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. Author manuscript; available in PMC: 2014 Jan 21.
Published in final edited form as: Phys Med Biol. 2012 Dec 21;58(2):335–349. doi: 10.1088/0031-9155/58/2/335

The Effect of Temporal Impulse Response on Experimental Reduction of Photon Scatter in Time-Resolved Diffuse Optical Tomography

Niksa Valim 1, James Brock 1, Miriam Leeser 1, Mark Niedre 1,*
PMCID: PMC3568978  NIHMSID: NIHMS432520  PMID: 23257349

Abstract

New fast detector technology has driven significant renewed interest in time-resolved measurement of early photons in improving imaging resolution in diffuse optical tomography and fluorescence mediated tomography in recent years. In practice, selection of early photons results in significantly narrower instrument photon density sensitivity functions (PDSFs) than the continuous wave case, resulting in a better conditioned reconstruction problem. In this work, we studied the quantitative impact of instrument temporal impulse response function (TIRF) on experimental PDSFs in tissue mimicking optical phantoms. We used a multi-mode fiber dispersion method to vary the system TIRF over a range of representative literature values. Substantial disagreement in PDSF width – by up to 40% - was observed between experimental measurements and Monte Carlo (MC) models of photon propagation over the range of TIRFs studied. On average, PDSFs were broadened by about 0.3 mm at the center plane of the 2 cm wide imaging chamber per 100 ps of instrument TIRF at early times. Further, this broadening was comparable on both the source and detector sides. Results were confirmed by convolution of instrument TIRFs with MC simulations. These data also underscore the importance of correcting imaging PDSFs for instrument TIRF when performing tomographic image reconstruction to ensure accurate data-model agreement.

1. Introduction

Diffuse optical tomography (DOT) and fluorescence mediated tomography (FMT) are in increasing use as clinical and pre-clinical imaging modalities, wherein three dimensional images of native tissue contrast, exogenous fluorescent markers or fluorescent proteins are reconstructed (Arridge, 1999; Ntziachristos et al., 2000; Boas et al., 2001; Ntziachristos et al., 2003; Graves et al., 2003; Ntziachristos V. et al., 2005). DOT and FMT instrumentation require measurement of photons that propagate through bulk biological tissue in the diffusive regime either in transmittance or reflectance modes (Culver et al., 2003; Hielscher, 2005). Images are obtained by relating measurements to the unknown quantity of interest (i.e. tissue optical properties or fluorescence concentrations) with physical models of light propagation in a system of linear equations, and then solution of the subsequent ill-posed inverse problem (Klose et al., 2005). The most important technical limitation of both DOT and FMT is arguably the relatively low imaging resolution due to the high degree of light scatter in biological tissue, which is generally understood to be about 10–20% of the imaging length (Wang and Wu, 2007). One well established approach to improve imaging resolution is time-resolved detection of “early-arriving” or “weakly-diffuse” photons. Photons that arrive on the rising edge of the transmitted intensity curve from a pulsed laser source have undergone significantly fewer total scattering events than the bulk photon population (i.e. time-integrated photons). As such, the width of the imaging photon density sensitivity function (PDSF; alternatively referred to as the “weight function”, “photon measurement density function”, Jacobian, or “banana function”) is correspondingly reduced, leading to a better conditioned image reconstruction problem and improved imaging resolution. Early work by Feld et. al. (Wu et al., 1997) demonstrated this in localizing fluorescent inclusions with a fiber-coupled streak camera in bulk diffusive media, and the approach was studied by other groups in trans-illumination imaging geometries (Andersson-Engels et al., 1990; Gandjbakhche et al., 1994; Grosenick et al., 1999). In recent years, there has been substantial renewed interest in the concept with the development of new classes of fast time-gated and time-resolved detectors (Turner et al., 2005; Leblond et al., 2009; Niedre and Ntziachristos, 2010; Valim et al., 2010; Venugopal et al., 2010b; Zhang et al., 2011; Tosi et al., 2011; LaPointe et al., 2012).

Recently, we experimentally measured time-dependent PDSFs for a particular pulsed light source (super-continuum laser) and detector (fast photomultiplier tube) combination (Valim et al., 2010). Very briefly, we demonstrated substantial disagreement with experimentally measured PDSFs and theoretical results computed with time resolved Monte Carlo simulations, particularly at very early arrival times, i.e. earlier than the 10%-of-maximum point on the rising edge of the curve. At the time, we hypothesized that this effect was chiefly due to the finite (non-ideal) temporal impulse response function (TIRF) of our system, which was 163 ps. Although we did not confirm this explicitly, it is generally understood that “faster is better” in the time-resolved (TR) tomography literature. For example, other authors have shown that non-ideal instrument TIRFs can yield incorrect estimation of tissue optical properties from full time curves (Ntziachristos and Chance, 2001; Liebert et al., 2003), or can introduce of imaging artifacts when performing time-resolved imaging with mean, variance and “skew” time-resolved data-types (Hillman et al., 2000). Other authors have similarly shown that this can lead to erroneous estimation of fluorescence lifetimes in TR-FMT (Luchowski et al., 2009; Gerega et al., 2011).

However, to our knowledge the quantitative effects of instrument TIRF on measurements in the early-photon regime have never been studied in detail previously. Of particular interest is the degree of experimental resolution improvement obtainable relative to the theoretical maximum. A survey of the literature published in time-resolved DOT and FMT shows that previously reported TR instrument configurations have intrinsic system TIRFs in the range of approximately 80–800ps. Multi-channel plate (MCP) based systems have reported TIRFs in the range of 80–200 ps (Andersson-Engels et al., 1990; Pifferi et al., 2004), photomultiplier tube (PMT) based systems in the range of 160–800ps (Grosenick et al., 2003; Liebert et al., 2004; Contini et al., 2006; Kepshire et al., 2009; Wabnitz et al., 2010; Kacprzak et al., 2012), intensified charge-coupled device camera (ICCD) based systems in the range of 200–400 ps (Niedre et al., 2006; Kumar et al., 2008; Venugopal et al., 2010a; Sawosz et al., 2010; Zhao et al., 2011), streak camera based systems in the range of 150–250 ps (Wu et al., 1997). Fast avalanche photodiode (APD) systems have recently been used in TR tomography with stated detector response time of about 40 ps (Bérubé-Lauzière and Robichaud, 2007; Tosi et al., 2011), although the overall system TIRF was not stated. Interferometric (i.e. transmission geometry optical coherence tomography) approaches have also been used to measure TR curves through diffusive media, but to date these have been limited to relatively small (~1–5 mm) imaging volumes (Hee et al., 1993; Li and Wang, 2007).

In this work we quantitatively studied the impact of instrument TIRF on imaging resolution by systematically modulating our instrument TIRF in approximately the range of reported literature values (specifically, between 163 and 716 ps) using a fiber dispersion method similar to previous authors (Ntziachristos et al., 1998; Liebert et al., 2003) and measured the impact of this on experimentally measured PDSFs. As we demonstrate, all instrument TIRFs that we tested resulted in significant disagreement (up to 40%) in PDSF width compared to time-resolved MC theory. Experimental disagreement was most pronounced at very early time points (for example, at the 10%-of-peak point on the rising edge of the curve or earlier), and varied by about 0.3 mm per 100 ps of instrument TIRF in this regime. We found that modulation of the system TIRF by elongating the response on either the source or detector side had comparable effects, implying that use of fast sources or detectors will yield approximately the same improvement in imaging resolution in a TR optical tomography system. This was also confirmed by temporal convolution of system TIRFs with the output of TR Monte Carlo (MC) simulations. Effects of detector sensitivity and noise properties – which are also known to affect imaging resolution in DOT and FMT – were not studied here. The results presented here have significant implications, i) in correct calculation of imaging PDSFs for time-resolved DOT and FMT, and ii) in design of time-resolved DOT and FMT systems that utilize the early-photon effect.

2. Methods and Materials

2.1 Instrumentation

The basic system used for these experiments is shown in figure 1 and is similar to the system we described previously (Valim et al., 2010). A fast, pulsed Super-continuum fiber laser with an 80 MHz repetition rate was used as the light source (KoherasSuper-K Power, NKT Photonics, Birkerod, Denmark). The output of the laser was passed through a 670 nm bandpass interference filter (30 nm FWHM; Chroma Technology, Bellows Falls, Vermont). As we discussed previously (Valim et al., 2010), the original broadband light pulse from the Super-continuum source was approximately 400 ps FWHM, but after wavelength filtering was reduced to about 30 ps. Light was coupled into a 0.37 NA, 600 μm core step-index multimode optical fiber (BFL37-600; Thorlabs) of varying length by illuminating a diffusing glass plate (220-Grit ground glass diffuser, Edmund Optics, Barrington, NJ) placed directly in front of the fiber end (see section 2.2 below). Light exiting the fiber was focused to a 1 mm diameter spot on the surface of a custom made 15 cm × 15 cm × 2 cm glass sample chamber with a pair of plano-convex lenses (Edmund Optics). The thickness of the glass walls was about 2 mm. Neutral density filters were placed in the beam path so that the illumination intensity at the chamber surface was 1.25 mW. The sample chamber was filled with liquid phantoms prepared from a stock solution of 10% Intralipid (Baxter Healthcare Corporation, Deerfield, Illinois) and India ink. These were diluted so that the final liquid optical properties at 670 nm were approximately as follows: reduced scattering coefficient μ’s = 10 cm−1 and absorption coefficient μa = 0.15 cm−1. Here, the optical properties were estimated based on characterization of liquid phantom optical properties by TR analysis of transmitted intensity curves performed previously by M. Niedre in (Niedre et al., 2006). Moreover, these optical properties were selected since they are approximately equivalent to typical literature values for biological tissue at red and near-infrared wavelengths (Tromberg et al., 1997; Torricelli et al., 2001; Niedre et al., 2006).

Figure 1.

Figure 1

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Schematic of instrument used to measure system PDSFs. Two multimode optical fibers of varying lengths were coupled at the system input and output to alter the instrument temporal impulse response function.

Light transmitted through the 2 cm wide chamber was collected with a second multimode optical fiber that was placed co-axially with the position of the input beam. Given the 0.37 NA of the fiber and the 2 mm thickness of the chamber glass, the detection spot size on the inner surface of the chamber was about 1.5 mm. Light exiting the detector optical fiber was collimated using a SMA-coupled achromatic plano-convex lens package (F260 SMA-B; Thorlabs) and then passed through a second 670-nm bandpass filter (Chroma). The transmitted light was measured with a 16–channel photomultiplier tube (PMT) array (PML-16-C, Becker and Hickl, Berlin, Germany). The gain on the multi-anode PMT was controlled with a detector control card (DCC; Becker and Hickl) that was operated at 90% of maximum. The output of the PMT array was amplified with internal preamplifiers and coupled into a time-correlated single photon counting card (SPC-130, Becker and Hickl) configured so that it had a 16.3 ps temporal resolution. Although the PMT array had 16 individual anodes, it was determined that the detected light illuminated approximately the middle 6 detection channels. Therefore, the time-correlated photon count signals from only the middle 6 channels were summed after correction of the minor inter-channel time skew (<25 ps in this case). The overall system TIRF full width at half maximum (FWHM) was measured to be 154 ps, which generally agreed well with performance specification of our light source (~30 ps) and multichannel PMT (~150 ps).

2.2 Modulation of Instrument Temporal Impulse Response Function

We used a fiber dispersion method similar to that described by (Ntziachristos et al., 1998) and (Liebert et al., 2003) to modulate the temporal impulse response function (TIRF) of our system. As shown in figure 1, this was achieved by varying the length of the multimode optical fibers on the source and detector sides. The intermodal dispersion properties of the fibers temporally broadened transmitted light pulses with the amount of dispersion directly proportional to fiber length. To ensure as complete modal filling as possible (and therefore to maximize this effect) we placed a diffusing glass plate in front of the source fiber. The overall TIRF of the instrument was then altered by using either, i) a source fiber length (SFL) of 1, 5, 10 or 20 m, with a constant detector fiber length (DFL) of 1 m, ii) a DFL either 1, 5, 10 or 20 m, with a constant SFL of 1 m, and, iii) SFL and DFL of equal lengths between 1 m and 20 m. The rationale here was that we were interested in separating the effects of either a slower source or detector on the measured instrument PDSF at early time points. Instrument TIRFs for each combination were determined by measuring the temporal transmitted light through non-scattering water filled chamber. For these measurements, the source intensity was attenuated by placing neutral density (ND) filters in front of source fiber to avoid saturation of the PMT array. A thin scattering layer (i.e. a piece of white paper) was placed in front of the detector fiber to ensure complete mode filling, although this was removed for subsequent experiments (section 2.3) where intralipid solution was used and performed a similar mode-filling function. We determined experimentally (as have other authors) that pulse broadening increased more than 2 times with the scattering layer in place than with a bare fiber. For our data analysis, equal pulse dispersion was assumed when either the paper or intralipid were in place. As we demonstrate, this fiber dispersion approach allowed broadening of the overall instrument TIRF FWHM from 163 ps to 716 ps.

2.3 Experimental Measurement of Instrument Photon Density Sensitivity Function

To measure the photon density sensitivity functions we used a similar method to our previous work (Valim et al., 2010) and translated a 10 cm × 10 cm absorbing sheet with a centered 2 mm × 50 mm vertical aperture stepwise through the intralipid media (in our previous work, we used an absorbing rod, but the aperture method was empirically found to reduce inter-experimental variability between measurements). The absorbing sheet was made from anodized foil (BKF12, Thorlabs) and was sufficiently large and opaque to absorb photons not passing through the aperture. The slit was translated through the liquid phantom using computer-controlled stepper motor stages (XSlide, Velmex Inc., Bloomfield, New York). A complete set of measurements were performed by translation of the slit through 121 steps with a step size of 0.25 mm in the lateral direction (for a total of 3 cm), and through 39 steps with a step size of 0.5 mm in the depth direction. The transmitted light was measured with the PMT array at each slit position. The maximum detected photon count rate was ~106 counts/s when the slit was positioned directly in front of the laser. By moving the slit to an off axis position, the count rate dropped to approximately 104 counts/s. We verified that no significant temporal shift of the transmitted photon intensity curve was measured over this range of photon count rates for otherwise identical conditions. We also showed numerically that the measured PDSF with a slit aperture had the identical FWHM as a point aperture at the center plane (data not shown for brevity) but had the benefit of increasing the photon count rate and improving the measurement SNR.

To compute the experimental PDSF, the full time curve through the liquid phantom media was first measured without the slit in place for each fiber combination. This temporal transmitted curve was analyzed to find the timing of the peak point and the 5, 10, 25 and 50%-of-peak time points on the early-time range (i.e. rising edge) of the detected signal. The total photon counts at each slit position were summed within a 32 ps time window (two time points) and the PDSF was then computed as follows:

PDSF(x,z,t)=I(x,z,t)Imax(t) (1)

, where Imax (t) was the maximum transmitted intensity at a given time t and I(x, z, t) was the transmitted intensity at position x, z at time t. As a metric of the overall breadth of the instrument PDSF, we considered the full-width at half maximum (FWHM) of the PDSF in the middle plane of the chamber (i.e. z = 1 cm) where the extent of photon diffusion was greatest. Broadening of the PDSF due to the finite detection area (~1.5 mm) at the exit plane of the chamber was minimal in the middle of the chamber. For all experiments, the acquisition time was 1 s per slit position. Each experiment was repeated at least 3 times for each source and detector fiber length combination. The PDSF FWHM was measured for each experiment individually, and the mean and standard deviation of all trials at each time point was calculated using standard formulas assuming normally distributed data.

2.4 Time-resolved Monte Carlo Simulations

Theoretical PDSFs were computed with a TR Monte Carlo simulation based on the publically available MCML code on the Oregon Medical Laser Center website (Wang et al., 1995). Monte Carlo is a gold standard method for modeling TR photon propagation – particularly at early times where the diffusion approximation to the Boltzmann Transport Equation (BTE) has been shown to be inaccurate – and is considered a numerical implementation of the BTE. We compiled the paths of photons from a pulsed (infinitely short) pencil beam emerging from a 2 cm semi-infinite slab media at a 1-mm by 1-mm square detector located directly opposite from the input position. The scattering coefficient μs was assumed to be 66.7 cm−1 based on our reduced scattering coefficient (μ’s) and literature values for the anisotropy coefficient of intralipid at 670 nm, g = 0.85 (Flock et al., 1992). The absorption coefficient was assumed to be μa = 0.15 cm−1.A total of 1 billion photons were tracked and the simulation yielded a temporal resolution of 30 ps.

Since time-resolved MC simulations characterize photon propagation through the media for an infinitely short incident pulse, it was also possible to calculate the effect of finite instrument temporal impulse response functions on the measured signal using a point-by-point temporal convolution as follows:

Measured(r,z,t)=∫-∞+∞MC(r,z,t-τ)TIRF(τ)dτ (2)

, where MC(r, z, t) is the photon density at position r and z at time t obtained from the MC simulation respectively, and τ is an integrating factor for time. Here TIRF(t) was first explicitly measured with our instrument for each of the SFL and DFL combinations as above.

3. Results

3.1 Varying System Temporal Impulse Response Function with Multimode Optical Fibers

We first used the multimode fiber pulse dispersion method to vary the overall system temporal impulse response function (TIRF) as described. Other authors have shown previously that the degree of pulse dispersion obtained with this method is highly dependent on complete modal filing of the multimode optical fiber (Liebert et al., 2003). Therefore, for these measurements we placed a thin scattering material (white paper) in front of the detection fiber to mimic the intralipid media. Figure 2a shows the broadening effect observed by increasing the source and detector fiber lengths (in this case, when the SFL and DFL were equal and were increased from 1 m to 20 m) on the measured system TIRFs. We quantified this effect by measuring the TIRF full width at half maximum (FWHM) when the fiber length was altered from 1 to 20 m on the source, detector and source and detector sides as summarized in figures 2b–d, respectively. As shown, increasing the fiber length increased the system TIRF by 5.6 ps/m on the source side, 11.6 ps/m on the detector side, and 14.5 ps/m when both the source and detector fiber lengths were increased equally. The longest TIRF FWHM we could produce with our method was 716 ps (for SFL = DFL = 20 m), which is on the upper end of instrument TIRF FWHMs reported in the TR literature. The difference in dispersion observed on the source and detector sides was most likely due to sub-optimal mode filling on the source side; it was found that placing a white paper diffuser (versus the glass diffuser) resulted in a further ~50% increase in TIRF (although paper was not used in the experiments here since it resulted in excessive light loss through the system). Our results indicate less temporal dispersion than reported by (Liebert et al., 2003) that was achieved using a similar method, although we attribute this difference to the lower numerical aperture of our fiber, specifically 0.37 versus 0.54. This effect can be estimated using the group delay relationship,

Figure 2.

Figure 2

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(a) Measured system TIRFs for 1, 5, 10, and 20 m source and detector fiber lengths. The effect of fiber length from 1 m to 20 m on system TIRF FWHM for varying (b) SFL, (c) DFL, and (d) SFL and DFL.

ΔtL=nc0(11-NA2n2-1) (3)

, where is the speed of light in vacuum, is the refractive index of the core material, the numerical aperture of the fiber, the length of fiber, and is the delay of highest order mode in comparison with the lowest order mode. Based on this relationship, use of an NA of 0.54 would have yielded about 2.25 times more dispersion for a fiber of identical length and core index of refraction, consistent with the observed differences. In summary, we verified that it was possible to use this approach to adjust the instrument TIRF without altering major system components (i.e. light sources and detectors).

3.2 Measurement of Instrument Photon Density Sensitivity Functions

We then investigated the effect of the altered TIRF on experimentally measured time-dependent instrument photon density sensitivity functions (PDSFs). We first filled our 2 cm wide sample chamber with a tissue mimicking intralipid solution and measured the TR transmitted light through the media. As expected, increasing the length of the source and detector fibers (and therefore the instrument TIRF) resulted in significant broadening of the transmitted TR curve. Example curves are shown in figure 3a when the source and detector fiber lengths were equally increased from 1 m (TIRF FWHM = 163 ps) to 20 m (TIRF FWHM = 716 ps). Here, curves were normalized to their maxima, both in amplitude and in time. Temporal normalization to the peak was performed since the true starting point is difficult to accurately measure in practice.

Figure 3.

Figure 3

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(a) Example measured diffusely transmitted light through the liquid phantom for 1, 5, 10, and 20 m source and detector fiber lengths. Example measured photon density sensitivity function for 1 m source and detector fiber lengths (b) at the 50% point on the rising edge of transmitted time-resolved curve, (c) quasi-CW photons, and (d) and at the 50% point on the falling edge.

We then translated a rectangular slit aperture through the media and measured the effect on the transmitted curve for each position as described in equation 1. Figures 3b-d show example PDSFs for the 50% time point on the rising edge, quasi-CW photons, and the 50% time point on the falling edge of the time-resolved curve, respectively, obtained for the fiber configuration corresponding to SFL = DFL = 1 m.

It is evident from the figures that the increase in photon scatter with time resulted in broadening of the measured instrument PDSFs. As expected, time points later than the meantime of the TR curve have PDSFs which are broader than the time-integrated, continuous wave case. For all configurations, we maintained identical photon count rate, laser power, detector gain and acquisition time to eliminate the possibility of artifacts due to these effects, although this resulted in relatively minor differences in peak transmitted intensities in the time-resolved curve for different TIRFs as discussed below.

We repeated these measurements for all of the source and detector fiber length combinations as shown in figure 4 and measured the FWHM of the PDSF in the middle plane of the sample chamber as shown. Since we were principally interested in the use of this approach to improve tomographic imaging resolution, we considered the PDSF FWHM on the “early” rising edge of the measured TR transmitted curve, i.e. where PDSFs were narrower than the un-gated CW case. Measured PDSF FWHM are shown in figs. 4a-c as a function of fraction of peak (i.e. position on the rising edge of the TR curve) for the case where the SFL (fig 4a), DFL (fig4b), and both SFL and DFL (fig 4c) were varied. Error bars are not shown on plots for clarity, but the measured FWHMs had uncertainty of less than ± 0.25 mm throughout the rising edge of the curve (N = 3 for all cases).

Figure 4.

Figure 4

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The FWHM of the middle plane of PDSF at early time points (fraction of the peak point) for varying (a) SFL, (b) DFL, and (c) SFL and DFL. (d) The FWHM of the middle plane of PDSF vs. the measured rise time for varying SFL and DFL.

As shown, the PDSF breadth increased significantly with longer instrument TIRFs, and the most significant broadening was observed to occur at very early times. Considering the fiber configuration with the fastest TIRF (SFL = DFL = 1 m; TIRF FWHM = 163ps), the PDSF FWHM at the 10%-of-peak point on the curve was 5.8 ± 0.25 mm compared to the quasi-CW width of 10.5 mm. This represents a relative reduction by about 44.7%, which agrees with our previous results of 40 to 60% over a wide range of conditions (Valim et al., 2010). In contrast, for the slowest overall TIRF, (SFL = DFL = 20m; TIRF FWHM = 716 ps) the PDSF FWHM at the 10%-of-peak point on the curve was about 7.6 ± 0.1 mm, which is equivalent to a relative reduction of only about 27.6%. For all fiber combinations, when considering the 10%-of-peak point on the transmitted curve, PDSFs broadened in the mid-plane by 0.2 ± 0.1 mm per 100 ps, 0.36 ± 0.06 mm per 100 ps and 0.29 ± 0.03 mm per 100 ps when the SFL, DFL, and both the SFL and DFL were varied, respectively. Similarly, when considering the 50%-of-peak point on the transmitted curve, the PDSFs broadened by 0.2 ± 0.1 mm per 100 ps, 0.24 ± 0.03 mm per 100 ps and 0.16 ± 0.07 mm per 100 ps when varying the SFL, DFL, and both the SFL and DFL, respectively. Therefore, our experimental data imply that altering the system TIRF by altering either the source or detector response (or both) results in approximately equivalent effect on the instrument PDSF breadth. We discuss the implications of this below. In the tomographic image reconstruction problem, this PDSF broadening due to instrument TIRF would directly result in loss of imaging resolution.

For figures 4a–c, we chose to use the “fraction of peak” as the x-axis, as opposed to the absolute photon arrival time. The rationale was that these represent points on the measured TR curve with approximately equivalent signal-to-noise ratios (SNR), which, in practice are most relevant for comparison of different instrument configurations. (It is noted that the peak value on the TR curve was slightly less for configurations with longer TIRFs, but that this effect was minimal. Between all configurations studied here the SNR of the peak varied by 3dB or less). However, we note that the 10%-of-peak point on the TR curve is actually “later” on a slower system than the equivalent point on a faster system, i.e. since the rise portion of the curve is broadened. We therefore re-plotted the data from fig 4c on an absolute time scale in fig 4d. As shown, PDSF broadening was still observed with the slower configurations (TIRF FWHM = 716 ps) relative to the faster configurations (TIRF FWHM = 163 ps). For example, PDSFs measured in the first 50ps were broadened by about 1.5 mm or 25%. Here, time curves were aligned at the 0.1%-of-peak point; experimentally, this represents a “very-early” point on the rising edge of the curve which was about 50 photon counts above background (assuming 5 × 104 photon count peak intensity). In principle, the very earliest points on the 716 ps curve should have PDSF widths that are equivalent to the fastest configuration, but in practice these time points have extremely low SNRs (or are below the system noise floor). Therefore, measurement of even the earliest detectable photons on the transmitted TR curve with the slower system would result in significant loss of tomographic imaging resolution at any early time-point.

3.4 Monte Carlo Analysis of Time-Resolved Instrument PDSFs

We then considered the effect of system TIRF on instrument PDSFs computed with time-resolved Monte Carlo (MC) simulations. To consider the effect of finite system TIRFs, we performed point-by-point convolutions of the MC-computed PDSFs (due to an infinitely short laser pulse) with experimentally measured instrument TIRFs. Example results are shown in fig 5a for the original output of the MC simulation, as well as convolutions with measured system TIRF FWHM of 163, 223, 375 and 716 ps (corresponding to source and detector fiber lengths of 1, 5, 10 and 20 m, respectively). In this figure the PDSF FWHM as a function of position on the rising edge of the TR curve is shown for each case. As indicated, the results of the MC simulations were qualitatively similar to our experimental results shown in fig 4c, with the PDSF FWHM increasing in width with increasing instrument TIRFs. For example, at the 10%-of-peak point on the rising edge of the TR curve, the TR MC simulation predicted a PDSF FWHM of 5.75 mm, representing a 45.2% relative reduction in width versus the un-gated CW case. When non-ideal system TIRFs were considered, this resulted in PDSF FWHM broadening up to 7.25 mm (for TIRF FWHM = 716 ps). Earlier time points resulted in even greater relative broadening; for example, at the 0.1%-of-peak-point, the PDSF FWHM broadened from 3.25 mm (MC) to 6 mm (for TIRF FWHM = 716 ps).

Figure 5.

Figure 5

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a) Convolution of the output of time-resolved MC simulations with experimentally measured instrument TIRFs at different points on the rising edge of the transmitted curve. The effect of TIRF on broadening of the instrument PDSF at the middle plane of the imaging chamber are shown for MC calculations, and experimental measurements at the (b) 10% and (c) 50% of maximum on the rising edge of the time-resolved curve.

Overall, time resolved MC computations yielded 0.2 ± 0.02 mm per 100 ps broadening of the PDSF FWHM for the 10%-of-peak point, and a 0.15 ± 0.01 mm per 100 ps broadening for the 50%-of-peak point as shown in figs 5b and c. Generally this analysis agrees well with our experimental data (plotted on the same figures for clarity) where an average of 0.3 ± 0.02 mm and 0.17 ± 0.03 mm per 100 ps broadening of the PDSF FWHM was observed for the 10%- and 50%-of-peak points, respectively. We extrapolated the experimental best-fit curves in figs 5b and c to the respective y-axes (corresponding to infinitely short TIRFs) and obtained 5.45 ± 0.08 mm and 6.8 ± 0.1 mm for the 10% and 50%-of peak points respectively. This was in good agreement with our original MC results (also corresponding to infinitely short TIRFs), of 5.75 and 6.75 mm for the two points. The slight discrepancy in slope of the 10%-of-peak curves may have been due to minor mismatch between the optical properties of our liquid phantoms and those used in the MC theory, or due to a slight mismatch in the measured system TIRFs with a white paper versus that with intralipid in place.

4. Discussion and Conclusions

In recent years, there has been significant renewed interest in the fields of DOT and FMT in the measurement of early-arriving photons to improve imaging resolution, partially driven by the development of new, high speed light sources and detectors (Turner et al., 2005; Leblond et al., 2009; Valim et al., 2010; Venugopal et al., 2010b; Zhang et al., 2011; LaPointe et al., 2012). Our previous work indicated that a reduction in the relative instrument PDSF width by approximately a factor of 2 could be realized over a wide range of conditions by measuring photons that arrived at the 10%-of-peak point on the transmitted TR curve using our super-continuum laser and PMT combination (Valim et al., 2010). It was also determined that further reduction in PDSF width could not be obtained at earlier time points (<10%-of-peak), even though MC simulations predicted that a reduction of 3–4 times should be obtainable. In other words, we experimentally observed a “leveling out” of the early photon effect that was not predicted by theory. As such, the experiments described in this work were performed to quantify the relationship between system TIRF and time-dependent instrument PDSF widths over a range of response times that were generally representative of previously reported literature values (from 163 ps to 716 ps).

Two important conclusions from this work are noted. First, while it is generally understood that “faster is better” for time resolved DOT and FMT systems, our data indicate that substantial disagreement was observed versus TR MC models for all experimental instrument TIRFs considered. Further, the effect was larger at earlier time points, where up to 19%, 32% and 40% error were observed at the 50%-, 10%-and 5%-of-peak points on the transmitted intensity curve, respectively. Experimental and Monte Carlo analysis showed that this broadening was fairly linear and was 0.15 to 0.3 mm per 100 ps for system TIRF (depending on the position on the time-resolved curve). As such, these results confirm that the “leveling out” of the early-photon effect observed in our previous work was most likely due to non-ideal system TIRF. Although systems with longer TIRFs resulted in temporally extending the rise portion of the transmitted intensity curve, in practice all usable points – in this case all points on the rising portion above the TR transmitted curve above 0.1% of the peak value - yielded broadened PDSFs versus faster systems. In a DOT or FMT imaging system, this would yield a corresponding loss in imaging resolution. Likewise, significantly narrower PDSFs and improved imaging resolution should be obtainable with faster systems, although this was not explicitly confirmed here. However, it is critical to re-iterate that the imaging resolution obtained in DOT and FMT is not only a function of instrument PDSF width, but also the noise and sensitivity properties of the measurements, i.e. two systems with identical TIRF but disparate signal to noise ratios will yield different overall imaging performance.

Second, our data indicates that PDSF broadening was approximately equal when the instrument TIRF was increased on either the source or detector sides (or both simultaneously). In terms of the design of time-resolved DOT or FMT systems, this implies that no particular advantage would be obtained by using a faster pulsed laser or detector, so that the net effect on the overall system TIRF should be considered (again, this neglects effects of detector sensitivity and noise performance in the DOT inverse problem). As noted in the introduction, several groups have attempted to measure early photons with a range of detector types, including MCPs, PMTs, ICCDs, streak cameras and fast APDs. One evident issue therefore is the generalizability of these results for systems with alternate component types. For example we cannot rule out the possibility that two detectors with identical TIRFs but with different sensitivity and noise properties would yield different instrument PDSF widths. However, the results of our studies with time-resolved MC data generally agree very well with our experimental data, and therefore our data indicates that simple convolution of the instrument TIRF appears to entirely account for the observed PDSF broadening effect. These data also underscore the importance of correcting PDSFs for instrument TIRFs via convolution when performing tomographic image reconstruction to ensure accurate data-model agreement.

Acknowledgments

This work was funded with a grant from the National Institutes of Health (R01EB012117-01) and from a Northeastern University laboratory startup grant.

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