Time Domain Diffuse Optical Spectroscopy (TD-DOS)

Time Domain Diffuse Optical Spectroscopy (TD-DOS) or Time resolved spectroscopic (TRS) techniques rely on introducing a brief pulse of light into a turbid medium and measuring the transit times of a distribu- tion of many photons. Together with the index of refraction of tissue, the mean of this transit time gives the average photon path length, which is related to the absorption and scattering of the tissue46. This average path length can be utilized directly to derive chromophore concentrations (see Equation3.3). Alternately, one can fit a diffusion model (e.g. Equation2.50) to the data.

A TD-DOS measurement consists of the following steps: 1) Introduce a very short (_∼300) pulse of light into tissue.

2) Collect diffuse output light using a high-speed photon counting PMT or APD.

3) Collect output pulses from detector to create a very high (_∼10 ps) temporal resolution histogram. 4) Measure the Instrument Response Function (IRF) and convolute with a theoretical model (e.g. Eqn.

2.37).

5) Fit the data to the above convolution to extractµa[λ] andµ0s[λ]; iterate parameters of theoretical

model.

Alternatively, one could deconvolute the IRF from the measured data and fit the result to the theoretical curve, but this is computationally more expensive; the IRF will be discussed in detail below. Roughly speaking, scattering determines the peak of the photon arrival time histogram and absorption determines the terminal log-slope. The latter is obvious by rewriting Eqn.2.37as

Ψ[r, t] = vS0 (4πDt)3/2e

−r2

4Dte−µavt_{. (2.86)}

The last term dominates at long times (i.e. ast _{→ ∞},Ψ _→ e−µavt_{). An example histogram is shown} in Figure2.16; Figure2.17shows the effect of titrating an absorbing dye into a liquid phantom on the TD histograms.

Measurements using short source-detector separations (e.g. < 10l∗_{) generally require a more careful} treatment of light propagation in scattering media, including realistic modeling of sources and detectors. In the context of DOS and DOT of the breast, sources and detectors are generally well separated (>2.5cm for hand-held systems,>6cm for parallel plate tomographic systems) and point approximations of the source and detector are reasonably accurate.

The output of a TD measurement is a histogram of photon arrival times, as shown in Figure2.16. If the light source and detector have an infinitely fast response, this histogram can be directly fit to a solution

0 2 4 6 8 10 12 10−2 10−1 100 Time [ns] Normalized Counts 0 2 4 6 8 10 12 10−2 10−1 100 Time [ns] Normalized Counts IRF Fitted Theory Convolution Data

IRF convolved with Theory

Convolution of IRF and Theory

Comparison with Data

Figure 2.16: Example of IRF, fitted theoretical curve, and data for subject 61003 (8.3 cm compression, BMI 41.5, bra size 44 DDD) at 798 nm. Fitted parameters:µa = 0.034cm−1,µ0s= 13.3cm−1,t0 = 0.029ns, and amplitude = 0.73. These correspond to a very fatty breast with fairly low signal (max 746) because of the thickness. Note, the IRF is clipped at 10% on the up slope and 5% on the down slope andt0is measured relative to a reference channel.

2

4

6

8

10

0.01

0.1

1

Time [ns]

Normalized Intensity

Figure 2.17: An example of the effect of increasing absorption on TD data histograms. Black is the IRF; as the colors change from blue to red, the absorption goes up from 0.02 cm−1_{to 0.2 cm}−1_{in steps of 0.024} cm−1_.

of the time domain diffusion equation in the appropriate geometry (e.g. Eqn. 2.37). As with FD and CW measurements, a TD source is generally modeled as a point source locatedz0=_µ10

s into the medium, where

µ0

sis the average reduced scattering44, 46. This extrapolated source positionz0 is shown schematically in Figure2.8and a schematic of a basic TD system is shown in Figure2.18.

Dealing with the assumption of an infinitely narrow input pulse of light is somewhat more complicated. As a practical matter, one must consider the convolution of the laser output pulse, the pulse broadening

Figure 2.18: Conceptual schematic of TD-DOS measurements. A narrow pulse of light is introduced to the tissue from a laser diode, driven by an oscillator. Several possible photon paths through the scattering and absorbing centers of the media are shown. Light is detected using a PMT in Geiger mode; counting electronics determine the transit time of the detected photon compared to the input pulse.

(dispersion) in any optical elements (fibers, etc.), the transit time spread (TTS) of the detector, and any delays in recording electronics. The convolution of these elements is referred to as the ‘Instrument Response Function’ (IRF). For DOT in the breast, an IRF with full-width-at-half-max (FWHM) of_∼400 is sufficient for recovery of optical properties139. An example of the convolution and fit is shown in Figure2.16.

Note: in order to properly replicate the detector fiber collecting light from a diffuse media, the IRF must be collected in such a manner that all of the fiber modes are excited. In practice, this consists of separating the source and detector fiber by_∼1 cm and inserting a plano-convex lens to quasi-collimate the beam from the source fiber. Additionally, one can insert a thin diffuser (e.g. a layer of tissue paper) after the plano-convex lens to further homogenize the input beam.

An offset t0 accounts for time delays not originating in the diffuse media and would be included in Equation2.86as an offset (i.e.,t_→(t₋t0)). This offsett0can change with time, for example, as a result of temperature changes in the opto-electronics139_.