Anal. Chem. 2005, 77, 4213-4218
Fractal Dimension Analysis of Time-Resolved Diffusely Scattered Light from Turbid Samples Claudia E. W. Gributs and David H. Burns*
Department of Chemistry, McGill University, 801 Sherbrooke Street West, Montre´ al, QC, H3A 2K6, Canada
To improve quantification of optical properties in highly scattering and absorbing samples, time-correlated single photon counting measurements were analyzed using quantities related to the correlation dimension. Photon time-of-flight (TOF) distributions were collected in reflection and transmission optical configurations from samples made of cream and water-soluble dye (0 < µa < 0.05 mm-1; 100 < µs < 250 mm-1). It was found that absorption and scattering properties of samples could be accurately quantified from information used to determine the correlation dimension. Scattering coefficients were estimated with less than 4% error for both optical configurations. Absorption estimates were made with CVs of 7.5 and 9.6% for reflection and transmission, respectively. Overall, fractal dimension analysis of TOF distributions provides a simple method of determining the optical properties of a sample. Spectroscopic analysis of highly scattering samples such as tissue or colloidal suspensions is problematic. Light attenuation cannot be related directly to concentration using the BeerLambert relation, as the distance traveled by individual photons through turbid samples increases with higher scattering levels. Theoretical methods such as the diffusion approximation to the radiative transport equation have sought to describe the path length distribution analytically.1,2 Fitting this equation to experimental time-resolved intensity measurements of an impulse of light traversing the sample leads to estimates of absorption and scattering.3,4 These optical properties are commonly expressed as the absorption coefficient (µa) and the scattering coefficient (µs), which represent the number of absorption and scattering events per millimeter. Absorption coefficients can be related to Beer’s law by A ) Lc ) 0.434µaL, where A is absorbance, is molar absorptivity, L represents path length, and c is molar concentration.5 However, approximations made in the derivation of the diffusion equation are known to lead to errors of up to 10% when this model is used to analyze experimental data.3 * To whom correspondence should be addressed. E-mail:
[email protected]. Tel: (514) 398-6933. Fax: (514) 398-3797. (1) Ishimaru, A. Wave Propagation and Scattering in Random Media; IEEE Press: New York, 1997. (2) Patterson, M. S.; Chance, B.; Wilson, B. C. Appl. Opt. 1989, 28, 23312336. (3) Madsen, S. J.; Wilson, B. C.; Patterson, M. S.; Park, Y. D.; Jacques, S. L.; Hefetz, Y. Appl. Opt. 1992, 31, 3509-3517. (4) Matcher, S. J.; Cope, M.; Delpy, D. T. Appl. Opt. 1997, 36, 386-396. (5) Haselgrove, J.; Leigh, J.; Yee, C.; Wang, N.-G.; Maris, M.; Chance, B. Proc. SPIE 1991, 1431, 30-41. 10.1021/ac048495o CCC: $30.25 Published on Web 06/07/2005
© 2005 American Chemical Society
Alternately, statistically based methods have been developed to obtain µa and µs values from photon time-of-flight (TOF) distributions. Statistical descriptors such as mean fall or rise time, peak maximum, and statistical moments have been shown to be useful to estimate optical properties.6 Haar wavelet analysis of TOF profiles has also been investigated.7 It was shown that a small number of wavelets selecting low-frequency features of the distributions could be used to accurately estimate µa and µs. Overall, descriptors of the shape of TOF distributions are therefore useful for quantification. However, better tools to describe the profiles are needed. Calculation of a fractal dimension can be used as a method of describing geometrical shapes. Much like wavelets, fractal dimension is often applied to self-similar data. Initially, the development of a family of fractal dimensions was prompted by the difficulty of determining statistical moments (area, mean, etc.) of curves with an infinite amount of detail such as the Koch curve or coastlines. However, dimensions have been found to be more generally useful than just in the context of fractal analysis. Fractal dimension D is a good measure of shape, i.e., the intricacy and space-filling properties of the data.8,9 For instance, a straight line (D ) 1), a square wave (D ≈ 1.5), and a saw-tooth wave (1 < D < 1.5) can be distinguished based on their dimension even though they are not fractals.8 For this reason, fractal dimension has been used to describe complex medical signals such as electrocardiograms and electroencephalograms.8,10 Fractal dimension has also recently been used as a descriptor of the shape of chromatograms for the classification of medicinal herbs.11 Furthermore, fractal dimension has been proposed as a criterion to direct feature selection algorithms for compression. Individual variables are retained or removed based on their effect on the dimension of a data set.12,13 In the present article, the use of fractal dimension to estimate scattering and absorption from photon TOF distributions was investigated. Since fractal dimension efficiently describes the (6) Leonardi, L.; Burns, D. H. Appl. Spectrosc. 1999, 53, 628-636. (7) Gributs, C. E. W.; Burns, D. H. Appl. Opt. 2003, 42, 2923-2930. (8) Katz, M. J. Comput. Biol. Med. 1988, 18, 145-156. (9) Katz, M. J. Comput. Biol. Med. 1989, 19, 291. (10) Galka, A. Topics in Nonlinear Time Series Analysis; World Scientific: Singapore, 2000. (11) Yiyu, C.; Minjun, C.; Welsh, W. J. J. Chem. Inf. Comput. Sci. 2003, 43, 1959-1965. (12) Traina, C., Jr.; Traina, A.; Wu, L.; Faloutsos, C. XV Brazilian Database Symposium, Joa˜o Pessoa, Brazil, 2000; pp 158-171. (13) de Sousa, E. P. M.; Traina Jr., C.; Traina, A. J. M. First Workshop on Fractals and Self-similarity in Data Mining: Issues and Approaches (in conjunction with 8th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining), Edmonton, Alberta, Canada, 2002; pp 26-30.
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Figure 1. Photon TOF instrument. Ti:Sapph. laser, titanium:sapphire laser; BS, beam splitter; PD, photodiode; CFD, constant fraction discriminator; DL, delay line; ND, neutral density filter; S, sample; MCP-PMT, microchannel plate photomultiplier tube; TAC, time-toamplitude converter; PC, personal computer. Dotted lines show the configuration of the detection system for collecting either reflected, R, or transmitted, T, photons.
shape of signals in one variable, it led to simple calibration models. To increase the robustness of estimations, additional parameters related to fractal dimension were also investigated. Accurate estimations of scattering and absorption coefficients were obtained, and it was shown that instrumental timing drift would not significantly affect the latter estimations. EXPERIMENTAL SECTION Instrument. Time-correlated single photon counting measurements were made in a reflection and a transmission optical configuration, as illustrated in Figure 1. A detailed description of this system has been given previously.6 Briefly, a mode-locked Ti:sapphire laser (Mira Basic, Coherent) pumped with an argon ion laser (Innova 300, Coherent) was used to generate 170-fs pulses of light at 780 nm with a repetition rate of 76 MHz. The output power of the laser was measured throughout the experiment to be 500 mW, corresponding to peak powers of 50 kW. Measured peak powers of the laser pulses had a 20% sinusoidal variation at 400 kHz. A PIN photodiode (ET 2000, Electro-Optics Technology) before the sample started a clock (model 2145, Canberra) for the time-to-amplitude converter. A threshold was set for the start pulse to select only the maximum pulse at the 400-kHz repetition rate, stabilizing measurements to one peak power. A neutral density filter was used to attenuate the laser before entering the sample for single-photon detection. The timeto-amplitude converter was stopped after detection of a single photon by a microchannel plate photomultiplier tube (R3808U01, Hamamatsu) at the output of the sample. Photon TOF distributions were digitized with 4.9-ps resolution. The instrument response with no scattering sample present was measured to be 150 ps (fwhm) before and after the experimental measurements. In reflection experiments, a 4.5 × 5.5 × 7.5 cm sample cell was used. Light was input and collected from the sample through an entrance and an exit aperture (1 × 5 mm) separated by 1.2 cm. Five minutes of integration was necessary to obtain TOF profiles with a total number of counts between 106 and 107. These distributions were referenced to a zero-time marker recorded by 4214 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
Figure 2. Photon time-of-flight distributions from reflection configuration (a) varying absorption from 0 to 0.025 mm-1 at constant low scattering (b) varying scattering from 150 to 250 mm-1 at constant low absorption. TOF distributions from transmission configuration (c) varying absorption from 0 to 0.05 mm-1 at constant low scattering (d) varying scattering from 150 to 250 mm-1 at constant low absorption.
placing a mirror at the entrance of the sample cell. Typical TOF distributions are shown in Figure 2a and b. In a transmission configuration, the sample was held by a 1 cm by 1 cm cuvette. Integrating over 3 min gave TOF distributions of 106-107 counts. The instrument response was measured with a nonscattering sample in the cuvette, and the peak maximum was used as a zero-time marker for all the profiles collected. TOF distributions collected in this configuration are illustrated in Figure 2c and d. Materials. Two independent sets of samples with a combination of seven scattering and eight absorption levels were analyzed at separate times and in random order.6 Scattering coefficients (µs) ranging from 100 to 250 mm-1 were obtained by diluting cream (35% milk fat). Appropriate dilution factors were determined based on previously reported µs values for milk and by assuming that scattering scales linearly with concentration.6,14 Absorption coefficient (µa) values spanning 0-0.025 mm-1 for reflection and 0-0.050 mm-1 for transmission were obtained by diluting watersoluble dye (Dr. Ph. Martin’s Dye, Salis Inc.), where the absorption coefficient was determined from near-infrared spectroscopy. Microscope observation verified that the dye was not soluble in the scatterers. Both the scattering and absorption ranges studied were comparable to the properties of highly scattering biological samples.6 For comparisons between these media, possible differences in anisotropy factor and refractive index must however be taken into account (g ) 0.9 and n ≈ 1.35 in this wavelength region for milk).14, 15 METHODS OF ANALYSIS Correlation Dimension. An entire family of generalized dimensions exists to describe geometrical shape. However, it is often impossible to distinguish among the different dimensions (14) Waterworth, M. D.; Tarte, B. J.; Joblin, A. J.; van Doorn, T.; Niesler, H. E. Australas. Phys. Eng. Sci. Med. 1995, 18, 39-44. (15) Ra¨ty, J. A.; Peiponen, K.-E. Appl. Spectrosc. 1999, 53, 1123-1127.
Figure 3. Calculation of the correlation sum Cr. The number of points less than r away from each given data point are counted and added over the entire TOF distribution; repeat varying r.
due to imperfections in experimental data,16 and therefore a single dimension is chosen. The correlation dimension, Dc, is the most widely used dimension to characterize fractal or nonfractal shapes. It can probe interpoint distances down to a much finer scale than other dimensions, and computation from experimental data is efficient.16,17 When a pointwise-distance algorithm is used, the correlation dimension is given by10,18
Dc ) δ log Cr/δ log r
(1)
where r is the interpoint distance investigated and Cr is the correlation sum defined as N
∑(total number of points within r of point x ) i
Cr )
i)1
N(N - 1)
(2)
with N being the total number of points in the data set. Figure 3 illustrates the calculations involved in computation of Cr for any given interpoint distance r. Starting with the first data point, x1, the number of points that lie closer than r away from x1 are counted. Schematically, this is done by counting the number of points that fall within a circle of radius r centered around x1. Once this number is found, the same procedure is applied to x2 and subsequent points. The number of points found is summed over all xi and normalized by N(N - 1) to give a Cr value for the chosen r. The above procedure is repeated for different radii r to give a series of Cr values. If the data set has one underlying fractal characteristic, then the plot of log(Cr) as a function of log(r) is linear, and the slope corresponds to Dc. In cases where the analyzed shape has multiple fractal properties, several linear regions appear in the correlation sum plot. Deviations from linearity usually appear at small r values and are due to limited experimental resolution. Artifacts at large r values also arise and are due to the limited data size.18 As the measuring radius r is (16) Galka, A. Topics in Nonlinear Time Series Analysis; World Scientific: Singapore, 2000; Vol. 14, pp 93-112. (17) Theiler, J. J. Opt. Soc. Am. A 1990, 7, 1055-1073. (18) Williams, G. P. Chaos Theory Tamed; Joseph Henry Press: Washington, DC, 1997.
increased, it eventually becomes so large that it starts to catch nearly all available points. Care must be taken to exclude these regions when the correlation dimension is determined by linear regression. From the above definition of Dc, it can be noted that Cr gives the probability of two points randomly chosen from the analyzed curve of having a distance smaller than r.19 The correlation dimension therefore provides information about the combined height and width of a shape. In chromatography, this has been useful for characterizing the number, shape, and size of peaks of unknown chromatograms, in view of sample classification. Since photon TOF distributions are similar in appearance to chromatographic peaks, Dc should also be a good descriptor of the shape of temporal point spread functions. It has previously6 been found that the second moment of the distribution (i.e., peak width) and peak height are well correlated to µa and µs. Therefore, a relation should exist that links the correlation dimension of a TOF distribution to values of µa and µs for a sample. Obtaining the correlation sum plot was the first step in analyzing experimental data. All software for the analysis was written in the Matlab (The MathWorks, Natick, MA) programming language. Due to matrix size constraints in the software used, TOF profiles were compressed to have a resolution of 9.8 ps before computing Cr for 80 logarithmically spaced values of r. To calculate the correlation dimension by linear regression, linear regions of the correlation sum plot had to be identified. Nonlinearity at large interpoint distances was excluded from the analysis by disregarding the region of log(Cr) ) 0, as well as points with log(r) within 0.2 of the latter plateau. To determine how far the linear region extended toward low log(r) values, linear regressions were computed over increasingly wide ranges. An initial regression was carried out using the 10 data points with the largest log(r) values (i.e., a window half a decade wide on the x-axis). Points at smaller log(r) were progressively added to the regression range until the standard error about the regression, sy,20 increased significantly based on an f-test at the 99% confidence level (CL). From this regression, the slope (m), intercept with the y-axis (by), and intercept with the x-axis (bx) were obtained. Rather than limiting the analysis to only the correlation dimension (Dc ) m), by and bx were also saved to maximize the information extracted from the plot. Any combination of two of these three parameters could be used to accurately describe the linear relation between log(Cr) and log(r). Estimating Scattering and Absorption. Scattering and absorption coefficients were independently estimated. For scattering, inverse least-squares (ILS) regression with cross-validation was used to relate µs to combinations of m ()Dc), by, and bx. All six possible models with up to two variables were successively tested. For a given combination of parameters, the calibration was validated using a “leave one scattering level out” cross-validation. An estimate of the scattering coefficient was obtained for each sample in the validation sets. Once cross-validation was completed, the µs estimates were used to compute an overall coefficient of variation (CV) and r2. The most parsimonious combination of (19) Galka, A. Topics in Nonlinear Time Series Analysis; World Scientific: Singapore, 2000; Vol. 14, pp 123-148. (20) Christian, G. D. In Analytical Chemistry; John Wiley & Sons: New York, 1994; p 50.
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Figure 5. Effect of instrumental drift on correlation sum plots. (a) TOF distribution with up to 1.22-ns drift in timing. (b) Correlation sum plots corresponding to the TOF distributions shown in (a).
Figure 4. Variation of log(Cr) vs log(r) for reflection measurements with (a) increasing scattering and (b) absorption, and for transmission measurements with varying (c) scattering and (d) absorption. The dashed line represents the behavior of noise.
regression parameters to estimate scattering was determined using an f-test at the 99% CL. Estimations of µa were carried out using the method described above, with a “leave one absorption level out” cross-validation. In addition to constructing linear models between µa and combinations of m, by, and bx, ILS regressions were also computed using normalized parameters from the correlation sum plot. This normalization consisted of scaling m, by, and bx to an estimated µs for each sample. Since µs was much larger than µa in the media analyzed, it can be assumed that optical path lengths were mostly affected by scattering. Therefore, scattering coefficients were directly related to the distance traveled by photons through the medium and could be used to normalize m, by, and bx. RESULTS AND DISCUSSION Properties of the Correlation Sum Plots. Characteristic correlation sum plots for both reflection and transmission data are shown in Figure 4. In all cases, two linear regions are clearly distinguishable, one for small interpoint distances (log(r) < 1.5) and the other for large distances (log(r) > 2.5). Analysis of a nonmodulated laser signal recorded by the photon TOF instrument shows that the variation of log(Cr) with log(r) for background noise is similar to the behavior of the TOF data at log(r) < 1.5 (see Figure 4). Therefore, as would be expected, noise in the data accounts for the slope at small interpoint distances. However, the trend in the log(r) > 2.5 region is linked to the overall shape of photon TOF distributions and varies as optical properties are changed. Time-correlated single photon counting experiments are not immune to instrumental drift. The effect of a change in t ) 0 on the correlation sum plots was therefore investigated. A photon TOF distribution was progressively shifted toward longer times, with a maximum time delay of 1.22 ns (Figure 5a). Corresponding correlation sum plots are shown in Figure 5b. From the latter figure, it is clear that drift in t ) 0 does not influence correlation sums at large r values. Since the time axis in a TOF distribution has a much smaller range than the y-axis (several nanoseconds compared to thousands of counts), changes in the x-axis do not significantly affect interpoint distances when only large r values are considered. Based on the above observations, trends in Cr at large interpoint distances are linked to the scattering and absorption 4216 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
Table 1. Scattering Coefficient Estimation Based on Different Combinations of Parameters Obtained from the log(Cr) vs log(r) Plot experimental configuration
parameters used in calibrationa
CV (%)
r2
reflection
m by bx m, by m, bx bx, by
19 21 3.9 3.9 3.3 3.4
0.25 0.06 0.97 0.98 0.98 0.98
transmission
m by bx m, by m, bx by, bx
12 14 3.3 4.0 3.4 3.4
0.64 0.54 0.98 0.98 0.98 0.98
a Abbreviations used: m, slope; b , intercept with the y-axis; b , y x intercept with the x-axis.
properties of highly scattering samples. Furthermore, these calibrations are resilient to instrumental drift. Scattering Coefficient Estimation. Scattering coefficients were estimated from parameters that described the linear trend in the region of log(r) > 2.5 of the correlation sum plots. Results for all six possible combinations of regression parameters are given in Table 1. When TOF distributions for diffusely reflected photons were analyzed, the best-fit model was based on a combination of the slope () Dc) and x-intercept of the correlation sum plot. A CV of 3.3% was obtained, with an r2 of 0.98. However, it was possible to estimate scattering from a single variable without significantly affecting the error of estimation, based on an f-test at the 99% CL. A model that used bx yielded a CV of 3.9% and an r2 of 0.97 and is illustrated in Figure 6a. In the latter calibration, the mean of the models obtained during cross-validation was: µs ) -207bx + 1.14 × 103. For transmitted photons, the best-fit solution was obtained using only the x-intercept of the correlation sum plot. Estimations were made with a CV of 3.3% and an r2 of 0.98, as shown in Figure 6b. The mean calibration µs ) -294bx + 1.66 × 103 was obtained during cross-validation to relate bx to scattering. Comparison of the above models shows that the same parameter, bx, was useful for estimating scattering in both optical configurations of the instrument. Similar relationships have not been shown before. Though the calibration relations were not identical, they were proportional. The mean coefficients determined from transmitted and reflected photons were related by a factor of 0.71. Since the dominant path length through the sample
Table 2. Absorption Coefficient Estimation Based on Different Combinations of Parameters Obtained from the log(Cr) vs log(r) Plot experimental configuration
Figure 6. Scattering estimates from (a) reflection experiments using the x-intercept of correlation sum plots (CV of 3.9% and r2 of 0.97) and (b) transmission experiments using the x-intercept of correlation sum plots (CV of 3.3% and r2 of 0.98).
parameters used in µa calibrationa
CV (%)
r2
reflection
m/µs by/µs bx/µs m/µs, by/µs m/µs, bx/µs by/µs, bx/µs
36 42 56 9.4 7.5 7.7
0.51 0.35 0.76 0.97 0.98 0.98
transmission
m/µs by/µs bx/µs m/µs, by/µs m/µs, bx/µs by/µs, bx/µs
48 51 56 19.6 11.9 9.8
0.18 0.07 0.61 0.84 0.94 0.96
a The abbreviations used represent: m, slope; b , intercept with the y y-axis; bx, intercept with the x-axis. µs is the scattering coefficient estimated for each sample using bx.
Figure 7. Significance of the x-intercept of the correlation sum plot. rbx is the distance between the end of the TOF distribution and the peak maximum and combines information of peak position and peak height.
is proportional to the time corresponding to maximum counts (tmax), this quantity should account for the difference in magnitude between the calibration coefficients obtained. Examination of the temporal point spread functions in Figure 2a and c shows that the ratio of tmax(transmission)/tmax(reflection) was 0.63. Therefore, changes in tmax accounted for 89% of the variation between calibrations. The unexplained variation can be attributed to differences in the method of light collection. When transmitted light is measured, a high proportion of nonscattered, ballistic photons are detected, whereas reflected photons are always highly scattered. Therefore, measurement of transmitted photons skews the distributions to shorter times and will contribute to the error observed. The significance of the x-intercept in a correlation sum plot has not been previously discussed. However, when the x-intercept of the log(Cr) versus log(r) plot is examined in relation to the TOF distributions, it is noted that bx represents the largest interpoint distance in the data. Any further increase in r above bx does not change the number of points counted, whereas a decrease of r below bx decreases log(Cr). In the TOF profiles collected, the largest interpoint distance rbx corresponds to the distance between the end of the TOF distribution and the peak maximum, as illustrated in Figure 7. This distance rbx contains peak position and peak height information. Thus, the x-intercept is a parameter that combines tmax and the maximum height of the distribution. Compared with previous quantification of scattering based on classical statistical descriptors6 or a Haar transform (HT) analysis of the same TOF distributions,7,21 the x-intercept of the correlation sum plot leads to comparable errors but uses fewer parameters. (21) Gributs, C. E. W.; Burns, D. H. Can. J. Anal. Sci. Spectrosc. 2004, 49.
Figure 8. Absorption estimates from (a) reflection experiments using the scattering-corrected slope and x-intercept of correlation sum plots (CV of 7.5% and r2 of 0.98) and (b) transmission experiments using the scattering-corrected y-intercept and x-intercept of correlation sum plots (CV of 9.8% and r2 of 0.96). Scattering estimates were obtained from the best fit illustrated in Figure 6.
An additional advantage of fractal dimension analysis is the lower sensitivity to drift between runs. Unlike HT analysis, for example, drift in t ) 0 does not affect the results since the correlation sum emphasizes the relative distance between points. Absorption Coefficient Estimation. Changes in the absorption coefficient were also reflected in the correlation sum plots by characteristic variations at large interpoint distances (Figure 4). Therefore, a simple method was sought to accurately link trends in the correlation sum of photon TOF distributions to µa values in scattering samples. To determine whether a simple relation exists between µa and a combination of m ()Dc), by, and bx from the correlation sum plot, all possible combinations of up to two parameters were tested. The best estimations of absorption from reflection measurements were obtained using a combination of by and bx. A CV of 15% with an r2 of 0.92 was obtained. Measurements of transmitted photons led to an error of 19% with an r2 of 0.85 using m and by. Estimation of absorption coefficients was improved when scattering was taken into account. Optical path length differences between samples were normalized by scaling m, by, and bx to the estimated µs for each sample. Models with a combination of up to two scatter-corrected variables were investigated. Table 2 shows the absorption coefficient results in the case where scattering estimates were obtained from the most parsimonious model (using bx). Analytical Chemistry, Vol. 77, No. 13, July 1, 2005
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The best-fit model for estimating µa from diffuse reflection measurements used normalized values of m and bx. A CV of 7.5% and an r2 of 0.98 were obtained (Figure 8a). For transmission measurements, the best fit was obtained when calibration relied on the normalized x- and y-intercepts. The latter estimation, shown in Figure 8b, had a CV of 9.8% and r2 of 0.96. Compared with reflection, it is possible that the increased error in transmission measurements is due to the much higher number of ballistic photons collected and the resulting complexity of the TOF distributions. Compared to previous results from Leonardi and Burns,6 the error obtained by using the correlation sum was similar and even 2% lower for reflection. Fractal analysis also outperformed Haar transform processing7,21 since it used fewer variables in the calibration models. Furthermore, since the analysis based on Cr is less dependent on drift than the previous two methods, it has a clear advantage for absorption estimations. CONCLUSION It has been shown that when large interpoint distances are considered in TOF distributions, the correlation sum contains information that can be directly used to estimate µs and µa. Both reflection and transmission time-resolved light measurements can be analyzed effectively by this method. A model that will work for reflection and transmission has not been shown previously. Scattering coefficient estimates were made with less than 4% error for both experimental configurations. These estimations depended on the x-intercept of the correlation sum plot, which can easily be related back to classical descriptors of peak height and position. Furthermore, results indicate that a universal calibration may exist that relates bx to µs, no matter what source-
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detector separation is used for photon collection. Further study of the latter question seems warranted. Absorption estimates required a normalization of the correlation sum plot parameters using an estimate of µs for each sample. Estimates from reflection measurements were made using the scatter-corrected correlation dimension and x-intercept of the plot (CV of 7.5%). For transmission, the best fit was obtained when the scaled x- and y-intercepts were used (CV of 9.8%). Compared to previously established methods of quantifying µa, errors obtained were statistically comparable in both optical configurations, with reflection being slightly improved. Overall, fractal-based analysis of photon TOF distributions provides a straightforward method of accurately quantifying analytes in highly scattering media. Few variables are required for good quantification, and the method is less sensitive to timing errors than Haar transform analysis for example. Therefore, fractal dimension analysis is well suited for use with simplified photon TOF instruments for quick and accurate determination of optical properties in media such as tissue. ACKNOWLEDGMENT The authors thank Lorenzo Leonardi for acquiring the data. Funding for this research was provided by the National Science and Engineering Research Council of Canada (NSERC) and the Fonds Que´becois de la Recherche sur la Nature et les Technologies (FQRNT).
Received for review November 24, 2004. AC048495O
October
12,
2004.
Accepted
