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How source/collector placement and

subsurface absorbing layer affect

time-resolved and phase/modulation-

resolved photon migration

Steven L. Jacques, Andreas H. Hielscher, Lihong V. Wang,

Frank K. Tittel

Steven L. Jacques, Andreas H. Hielscher, Lihong V. Wang, Frank K. Tittel,

"How source/collector placement and subsurface absorbing layer affect time-

resolved and phase/modulation-resolved photon migration," Proc. SPIE 1888,

Photon Migration and Imaging in Random Media and Tissues, (14 September

1993); doi: 10.1117/12.154649

Event: OE/LASE'93: Optics, Electro-Optics, and Laser Applications in

Scienceand Engineering, 1993, Los Angeles, CA, United States

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How source/collector placement and subsurface absorbing layer affect

timeresolved and phase/modulation-resolved photon migration.

Steven L. Jacques1'

Andreas H. Hielscher2

Lihong Wang1

Frank Tittel2

1Laser Biology Research Laboratory

University of Texas M. D. Anderson Cancer Center

1515 Holcombe Blvd., Houston, TX 77030

2Department of Electrical and Computer Engineering

Rice University, Houston, TX 77030

1. ABSTRACT

The time-resolved reflectance of photons from a homogeneous tissue was modeled using a

Monte Carlo simulation. The data was then converted by fast Fourier transform (FFT) into the

frequency domain. In the frequency domain, the phase, c1, and modulation, M, of collected light

from a frequency-modulated light source was determined. A comparison of Monte Carlo and

ciffusion theory was made for various separation distances between the source and collector on the

tissue surface. The results showed that Monte Carlo and diffusion theory agreed in the time domain

only for times Larger than 500 ps after injection of an impulse of photons. In the frequency domain,

Monte Carlo and diffusion theory agreed only if the probe separation, r, was at least 2 cm apart for

= p.(1-g)

5 cm1,

or in dimensionless units r' >

10.

The effect of buried absorbed is also tested in the time and frequency domains. A semi-infinite

volume of absorber is placed at 0, 3 mm, 6 mm, or oo

from

the surface of a nonabsorbing tissue. The

presence of a deep absorber on the time and frequency domain data show that attenuation of longer

pathlength photons causes the phase of collected photons to reduce and the modulation of collected

photons to increase. Both effects are indicative of the net shorter pathlength of the ensemble of

collected photons.

INTRODUCTION

Photon migration in tissues can be described in either the time or frequency domains. We

consider here the case of light escaping at the surface of a semi-infinite tissue. The time domain

indicates the timecourse of collected light, called reflectance, by a collector at the surface in response

to an impulse of injected light. The frequency domain indicates the phase and modulation of

collected light from a a frequency-modulated source.

Several groups have introduced diffusion theory descriptions for photon migration in both the

time and frequency domains [1,2,3 ,4]. Jacques [5]

reported

that time-resolved Monte Carlo

simulations of photon migration did not agree with diffusion theory until after a time delay had

elapsed. Sufficient time must pass to allow some scattering before photons behave according to

diffusion theory. Therefore, in this study we sought to understand how this restriction in the time

domain might translate into the frequency domain.

310

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Furthermore, a second issue addressed in this paper is how an absorbing layer deep in the

tissue wifi affect photon migration in both the time and frequency domains.

3. METHODS

3 .1 Monte Carlo

A well-tested Monte Carlo simulation [6] was adapted to provide time-resolved photon

propagation. Propagation was three-dimensional but photon escape as reflectance was collected in a

cylindrically symmetric array. Photons escaping at radial position, r, within an incremental Ar of

1-mm were collected in an array R(r, t) with spatial resolution of 0. 1 mm between 0-4 cm (40 intervals)

and temporal resolution, At, of 25 ps over the range of 0-3.2 ns (or 128 time intervals). The results

were expressed as the time-resolved local reflectance in mnr1ns1.

3 .2 Fast Fourier Transform (FF1')

To convert the data to the frequency domain, a fast fourier transform (FF1') [71 was used to

convert a list of time-domain data, U, h(t)), into a list of freqency-domain data, (Re(H(f), Im(H(f)),

which were the real and imaginary components of the discrete Fourier transform, H(f). The phase

shift, 4, and the demoduhition, M, for a photon density wave were then calculated:

Tm (H(fn))

Im (Tin)

4 =

arctan(

Re (H(fn)) )

arctan(

Re (Hn) )

(1)

(H(fn)) +Jfl2 (H(f))

Re2(Hn)

+1m2 (Tin)

M =

Re2 (H(O))

Re2

(He)

(2)

The 25-ps resolution, At, in the time domain determines the maximum frequency, fmax

1/(2t) =

GHz, available in the frequency domain. For n data points in the time domain, the FF1'

yields n data points in the frequency domain. The resolution in the frequency domain is given by M =

fmax/n.

In order to increase the frequency resolution the common technique of "zero adding" was

applied [8] The time domain array with 128 points (3.2 as) was expanded up to 1024 points (265

ns) by adding zeros into the R(r, t) array beyond 3.2 ns. The procedure is justified in this case by the

fact that after 3 .2 ns the time signal has decayed by several orders of magnitude. The frequency

resolution achieved was M =

39.

1 MHz.

When tissue absorption exerted sufficiently high attenuation on photon density wave

propagation at high frequencies of modulation, the frequency-domain data fluctuated abnormally.

Therefore, in the graphs of this report, we have truncated the upper range of the frequency.domain

results when the behavior abnormally fluctuated.

3 3 Diffusion theory

The time-resolved diffusion theory of Patterson et al. [1], with a scaling factor k, was used to

predict the behavior of photon migration in the model:

r2+ 2

R(r, t) =

(4itDc)312 z0t5'2 exp[-

exp(-p.ct)

(3)

SPIE

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