Optical dosimetry probes to validate Monte Carlo and empirical-method-based NIR dose planning in the brain

Abstract

A three-dimensional photon dosimetry in tissues is critical in designing optical therapeutic protocols to trigger light-activated drug release. The objective of this study is to investigate the feasibility of a Monte Carlo-based optical therapy planning software by developing dosimetry tools to characterize and cross-validate the local photon fluence in brain tissue, as part of a long-term strategy to quantify the effects of photoactivated drug release in brain tumors. An existing GPU-based 3D Monte Carlo (MC) code was modified to simulate near-infrared photon transport with differing laser beam profiles within phantoms of skull bone (B), white matter (WM), and gray matter (GM). A novel titanium-based optical dosimetry probe with isotropic acceptance was used to validate the local photon fluence, and an empirical model of photon transport was developed to significantly decrease execution time for clinical application. Comparisons between the MC and the dosimetry probe measurements were on an average 11.27%, 13.25%, and 11.81% along the illumination beam axis, and 9.4%, 12.06%, 8.91% perpendicular to the beam axis for WM, GM, and B phantoms, respectively. For a heterogeneous head phantom, the measured % errors were 17.71% and 18.04% along and perpendicular to beam axis. The empirical algorithm was validated by probe measurements and matched the MC results (R² > 0.99), with average % error of 10.1%, 45.2%, and 22.1% relative to probe measurements, and 22.6%, 35.8%, and 21.9% relative to the MC, for WM, GM, and B phantoms, respectively. The simulation time for the empirical model was 6 s versus 8 h for the GPU-based Monte Carlo for a head phantom simulation. These tools provide the capability to develop and optimize treatment plans for optimal release of pharmaceuticals in the treatment of cancer. Future work will test and validate these novel delivery and release mechanisms in vivo.

1. INTRODUCTION

Optical therapy is a non-invasive technique that can be utilized to treat cancer through photodynamic therapy (PDT) techniques [1] and the triggered release of drugs from nanocomplexes in metastatic tissues [2,3]. Near-infrared (NIR) light is preferred for deep tissue activation due to its low optical attenuation and thus high penetration. This study is part of a project which seeks to develop optically activated drug release in order to implement macrophage-based “Trojan horse” drug delivery to brain metastasis [2,3]. Given that the rate of release and quantity of the released drug are likely to be affected by the optical power delivered [2,3], the development of optical simulation and dosimetry tools that can accurately determine the photon fluence in the brain and ultimately be integrated into a clinical environment is critical. In our study a GPU-based 3D Monte Carlo (MC) code is used to simulate the photon energy deposition in the brain [4–10], and an optical dosimetry probe was fabricated to validate the photon fluence in heterogeneous media consistent with the skull and brain. While the primary application is in the field of controlled drug release, these tools can also be applied to a wide range of applications, including NIR-based imaging techniques, such as photoacoustic tomography[11] and near-infrared spectroscopy [12], and in infrared neural stimulation [13] of the cochlea and neuroprosthesis.

Monte Carlo simulation methods have been used as the gold standard for photon propagation studies in heterogeneous media [4–8]. Traditional CPU-based Monte Carlo models suffer from slow computation speeds, which makes them nonviable for clinical applications, which require the simulation of a large number of photons (excess of 10⁷) especially in highly attenuating (scattering) brain tissues, such as the skull and white matter. The GPU-based 3D Monte Carlo used in this study, developed by the Fang group [9], can achieve complete brain simulation of a high-fluence broad-beam source within hours compared to days for CPU-based models (300 × faster) [9]. This new model features a voxel-based 3D fluence and energy distribution with significant improvements, such as efficient parallel random number generators, reflection at 3D boundaries, time-resolved simulations, fluence normalizations, and ease of integration with anatomical imaging modalities (e.g., CT, MR, ultrasound, etc.), thus improving its clinical viability and efficacy for diagnosis and therapy. To date, this code has not been validated through 3D measurements under various illumination conditions.

Application of a 3D Monte Carlo as an optical therapy planning tool in clinical practice requires rigorous validation of its physics-based model, so as to reduce the uncertainty in light dose delivery and improve diagnostic accuracy. Past studies have validated the Monte Carlo using diffusion approximation simulations in slab geometry and under limited heterogeneity conditions [8,9]. Only a few studies have attempted to rigorously validate the 3D MC fluence using dosimetry measurements in phantoms under varying illuminations with finite beam width and heterogeneity conditions [14,15]. The varying degree of accuracy can be attributed to differences in estimation of optical properties of the medium, illumination and object boundary conditions, and the ability of the Monte Carlo to accurately estimate photon distribution in complex heterogeneous boundary conditions. All of these differences need to be analyzed and optimized in order to get maximum efficiency from the Monte Carlo simulations. The use of optical dosimetry for pre-clinical validation, along with the ability to verify optical properties of phantoms used in the measurements, is essential in order to ensure therapeutic accuracy of the Monte Carlo. Additionally, any mismatch in simulation and actual measurement conditions can be addressed by this exercise to improve therapeutic confidence in the tool.

To validate the MC, the localized photon distribution in tissues was measured using a novel titanium dioxide-based optical dosimetry probe (ODP). ODPs have been extensively used in PDT to measure photon fluence as well as to deliver light locally to tissues [1,16–21]. The small size of the dosimetry probe (1.5 mm diameter) and near-isotropic response at NIR wavelength (800 nm) improves the accuracy of localized light measurement in highly scattering tissues/phantoms, compared to other dosimetry probes. It is also advantageous over standard photon measurement devices, such as calorimeters, which only measure fluence in air. In this study, the novel titanium-based probe was designed and evaluated against a standard nylon-based probe [17,20] and their linear and isotropic response analyzed. Its linear response in air was calibrated using a calorimeter (gold standard) for absolute fluence quantification.

Translating an optical therapy planning tool into the clinic requires that simulations be performed in feasible time frames. While the GPU-based Monte Carlo takes a few hours for a complete head simulation under a single illumination condition, optimizing a treatment plan would require iterations with multiple laser sources. While a cluster of GPUs could be used to solve this problem, thereby significantly increasing costs, it prompted us to design an empirical model of photon propagation [22] to approximate 3D photon fluence in tissue-like media at a fraction of time taken by the GPU-based Monte Carlo. This model assumes the photon fluence in a voxel is the weighted sum of the fluences in the neighboring voxels in the preceding layer [22], where the weights are determined using the Monte Carlo for typical brain tissues, such as gray matter, white matter, skull bone, and astrocytoma. Previous studies have used approximate methods, such as the diffusion approximation (DE) [23,24], hybrid (MC + DE) [25], radiative transport equation [26,27], and the adding-doubling method [28], to approximate 3D photon distribution. The empirical method (EM) is an addition to the existing list of methods and mainly differs in the means by which photon scatter is calculated and has a similar speed as the diffusion approximation [29] (in seconds). While other methods rely on physics-based models based on approximations to the radiative transport equation, the empirical approach is designed to be calibrated against the Monte Carlo in a homogeneous medium for a fixed set of optical properties, before being applied in a complex medium under different illumination conditions. Hence a calibration step with a fixed set of optical properties is necessary before applying this model. It is important to note that once calibrated, this model does not require additional calibrations, which allows for its repetitive use for standard clinical applications with a fixed set of optical properties (e.g., head simulation) under differing anatomical geometries and illumination conditions. The significant improvement in the speed and reasonable accuracy inspired us to integrate it into the optical therapy planning tool along with the Monte Carlo and the titanium-based optical dosimetry probe.

In this paper, a titanium-based optical dosimetry probe was fabricated and calibrated to provide localized energy fluence measurements, and used to validate the 3D GPU-based Monte Carlo code under differing beam profiles and heterogeneous tissue phantoms. To provide an accelerated, real-time solution to photon transport, an empirical model was proposed and evaluated against MC simulated in tissue phantoms. Finally, the role of these new and/or improved tools in optically stimulated drug release will be discussed.

2. MATERIALS AND METHODS

A. Optical Brain Phantoms

Brain phantoms were designed to mimic the optical absorption and reduced scattering coefficients of white matter, gray matter, and skull bone (at 800 nm) [29–31] using India ink (absorber) and intralipid (scatterer) [32–35] in water (μ_a = 0.020 cm⁻¹) [36]. The absorption coefficient of India ink (Higgins non-water-proof black ink) and the scattering coefficient of intralipid (Intralipid 20% from Fresenius Kabi), at 800 nm wavelength, were determined and validated using a spectrophotometer (Shimadzu UVmini 1240 Spectrophotometer), which served as the gold standard for absorbance. The scattering properties of intralipid were determined using the spectrophotometer for low concentrations (0.003%–0.012%), where μ_s = 9.52 cm⁻¹ %concentration⁻¹, compared to 9.95 cm⁻¹ %concentration⁻¹ as shown in previous studies by Van Staveren [33], which is within the 4.32% relative error. Figures 1 and 2 show the calibration curves of absorbance (or attenuation) versus concentration of India ink and intralipid obtained using the spectrophotometer. The linear relationship between concentration and absorbance was used to design optical phantoms with specific absorption and scattering properties of white matter, gray matter, and skull bone [29,30]. These optical properties (absorption and scattering coefficients and anisotropy factor) are shown in Table 1 and were used to design the optical brain phantoms (Table 2) using the equations derived from Figs. 1 and 2.

Fig. 1.

Calibration curve of spectrophotometer measured absorbance versus % concentration of India ink. The error bars represent minimum and maximum absorbance.

Fig. 2.

Intralipid scattering: upper figure represents measured values of scattering coefficients (μ_s and ${\mu }_s^1$) versus % concentration of intralipid using a spectrophotometer. The error bars represent minimum and maximum values. Lower figure shows extrapolated values from the linear relationship derived from the measured values (with relative and statistical uncertainties of 4.32% and 1.8%, respectively).

Table 1.

Optical Properties of Brain Tissues [29,30]

Tissue Type	Absorption Coefficient μa (cm−1)	Scattering Coefficient μs (cm−1)	Anisotropy Factor g	Reduced Scattering Coefficient ${{\mu }^{\prime }}_s({\text{cm}}^{-1})$
White matter	0.05	550	0.85	82.5
Gray matter	0.35	700	0.965	24.5
Skull bone	0.24	184	0.9	18.4

Table 2.

Phantom Composition for MC Versus Probe Comparison

Phantom Type	Tissue Type	Intralipid %	Ink %	Absorption Coefficient μa (cm−1)	Reduced Scattering Coefficient ${{\mu }^{\prime }}_s({\text{cm}}^{-1})$
Liquid phantomab	White matter	7.5	0	0.002	71.84
Liquid phantoma	Gray matter	2.36	0.023	0.379	22.64
Liquid phantoma	Skull bone	1.82	0.0156	0.26	17.44
Solid phantomb	Gray matter	2.56	0.0195	0.35	24.5
Solid phantomb	Skull bone	1.92	0.0134	0.25	18.399

^aPhantom composition used in homogeneous phantom studies.

^bPhantom composition used in heterogeneous (layered) phantom studies.

Notes: The anisotropy factor of intralipid (g = 0.636) was used for the phantoms.

The linear relationship of concentration versus total attenuation is particularly useful in determining intralipid concentrations for high scattering tissues in the brain. Both homogeneous as well as heterogeneous phantoms were made using various concentrations of India ink and intralipid. The phantom composition for the homogeneous phantom was a solution of India ink and intralipid in water, while the heterogeneous phantoms were comprised of layers of 1% agar (Fisher Scientific Laboratory Grade Agar A3600–500) doped with India ink and intralipid to represent skull bone and gray matter, with a layer of white matter in aqueous solution. The scattering properties of agar and the absorption properties of India ink were also measured using the spectrophotometer for each individual sample used in the study while the scattering properties of intralipid were derived from the linear relationship in Fig. 2.

B. Optical Dosimetry Probe Design and Calibration

The optical dosimetry probe consists of a spherical bulb of highly scattering material attached to an optical fiber [1,16,17,19]. The construction and fluence measurement by the probe are shown in Fig. 3. The optical fiber is connected to a PIN photodiode circuit (ET-2030, Electro Optics Technology Inc.) whose output voltage is read using an oscilloscope (Tektronix TDS3052B Digital Oscilloscope). Two models of optical dosimetry probes with tips made of nylon and titanium dioxide were designed [37]. The performance of the traditionally used nylon-based probes [19] was compared to that of the titanium dioxide-based probes by measuring the response to different angles of incidence in the equatorial and azimuthal planes, e.g., isotropicity response function.

Fig. 3.

Design and working of the optical dosimetry probe.

The nylon spheres have been widely used for photodynamic therapy and were made of nylon due to their isotropic response [19]. These were used as the standard of reference to compare the performance of the novel titanium-based probes. The titanium-based probes have the scattering tips made of a mixture titanium dioxide (Du Pont Ti-Pure R-900) and a clear two-part epoxy (Tra-Con BA-F114) mixed in a ratio of 9.1 mg TiO₂ to 1 ml of the epoxy (mixed) [37,38]. The use of epoxy as a medium for TiO₂ is ideal as it is transparent to NIR light and ensures that light scatter in the probe tip is entirely due to TiO₂ particles. The titanium probes were selected since the particle size of titanium dioxide is around 410 nm, which results in optimal scattering at 820 nm or when the wavelength is approximately twice the particle size. Since the wavelength used in our studies is 800 nm, the titanium probes have been designed to show a higher sensitivity than the nylon probes. The titanium-based probe tips were molded within a specially designed caste and a fiber optic cable (BFL48-400, 4 mm diameter, 0.48 numerical aperture) was inserted within the cured spheres at a depth of approximately 1/3 of the sphere’s radius [37]. The nylon-based probes were approximately 3.175 mm in diameter and were machined in a multistage process [37]. The other end of the fiber optic cable was connected to a PIN photodiode (ET-2030, Electro Optics Technology Inc.) through a SMA-905 connector and the voltage output was read using an oscilloscope (Tektronix TDS3052B).

Figures 4 and 5 show the setup used to test the isotropicity of the dosimetry probes in the equatorial and azimuthal planes. The probe tip is illuminated by laser light emanating from an integrating sphere (Melles Griot two-port integrating sphere), which produces a uniform broad beam of diffuse light, which is used to characterize the probe response. The probe was rotated with angular increments of 15° in the equatorial and 10° in the azimuthal planes, and its response on the oscilloscope was measured.

Fig. 4.

Setup for calibration of the optical dosimetry probe: measuring isotropicity (in equatorial and azimuthal planes) of the probe.

Fig. 5.

Setup for calibration of the optical dosimetry probe: measuring the linearity of the probe to incident laser fluence.

The center of the probe tip was maintained at the center of rotation during the entire procedure. The isotropicity of the probe, measured by calculating its coefficient of variation, defined as the ratio of standard deviation to the mean of the rms value over all the angles, is shown in Fig. 6(a). The range of angles was 0 to 360 deg for equatorial measurements and 0 to 150 deg for azimuthal measurements. The angular span for azimuthal measurements was limited due to physical limitations of the experimental setup. The individual rms voltage measured by the probe was normalized to the power measured using a calorimeter.

Fig. 6.

(a) Probe isotropicity of titanium versus nylon probes. The coefficient of variation (stdev/mean rms voltage) was calculated for probes made of nylon and titanium. The error bars show minimum and maximum variation in individual probe response. The TiO₂-based probes showed more isotropicity in equatorial and azimuthal directions. (b) Probe linearity: graph of total voltage (area under pulse) (mV/mm²) measured by a 1.5 mm diameter TiO₂ probe versus the power measured by a calorimeter per unit surface area (mW/mm²). The probe response is linear over a dynamic range of 0.01074 mW/mm² to 0.031316 mW/mm² aserror bars show a 3% variation (systematic error) in pulsed laser power fluctuation.

In the test of linearity (Fig. 5), the dosimetry probe response is validated using a calorimeter (Ophir CE, Nova 2 Ophir: 3A-P-SH-VI, aperture 12 mm) by illuminating with a pulsed laser beam of wavelength 800 nm, frequency 10 Hz, output fluence 17.64 mW/cm². The probe and the calorimeter are translated along the beam axis (in air) and the photon fluence is measured at different positions. The linearity of response of the 1.5 mm titanium probe (selected due to its isotropicity) is shown in Fig. 6(b). The linearity of the probe measurement (Φ_probe, mV/mm²) with respect to the calorimeter measurement (Φ_calorimeter, mV/mm²) was derived as follows by fitting a line on the measured data:

$${\varphi }_{\text{calorimeter}}=\frac{{\varphi }_{\text{probe}}+1.7296}{590.46}.$$ (1)

The probe voltage signal was filtered using a bandpass Butterworth filter to remove high-frequency noise in the probe response. The photon fluence is proportional to integral of the probe voltage output over the time of the laser pulse and was calculated by summing over the measured voltages over the pulse duration (Fig. 3). This is a novel technique used in this study and was found to be a much better representation of the probe output as compared to measuring the absolute voltage.

The probe voltage and calorimeter power were normalized to the respective surface areas [probe diameter = 1.5 mm, probe surface area (sphere) = 7.065 mm², calorimeter aperture = 12 mm, calorimeter surface area (circle) = 113.04 mm²].

C. Absolute Photon Quantification in Tissue-Like Media

The linear relationship in Eq. (1) was modified to derive the optical fluence in liquid brain phantoms. The light entering the fiber within the probe depends on the amount of multiple photon scattering within the spherical probe tip, which in turn depends on the ratio of the refractive index of the probe with respect to the medium [19]. The refractive index of the titanium probe tip should lie close to that of the epoxy (n = 1.53) [39] since the concentration of titanium (n = 2.73) [38] is low (<20 mg/mL). Since the refractive index of air is less than that of water, the probe response in water is smaller as more light leaves the probe in water compared to air. To account for the difference in the reflectance between probe–air and probe–water interfaces a calibration factor was measured. The probe was placed inside a cuvette, 0.2 cm from the surface. First the probe response voltage was measured in air (φ_air) and then measured again by filling the cuvette with water (φ_water), without changing the position of the probe with respect to the cuvette and the laser source. By accounting for the attenuation by water (μ_water = 0.020 cm⁻¹; 0.2 cm path length) and the difference in the beam spread (change in refractive index for cuvette-to-air versus cuvette-to-water), the calibration factor (CF) was determined experimentally to be 1.32, using the following equation:

$$\text{CF}=\frac{{\varphi }_{\mathrm{air}}\left(e^{-{\mu }_{{\text{water}}^l}}\right)\left(\frac{{\text{FWHM}}_{\text{air}}}{{\text{FWHM}}_{{\text{water}}_l}}\right)}{{\varphi }_{\text{water}}}.$$ (2)

The FWHM values were measured for both these setups using the probe (FWHM_air = 40 mm, FWHM_water = 42 mm) and their ratio (0.9091) denotes the proportional change in fluence due to beam spread. The experimentally derived calibration factor is then used for photon quantification within tissue-like media:

$${\varphi }_{\text{calorimeter}}=\frac{\left(CF\ast {\varphi }_{\text{probe}}\right)+1.7296}{590.46},$$ (3)

where Φ_calorimeter is the absolute energy fluence rate in mW/mm² and Φ_probe is the fluence measured by the probe (mW/mm²). The absolute energy fluence rate can be converted to energy fluence (Φ′_absolute) (J/mm²) by integrating it over the illumination time.

D. Beam Profile Modeling Using the Monte Carlo

The original code of the GPU-based 3D MC was advanced to emit a broad-beam super-Gaussian beam profile emanating from an optical fiberguide: ${\varphi }_{\text{SG}}=e^{-0.5\left(\frac{r}{\sigma }\right)n}$, where r = z tan θ, θ is the divergence angle, z is the depth along the beam axis, σ is the standard deviation of the super-Gaussian beam, n is the order of the super-Gaussian distribution. For these studies, the measured broad super-Gaussian beam was simulated using multiple point sources, each with a super-Gaussian output (σ = 3.8 and n = 2.9) for a source (fiberguide) of radius 0.3 cm. In addition to the super-Gaussian beam, a flat broad beam with radius equal to the output port of the integrating sphere (port radius 0.9 cm) was implemented, similar to other studies. The beam characteristics, phantom dimensions, and optical properties of the phantom were inputted to the MC code (and empirical model code) to emulate the illumination conditions used in the optical dosimetry setup of Fig. 7(a). A total of 10⁸ photons were simulated per beam.

Fig. 7.

(a) Setup for optical dosimetry in a liquid phantom. The optical dosimetry probe is translated within a Plexiglas cuvette filled with liquid phantom, and illuminated by an NIR source. This setup is used for intralipid calibration as well as for actual phantom studies. (b) Pictures of homogeneous liquid phantoms (left) and heterogeneous layered agar phantoms (right). In the heterogeneous phantom a white matter liquid phantom (not shown) is added on top of the gray matter solid agar layer for dosimetry measurements.

E. Validation of the 3D Monte Carlo in Brain Phantoms Using Optical Dosimetry

As described in Section 2.A, liquid optical phantoms, resembling white matter, gray matter, and skull bone, were designed using predetermined concentrations of India ink and intralipid. The setup for the light dosimetry in the phantoms is similar to that shown in Fig. 7(a). Cuvette dimensions were 5.3 cm × 4.6 cm × 3.1 cm, thickness = 0.4 cm. Two beam profiles—flat-top and super-Gaussian—were used to illuminate these phantoms with the laser/optical parametric oscillator (OPO) system tuned to 800 nm. The 3D energy fluence rate distribution was determined by translating the optical dosimetry probe positioned within the brain phantom using a three-axis micrometer stage. These measurements were compared to Monte Carlo generated data to test the validity and accuracy of the Monte Carlo code (Figs. 8–11). The error% was averaged over profiles measured and calculated at different depths within the phantom along and perpendicular to the beam axis.

Fig. 8.

Optical dosimetry in white and gray matter phantoms (cuvette dimensions: 5.3 cm × 4.6 cm × 3.1 cm, thickness = 0.4 cm). The photon source used is an integrating sphere connected to a pulsating laser source. The graphs show the exponential decrease of photon fluence along the beam axis in white and gray matter mimicking phantoms. The Monte Carlo generated fluence closely matches the fluence measured by the probe.

Fig. 11.

Validation of Monte Carlo in a heterogeneous phantom (1.1 cm skull agar, 1.1 cm gray matter agar, and 5.9 cm white matter liquid). Lateral profile measurements (in the white matter phantom solution) are shown on the left while axial comparison is on the right. The depth shown by z is the distance from the gray matter agar surface.

In order to quantify the photon energy fluence (in J/mm²) brain tissues, we converted the output of the Monte Carlo fluence (Φ_photons/mm²) as follows:

$$\varphi \left(\mathrm{J}/{\text{mm}}^2\right)={\varphi }_{\text{MC}}\ast \frac{hc}{\lambda },$$ (4)

where h is Planck’s constant, c and λ are the speed and wavelength of light in the medium. The energy deposited (Edose in J/mm³) in the tissue is obtained by multiplying the photon dose with the absorption coefficient μ_a:

$$E_{\text{dose}}\left(\mathrm{J}/{\text{mm}}^3\right)={\varphi }_{\text{MC}}\ast {\mu }_a\ast \frac{hc}{\lambda }.$$ (5)

F. Empirical Algorithm for Photon Propagation

The empirical model approach is based on the assumption that the photon fluence in a voxel in a particular layer is a weighted sum of the fluences of neighboring voxels in the previous layer. The schematic of this algorithm is shown in Fig. 12. The steps of the algorithm can be summarized as follows:

1. Assign the incident photon beam distribution (e.g., super-Gaussian beam) to the first layer of voxels entering the tissue phantom. In this case, the direction of photon propagation is along the +z direction.

2. For every voxel in a layer, calculate the 14 directional scatter components and accumulate total fluence by summing the directional fluences in each voxel:

   $$\begin{array}{l}\text{If}\left(i,j,k\right)=\text{If}\left(i,j,k-1\right)+\text{Idr}\left(i-1,j,k-1\right)\text{wt}2\\ +\text{Idf}\left(i,j-1,k-1\right)\text{wt}2+\text{Idb}\left(i,j+1,k-1\right)\text{wt}2\\ +\text{Idl}\left(i+1,j,k-1\right)\text{wt}2,\end{array}$$ (6)

   $$\begin{array}{l}I\left(i,j,k\right)=\text{If}\left(i,j,k\right)+\text{Idr}\left(i,j,k\right)\\ +\text{Idl}\left(i,j,k\right)+\text{Idf}\left(i,j,k\right)+\text{Idb}\left(i,j,k\right)\\ +\text{Isr}\left(i,j,k\right)+\text{Isl}\left(i,j,k\right)+\text{Isf}\left(i,j,k\right)\\ +\text{Isb}\left(i,j,k\right)+\text{Idbr}\left(i,j,k\right)+\text{Idbl}\left(i,j,k\right)\\ +\text{Idbf}\left(i,j,k\right)+\text{Idbb}\left(i,j,k\right)+\text{Id}\left(i,j,k\right).\end{array}$$ (7)

   The nomenclature for the 14 directional fluence vectors is shown in Table 3. Equation (6) represents an example of how the forward scatter component is calculated. The coordinates i, j, k denote the x, y, and z location of the voxel. The weights wt1 and wt2 are forward and diagonal scattering weights. The sum of all weights is always equal to one (i.e., wt1 + 4 ∗ wt2 = 1), so that the total fluence entering a voxel is equal to the sum of fluence absorbed and exiting the voxel. The equations used to calculate the other components can be similarly determined. The sum of all the scatter components in a voxel equals the total fluence in that voxel (arbitrary units: photons/unit volume), as shown by Eq. (7).

3. Calculate absorbed fluence (Iabs) using the Beer–Lambert law for a path length dl equal to the voxel length, based on the absorption coefficient (μ_a) of the voxel:

   $$I_{\text{abs}\left(i,j,k\right)}=I\left(i,j,k\right)\ast \left(1-e^{-{\mu }_a\ast dl}\right).$$ (8)

4. Calculate and accumulate the fluence over all voxels in the current layer and increment z to proceed to the next layer. Continue propagation until the fluence exits the volume.

Fig. 12.

Empirical method for 3D photon propagation. The fluence at each voxel in a layer is a weighted sum of the neighboring fluences in adjoining voxels. Part A shows the simplified schematic of fluence accumulation (in 2D), while the actual 3D model with directional fluence components is shown in part B of the figure. The fluence in each voxel consists of forward, backward, side, and diagonal fluence components (mentioned in Table 3). The directional fluences are first calculated in each voxel along with the total fluence, before being propagated to neighboring voxels.

Table 3.

Vector Directions and Nomenclature Empirical Algorithm

Vector Name	Vector Component s (x, y, z)	Description
If	(0 0 1)	Upward vector in +z direction
Ib	(0 0 −1)	Downward vector in −z direction
Isf	(0 1 0)	Side vector in +y direction
Isb	(0 −1 0)	Side vector in y direction
Isr	(1 0 0)	Side vector in +x direction
Isl	(−1 0 0)	Side vector in −x direction
Idf	(0 0.5 0.5)	Upward diagonal vector in +z and +y direction
Idb	(0 −0.5 0.5)	Upward diagonal vector in +z and −y direction
Idr	(0.5 0 0.5)	Upward diagonal vector in +z and +x direction
Idl	(−0.5 0 0.5)	Upward diagonal vector in +z and −x direction
Idbf	(0 0.5 −0.5)	Downward diagonal vector in −z and +y direction
Idbb	(0 −0.5 −0.5)	Downward diagonal vector in −z and −y direction
Idbr	(0.5 0 −0.5)	Downward diagonal vector in −z and +x direction
Idbl	(−0.5 0 −0.5)	Downward diagonal vector in −z and −x direction

The weights to be assigned for scatter were obtained by an iterative optimization phase, which compares the output fluence of the empirical algorithm to that generated by the Monte Carlo routine for voxels along the beam axis, as calculated by the R-square coefficient (R²). Table 4 shows the weights (wt1 and wt2) obtained for a pencil beam simulation within different tissue types. Figure 11 shows an example of the match in fluence profiles between the Monte Carlo and the empirical approach.

Table 4.

Optimization of Empirical Algorithm for Brain Tissue Phantoms

Tissue Phantom	Resolution (mm)	Wt1	Wt2	R2 Coefficient
White matter	0.1	0.949461	0.012635	0.99999
Gray matter	0.1	0.87469	0.031328	0.999969
Skull bone	0.1	0.914828	0.021293	0.99979
White matter	1	0.474876	0.131281	0.99992
Gray matter	1	0.062097	0.234476	0.999828
Skull bone	1	0.245949	0.188513	0.999871

3. RESULTS AND DISCUSSION

A. Optical Dosimetry Probe Calibration

The isotropicity (coefficient of variation) of nylon- and titanium-based optical dosimetry probes are shown in Fig. 5(a). The titanium-based probes demonstrated a significantly improved isotropicity with an equatorial coefficient of variation 0.038–0.072 versus 0.193 for the nylon and an azimuthal coefficient of variation ranging from 0.21–0.29 versus 0.44 for the nylon. This is a 2–5 × overall better isotropicity, depending on the concentration of TiO₂. Figure 6(b) shows the linearity of response of the 1.5 mm titanium probe. The measurement error in probe response was 1.7%, and 1.1% in the calorimeter readings, where the majority of this error was due to variation in the output power of the laser (3% absolute deviations). Thus, the titanium probes proved to be better suited to optical dosimetry studies compared to nylon probes and hence were used in brain phantom studies. While nylon probes have been used before, this is the first time that titanium probes were designed and shown to have a better isotropic response. Previous studies have measured the isotropicity of dosimetry probes made of Arnite (11%), nylon (11%), Helioseal dental sealant (20%) [17,19]. Our study measured the % variation of the best titanium probe to be 13% and the best nylon probe to be 17%. The differences in probe construction and calibration setup contribute to different measured isotropicity of the same probe material (e.g., nylon) across various studies.

To quantify the photon fluence in tissues and tissue-like media, the probe voltage measured within the medium must be multiplied by a set of calibration factors to account for the change in photon acceptance and loss in the probe–medium interface with respect to the probe–air interface. Since the refractive index of tissues is nearly the same as that of water, the calibration factor obtained for water can be used for biological media. The probe calibration factor was determined to be 1.32. The technique presented in Section 2.C [using Eqs. (1)–(3)] provides a simple and new method to determine the calibration factor experimentally.

B. Validation of Monte Carlo in Homogeneous Phantom Using the Optical Dosimetry Probe

The optical fluence measured by the ODPs in white and gray matter phantoms was compared to the Monte Carlo generated fluence maps to validate the Monte Carlo. These experiments necessitate an accurate determination of the phantom’s optical properties. Figures 8 and 9 show the comparison between the probe measurements and the Monte Carlo in brain phantoms for two different illumination profiles, flat diffuse beam (integrating sphere) and super-Gaussian beam (optical fiberguide). A comparison was also made in the direction perpendicular to the beam axis, as shown in Fig. 10. From these data, the average relative % error of the Monte Carlo simulation relative to the probe measurements was 11.2% for white matter, 13.2% for gray matter, and 11.8% for skull bone along the beam axis for a range of 0.2–2.0 cm within the phantom, while perpendicular to the beam axis, the % relative error in the lateral profile was 9.4%, 12.0%, and 8.9% for white matter, gray matter, and skull bone phantoms, respectively. These systematic errors represent the level of confidence with which the Monte Carlo matches the probe measurements and validate their use in simulating optical energy distribution in media (Tables 5 and 6).

Fig. 9.

Optical dosimetry in white matter, gray matter, and skull bone phantoms (cuvette dimensions: 5.3 cm × 4.6 cm × 3.1 cm, thickness = 0.4 cm). The photon source used is an optical fiberguide (connected to a pulsating laser source) with a super-Gaussian beam distribution. The Monte Carlo generated fluence closely matches the fluence measured by the probe. This shows that the Monte Carlo beam modeling can be reliably used as a reliable estimator of photon energy distribution in tissue phantoms.

Fig. 10.

Lateral beam profile measurement using optical dosimetry probe and validation by Monte Carlo in homogeneous phantoms. The laser source used was an optical fiberguide (0.3 cm diameter) connected to a pulsed NIR laser emitting at a wavelength of 800 nm. Cuvette dimensions: 5.3 cm × 4.6 cm × 3.1 cm, thickness = 0.4 cm. (a) Beam characterization in air using calorimeter. Optical dosimetry to measure and validate lateral beam profiles in phantoms resembling: (b) white matter, (c) gray matter, and (d) skull bone, i.e., perpendicular to the beam axis. The measurements have been normalized to the maximum value measured by the dosimetry probe in each of the respective phantoms. The Monte Carlo generated fluence closely matches the probe-measured fluence and effectively models the beam spread in tissue phantoms.

Table 5.

% Mismatch Error in Homogeneous Phantom Along Beam Axis (z)

Phantom	Error% (0.2–1 cm)	Error% (1–2 cm)	Average Error%
White matter	12.13	10.5	11.27
Gray matter	13.39	30.12	13.25
Skull bone	11.39	12.28	11.81

Table 6.

% Mismatch Error in Homogeneous Phantom Perpendicular to Beam Axis

Phantom	Error% z = 0.2 cm	Error% z = 0.6 cm	Error% z = 1.0 cm	Error% z = 1.4 cm	Average Error%
White matter	10.34	13.34	9.14	4.79	9.4
Phantom	Error% z = 0.2 cm	Error% z = 0.4 cm	Error% z = 0.6 cm	Error% z = 1 cm	Average error%
Gray matter	13.45	11.52	14.29	8.97	12.06
Phantom	Error% z = 0.6 cm	Error% z = 0.7 cm	Error% z = 0.8 cm		Average error%
Skull bone	8.41	5.69	12.64		8.91

An analysis of the fluence profile within the skull phantom showed that the super-Gaussian beam fluence decreases with an effective attenuation coefficient of 4.81 cm⁻¹. This factor was used to estimate the effectiveness of therapy for different skull thicknesses. Based on Monte Carlo simulated data, the powers transmitted by the fiberguide source for various skull thicknesses are: 5.9 mW/cm² (0.4 cm thick), 2.16 mW/cm² (0.6 cm thick), 0.83 mW/cm² (0.8 cm thick), 0.33 mW/cm² (1 cm thick). Thus, the % power transmitted in a 1 cm thick adult skull is 1.8%, while in a pediatric skull with a thickness of 4.0 mm, 33.5% optical power would reach the soft tissue, representing 18.5 times more fluence.

C. Validation of Monte Carlo in Heterogeneous Phantom Using the Optical Dosimetry Probe

Energy fluence distribution was measured in the white matter volume of a heterogeneous head phantom, which consisted of a 1.1 cm thick skull agar phantom, a 1.1 cm thick gray matter agar phantom, and a 5.9 cm thick white matter liquid phantom (Fig. 7). The exponential decreases in energy fluence along the beam axis and the lateral beam profiles at different depths above the gray matter agar surface are shown in the right and left plots of Fig. 11. The lateral fluence profiles have been normalized to the intensity at a depth of 0.2 cm above the gray matter agar surface, while the fluence profiles along the beam axis were normalized to the fluence at z = 0.1 cm above the gray matter agar surface. The average error (relative % error) between the probe and Monte Carlo fluence was 17.7% along the beam axis and 18.0% perpendicular to the beam axis (Tables 7 and 8). These results demonstrate that the beam spread can be determined accurately by the Monte Carlo to an overall depth of nearly 4 cm given the optical coefficients of the head phantom.

Table 7.

% Mismatch Error in Heterogeneous Phantom Along Beam Axis (z)

Phantom	Error% (0.1–1 cm)	Error% (1–2 cm)	Average Error%
White matter (over skull and gray matter layers)	2.11	33.32	17.71

Table 8.

% Mismatch Error in Heterogeneous Phantom Perpendicular to Beam Axis

Phantom	Error% z= 0.2 cm	Error% z= 0.4 cm	Error% z= 0.6 cm	Error% z= 1 cm	Average Error%
White matter (over skull and gray matter layers)	18.52	15.09	15.37	23.17	18.04

Past studies have compared the Monte Carlo simulations to dosimetry probe measurements for simple beam profiles, such as flat beams or isotropic sources [14,15]. One study compared the angular radiance measured by a flat cleaved dosimetry probe to that simulated by the Monte Carlo and determined an accuracy of around 25% (see Fig. 6 in referenced publication) [14]. Another study validated the Monte Carlo in pig bronchus tissue using dosimetry probe measurements with errors up to 50% at 4 cm depth and systematic variation of 15% [15]. In past studies, the long computation times (>24 h) associated with the CPU-based Monte Carlo prevented a rigorous approach to Monte Carlo validation across diverse tissue types and under different illuminations. Another source of error in these studies was the uncertainty in the estimation of tissue/phantom optical properties. In this study, we have minimized this uncertainty by independently validating the optical properties. The use of a GPU-based Monte Carlo reduces the development time (few hours) associated with optimizing the Monte Carlo for complex illumination and boundary conditions.

The use of pulsed laser beams has been shown to improve the instantaneous temperature rise and release of drug molecules tethered to nanoparticles, compared to continuous-wave lasers [42]. This will allow more instantaneous power to be coupled to the nanomolecules while reducing the chances of laser-induced adverse tissue heating. Assuming nanosecond pulse duration, the maximum permissible exposure limit of laser for skin is 3.17 × 10⁷ W cm⁻² [41,42]. Based on our MC simulations with a flat 2 cm diameter beam, the amount of fluence coupled to the soft tissue for a 1.1 cm thick skull is 4.3 × 10⁵ W cm⁻² and 1.5 × 10⁷ W cm⁻² for neonatal heads with a 0.2 cm thick skull. Thus, a drug needs to be designed to activate at a fluence threshold, less than the maximum fluence coupled to the tissues at a specific depth where the metastasis is located. For example, a drug molecule with an activation threshold of 1 W cm⁻² can be activated up to 3.8 cm in the white matter of a 20 cm diameter adult brain (1.1 cm thick skull plus 2 cm gray matter) and up to 4.6 cm in a 8 cm diameter neonatal brain (0.2 cm skull plus 0.2 cm gray matter). These results demonstrate that NIR activated drug release is more feasible in neonatal brains where 4.6 cm is greater than half of the brain diameter. The temporal widening of the laser pulse with depth in the brain is also seen to affect the instantaneous fluence (W · cm⁻² · s) coupled to tissues. Based on our Monte Carlo simulations, the pulse width is seen to be 4–5 times at depths of 3–5 cm for a 1 ns wide input laser pulse. This implies a decrease in instantaneous fluence by that amount. The Monte Carlo can be used to predict the temporal beam widening in order to determine the instantaneous fluence at specific depths.

D. Empirical Versus Monte Carlo

In Figs. 12 and 13, a schematic of the EM used in these studies is presented and the corresponding weights determined by matching photon fluence results to MC data. As seen in Table 4, the weights obtained during optimization depend on the voxel size and the scattering coefficient of the medium. The scattering coefficient and the voxel size determine the number of scattering events per unit volume, which affects the percentage scatter in the forward versus diagonal directions, thus impacting the weights (wt1 and wt2). Based on these weights, EM simulated data for homogeneous and heterogeneous phantoms were compared to Monte Carlo and ODP measurements. This comparison was performed along the beam axis, see Fig. 13(b), in homogeneous brain tissue phantoms. The average relative % error between the empirical and probe measurements for the super-Gaussian beam simulation is 10.1% for white matter, 45.2% for gray matter, and 22.1% for skull bone phantoms; while the comparison between empirical and MC is 22.6% for white matter, 35.8% for gray matter, and 21.9% for skull bone measurements. The errors between probe measurements and the empirical algorithm are cumulative in nature, and can be reduced with better optimization methods. No systemic trend was observed in these errors with % differences having both positive and negative values. Figure 14 shows a comparison between the lateral fluence profiles generated by a Monte Carlo and the empirical models for a flat beam of 2 cm diameter in homogenous white and gray matter tissues. A significant match is seen between the normalized fluence profiles at two different depths. Figure 15 shows a complete head simulation and the comparison between the MC and empirical fluence versus depth along the beam axis. The empirical fluence closely follows the Monte Carlo but has significant errors at object boundaries.

Fig. 13.

(a) Optimization phase: a flat broad beam (2 cm diameter) is simulated using Monte Carlo and the empirical algorithm is optimized to match the Monte Carlo with the identical illumination conditions. (b) Comparison of the empirical algorithm with Monte Carlo and probe-measured fluence in brain phantoms resembling white matter, gray matter, and skull bone. The illumination source is a super-Gaussian beam. The empirical approach is a close approximation of the Monte Carlo and the probe over the range of optical properties of brain tissues.

Fig. 14.

Lateral beam profile comparison between normalized lateral fluence profiles of empirical (red line) and Monte Carlo (blue line) for gray matter and white matter tissue at depths of 5 and 10 mm in a homogeneous phantom. A flat broad beam (20 mm diameter) is simulated. A close match is seen between the Monte Carlo and empirical lateral beam profiles.

Fig. 15.

(a) Simulation of a spherical head phantom with brain tissue properties shown in Table 1. Skull thickness = 4 mm; gray matter thickness = 10 mm, white matter diameter = 72 mm. Uniform illumination beam diameter = 20 mm (800 nm wavelength). (b) Comparison of fluence along the beam axis between empirical (red line) and MC (blue line) models. The empirical model achieves this simulation in 6 s while the MC takes 8 h for a total of 10¹⁰ photons.

The Monte Carlo code is computationally expensive in spite of being based on a GPU-based platform. In order to simulate the optical properties and fine structural boundaries of brain tissues, a resolution equal to or less than 1 mm is necessary. Even though an optical map of the brain can be obtained through segmenting and translating CT and/or MR images, the dataset size of the brain becomes very large, e.g., a CT scan requires a 512 × 512 × 512 matrix. This significantly increases the execution time of the Monte Carlo routine to iterate over various illumination conditions. The empirical approach provides an alternative to the computationally expensive Monte Carlo, and can be used to select the best illumination conditions for therapy. The speed-up achieved by the empirical algorithm was approximately 700 × for a pencil beam simulation and 16, 000 × for a broad-beam simulation in a head phantom. Thus while a broad-beam head simulation takes 6 s with the empirical, it can take in excess of 8 h with the GPU-based Monte Carlo for a head phantom simulation of volume 1000 cm³. It is important to note that the number of photons to be simulated depends on the depth at which the photon distribution is to be evaluated, and the % fluence penetration (and systematic errors between MC, probe, and empirical) is independent of the number of photons simulated.

4. CONCLUSION

We validated the 3D Monte Carlo using a novel optical dosimetry probe in phantoms resembling white matter, gray matter, and skull bone. The TiO₂-based dosimetry probe was shown to have superior linearity and isotropicity of response, and was better suited to validate the Monte Carlo using localized 3D measurement (<25% error) in tissue phantoms. The independent characterization of the probes and the optical phantoms using the calorimeter “gold standard” and spectrophotometer (respectively) further validate our results. The 3D validation of the Monte Carlo generated fluence for two different types of NIR light sources in different optical phantoms, and the ease of photon quantification to absolute dose (in mW/mm² or J/mm²) demonstrates the fidelity of the Monte Carlo as a predictive tool to accurately estimate 3D photon energy distribution in complex media. The measurement and prediction accuracy can be improved with better instrumentation for probe localization and better laser beam positioning setups. The voxel-based Monte Carlo can emulate the complex geometrical shape of tissues using imaging datasets such as CT/MRI and can now be used as a predictive clinical tool to estimate photon dose in heterogeneous brain tissues.

The empirical algorithm, validated by the 3D Monte Carlo, increases the clinical feasibility of optical therapeutic planning to narrow down the complex possibilities of illumination conditions, further compounded by the heterogeneous structure of the brain (e.g., varying skull thicknesses and densities). A limitation of using the empirical approach is that the weights assigned depend on the resolution of the medium and changing resolution requires an optimization phase to derive the weights. A rigorous validation of the empirical approach is currently under development, which is necessary for its applicability in heterogeneous tissue environments in clinical settings. Despite its limitations, it can be used as a preliminary step, with the final estimation done by the Monte Carlo to ensure high predictive accuracy.

Our ultimate goal is to design a fast Monte Carlo-based optical therapeutic protocol to treat brain metastasis. This requires voxel by voxel determination of photon energy distribution to accurately predict the rate/quantity of drug release. This study achieves the first step in the design of a tool to validate the NIR photon distribution in three dimensions in optical phantoms, where tissue structure and boundaries can be determined from imaging modalities such as MRI or CT.