Application limits of the scaling relations for Monte Carlo simulations in diffuse optics. Part 2: results

Abstract

The limits of applicability of scaling relations to generate new simulations of photon migration in scattering media by re-scaling an existing Monte Carlo simulation are investigated both for the continuous wave and the time domain case. We analyzed the convergence properties in various scenarios by numerical methods, trying to derive practical guidelines for the judicious use of this approach, as well as a deeper understanding of the physics behind such relations. In the case of scaling of the absorption coefficient, the convergence is always rigorous both for the forward and inverse problems, relying on the derivatives with respect to the absorption coefficient. Also, the regenerated simulation inherits the very same noise of the original Monte Carlo simulation. In the case of scaling of the scattering coefficient, the situation is more critical. For forward problems, even for just a 10% uniform increase in scattering, appreciable deviations are observed whenever a high number of scattering interactions is involved. We tested a practical criterion based on the number of scattering events in the original simulation to judge the convergence of the scaling factors. For inverse problems, the scaling relations provide accurate regenerated simulations apart from the noise level that is increased with respect to the initial simulation, although anyway lower than the noise level obtained by implementing the direct calculation. The results of this study are important whenever an increase of Monte Carlo code throughput is mandatory, e.g., for fast data analysis of diffuse data, or in machine-learning scenarios, when generating huge datasets is needed.

1. Introduction

The study of photon propagation in highly scattering media, with a wealth of applications ranging from biomedical optics to astrophysics, is usually supported by the Radiative Transfer Equation (RTE), based on photon-particle description of light propagation governed by the energy conservation laws [1]. Given the difficulties in deriving closed-form solutions in complex structures – e.g., the human head – numerical approaches based on Monte Carlo (MC) simulations are often adopted both for forward and inverse problems. By generating many possible photon trajectories on a statistical basis, MC methods permit to simulate any geometry and combination of optical properties – namely, the absorption coefficient ${\mu }_a$ , the scattering coefficient ${\mu }_s$ , the scattering phase function $p(\theta )$ , the refractive index $n$ . In particular, when photon propagation is dominated by multiple scattering, it is also useful to introduce the anisotropy factor $g$ , defined as the average cosine of the scattering angle, and the reduced scattering coefficient ${\mu }_s^{\prime }$ , defined as ${\mu }_s^{\prime }={\mu }_s(1-g)$ [2]. For this reason, MC methods became the gold standard for modeling light propagation in tissues or in diffusive media and an essential tool in biomedical optics [3]. To this extent, many MC codes have been developed for the purposes of several applications [4–16].

In this context, a set of rigorously derived scaling relations (SR) provides an efficient way to generate a new simulation with a perturbed set of optical properties ( ${\mu }_a={\mu }_{a0}+\mathrm{\Delta }{\mu }_a$ , ${\mu }_s={\mu }_{s0}+\mathrm{\Delta }{\mu }_s$ ) starting from an initial simulation obtained with a given set of optical properties ( ${\mu }_{a0}$ , ${\mu }_{s0}$ ) [4]. Indeed, even with parallel computing and GPUs, the computational burden of MC can be a limitation in realistic scenarios, particularly for inverse problems, which require many attempts to be tested. Then, MC codes could benefit from the use of SR, namely to reduce computational time.

Yet, there is more in SR than just that. The elegance of the SR and the beautiful physics behind them can unveil general properties of the RTE to ignite new findings.

In the previous companion paper [17], we briefly recalled the origin of the SR mostly in the neutron transport field [18–20], and their further extension to the biomedical optics field [4,21], encompassing also polarization [22], heterogeneous and inverse problems [23–26], photoacoustics [27]. We presented the derivation of the SR, for a homogeneous and a non-homogeneous medium, discussing the convergence properties of the method for absorption and for scattering. An insightful analysis led to the study of the probability function $p_{\ell }(k,{\mu }_s)$ for the number of scattering events $k$ undergone by a photon along a path of length $\ell$ for a given trajectory, which permitted to understand the frailty of the scaling factor for ${\mu }_s$ . Furthermore, the implications when using the derivative of the SR with respect to ${\mu }_a$ and ${\mu }_s$ in inverse problems were addressed.

This paper complements the previous contribution [17], where the theoretical bases of the method were discussed, presenting an extensive numerical study on the validity and convergence properties of the SR. This study faces many different cases and permits to gain both a general understanding of the formulas as well as its practical utility. In particular, the probability function $p_{\ell }(k,{\mu }_s)$ is derived numerically in different scenarios, to understand what determines the corrective factor for a change in scattering. Section 2 briefly recalls the key formulas, Section 3 presents the methods used in MC simulations and the metrics to grade convergence, Section 4 displays the key findings on the convergence properties and the shape of the the probability function $p_{\ell }(k,{\mu }_s)$ in various scenarios, encompassing continuous wave (CW) and time domain (TD), homogeneous and heterogeneous cases, finite perturbations and calculus of derivatives. Finally, Section 5 draws the general conclusions and presents future directions.

2. Theory

In this Section, we briefly report the main theoretical results about the scaling approach to MC simulations in a homogeneous medium, whose extensive demonstration can be found in the companion paper [17].

Firstly, we report the scaling factors for the absorption ( $F_a$ ) and scattering ( $F_s$ ) coefficients in a homogeneous medium:

$$F_a=\frac{W({\mu }_a,{\mu }_{s0})}{W({\mu }_{a0},{\mu }_{s0})}=e^{-({\mu }_a-{\mu }_{a0})\ell },$$ (1)

$$F_s=\frac{W({\mu }_{a0},{\mu }_s)}{W({\mu }_{a0},{\mu }_{s0})}={\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^k{e}^{-({\mu }_s-{\mu }_{s0})\ell },$$ (2)

where $W({\mu }_{a0},{\mu }_{s0})$ is the weight of each detected photon in the original simulation, while $W({\mu }_a,{\mu }_{s0})$ and $W({\mu }_{a0},{\mu }_s)$ are the weights of the same photon (detected after traveling along the same trajectory) in the scenarios where either the absorption coefficient or the scattering coefficient is changed.

In the time-resolved approach, the weights of the received photons are classified into a histogram of time-of-flights. Then, if the time bin width is assumed to be sufficiently narrow, all photons within the $i$ -th time bin have the same pathlength ${\ell }_i=v{t}_i$ and we can write:

$$TPSF({\mu }_a,t_i)=TPSF({\mu }_{a0}=0,t_i)e^{-{\mu }_{a}v{t}_i},$$ (3)

where $t_i$ is the average time of the $i$ -th time bin, $TPSF$ is the Temporal Point Spread Function, i.e., the probability that a photon is received in the $i$ -th time bin, $v$ is the speed of light in the diffusive medium, and the basic MC simulation is obtained for ${\mu }_{a0}=0$ .

On the contrary, due to the presence of the number $k$ of scattering interactions in Eq. (2), a straightforward calculation of the scaling factor caused by a variation of the scattering coefficient is not possible. However, if the probability function $p_{\ell }(k,{\mu }_{s0})$ for the number of scattering events $k$ undergone by the received photons with pathlength $\ell$ is known, the scaling factor for $TPSF({\mu }_s,t)$ is:

$$\begin{array}{rl}{F}_{TPSF,s}({\mu }_{s0}\to {\mu }_s,\ell )& =\sum _{k=0}^{\mathrm{\infty }}{p}_{\ell }(k,{\mu }_{s0})F_s(k,{\mu }_s)=\\ & =e^{-({\mu }_s-{\mu }_{s0})\ell }\sum _{k=0}^{\mathrm{\infty }}{p}_{\ell }(k,{\mu }_{s0}){\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^k,\end{array}$$ (4)

where $F_s(k,{\mu }_s)$ is given in from Eq. (2).

In particular, in the case of an infinite non-absorbing homogeneous medium with no detectors, the function $p_{\ell }(k,{\mu }_{s0})$ can be explicitly calculated provided that all propagating photons are considered:

$$p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})={\displaystyle \frac{{({\mu }_{s0}\ell )}^k}{k!}}{e}^{-{\mu }_{s0}\ell },$$ (5)

resulting in a Poisson distribution with mean value $\lambda ={\mu }_{s0}\ell$ . In the case of finite geometries and a receiving detector the probability function $p_{\ell }(k,{\mu }_{s0})$ has to be reconstructed by exploiting MC simulations, where both the trajectory length $\ell$ , and the number of scattering events $k$ are recorded for each received photon. In the following Section 4.1 we report some examples of these probability functions, named $p_{\ell ,MC}(k,{\mu }_{s0})$ .

In order to understand the limit of applicability of Eq. (4), we proposed the following approximation for $F_{TPSF,s}$ [17]:

$$F_{MC,s}({\mu }_{s0}\to {\mu }_s,\ell )=\sum _{k=k_{min}({\mu }_{s0})}^{k_{max}({\mu }_{s0})}{p}_{\ell ,\mathrm{\infty }}(k,{\mu }_s),$$ (6)

where $k_{min}({\mu }_{s0})$ and $k_{max}({\mu }_{s0})$ are the smallest and the largest number of scattering events of the simulated photons with pathlength $\ell$ , while the $p_{\ell ,MC}(k,{\mu }_{s0})$ of the initial simulation was approximated as:

$$p_{\ell ,MC}(k,{\mu }_{s0})\approx \{\begin{array}{ll}{p}_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})={\displaystyle \frac{{({\mu }_{s0}\ell )}^k}{k!}}{e}^{-{\mu }_{s0}\ell },& k_{min}({\mu }_{s0})\le k\le k_{max}({\mu }_{s0})\\ 0,& \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\end{array}$$ (7)

The calculation of $F_{MC,s}$ through Eq. (6) provides a criterion of convergence for the method: it can be used safely when $F_{MC,s}({\mu }_{s0}\to {\mu }_s,\ell )\approx 1$ .

As for the CW approach, if the time window spanned by the $TPSF$ is large enough, the collected photons can be obtained by integrating the $TPSF$ over time:

$$CW({\mu }_a,{\mu }_s)=\sum _{i=1}^{M}TPSF({\mu }_a,{\mu }_s,t_i)\mathrm{\Delta }t,$$ (8)

where $M$ is the number of time bins and $\mathrm{\Delta }t$ their width.

Finally, the SR for absorption and scattering coefficients can be used to evaluate the derivative with respect to the optical properties of the medium:

$${\frac{\mathrm{\partial }TPSF({\mu }_s,{\mu }_a,t_i)}{\mathrm{\partial }{\mu }_a}|}_{{\mu }_s={\mu }_{s0}}=-⟨\ell {⟩}_{i}TPSF({\mu }_{s0},{\mu }_a,t_i),$$ (9)

$${\frac{\mathrm{\partial }TPSF({\mu }_s,{\mu }_a,t_i)}{\mathrm{\partial }{\mu }_s}|}_{{\mu }_s={\mu }_{s0}}=⟨\frac{k}{{\mu }_{s0}}-\ell {⟩}_{i}TPSF({\mu }_{s0},{\mu }_a,t_i).$$ (10)

Moreover, for $\mathrm{\Delta }t$ small enough, we can set $⟨\ell {⟩}_i=v{t}_i$ , while the term $⟨k/{\mu }_{s0}-\ell {⟩}_i$ is almost nil, except for those time bins, i.e., pathlengths, where the probability functions $p_{\ell }(k,{\mu }_{s0})$ differ substantially from the correspondent Poisson distributions.

In the CW case, we obtain:

$${\frac{\mathrm{\partial }CW({\mu }_s,{\mu }_a)}{\mathrm{\partial }{\mu }_a}|}_{{\mu }_s={\mu }_{s0}}=-⟨\ell ⟩CW({\mu }_{s0},{\mu }_a),$$ (11)

$${\frac{\mathrm{\partial }CW({\mu }_s,{\mu }_a)}{\mathrm{\partial }{\mu }_s}|}_{{\mu }_s={\mu }_{s0}}=⟨\frac{k}{{\mu }_{s0}}-\ell ⟩CW({\mu }_{s0},{\mu }_a),$$ (12)

where the average values are calculated using all the received photons.

3. Methods

The MC simulations shown in this work were generated using three different codes: 1) an elementary and extensively validated MC code [2,3,28–31] that stands in continuity with the first MC code used to implement this approach for biomedical optics applications, i.e., Ref. [4]; 2) the open source GPU-accelerated MCX code [6,7]; and 3) the open source GPU-accelerated MCML code [8,9]. The use of multiple platforms to generate MC results is motivated on one side by the need to provide examples of applicability of the presented approach in realistic 3D geometries on an efficient timescale (and this is only possible through MC codes capable of using GPU resources), and on the other side by the need to easily modify the code for implementation of the quantity of interest. The equivalence and verification of different MC codes have already been extensively reported [5–7]. Then, for the simulations in this work, we used the most convenient code depending on the considered case. In addition, the scattering function used to extract the scattering angles was the Henyey-Greenstein model [3]. Finally, all MC simulations were carried with at least ${10}^6$ detected photons.

All the quantities of interest are calculated inside the MC codes together with their standard errors. The standard error was evaluated by a single simulation, where for each quantity of interest $x$ , relative to each trajectory, e.g., the scaled weight $W({\mu }_a,{\mu }_s)$ , its average $\overline{x}$ and mean squared value $\overline{x^2}$ were calculated. Then, the standard error on the calculated quantity, ${\sigma }_{\overline{x}}$ , was evaluated by its intrinsic definition, i.e.,

$${\sigma }_{\overline{x}}=\sqrt{\frac{(\overline{x^2}-{\overline{x}}^2)}{N_e-1}},$$ (13)

where $N_e$ is number of simulated trajectories [19,32].

For the purpose of this work, the evaluation of the standard error on the absorbing or scattering perturbation is critical to determine whether the reconstructed quantities can be applied with sufficient reliability. Indeed, to be reliable, a MC result related to a calculated quantity $\overline{x}$ should exhibit a relative error, ${\sigma }_{\overline{x}}/\overline{x}$ , of less than 5% [19]. The information provided by the standard error is therefore also an index of convergence of the weight factors.

4. Results

4.1. Examples of $p_{\ell ,MC}(k,{\mu }_s)$ in finite geometries

In this Section, we report some examples of the probability functions $p_{\ell ,MC}(k,{\mu }_s)$ calculated for the case of a non-absorbing homogeneous diffusive infinite slab in reflectance geometry. The methodology adopted to calculate $p_{\ell ,MC}(k,{\mu }_s)$ is reported in Appendix A.

As an example, the cases for a non-absorbing slab of thickness $s=4\mathrm{c}\mathrm{m}$ , ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ , with $g=0$ and $g=0.8$ are depicted in Fig. 1 and Fig. 2, respectively. In each panel, for a specific pathlength $\ell$ , we display the $p_{\ell ,MC}(k,{\mu }_{s0})$ relative to different source-detector distances $\rho$ . Moreover, as a comparison, the correspondent $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ , i.e., the Poisson function obtained in the case of the infinite medium, is depicted as well.

Fig. 1.

Examples of probability functions $p_{\ell ,MC}(k,{\mu }_{s0})$ for different pathlengths $\ell$ are reported for a non-absorbing infinite slab. In each plot, different source-detector distances $\rho$ are considered: $\rho =0.3\mathrm{c}\mathrm{m}$ (magenta line), $\rho =0.5\mathrm{c}\mathrm{m}$ (cyan line), $\rho =1\mathrm{c}\mathrm{m}$ (red line), $\rho =2\mathrm{c}\mathrm{m}$ (green line), $\rho =3\mathrm{c}\mathrm{m}$ (blue line). As a reference, the corresponding function $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ is also reported in each plot (black line). Parameters for MC simulations: slab thickness $s=4\mathrm{c}\mathrm{m}$ , internal and external refractive indexes $n_i=n_e=1$ , ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ , and $g=0$ , resulting in ${\mu }_{s0}=10{\mathrm{c}\mathrm{m}}^{-1}$ .

Fig. 2.

The same as Fig. 1, but for $g=0.8$ , resulting in ${\mu }_{s0}=50{\mathrm{c}\mathrm{m}}^{-1}$ .

We note that when $\rho >\ell$ , the $p_{\ell ,MC}(k,{\mu }_{s0})$ is not reported: as a matter of fact, no photon with such a pathlength can reach the detector and the corresponding $p_{\ell ,MC}(k,{\mu }_{s0})$ is undefined. On the other side, when $\rho \ll \ell$ , the probability functions $p_{\ell ,MC}(k,{\mu }_{s0})$ are practically the same as the corresponding Poisson function $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ .

The most interesting behavior is observed when $\ell \gtrsim \rho$ : in this case, the probability functions $p_{\ell ,MC}(k,{\mu }_{s0})$ are shifted towards smaller values of $k$ with respect to the corresponding $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ . This can be explained considering the fact that, since the length of the trajectories is similar to the source-detector distance, the few received trajectories can only slightly deviate from the straight line connecting source and detector, and this is possibly favored if few scattering events occur with rather long steps. Furthermore, we explicitly note that the two probability functions $p_{\ell ,MC}(k,{\mu }_{s0})$ for $\rho =2\mathrm{c}\mathrm{m}$ when $\ell =2.5\mathrm{c}\mathrm{m}$ (green line) and for $\rho =3\mathrm{c}\mathrm{m}$ when $\ell =5\mathrm{c}\mathrm{m}$ (blue line) are not reported in Figs. 1 and 2, because they are comprehensive of few tens of photons. As a matter of fact such a kind of trajectories are possible ( $\ell \gtrsim \rho$ ), but very unlikely.

As a general remark, these results suggest that only for length $\rho \lesssim \ell \lesssim 5\rho$ the probability functions $p_{\ell ,MC}(k,{\mu }_{s0})$ related to a specific detector are different from the corresponding Poisson function $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ for the infinite medium. Then, in the time-resolved approach, the scaling factor for scattering $F_{TPSF,s}$ [Eq. (2)] results different from 1 (as for the Poisson distribution) and the derivative with respect to scattering [Eq. (10)] is substantially different from zero only for arrival times $t\lesssim 5\rho /v$ .

Finally, we can recall that the probability function $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ represents a limiting case for the distribution of a specific detector in a given geometry, provided that the pathlengths $\ell$ of the received photons are much greater than the source-detector distance. In that case, in fact, the photons received by the detector would propagate in a diffusive manner, similarly to that established in free propagation in the infinite medium, albeit geometry and detectors still play some effect. This fact can be exploited to identify those ranges of $\ell$ where $p_{\ell }(k,{\mu }_{s0})$ can be replaced by a Poisson distribution.

4.2. Convergence of the scattering scaling factor

The convergence, i.e., the applicability, of the scaling method to obtain the $TPSF$ at ${\mu }_s$ starting from $TPSF$ at ${\mu }_{s0}$ is the main issue for the scattering SR. The main requirement to have a good convergence of the $TPSF$ at ${\mu }_s$ is that the trajectory ensemble included in the initial MC simulation is representative also for the new condition. This is fulfilled when the probability functions $p_{\ell ,MC}(k,{\mu }_{s0})$ and $p_{\ell ,MC}(k,{\mu }_s)$ are considerably overlapped [17].

In order to quantify the overlap of the two probability functions, we can approximate them as Poisson distributions with mean values ${\overline{k}}_0={\mu }_{s0}\ell$ and $\overline{k}={\mu }_s\ell$ , respectively. Then, it is reasonable to assume that the two distributions are overlapped if:

$$|\overline{k}-{\overline{k}}_0|\lesssim {\sigma }_{k_0}=\sqrt{{\overline{k}}_0},$$ (14)

that means, after trivial algebra:

$$\left|{\displaystyle \frac{\mathrm{\Delta }{\mu }_s}{{\mu }_{s0}}}\right|\lesssim {\displaystyle \frac{1}{\sqrt{{\overline{k}}_0}}}={\displaystyle \frac{1}{\sqrt{{\mu }_{s0}\ell }}}.$$ (15)

Then, if we consider pathlenghts $\ell$ up to about $100/{\mu }_{s0}$ as an example, from Eq. (15) we can assume that convergence is granted for a change in ${\mu }_s$ up to 10%.

On the other side, one expects that the effect of a scattering variation is minimal on photons that have travelled long pathlengths before being detected, i.e., associated to late arrival times. Then, the scattering scaling factor $F_{TPSF,s}$ is expected to be close to 1 for late time bins.

In order to estimate the convergence of the scaling method, we can consider the approximated value $F_{MC,s}$ of the scattering scaling factor $F_{TPSF,s}$ , calculated through Eq. (6). When $F_{MC,s}$ is far from 1, the calculation of the scaled $TPSF$ is incorrect because of the lack of useful trajectories (i.e., representative of the new simulation) in the initial MC simulation. As a matter of fact, $F_{MC,s}$ represents a measure of the separation between the probability functions $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ and $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ [17]: when its value is closed to 1, the two distributions are significantly overlapping and convergence is guaranteed.

Tables 1 and 2 quantify $F_{MC,s}$ for some example according to the approximated expression of Eq. (6). A variation of 10% for the reduced scattering coefficient is considered for both $g=0$ (Table 1) and $g=0.8$ (Table 2). The values $k_{min}({\mu }_{s0})$ and $k_{max}({\mu }_{s0})$ obtained from the probability functions $p\ell ,MCk,{\mu }_{s0}$ (plotted in Figs. 1 and 2) are shown as well.

Table 1. Scaling factors $F_{MC,s}$ calculated exploiting Eq. (6), for $g=0$ and different values for pathlengths $\ell$ , $k_{min}({\mu }_{s0})$ and $k_{max}({\mu }_{s0})$ . The table pertains to a non-absorbing infinite slab of thickness $s=4\mathrm{c}\mathrm{m}$ , ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ and ${\mu }_s^{\prime }=11{\mathrm{c}\mathrm{m}}^{-1}$ .

$\ell$ [cm]	2.5	5	10	15	20	40	60	80
$k_{min}({\mu }_{s0})$	9	25	65	110	150	330	520	720
$k_{max}({\mu }_{s0})$	45	75	135	190	250	460	680	880
$F_{MC,s}$	0.999	0.996	0.991	0.974	0.978	0.836	0.788	0.509

Table 2. As in Table 1, but for $g=0.8$ .

$\ell$ [cm]	2.5	5	10	15	20	40	60	80
$k_{min}({\mu }_{s0})$	75	200	430	660	900	1850	2850	3850
$k_{max}({\mu }_{s0})$	170	300	575	840	1100	2150	3150	4150
$F_{MC,s}$	0.997	0.936	0.861	0.707	0.508	0.146	$4.40\times {10}^{-3}$	$7.38\times {10}^{-5}$

Some values for the scaling factor $F_{MC,s}$ reported in Tables 1 and 2 are much less than 1. The discrepancies essentially depend on the values of $k_{min}({\mu }_{s0})$ and $k_{max}({\mu }_{s0})$ : the higher they are, the greater the discrepancy.

Furthermore, we note that by maintaining the same scaling factor ${\mu }_s/{\mu }_{s0}=1.1$ , convergence extends to longer $\ell$ when $g=0$ . For $g=0.8$ , instead, results are more critical, as one can see in Table 2: as an example, for $\ell =80$ cm, $F_{MC,s}$ reduces to $\approx 7\times {10}^{-5}$ , indicating the total inadequacy of the SR applied to the MC results. This is clearly due to the fact that, for fixed $\ell$ and ${\mu }_s^{\prime }$ , the number of scattering interactions for $g=0.8$ is higher than for $g=0$ . In fact, because the scattering coefficient ${\mu }_s$ is $1/(1-g)=5$ times higher in the former case, much more interactions with short path are needed to fill a fixed pathlength.

The results presented in this Section, even with all the approximations of interest (i.e., having approximated the $p_{\ell ,MC}(k,{\mu }_{s0})$ with the truncated $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ and having chosen the $k_{min}$ and $k_{max}$ values on the basis of specific initial MC simulations), allow to establish a criterion to predict for which values of ${\mu }_s$ it is reasonable to expect sufficiently accurate scaled results, given a ${\mu }_{s0}$ value for the initial MC simulation and the pathlengths $\ell$ of interest in the specific application. In particular, in the perturbative framework that we are considering, a deviation from 1 of $F_{MC,s}$ of the order of 10% is indicative for a successful application of SR. In all the other cases, it is difficult to provide a general a priori statement about the convergence of the scattering scaling factor. It is worth noting also that, whenever the calculation of $F_{MC,s}$ assures the convergence of SR for scattering, the calculation of the scaled $TPSF$ should be performed exploiting straightforwardly Eq. (2).

Finally, we underline that all the examples here presented refer to a specific geometry: a non-absorbing homogeneous slab. Nevertheless, the one reported is a general criterion and all considerations can be extended to complex, heterogeneous media, even with a partial scattering scaling: the calculation of the approximated value of the scattering scaling factor by exploiting Eq. (6) is still an efficient convergence test, provided that the pathlengths are much greater than the source-detector distance, and the perturbative approach is preserved.

4.3. Applications of the scaling factor for the scattering coefficient

In this Section, the results obtained applying the SR for scattering on both TD e CW cases are shown. In particular, for TD case, both homogeneous and heterogeneous media are considered.

4.3.1. Time domain results

Homogeneous case

Reflectance is calculated at four source-detector distances $\rho$ , ranging from 0.5 to 3 cm, for a homogeneous slab with thickness $s=4\mathrm{c}\mathrm{m}$ and refractive indices $n_{int}=n_{ext}=1$ . We also considered the effect of the anisotropy factor $g$ , by assuming $g=0$ and $g=0.8$ . In compliance with the expectations retrieved with Tables 1 and 2 and the considerations at Section 4.1, the scaling has been applied for a small scattering variation, from a MC simulation run for ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ to derive results for ${\mu }_s^{\prime }=11{\mathrm{c}\mathrm{m}}^{-1}$ .

Figures 3 and 4 compare the $TPSF$ obtained with the SR, from now on referred to as “scaled $TPSF$ ”, and the direct simulation, referred to as “direct $TPSF$ ”.

Fig. 3.

First row: comparison between $TPSFs$ as a function of time obtained by a direct MC simulation for ${\mu }_s^{\prime }=11{\mathrm{c}\mathrm{m}}^{-1}$ and by the scaling approach from ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ to ${\mu }_s^{\prime }=11{\mathrm{c}\mathrm{m}}^{-1}$ in a homogeneous non-absorbing slab with thickness $s=4\mathrm{c}\mathrm{m}$ , refractive indices $n_{int}=n_{ext}=1$ , and $g=0$ , considering the source-detector distances $\rho$ indicated on the top label. Second row: ratio between scaled and direct $TPSF$ . Third row: relative standard error ${\sigma }_r$ of scaled $TPSF$ . Scaled $TPSFs$ are relative to about $N\approx {10}^6$ detected photons.

Fig. 4.

As in Fig. 3, but for $g=0.8$ . Scaled $TPSFs$ are relative to $N\approx {10}^6$ (blue lines) and $N\approx {10}^9$ (green lines) detected photons.

We remind that the effect of ${\mu }_a$ on the $TPSF$ in case of homogeneous media is an exponential attenuation, which depends only on the time-of-flight. Thus, the ratio between the scaled and the corresponding direct $TPSFs$ with the same absorption is independent of ${\mu }_a$ . For this reason, we considered only the case of a non-absorbing medium.

Figure 3 is relative to the case $g=0$ . The scaled $TPSFs$ are obtained considering about $N\approx {10}^6$ detected photons. Despite noise, the superposition of the direct and scaled $TPSFs$ in the first row indicates that scaling in TD is very efficient. In fact, their ratios (second row) fluctuate around 1 at all distances and times considered. Moreover, we note that the relative error ${\sigma }_r$ of the scaled $TPSF$ (third row) increases with time, becoming relevant only at long times.

Figure 4 summarizes the results obtained for $g=0.8$ . The main difference with respect to the previous case emerges in the comparison of the scaled $TPSFs$ and the direct calculations (first row), which highlights the underestimation for scaled results at long times-of-flight, if the same number of detected photons ( $N\approx {10}^6$ ) as in Fig. 3 is considered. This is confirmed by the ratios, that drop after some hundreds of picoseconds (second row). As for the relative error ${\sigma }_r$ (third row), it increases with time faster than the previous case, becoming relevant at earlier times. The results obtained by applying SR improve both in terms of accuracy and precision. In fact, by increasing the number $N$ of detected photons, the time range where the scaled $TPSFs$ overlap the direct ones becomes larger with lower relative errors ${\sigma }_r$ . As an example, Fig. 4 shows the case for the scaled $TPSFs$ obtained with about $N\approx {10}^9$ detected photons.

Based on published literature [19], it is expected that whenever the relative standard error is less than 5%, convergence is generally highly probable. Indeed, it is immediate to observe that scaled and direct $TPSFs$ begin to drift apart when the relative error on the scaled $TPSF$ increases. In particular, for the cases reported in Figs. 3 and 4, this approximately happens when the relative error overcomes the threshold of about 25%. Thus, we can conclude that a simple check of the relative standard error on the scaled $TPSFs$ can provide an effective a posteriori test to assess its level of convergence, additional to the test previously considered in Section 4.2.

Heterogeneous case

In addition to what we reported in the previous section, we expect that the maximum perturbation on the initial case is caused when the change in the scattering coefficient is applied at the whole diffuse medium. Therefore, considering the actual possibility of application of the SR, we expect the homogeneous case to be the worst in terms of convergence of the procedure.

For this reason, we report in Fig. 5 two examples of a heterogeneous diffuse medium, where the change in the scattering coefficient happens only in one sub-region of the medium considered. For both examples, the source-detector distance $\rho$ is 2 cm, the refractive indices are $n_{int}=1.4$ and $n_{ext}=1$ , the anisotropy factor $g$ is 0.8, the detected photons $N$ are about ${10}^6$ , and a scattering variation of 20% is considered.

Fig. 5.

As in Fig. 3, but for two examples of a heterogeneous diffuse medium. Left column: 3-layered slab, thicknesses $s_1=1\mathrm{c}\mathrm{m}$ , $s_2=0.3\mathrm{c}\mathrm{m}$ , $s_3=4\mathrm{c}\mathrm{m}$ ; absorption coefficients ${\mu }_{a0,1}=0.05{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{a0,2}={\mu }_{a0,3}=0.15{\mathrm{c}\mathrm{m}}^{-1}$ ; reduced scattering coefficients ${\mu }_{s0,1}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{s0,2}^{\prime }={\mu }_{s0,3}^{\prime }=5{\mathrm{c}\mathrm{m}}^{-1}$ . The reduced scattering coefficient of the second layer is changed to ${\mu }_{s,2}^{\prime }=6{\mathrm{c}\mathrm{m}}^{-1}$ . Right column: cylindrical inclusion in a homogeneous slab: thickness $s=5\mathrm{c}\mathrm{m}$ , inclusion radius $r=1\mathrm{c}\mathrm{m}$ , height $h=1\mathrm{c}\mathrm{m}$ , depth $z=1\mathrm{c}\mathrm{m}$ ; absorption coefficients ${\mu }_{a0}={\mu }_{a0,i}=0.1{\mathrm{c}\mathrm{m}}^{-1}$ ; reduced scattering coefficients ${\mu }_{s0}^{\prime }={\mu }_{s0,i}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ . The reduced scattering coefficient of the inclusion is changed to ${\mu }_{s,i}^{\prime }=8{\mathrm{c}\mathrm{m}}^{-1}$ . Other common parameters: source-detector distance $\rho =2\mathrm{c}\mathrm{m}$ , refractive indices $n_{int}=1.4$ and $n_{ext}=1$ , anisotropy factor $g=0.8$ , detected photons $N\approx {10}^6$ .

The left column of Fig. 5 reports the case of a 3-layered slab, where only the optical properties of the second layer is changed. Specifically, the optical and geometric properties are: first layer, thickness $s_1=1\mathrm{c}\mathrm{m}$ , ${\mu }_{a,1}=0.05{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{s0,1}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ ; second layer, thickness $s=0.3\mathrm{c}\mathrm{m}$ , ${\mu }_{a,2}=0.15{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{s0,2}^{\prime }=5{\mathrm{c}\mathrm{m}}^{-1}$ ; third layer, thickness $s=4\mathrm{c}\mathrm{m}$ , ${\mu }_{a,3}=0.15{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{s0,3}^{\prime }=5{\mathrm{c}\mathrm{m}}^{-1}$ . The reduced scattering coefficient of the second layer is then changed to ${\mu }_{s,2}^{\prime }=6{\mathrm{c}\mathrm{m}}^{-1}$ . This case can be assumed as a scheme of the head, with the second layer represents the brain cortex.

In the right column of Fig. 5, instead, is reported the case of an inclusion embedded in an otherwise homogeneous slab. Specifically, the optical and geometric properties in this case are: background medium, thickness $s=5\mathrm{c}\mathrm{m}$ , ${\mu }_{a0}=0.1{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ ; cylindrical inclusion located at halfway between source and detector, radius $r=1\mathrm{c}\mathrm{m}$ , height $h=1\mathrm{c}\mathrm{m}$ , depth of the top base $z=1\mathrm{c}\mathrm{m}$ , ${\mu }_{a0,i}=0.1{\mathrm{c}\mathrm{m}}^{-1}$ , ${\mu }_{s0,i}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ . For both background and inclusion we have $g=0.8$ . The reduced scattering coefficient of the inclusion is then changed to ${\mu }_{s,i}^{\prime }=8{\mathrm{c}\mathrm{m}}^{-1}$ . This case can mimic the case of a cyst inside the breast.

From the plots reported in Fig. 5 we can note that the SR can be applied very efficiently in these examples of heterogeneous media, being the results similar to those reported in Fig. 3, where, differently from here, we considered $g=0$ and a scattering variation of 10%.

Finally, we can note that the scaled $TPSFs$ reported in Figs. 3, 4 and 5 are underestimated. As a general remark, SR applied to an initial MC simulation in a medium with scattering coefficient ${\mu }_{s0}$ will yield a scaled $TPSF$ that more likely underestimates the true one at a different value ${\mu }_s$ . The reason is that, for practical time spans used to run the initial MC simulation at ${\mu }_{s0}$ , detected trajectories will not include those that are more relevant (i.e., have larger weights) for the new medium with a different scattering coefficient ${\mu }_s$ . The trajectories in the new medium, indeed, are characterized by a different number of scattering events $k$ for a fixed pathlength of detected photons; in other words, the distribution $p_{\ell }(k,{\mu }_s)$ is shifted with respect to $p_{\ell }(k,{\mu }_{s0})$ [17]. As demonstrated in Fig. 4, by increasing the sampling of the trajectories in the initial MC simulation, the convergence properties of scaled $TPSFs$ improve. However, we stress that to obtain the improvement shown in Fig. 4 we needed to increase the number of detected trajectories by 3 orders of magnitude, and this has a remarkable impact on the computation time.

4.3.2. CW results

Here we report the results obtained for the CW approach, considering as an example a homogeneous slab with the same optical and geometric parameters we assumed in the homogeneous TD case (see Sec. 4.3.1). Then, according to Eq. (8), we calculated reflectance at four source-detector distances $\rho$ , ranging from 0.5 to 3 cm, for a homogeneous slab with thickness $s=4\mathrm{c}\mathrm{m}$ and refractive indices $n_{int}=n_{ext}=1$ . We also considered the effect of ${\mu }_{a0}$ and the anisotropy factor $g$ . Finally, the scaling has been applied for a small scattering variation, from a MC simulation run for ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ to derive results for ${\mu }_s^{\prime }=11{\mathrm{c}\mathrm{m}}^{-1}$ .

The results obtained for the CW approach are summarized in Tables 3, 4, 5, and 6. All tables have the same structure: the first column reports source-detector distances of interest, the second and third columns report the results for the direct MC simulation for ${\mu }_{s0}^{\prime }$ with the corresponding relative errors, the fourth and fifth columns report the results for the direct MC simulation for ${\mu }_s^{\prime }$ with the corresponding relative errors, the sixth and seventh columns report the results obtained by applying the scaling method from ${\mu }_{s0}^{\prime }$ to ${\mu }_s^{\prime }$ with the corresponding relative errors, and the eighth column shows the ratios between the scaled result and the direct result for ${\mu }_s^{\prime }$ .

Table 3. CW values and percent relative standard error ${\sigma }_r$ obtained both with direct MC simulations and by applying the scaling method. Parameters: ${\mu }_{s0}^{\prime }=10$ ${\mathrm{cm}}^{-1}$ , ${\mu }_s^{\prime }=11$ ${\mathrm{cm}}^{-1}$ , ${\mu }_a=0$ ${\mathrm{cm}}^{-1}$ , $g=0$ .

$\rho$ [cm]	Direct ( ${\mu }_{s0}^{\prime }$ )		Direct ( ${\mu }_s^{\prime }$ )		Scaled ( ${\mu }_{s0}^{\prime }\to {\mu }_s^{\prime }$ )		${\displaystyle \mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}({\mu }_{s0}^{\prime }\to {\mu }_s^{\prime })}$
	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	${\displaystyle \mathrm{Direct}({\mu }_s^{\prime })}$
0.5	1.46 $\times {10}^{-1}$	9.91 $\times {10}^{-2}$	1.39 $\times {10}^{-1}$	9.91 $\times {10}^{-2}$	1.39 $\times {10}^{-1}$	1.24 $\times {10}^{-1}$	1.000
1	2.25 $\times {10}^{-2}$	9.93 $\times {10}^{-2}$	2.06 $\times {10}^{-2}$	9.94 $\times {10}^{-2}$	2.06 $\times {10}^{-2}$	2.00 $\times {10}^{-1}$	0.998
2	2.91 $\times {10}^{-3}$	9.93 $\times {10}^{-2}$	2.63 $\times {10}^{-3}$	$9.93\times {10}^{-2}$	2.62 $\times {10}^{-3}$	1.64 $\times {10}^0$	0.999
3	6.99 $\times {10}^{-4}$	9.96 $\times {10}^{-2}$	6.13 $\times {10}^{-4}$	9.97 $\times {10}^{-2}$	6.02 $\times {10}^{-4}$	1.27 $\times {10}^0$	0.983

Table 4. As in Table 3, but for ${\mu }_a=0.2{\mathrm{c}\mathrm{m}}^{-1}$ and $g=0$ .

$\rho$ [cm]	Direct ( ${\mu }_{s0}^{\prime }$ )		Direct ( ${\mu }_s^{\prime }$ )		Scaled ( ${\mu }_{s0}^{\prime }\to {\mu }_s^{\prime }$ )		${\displaystyle \mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}({\mu }_{s0}^{\prime }\to {\mu }_s^{\prime })}$
	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	${\displaystyle \mathrm{Direct}({\mu }_s^{\prime })}$
0.5	8.65 $\times {10}^{-2}$	1.06 $\times {10}^{-1}$	8.08 $\times {10}^{-2}$	1.07 $\times {10}^{-1}$	8.09 $\times {10}^{-2}$	1.15 $\times {10}^{-1}$	1.000
1	6.43 $\times {10}^{-3}$	1.23 $\times {10}^{-1}$	5.52 $\times {10}^{-3}$	1.25 $\times {10}^{-1}$	5.52 $\times {10}^{-3}$	1.45 $\times {10}^{-1}$	1.001
2	1.38 $\times {10}^{-4}$	1.90 $\times {10}^{-1}$	1.05 $\times {10}^{-4}$	1.98 $\times {10}^{-1}$	1.05 $\times {10}^{-4}$	2.71 $\times {10}^{-1}$	0.999
3	5.43 $\times {10}^{-6}$	2.92 $\times {10}^{-1}$	3.68 $\times {10}^{-6}$	3.11 $\times {10}^{-1}$	3.66 $\times {10}^{-6}$	4.99 $\times {10}^{-1}$	0.996

Table 5. As in Table 3, but for ${\mu }_a=0{\mathrm{c}\mathrm{m}}^{-1}$ and $g=0.8$ .

$\rho$ [cm]	Direct ( ${\mu }_{s0}^{\prime }$ )		Direct ( ${\mu }_s^{\prime }$ )		Scaled ( ${\mu }_{s0}^{\prime }\to {\mu }_s^{\prime }$ )		${\displaystyle \mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}({\mu }_{s0}^{\prime }\to {\mu }_s^{\prime })}$
	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	${\displaystyle \mathrm{Direct}({\mu }_s^{\prime })}$
0.5	1.63 $\times {10}^{-1}$	9.90 $\times {10}^{-2}$	1.54 $\times {10}^{-1}$	9.90 $\times {10}^{-2}$	1.53 $\times {10}^{-1}$	4.77 $\times {10}^{-1}$	0.996
1	2.36 $\times {10}^{-2}$	9.93 $\times {10}^{-2}$	2.15 $\times {10}^{-2}$	9.93 $\times {10}^{-2}$	2.05 $\times {10}^{-2}$	1.03 $\times {10}^0$	0.951
2	2.98 $\times {10}^{-3}$	9.92 $\times {10}^{-2}$	2.68 $\times {10}^{-3}$	9.93 $\times {10}^{-2}$	2.56 $\times {10}^{-3}$	1.53 $\times {10}^{+1}$	0.955
3	7.10 $\times {10}^{-4}$	9.96 $\times {10}^{-2}$	6.21 $\times {10}^{-4}$	9.96 $\times {10}^{-2}$	3.17 $\times {10}^{-4}$	1.46 $\times {10}^{+1}$	0.510

Table 6. As in Table 3, but for ${\mu }_a=0.2{\mathrm{c}\mathrm{m}}^{-1}$ and $g=0.8$ .

$\rho$ [cm]	Direct ( ${\mu }_{s0}^{\prime }$ )		Direct ( ${\mu }_s^{\prime }$ )		Scaled ( ${\mu }_{s0}^{\prime }\to {\mu }_s^{\prime }$ )		${\displaystyle \mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}({\mu }_{s0}^{\prime }\to {\mu }_s^{\prime })}$
	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	CW [ ${\mathrm{cm}}^{-2}$ ]	${\sigma }_r$ [%]	${\displaystyle \mathrm{Direct}({\mu }_s^{\prime })}$
0.5	9.96 $\times {10}^{-2}$	1.05 $\times {10}^{-1}$	- 9.14 $\times {10}^{-2}$	1.06 $\times {10}^{-1}$	9.15 $\times {10}^{-2}$	1.88 $\times {10}^{-1}$	1.000
1	6.90 $\times {10}^{-3}$	1.22 $\times {10}^{-1}$	5.88 $\times {10}^{-3}$	1.24 $\times {10}^{-1}$	5.84 $\times {10}^{-3}$	4.16 $\times {10}^{-1}$	0.994
2	1.43 $\times {10}^{-4}$	1.88 $\times {10}^{-1}$	1.08 $\times {10}^{-4}$	1.96 $\times {10}^{-1}$	1.06 $\times {10}^{-4}$	1.84 $\times {10}^0$	0.979
3	5.51 $\times {10}^{-6}$	2.89 $\times {10}^{-1}$	3.69 $\times {10}^{-6}$	3.07 $\times {10}^{-1}$	3.34 $\times {10}^{-6}$	4.00 $\times {10}^0$	0.904

Case: ${\mu }_a=0$ ${\mathrm{cm}}^{-1}$ and $g=0$

Table 3 shows how the scaled CW results are in good agreement with the direct simulation: in fact, the ratio between the scaled and the direct values is very similar to 1 for all the considered source-detector distances (forth column). Furthermore, the relative errors affecting the scaled values are very small, less than 2%. These small errors are possible thanks to the high accuracy of elementary MC data ( ${10}^6$ collected photons for each detector).

The worsening of CW results as the source-detector distance increases is compliant with the higher contribution of photons associated to long time-of-flight, which coincides with many scattering events. In particular, this reflects the higher noise affecting the scaled $TPSF$ at late times (compare Fig. 3). Since the weight of these photons reduces with increasing absorption, we expect scaling to work even better for absorbing media. On the contrary, we expect larger errors as the anisotropy parameter grows (maintaining the same ${\mu }_s^{\prime }$ ), due to the greater number of scattering events with the same time-of-flight.

Case: ${\mu }_a=0.2{\mathrm{cm}}^{-1}$ and $g=0$

By inspecting values reported in the forth column of Table 4, one can confirm the hypothesis of an accuracy improvement in the case of absorbing media. In compliance with the selective reduction of the contribution of photons based on the time-of-flight, it is also observed that the effect of absorption on CW values strongly depends on the source-detector distance, with a reduction of more than two orders of magnitude with respect to the non-absorbing case of the collected photons at $\rho =3\mathrm{c}\mathrm{m}$ . However, the most interesting result is that the relative standard error affecting the scaled result is only slightly larger than the one affecting the direct simulation for all values of $\rho$ , and smaller than the non-absorbing case.

Case: ${\mu }_a=0{\mathrm{cm}}^{-1}$ and $g=0.8$

Let us now consider a non-absorbing medium with $g=0.8$ (Table 5). As expected, the SR performance degrades: at $\rho =1\mathrm{c}\mathrm{m}$ there is already a discrepancy of about 5% with respect to the expected value (compare fourth column). The situation worsens at larger distances, with an increasing underestimation of the CW values and a corresponding growth of the relative standard errors of the scaled results (compare second and third columns). In particular, the latter consideration indicates that the convergence of the scaled results is no longer reliable, as we observed for the TD case.

Case: ${\mu }_a=0.2$ ${\mathrm{cm}}^{-1}$ and $g=0.8$

As reported in Table 6, the addition of absorption significantly extends the applicability of scattering scaling also for this $g$ value: the scaled values for $\rho =1\mathrm{c}\mathrm{m}$ and $\rho =2\mathrm{c}\mathrm{m}$ are still in agreement with those of the direct simulation, with an error of about 2% (see fourth column). On the contrary, for $\rho =3\mathrm{c}\mathrm{m}$ an underestimation of the scaled value of about 10% takes place. Finally, it is worth noting that the absorption reduces the relative standard error of the scaled values with respect to the non-absorbing case, as already observed for the $g=0$ case.

4.4. Derivatives

In this Section we will study the convergence of the derivatives with respect to absorption and scattering obtained by applying the SR according to Eqs. (9) and (10) for the TD case, and Eqs. (11) and (12) for the CW case. In particular, a 3-layered non-absorbing slab of thicknesses $s_1=0.5$ cm, $s_2=0.5$ cm and $s_3=3$ cm, and initial reduced scattering ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ for all the three layers has been considered. The refractive index was fixed to 1 both in the medium and outside, while two values for the anisotropy factor ( $g=0$ and $g=0.8$ ) were discussed. Moreover, four source-detector distances have been considered: 0.5, 1, 2 and 3 cm. For the sake of comparison, derivatives were also directly calculated according to the definition of incremental ratio, assuming $\mathrm{\Delta }{\mu }_a={10}^{-3}{\mathrm{c}\mathrm{m}}^{-1}$ and $\mathrm{\Delta }{\mu }_s=0.1{\mathrm{c}\mathrm{m}}^{-1}$ as absorption and scattering increment, respectively.

4.4.1. Time domain results

The derivatives with respect to absorption can be calculated by applying Eq. (9). Some examples of absorption derivatives are reported in Fig. 6. As a matter of fact, SR do not show critical aspects in this case and can be used without any special precaution, resulting in smooth (i.e., low-noise) TD curves of the absorption derivatives. Indeed, Eq. (9) is the common method adopted to compute derivatives with respect to absorption [25]. For the sake of completeness, in Fig. 6 are also reported the derivatives with respect to absorption calculated with the incremental ratio in accordance to the basic definition

Fig. 6.

Absorption derivatives, expressed in $\mathrm{c}{\mathrm{m}}^{-1}\mathrm{p}{\mathrm{s}}^{-1}$ , are shown for a non-absorbing 3-layered slab of thicknesses $s_1=0.5$ cm, $s_2=0.5$ cm and $s_3=3$ cm, ${\mu }_{s0}=10{\mathrm{c}\mathrm{m}}^{-1}$ , and $g=0$ , thus resulting ${\mu }_{s0}^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ , for all the three layers. The refractive index is 1 both in the medium and outside. Subscripts 1, 2, and 3 denote the first layer, second layer, and third layer, respectively. In absence of subscript, the whole slab is considered. The results are shown for four source-detector distances from the first to the last column: 0.5, 1, 2 and 3 cm. Blue curves pertain to the derivatives calculated by the scaling formula Eq. (9), while red curves are the derivatives obtained by the direct calculation of the incremental ratio.

More precisely, starting from Eq. (9), we can calculate the relative standard error of the absorption derivatives, yielding to the following expression:

$${\sigma }_r\left({\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_a}|}_{{\mu }_s={\mu }_{s0}}\right)=\sqrt{{\sigma }_r^2(⟨\ell {⟩}_i)+{\sigma }_r^2\left(TPS{F}_0\right)},$$ (16)

where we set $TPS{F}_0=TPSF({\mu }_{s0},{\mu }_a,t_i)$ . One can note that this error is only slightly larger than ${\sigma }_r^2\left(TPS{F}_0\right)$ , i.e., that on the initial $TPSF$ . Indeed, for the additional term ${\sigma }_r(⟨\ell {⟩}_i)$ , which is the relative error on the mean path of the photons received in the i-th time bin, it results:

$${\sigma }_r(⟨\ell {⟩}_i)={\displaystyle \frac{\sigma (⟨\ell {⟩}_i)}{⟨\ell {⟩}_i}}\approx {\displaystyle \frac{v\mathrm{\Delta }t}{v{t}_i}}\approx {\displaystyle \frac{\mathrm{\Delta }t}{t_i}}\approx 1\mathrm{\%},$$ (17)

where we assumed a small time bin width $\mathrm{\Delta }t$ , e.g., $\mathrm{\Delta }t\approx 10\mathrm{p}\mathrm{s}$ , as it is usually the case, and $t_i\approx 1\mathrm{n}\mathrm{s}$ . This means that the error on the absorption derivative is comparable to the error on the initial simulation.

The results for the TD derivative with respect to the scattering coefficient are very different from those for absorption. In Fig. 7, some examples of scattering derivatives are reported for the case $g=0$ . Each plot shows the TD derivatives calculated both with the scaling formula Eq. (10) and with the incremental ratio in accordance to the basic definition. From Fig. 7, we can note that the absolute values for the derivative with respect to ${\mu }_s$ are significantly smaller than those for the derivative with respect to ${\mu }_a$ , especially for larger source-detector distances. In particular, the scattering derivatives increase rapidly at early times, reach a maximum, and, then, decreases rapidly towards 0.

Fig. 7.

Scattering derivatives, expressed in $\mathrm{c}{\mathrm{m}}^{-1}\mathrm{p}{\mathrm{s}}^{-1}$ , are shown for the same geometry as in Fig. 6. Blue curves pertain to the derivatives calculated by the scaling formula Eq. (10), while red curves are the derivatives obtained by the direct calculation of the incremental ratio.

The relative standard error affecting the scattering derivative can be evaluated by exploiting Eq. (10):

$${\sigma }_r\left({\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s}|}_{{\mu }_s={\mu }_{s0}}\right)=\sqrt{{\sigma }_r^2(⟨\frac{k}{{\mu }_{s0}}-\ell {⟩}_i)+{\sigma }_r^2\left(TPS{F}_0\right)}.$$ (18)

The behavior of ${\sigma }_r\left({\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s}|}_{{\mu }_s={\mu }_{s0}}\right)$ is basically determined by the first term inside the square root in Eq. (18), resulting much greater than that for the derivative with respect to ${\mu }_a$ . It can be inferred, indeed, that for late propagation times, i.e., for times much larger than ballistic time, we expect that $⟨k/{\mu }_{s0}-\ell {⟩}_i\approx 1/{\mu }_{s0}$ . Moreover, the standard error affecting $⟨k/{\mu }_{s0}-\ell {⟩}_i$ can be calculated as:

$$\sigma (⟨\frac{k}{{\mu }_{s0}}-\ell {⟩}_i)\approx \sqrt{{\sigma }^2(⟨{\displaystyle \frac{k}{{\mu }_{s0}}}{⟩}_i)+{\sigma }^2(⟨\ell {⟩}_i)}.$$ (19)

It results that $\sigma (⟨\ell {⟩}_i)\approx v\mathrm{\Delta }t$ (compare Eq. (17)). As for the standard error on $⟨k/{\mu }_{s0}{⟩}_i$ , it results:

$$\sigma (⟨{\displaystyle \frac{k}{{\mu }_{s0}}}{⟩}_i)={\displaystyle \frac{\sigma (⟨k{⟩}_i)}{{\mu }_{s0}}}\approx {\displaystyle \frac{\sqrt{{\mu }_{s0}⟨\ell {⟩}_i}}{{\mu }_{s0}}}\approx \sqrt{{\displaystyle \frac{v{t}_i}{{\mu }_{s0}}}},$$ (20)

where we assumed that the values $k$ were distributed according to the Poisson distribution $p_{⟨\ell {⟩}_i,\mathrm{\infty }}(k,{\mu }_{s0})$ with mean value ${\mu }_{s0}⟨\ell {⟩}_i$ and standard error $\sqrt{{\mu }_{s0}⟨\ell {⟩}_i}$ . Moreover, we set $⟨\ell {⟩}_i\approx v{t}_i$ as already done in Eq. (17).

We note that the standard error reported in Eq. (20) increases with time, while $\sigma (⟨\ell {⟩}_i)$ remains constant. In particular, if we assume $\mathrm{\Delta }t\approx 10\mathrm{p}\mathrm{s}$ and $t_i\approx 1\mathrm{n}\mathrm{s}$ as before, together with ${\mu }_{s0}=10{\mathrm{c}\mathrm{m}}^{-1}$ and $v=0.03\mathrm{c}\mathrm{m}/\mathrm{p}\mathrm{s}$ , we obtain:

$$\sigma (⟨\ell {⟩}_i)\approx 0.3\mathrm{c}\mathrm{m},\sigma (⟨{\displaystyle \frac{k}{{\mu }_{s0}}}{⟩}_i)\approx 2\mathrm{c}\mathrm{m}.$$ (21)

Then, for late propagation times, the term $\sigma (⟨\ell {⟩}_i)$ in Eq. (19) can be neglected, and the standard error affecting $⟨k/{\mu }_{s0}-\ell {⟩}_i$ can be approximated as:

$$\sigma (⟨\frac{k}{{\mu }_{s0}}-\ell {⟩}_i)\approx \sigma (⟨{\displaystyle \frac{k}{{\mu }_{s0}}}{⟩}_i)\approx \sqrt{{\displaystyle \frac{v{t}_i}{{\mu }_{s0}}}},$$ (22)

leading to the following expression for relative standard error:

$${\sigma }_r(⟨\frac{k}{{\mu }_{s0}}-\ell {⟩}_i)={\displaystyle \frac{\sigma (⟨k/{\mu }_{s0}-\ell {⟩}_i)}{⟨k/{\mu }_{s0}-\ell {⟩}_i}}\approx {\displaystyle \frac{\sqrt{v{t}_i/{\mu }_{s0}}}{1/{\mu }_{s0}}}=\sqrt{{\mu }_{s0}v{t}_i}.$$ (23)

Finally, the relative standard error affecting the scattering derivative [Eq. (18)] is dominated by the first term under the square root and it increases with time; in particular, for the example values of the parameters considered above we obtain:

$${\sigma }_r(⟨\frac{k}{{\mu }_{s0}}-\ell {⟩}_i)\approx 20,$$ (24)

to be compared with the relative error on $TPS{F}_0$ , that is usually of the order of some percent or less. It is worth noting, however, that for late times the value of the derivative ${{\displaystyle \frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s}}|}_{{\mu }_s={\mu }_{s0}}$ is close to 0. Then, in this situation, the huge value of the relative standard error calculated in Eq. (24) is not a suitable quantity to describe the accuracy and its use may be misleading.

In general, it is true that for the derivative with respect to the scattering coefficient the convergence is slower compared to that of the $TPSF$ and may show criticalities. However, plots reported in Fig. 7 offer a clear evidence of the advantage in terms of convergence in using the SR for the calculation of the scattering derivatives with respect to the definition of incremental ratio. Although the two calculations are statistically equivalent, the SR return significantly smoother values, so that their use must be preferred. Simulations for the asymmetry parameter $g=0.8$ were also carried out, confirming the advantage to use the calculation of the scattering derivative with the scaling formula (data not shown).

The scattering effects on the scaled $TPSF$ are also related to the asymmetry parameter of the scattering function at exam, $g$ . The effect of $g$ on the calculation of the time derivative can be simply described by means of similarity relations. If ${\mu }_s^{\prime }$ is constant as $g$ varies, given the similarity relation valid for multiple scattering problems [2,3], almost identical results should be obtained for the derivatives with respect to ${\mu }_s^{\prime }$ , i.e.:

$${\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s^{\prime }}|}_{g=g_1}\approx {\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s^{\prime }}|}_{g=g_2}.$$ (25)

Using the definition of the reduced scattering coefficient ${\mu }_s^{\prime }={\mu }_s(1-g)$ , we obtain:

$$\begin{array}{rl}{\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s}|}_{g=g_1}=(1-g_1){\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s^{\prime }}|}_{g=g_1}& \approx \\ & \approx (1-g_1){\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s^{\prime }}|}_{g=g_2}=\frac{1-g_1}{1-g_2}{\frac{\mathrm{\partial }TPSF}{\mathrm{\partial }{\mu }_s}|}_{g=g_2}.\end{array}$$ (26)

For the derivatives with respect to scattering, in the case $g_1=0.8$ we can then expect smaller values than in the case $g_2=0$ by about a factor of 5. These similarity relations can be automatically transferred to the CW domain.

In conclusion, the TD derivatives calculated in this Section by exploiting the SR have shown excellent property of convergence compared to the direct calculation. In particular, concerning the derivative with respect to absorption, SR does not present critical aspects, being governed by a convergence behavior similar to that of the direct simulation. As for the derivative with respect to scattering, although a larger relative error is observed, this occurs at late times, when the absolute value of the derivative is almost zero and the dependence on scattering vanishes. Therefore, scaled results are significantly better than those for direct calculations (see Fig. 7).

4.4.2. CW results

We expect the CW derivatives to reflect the behavior observed for TD. Tables 7 and 8 report the CW derivatives with respect to absorption and scattering for two values of the anisotropy factor, $g=0$ and $g=0.8$ , calculated by exploiting Eqs. (11) and (12), respectively. We can note that the values of the absorption derivatives for $g=0$ and $g=0.8$ are almost the same, i.e., their ratio is $\approx$ 1, as expected (see Table 7). Furthermore, we observe that:

$$\frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_a}=\sum _{i=1}^3{\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{ai}}},$$ (27)

again as expected.

Table 7. CW absorption derivatives calculated by Eq. (11). It was considered a 3-layer non-absorbing slab with thicknesses $s_1=s_2=0.5$ cm, $s_2=3$ cm, ${\mu }_s^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ in all the layers, and refractive index 1 both in the medium and outside. Subscripts 1, 2, and 3 denote the first layer, second layer, and third layer, respectively. In absence of subscript, the whole slab is considered. The units are ${\mathrm{cm}}^{-1}$ for all the derivatives.

	$g=0$				$g=0.8$
$\rho$ [cm]	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a1}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a2}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a3}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_a}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a1}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a2}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a3}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_a}}$
0.5	-3.8 $\times {10}^{-1}$	-9.3 $\times {10}^{-2}$	-1.6 $\times {10}^{-2}$	-4.9 $\times {10}^{-1}$	-4.1 $\times {10}^{-1}$	-9.3 $\times {10}^{-2}$	-1.6 $\times {10}^{-2}$	-5.2 $\times {10}^{-1}$
1	-1.1 $\times {10}^{-1}$	-6.8 $\times {10}^{-2}$	-2.5 $\times {10}^{-2}$	-2.1 $\times {10}^{-1}$	-1.2 $\times {10}^{-1}$	-6.9 $\times {10}^{-2}$	-2.5 $\times {10}^{-2}$	-2.1 $\times {10}^{-1}$
2	-2.2 $\times {10}^{-2}$	-3.0 $\times {10}^{-2}$	-3.4 $\times {10}^{-2}$	-8.6 $\times {10}^{-2}$	-2.3 $\times {10}^{-2}$	-3.0 $\times {10}^{-2}$	-3.5 $\times {10}^{-3}$	-8.7 $\times {10}^{-3}$
3	-5.8 $\times {10}^{-3}$	-9.9 $\times {10}^{-3}$	-1.6 $\times {10}^{-2}$	-3.2 $\times {10}^{-2}$	-6.0 $\times {10}^{-3}$	-1.0 $\times {10}^{-2}$	-1.6 $\times {10}^{-2}$	-3.2 $\times {10}^{-2}$

Table 8. CW scattering derivatives calculated by Eq. (12). Parameters are the same as in Table 7. Subscripts 1, 2, and 3 denote the first layer, second layer, and third layer, respectively. In absence of subscript, the whole slab is considered. The units are ${\mathrm{cm}}^{-1}$ for all the derivatives.

	$g=0$				$g=0.8$
$\rho$ [cm]	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s1}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s2}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s3}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_s}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s1}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s2}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s3}}}$	${\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_s}}$
0.5	-8.6 $\times {10}^{-3}$	1.6 $\times {10}^{-3}$	3.2 $\times {10}^{-4}$	-6.6 $\times {10}^{-3}$	-2.4 $\times {10}^{-3}$	3.0 $\times {10}^{-4}$	7.2 $\times {10}^{-5}$	-1.97 $\times {10}^{-3}$
1	-2.4 $\times {10}^{-3}$	2.2 $\times {10}^{-4}$	2.3 $\times {10}^{-4}$	-2.0 $\times {10}^{-3}$	-5.3 $\times {10}^{-4}$	4.9 $\times {10}^{-5}$	4.3 $\times {10}^{-5}$	-4.4 $\times {10}^{-4}$
2	-2.5 $\times {10}^{-4}$	-1.2 $\times {10}^{-4}$	5.5 $\times {10}^{-5}$	-3.1 $\times {10}^{-4}$	-5.4 $\times {10}^{-5}$	-2.4 $\times {10}^{-5}$	1.3 $\times {10}^{-5}$	-6.5 $\times {10}^{-5}$
3	-4.4 $\times {10}^{-5}$	-4.7 $\times {10}^{-5}$	-4.1 $\times {10}^{-6}$	-9.5 $\times {10}^{-5}$	-8.8 $\times {10}^{-6}$	-9.9 $\times {10}^{-6}$	-4.6 $\times {10}^{-7}$	-1.9 $\times {10}^{-5}$

As for the scattering derivatives, from Table 8 we can observe that the values for $g=0$ are larger that those for $g=0.8$ , their ratios ranging mostly between 2.7 and 5.7, in agreement with the expectations set by the SR in Eq. (26) (values of the ratios of the scattering derivatives for the two $g$ cases are not explicitly reported for the sake of brevity).

In order to characterize the convergence of the CW derivatives with respect to absorption and scattering, Tables 9 and 10 display the relative standard errors corresponding to the conditions of Tables 7 and 8, respectively. The calculus of the relative standard errors of the derivative is based on the formula (13) applied to Eqs. (11) and (12). All these relative standard errors are consistent with the convergence of the corresponding CW derivatives (relative standard error lower than 5%). For the sake of comparison, we also list in Table 11 the relative standard errors affecting the CW values. An inspection of values reported in Tables 9, 10, and 11 shows that for the derivatives with respect to absorption the relative standard errors are most of the times only slightly larger than those on CW values. Exceptions can be noted, such as derivatives with respect to absorption in the second and third layers for the nearest detector, due to the few received photons that were able to penetrate deep into these layers. For derivatives with respect to scattering, however, the relative standard errors on derivatives are markedly larger than those on CW value, by a factor of ten ore more, as we expected from the previous discussion.

Table 9. Percent relative standard error on the CW absorption derivatives reported in Table 7.

	$g=0$				$g=0.8$
$\rho$ [cm]	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a1}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a2}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a3}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_a}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a1}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a2}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{a3}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_a}}\right)$
$0.5$	1.2 $\times {10}^{-1}$	2.9 $\times {10}^{-1}$	7.3 $\times {10}^{-1}$	1.5 $\times {10}^{-1}$	1.2 $\times {10}^{-1}$	3.1 $\times {10}^{-1}$	7.6 $\times {10}^{-1}$	1.5 $\times {10}^{-1}$
$1$	1.1 $\times {10}^{-1}$	1.7 $\times {10}^{-1}$	3.2 $\times {10}^{-1}$	1.3 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$	1.7 $\times {10}^{-1}$	3.2 $\times {10}^{-1}$	1.3 $\times {10}^{-1}$
$2$	1.1 $\times {10}^{-1}$	1.2 $\times {10}^{-1}$	1.7 $\times {10}^{-1}$	1.2 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$	1.2 $\times {10}^{-1}$	1.7 $\times {10}^{-1}$	1.2 $\times {10}^{-1}$
$3$	1.1 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$	1.3 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$	1.3 $\times {10}^{-1}$	1.1 $\times {10}^{-1}$

Table 10. Percent relative standard error on the CW scattering derivatives reported in Table 8.

	$g=0$				$g=0.8$
$\rho$ [cm]	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s1}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s2}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s3}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_s}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s1}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s2}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_{s3}}}\right)$	${\sigma }_r\left({\displaystyle \frac{\mathrm{\partial }CW}{\mathrm{\partial }{\mu }_s}}\right)$
$0.5$	8.7 $\times {10}^{-1}$	2.3 $\times {10}^0$	4.9 $\times {10}^0$	1.3 $\times {10}^0$	1.6 $\times {10}^0$	5.7 $\times {10}^0$	1.0 $\times {10}^1$	2.1 $\times {10}^0$
$1$	6.6 $\times {10}^{-1}$	5.8 $\times {10}^0$	3.3 $\times {10}^0$	1.1 $\times {10}^0$	1.4 $\times {10}^0$	1.8 $\times {10}^1$	7.9 $\times {10}^0$	2.3 $\times {10}^0$
$2$	1.0 $\times {10}^0$	2.5 $\times {10}^0$	5.8 $\times {10}^0$	1.6 $\times {10}^0$	2.2 $\times {10}^0$	5.5 $\times {10}^0$	1.1 $\times {10}^1$	3.5 $\times {10}^0$
$3$	1.4 $\times {10}^0$	1.8 $\times {10}^0$	2.6 $\times {10}^1$	1.6 $\times {10}^0$	3.3 $\times {10}^0$	3.8 $\times {10}^0$	1.1 $\times {10}^2$	3.5 $\times {10}^0$

Table 11. CW reflectance and its percent relative standard error, ${\sigma }_r(CW)$ , calculated using MC simulations with same parameters as in Table 7.

	$g=0$		$g=0.8$
$\rho$ [cm]	CW $\left[{\mathrm{c}\mathrm{m}}^{-2}\right]$	${\sigma }_r(CW)$	CW $\left[{\mathrm{c}\mathrm{m}}^{-2}\right]$	${\sigma }_r(CW)$
0.5	1.5 $\times {10}^{-1}$	9.9 $\times {10}^{-2}$	1.6 $\times {10}^{-1}$	9.9 $\times {10}^{-2}$
1	2.3 $\times {10}^{-2}$	9.9 $\times {10}^{-2}$	2.4 $\times {10}^{-2}$	9.9 $\times {10}^{-2}$
2	2.9 $\times {10}^{-3}$	9.9 $\times {10}^{-2}$	3.0 $\times {10}^{-3}$	9.9 $\times {10}^{-2}$
3	7.0 $\times {10}^{-4}$	1.0 $\times {10}^{-1}$	7.1 $\times {10}^{-4}$	1.0 $\times {10}^{-1}$

A final remark can be done by comparing the relative errors for $g=0$ and $g=0.8$ . From Table 9 it results that the values obtained in the case of absorption derivatives are practically the same for $g=0$ and $g=0.8$ . As for the scattering derivatives, the relative error for $g=0.8$ is about twice that for $g=0$ (see Table 10). This information is in agreement with the convergence of scaled $TPSFs$ , which decreases as $g$ increases.

The results presented in this Section for the calculation of CW derivatives with scaling rules are similar to those obtained for TD in the previous Section, thus leading to similar conclusions. The weakest point, as for TD, is the scattering derivative. However, the relative standard error produced by these calculations, again with a reasonably low number of detected trajectories, ensures smooth convergence of the scattering derivatives as well.

5. Conclusion

In the companion paper [17], we analyzed the origin of the SR to derive a new MC simulation with a change in absorption and scattering properties by re-processing the collected trajectories of a reference simulation with initial ${\mu }_{a0}$ and ${\mu }_{s0}$ values. Therein, we also presented the crucial aspects at the basis of the convergence criticalities, that are ultimately related to how well the sampled trajectories for the reference simulation are sufficiently representative also of the new problem. In the present paper, we analysed the convergence properties in various scenarios with numerical methods, trying to derive a practical guide for the judicious use of this approach, as well as a deeper understanding of the peculiarities of the SR.

In the case of absorption, the convergence is always optimal both for forward problems and inverse problems - i.e., for the calculus of the derivatives of the $TPSF$ with respect to ${\mu }_a$ . As a matter of fact, the exploitation of Eq. (1) in MC simulations to model the effect of changes in the absorption coefficient is a well-established procedure [25,26]. Practically, the absorption properties, whatever they are, are added in post-processing, after running a white, i.e., for a non-absorbing medium, MC simulation. The key point is that absorption does not alter the photon trajectories, which can adequately describe also the perturbed situation with proper weighting factors - the Lambert-Beer factors. Also, in terms of noise in the scaled simulation, this is solely due to the noise in the original $TPSF$ .

In the case of scattering, the situation is quite different. In particular, we analysed the distribution $p_{\ell }(k,{\mu }_s)$ for the number of scattering events $k$ undergone by photons along a path of length $\ell$ over all recorded trajectories. We observe that $p_{\ell }(k,{\mu }_s)$ is very close to $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ , that is the distribution of the scattering events for a free propagation in an infinite medium, well described by a Poisson distribution. The main deviations of $p_{\ell }(k,{\mu }_s)$ from $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ in finite geometries are related to a shift of the distributions towards lower $k$ for pathlengths $\ell$ close to the source-detector distance $\rho$ . In addition to that, for a change in ${\mu }_s$ , the whole $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ is shifted along the $k$ -axis.

For forward problems, with a finite change in ${\mu }_s$ , this latter shift is the key source of instabilities, causing a possible undersamplig of one side of the distribution of scattering events. While for short enough pathlenghts $\ell$ the convergence is granted, for long $\ell$ we identified a simple criterion for a first-order estimate of convergence properties based on $k_{min}$ and $k_{max}$ values of the $p_{\ell }(k,{\mu }_{s0})$ relative to the reference MC simulation, that are used to calculate an approximated scaling factor $F_{MC}$ . Values of $F_{MC,s}$ close to 1 are an index of convergence for the method. As a matter of fact, $F_{MC,s}$ represents a measure of the overlap between the probability functions $p_{\ell ,\mathrm{\infty }}(k,{\mu }_{s0})$ and $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ , and therefore it indicates if the trajectory ensemble of the reference simulation is representative also for the scaled case. This criterion could be used – for instance – in machine-learning training sessions involving large amount of simulations to exclude wrong datasets which could negatively polarise the model.

From the analysis of the forward problem in few homogeneous exemplary cases, we observed that the main inaccuracies in the generation of scaled simulations occur for a high number $k$ of scattering events. For instance, in CW, for a 10% increase in ${\mu }_s^{\prime }=10{\mathrm{c}\mathrm{m}}^{-1}$ there is more than $5\mathrm{\%}$ inaccuracy on the scaled simulation only for the anisotropic case $g=0.8$ (a 5-fold higher $k$ values than for the case $g=0$ for the same ${\mu }_s^{\prime }$ ) combined with a large $\rho =3$ cm. Absorption helps improve the accuracy since it reduces the weight of trajectories with a higher number of scattering events. It is worth noting at this point that SR for absorption and scattering are independent from one another and, then, they can be applied together to consider a simultaneous change in both the optical parameters. The TD case is more delicate, since long trajectories can still be extracted, independently of ${\mu }_a$ and $\rho$ . As an example, for the same $10\mathrm{\%}$ increase in ${\mu }_s^{\prime }$ , strong deviations are observed for $t>1$ ns and $g=0.8$ at $\rho =2$ or 3 cm, while, for shorter source-detector distances $\rho =0.5$ or 1 cm, inaccuracies are found for $t>0.5$ ns.

For heterogeneous cases, where the perturbed region is much smaller than the whole medium, the number of $k$ scattering events in the altered region is smaller, and therefore convergence of SR improves. This was shown in two exemplary cases of a head and a breast model, where a 20% ${\mu }_s$ change in the brain cortex or in an optical inhomogeneity of $\approx 1$ ${\mathrm{cm}}^3$ at 1 cm depth was properly described by the SR.

It is worth to note that, although the SR is an unbiased estimator which, in the limit of an infinite number $N_e$ of simulated trajectories in the initial MC simulation, converges to the true value, the scaled $TPSFs$ obtained in this work show a underestimation of the corresponding direct simulation. This fact is caused by the set of trajectories considered in the initial simulation at ${\mu }_{s0}$ , which do not uniformly sample the distribution of the scattering events relative to the new simulation at ${\mu }_s$ , in the different time windows of the scaled $TPSF$ . The observed significant increase in the standard error affecting the scaled $TPSFs$ is indicative of this undersampling.

For the calculus of the derivatives with respect to ${\mu }_s$ , the situation is much better. Since the change in ${\mu }_s$ is infinitesimal, the probability distributions of $p_{\ell }(k,{\mu }_s)$ are identical for the reference and the perturbed cases. It is known that for the distribution $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ the derivative of the $TPSF$ with respect to ${\mu }_s$ is null. Therefore, scattering information is encoded in the deviations of $p_{\ell }(k,{\mu }_s)$ from $p_{\ell ,\mathrm{\infty }}(k,{\mu }_s)$ occurring for short pathlengths, i.e., low $t$ .

In terms of noise, in addition to the noise affecting the $TPSF$ as for the absorption case, also the noise affecting the estimate of the mean number of scattering events $k$ must be considered. In particular, we evaluated the latter contribution for a typical set of parameter values, demonstrating that it increases dramatically for large $t$ . At the same time, however, also the scattering derivative rapidly goes to zero. Then, it could also be safely set to $0$ after a proper time, preventing noise to create artificial inaccuracies. In general, we observe that the calculus of derivatives using the SR yields less noisy results as compared to the use of the incremental ratio, and therefore should always be preferred - as in the case of inverse problems.

The key message of this work is that SR can be always used for ${\mu }_a$ . For ${\mu }_s$ , they can be safely used for calculating the scattering derivatives, but in general not for generating forward simulations, unless the ${\mu }_s$ perturbation is small (e.g., < 10%) or the perturbed volume is small (e.g., brain cortex, breast lesion).

In this paper, we deciphered the complex physics of the SR, ultimately related to the distribution of scattering events $p_{\ell }(k,{\mu }_s)$ . This could foster further studies to advance on the modelling of photon migration and the acceleration of MC simulations, which are gaining a crucial role in applications.

Appendix. Calculation of $p_{\ell ,MC}(k,{\mu }_s)$ distributions

In this Appendix, we show how to calculate the probability distribution $p_{\ell ,MC}(k,{\mu }_s)$ starting from the trajectory information saved in the MC simulation.

In the companion paper [17], we defined the infinitesimal probability to travel along a trajectory $\mathrm{\Gamma }$ in a homogeneous medium with fixed ${\mu }_a$ , ${\mu }_s$ and $p(\theta )$ :

$$\mathrm{d}P(\mathrm{\Gamma },{\mu }_a,{\mu }_s,p)=\{\begin{array}{lcr}{\mu }_s^k{e}^{-{\mu }_t\ell }{\displaystyle \prod _{m=1}^k\mathrm{d}{\ell }_{m}p({\theta }_m)\mathrm{d}{\mathrm{\Omega }}_m}& \mathrm{f}\mathrm{o}\mathrm{r}& k\ge 1\\ e^{-{\mu }_t\ell }& \mathrm{f}\mathrm{o}\mathrm{r}& k=0\end{array}$$ (28)

where ${\ell }_m$ is the path travelled by the photon before being scattered at an angle ${\theta }_m$ within a solid angle $\mathrm{d}{\mathrm{\Omega }}_m$ , with $m=1,\dots ,k$ being $k$ the number of scattering events of the trajectory, while ${\mu }_t={\mu }_a+{\mu }_s$ is the extinction coefficient.

We remember that the heart of the scaling relations method is to estimate the probability to travel along a given trajectory in a medium, characterized by different optical properties with respect to an initial or background medium. Let us now consider the background medium with optical properties ( ${\mu }_{a0}$ , ${\mu }_{s0}$ , $p_0$ ) and a new medium with optical properties ( ${\mu }_a$ , ${\mu }_s$ , $p$ ); the main principle of the method is that the ratio of the weights of each trajectory $\mathrm{\Gamma }$ in the two media, $W_0$ and $W$ , is the same as the ratio of the theoretical probabilities reported in Eq. (28), that can be derived from first principles of photon migration:

$${\displaystyle \frac{W({\mu }_a,{\mu }_s,p(\theta ))}{W({\mu }_{a0},{\mu }_{s0},p_0(\theta ))}}={\displaystyle \frac{dP(\mathrm{\Gamma },{\mu }_a,{\mu }_s,p(\theta ))}{dP(\mathrm{\Gamma },{\mu }_{a0},{\mu }_{s0},p_0(\theta ))}}.$$ (29)

By using Eq. (29), we have:

$${\displaystyle \frac{W({\mu }_a,{\mu }_s)}{W({\mu }_{a0},{\mu }_{s0})}}={\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^k{e}^{-({\mu }_t-{\mu }_{t0})\ell },$$ (30)

where we considered the further simplifying assumption $p(\theta )=p_0(\theta )$ .

The distribution of the detected trajectories in the background medium is calculated with a MC simulation. Let us consider an initial MC simulation with $N_e$ launched photons: by definition the weight of each detected trajectory $W({\mu }_{a0},{\mu }_{s0})$ is equal to 1, while the weight $W({\mu }_a,{\mu }_s)$ of the same trajectory in the new medium with different optical properties is given by Eq. (30).

According to the companion paper [17], an estimate of $TPSF(t_i,\mathrm{\Delta }t,{\mu }_a,{\mu }_s)$ , i.e., of the photons detected within the time range $(t_i-\mathrm{\Delta }t/2,t_i+\mathrm{\Delta }t/2)$ , is given by:

$$TPSF(t_i,\mathrm{\Delta }t,{\mu }_a,{\mu }_s)=\frac{1}{N_{e}S\mathrm{\Delta }t}\sum _{j\in J_i}{W}_j({\mu }_a,{\mu }_s)=\frac{1}{N_{e}S\mathrm{\Delta }t}\sum _{j\in J_i}{\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^{k_j}{e}^{-({\mu }_t-{\mu }_{t0}){\ell }_j},$$ (31)

where $S$ is the detector area, $\mathrm{\Delta }t$ is the width of the time bin, $J_i$ is the ensemble of photons received in the $i$ -th time bin, $k_j$ and ${\ell }_j$ are the number of scattering events undergone and the length travelled by the $j$ -th photon, respectively. In particular, if the width of the time bin is small enough, we can assume ${\ell }_j\approx v{t}_i$ , being $v$ the speed of light in the medium. In Eq. (31) we also exploited Eq. (30) and the fact that $W_j({\mu }_{a0},{\mu }_{s0})=1$ for all the received photons.

Equation 31 can be rewritten by using the distribution of scattering events $k_j$ of the detected photons, with $j\in J_i$ . In other words, we can rewrite Eq. (31) as:

$$\begin{array}{rl}TPSF(t_i,\mathrm{\Delta }t,{\mu }_a,{\mu }_s)& =\frac{N_i}{N_{e}S\mathrm{\Delta }t}{e}^{-({\mu }_t-{\mu }_{t0})\ell }\times \\ & \times [\frac{n_0}{N_i}{\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^0+\frac{n_1}{N_i}{\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^1+\frac{n_2}{N_i}{\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^2+\cdots +\frac{n_k}{N_i}{\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^k+\dots ],\end{array}$$ (32)

where $N_i$ is the number of detected photons in the $i$ -th time bin, $\ell =v{t}_i$ , and $n_0,n_1,n_2,\dots ,n_k,\dots$ are the numbers of detected photons that underwent $0,1,2,\dots ,k,\dots$ scattering events, respectively. We explicitly note that in a MC simulation some of the frequencies $n_k/N_i$ can be zero: as a matter of fact, the distribution of scattering events will range between a minimum and a maximum value for $k$ .

Now, if we remember that the $TPSF$ for the background medium can be calculated from the MC simulation as:

$$TPSF(t_i,\mathrm{\Delta }t,{\mu }_{a0},{\mu }_{s0})=\frac{1}{N_{e}S\mathrm{\Delta }t}\sum _{j\in J_i}{W}_j({\mu }_{a0},{\mu }_{s0})=\frac{N_i}{N_{e}S\mathrm{\Delta }t},$$ (33)

and by considering all possible values of $k$ , from Eq. (32) it results:

$$TPSF(t_i,\mathrm{\Delta }t,{\mu }_a,{\mu }_s)=TPSF(t_i,\mathrm{\Delta }t,{\mu }_{a0},{\mu }_{s0})e^{-({\mu }_t-{\mu }_{t0})\ell }\sum _{k=0}^{\mathrm{\infty }}\frac{n_k}{N_i}{\left(\frac{{\mu }_s}{{\mu }_{s0}}\right)}^k$$ (34)

Then, we can construct the probability distribution $p_{\ell ,MC}(k,{\mu }_{s0})$ by means of the frequencies $n_k/N_i$ obtained from the MC simulation, by identifying:

$$p_{\ell ,MC}(k,{\mu }_{s0})\equiv \frac{n_k}{N_i}\mathrm{f}\mathrm{o}\mathrm{r}k=0,1,2,\dots$$ (35)

In this way, indeed, when we consider the limit for $N_e\to \mathrm{\infty }$ , the frequencies $p_{\ell ,MC}(k,{\mu }_{s0})$ converge to the probability function $p_{\ell }(k,{\mu }_s)$ , that represents the probability of detecting a photon that have travelled a length $\ell =v{t}_i$ with $k$ scattering events, in the case of the background medium, and from Eq. (34) we get the scaling factor for $TPSF$ reported in Eq. (4).

Funding

Ministero dell'Università e della Ricerca10.13039/501100021856 (2020477RW5PRIN, 20227EPKW2, 2022EB4B7E, CN00000022, PE0000006, PE0000015); National Institutes of Health10.13039/100000002 (R01 EB029414); Horizon 2020 Framework Programme10.13039/100010661 (863087); HORIZON EUROPE European Innovation Council10.13039/100018703 (101099093).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.