captured from the mouth phantom before the India ink and water were added. Then, the left and right uncalibrated measurement sets were calibrated according to $y_n=y_n^{\text{uncal}}[y_n^{\text{syn}}/y_n^{\text{base}}]$, where $y_n^{\text{uncal}}$ are uncalibrated data from either the left or right measurement set and $y_n^{\text{syn}}$ are corresponding simulated data (in this case equal to y_o). For a simple geometry, [ $y_n^{\text{syn}}/y_n^{\text{base}}$] may be approximately constant for each n. In this case, the calibration becomes $y_n=y_n^{\text{uncal}}\beta$, where β is a constant that can be absorbed into Δμ̂_a, in Eq. (9), removing the need for measuring $y_n^{\text{base}}$. However, we found that calibration was necessary for localization in the phantom of Fig. 2. In the clinic, data from the phantom in Fig. 2 could be captured a priori and used for the calibration.

3. RESULTS

A. Constrained Localization

In order to localize the absorption inhomogeneities in the mouth phantom, we constrained the problem to the localization of absorbing line sources. Each line source is represented by L = 10 point absorbers and is characterized by its starting position (x_a; y_a; z_a), its azimuthal angle ψ, and its length R, as shown in Fig. 4. We further constrained the problem by selecting y_a = 0 and R = 30 mm, and used the left and right calibrated data sets to localize the left and right absorbing line sources, respectively. Specifically, Eq. (6) and then Eq. (7) were calculated for each (x_a; z_a; ψ) in a region of interest for each absorbing line source. The position of each absorbing line source was then estimated using Eq. (8). The result of this localization procedure is shown in Fig. 5. Figures 5(a) and 5(b) show plots of Eq. (7), over the region of interest, for the left and right absorbing line sources, respectively. In order to show two dimensional plots, ψ was fixed such that the x–z planes contained the minimum of Eq. (7). Figures 5(c) and 5(d) show the localized absorbing line sources plotted with the surface profile used to fabricate the mouth phantom. The minimum distance between the estimated and actual positions is about 0.8 mm, and the maximum distance is about 3.9 mm. The better quality of the data set involved with Fig. 5(a) produced a better localization result than that for Fig. 5(b), consistently the case, possibly due to the accuracy of the laser line scan profile used to determine the detector positions. The result is promising given the approximations made in the forward model, the complicated geometry, and the high absorption of the simulated blood.

Fig. 4.

Constrained geometry for the localization of an absorbing line source represented by L = 10 point absorbers. The positions r_alof the point absorbers (●) are separated by Δr along a line of length R starting at (x_a; y_a; z_a) with azimuthal angle ψ. The polar angle (not shown) is fixed at π/2 rad.

Fig. 5.

Localizations using data captured above the left and right absorption inhomogeneities in the printed mouth phantom. (a) Plot of Eq. (7) with ψ = 1.14 rad using the left data set. (b) Plot of Eq. (7) with ψ = 1.73 rad using the right data set. The plots show the x–z planes that contained the minimum of Eq. (7). From Eq. (8), the position that minimizes Eq. (7) is the estimated location. The black dots show the actual positions. For each (x_a; z_a) shown, Eq. (7) was evaluated for ψ from 0 to π/2 rad in (a) and π/2 to π in (b). Note that, from Fig. 4, we have selected y_a = 0. (c)Mouth phantom boundary plotted with the localized absorption line sources (red dots) and the true locations (dashed black lines). (d) Another view. The localization error is likely due to assumptions in the forward model and the high μ_a of the simulated blood.

The localization in Fig. 5 is based on a model constrained to an absorbing line source, but the constraint could be relaxed or modified to extract the more complicated vascular structure of the GPA in a real patient. Alternatively, point absorbers within the vascular structure could be localized individually [8,26], for example, by using a borescope.

B. Printed Surgical Guide

Surgical guides can improve the success rate of surgery by transferring anatomical information to the operating theater. For example, surgical guides have found widespread use in the placement of dental implants, where they facilitate proper positioning in the bone and can assist in diagnosis and treatment planning [27]. Methods include 3D printing of surgical guides from CT scans using stereolithography, which was found to improve implant placement compared with conventional methods [28]. However, to our knowledge, surgical guides have not been fabricated for protecting the GPA. We demonstrate in Fig. 6 that the data extracted using localization can be used to fabricate such a guide. The guide was designed by projecting the localized absorption line sources to the mouth phantom surface and selecting points on the surface less than the radial distance of the GPA away from the projected line. Additional points were added in the +z direction above these points to increase the thickness, and the resulting surface profile was converted into a stereolithography (STL) file, which was used to 3D print the design. This surgical guide could assist with oral surgery by providing information on the location of the GPA and obstructing its location so that it and the GPN cannot be damaged. Practically, the guide should be improved so that it can be fastened to the roof of the mouth in a patient.

Fig. 6.

3D printed surgical guide to protect the GPA. (a) Surgical guide design. Blue lines are the estimated locations of the absorption line sources, and the dark blue isoimage is the surgical guide. The isoimage was formed by projecting the blue lines to the phantom surface. (b) Photo of the mouth phantom and 3D printed surgical guide. The lateral space in the roof of the mouth is occluded by the guide so that the surgeon does not accidentally cut the GPA/GPN.

C. Conclusion

We have presented a DOI method for fast 3D localization of obscured absorption inhomogeneities. The inhomogeneities are treated as a superposition of point absorbers with locations that can be constrained using prior information in order to reduce the computational burden. We showed that it is possible to localize absorbing line inhomogeneities within a highly scattering 3D printed mouth phantom from calibrated experimental data, which has relevance for oral surgery. Furthermore, we showed that the position information can be used to fabricate a surgical guide using 3D printing to protect the GPA during surgery. In principle, the method could be employed in a clinic using a bright light source and a camera or small probe. Images of the arteries could be viewed in real time, and a custom surgical guide could be fabricated for a return visit. It is possible to localize lesions in addition to blood vessels, and the specificity could be improved with topical fluorescence or absorption contrast agents. The method also may be useful for identifying operative and postoperative bleeding [29].