Determination of fluence rate and temperature distributions in the rat brain; implications for photodynamic therapy

Even Angell-Petersen; Henry Hirschberg M.D.; Steen J. Madsen

doi:10.1117/1.2709882

1 January 2007 Determination of fluence rate and temperature distributions in the rat brain; implications for photodynamic therapy

Even Angell-Petersen, Henry Hirschberg M.D., Steen J. Madsen

Author Affiliations +

Journal of Biomedical Optics, Vol. 12, Issue 1, 014003 (January 2007). https://doi.org/10.1117/1.2709882

Abstract

Light and heat distributions are measured in a rat glioma model used in photodynamic therapy. A fiber delivering 632-nm light is fixed in the brain of anesthetized BDIX rats. Fluence rates are measured using calibrated isotropic probes that are positioned stereotactically. Mathematical models are then used to derive tissue optical properties, enabling calculation of fluence rate distributions for general tumor and light application geometries. The fluence rates in tumor-free brains agree well with the models based on diffusion theory and Monte Carlo simulation. In both cases, the best fit is found for absorption and reduced scattering coefficients of 0.57 and 28 cm^-1, respectively. In brains with implanted BT₄C tumors, a discrepancy between diffusion and Monte Carlo–derived two-layer models is noted. Both models suggest that tumor tissue has higher absorption and less scattering than normal brain. Temperatures are measured by inserting thermocouples directly into tumor-free brains. A model based on diffusion theory and the bioheat equation is found to be in good agreement with the experimental data and predict a thermal penetration depth of 0.60 cm in normal rat brain. The predicted parameters can be used to estimate the fluences, fluence rates, and temperatures achieved during photodynamic therapy.

1. Introduction

Intracranial light delivery via fiber optics or implanted LED light sources has been used for photodynamic therapy (PDT) in several animal studies. ^{1, 2, 3, 4} PDT is a local treatment involving the administration of a tumor-localizing photosensitizing drug⁵ and has several features that make it a potentially attractive adjuvant local therapy for brain tumors. The treatment volume is limited by the attenuation of light in brain tissues, and repeated application is an option due to low long-term morbidity. The probability of achieving local control with PDT therefore depends strongly on the ability to achieve adequate light fluences 1 to $2 cm$ from the resection margin, where the majority of tumor recurrences occur.⁶ PDT has been tested for glioma patients in several clinical trials ^{7, 8, 9, 10, 11, 12} using photosensitizers activated by light with a wavelength around $630 nm$ . At this wavelength, the light fluence rate decreases¹³ exponentially with distance from the resection margin with a light penetration depth of 1 to $3 mm$ . The penetration depth (distance required to reduce the fluence rate by a factor $e^{- 1}$ ) of light in brain tissue has been measured in vivo for several mammalian species ^{13, 14, 15, 16, 17} and is sensitively dependent on the scattering and absorption properties of the tissue. Accurate knowledge of scattering and absorption coefficients is required to calculate the light distribution for a given geometry in terms of absolute fluence rates. A search of the literature yields large variations in the scattering and absorption coefficients reported for $630 - nm$ light in mammalian brain tissues.^{18, 19} These discrepancies are likely due to differences in measurement conditions. Furthermore, the accuracy of these values must be questioned since measurements have been performed exclusively on postmortem tissue. Accurate determination of the optical properties of brain tissue requires direct in situ measurements.¹⁴

PDT dosimetry requires knowledge of a number of parameters including light fluence rate. This is particularly important in the brain where damage to eloquent areas can result in severe morbidity. It is possible to measure fluence rates interstitially by inserting isotropic fiber optic probes directly into the brain. Studies using either scattering or fluorescent tip probes have been reported in a number of organs,^{20, 21, 22} but data on absolute fluence rates during interstitial light delivery in mammalian brains are limited. We currently use intracranial light delivery in a rat glioma model to study PDT with the endogenous photosensitizer aminolevulinic acid.^{4, 23} In this paper, we present in vivo measurements of fluence rates around a bare fiber delivering $632 - nm$ light to rat brain and glioma tissue. The measured values in tumor and normal brain were compared to calculated fluence rates obtained from diffusion theory and Monte Carlo simulations, and the comparison yielded estimates of scattering and absorption coefficients. Additionally, temperature profiles were measured by inserting thermocouples directly into the brain, and these measurements were compared to predictions of diffusion theory.

2. Materials and Methods

2.1.

Animal Care

Animal care and protocol were in accordance with national legislation and institutional guidelines. All animals tested negative for parasitic, bacterial, and viral agents according to Federation of European Laboratory Animal Science Assocations (FELASA) recommendations.²⁴ Inbred BDIX rats (Charles River, Lyon, France) of both sexes weighing 200 to $400 g$ were caged in pairs or groups of three in Macrolon III cages on a 12-h light and dark schedule. The animal holding rooms and laboratories were maintained at constant temperature and humidity at an air-exchange rate of 18 changes per hour. A total of eight animals were used. Prior to surgery, rats were anesthetized with a standard 1:1 combination of fentanyl/fluanisone (Hypnorm, Janssen, Beerse, Belgium) and midazolam (Dormicum, Roche, Basel, Switzerland). Intracranial tumors were established, as described by Sørensen, ²⁵ by injecting 10,000 ${BT}_{4} C$ cells $3 mm$ into the brain via a burr hole in the skull $3 mm$ posterior to and $2 mm$ to the right of the bregma. Tumor-bearing rats were used for experiments 17 days after tumor induction. At that time the tumors had a size of about $4.5 \times 6.5 mm$ centered 3 to $4 mm$ into the brain, and the animals did not display clinical signs of tumor development. Histopathology of more than 10 rats that were part of other studies^{4, 23} confirmed that clinical signs occur when the largest tumor diameter is at least $7 mm$ , and that the tumors at that state are entirely without necrosis. The rats were euthanized immediately after each procedure with a lethal dose of pentobarbitone [ $100 mg ∕ kg$ ip (intraperitoneal)].

2.2.

Measurement and Modeling of Light Distributions

2.2.1.

Isotropic probe for in situ fluence rate measurements

Small fiber optic isotropic probes, similar to those previously described by Bays, ²¹ were developed [Fig. 1a ]. A $250 - μ m$ ruby sphere (Sandoz fils SA, Cugy, Switzerland), which is fluorescent under $632 - nm$ light, was glued to a $400 μ m ∕$ numerical aperture (NA) 0.37 bare flat-end quartz fiber (Biolitec AG, Jena, Germany) with a UV curable adhesive (Loctite 350, Loctite, Dublin, Ireland). The fiber was covered with thin black polyester tubing (018050 BST, Advanced Polymers Inc., Salem, New Hampshire). Fluorescence from the ruby sphere collected by the fiber was measured with a CCD spectrometer (S2000, OceanOptics Inc., Orlando, Florida) equipped with a $665 - nm$ long-pass filter. A linear combination of the ruby fluorescence spectrum and a third-degree background function, weighted by the spectral response of the system, was fitted to each observed spectrum to quantify the ruby fluorescence and hence the fluence rate. Calibration was done with a setup similar to that described by Marijnissen and Star,²⁶ with a flat photodiode (S2387-33R, Hamamatsu, Japan) positioned next to the probe in uniform isotropic $632 - nm$ light inside an integrating sphere (Avasphere-50-IRRAD-S, Avantes, Eerbeek, Netherlands) coupled to a diode laser (CeramOptec, Bonn, Germany). The sphere was filled with a 23% sucrose/water solution with a refractive index of 1.37. The spectral response and active area of the photodiode was measured in a National Physical Laboratory (Middlesex, United Kingdom) traceable setup by the Norwegian Metrology Service (Kjeller, Norway), and was corrected for the effects of reflection by the window material $(+ 1.7 %)$ and of measured deviation from the ideal cosine angular response $(+ 1.0 %)$ .

Fig. 1

Image of (a) the fluorescent isotropic probe used for fluence rate measurements and (b) the thermocouple probe used for temperature measurements.

2.2.2.

In vivo measurements of intracranial fluence rate distribution

A plastic template with parallel channels, fitting the light delivery and probe fibers, was used together with a stereotactic frame to control the fiber positions with $0.1 - mm$ precision. The setup is shown in Fig. 2 . Anesthetized animals were fixed in the stereotactic frame, the skin was incised, and the frame was used to position the template with the source fiber channel $3 mm$ posterior to and $2 mm$ to the right of the bregma. The template was then fixed to the skull using dental cement (Fuji plus, GC, Tokyo, Japan), and the skin closed around the template with sutures to prevent local hypothermia. Holes were drilled through the template channels and into the skull. The light delivery fiber was then inserted into the brain and fixed in the template with histoacryl adhesive (Braun, Melsungen, Germany). The fiber tip delivered $632 - nm$ light from the diode laser at a radiant power of $15 mW$ . Initial tests of light transmission from the fiber into whole blood and rat brain showed that this radiant power was at least four and seven times, respectively, below the thresholds for heat-induced coagulation. The fluence rate probe was mounted in the stereotactic frame’s electrode holder and inserted through the template to record fluence rates along four parallel tracks positioned 2 or $4 mm$ in front of and behind the light delivery fiber, as shown in Fig. 2. Fluence rates were measured at $0.5 - mm$ increments as the probe was moved into the brain, starting with the tracks $4 mm$ from the source. The procedure lasted 1 to $2 h$ , which is comparable to the duration of light exposures in our PDT protocols using BDIX rats.⁴

Fig. 2

Setup for the in vivo fluence rate measurements. The measurement tracks were positioned within one sagittal plane through the rat brain.

It was recognized that the probe’s black polyester tubing [Fig. 1a] could act as a photon sink, resulting in a reduction in the measured signal. To estimate the extent of this perturbation, Monte Carlo simulations were performed for the relevant probe geometry, light penetration depths, and fluence rate gradients. The results show that the black cladding reduces the measured signal by 3 to 25% depending on the fluence rate gradient along the long axis of the probe. These results were used to calculate a general correction factor that was a function of the logarithmic local fluence rate gradient. The maximum correction (corresponding to 25% loss) occurred at the positions where the fluence rate at the cladding was largest compared to the fluence rate at the sphere. For each measurement, the local gradient was estimated using the ratio to the previous measurement in the same measurement series. All measurements were subsequently corrected using their resulting correction factors. The calculation of the corrected fluence rate $ϕ$ can be summarized as

ϕ (z) = CF [\frac{\tilde{ϕ} (z)}{\tilde{ϕ} (z - 0.5 mm)}] \tilde{ϕ} (z)

where

\tilde{ϕ} (z)

is the uncorrected value measured at position

z

(the geometry is illustrated in Fig. 2), and CF is the correction factor expressed as a function of the ratio.

2.2.3.

Diffusion models of the fluence rate distribution

According to diffusion theory, the fluence rate $ϕ$ at distance $r$ around an isotropic point source delivering a radiant power $P$ into an infinite turbid medium is approximated by²⁷

Eq. 1

ϕ (r) = \frac{P}{4 π} \frac{3 (μ_{s}^{'} + μ_{a}) \exp (- r ∕ δ)}{r},

where

μ_{s}^{'}

is the reduced scattering coefficient,

μ_{a}

is the absorption coefficient, and

δ

is the light penetration depth:

Eq. 2

δ = {[3 μ_{a} (μ_{s}^{'} + μ_{a})]}^{- 1 ∕ 2} .

The presence of a spherical body with differing scattering and absorption coefficients

μ_{s, tumor}^{'}

and

μ_{a, tumor}

and radius

R

leads to the following solution of the diffusion equation:

Eq. 3

ϕ (r) = {\begin{matrix} \frac{P}{4 π} [C_{1} \frac{\exp (r ∕ δ_{tumor})}{r} + C_{2} \frac{\exp (- r ∕ δ_{tumor})}{r}] & r \leq R, \\ \frac{P}{4 π} [C_{3} \frac{\exp (- r ∕ δ)}{r}] & r > R, \end{matrix}

assuming a zero fluence rate at an infinite distance from the source. The constants

C_{1}

,

C_{2}

, and

C_{3}

follow from conservation of energy, which implies that the output power from the source equals the sum of all optical power absorbed in the medium:

P = \int_{0}^{\infty} ϕ (r) μ_{a} 4 π r^{2} d r,

and from continuity of fluence rate and flux:

ϕ ∣_{r = R^{-}} = ϕ ∣_{r = R^{+}},

j ∣_{r = R^{-}} = j ∣_{r = R^{+}},

where the flux

j (r)

is given by the fluence rate²⁸ as

- [3 (μ_{s}^{'} + μ_{a})]^{- 1} \nabla ϕ (r)

. This yields all three constants

C_{1}

,

C_{2}

, and

C_{3}

as functions of the radius

R

and the scattering and absorption coefficients of both the spherical body (

μ_{s, tumor}^{'}

,

μ_{a, tumor}

) and the surrounding medium (

μ_{s}^{'}

,

μ_{a}

).

An anisotropic source can be approximated as an isotropic source displaced a distance $1 ∕ (μ_{s}^{'} + μ_{a})$ along the direction of the emitted light.²⁹ In this manner, Eqs. 1, 3, for tumor-free and tumor-bearing animals, respectively, with $r$ replaced with the distance

Eq. 4

r = {[x^{2} + {(z - \frac{1}{μ_{s}^{'} + μ_{a}})}^{2}]}^{1 ∕ 2}

were used to model the fluence rate at horizontal distance

x

and vertical distance

z

from the fiber tip. These models for the fluence rate

ϕ (z, x)

were fitted to the in vivo fluence rate measurements by log-weighted least-squares estimation (Marquardt-Levenberg algorithm), to estimate the tissue optical properties

μ_{s}^{'}

,

μ_{a}

,

μ_{s, tumor}^{'}

, and

μ_{a, tumor}

.

2.2.4.

Monte Carlo model of the fluence rate distribution

A commonly used Monte Carlo code ³ ³⁰ was modified for the light delivery geometry in the animals. The fiber tip was modeled as a $400 - μ m$ disk emitting a conical photon beam of $\pm 9 \deg$ , and the brain was considered to be an infinite medium with uniform scattering and absorption properties. Anisotropic scattering was modeled using a Henyey-Greenstein phase function with scattering anisotropy $(g)$ values of between 0.8 and 0.98, which is the range of values reported for tissues. ^{18, 19, 31, 32} Tumors were modeled as ellipsoids with varying scattering and absorption coefficients, and with their size and position determined from histological sections of brains from tumor-bearing animals. The Monte Carlo simulations were fit to measured fluence rates using an optimization algorithm that systematically varied the absorption and scattering coefficients to find the best least-squares estimate. About 500,000 runs, requiring $30 \min$ on a $3.6 - GHz$ Pentium 4 processor, were necessary for each set of coefficients to achieve an accuracy of 2%.

2.3.

Measurement and Modeling of Temperature Distributions

2.3.1.

In vivo measurements of intracranial temperature distributions

Temperature measurements in tumor-free animals were accomplished by inserting thermocouples into the rat brain. The thermocouple probes are shown in Fig. 1b, and were assembled from K-type $75 - μ m$ wire thermocouples (CHAL-003, Omega Engineering, Stamford, Connecticut) mounted in $800 - μ m$ ceramic tubing (TRA-005132, Omega Engineering, Stamford, Connecticut). The probes were tested immersed in water under a laser beam to verify that direct light absorption in the thermocouple did not affect the readings. No noticeable increase related to direct absorption was observed for fluence rates below $0.25 W {cm}^{- 2}$ , and at $1.2 W {cm}^{- 2}$ , the increase was only $0.2 K$ . To account for subtle changes in body temperature, all measurements were normalized to rectal body temperature monitored with a $0.5 - mm$ insulated thermocouple. In the rat brain, positioning of the thermocouples with respect to the source fiber was accomplished in a similar manner as that described for the fluence rate measurements in Sec. 2.2.2. Two distinct data sets were collected from each animal: First, steady state temperatures elevated by the light delivery were measured at $0.5 - mm$ increments as the probe was moved into the brain. The temperature increase caused by light absorption was then estimated by subtracting steady state temperatures measured while no light was delivered from the fiber tip during the withdrawal of the temperature probe. Second, temporal measurements of temperature increase and decrease (as the laser light was switched on and off, respectively) were made during the withdrawal of the probe, with the probe fixed $2 mm$ lateral to the source fiber tip.

2.3.2.

Bioheat model of the temperature distribution

From the bioheat equation³³ and Eq. 1, the temperature increase $T$ in tissue around a point source is described by

Eq. 5

\frac{\partial T}{\partial t} = \frac{κ}{ρ c} \nabla^{2} T - Q T + \frac{P}{4 π} \frac{\exp (- r ∕ δ)}{κ δ^{2} r},

where

κ

is the heat conductivity,

ρ

is the density,

c

is the specific heat capacity, and

Q

is the blood perfusion rate. To model the spatial temperature distribution, the steady state solution to Eq. 5,

Eq. 6

T (r) = \frac{P}{4 π} \frac{\exp (- r ∕ δ_{v}) - \exp (- r ∕ δ)}{κ (1 - \frac{δ^{2}}{δ_{v}^{2}}) r},

was fitted to the spatial measurement series. The thermal penetration depth

δ_{v} = [κ ∕ (Q ρ c)]^{1 ∕ 2}

was used as a free parameter, while a heat conductivity of

0.53 W m^{- 1} K^{- 1}

(Ref. 34) was assumed. Equation 4 was substituted into Eq. 6 to correct for the distance to the anisotropic source. The temporal temperature variation was modeled using the time-dependent Eq. 5. For this purpose, Eq. 5 was solved numerically for different

δ_{v}

values, using a finite difference method with explicit time stepping,³⁵ to find the best least-squares fit to the temporal measurements.

3. Results

3.1.

Light Distribution

3.1.1.

Measured fluence rates in the rat brain

Fluence rate measurements from four tumor-free rats are shown in Fig. 3a . (A preliminary analysis of the results from two of the rats was published in a previous report.³⁶) Inter- and intraanimal variation was of similar magnitude. As seen from the data series in the diagrams in Fig. 3, the measured fluence rates throughout the brains were consistent within a total variability of a factor of 2. There was relatively good agreement between diffusion theory [Eqs. 1, 2, 4] and the measured data. The best fit was found for $μ_{a} = 0.570 {cm}^{- 1}$ (standard error $0.021 {cm}^{- 1}$ ) and $μ_{s}^{'} = 27.9 {cm}^{- 1}$ (standard error $2.4 {cm}^{- 1}$ ), and is shown as solid lines in Fig. 3a. These values correspond to a light penetration depth of $0.143 cm$ [Eq. 2]. Measurements made closer than $2 mm$ from the skull were excluded from the analysis.

Fig. 3

Interstitial fluence rate $ϕ$ relative to radiant power $P$ plotted as a function of vertical distance $z$ and lateral distance $x$ from the source fiber tip. The geometry and coordinates are illustrated in Fig. 2. (a) Data from four tumor-free rats with the source fiber fixed 4 to $4.5 mm$ into the brain and (b) data from two tumor-bearing rats with the source fiber fixed $2 mm$ into the brain. Measurement series were recorded along tracks positioned parallel to the source fiber at $x = \pm 2 mm$ $(◻)$ and $x = \pm 4 mm$ $(엯)$ . The thick solid lines represent diffusion theory predictions [Eqs. 1 to 4] fitted to the measured fluence rates, omitting measurements made less than $2 mm$ from the skull. Correspondingly, the dotted lines are the best fit of the Monte Carlo model. The fluorescent probes were somewhat fragile, and therefore three measurement series in (a) and one (in part) in (b) are absent. The data in the diagrams are corrected using the method described in Sec. 2.2.2.

Results of measurements from two tumor-bearing rats with fibers positioned 1 to $2 mm$ above the tumor centers are shown in Fig. 3b. The recorded fluence rates were significantly lower than in the tumor-free brains, particularly in the area closest to the fiber tip where the values were reduced by a factor of 2. Furthermore, the light distribution was shifted about one millimeter away from the fiber tip (i.e., $z$ direction in Fig. 3), suggesting that the tumor tissue in front of the fiber tip had a lower scattering coefficient than normal brain. The diffusion theory model [Eq. 3], with tumor radius $R = 2.5 mm$ and the already mentioned scattering and absorption coefficients from tumor-free animals, was fitted to the measured fluence rates. As we see in Fig. 3b there is good agreement between the model and the experimental data. The best fit was found for $μ_{a, tumor} = 1.46 {cm}^{- 1}$ (standard error $0.094 {cm}^{- 1}$ ) and $μ_{s, tumor}^{'} = 5.45 {cm}^{- 1}$ (standard error $0.44 {cm}^{- 1}$ ). This corresponds to a light penetration depth of $0.181 cm$ [Eq. 2].

3.1.2.

Fluence rates estimated by Monte Carlo simulation

As expected, in media where scattering dominates absorption, the distributions calculated by Monte Carlo simulation were closely related to the diffusion theory model. For tumor-free rat brain, the absorption and scattering coefficients yielding the best least-squares fit to the measured fluence rates did not deviate from the diffusion model estimates (i.e., the deviation was less than the 2% optimization accuracy described in Sec. 2.2.4). At positions along the measurement tracks (Fig. 2), the Monte Carlo model predicted fluence rates almost identical to diffusion theory. Likewise, as expected when diffusion theory is valid, variation of the scattering anisotropy had no significant effect. In general, major fluence rate discrepancies between Monte Carlo simulations and diffusion theory were seen exclusively at positions within $0.5 mm$ from the source fiber tip, where no experimental data were recorded.

More pronounced differences between the diffusion model and Monte Carlo simulations were observed in tumor-bearing animals. The tumor was modeled as a $6.5 \times 4.5 - mm$ ellipsoid centered $1.5 mm$ below the fiber tip, embedded in normal brain with the scattering and absorption coefficients estimated from measurements in tumor-free animals. The best least-squares fit to the measured fluence rates was found for $μ_{a, tumor} = 1.39 {cm}^{- 1}$ and $μ_{s, tumor}^{'} = 7.3 {cm}^{- 1}$ , and is shown in Fig. 3b. These values deviate from the diffusion theory results already determined and correspond to a shorter light penetration depth of $0.166 cm$ . A map of the light distributions predicted by Monte Carlo simulation and diffusion theory for a given tumor shape and set of optical properties is shown in Fig. 4 . Such comparisons showed that, although the models yielded similar results in the normal brain at distances in excess of 3 to $4 mm$ from the fiber tip, the diffusion model overestimated the fluence rate in large parts of the tumor by 30 to $50 %$ compared to the Monte Carlo model. The scattering anisotropy did not have much effect on the Monte Carlo model. Variation of the anisotropy value $g$ within the range 0.8 and 0.99 shifted the isodose contours shown in Fig. 4 by less than $0.1 mm$ .

Fig. 4

Distribution of light in tumor-bearing rat brain. The contours show fluence rate $ϕ$ relative to the radiant power $P$ emitted from a bare fiber tip (unit: inverse centimeters squared). The solid lines are calculated by Monte Carlo simulations for an ellipsoid $4.5 \times 6.5 - mm$ tumor geometry and optical properties $μ_{a} = 0.62 {cm}^{- 1}$ , $μ_{s}^{'} = 31 {cm}^{- 1}$ , $μ_{a, tumor} = 1.6 {cm}^{- 1}$ , $μ_{s, tumor}^{'} = 7.7 {cm}^{- 1}$ , and $g = 0.90$ . The dotted lines are the corresponding contours according to the diffusion theory model [Eqs. 3, 4] with the same optical properties and the tumor approximated as a $2.5 - mm$ -radius sphere.

Monte Carlo simulations featuring 100% reflecting or absorbing surfaces at the positions of the skull were performed to investigate finite volume effects. As shown in Fig. 5 , the fluence rate at the measured positions was altered by less than 5% as long as sites closer than $2 mm$ to the surface were omitted. The skull and its surrounding soft tissues have optical properties of similar order of magnitude as the brain,¹⁸ and should represent far lower perturbations than the 100% reflecting or absorbing surfaces.

Fig. 5

Fluence rates calculated by the Monte Carlo model featuring worst-case finite volume effects. The lines represent 100% reflecting (—) and absorbing (- - -) surfaces added at $z = - 4 mm$ to the geometry presented in Fig. 3. The dotted lines represent the ideal infinite geometry.

3.2.

Measured Temperatures in the Rat Brain

Temperatures around the source fiber were measured in two tumor-free rats. Background measurements made in the absence of laser light showed a significant gradient, with temperatures starting at 2.1 to $3.5 K$ below rectal body temperature (33.1 to $35.1 ° C$ ) in the most superficial layers of the brain, and reaching rectal body temperature 7 to $9 mm$ into the brain. Interstitial delivery of $100 mW$ of light caused elevated temperatures detectable more than $6 mm$ from the source fiber tip. At positions $2 mm$ lateral to the fiber tip the increase was 3.5 to $4.0 K$ at steady state. The variation within measurements made at equivalent positions was mostly less than $0.5 K$ . As we see in Fig. 6 , the steady state model described by Eq. 6 was in good agreement with measured temperature increases. The best fit was found for a thermal penetration depth of $δ_{v} = 0.60 cm$ (standard error $0.02 cm$ ) if $δ$ was fixed at $0.143 cm$ as estimated from the fluence rate measurements (see Sec. 3.1.1). Similar values of $δ_{v} = 0.58 cm$ and $δ = 0.137 cm$ were found if both variables were used as free parameters. Extrapolation of Eq. 6 using these parameters yields a temperature increase of $8.5 K$ at the fiber tip.

Fig. 6

Spatial distribution of steady state temperature increase during light exposure. The geometry and coordinates are illustrated in Fig. 2. The symbols show the difference between the temperature measured with $100 mW$ of input power and the local temperature measured with no light input. Measurement series were recorded in two rats along tracks positioned parallel to the source fiber at $x = \pm 2 mm$ $(◻)$ and $x = \pm 4 mm$ $(엯)$ . The source fibers were fixed $4 mm$ into the brain. The thick solid lines are the steady state bioheat model [Eq. 6] fitted to the measured temperature.

An alternative estimate of the thermal penetration depth was found by fitting the time dependent bioheat Eq. 5 to the temporal temperature measurements recorded at fixed positions $2 mm$ lateral to the fiber tip. These temporal temperature changes following onset and termination of light input are shown in Fig. 7 . Numerical solutions of the bioheat equation [Eq. 5] were in good agreement with the data, particularly for the temperature decrease following light termination [Fig. 7b], and the best fit was found for a thermal penetration depth $δ_{v} = 0.51 cm$ .

Fig. 7

Temperature change after (a) onset and (b) termination of $100 mW$ of light input. The diagrams show the increase above steady state temperature with no light input. The temperature was recorded at perpendicular positions $2 mm$ lateral to the fiber tip ( $x = \pm 2 mm$ and $z = 0$ in Fig. 2) in two rats. Each solid line shows data from one measurement series. The thick dashed lines are the bioheat model [Eq. 5] fitted to all measurements.

4. Discussion

The in vivo measurements yielded the tissue optical parameters: $μ_{a} = 0.57 {cm}^{- 1}$ , $μ_{s}^{'} = 28 {cm}^{- 1}$ , and $δ = 0.14 cm$ for $632 - nm$ light in rat brain tissue. This is in good agreement with other in vivo measurements of $δ$ in mammalian brains, which have been found ^{13, 14, 15, 16, 17} to be within the range 0.10 to $0.27 cm$ . To our knowledge, in vivo measurements of the absorption and scattering coefficients at this wavelength have not been reported, but values determined from extracted tissues have been ^{18, 19, 32, 37, 38, 39, 40} within the ranges $μ_{a} = 0.20$ to $2.7 {cm}^{- 1}$ and $μ_{s}^{'} = 6.4$ to $100 {cm}^{- 1}$ . Unlike absorption, scattering does not depend strongly on wavelength: $μ_{s}^{'}$ has been reported ^{19, 38, 39, 40} to decrease by less than 20% as the wavelength increases from $632 nm$ and into the near-IR, where in vivo measurements have been performed with diffuse reflectance probes. In the human brain Bevilacqua have found⁴¹ $μ_{s}^{'} = 10 {cm}^{- 1}$ for gray matter and $μ_{s}^{'} = 14 {cm}^{- 1}$ for white matter at $674 nm$ . In better agreement with this study, higher values were reported for rat brain by Johns, ⁴² who found $μ_{s}^{'} = 12$ to $25 {cm}^{- 1}$ in gray matter and $μ_{s}^{'} = 35$ to $60 {cm}^{- 1}$ in white matter at $750 nm$ . The white matter structures in the rat brain are typically less than $1 mm$ thick and therefore constitute just 7 to 10% of the volume interrogated by the light fields in this study.⁴³ Hence, the presented estimates are best interpreted as optical properties of gray matter. The estimated $δ$ correspond to a 1000-fold decrease in fluence rate over a $1 - cm$ distance through rat gray matter. However, since most tumors in patients arise and infiltrate in white matter the high attenuation of light at $632 nm$ in gray matter does not necessarily represent a clinically imposing problem. The measurements by Muller and Wilson¹³ indicate a more favorable penetration in actual resection margins where the tissue is a mixture of brain and tumor tissue.

For tumor tissue, there was some discrepancy between the diffusion and Monte Carlo models. The Monte Carlo estimates are considered to be more accurate, since the exact geometry (tumor shape, fiber position, and the nonisotropic source) can easily be represented, and since discrepancies due to limitations of the diffusion approximation are expected for the relatively small difference between $μ_{a, tumor}$ and $μ_{s, tumor}^{'}$ . The optical properties were estimated from only two animals, and this makes the values less reliable than those measured in tumor-free brain. Still, both the observed displacement along the $z$ axis and the average reduction in fluence rate (as seen in Fig. 3) were distinct compared to the interanimal variability. This suggests that there are differences in the optical properties of tumor tissue and normal brain. As illustrated in Fig. 4, the estimates can be useful when evaluating light dosimetry for the ${BT}_{4} C$ tumor model. However, measurements in additional animals will be required for a more accurate determination of the optical properties of glioma tissue.

The large number of measurements should rule out the possibility that small structures, like blood vessels or ventricles, affected the predicted light distributions. Still, inhomogeneities may have been the major cause of the twofold variability seen in Fig. 3. The most noteworthy deviation between the measured and the modeled fluence rate is seen in Fig. 3 for the region $4 mm$ lateral to the source fiber tip in both tumor-free [Fig. 3a] and tumor-bearing [Fig. 3b] brains. The predicted fluence rates (the lower solid lines in the diagrams) around the peak positions seem low compared to the measurements, implying an overestimation of light attenuation. There is no obvious explanation for this discrepancy, but it could be related to local variations in optical properties.

The sources of error are complex and therefore it is difficult to evaluate the uncertainty in the estimates of $μ_{a}$ , $μ_{s}^{'}$ , and $δ$ . Calibration of the fluence rate probes can be a major source of error. Ideally, the probes should be calibrated in a medium similar to rat brain, rather than in a clear solution, because the probe may perturb the fluence rate of its surroundings in a scattering- and absorption-dependent manner.⁴⁴ In the presented case, such perturbation was probably minor due to the small probe dimensions. Furthermore, calculations using the multilayer Monte Carlo software⁴⁵ MCML indicated that infinite layers of ruby or silica, with thicknesses comparable to the respective dimensions in the probes, would alter the fluence rate in adjacent brain tissue by less than 5%. The small ruby and silica structures in the probes would without doubt perturb far less than infinite layers. As described in Sec. 2.2.2, the black fiber tubing close to the fluorescent sphere affected the measurements, but the applied correction factor based on Monte Carlo simulations essentially eliminated this artifact. Additional simulations featuring relevant variations in the absorption of the tubing and the light penetration depth in rat brain suggested that those factors do not alter the correction factor significantly. It is clear from Eq. 1 that both $μ_{s}^{'}$ and $μ_{a}$ are sensitive to probe calibration errors, but fortunately they are estimated by utilizing both the absolute magnitude of the fluence rate [Eq. 1] and the displacement of the light distribution along the $z$ axis [Eq. 4]. The fact that the model fits the data in terms of both absolute fluence rate and $z$ offset simultaneously [Fig. 3a] strengthens the reliability of the estimated $μ_{s}^{'}$ and $μ_{a}$ .

The measured temperature distribution shown in Fig. 6 is in good agreement with the predictions of the steady state bioheat model [Eq. 6]. The model yielded similar thermal penetration depths $(δ_{v})$ when fitted to the experimental data in both spatial (Fig. 6) and time domains (Fig. 7). This implies that this simple model is adequate for describing the temperature increase during interstitial light delivery with a bare fiber tip. Furthermore, sound estimates of both $δ$ and $δ_{v}$ were found when $δ$ as used as a free parameter instead of being fixed at the value found from the fluence rate measurements. The thermal penetration depth is directly related to blood flow (as defined in Sec. 2.3.2). The measured thermal penetration depth of 0.51 to $0.60 cm$ corresponds to a blood flow of 0.23 to $0.32 mL \min^{- 1} g^{- 1}$ , if a heat capacity of $3.7 kJ {kg}^{- 1} K^{- 1}$ and a density of $1.03 g {mL}^{- 1}$ are assumed.⁴⁶ This flow rate is 2 to 4 times less than that reported elsewhere for rat gray matter,⁴⁷ which means that both the gradient in the spatial measurements and the rate of change in the temporal measurements were lower than one would expect from the literature. The reduction can, at least in part, be explained by well-known effects of the applied anesthesia on cerebral perfusion.⁴⁸ In addition, finite volume effects from ambient air, bone, muscle, and skin, which is less perfused,^{49, 50} may have elevated the temperature in the brain, resulting in an overestimation of the thermal penetration depth and hence, underestimation of perfusion. The finite volume effect of the top tissue-to-air surface is also a plausible cause for the distinct asymmetry seen in Fig. 6, where the temperature is elevated closest to the top surface (i.e., for $z < 0$ ). The initial tests of direct light absorption in the thermocouple (Sec. 2.3.1) suggests that this artifact increased the temperature readings closest to the source by $0.2 K$ , which is a minor aberration compared to the observed spatial and temporal temperature spans.

In conclusion, the distributions of light and heat during interstitial light delivery were successfully mapped using in situ measurements and mathematical modeling. Optical properties were derived, and can be used to predict fluence rate distributions during PDT in rat brain. The fluence rate distribution was found to be predictable, and could be estimated by diffusion theory in tumor-free brains and by Monte Carlo simulations in tumor-bearing brains. The presence of a tumor altered the light distribution significantly, and the models suggested that the tumor had higher absorption and lower scattering than normal brain. Rat brain was found to have absorption and reduced scattering coefficients of $0.57$ and $28 {cm}^{- 1}$ , respectively. These coefficients have, to our knowledge, not been reported for comparable wavelengths in vivo, but the resulting penetration depth of $0.14 cm$ is in agreement with numerous in vivo studies in the literature. The temperature measurements showed that the extent of heating during interstitial light delivery could be modeled with a regular point source diffusion model. The findings of the study make it feasible to predict the fluences, fluence rates, and temperatures achieved during intracranial photodynamic therapy.

Acknowledgments

We would like to thank Jørn Iversen for his work at the instrument workshop; Roland Bays and Lise Lyngsnes Randeberg for useful tips regarding the fluence rate probes and the temperature measurements, respectively; and Dag Sørensen at the Centre for Comparative Medicine at Rikshospitalet for help with the animal work. The ${BT}_{4} C$ cell line was provided by Rolf Bjerkvig at the University of Bergen.

References

1.

L. Lilge, M. C. Olivo, S. W. Schatz, J. A. MaGuire, M. S. Patterson, and B. C. Wilson, “The sensitivity of normal brain and intracranially implanted VX2 tumour to interstitial photodynamic therapy,” Br. J. Cancer, 73 332 –343 (1996). https://doi.org/10.1016/0893-133X(95)00237-8 0007-0920 Google Scholar

2.

J. Lobel, I. J. MacDonald, M. J. Ciesielski, T. Barone, W. R. Potter, J. Pollina, R. J. Plunkett, R. A. Fenstermaker, and T. J. Dougherty, “2-[1-hexyloxyethyl]-2-devinyl pyropheophorbide-a (HPPH) in a nude rat glioma model: implications for photodynamic therapy,” Lasers Surg. Med., 29 397 –405 (2001). https://doi.org/10.1002/lsm.10001 0196-8092 Google Scholar

3.

S. K. Bisland, L. Lilge, A. Lin, R. Rusnov, and B. C. Wilson, “Metronomic photodynamic therapy as a new paradigm for photodynamic therapy: rationale and preclinical evaluation of technical feasibility for treating malignant brain tumors,” Photochem. Photobiol., 80 22 –30 (2004). https://doi.org/10.1562/2004-03-05-RA-100.1 0031-8655 Google Scholar

4.

E. Angell-Petersen, S. Spetalen, S. J. Madsen, C.-H. Sun, Q. Peng, S. W. Carper, M. Sioud, and H. Hirschberg, “Influence of light fluence rate on the effects of photodynamic therapy in an orthotopic rat glioma model,” J. Neurosurg., 104 109 –117 (2006). 0022-3085 Google Scholar

5.

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, D. Kessel, M. Korbelik, J. Moan, and Q. Peng, “Photodynamic therapy,” J. Natl. Cancer Inst., 90 889 –905 (1998). https://doi.org/10.1093/jnci/90.12.889 0027-8874 Google Scholar

6.

K. E. Wallner, J. H. Galicich, G. Krol, E. Arbit, and M. G. Malkin, “Patterns of failure following treatment for glioblastoma multiforme and anaplastic astrocytoma,” Int. J. Radiat. Oncol., Biol., Phys., 16 1405 –1409 (1989). 0360-3016 Google Scholar

7.

E. A. Popovic, A. H. Kaye, and J. S. Hill, “Photodynamic therapy of brain tumors,” Semin Surg. Oncol., 11 335 –345 (1995). 8756-0437 Google Scholar

8.

P. J. Muller and B. C. Wilson, “Photodynamic therapy for recurrent supratentorial gliomas,” Semin Surg. Oncol., 11 346 –354 (1995). 8756-0437 Google Scholar

9.

P. J. Muller and B. C. Wilson, “Photodynamic therapy for malignant newly diagnosed supratentorial gliomas,” J. Clin. Laser Med. Surg., 14 263 –270 (1996). 1044-5471 Google Scholar

10.

S. Krishnamurthy, S. K. Powers, P. Witmer, and T. Brown, “Optimal light dose for interstitial photodynamic therapy in treatment for malignant brain tumors,” Lasers Surg. Med., 27 224 –234 (2000). https://doi.org/10.1002/1096-9101(2000)27:3<224::AID-LSM4>3.0.CO;2-# 0196-8092 Google Scholar

11.

M. A. Rosenthal, B. Kavar, S. Uren, and A. H. Kaye, “Promising survival in patients with high-grade gliomas following therapy with a novel boronated porphyrin,” Clin. Neurosci., 10 425 –427 (2003). 1065-6766 Google Scholar

12.

S. S. Stylli, M. Howes, L. MacGregor, P. Rajendra, and A. H. Kaye, “Photodynamic therapy of brain tumours: evaluation of porphyrin uptake versus clinical outcome,” Clin. Neurosci., 11 584 –596 (2004). 1065-6766 Google Scholar

13.

P. J. Muller and B. C. Wilson, “An update on the penetration depth of

630 nm

light in normal and malignant human brain tissue in vivo,” Phys. Med. Biol., 31 1295 –1297 (1986). https://doi.org/10.1088/0031-9155/31/11/012 0031-9155 Google Scholar

14.

B. C. Wilson, W. P. Jeeves, and D. M. Lowe, “In vivo and post mortem measurements of the attenuation spectra of light in mammalian tissues,” Photochem. Photobiol., 42 153 –162 (1985). 0031-8655 Google Scholar

15.

D. R. Doiron, L. O. Svaasand, and A. E. Profio, “Light dosimetry in tissue applications to photoradiation therapy,” Porphyrin Photosensitization, 63 –75 Plenum Press, New York (1983). Google Scholar

16.

S. K. Powers and J. T. Brown, “Light dosimetry in brain tissue: an in vivo model applicable to photodynamic therapy,” Lasers Surg. Med., 6 318 –322 (1986). 0196-8092 Google Scholar

17.

Q. Chen, M. Chopp, L. Madigan, M. O. Dereski, and F. W. Hetzel, “Damage threshold of normal rat brain in photodynamic therapy,” Photochem. Photobiol., 64 163 –167 (1996). 0031-8655 Google Scholar

18.

W. F. Cheong, “Summary of optical properties,” Optical-Thermal Response of Laser-Irradiated Tissue, 275 –303 Plenum Press, New York (1995). Google Scholar

19.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, “Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range,” Phys. Med. Biol., 47 2059 –2073 (2002). https://doi.org/10.1088/0031-9155/47/12/305 0031-9155 Google Scholar

20.

L. Lilge, T. Haw, and B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissue,” Phys. Med. Biol., 38 215 –230 (1993). https://doi.org/10.1088/0031-9155/38/2/001 0031-9155 Google Scholar

21.

R. Bays, G. Wagnieres, D. Robert, J. F. Theumann, A. Vitkin, J. F. Savary, P. Monnier, and H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Lasers Surg. Med., 21 227 –234 (1997). https://doi.org/10.1002/(SICI)1096-9101(1997)21:3<227::AID-LSM2>3.0.CO;2-S 0196-8092 Google Scholar

22.

T. C. Zhu, A. Dimofte, J. C. Finlay, D. Stripp, T. Busch, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, “Optical properties of human prostate at

732 nm

measured in mediated photodynamic therapy,” Photochem. Photobiol., 81 96 –105 (2005). https://doi.org/10.1562/2004-06-25-RA-216.1 0031-8655 Google Scholar

23.

S. J. Madsen, E. Angell-Petersen, S. Spetalen, S. W. Carper, S. A. Ziegler, and H. Hirschberg, “Photodynamic therapy of newly implanted glioma cells in the rat brain,” Lasers Surg. Med., 38 540 –548 (2006). https://doi.org/10.1002/lsm.20274 0196-8092 Google Scholar

24.

C. Rehbinder, P. Baneux, D. Forbes, H. van Herck, W. Nicklas, Z. Rugaya, and G. Winkler, “FELASA recommendations for the health monitoring of mouse, rat, hamster, gerbil, guinea pig and rabbit experimental units. Report of the Federation of European Laboratory Animal Science Associations (FELASA) Working Group on Animal Health accepted by the FELASA Board of Management, November 1995,” Lab Anim., 30 193 –208 (1996). 0023-6772 Google Scholar

25.

D. R. Sørensen, T. A. Read, T. Porwol, B. R. Olsen, R. Timpl, T. Sasaki, P. O. Iversen, H. B. Benestad, B. K. Sim, and R. Bjerkvig, “Endostatin reduces vascularization, blood flow, and growth in a rat gliosarcoma,” Neuro.-Oncol., 4 1 –8 (2002). Google Scholar

26.

J. P. Marijnissen and W. M. Star, “Calibration of isotropic light dosimetry probes based on scattering bulbs in clear media,” Phys. Med. Biol., 41 1191 –1208 (1996). https://doi.org/10.1088/0031-9155/41/7/008 0031-9155 Google Scholar

27.

I. Driver, C. P. Lowdell, and D. V. Ash, “In vivo measurement of the optical interaction coefficients of human tumours at

630 nm

,” Phys. Med. Biol., 36 805 –813 (1991). https://doi.org/10.1088/0031-9155/36/6/008 0031-9155 Google Scholar

28.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A, 11 2727 –2741 (1994). 0740-3232 Google Scholar

29.

S. L. Jacques, “Light distributions from point, line and plane sources for photochemical reactions and fluorescence in turbid biological tissues,” Photochem. Photobiol., 67 23 –32 (1998). https://doi.org/10.1562/0031-8655(1998)067<0023:LDFPLA>2.3.CO;2 0031-8655 Google Scholar

30.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE, IS 5 102 –111 (1989). 0277-786X Google Scholar

31.

R. Splinter, W. F. Cheong, M. J. van Gemert, and A. J. Welch, “In vitro optical properties of human and canine brain and urinary bladder tissues at

633 nm

,” Lasers Surg. Med., 9 37 –41 (1989). 0196-8092 Google Scholar

32.

B. C. Wilson, M. S. Patterson, and D. M. Burns, “Effect of photsensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci., 1 235 –244 (1986). 0268-8921 Google Scholar

33.

H. H. Pennes, “Analysis of tissue and arterial blood temperature in the resting human forearm,” J. Appl. Physiol., 1 93 –122 (1948). 0021-8987 Google Scholar

34.

T. E. Cooper and G. J. Trezek, “Correlation of thermal properties of some human tissue with water content,” Aerosp. Med., 42 24 –27 (1971). 0001-9402 Google Scholar

35.

W. J. Minkowycz, E. M. Sparrow, G. E. Schneider, and R. H. Pletcher, Handbook of Numerical Heat Transfer, Wiley, New York (1988). Google Scholar

36.

E. Angell-Petersen, D. R. Sørensen, S. J. Madsen, and H. Hirschberg, “Interstitial light application for photodynamic therapy in a rat brain tumor model,” Proc. SPIE, 5312 415 –423 (2004). 0277-786X Google Scholar

37.

S. T. Flock, B. C. Wilson, and M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at

633 nm

,” Med. Phys., 14 835 –841 (1987). https://doi.org/10.1118/1.596010 0094-2405 Google Scholar

38.

J. L. Karagiannes, Z. Zhang, and L. I. Grossweiner, “Applications of the 1D diffusion approximation to the optics of tissues and tissue phantoms,” Appl. Opt., 28 2311 –2317 (1989). 0003-6935 Google Scholar

39.

H. J. C. M. Sterenborg, M. J. van Gemert, W. Kamphorst, J. G. Wolbers, and W. Hogervorst, “The spectral dependence of the optical properties of human brain,” Lasers Med. Sci., 4 221 –227 (1989). 0268-8921 Google Scholar

40.

P. van der Zee, M. Essenpreis, and D. T. Delpy, “Optical properties of brain tissue,” Proc. SPIE, 1888 454 –465 (1993). https://doi.org/10.1117/12.154665 0277-786X Google Scholar

41.

F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, and C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt., 38 4939 –4950 (1999). 0003-6935 Google Scholar

42.

M. Johns, C. A. Giller, D. C. German, and H. L. Liu, “Determination of reduced scattering coefficient of biological tissue from a needle-like probe,” Opt. Express, 13 4828 –4842 (2005). https://doi.org/10.1364/OPEX.13.004828 1094-4087 Google Scholar

43.

G. Paxinos and C. Watson, The Rat Brain in Stereotaxic Coordinates, Academic Press, San Diego (1998). Google Scholar

44.

L. D. Lilge and B. C. Wilson, “Accuracy of interstitial measurements of absolute light fluence rate in the determination of tissue optical properties,” Proc. SPIE, 1882 291 –304 (1993). https://doi.org/10.1117/12.147668 0277-786X Google Scholar

45.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed., 47 131 –146 (1995). https://doi.org/10.1016/0169-2607(95)01640-F 0169-2607 Google Scholar

46.

F. A. Duck, Physical Properties of Tissue: A Comprehensive Reference Book, Academic Press, London (1990). Google Scholar

47.

R. F. Tuma, G. L. Irion, U. S. Vasthare, and L. A. Heinel, “Age-related changes in regional blood flow in the rat,” Am. J. Physiol., 249 H485 –H491 (1985). 0002-9513 Google Scholar

48.

K. S. Hendrich, P. M. Kochanek, J. A. Melick, J. K. Schiding, K. D. Statler, D. S. Williams, D. W. Marion, and C. Ho, “Cerebral perfusion during anesthesia with fentanyl, isoflurane, or pentobarbital in normal rats studied by arterial spin-labeled MRI,” Magn. Reson. Med., 46 202 –206 (2001). https://doi.org/10.1002/mrm.1178 0740-3194 Google Scholar

49.

A. Schoutens, P. Bergmann, and M. Verhas, “Bone blood flow measured by 85 Sr microspheres and bone seeker clearances in the rat,” Am. J. Physiol., 236 H1 –H6 (1979). 0002-9513 Google Scholar

50.

W. C. Seyde, L. McGowan, N. Lund, B. Duling, and D. E. Longnecker, “Effects of anesthetics on regional hemodynamics in normovolemic and hemorrhaged rats,” Am. J. Physiol., 249 H164 –H173 (1985). 0002-9513 Google Scholar

Notes

³Available online at http://omlc.ogi.edu/software/mc/small_mc.c.

Citation Download Citation

Even Angell-Petersen, Henry Hirschberg M.D., and Steen J. Madsen "Determination of fluence rate and temperature distributions in the rat brain; implications for photodynamic therapy," Journal of Biomedical Optics 12(1), 014003 (1 January 2007). https://doi.org/10.1117/1.2709882

Published: 1 January 2007

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KEYWORDS

Brain

Tumors

Temperature metrology

Diffusion

Monte Carlo methods

Scattering

Absorption

1.

Introduction

2.

Materials and Methods

2.1.

Animal Care

2.2.

Measurement and Modeling of Light Distributions

2.2.1.

Isotropic probe for in situ fluence rate measurements

Fig. 1

2.2.2.

In vivo measurements of intracranial fluence rate distribution

Fig. 2

2.2.3.

Diffusion models of the fluence rate distribution

Eq. 1

Eq. 2

Eq. 3

Eq. 4

2.2.4.

Monte Carlo model of the fluence rate distribution

2.3.

Measurement and Modeling of Temperature Distributions

2.3.1.

In vivo measurements of intracranial temperature distributions

2.3.2.

Bioheat model of the temperature distribution

Eq. 5

Eq. 6

3.

Results

3.1.

Light Distribution

3.1.1.

Measured fluence rates in the rat brain

Fig. 3

3.1.2.

Fluence rates estimated by Monte Carlo simulation

Fig. 4

Fig. 5

3.2.

Measured Temperatures in the Rat Brain

Fig. 6

Fig. 7

4.

Discussion

Acknowledgments

References

Notes

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