1. Introduction

Optical biomedical diagnosis is a growing field mainly due to the non-invasiveness of devices and non-ionizing radiation. In the near infrared range, where tissues are less absorbing, there are many potential applications such as small animal imaging [1], oximetry [2, 3], photodynamic therapy [4], diffuse optical tomography [5, 6], or fluorescence optical tomography (fDOT) [7, 8]. In tissues, propagating photons interact mostly by means of elastic scattering and/or absorption. In general, biological media can thus be characterized by their reduced scattering coefficient μ_s’ and absorption coefficient μ_a. Knowledge of these optical properties is necessary for the above applications. As examples, it has been shown that errors on background optical properties can result in misestimation of other optical properties or the position of an absorbing inclusion [9], but knowledge of scattering is also of particular importance for reconstructing the depth and concentration of fluorescent inclusions during fDOT [10].

Time-resolved spectroscopy (TRS) is one of the well established techniques for μ_a and μ_s’ estimation [11]. It has been introduced for biomedical investigations since the late 1980s [12–15], and since then, it has been widely used. For instance, TRS has been applied in a variety of in-vivo optical property estimates, such as for muscle [16], human brain [12], human breast [17, 18] or more recently for prostate [19]. Schematically, the method consists in illuminating the medium to be probed with a subnanosecond pulse, then collecting the temporal profile, after dispersion in the medium. An inversion algorithm is used to estimate the optical parameters, μ_a and μ_s’, by fitting the data with a photon propagation model. The algorithm seeks to minimize a χ² error function between the temporal profile and the model. An important issue, which has been discussed for many years, is then the choice of an appropriate model, and fast enough to compute for estimation algorithm [20]. For example, the diffusion equation can be computed in a very short time, a fact that is of great advantage for inversion algorithm, but has the drawback of a limited range of applications [21], in particular μ_s’ has to be large compared to μ_a, the source-to-detector distance has to be large enough and early arriving photons fail to fit the model [22]. Moreover, when using an infinite medium, there has to be sufficient distance between air and medium or to other boundaries [23]. In all these situations, where the diffusion model is incorrect, μ_a and μ_s’ estimates are inaccurate. Many works have thus been performed to assess accuracy of μ_a and μ_s’ estimates when using diffusion approximation and TRS measurements [21]. For example, it has been shown that for strong absorption tissues and weak scattering (μ_a>0.5cm⁻¹ and μ_s’<5 cm⁻¹), errors in both parameters can exceed 30% [24–26].

To overcome these limitations, Monte Carlo (MC) simulation of radiative transport theory has appeared as a gold standard, but faced time consuming computation [27–29]. More recently, thanks to the scalability of MC simulation, Alerstam et al. [24, 25] computed a fast version of MC called White Monte Carlo to overcome both incorrect model and computation time issues. This new method may appear to be a reference for TRS community, especially for strong absorption tissue characterization; the method has already been applied to accurately evaluate optical properties of human prostate [30].

However, although the great efforts made in recent years, estimating the accuracy of optical properties in TRS remains an issue for experimenters. In order to judge the goodness of fit, and afterwards the accuracy of estimates, it is proposed to consider the χ² criterion together with the weighted residuals between model and data. This fitting method has proven efficiency for different time-resolved photon counting applications, notably for TCSPC lifetime measurements for choosing the right number of parameters in multi-exponential decay fluorescence. Fitting values are correct inasmuch that data follow a certain set of assumptions. According to [31], these assumptions are:

This article examines the case of using an incorrect model (depending on experimental conditions and investigated optical ranges), in this instance the infinite diffusion model, for estimating absorption and scattering in turbid media. The reduced χ² value (${\chi }_R{}^2$) is known to judge efficiently goodness of fit, but is unable to judge accuracy of estimates in all situations; an incorrect model can fit the data leading to a good ${\chi }_R{}^2$ value, but provide incorrect estimates (see [31] and subsections 3.1 and 3.7).

The aim of this work is to demonstrate the strengths and the limits of using the ${\chi }_R{}^2$ value and weighted residuals (WRes) to judge the accuracy of μ_a and μ_s’ estimation in TRS. As proof of the principle of using ${\chi }_R{}^2$ to judge this accuracy, the study is restricted to a homogeneous medium characterized using the infinite diffusion equation. The paper is organized as follows. The experimental materials and numerical methods are described in Sect. 2, while the results are presented in Sect. 3. How ${\chi }_R{}^2$ value is affected by the well known sources of misestimation are investigated: influence of high absorption, distance between fibers, Instrument Response Function (IRF), and boundary effects, all of which are presented in Sect. 3.

2. Materials and methods

2.1 Experimental system

The experimental setup (Fig. 1 ) schematically consists of a subnanosecond excitation laser, a medium to be characterized, and a time-resolved detection system. A femtosecond laser is used, producing 775 nm pulses, with 80 MHz repetition rate (Tsunami, Spectra Physics). Photons are brought to the turbid medium to be characterized via an optical fiber (62.5 μm core diameter), providing an average typical power output of 2 mW; the pulses are broadened to a few picoseconds due to dispersion in the fiber. After interaction in the medium, photons are collected via another fiber (plastic-glass, 600 μm core diameter) and conveyed to a fast photomultiplier tube (PMT, R3809U-50, Hamamatsu Photonics). The electronic acquisition chain is a time-correlated single photon counting (TCSPC) card (SPC-630, Becker&Hickl). The time reference signal for TCSPC is generated by a fast photodiode from energy leakage of the laser. Recorded files provide a number of collected photons as a function of time, divided into 4096 channels of 3 ps. The absence of any correlation between channels was checked by repeating the same acquisition 50 times and measuring the correlation coefficients.

 

Fig. 1 Experimental setup. Sync stands for synchronization. Fibers are separated by a distance r and are both at depth z.

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The Instrument Response Function (IRF) is measured by aligning emission and collection fibers separated by a known distance, in order to compensate for the induced time lag. The measured IRF has a temporal width of 50 ps (FWMH).

The turbid medium is made of an aqueous solution of Acronal (BASF, aqueous dispersion of styrene-acrylic copolymer) and black India ink (Dalbe, France). The dilution of each controls respectively scattering and absorption properties of this tissue-mimicking phantom. Due to high viscosity of both Acronal and ink, an intermediate solution is made in order to pipette these stock solutions precisely. The scattering properties of the Acronal stock solution can be evaluated with the time-resolved measurements at μ_s’=180+/−20cm⁻¹. The absorption property of the ink stock solution is μ_a(ink)=5.35+/−0.15cm⁻¹, determined with an absorption UV-Visible spectrophotometer (Cary 300 Scan, Varian). Note that this is only the ink contribution to absorption as a water tank is used in the second line of the spectrometer. This measurement is used to cross-check the μa estimate by the time-resolved measurements using the following formula: ${\mu }_{\text{a}}={\mu }_{\text{a}}(ink)\ast V(ink)/V(total)+{\mu }_{\text{a}}(water,775nm)$, whereas V(ink) and V(total) are respectively the ink volume added to the solution and the volume of the final solution. Absorption of water at 775nm is μ_a(water,775nm)=0.027cm⁻¹ and depends only slightly on temperature and purity [32].

Acronal solutions are used instead of conventional Intralipid because low absorption and diluted solutions are more stable: the same optical properties have been measured for a solution more than 6 months after its preparation. The fibers are immersed in the liquid at a controlled depth z and distance between fibers r, both typically approximately ten millimeters. Other distances, from fibers to boundaries, typically 5 cm, are kept sufficiently large compared to z, so that any deviation compared to the infinite model is controlled only by the fiber depth z.

This study explores the typical absorption and scattering of thick biological tissues, between 0.1 and 0.6 cm⁻¹ for absorption, and 1 to 15 cm⁻¹ for isotropic scattering. Values of 0.1 cm⁻¹ for absorption, and 10 cm⁻¹ for isotropic scattering are considered as reference values. Therefore, strong absorption is defined by values greater than 0.1 cm⁻¹ and weak scattering by values smaller than 10 cm⁻¹.

2.2 Monte Carlo simulations

When experimental data appear to deviate from expected behavior, in particular for the case of strong absorption, they are compared with MC data. The C written program used for simulation is derived from the Internet available version of Prahl [33]. The original program is described in [28,34], and has been adapted to record temporal profile at a given source-detector distance. Briefly, an isotropic point source launches photons, which propagate in isotropic and infinite geometry. In order to decrease computation time, photons are not terminated due to the absorption in the medium or due to the acquisition of the detector fiber, but only after a certain time t_max=7.6ns. It means that a photon can generate multiple detection events implying entangled photon events [24]. In the case of μ_s’=14.2cm⁻¹, source-detector distance of r=1.7cm, and time channel resolution of 3ps, it leads to correlation between typically five adjacent time channels (data not shown), the two first neighbors having a correlation coefficient of 0.5. This seems not to be a problem in this configuration as the number of fitting channels T is generally a few hundreds, large enough compared to 5. However, this point would require a more exhaustive study. In the case of strong absorption (less fitting channels) discussed in subsection 3.6, the effects of these correlations start to be visible.

To take absorption μ_a into account, data are further multiplied by $\exp (-{\mu }_{a}ct)$, where c is the speed of light in the medium. We call such simulation scalable MC, or MCs (even if the space is not scaled as in [24]). The drawback of this method for producing “artificial experimental data” is that the Poisson statistic of each channel is not kept – each channel has a standard deviation multiplied by$\exp (-{\mu }_{a}ct)$. This issue is important when fitting these data with any model, in particular the diffusion model (see subsection 2.4).

To overcome this issue, MC simulations taking absorption into account (we call these other simulations MCa, subscript a stands for absorption) have also been performed. After each scattering event, MCa has classically the probability (1-albedo) to kill a photon, where albedo=(μ_s/(μ_s+μ_a)). However, for a question of computation time, MCa simulations were used only when the statistic issue is at stake.

2.3 Model

Data modeling is based on the diffusion equation transport theory in an infinite homogeneous medium. Validity of the diffusion equation implies, in particular, that multiple scattering occurs, that is to say the mean free path of scattering is small compared both to r and the mean free path of absorption. Moreover, as the first arriving photons have not undergone sufficient scattering events, the first-time channels after the excitation pulse, which do not follow the diffusion equation [22], are excluded. Homogeneity of the medium is ensured by stirring the liquid phantom before acquisition. Measured quantities are assumed to be proportional to the fluence rate φ which is the integrated radiance over all solid angles expressed in watts per square meter. The flux term$\genfrac{}{}{0.1ex}{}{\partial \varphi }{\partial z}$ in the measured quantities is neglected, which is appropriate for an infinite medium [23].

Following the diffusion assumptions, and assuming a point source at the origin in time and space, the measured quantities are proportional to the following model m [15,35]:

2.4 Fitting procedure

The fitting procedure goal is to estimate two optical parameters, μ_a and μ_s’, from the temporal profile recorded, containing numerous data points. The Nonlinear Least Squares method is used for the estimation; more precisely, a Nelder-Mead simplex algorithm drives minimization of a reduced χ_R² error function which measures the deviation with respect to the above model. The χ_R² error value is expressed as:

Before running the fitting procedure, background (corresponding to the average random obscurity noise from the PMT) is subtracted, and the model is then convolved with the measured IRF and multiplied by a constant to obtain the same energy (integral over time) as the signal. This latter operation, called normalization, following Cubeddu et al. [21], permits to fit only the shape of the curve regardless of amplitude variation. Thus, only two parameters instead of classically three are used for the fitting algorithm, which becomes faster and more robust. By normalizing the model without changing the signal, the fit covers only the temporal profile of the curve, and the Poisson noise distribution of the data is kept. The fit is carried out between t₁ corresponding to 95% of the maximum on the rising edge (because the first arriving photons do not follow the diffusion equation) and t₂ corresponding to 5% of the maximum on the trailing edge (noise limitation), except where otherwise stated.

When using MCs data, the fitting procedure has to be modified. Indeed, the statistic of the raw MCs signal s_i is modified by the absorption weighting $s_i\text{'}=s_i\exp (-{\mu }_{a}ct)$. The distribution can still be approximated by a Gaussian statistic but with a standard deviation of ${\sigma }_i\text{'}=\sqrt{s_i}\exp (-{\mu }_{a}ct)=\sqrt{s_i\text{'}}\exp (0.5*{\mu }_{a}ct)$. Technically, residuals of the fitting procedure have to be weighted differently as all other data (including MCa data), due to this modified statistic. Figure 2 clearly emphasizes the need of the correction. When the correction is omitted, weighted residuals are no longer randomly distributed around 0, and the associated ${\chi }_R{}^2$ value has no signification. Besides, late arriving photons contribute less to the fit, leading to incorrect estimation, especially for μ_a – but surprisingly, the error due to the omitted correction is very small (just a few percents).

 

Fig. 2 Weighted residuals, (a) uncorrected and (b) corrected, as a function of time (scalable MC simulation, μ_s’=7cm⁻¹, μ_a=0.03cm⁻¹, fitted by the diffusion equation). The fitting range 95/1 has been extended to 1% of the trailing edge in order to better emphasize the effect of the correction.

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Due to the correction, a good fit still corresponds to a ${\chi }_R{}^2$ value of 1, whereas a bad fit corresponds to a significantly higher value.

2.5 Goodness of fit

A high value of χ_R², together with a non-random pattern of weighted residual, is indicative of a bad fit, thus providing poor confidence in estimated parameters. Assuming both the model is correct and there are 200 independent measurements (in this case time points), there are less than five chances in 100 of finding a χ_R² value above 1.17 [31]. Such probabilities can be found using χ² distribution which tends towards normal distribution for high degrees of liberty (in practice, a normal distribution is used for T>200). Table 1 gives a few probabilities for usual situations in TCSPC when all the above assumptions are verified, in particular model validity. Moreover, properties of normal distribution imply that less than 5% of the absolute weighted residual points should be above 2, and less than 1% above 3.

Table 1. Probability of finding a certain value of χ_R² or higher, as a function of T, number of degrees of liberty, due to random distribution of photons. Deviation from the values in this table is indicative that the model is false.

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A deviation in these characteristics can be indicative of a systematic error, or a model deviation of the data. On the other hand, a good value of χ_R² and behavior of weighted residuals is not proof that the fit is good and that the estimated parameters are correct. This study aims at giving an insight into what can be expected from these criteria to estimate the optical parameters, μ_a and μ_s’.

3. Results

3.1 χ_R² and number of photons

To take model deviation into account, a bias term is added. This term takes into account discrepancies between data and the diffusion model. Bias b_i is greater for early times, because the first arriving photons do not follow the diffusion model as they do not undergo sufficient scattering events. Both m_i and b_i depend on the experimental conditions, such as optical properties of the medium, and distance between fibers r. For a particular time channel ${\text{t}}_{\text{i}}$, the signal $s_i$ is $s_i={\text{m}}_{\text{i}}+{\text{b}}_{\text{i}}+{\epsilon }_i$, where $s_i$ follows a Poisson distribution; thus ${\epsilon }_i$ is approximated by a random zero-centered Gaussian noise for more than a few tens of photons, with amplitude${\sigma }_i=\sqrt{{\text{m}}_{\text{i}}+{\text{b}}_{\text{i}}}\approx \sqrt{s_i}$. In the following, random variables ($s_i,{\epsilon }_i,{\chi }_R^2$) will be denoted with italic letters, whereas realization-independent quantities (${\text{b}}_{\text{i}}{\text{,m}}_{\text{i}}$) will be denoted with roman letters. The χ_R² components then become:

If the experiment is repeated many times with identical experimental conditions, a distribution is obtained for each of the three terms and for each time channel. Averaging the first term of Eq. (3) tends to one when the number of experiments increases, the second term tends to zero due to the zero centered ${\epsilon }_i$ distribution and the fact that ${\epsilon }_i$ is not correlated with ${\text{b}}_i$. The last term tends to zero when the model fits the data (${\text{b}}_i=0$), but will stay strictly positive when model deviation occurs.

When increasing the acquired number of photons $N={\displaystyle \colorbox[rgb]{}{${\sum }_i{s}_i$}}$ (by increasing experimental acquisition time τ for example), signal-to-noise increases and a biased model is more easily detected with${\chi }_R^2$. Note that summation for N is performed between the two time channel boundaries used for fitting, t₁ and t₂, and after background noise is subtracted. More precisely, the signal, the model and the bias also increase linearly with N, with the standard deviation of the signal increasing with $\sqrt{N}$. Indeed, if quantities are introduced with a reduced number of photons dependence, $s_i=N{\stackrel{⌣}{s}}_i$ and $b_i=N{\stackrel{⌣}{b}}_i$, the last term of Eq. (3) becomes $N\genfrac{}{}{0.1ex}{}{{\stackrel{⌣}{b}}_i^2}{{\stackrel{⌣}{s}}_i}$. It can then be expected that ${\chi }_R^2$will increase linearly with the total number of photons N, ${\chi }_R{}^2=1+N\Delta$, where $\Delta ={\displaystyle \colorbox[rgb]{}{${\sum }_i\genfrac{}{}{0.1ex}{}{{\stackrel{⌣}{b}}_i^2}{{\stackrel{⌣}{s}}_i}$}}$ is strictly positive when there is a bias and this gives an indication of the model error. It was also found that ${\chi }_R{}^2$ tends to one in the limit of a small number of acquired photons, regardless the model and its bias with respect to experimental values. This means that the ${\chi }_R^2$ value (and weighted residual distribution) has no meaning without an indication of the total number of acquired photons. A ${\chi }_R{}^2$ value of one is easily found, regardless of whether the model is correct or not, if just a few photons are acquired. This also means that a large number of photons is necessary in order to choose between two distinct models: this property is known to lifetime measurers when choosing between one, two, or three decay components [31].

Experimentally, due to the large IRF (>50ps) and slowly varying diffusion intensities compared to time channels (3ps with our TSCPC card), averaging is already performed in ${\chi }_R{}^2$ summation between adjacent channels. Indeed, the experimental data shown in Fig. 3(b) illustrate perfectly the linear increase in ${\chi }_R{}^2$ with the total number of acquired photons, and also the limit of one when the number of photons tends to zero. This confirms that in order for ${\chi }_R{}^2$ to be a good criterion to judge the fit, the number of experimentally acquired photons must be increased. In the following, care is taken to keep this number to approximately 1.5*10⁷, and a corrected ${\chi }_R{}^2$ is introduced to remove the effect of the remaining number of photons: ${\chi }_{R-C}{}^2=1+1.5*{10}^7({\chi }_R{}^2-1)/N$.

 

Fig. 3 (a) χ_R² as a function of acquired photons for two solutions, varying ink concentrations and with same scatterer concentration (μ_s’=16cm⁻¹). Linear fits are also plotted, emphasizing the behavior of χ_R²; for each fit a typical R² value of 0.98 and an intercept of 1.01 ± 0.04 are obtained. Each data point corresponds to a different experiment, except in the case of a high number of photons for which independent acquisitions were added. (b) Weighted residuals for increasing numbers of acquired photons when there is only little model bias (expected μ_a=0.1cm⁻¹) and (c) worse model bias (expected μ_a=0.3cm⁻¹).

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Figure 3(b) and 3(c) show that weighted residuals behave similarly as the ${\chi }_R{}^2$ value when increasing the number of photons: weighted residuals randomly distributed around 0 implies a ${\chi }_R{}^2$ value around 1, even when there is a model bias, if the number of acquired photons N is too small. On the contrary, a high ${\chi }_R{}^2$ value corresponds to un-random distribution of weighted residuals. Thus, in the case of characterization of turbid media, this number is as high as a few millions (Fig. 3). Such high number may require enough time acquisition which could be in contradiction with the medical needs.

3.2 Influence of the distance between fibers

At a fiber depth of 28 mm, where boundary effects can be ignored (see paragraph 3.7), temporal intensity profiles were measured as a function of r, distance between source and detector fibers. For each of these temporal profiles, a fitting procedure is performed providing μ_a and μ_s’ estimates as well as the ${\chi }_{R-C}{}^2$ parameter (corrected by taking N into account) and weighted residuals for each point used in the fitting procedure. Figure 4(a) shows these estimates and ${\chi }_{R-C}{}^2$ as a function of r. Both μ_a and μ_s’ are overestimated when r is too short (respectively 30% and 45% error for r=0.5cm), before reaching a plateau for μ_s’=7.9 cm⁻¹ and μ_a=0.106 cm⁻¹, considered as “the true values”, the latter being checked by the absorption spectroscopy method. Measurements at the two first distances lead to a high value of ${\chi }_{R-C}{}^2$ (${\chi }_{R-C}{}^2=7.3$ for r=0.5cm), and non-random weighted residuals, which warn experimenters about the problem. For r=1.1cm and above, μ_a has already reached the plateau and μ_s’ is overestimated by 10% or less, and ${\chi }_{R-C}{}^2$ also reaches a “plateau” at 1.5±0.5. In the following, distance between fibers is kept to r=1.7cm when not specified. We attribute the high value of 1.5 at the plateau to unknown systematic errors (light leakage, slightly incorrect IRF) together with a high number of acquired photons which exhibits any little bias. Note that when the distance between fibers is increased, the collected intensity strongly decreases: as a result, the optical density before the PMT and the gain voltage applied to the PMT had to be changed. For each applied voltage, the IRF with the same gain was performed and used for deconvolution. The deviation due to the short detection distance can be explained by the fact that early arriving photons do not follow the diffusion behavior – these photons predominate because source and detector are close to each other. ${\chi }_{R-C}{}^2$ is thus a useful value for indicating parameter overestimation due to the short distance r between fibers, especially in the case of μ_a estimation.

 

Fig. 4 (a) Estimated parameters, μ_a and μ_s’, as a function of interfiber distance r. The corresponding calculated parameter χ_R-C ² is plotted. (b) Measured temporal profile, together with the estimated diffusion profile in an infinite medium for r = 0.8cm. The associated profile of weighted residuals over the fitting zone is clearly not randomly distributed around zero, emphasizing how wrong this fit is.

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3.3 Influence of the IRF used for deconvolution

When performing photon counting experiments, the voltage of the PMT can be varied, typically between 2800 and 3400V. This changes the gain, but also the IRF because electrons are accelerated differently inside the PMT: with higher voltages, the IRF peak arrives earlier and is narrower. Deconvolution by an incorrect IRF can lead to misestimating of parameters by more than 20% (data not shown), and this can happen when IRF is measured in inappropriate conditions, such as after reflection on a white support or using a different gain than for acquisition. IRF has been measured by separating the excitation and collection fibers by a known distance l (with further correction by a time shift $\Delta t=l/c$), in order to decrease light intensity and to keep all other measurement parameters unchanged (same gain, same filters). The manner in which ${\chi }_{R-C}{}^2$ is affected by an incorrect IRF was explored by changing the PMT voltage. Table 2 (and other data not presented) illustrates how a voltage change increases ${\chi }_{R-C}{}^2$ (and degrades the associated weighted residual profile): ${\chi }_{R-C}{}^2$ is also an ideal means of improving confidence in the IRF measurement.

Table 2. Effect of deconvolution by an incorrect IRF. PMT voltage during the experiment was 2.83 kV.

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Different IRF have also been measured using or not a piece of paper to excite all the modes of the collection fiber, as proposed by [37]. The white paper placed before the collection fiber slightly widen the IRF, but only little changes have been noticed in both optical properties estimates and ${\chi }_{R-C}{}^2$ (data not shown).

3.4 Weak scattering tissues or phantom

The validity of the diffusion equation is restricted to the condition ${{\mu }^{\prime }}_s>>{\mu }_a$. To test this condition, the temporal profile was measured for solutions with an increasing concentration of Acronal scatterers. As shown on Fig. 5 , the high ${\chi }_{R-C}{}^2$ values at weak scattering highlight the limits of the model, μ_a and μ_s’ estimates can thus be rejected. Figure 5 was drawn up using a fit interval t₂ corresponding to 0.5% of the maximum instead of 5% in order to show the slight decreasing trend of μ_a due to dilution, which is expected. For t₂ taken at 5% of the maximum (data not shown), estimates are modified by less than 2% for μ_a but are then not able to show the decrease, and less than 3% for μ_s’.

 

Fig. 5 Estimated parameters, μ_a and μ_s’, as a function of Acronal concentration, for t₂ taken at 0.5% of the maximum. The corresponding calculated parameter χ_R-C ² is plotted. For comparison, the expected μ_a values provided by absorption spectrometer measurements (the error bar arises from μ_a estimation error of the stock solution, but is identical for each point), and the linear increase in μ_s’ were plotted assuming the value of the most concentrated solution to be correct. Number of time channels used for fitting are between T=425 for the smallest and T=700 for the strongest concentration of Acronal.

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3.5 Strong absorption tissues or phantom: experimental artifacts

Strong absorption is another situation widely encountered in biological tissues, such as for prostate [24]. Multiplying the zero-absorption equation by $\exp (-{\mu }_{a}ct)$ obviously increases the relative amplitude of the early part of the signal. As the diffusion equation fails to fit correctly the first arriving photons, it leads to overestimation of both μ_a and μ_s’ [30].

In this subsection, the case of strong absorption is investigated starting with μ_a=0.1cm⁻¹ and μ_s’=15cm⁻¹, and further increasing ink concentration. Figure 6(a) shows how μ_a and μ_s’ are both underestimated for strongly absorbing media, which is in contradiction with previous results comparing experimental data, diffusion equation and MC simulations [24, 25, 30]. The first three measurements, corresponding to a “small” absorption value and leading to parameter underestimation of less than 10%, are associated with a ${\chi }_{R-C}{}^2$ value below 2. With increasing ink concentration, the error increases and ${\chi }_{R-C}{}^2$ also increases considerably, reaching ${\chi }_{R-C}{}^2$ =85 for expected μ_a=0.53cm⁻¹ (26% underestimation).

 

Fig. 6 (a) Estimated parameters, μ_a and μ_s’, as a function of ink concentration. The corresponding calculated parameter ${\chi }_{R-C}{}^2$ is plotted. After the first three measurements, the ${\chi }_{R-C}{}^2$ values deviate from their plateau, thus highlighting incorrect fit. Number of channels used for fitting are T=470 for the lowest and T=190 for the strongest ink concentration. (b) Comparison between measured temporal profiles and MC simulation convolved with the experimental IRF, for excepted μ_a=0.4cm⁻¹ and μ_s'=14.2cm⁻¹ ([ink]=70 arbitrary units). The circle emphasizes the unexpected behavior of the data.

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In order to understand the controversy between the underestimation found here, and the reported overestimation known for strong absorption, Fig. 6(b) compares the experimental data and the simulated MC data convolved with the measured IRF. Underestimation of μ_a is explained by the tail which is higher than expected. We checked that this behavior cannot be explained by different IRF measurements, in particular with or without using a white paper to excite the different modes of the collection fiber [37]. Often, later photons are excluded of the fitting range [19] in order to eliminate low signal level were excitation leakage or other experimental artifacts could explain such behavior.

Note that underestimation behavior was also already found with time resolved experiments, probably for the same small signal to artifacts reasons (see Fig. 4(a) and 4(c) in [38]). Consequently, when facing experimental artifacts in the case of small signal-to-noise ratio appearing in strong absorption media, ${\chi }_{R-C}{}^2$ is a pertinent value to analyze to warn experimenters about potential problems.

3.6 Strong absorption: MC simulation

In order to investigate strong absorption, and because we were not able to eliminate experimental artifacts described in previous subsection, MC data were simulated to replace experimental data. In all cases investigated (μ_s’ between 5 and 15 cm⁻¹), strong absorption results in overestimation of both μ_a and μ_s’, together with elevation of the ${\chi }_R{}^2$ value providing a sufficient number of photons is launched in the MC simulation (Fig. 7(a) ). Scalable MC (MCs) permits to demonstrate that ${\chi }_R{}^2$ significantly elevates when absorption increases due to the diffusion model bias at early times (Fig. 7(a) and data not shown). Indeed, with high enough number of acquired photons, estimation error is limited to 3% before ${\chi }_R{}^2$ elevates.

 

Fig. 7 (a) Estimation errors and the ${\chi }_R{}^2$ value are plotted as a function of absorption for μ_s’=7cm⁻¹ (scalable MC). (b) The ${\chi }_R{}^2$ value as a function of N for MCa in the case of a prostate simulated medium. (c)(d) Weighted residuals corresponding to small and large number of photons of the Fig. 7(b) (note the WRes scale). Number of channels used for the fit is T=80.

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However, the linear increase in ${\chi }_R{}^2$ as a function of N in the presence of a model bias, as shown in subsection 3.1, is lost with scalable MC, even after correction of the weighted residuals (data not shown). Consequently, MCa simulations (MC including the probability of absorbing a photon) have also been computed. Optical properties of the prostate have been chosen for the simulation (μ_a=0.3cm⁻¹ and μ_s’=5cm⁻¹ at 786nm [30]) but for a reduced fibers distance of r=1.2cm due to computational time issue. In this situation where diffusion approximation is known to fail due to the predominance of first arriving photons, the model bias is evidenced by high ${\chi }_R{}^2$ value providing a sufficient number of photons is acquired, typically a few millions. Indeed, Fig. 7(b) shows the perfect linear increase of the ${\chi }_R{}^2$ as a function of the number of photons. When including more early photons, in the 80/20 fitting range, the ${\chi }_R{}^2$ is higher, evidencing a worse estimation (13% and 11% overestimation for μ_a and μ_s’ compared to respectively 2% and 1% for the 95/5 fitting range). Figure 7(c) shows that weighted residuals are almost randomly distributed for a small number of photons but Fig. 7(d) clearly demonstrates the short time bias for a large number. Again, for a given model bias and a low number of photons, both the ${\chi }_R{}^2$ value and the weighted residuals behavior are misleading: both quantities require high enough photons to assess accuracy of the estimates.

In Fig. 7(b) and 7(c), correlation between channels is expected to be responsible for the ${\chi }_R{}^2$ value clearly below one (0.87+/−0.05) for a small number of photons, (channels are typically correlated with their 5 first neighbors, which is no more negligible compared to T=80).

3.7 Boundary effects

Surface boundaries with air are responsible for erroneous estimation of μ_a and μ_s’ [23], when using the infinite diffusion model. For example, photons which pass the surface escape definitively from the diffusing medium; the fitting algorithm compensates for these losses by overestimating the μ_a parameter. This implies that the curve tail is steeper, so that the time measured to obtain the peak intensity, t_max, is underestimated. And given that μ_s’ increases with t_max [15,39], μ_s’ is also underestimated. These deviations from an infinite medium as a function of depth are shown in Fig. 8 ; the plateau of an infinite medium is almost reached at z = 2cm. Except for the first two depth values, where the ${\chi }_{R-C}{}^2$ value is higher than the others, there is no clue given by either the weighted residuals or ${\chi }_{R-C}{}^2$ of the deviation due to the surface boundary. This means that a plane surface boundary with air modifies the temporal profile but in such a way that other optical parameters give a similar profile; the algorithm can compensate for the air-boundary effect without any significant degradation of the fit criteria. Consequently, the main drawback of the ${\chi }_{R-C}{}^2$ and weighted residual criteria for judging the accuracy of the estimators is the presence of a plane boundary with air: an error of 17% in μ_a is reached without any clear indication on ${\chi }_{R-C}{}^2$ (whereas an error of less than 5% is obtained for μ_s’). The distance to all boundaries has to be chosen precisely by experimenters in order to avoid any such misestimation which cannot be highlighted by ${\chi }_{R-C}{}^2$.

 

Fig. 8 Estimated parameters, μ_a and μ_s’, as a function of fiber depth z for r = 1.7cm. The corresponding ${\chi }_{R-C}{}^2$ parameter is plotted.

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The presence of a semi-reflective element also imposes different boundary conditions for propagating photons and, obviously, the infinite model does not take these into account. A possible cause of artifact could, for example, be a metal support for an excitation fiber where the end of the optical fiber is located too close to the metal surface. Figure 9 shows the comparison between different holders for the excitation fiber: the optical fiber is either both stripped and inserted in a thin metallic needle, or the fiber end with its commercial connector is just inserted in a large metallic holder. Note that the collection fiber is rigid enough so that no specific holder close to the fiber end is necessary. The large holder modifies data so that neither the infinite model nor the μ_a and μ_s’ estimates are correct anymore. It is noticed that early photons are more affected by the large holder. Consequently, Fig. 9(a) shows weighted residuals and ${\chi }_{R-C}{}^2$ for a 80/5 fitting range. Both statistical quantities warn the experimenters about the artifact. As the validity of the model and the ${\chi }_{R-C}{}^2$ value depend together on the fitting range, Table 3 summarizes the results.

 

Fig. 9 (a) Weighted residuals and ${\chi }_{R-C}{}^2$ are plotted for both holders (same solution). The fitting range has been extended to 80/5 in order to include more early photons to evidence the failure of the model in the presence of the large metallic piece. (b) Pictures of the holders. The needle holder is used in all the other data presented in the article.

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Table 3. Estimates of the Acronal solution as a function of fiber holders and fitting range. The error introduced by the presence of the large metallic holder is also indicated.

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3.8 Experimental uncertainties

Looking at the diffusion equation, an erroneous measurement of the distance r is completely compensated by the diffusion coefficient D, leading to an incorrect estimate of μ_s’, but the same estimate of μ_a and the same residuals between model and data: there is no clue given to experimenters of this potential error. Due to the squared dependence of r in the exponential, a 5% error in r will be responsible for a 10% error in μ_s’.

Another possible error could come from the IRF distance measurement which has to be compensated; a time shift could thus be introduced. Figure 10 shows the error in estimated parameters if a small time shift is introduced, without any substantial change in ${\chi }_{R-C}{}^2$, between plus 30ps and minus 15ps. Note that a time shift of 17ps, corresponding to a 5mm error in distance l between fibers when measuring IRF, leads to an error of around 6% in μ_s’ and 2% in μ_a. As the μ_s’ estimation is directly linked with t_max, it is normal that μ_s’ is more affected than μ_a by a time shift. On the other hand, for abnormally big time shifts (such as 100ps), a high ${\chi }_{R-C}{}^2$ (such as 3) warns experimenters about the problem.

 

Fig. 10 Estimated parameters, μ_a and μ_s’, as a function of an artificial introduced time shift. The corresponding calculated parameter ${\chi }_{R-C}{}^2$ is plotted. Each grid line corresponds to a 5% error. Note that for a reasonable r measurement error (less than 5 mm) during the IRF, the time shift is very small (less than 17 ps).

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4. Conclusion and summary

The infinite diffusion model for light propagation in a diffusing medium is attractive given its computational rapidity and ease of use. However, model inversion for estimating optical parameters presents limitations due to the model itself. This article shows that when there is a model bias, ${\chi }_R{}^2$ increases linearly with the number of acquired photons: a high number of acquired photons is thus required to discriminate between models and/or to test model validity. Moreover, regardless of the model bias, ${\chi }_R{}^2$ always tends to one when the number of acquired photons is too low. The authors therefore recommend that this number should always be provided together with ${\chi }_R{}^2$ and weighted residuals to judge goodness of fit. These two conclusions are not specific to time-resolved diffuse optical parameter estimation, but also apply to different ${\chi }_R{}^2$ estimations, in particular for estimating dye lifetime.

Weighted residuals and the ${\chi }_R{}^2$ value can both judge a model bias providing enough photons are acquired: for a ${\chi }_R{}^2$ value around one is obtained when weighted residuals are randomly distributed around zero, and weighted residuals are correlated (which could be checked by autocorrelation) for ${\chi }_R{}^2$ above one. However, as the ${\chi }_R{}^2$ value can sometimes be ambiguous, weighted residuals should always be shown and analyzed. If the ambiguity remains, the experiment could be made again with more photons, ideally with a longer acquisition time.

In the field of turbid media characterization, experimenters should at least acquire a few million photons to assess accuracy of estimation with the ${\chi }_R{}^2$ value and weighted residuals. If significantly fewer photons are acquired, only big artifacts can be detected with these statistical quantities; accuracy of μ_a and μ_s’ estimation cannot be assessed.

In this article, a correct situation (with satisfactory estimation) corresponds to a ${\chi }_{R-C}{}^2$ value of less than 2, which is already a high value. This is due to the large number of photons, N=1.5*10⁷, and the fact that both the diffusion equation is not perfectly applicable and there are remaining little experimental artifacts such as small light leakage or slightly incorrect IRF measurements. For the situations reviewed in this article and summarized in Table 4 , ${\chi }_{R-C}{}^2$ and the associated weighted residuals are powerful quantities for judging the accuracy of estimation except for μ_a in the presence of a flat air boundary and μ_s’ when there is an error in fiber distance r. More precisely, with regard to the influence of fiber distance, IRF, weak scattering, strong absorption, and the presence of semi-reflective elements, these criteria are pertinent for judging the accuracy of estimation. With regard to surface effects, the model compensates for photon loss by overestimating μ_a without significant degradation of ${\chi }_{R-C}{}^2$: in this case, the criteria are less pertinent for judging the accuracy of μ_a estimation but are still appropriate for μ_s’. The table should be read as follows: when one obtains a ${\chi }_{R-C}{}^2$ value of around 1 (or even 2 as N_ref=1.5*10⁷), the estimates are reliable providing surface interface is far enough and distance r between fibers is precise. On the contrary, if ${\chi }_{R-C}{}^2$ is really high (more than 2), one can expect incorrect estimates. The explanation can be found in the table as being incorrect IRF, experimental artifacts, weak scattering, strong absorption, or too short fiber distance. The last three explanations are related to model error and are specific to the use of diffusion equation. However, the presence of a plane air boundary interface or a wrong fiber distance measurement cannot explain the bad fit.

Table 4. Summary of the role of ${\chi }_{R-C}{}^2$ for judging the accuracy of estimation together with typical situations encountered by experimenters. When not specified, this table refers to values of μ_a~0.1cm⁻¹ and μ_s’~10cm⁻¹. Significant increase of ${\chi }_{R-C}{}^2$ means ${\chi }_{R-C}{}^2>2$ (due to the large number of photons N used as a reference).

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Monte Carlo simulations have been made in order to analyze our results and to track any experimental artifacts, in particular in the case of strong absorption. Both MC scalable with absorption and MC taking absorption into account have been performed. The modified statistic of the scalable MC has been evidenced. The linear increase in ${\chi }_R{}^2$ with the number of photons in the presence of a model bias has been confirmed with the MC simulations taking absorption into account.

To provide more exhaustive results, a future study will consider the case of inhomogeneous media, more particularly the presence of inclusions with different optical properties. The use of a semi-infinite model, or other models closer to radiative transport theory, will also be investigated.