Time resolved propagation of ultrashort laser pulses ... - OSA Publishing

Time resolved propagation of ultrashort laser pulses within turbid tissues Steven L. Jacques

The time resolved propagation of femtosecond and picosecond laser pulses within turbid tissues is simulated by a Monte Carlo model. The internal distribution of irradiance for an impulse vs a 4-ps pulse is specified at

different times for various scattering coefficients and scattering phase functions. Such simulations provide time resolved dosimetry for predicting the distribution of single- and two-photon chemical reactions in turbid tissues. For femtosecond pulses in highly scattering tissues, two-photon reactions are dominated by the initial primary (unscattered, unabsorbed) pulse, and single-photon reactions are dominated by the scattered diffuse irradiance. For picosecond pulses in highly scattering tissues, both single- and two-photon reactions are dominated by the scattered irradiance.

1.

Introduction

Ultrashort laser pulses in the femtosecond and picosecond range will probably

enjoy increased use in

medicine. One application is the disruption of tissue by laser-induced plasma, for example, in ophthalmology to disrupt the residual lens membrane following ocular lens surgery,' in urology to fragment kidney stones,2 and in vascular surgery to ablate calcified plaque. 3 A second application is ablation of normally

nonabsorbing tissue by initiating nonlinear absorption with femtosecond pulses.4 A third application is the production of laser-induced photochemical reactions, which have been developed principally around the technique of photodynamic therapy.5 An ultrashort pulse provides a brief intense irradiance that can allow two-photon chemical reactions in tissues. New strategies of photochemistry are envisioned that involve (1) simultaneous absorption of two

photons and (2) sequential absorption of two photons wherein the initial absorption event yields an excited chemical species that absorbs the second photon. An important aspect of the problem is the time resolved internal light dosimetry within turbid tissues that drives such chemical reactions. Studies of the time course of transmitted light in turbid media have been performed.6 Diagnostic ap-

The author is with University of Texas, M. D. Anderson Cancer Center, Laser Biology Research Laboratory, Houston, Texas 77030. Received 13 January 1989. 0003-6935/89/122223-07$02.00/0. © 1989 Optical Society of America.

plications of time resolved measurements of reflectance and transmission for spectroscopy7 8 and specification of tissue optical properties9 1 0 have both been considered. This special journal issue includes papers on time resolved reflectancell and fluorescence'2 measurements, which may become important in medical diagnosis. In this paper, the time resolved distribution of irradiance within turbid tissues is specified by Monte Carlo simulations. The behavior of femtosecond vs picosecond laser pulses is described. The significance for the distribution of single- and two-photon chemistry is discussed. II.

Methods

A.

Simulation of Laser Impulse

The Monte Carlo technique used in these simulations has been adapted from the general methods outlined in Carter and Cashwell'3 and Cashwell and Everett' 4 with modifications by Keijzer et al.' 5 and Prahl' 6 that treat anisotropic light scattering functions and mismatched boundary conditions, yielding time resolved distributions of light within tissues of homogeneous optical properties. The Monte Carlo method has also been used by Wilson and Adam,' 7 Wilksch et al.18 and Flock et al.'9 to discuss steady state light distributions in tissue. The Monte Carlo method is based on radiative transport theory; it improves on diffusion theory which depends on approximate solutions to the radiative transport equation.2 0'2' Monte Carlo simulations can specify the behavior of photons during the transition from a collimated incident irradiance to a diffuse internal irradiance when the most intense laser-tissue interactions occur. Diffusion the15 June 1989 / Vol. 28, No. 12 / APPLIEDOPTICS

2223

ory becomes useful only after many scattering events. Time resolved diffusion theory, which treats time resolved escape of light as measurable reflectance,9"l0 has been introduced but is applicable only many picoseconds after an impulse. Monte Carlo simulations are well suited for modeling the early reflectance from turbid tissues followingultrashort laser pulses.2 2 The Monte Carlo simulation does not consider wave coherence and interference effects, but this special issue includes a paper2 3 on such interference effects during time resolved reflectance measurements. The tissue optical properties are described by the absorption coefficient A,,in cm-', the scattering coefficient iu in cm-', and the anisotropy g, which equals the expectation value , where

0

is the angle of de-

been subdivided into incremental thicknesses of Az (14Ambins). At time t, the photon weight had attenuated to a value of (1 - Rt) exp(-Aact/n), where Rt was the total observed reflectance up to time t. The photon weight, normalized by the total number of photons to be eventually launched, N, was added to the accumulated total weight in the ith bin at the jth time step, qimp[ij]. As each of Nphotons (N= 5 X 104)propagated through the tissue, after each time step j the photon was located at position i, and the array qimp[ij] was incremented according to the followingprogram statement: qimp[ij]- q[ij] + (1 - Rt)

exp(-Aact/n)

N

flection during a single scattering event. First, the light distribution within a tissue after an impulse of laser irradiance was calculated. The radiant energy fluence rate distribution

(z) in W/cm 2 , was

calculated in terms of the radiant energy fluence distribution, 0(z) = C(z)At in J/cm 2 . The temporal resolu-

I.aAl

where i =

where j =

Az t Aln/c (1)

Similarly, for each of the N photons, RtU] was incremented after each time step j by the program statement:

tion At was 4.6 fs, which corresponds to a 1-,um path

length in a tissue with a refractive index n equal to 1.37 (approximate n for a tissue with 80%water content). Each photon with initial weight equal to one was introduced perpendicular to the surface. A fraction Rp was reflected as specular reflectance at the airtissue interface, and 1-R5 p proceeded to propagate through the tissue. The photon trajectory was directed along a variable step of free path between scattering sites:

step = -ln(RND)/yu,

where RND is a random

number between 0 and 1. (The steps of free path between scattering sites are exponentially distributed in size.) The photon moved along this trajectory in incremental steps of Al (1 Aim),depositing a fraction of its weight with every Al step (see below). After sufficient Al steps to arrive at the next scattering site, the photon trajectory was deflected into a new direction according to a scattering phase function specified by the Henyey-Greenstein function2 4 ; its g parameter was

identical to the anisotropy parameter, g equals . The Henyey-Greenstein function approximates Mie scattering and has been experimentally shown to represent well the angular dependence of a single scattering event in turbid tissues.2 5 26 The total length of the photon path through the tissue was the accumulated sum of all steps, L = Al, and the total time of travel was specified by t = Ln/c, where c is the

vacuum speed of light. Throughout the simulation of each photon's path, whenever the photon struck the air-tissue interface at a given angle, a fraction of the photon weight defined

by Fresnel's law escaped as observable reflectance, and accumulated as nr, The remaining photon weight was internally reflected to continue propagation. At any time during each photon simulation, the total reflectance was specified by Rt = nr + R5p. With eachAlstep, a fraction [1- exp(-gA\l)] taAl of the current photon weight was deposited as absorbed energy in an array qimp[ij], where i = z/Az and

j = t/(Aln/c). In the computer model, the tissue had 2224

APPLIEDOPTICS / Vol. 28, No. 12 / 15 June 1989

Rt] - RtN]+

(2)

The photon was allowed to propagate until the last desired time of a snapshot. Then a new photon was launched. After all Nphotons had been launched, the conservation of photon energy was checked by ensuring that the sum of all reflected light, absorbed light, and unabsorbed light still within the tissue equaled unity. Finally, the photon deposition qimp[ij]was converted to the fluence impulse response impri,]j: oi'p[iil

= (J/cm

2

) qimp[ij 1 . AZ /la

(3)

The term (1 J/cm2 )qimp[ij]/Az corresponds to the absorbed energy density in J/cc at the jth time step, which is proportional to oimp[ij] by the factor 1/a. The Monte Carlo simulation can yield the 3-D spatiotemporal impulse response imp(x,y,z,t). Because the simulations in this paper ignore the x- and ydependence of photon absorption, the array imp[ij] constitutes the convolution of the impulse response over an infinitely broad beam and specifies the 1-D solution appropriate to the central axis under the beam. Edge effects due to light scattering at the boundary of the beam perimeter have no influence [see Keijzer et al.' 5 for a treatment of the dependence of O(xyz) on beam diameter for steady state irradiance and the influence of edge effects on beam penetration into turbid tissue]. A personal computer with limited memory and speed was used for the calculations in this paper. Therefore, the impulse response was not computed for all j steps of Al equal to 1 m. Rather, the impulse responses for particular values of j equal to tsnapshot/ (An/c) were calculated. The photon was allowed to propagate efficiently by large steps that equaled the free path, -ln(RND)/,u,, between scattering events until a step would cause the total L to exceed ctsnapshot/n.

2ps

10ps

46 ps

20 ps

where i =,

q[i]

-

q[i] + (1 - R )

N

uLaA1Ei]

wherej = tsnapshot-t Aln/c

(4) cw_

Depth (mm) Fig. 1. Snapshots of light distributions following the laser impulse. The fluence / is plotted at several times (2, 10, 20, 46 ps) after a 1-J/ cm2 impulse (At resolution 4.6 fs). The dashed line indicates the

The additional factor EU] weighted the incremental contribution to q [i] during a given step Al by the portion of EU] that would have arrived at the ith position at tsnapshot. In this trial, tnapshot was equal to the pulse duration, and the snapshot occurred just as the tail end of the pulse entered the tissue. After N photons (N = 104) had propagated and contributed to q[i], Eq. (3) converted q[i] to 0[i]. Tests for the conservation of photons were conducted. The computer time required for each of the simulationsislongbutnotprohibitive. On apersonalcomputer using floating point algorithms, each simulation requiredanovernightrunof'-12 h. This run time allowed -50,000 photons to be launched for snapshots of an impulse and -10,000 photons for snapshots of a pulse.

magnitude of the primary impulse as it moves through the tissue and suffers total attenuation due to absorption plus scattering.

Ill.

Results

A.

Laser Impulse Responses

E

C.)

II-

0

5

10

The fluence distributions 0imp[ij] at various times This last step was truncated so that L exactly matched At that jth time and ith position, qimp[ii] Ctsnapshot/n. and RtU] were updated by Eqs. (2) and (3), which assume a small incremental step of Al equal to 1 m (At

equals 4.6 fs). Then the remainder of the truncated larger step was completed, and the photon continued its propagation using large steps until the next desired A qimp[ij] and RtU] were computed for each desired tsnapshot. Equation (3) converted qimp[ij] to Oimplij] for each tsnapshot after all N tsnapshot was reached.

photons had been launched. In this fashion, snapshots of the fluence distribution at only a few desired

times following an impulse of 1 J/cm 2 were efficiently

obtained. B.

Simulation of Laser Pulse

The full impulse response Oimp[ij] can be convolved

over any arbitrary laser pulse shape to yield the fluence distribution 4[i] due to a laser pulse of arbitrary shape and duration time tsnapshot. With the limited computer available for this project, it was difficult to store the full impulse response 0imp[iyj]before executing the convolution. Therefore, the impulse response and convolution were calculated simultaneously for a particular laser pulse. The laser pulse source was characterized by a source array EUj] whose total energy content was 1 J/cm 2 with

time resolution At equal to 4.6 fs. For this paper, a Gaussian pulse shape was chosen, which was truncated at its 1/e2 points. The pulse width between the lie 2 points was set to 4 ps. Photon propagation used incremental steps of Al equal to 1,um along each step of the free path, and the array of photon deposition q[i]was incremented after each time step j by the followingprogram statement:

after an impulse of irradiance are shown in Fig. 1. These snapshots are at 2, 10, 20, and 46 ps, which correspond to total path lengths L of 435 Am,2.175mm,

4.35 mm, and 1 cm, respectively. The optical properties of the model tissue are Ma = 1 cm-', Ms = 100 cm-', and g = 0.9. For reference, the dashed line in Fig. 1

indicates the expected magnitude of the primary impulse (i.e.,unabsorbed unscattered impulse) as it moves into the tissue if the impulse suffers attenuation due to the total attenuation coefficient,At = Maa+ Ms. At time t, the primary impulse would have reached a depth of ct/n and have a magnitude exp(-Atct/n). At 2 ps, the primary impulse is not as strongly attenuated as would be

expected, since the scattering is forward directed. Multiple scattering events are required to cause the photons to lag behind the primary impulse. At longer times (10-46 ps) the primary impulse is very strongly attenuated. However, a trail of scattered light lags behind the primary impulse and diffuses slowlyinto the tissue. As this wave of diffuse light penetrates deeper, its concentration is (1) diluted over a larger volume, (2)

extinguished by absorption [exp(-Aact/n) = le at 46 ps], and (3) depleted by surface losses due to Rt.

The influence of the anisotropyg is illustrated in Fig. 2(A). The dashed line is the same as in Fig. 1. At 2 ps after delivery, oimp[ij]is plotted forgvalues of 0.95,0.9, 0.8, 0.7, 0.5, 0.3, and

O(Ala =

1 cm-',

Mts =

100 cm-').

Note how anisotropy affects the penetration of the primary impulse.

There is a discontinuity in q5imp[ij]be-

tween the primary impulse (i = 435) and the trail of scattered light that lags behind (i