Instrumental, optical and geometrical parameters affecting time-gated diffuse optical measurements: a systematic study

: In time-domain diffuse optics the sensitivity to localized absorption changes buried inside a diffusive medium depends strongly on the interplay between instrumental, optical and geometrical parameters, which can hinder the theoretical advantages of novel measurement strategies like the short source-detector distance approach. Here, we present a study based on experimental measurements and simulations to comprehensively evaluate the effect of all different parameters. Results are evaluated exploiting standardized figures of merit, like contrast and contrast-to-noise ratio, to quantify the system sensitivity to deep localized absorption perturbations. Key findings show that the most critical hardware parameter is the memory effect which ultimately limits the dynamic range. Further, a choice of the source-detector distance around 10 mm seems to be a good compromise to compensate non-idealities in practical systems still preserving the advantages of short distances. This work provides both indications for users about the best measurement conditions and strategies, and for technology developers to identify the most crucial hardware features in view of next generation diffuse optics systems.


Introduction
Since decades light has been considered a powerful tool to analyze diffusive media [1] such as biological tissues [2], wood [3], fruits [4], etc.Indeed, both medium composition and microstructure are strictly connected to its optical properties (absorptiona μ -and reduced scattering -' s μ -coefficient, respectively).Therefore, light in the so-called "therapeutic window" (from 600 to 1100 nm, where tissues behave as low absorbing and highly scattering media) can be exploited to non-invasively gather information about relative concentration of tissue constituents for several diagnostic purposes [5][6][7][8], to monitor blood oxygenation in muscles [9,10], or to detect tumors [11] or localized functional activations in the brain [12,13], just to give a few examples.
Within this framework, time-domain (TD) diffuse optics relies on the injection of fast (~100 ps) light pulses into the medium and on the collection of the distribution of time of flight (DTOF) of the re-emitted photons at a certain distance (i.e.source-detector separationρ -) [14].Since a μ and ' s μ affect the DTOF in different ways, they can be recovered independently, provided that a proper model of photon migration is given.Another important feature of the TD technique, for measurements performed in reflectance geometry, is the possibility to exploit the information about arrival time of photons to probe different depths into the investigated medium [15].For its high information content, TD technique is considered attractive even if it requires more complex and expensive instrumentation with respect to systems operating in the continuous wave (CW) regime [16] or in the frequencydomain (FD) [17].In particular, it has been recently demonstrated that CW instruments, at the level of ultimate performances, can reach a significantly lower penetration depth (~3 cm instead of ~6 cm of TD technique for the same paradigmatic case) and have a sensitivity to localized heterogeneities (e.g.localized brain functional activation) orders of magnitude lower [18].Further, during the last decade, many technological advancements for sources, detectors and time-tagging electronics had been achieved [19][20][21], thus allowing new TD systems [22] to reach a level of complexity, dimension and cost closer to CW devices [23].
To go in the direction of wearable devices and to simultaneously achieve the ultimate performances from TD technique, two requirements are needed [18]: availability of dense grids of fast time-gated (~100 ps) detectors to maximize the photon collection area (~1 cm 2 ) and similar dense grids of laser injection points to maximize the amount of light that can be injected into the tissue without exceeding the safety limits for skin.For this purpose, source and detectors have to be placed close to each other (i.e. in the so called small ρ approach) to maximize the tissue coverage.In principle, the small ρ brings advantages that have been studied in previous papers [24,25] that are: i) improved contrast at all photon arrival times along the DTOF; ii) improved spatial resolution due to higher photons confinement; iii) increased signal at all times.However, at small ρ (<1 cm), the number of early photons overcomes the number of late photons by orders of magnitude, thus saturating the detection chain with photons that only had the time to probe the tissue surface [24].To avoid this, a detector with a wide dynamic range of 5-8 orders of magnitude is required, thus being able to detect the few late photons arriving after the burst of early ones.It was demonstrated that this is feasible by using a time-gated Single-Photon Avalanche Diode (SPAD) [26] in combination with suitable measurement strategies [27].This allowed the use of the technique in several applications ranging from non-contact or single fiber spectroscopy to tomography and its application for medical imaging [6,[28][29][30].However, with respect to the estimated theoretical advantages using ideal systems [24], real measurements performed at small ρ showed non-idealities and limitations.In particular, these are due to: i) system characteristics, ii) geometrical and optical conditions; iii) measurements strategy.
Most of the relevant system parameters are related to the shape of the Instrument Response Function -IRF -(i.e. the temporal response of the system taken as a whole) [31].In particular, the key features are: i) Full-Width at Half-Maximum (FWHM); ii) fast (tens of ps) or slow (ns) decay tails [32]; iii) a specific source of noise termed memory effect [33,34], which sets a limit to the maximum achievable dynamic range.In addition, other system characteristics (such as the detector noise, the available laser power and the overall collection efficiency) can surely affect the sensitivity to small optical objects.
The geometrical conditions that can affect the measurement performance are the position and size of the optical perturbation to be detected, while the optical conditions are the background optical properties and the amount of perturbation with respect to the background.Of course, these are external factors related to the specific problem under study and out of the control of the operator.Among the controllable parameters, the source-detector distance ρ is crucial since it can severely affect the measurement outcome [24,32].Other relevant issues to be considered are the temporal position of the enabling gating windows applied to the detector and the integration time of the acquisition.
In this work, a systematic experimental and theoretical study on the optimal conditions for short-distance measurements is carried out with a double aim: i) to provide a guide to scientists to choose the best conditions for measurements protocols and ii) to understand which are the critical system parameters so as to provide to technology developers a set of relevant hardware features which can be considered key factors to maximize the performance of future generation of diffuse optics instruments.For what concerns the first aim, it is clearly of the utmost importance to get the best measurement results from existing systems.Just to give an idea, in many biomedical applications of diffuse optics (e.g.optical mammography, functional brain imaging, etc.) it is well known that the placement of optical fibers and the choice of the source-detector distance plays a fundamental role.Indeed, as discussed above, the use of a small ρ coupled to fast-gated acquisition is expected to improve the spatial resolution in measuring heterogeneous media.This can be relevant in applications like brain imaging or tumor characterization, where the spatial selectivity is important to reduce the contamination of the information from surrounding tissues.However, possible problems arising at short ρ (e.g.memory effect) can strongly affect the measurement, thus resulting into a dramatic loss of performance.Therefore, many times, there is the need to find a tradeoff, which is done on the basis of the scientist's experience.Another crucial point when measuring heterogeneous biological tissues is the fine tuning of the system.For example, pulsed laser diodes often allow the user to optimize the power tune, forcing the choice between a condition with a more ideal shape of the IRF with low emitted power and a condition with broad laser pulses with high emitted power.Also in this case, a trade-off based on the experience of the operator is usually applied, often without a solid theoretical study.
The study presented here is restricted to absorption perturbations (thus with a reduced scattering coefficient in the inhomogeneity equal to the background).However, this case is paradigmatic for several medical applications such as functional near-infrared spectroscopy or tumor detection, where mainly variations in absorption are expected.
In this study, the effect of some of the above listed relevant parameters have been systematically studied both with phantom measurements (where a particular instrument has been tested in comparison to simulations) and with simulations (thus enabling to extend the previous results to other possible instruments, e.g.featuring a different IRF shape).Conclusions are drawn considering the standardized figure of merits (Contrast and Contrastto-Noise Ratio) defined in the "nEUROPt" protocol [35], which is an internationally agreed procedure for the performance assessment of TD instruments.In this way, scientists can be supported in the choice of the best experimental conditions and measurement strategies to exploit at the best their system depending on the problem under study (e.g.detection of localized brain activation, tumor tissue characterization, etc.).
The paper is organized as follows: in Sec. 2 we describe the experimental setup, the simulation tool and the data analysis; Sec. 3 deals with the results obtained with phantom measurements and simulations; finally, in Sec. 4 we draw the conclusions of the work and discuss future perspectives.

Materials
Fig. 1   ps in order to allow the detection of photons within a 5 ns hardware gate.The different delays of the opening of the hardware gate were achieved by delaying the triggering signal through a home-made transmission-line based delayer (minimum step: 25 ps) [37].Following the procedure defined in the nEUROPt protocol, the optical perturbation (details in section 2.2.1) was immersed in the liquid medium in upright position thanks to a white-painted thin rigid wire, which was inserted from underneath. Figure 1(b) shows the reconstruction of the IRF of the setup acquired using the time-gated acquisition strategy as demonstrated in [27].The IRF has two distinct exponentially decaying tails.First, the well know diffusion tail, with typical time constants (τ 1 ) of 80 ps.The second tail is the so called "Memory tail", a slowly decaying tail that has been well described in [34].It is due to an increase in the background noise during the hardware gate, which is proportional to number of photons impinging on the detector before its opening.For this reason, the contribution of the memory effect is higher when a short source-detector distance is used, due to the increase of early photons that are rejected -i.e.not detected -by the time-gating technique.
The memory effect is a signal dependent phenomenon, since its contribution to the background noise increases with increase in the number of photons impinging on the detector.It has a typical time constant (τ 2 ) of around 1800 ps.The memory amplitude (A 2 ) is strongly dependent on the number of impinging photons.

Phantom and optical perturbation (equivalent black volume)
The liquid phantom hosting the optical perturbation was dilution in water of Intralipid 20% and Higgins India ink [38].The desired optical properties of the phantom were targeted by a precise proportion of Intralipid and India ink and by using their calibrated values for the scattering and absorption properties, respectively.
For what concerns the optical perturbation, we used totally absorbing objects, i.e. black cylinders made of Polyvinyl chloride (PVC) with different volumes.It was demonstrated in [39,40] that small totally absorbing objects are suitable to reproduce with any realistic absorption inhomogeneity with high reliability and reproducibility.More in detail, in a medium with a background characterized by a given ' s μ , an Equivalence Class of absorbing perturbations characterized by different shapes and volumes can be defined.All perturbations under this Equivalent Class affect the DTOF in the same way for any choice of geometry (e.g.ρ), a μ of the background and position of the inhomogeneity within the medium.For this reason, a totally absorbing object with a peculiar volume (Equivalent Black Volume -EBV -) can be used as a representative of an equivalent class of absorbing perturbations.It is worth to note that the use of the EBV is not perfectly valid for black objects with a very small volume (≤ 50 mm 3 ) and for shallower depths (≤ 10 mm).We adopted the EBV approach for its simplicity, reliability and high reproducibility of the measurements among different laboratories.Hence, we made use of black cylinders (height equal to diameter) with a volume of 25, 50, 100, 250 and 500 mm 3 , which are, for a medium with a ' s μ = 10 cm −1 , equivalent to realistic inclusions of volume 1000 mm 3 with a Δ a μ of 0.0065, 0.0100, 0.0162, 0.0362 and 0.0952 mm −1 , respectively [40].

Measurement strategy and data analysis
For each background medium, measurements were carried out at several ρ (namely 2, 4, 6, 8, 10, 15, 20, 25 and 30 mm) so as to compare results achieved with both the short distance approach (i.e.< 15 mm) and the classical ones.Once the ρ was set, the perturbation was aligned half way between the source and injection fibers and its depth (Z) was taken as the distance between half of the cylinder height and the surface of the liquid.The inclusion was moved in depth (Z from 5 to 30 mm at step of 2.5 mm) through a motorized axis controlled by the acquisition software.For each depth, 8 different portions (hardware gates) of the DTOF curves were acquired.This was done by changing the delay of the signal that triggers the generation of the 5 ns window hardware gate.For each delay, 5 repetitions of 1 s were recorded.
In the measurements, the effect of the optical perturbation was studied only on the phantom with a μ = 0.1 cm −1 and ' For what concerns data analysis, the repetitions were summed up to increase the signal to noise ratio.

Simulations
The effect of a localized absorption perturbation was simulated using an 8th-order perturbative solution of the Radiative Transfer Equation under the diffusion approximation and the Extrapolated Boundary conditions [41].The theoretical DTOF was convoluted with different IRFs to simulate the effect of the measurement apparatus.The finite count rate of the measurement system was taken into account by limiting the maximum number of counts per DTOF to 10 6 counts/s.Also, hardware time-gating used to suppress early photons and avoid saturation of the count rate at short ρ was simulated by slicing the DTOF before applying the count rate limitations.A realistic estimate of the DTOF amplitude (i.e.number of counts) was derived from the objective assessment of the system responsivity -as defined in the BIP Protocol [31] -converting the reflectance signal exiting the medium into detected counts.This information was then used to properly add shot noise to the simulated DTOFs.
The simulator consisted in an external framework with 4 iterators permitting to arbitrarily change different parameters in a range (e.g.background optical properties, absorption perturbation, features of the IRF, location of the inhomogeneity) and a central kernel performing the calculus of the homogeneous and perturbed DTOF.The external iteration framework was written in Matlab, while the kernel was compiled in C for computational efficiency.A typical simulation scenario, involving around 25000 points, is run automatically in about 10 minutes on a standard PC.
The standard value of the several parameters considered in the simulations are listed in Table 1 and, unless differently stated, they are valid for all simulations presented.

Figures of merit
To evaluate the performances obtained both in measurements and simulations, we made use of two standardized figures of merits (contrast -C -and contrast-to-noise ratio -CNR -), which were defined in the nEUROPt protocol [35] to assess the sensitivity (i.e.detection capabilities of a small absorption change) of a TD instrument.This choice is justified by the fact, as stated above, that this study is focused on measurements of heterogeneous media.Indeed, contrast and CNR are figures of merit devised to assess the capability of a system in detecting perturbations inside diffusive media (e.g. to distinguish a region where there is a brain activation or a tumor within a surrounding tissue).More in detail, the contrast is an indicator of the effect of perturbations (in this case only in absorption) on photon counts.However, photon counting is dominated by Poisson statistics and other effects due to instabilities of the system.Hence, the contrast is not sufficient to evaluate the detectability of perturbations.This is the reason why also CNR has been devised, as it measures the reliability of the contrast by comparing the variation of the number of photon counts with the fluctuations of the counts.
For the calculation of contrast and CNR, we made use of the so-called "time windowing" of the curve, meaning that contrast and CNR were computed for portions of the DTOF curve ("software gate").For both measurements and simulations, the width of the software gates used was set at 500 ps, while their position (i.e. the delay of the rising edge of the gate, reported as t g in the following equation) is referred to the peak of the IRF.Due to the structure of the simulator code, in the simulation study the software gate was opened at the beginning of the hardware one.For comparison purposes, the same strategy was applied also to experimental data analysis.
The contrast is defined as the absolute difference between the unperturbed (i.e.homogeneous medium) and perturbed (i.e.optical perturbation inside the medium) states, normalized by the unperturbed state.It was computed for all depths of the inclusions following Eq.( 1) [35]: where, N 0 is the overall number of counts in the software gate in the unperturbed case, while N is referred to the perturbed state.For the measurements, the acquisition of the homogeneous case was performed moving the perturbation 4 cm in depth, were its effect was completely negligible.
The CNR was computed following the definition given in [35] and reported in Eq. ( 2): where the difference at the numerator is calculated over the 5 repetitions while 0 ( ( )) is the standard deviation of the number of counts in the homogenous case computed over the repetitions.It has to be noted that in simulation the only noise source is the photonic one [42] thus while in measurements the standard deviation term can be affected also by other elements such as instabilities of the system (e.g.laser power fluctuations).

Results and discussion
The study consists of two parts.First, a comparative study between contrast calculated from experimental data and simulated data is presented.Secondly, a comprehensive simulation study addressed to different parameters is presented.In particular, we extend the simulation to study the effects of the parameters which can be classified under three categories, namely i) system parameters, which includes the IRF features (e.g.diffusion tail amplitude and time constant, FWHM, etc.), average laser power, system responsivity and noise sources (dark counts and afterpulsing probability); ii) geometrical and optical parameters such as the amount of the perturbation and its depth, the background optical properties of the medium and iii) measurement strategies namely source-detector separation, acquisition time and hardware gate opening delay.
For the sake of brevity, we discuss here two cases: i) the fast-tail slope and ii) the memory tail amplitude.These parameters, as reported in [32,33], represent the main limitation to the achievable performance in terms of contrast and maximum penetration depth inside the tissue.More in detail, the fast tail is responsible for the temporal spreading of photons which dilute the contrast, while the memory effect sets a limit to the dynamic range of the measurement.Additionally, those two parameters are intrinsic of the detector (they depend on the fabrication process), therefore it is not possible to tune them in real systems in order to appreciate their effect on the measurement performance.All the other dependences can be found in Dataset 1, Dataset 2, Dataset 3, Dataset 4, Dataset 5, Dataset 6, Dataset 7, Dataset 8, Dataset 9, and Dataset 10 reported at [43][44][45][46][47][48][49][50][51][52] and the main conclusions drawn are reported in Table 2.

reports plots of contrast vs time, at different depths of inclusion (rows) and absorption perturbation (columns).
For each situation, the contrast for 5 different sourcedetector separations is reported (colors in each plot).Figure 3 reports the CNR in the same conditions.In each picture, contrast or CNR obtained from measurements (dots) and simulations (continuous curves) are compared.As it can be observed, there is a fair agreement between measurements and simulations in the case of contrast.Some discrepancies (mainly evident for larger source-detector distances) are most probably due to loss in counts which is experienced at late times for larger ρ and Z values.In the case of CNR however, the agreement is much worse as simulations only take into account fluctuations due to Poisson noise and not all the sources of instability that can affect real systems (such as fluctuations in laser power, spurious leakage of room light, etc.).Hence, the quantification of CNR from simulations should be only considered as the maximum CNR one should expect by operating a system with perfect stability.
For what concerns the CNR, we can see a trend similar to the one of contrast, meaning that the as the optical perturbation increases its value, we can observe a peaking of the CNR curve, as well as a (slight) increase in the amplitude.On the other hand, the peak of the CNR moves to later times when the inclusion is posed deeper.It is worth noting that CNR curves are not as smooth as contrast ones.This behavior is due to the implementation of the timegated technique, which allows to adjust the optical power injected into the medium under investigation in each hardware gate, depending on its delay with respect to t = 0 ns.When the hardware gate is opened at early delays, the signal in the DTOF is so high that it must be attenuated not to exceed the maximum count rate allowed by the system ("count rate limitation" in Sect.2.3).On the contrary, at later delays, the signal in the DTOF is orders of magnitude lower (in particular at short ρ), thus reaching a condition where the attenuation of the signal is no more needed to fit the maximum count rate.Hence, the sharp corner in the CNR curves is due to the transition between two different regimes of operation, on the left (early times) the signal is adjusted at each delay of the hardware gate not to exceed the maximum count rate, while on the right (late times) the signal is kept constant.This effect is not present in contrast curves just because they do not depend on the number of counts.
It can also be noticed that the change in contrast upon increasing ∆µ a (moving along columns) is dependent not only on Z (rows) but also on ρ.By moving along columns, we can see that contrast increases with increasing ∆µ a , as expected.Indeed, the probability of a photon having encountered the inclusion increases with increasing absorption perturbation.
For this reason, keeping the other conditions equal (i.e.depth and source-detector distance), the contrast curves are supposed to be rescaled by a factor given by the different Δμ a .If moving along the rows (i.e.perturbation buried more in depth), the contrast decreases due to the spreading of the sensitivity profile of photons.Additionally, the peak corresponding to maximum contrast shifts towards longer time.This is due to the fact that reemission time of photons encodes the mean probed depth.Thus, as it is well known working in the TD, for superficial inclusions the maximum contrast is provided by photons re-emitted earlier in time compared to those for the deeper inclusions.The contrast decrease for largest arrival times is mainly due to the memory tail (in particular for shorter ρ) and to the background noise floor given by both DCR and AP (in particular for larger ρ).
Crucially, under the effect of the memory tail, we observe that the maximum contrast is not obtained for small ρ values, as it was theorized in [24] for ideal systems, but for intermediate values (i.e., ρ = 10-15 mm).This can be ascribed to the fact that memory tail limits the dynamic range of the system and hence clamps down the contrast at late times.However, since memory is a signal-dependent phenomenon [34], the reduction in contrast is maximum for small ρ values, where the early arriving photons signal is maximum [24].
As stated above, the decrease in contrast for larger ρ is due to the detector background noise.Indeed, it is not possible to increase the power injected endlessly.Thus, the ratio between the curve and the noise floor decreases at later delays, demoting the contrast in particular at large ρ.Such an effect of the noise floor can be better appreciated in the figures reporting the simulations varying the noise floor (see [49,50].) and also in those reporting the variations in contrast upon a variation of the injected power (see [47].).
Tables 3-7 in the Appendix report the relative errors between measurements and simulations for each point of Fig. 2 and Fig. 3.In general, apart from the discrepancies discussed above, it is possible to verify a fair concordance, having in many cases errors in the order of 10-20%.
Measured and simulated data have been also compared by changing 5 different optical properties of the background medium, keeping constant this time the black cylinder perturbation (i.e. the one with a 100 mm 3 volume, which features, for a medium with ' s μ = 10 cm −1 , a Δ a μ = 0.0162 mm −1 in a volume 1000 mm 3 ).Errors are consistent with the previously discussed study performed with different perturbations.Results, not reported here for brevity, are available at [52].    5) shows the simulated contrast (CNR) vs time for different source-detector separations upon a variation of the depth (rows) of the inclusion (with Δµ a = 0.0162 mm −1 ) and diffusion tail slope (τ 1 ) of the IRF (columns).As the diffusion tail slope decreases (i.e. the IRF get sharper with a fast decrease of the first diffusion tail), we observe a narrowing of the contrast curves due to the reduced contamination of the early photons in a temporal region of the curve where only contribution of late photons are expected, as predicted in [32].Indeed, for shorter ρ, the reflectance curve is less broadened and so, in the convolution process, a large diffusion tail can dominate when compared to the ideal (i.e.IRF equal to a delta Dirac) response of the medium.A reasonable upper threshold for τ 1 is around 200 ps where there are no significant effects at least for ρ≥10 mm.Even in the almost ideal case of τ 1 = 1ps (last column) the contrast at the shorter ρ is suboptimal because of the memory effect.We recall here that simulations are performed varying specific parameters at a time (here ρ, Z and τ 1 ) while keeping the others (e.g.memory effect) fixed at a given pivotal position, reported in Table 1.  and Δµ a can be found at [45].The threshold condition for τ 1 is changing (and getting stricter) for more harsh environments (higher a μ , lower ' s μ ).Partially, this can be compensated by increasing ρ.
In Fig. 6 (Fig. 7) we see the variation of contrast (CNR) with Z (rows) and memory tail amplitude (A 2 , columns), keeping the memory tail slope fixed at 1800 ps, which is the experimental value recovered using the fast-gated SPAD detector in the system in Fig. 1(a).
As it is evident from the figure, the contrast improves by lowering the memory tail.The improvement is largest for small ρ, as the contribution of memory effect to the background noise increases with increase in the number of early photons impinging on the detector.Hence, generally speaking, we do not obtain a higher contrast at null source-detector separation by using detectors which have a memory tail amplitude even 7 decades below the peak.Anyway, it has to be noted that the larger the memory tail amplitude the more detrimental the effect on the contrast.This fact shows therefore that the memory effect sets the main limitation to the measurement performance in time gated systems.Here, a reasonable threshold is observed with a memory effect at least 6 decades below the peak.Unfortunately, this parameter can hardly be controlled even at the level of device fabrication.The same conclusions can be drawn looking at the CNR trend upon a variation of the memory tail amplitude, which are shown in Fig. 7.
For the sake of the synthesis, the key findings including also these additional parameters are summarized in Table 2.

Conclusions
The null or small source-detector separation approach (ρ ~0) has been proposed in timedomain diffuse optics more than ten years ago as a promising way to maximize measurement performances in terms of overall amount of detected signal, contrast produced by localized absorption perturbations inside the scattering medium and spatial resolution related to the perturbation localization.This could be achieved under the hypothesis to make use of a measurement system characterized by and ideal temporal response, free from slow tails or any source of background noise [24].However, implementations in real systems always showed limited advantages [25].In this work, using the state of the art system for the implementation of the small source-detector separation approach, we experimentally studied the effect of the source-detector distance choice and perturbation size and depth.The response function of our system, far different from a Dirac delta, as expected, gave rise to a compression of the contrast produced by the perturbation when the system is used at the smallest ρ (2 mm).Since also the contrast at the largest ρ (30 mm) is limited due to the broadening of the sensitivity shape, it turns out that in most cases the choice of a ρ of 10 or 15 mm represents the best condition to maximize both contrast and contrast-to-noise ratio.
The same scenarios were also studied in-silico, confirming the previous findings about optimal values for the source-detector distance.Hence, simulations allowed us to extend the study and understand the effects of different non-idealities in the instrument response function, simulating changes in different regions of the system temporal response (e.g.fullwidth at half maximum, diffusion tail amplitude and slope and memory effect amplitude).Similarly, other systems features were studied (e.g.dark count rate), as well as effects of the measurement geometries, optical properties and measurements strategies.
Among the main findings, it is evident that the memory effect represents the main limiting factor for the use of a small source-detector separation approach.Indeed, when the memory effect is particularly high (A 2 > 10 −7 ), it is capable to force the choice of an intermediate ρ (e.g.10-15 mm) to obtain the best contrast, thus completely reverting the direction of previous findings of [24], whose validity is not questioned here, since the study was intentionally restricted to ideal systems.A summary of the key findings of this work, and the relevance of different parameters is reported in Table 2.
Thanks to the broad spectrum of cases reported here, this work can represent a reference for scientists interested in the use of time-domain diffuse optics systems, allowing them to select the best measurement strategy and probe geometry depending on their specific problem under study and on the specific features of their hardware.Even more interestingly, this paper provides figures of merit for the evaluation of the performance of novel technologies under development (lasers, detectors, timing circuits) in the field of time-domain diffuse optics, permitting the estimation of the effects of hardware features on measurements results.

Appendix
In this section, the errors between measured and simulated data of Figure 2 (for contrast) and Figure 3 (for CNR) are reported.They have been computed as: where  x is the measured value (of either contrast or CNR) and x is the simulated one.Table 3-7 report all the errors, depending on the depth of the perturbation.The value is not reported, being not meaningful, in conditions where the measured CNR is lower than 1 or where the number of counts inside the software gate is lower than 1000 for measured data. 200

Fig. 2 .
Fig. 2. Plot of contrast vs time (colors represent different source-detector distances), with change in depth of inclusion (Z) along the rows and absorption perturbation (∆µ a ) along the columns.The experimental points are reported as stars, while the continuous curves correspond to simulated values.

Fig. 3 .
Fig. 3. Plot of CNR vs time (colors represent different source-detector distances), with change in depth of inclusion (Z) along the rows and absorption perturbation (∆µ a ) along the columns.The experimental points are reported as dots, while the continuous curves correspond to simulated values.

Figure
Figure2reports plots of contrast vs time, at different depths of inclusion (rows) and absorption perturbation (columns).For each situation, the contrast for 5 different sourcedetector separations is reported (colors in each plot).Figure3reports the CNR in the same conditions.In each picture, contrast or CNR obtained from measurements (dots) and simulations (continuous curves) are compared.As it can be observed, there is a fair agreement between measurements and simulations in the case of contrast.Some discrepancies (mainly evident for larger source-detector distances) are most probably due to loss in counts which is experienced at late times for larger ρ and Z values.In the case of CNR however, the agreement is much worse as simulations only take into account fluctuations due to Poisson noise and not all the sources of instability that can affect real systems (such as fluctuations in laser power, spurious leakage of room light, etc.).Hence, the quantification of CNR from simulations should be only considered as the maximum CNR one should expect by operating a system with perfect stability.For what concerns the CNR, we can see a trend similar to the one of contrast, meaning that the as the optical perturbation increases its value, we can observe a peaking of the CNR curve, as well as a (slight) increase in the amplitude.On the other hand, the peak of the CNR moves to later times when the inclusion is posed deeper.It is worth noting that CNR curves are not as smooth as contrast ones.This behavior is due to the implementation of the timegated technique, which allows to adjust the optical power injected into the medium under investigation in each hardware gate, depending on its delay with respect to t = 0 ns.When the hardware gate is opened at early delays, the signal in the DTOF is so high that it must be attenuated not to exceed the maximum count rate allowed by the system ("count rate limitation" in Sect.2.3).On the contrary, at later delays, the signal in the DTOF is orders of magnitude lower (in particular at short ρ), thus reaching a condition where the attenuation of the signal is no more needed to fit the maximum count rate.Hence, the sharp corner in the CNR curves is due to the transition between two different regimes of operation, on the left (early times) the signal is adjusted at each delay of the hardware gate not to exceed the maximum count rate, while on the right (late times) the signal is kept constant.This effect is not present in contrast curves just because they do not depend on the number of counts.It can also be noticed that the change in contrast upon increasing ∆µ a (moving along columns) is dependent not only on Z (rows) but also on ρ.By moving along columns, we can see that contrast increases with increasing ∆µ a , as expected.Indeed, the probability of a photon having encountered the inclusion increases with increasing absorption perturbation.For this reason, keeping the other conditions equal (i.e.depth and source-detector distance), the contrast curves are supposed to be rescaled by a factor given by the different Δμ a .If moving along the rows (i.e.perturbation buried more in depth), the contrast decreases due to the spreading of the sensitivity profile of photons.Additionally, the peak corresponding to maximum contrast shifts towards longer time.This is due to the fact that reemission time of photons encodes the mean probed depth.Thus, as it is well known working in the TD, for superficial inclusions the maximum contrast is provided by photons re-emitted earlier in time compared to those for the deeper inclusions.The contrast decrease for largest arrival times is mainly due to the memory tail (in particular for shorter ρ) and to the background noise floor given by both DCR and AP (in particular for larger ρ).Crucially, under the effect of the memory tail, we observe that the maximum contrast is not obtained for small ρ values, as it was theorized in[24] for ideal systems, but for intermediate values (i.e., ρ = 10-15 mm).This can be ascribed to the fact that memory tail limits the dynamic range of the system and hence clamps down the contrast at late times.However, since memory is a signal-dependent phenomenon[34], the reduction in contrast is maximum for small ρ values, where the early arriving photons signal is maximum[24].As stated above, the decrease in contrast for larger ρ is due to the detector background noise.Indeed, it is not possible to increase the power injected endlessly.Thus, the ratio between the curve and the noise floor decreases at later delays, demoting the contrast in

Fig. 4 .
Fig. 4. Plot of Contrast vs time, with change in depth of inclusion (Z) along the rows and diffusion tail slope (τ 1 ) along the columns.

Fig. 5 .
Fig. 5. Plot of CNR vs time, with change in depth of inclusion (Z) along the rows and diffusion tail slope (τ 1 ) along the columns.

Figure 4 (
Figure4(Fig.5)shows the simulated contrast (CNR) vs time for different source-detector separations upon a variation of the depth (rows) of the inclusion (with Δµ a = 0.0162 mm −1 ) and diffusion tail slope (τ 1 ) of the IRF (columns).As the diffusion tail slope decreases (i.e. the IRF get sharper with a fast decrease of the first diffusion tail), we observe a narrowing of the contrast curves due to the reduced contamination of the early photons in a temporal region of the curve where only contribution of late photons are expected, as predicted in[32].Indeed, for shorter ρ, the reflectance curve is less broadened and so, in the convolution process, a large diffusion tail can dominate when compared to the ideal (i.e.IRF equal to a delta Dirac) response of the medium.A reasonable upper threshold for τ 1 is around 200 ps where there are no significant effects at least for ρ≥10 mm.Even in the almost ideal case of τ 1 = 1ps (last column) the contrast at the shorter ρ is suboptimal because of the memory effect.We recall here that simulations are performed varying specific parameters at a time (here ρ, Z and τ 1 ) while keeping the others (e.g.memory effect) fixed at a given pivotal position, reported in Table1.

Fig. 6 .
Fig. 6.Plot of Contrast vs time, with change in depth of inclusion (Z) along the rows and memory amplitude (A 2 ) along the columns.

Fig. 7 .
Fig. 7. Plot of CNR vs time, with change in depth of inclusion (Z) along the rows and memory amplitude (A 2 ) along the columns.The effect of diffusion tail slope with different background optical properties ( a μ , ' s μ )