ultrafast laser pulse train radiation …coe papers/jp_88.pdf · hamel’s superposition theorem to...

Heat Transfer Research 46(9), 861–879 (2015)

1064-2285/15/$35.00 © 2015 by Begell House, Inc. 861

1. INTRODUCTION

The advent of ultrafast lasers has opened the window for many emerging applications. Specifi c applications in biomedicine include optical tomography (Yamada, 1995), tissue ablation (Sajjadi et al., 2013), tumor diagnostics (Bhowmik et al., 2014), thermal treat-ment (Jiao and Guo, 2009), to name a few. Recently, an excellent review on advances in the computational modeling of ultrafast radiative transfer was provided by Guo and Hunter (2013), in which advantages and disadvantages of various numerical solution methodologies have been summarized. The importance of studying the interaction of ul-trashort-pulsed lasers with biological tissues and the corresponding mathematical mod-eling of possible thermal damage was also reviewed by Sajjadi et al. (2013). Some prior numerical investigations relevant to the present study are reviewed as follows.

ULTRAFAST LASER PULSE TRAIN RADIATION TRANSFER IN A SCATTERING-ABSORBING 3D MEDIUM WITH AN INHOMOGENEITY

Masato Akamatsu1,* & Zhixiong Guo2

1Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa, Yamagata 992-8510, Japan2Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA*Address all correspondence to Masato Akamatsu Email: [email protected]

In this study, we continue investigating the ultrafast laser pulse train irradiation of inhomo-geneous media, focusing on diff erent degrees of inhomogeneity in a base scatt ering-absorbing medium. We used the transient discrete-ordinates method to acquire basic solutions of radiation transfer in the 3D inhomogeneous media subjected to a unit step pulse and then adopted the Du-hamel’s superposition theorem to construct series solutions subjected to a pulse of diff erent pulse widths or a pulse train consisting of many such pulses. The eff ects of scatt ering albedo, inhomoge-neity size and location, pulse width and interval, and detector position on the temporal refl ectance and transmitt ance signals are characterized and revealed.

KEY WORDS: discrete-ordinates method, superposition, short-pulse train, radiative transfer, inhomogeneity, heterogeneous medium

Heat Transfer Research

862 Akamatsu & Guo

In early studies, many researchers adopted the simplifi ed diffusion approximation to model light transport in turbid tissues (Yamada, 1995). The diffusion approxima-tion cannot describe the physical condition at the boundary such as light refl ection and refraction, neither it is accurate when absorption is strong inside the medium (Guo et al., 2003). The transport of short light pulses through one-dimensional (1D) scattering-absorbing media using different approximate mathematical models, includ-ing the discrete-ordinates method (DOM), was examined by Mitra and Kumar (1999). The transient DOM for ultrashort-pulsed laser radiation transfer in 2D anisotropically scattering, absorbing, and emitting media with properties similar to those of the bio-logical tissue was fi rst formulated by Guo and Kumar (2001). The same authors later extended their transient DOM method to the 3D geometry (Guo and Kumar, 2002). The Fresnel effect and the heterogeneous effect under specularly and/or diffusely re-fl ecting boundary conditions were further incorporated by Guo and Kim (2003).

The transport of a pulse train through a 2D rectangular participating medium con-sisting of a local inhomogeneity, by means of the fi nite-volume method (FVM) was numerically investigated by Muthukumaran and Mishra (2008). A complete thermal analysis combining the pulse train ultrafast radiative transfer and transient Pennes’ bioheat transfer in a model with a small tumor embedded in the skin tissue, using the transient DOM with an S10 scheme for radiation transfer and the FVM for bioheat transfer was developed by Jiao and Guo (2009). Recently, the transient heating on superfi cial tumors using the Monte Carlo method for solving the 3D radiative transfer problem was also investigated by Randrianalisoa et al. (2014). A numerical investiga-tion of the thermal response of the biological tissue phantoms during laser-based pho-

NOMENCLATURE

c speed of light, m/s x, y, z Cartesian coordinates, mdc width of collimated laser

Greek symbols sheet, mL side length of a cube, σa absorption coeffi cient, m 1/mN angular discrete order in SN σe extinction coeffi cient, 1/m approximation σs effective scattering t time, s coeffi cient, 1/mt* nondimensional time, ω scattering albedo = ct/L

Superscriptstd pulse interval, stp pulse width, s * dimensionless quantity

863Pulse Train Radiation Transfer in a 3D Medium with Inhomogeneity

Volume 46, Number 9, 2015

to-thermal therapy for destroying cancerous/abnormal cells was conducted by Kumar and Srivastava (2014). The presence of a tumor inside a breast tissue and its charac-teristics (e.g., location, size, and properties) with the skin surface temperature were estimated by Das and Mishra (2013). The temporal variations of transmittance and refl ectance from a normal skin and a malignant human skin subjected to a low-power short-pulse laser using the modifi ed DOM were analyzed by Bhowmik et al. (2014).

Duhamel’s superposition theorem was fi rst introduced into the analysis of radiation transfer by Guo and Kumar (2002) to analyze the pulse feature of ultrafast radiative transfer, in which a basic solution to pulse irradiation was fi rst estimated and then the superposition law was used to construct the responses of various pulses. Later, the DOM and Duhamel’s superposition theorem were applied by Akamatsu and Guo (2011) to scrutinize the transient characteristics of ultrafast radiative heat transfer in a homogeneous participating medium subjected to a diffuse short square pulse train. The transient characteristics of ultrafast laser radiative transfer in a homogeneous par-ticipating medium subjected to collimated irradiation of various square pulse trains by the DOM and Duhamel’s superposition theorem were clarifi ed by Akamatsu and Guo (2013b). Ultrafast radiative transfer characteristics in multilayer inhomogeneous media subjected to a collimated short square pulse train by the DOM and Duhamel's superposition theorem were reported by Akamatsu and Guo (2014). Akamatsu and Guo (2013a) also elucidated that Duhamel’s superposition theorem provides better accuracy and computational effi ciency than direct pulse simulation for the radiative transfer analysis of multiple pulses.

The investigation and elucidation of ultrafast radiative transfer characteristics in a medium with an embedded inhomogeneity subjected to a pulse train are very import-ant and practical to advance the applications of ultrafast laser technology in the bio-medical fi eld. Examples include laser cancer diagnosis, photodynamic therapy, laser hair removal, laser-induced interstitial thermotherapy, etc. To this end, we carried out a complete transient 3D numerical computations in the present study to understand the pulse train radiative transfer characteristics in scattering-absorbing media with an embedded inhomogeneity. The embedded medium may also be assumed to have the same optical properties like the surrounding medium to form a homogeneous medi-um for comparison purposes. Different pulse forms are considered, including a single collimated unit step pulse, a single collimated square pulse, and a train of collimated square pulses. We solved the problem by combining the DOM and Duhamel’s super-position theorem. The effects of the pulse width, pulse interval, scattering albedo, de-tecting position, and inhomogeneity size and location on the refl ected and transmitted temporal signals were analyzed.

2. TISSUE MODELS CONSIDERED

The 3D geometry of the models considered in the present numerical computations is shown in Fig. 1a. The side length of the cubic medium is L, which is assumed to be


864 Akamatsu & Guo

10 mm. The scattering albedo (ω = σs/σe) of the base medium is set at either ω1 = 0.9 (highly scattering) or ω1 = 0.1 (highly absorbing). An inhomogeneity with scattering albedo 2ω varying from 0.1 to 0.9 is embedded inside the base medium. When 1ω is equal to 2ω , the model tissue is homogeneous, i.e., no presence of inhomogeneity. When 1ω differs from 2ω , the model tissue is embedded with an heterogeneous me-dium (such as a tumor). The extinction coeffi cient σe (= σ + σa s) of both the base and embedded media is the same, equal to 1 mm–1. The tissue scattering is reduced to isotropic (Guo and Kumar, 2000), since an effective scattering coeffi cient is used.

In the case of 2 0.1ω = with 1 0.9ω = , the embedded medium has greater ab-sorption than the base medium. On the other hand, when ω2 = 0.9 with ω1 = 0.1, the embedded medium has greater scattering than the surrounding medium. In general,

FIG. 1: 3D geometry embedded with an inhomogeneity (a) and the locations of detectors and locations and sizes of inhomogeneity projected in the y–z plane (b–d)



the biological tissue is highly scattering and very weakly absorbing against near-infra-red light, but a heterogeneous medium such as that of a malignant tumor is relatively highly absorbing and low-scattering. In the present numerical computations, we ne-glected the mismatch of refractive indices between two different media and both the media are assumed to be cold (i.e., medium emission is negligible).

The collimated laser square sheet with a width of dc, and dc = L/49 is normally incident at the central spot in the plane at x = 0. Three pulse forms are considered: a single collimated unit step pulse, a single collimated square pulse, and a pulse train consisting of fi ve square pulses with pulse width tp and time interval td between two successive pulses. Three different inhomogeneity scenarios are also considered. In Fig. 1b, a small inhomogeneity with a side length of L/7 is embedded at the center of the cubic medium. The refl ected signals are detected at locations 2–7 on the incident wall of x = 0, and the transmitted signals are detected at locations 1–7 on the opposite wall of x = L. To examine the effects of the location and size of the embedded inho-mogeneity, Fig. 1c considers an inhomogeneity of the same size like in Fig. 1b, but placed off the center, while Fig. 1d considers a larger inhomogeneity of side length 21L/49 placed at the cube center.

The governing equations and numerical schemes were well documented in our pri-or publications (Akamatsu and Guo, 2011, 2014) and thus, the details are not repeated here. The present method was previously intensively verifi ed by comparison with ex-isting published results including the Monte Carlo simulation in a variety of exempla-ry problems in 2- and 3D systems (Guo and Kumar, 2001, 2002). The present study continues our previous study on inhomogeneous media (Akamatsu and Guo, 2014), with focusing (and difference) on the inclusion of an inhomogeneity to mimic the sit-uation of a tumor surrounded by normal tissues. It is of practical importance in cancer optical diagnostics and imaging, and in photodynamic therapy.

3. RESULTS AND DISCUSSION

Figure 2 shows the temporal profi les of the refl ectance detected by detectors 2–7 on the incident surface at x = 0 of various situations when the small inhonogeneity is placed in the cube center (Fig. 1b). The incident laser pulse was a single collimated unit step pulse. The results were computed by the transient DOM with an S12 scheme. Because of the location symmetry, the signals in detectors 2–5 are identical. Signals at detectors 6 and 7 are different, with the strongest signal being at detector 6 and weakest at detector 7. It is seen that the refl ectance signals are not affected by the inhomoge-neity scattering albedo, but are determined by the base medium scattering albedo. The signal strength for the highly scattering base medium (ω1 = 0.9) is one to two orders of magnitude greater than that for the highly absorbing base medium (ω1 = 0.1).

Correspondingly, Fig. 3 shows the refl ectance results when the small inhomogene-ity is placed off the central position (see Fig. 1c). At all detector positions for either

1 0.1ω = or 1 0.9ω = , there is no visible difference between Figs. 2 and 3.


866 Akamatsu & Guo

FIG. 2: Temporal profi les of the refl ectance detected by detectors 2–7 when the small inhomo-geneity is located at the cube center: a single collimated unit step pulse: a–c) ω1 = 0.9; d–f) ω1 = 0.1

(a)

(b)

(c)

(d)

(e)

(f)



FIG. 3: Temporal profi les of the refl ectance detected by detectors 2–7 when the small in-homogeneity is located at the upper region: a single collimated unit step pulse: a–c) ω1 = 0.9; d–f) ω1 = 0.1

(a)

(b)

(c)

(d)

(e)

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868 Akamatsu & Guo

The corresponding refl ectance responses for the case with a large inhomogeneity at the central region (Fig. 1d) are displayed in Fig. 4. Apparently the scattering albedo of the large inhomogeneity strongly affects the signal strength, which increases with increas-ing inhomogeneity scattering albedo. In a highly scattering base medium ( 1 0.9ω = ), for instance, the ratio of the steady-state refl ectance value computed for 2 0.9ω = to that computed for 2 0.1ω = was 1.023 in Fig. 4a, 1.098 in Fig. 4b, and 1.172 in Fig. 4c. In a relatively highly absorbing base medium ( 1 0.1ω = ), the corresponding ratio val-ues were 1.028 in Fig. 4d, 1.452 in Fig. 4e, and 2.698 in Fig. 4f.

Figure 5 shows the temporal profi les of the transmittance detected by detectors 1–7 on the exit surface at x = L for the small inhomogeneity shown in Fig. 1b. The com-putational conditions are same like in Fig. 2. The magnitude of the transmittance de-creased with increase in the distance between the laser-incident point and the detector location. The signals at detectors 2–5 are identical, but differences exist for detectors 1, 6, and 7. For transmittance, even with inclusion of a small inhomogeneity, the in-homogeneity scattering albedo affects the signals. The ratio of the steady-state trans-mittance value computed for 2 0.9ω = to that computed for 2 0.1ω = was 1.014 in Fig. 5a, 1.280 in Fig. 5b, 1.266 in Fig. 5c, 1.237 in Fig. 5d, 1.000 in Fig. 5e, 1.847 in Fig. 5f, 4.003 in Fig. 5g, and 5.918 in Fig. 5h.

The corresponding transmittance responses for the small inhomogeneity placed in the upper region (Fig. 1c) are illustrated in Figs. 6 (ω1 = 0.9) and 7 (ω1 = 0.1). For the tissue model in Fig. 1c, symmetry exists only for detectors 3 and 5, and thus, only the signals from these two detectors are identical. In Fig. 6, the ratio of the steady state transmittance value computed for ω2 = 0.9 to that computed for

2 0.1ω = was 1.006 in Fig. 6a, 1.131 in Fig. 6b, 1.162 in Fig. 6c, 1.101 in Fig. 6d, 1.066 in Fig. 6e, and 1.168 in Fig. 6f. In Fig. 7, the ratio was 1.000 in Fig. 7a, 1.020 in Fig. 7b, 1.117 in Fig. 7c, 1.120 in Fig. 7d, 1.036 in Fig. 7e, and 1.280 in Fig. 7f. From Figs. 2 to 7 it is seen that the refl ectance and transmittance will reach steady-state values within several dimensionless times when the incident laser pulse is an instanta-neous unit step pulse.

Next, the characteristics of a single collimated square pulse with various pulse widths and a pulse train consisting of several square pulses are investigated. Their responses can be reconstructed using Duhamel's superposition theorem based on the results given in Figs. 2–7 for a unit step pulse. Figure 8 shows the temporal profi les of the refl ectance in the medium with a large inhomogeneity subjected to a single col-limated square pulse or a pulse train of fi ve such pulses. The results are reconstructed using the basic solution shown in Fig. 4a. The corresponding fi gures reconstructed using the basic solution shown in Fig. 4e are shown in Fig. 9. The pulse width values were tp* = 0.03, 0.3, and 3.0, respectively. The pulse train intervals were td* = tp*, 10tp*, and 100tp*, respectively. Figures 8a and 8e and Figs. 9a and 9e show the temporal pro-fi les for a single collimated square pulse. The rest of the panels in Figs. 8 and 9 show the temporal profi les for the pulse train.



FIG. 4: Temporal profi les of the refl ectance detected by detectors 2–7 when the large inho-mogeneity is placed at the cube center: a single collimated unit step pulse: a–c) ω= = 0.9; d–f) ω1 = 0.1

(a)

(b)

(c)

(d)

(e)

(f)


870 Akamatsu & Guo

FIG. 5: Temporal profi les of the transmittance detected by detectors 1–7 on the surface at x = L when the small inhomogeneity is placed at the cube center: a single collimated unit step pulse: a–d) ω1 = 0.9; e–h) ω1 = 0.1

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)



FIG. 6: Temporal profi les of the transmittance detected by detectors 1–7 when the small in-homogeneity is placed at the upper region with ω1 = 0.9: a single collimated unit step pulse

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(b)

(c)

(d)

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872 Akamatsu & Guo

FIG. 7: Temporal profi les of the transmittance detected by detectors 1–7 when the small in-homogeneity is placed at the upper region with ω1 = 0.1: a single collimated unit step pulse

(a)

(b)

(c)

(d)

(e)

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As seen in Figs. 8b, 8c, 8f, 8g, and Figs. 9b, 9c, 9f, 9g, the temporal refl ectance signals from the irradiation of a pulse train differ greatly from those of a single pulse, due to the overlap effect. Such an overlap effect of the pulse train depends on the magnitude of the pulse broadening in response to the single square pulse. Therefore, the overlap effect in the refl ectance signals shown in Fig. 8 is greater than those shown in Fig. 9, since the media had a higher scattering property in Fig. 8 than in Fig. 9. When the pulse train interval increased, however, the temporal refl ectance signals produced by the irradiation of the pulse train completely agreed with those produced by the irradiation of a single pulse.

The temporal transmittance signals shown in Figs. 5b and 5f were also reconstruct-ed by Duhamel’s theorem to get the responses to a square pulse or a pulse train. The results are shown in Figs. 10 and 11, respectively. A similar trend was observed in the transmittance signals as in the refl ectance. Namely, with increase in the pulse train interval, the overlap effect of the transmitted pulse train gradually vanished, and the signals with fi ve peaks were detected. It is seen that, when the pulse train interval is long enough compared to the pulse broadening, there is little overlap between the responses of successive pulses, and the characteristics of a pulse train are similar to those of a single pulse.

When we compare the refl ectance signals shown in Fig. 8 to the transmittance sig-nals shown in Fig. 10 for the inhomogeneous media with a relatively high scattering characteristic, the overlap effect in the transmittance is stronger. In addition, the dif-ferences of the peak value between the medium with an embedded heterogeneous medium (fi gures a to d) and the homogeneous participating medium (fi gures e to h) in Figs. 10 and 11 are signifi cant compared to those shown in Figs. 8 and 9. When the inhomogeneity with a relatively high absorbing property is embedded in the highly scattering medium, its peak value in the left column is smaller than that in the right column under the same pulse width and the same pulse train interval, as seen in Fig. 10. On the contrary, when the inhomogeneity with a relatively high scattering property is embedded in a highly absorbing medium, its peak value in the left column is larger than that in the right column under the same pulse width and the same pulse train interval, as seen in Fig. 11.

4. CONCLUSIONS

We carried out 3D numerical computations to examine the transient radiative transfer in scattering-absorbing media with an embedded heterogeneous medium exposed to a collimated short pulse and/or pulse train by combining the transient DOM radiative transfer computation and Duhamel’s superposition theorem. The following conclu-sions can be drawn through comparison studies.

1) The difference in the refl ectance signals between a medium with an inhomogene-ity and a homogeneous medium is not clearly visible when the embedded inhomoge-


874 Akamatsu & Guo

FIG. 8: Temporal profi les of the refl ectance at detector 6 when the large inhomogeneity is placed at the cube center: a single collimated square pulse or a pulse train of 5 collimated square pulses with base solution shown in Fig. 4a

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(b)

(c)

(d)

(e)

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FIG. 9: Temporal profi les of the refl ectance at detectors 2–5 when the large inhomogeneity is placed at the cube center: a single collimated square pulse or a pulse train of 5 collimated square pulses with base solution shown in Fig. 4e

(a)

(b)

(c)

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876 Akamatsu & Guo

FIG. 10: Temporal profi les of the transmittance at detector 6 when the small inhomogeneity is at placed the cube center: a single collimated square pulse or a pulse train of 5 collimated square pulses with base solution shown in Fig. 5b

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(b)

(c)

(d)

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FIG. 11: Temporal profi les of the transmittance at detector 6 when the small inhomogeneity is placed at the cube center: a single collimated square pulse or a pulse train of 5 collimated square pulses with base solution shown in Fig. 5f

(a)

(b)

(c)

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878 Akamatsu & Guo

neity is small. The refl ectance signals are not affected by the inhomogeneity scattering albedo, but are determined by the base medium scattering albedo. The signal strength for the highly scattering base medium (ω1 = 0.9) is by one to two orders of magni-tude greater than that for the highly absorbing base medium (ω1 = 0.1). However, the difference in refl ectance is obvious when the inhomogeneity is large. Apparently the scattering albedo of the large inhomogeneity affects strongly the signal strength, which increases with increasing inhomogeneity scattering albedo, and this effect is more signifi cant in highly absorbing base medium.

2) The difference in the transmittance signals between the medium with an inho-mogeneity and a homogeneous medium is large even when the inhomogeneity is rela-tively small. The inhomogeneity scattering albedo strongly affects the signals.

3) The temporal refl ectance signals due to the irradiation of a pulse train differ greatly from those of a single pulse, due to the overlap effect. Such an overlap effect of the pulse train depends on the magnitude of the pulse broadening in response to the sin-gle pulse as well as the pulse interval. When the medium has a scattering property, the pulse broadening is larger, and so is the overlap effect. The overlap effect in the trans-mittance is stronger than that in the refl ectance. With increasing the pulse train inter-val, however, the overlap effect weakens. The temporal refl ectance and transmittance signals produced by the irradiation of the pulse train may completely agree with those produced by the irradiation of a single pulse when the pulse interval is big enough.

ACKNOWLEDGMENTS

Under the Invitation Fellowship for Research in Japan (Short-term) of the Japan So-ciety for Promotion of Science (JSPS), the authors did the present work during the second author’s stay at the Graduate School of Science and Engineering, Yamagata University, Yonezawa, Japan. The authors gratefully acknowledge the fi nancial sup-port of the JSPS.

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