Anal. Chem. 2007, 79, 6792-6798
Power Law Analysis Estimates of Analyte Concentration and Particle Size in Highly Scattering Granular Samples from Photon Time-of-Flight Measurements Fabiano Pandozzi and David H. Burns*
Department of Chemistry, McGill University, 801 Sherbrooke Street West, Otto Maass Building Room 205, Montreal, Quebec, Canada H3A 2K6
Optical measurements of particle size and composition in granular samples are difficult to make due to complex light scattering from particles. These multiple scattering events bias absorption estimates and complicate the calculation of scattering and absorption coefficients used to estimate sample properties. Time series data, such as chromatograms and photon time-of-flight (TOF) profiles, contain self-repeating (fractal) characteristics. Power law analysis of photon TOF profiles allows the determination of absorption coefficients and particle sizes in a single experiment. A correlation dimension algorithm was used on photon TOF data from scattering samples. MLR models were then obtained from correlation dimension plots for the estimation of sample properties. Estimates of particle sizes and absorption coefficients were shown to agree well with theoretical values when compared using independent validation sets. Results show close to a 3-fold and up to a 5-fold decrease in the errors of estimation of dye concentration and particle size, respectively, as compared to steady-state measurements. The power law approach provides a useful means of determining sample properties in highly scattering media. Optical measurements of granular samples are commonly made. However, it is difficult to obtain quantitative estimates of sample properties from these measures. This complication is a result of multiple light paths due to reflections from irregular surfaces of particles, which are dependent upon particle size. As an example, for a dye-coated particle in air, absorption and scattering events are linked since both occur predominantly at the particle surface, while neither process occurs in the medium between particles (for example, dry thin-layer plates). For these measurements, classical models such as the Beer-Lambert law do not hold, so methods to characterize these samples are needed. Considerable research has been done on samples where scattering and absorption are independent, that is, where they do not occur at the same time. For example, a solution of scattering, but nonabsorbing oil dispersed in an absorbing, but nonscattering dye would represent independent absorption and * Corresponding author. E-mail:
[email protected]. Tel: 1-514-398-6933. Fax: 1-514-398-3797.
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scattering in a turbid sample. Models have been used to separate the number of scattering and absorption processes that occur per unit length through a sample and defined them as scattering (µs) and absorption (µa) coefficients, respectively. For a comparison to the classic Beer-Lambert relation, µa is proportional to the product of the molar absorptivity () and concentration of analyte (c).1 Measurements of turbid media have led to various methods for estimating sample properties. Both theoretical models, such as the radiative transport equation,2 as well as empirical methods, such as the Kubelka-Munk3 and Dahm model,4 have been developed to describe these types of samples. In addition, experiments employing time-resolved pulses of light passing through samples have been developed for the estimation of µs and µa.5,6 Recently, experimental photon time-of-flight (TOF) characteristics have been shown to provide reasonable estimates and reduced error of estimated sample properties in media where scattering and absorption are independent.7-9 An examination of photon TOF profiles from colloidal systems using a power law analysis technique has provided insight into the fractal (selfsimilar) characteristics present within the data.10 This type of analysis was shown to provide straightforward interpretation of photon TOF measurements with independent absorption and scattering estimates. Yet, although this power law analysis technique has been shown to be effective for samples where absorption and scattering are independent, there has not yet been any work where the two processes are dependent. Power law analysis has been shown to be a useful technique where complex systems and structures can be investigated in a simplified manner. The power law approach has been used in a broad range of applications in the literature with a great deal (1) Haselgrove, J.; Leigh, J.; Yee, C.; Wang, N.-G.; Maris, M.; Chance, B. Proc. SPIE 1991, 1431, 30-41. (2) Ishimaru, A. Wave Propagation and Scattering in Random Media; Academic Press: New York, 1978. (3) Kubelka, P.; Munk, F. Z. Technol. Phys. 1931, 12, 593-601. (4) Dahm, D. J.; Dahm, K. D. J. Near Infrared Spectrosc. 1999, 7, 47-53. (5) Patterson, M. S.; Chance, B.; Wilson, B. C. Appl. Opt. 1989, 28, 23312336. (6) Madsen, S. J.; Wilson, B. C.; Patterson, M. J.; Park, Y. D. Appl. Opt. 1992, 31, 3509-3517. (7) Leonardi, L.; Burns, D. H. Anal. Chim. Acta 1997, 348, 543-551. (8) Leonardi, L.; Burns, D. H. Appl. Spectrosc. 1999, 53, 628-636. (9) Gributs, C. E. W.; Burns, D. H. Anal. Chim. Acta 2003, 490, 185-195. (10) Gributs, C. E. W.; Burns, D. H. Anal. Chem. 2005, 77, 4213-4218. 10.1021/ac070961x CCC: $37.00
© 2007 American Chemical Society Published on Web 08/08/2007
Figure 1. Experimental setup used to acquire photon time-of-flight data from granular samples.
directed at effective sample characterization of particulate matter. For example, adsorption of gases or polystyrene chains of varying lengths to the surfaces of particles has been shown to have a fractal relationship with particle size.11 Results from angle-resolved scattered intensity measurements show fractal structure of surface irregularities of ground glass.12 Although particles and particle surfaces may appear to have random features, there are selfsimilarities at varying scales. This provides useful information for samples where surface irregularity is important, such as for catalysts or separation packings. Coherent illumination of fractal objects has shown that the intensity profiles exhibit fractional power law relationships.13 Likewise, characterization of cloud structure by power law analysis over large-scale variations has been proposed as a way in which to more effectively model light transmittance.14 From the above range of work published on light interaction with particles, the potential of power law analysis for the characterization of complex systems is evident. However, to date, extension of power law analysis to estimation of µa and particle size in granular samples has not been done. Power law analysis of photon TOF data from granular particles is proposed for the estimation of sample properties. In these samples, both absorption and scattering are allowed to change significantly. This type of analysis allows the identification of two distinct regions, which provide quantitative information for both particle size and absorption estimates. Compared to steady-state measurements, significant improvements in the error of estimation for both µa and particle size are observed. EXPERIMENTAL SECTION Instrument. Time-resolved diffuse reflectance experiments were conducted using the optical configuration shown in Figure 1.15 Briefly, a mode-locked Ti:sapphire laser (Mira 900B, Coherent, Santa Clara, CA) pumped by an argon ion laser (Innova 310, Coherent) was tuned to 780 nm with 170-fs pulses with a repetition rate of 76 MHz. Average output power was 510 mW with a peak (11) Avnir, D.; Farin, D. Nature 1984, 308, 261-263. (12) Popov, I. A.; Glushchenko, L. A.; Uozumi, J. Opt. Commun. 2002, 203, 191-196. (13) Uozumi, J.; Ibrahim, M.; Asakura, T. Opt. Commun. 1998, 156, 350-358. (14) Sachs, D.; Lovejoy, S.; Schertzer, D. Fractals 2002, 10, 1-12. (15) Long, W. F.; Burns, D. H. Anal. Chim. Acta 2001, 434, 113-123.
power of 51 kW. Before entering the sample, light from the laser was split so that a portion of the light hit a fast photodiode (ET2000, Electro-Optics Technology Inc., Traverse City, MI) starting a time-to-amplitude converter (TAC) clock. The remainder of the beam was attenuated by a neutral density filter before entering the sample so that single-photon events would be measured. Light was directed to the sample through the use of a computer-controlled mirror galvanometer (CX660, General Scanning Inc., Watertown, MA). The second mirror was used to guide the light to strike perpendicular to the face of the sample cell. A black reservoir 20 mm deep and having a 40 × 50 mm glass window, 0.15 mm thick, fitted to it, was used as a sample cell. Light emitted from the sample was focused by a pair of lenses onto a microchannel plate photomultiplier tube (MCP-PMT; R3808U, Hamamatsu Corp., Bridgewater, NJ). Photons detected by the MCP-PMT were used to control the timer (model 2145, Canberra). This arrangement maximized photon counting through the 400-kHz, 12-bit A/D converter (AT2000, National Instruments, Austin, TX). Temporal sampling was 4.9 ps. Overall instrument response was 280 ps fwhm when a nonscattering sample was monitored. Photon TOF distributions were obtained with source/ detector separations of 5, 10, and 15 mm. Integration time was set to 6 min so that the total counts obtained were between 1.8 × 105 and 9.6 × 105 for each curve. Smoothing with a Gaussian window of 25-ps standard deviation was used in order to reduce the periodic noise introduced by the TAC. Photon TOF profiles contained 520 data points and were initially processed using power law analysis. All data processing techniques were accomplished using routines generated with the Matlab programming language (The Mathworks Inc., Natick, MA). Materials. Samples of silica particles having varying sizes were prepared with a dye adsorbed to their surface as described in Long and Burns.15 Mean particle diameters used were 156, 293, 930, and 1612 µm (ICN Biomaterials). This corresponded to µs values ranging from 1.86 to 20.01 mm-1. A dye (Dr. Ph. Martin’s Transparent Water Color No. 33 Black, Hollywood, FL) was added to the samples to produce varying levels of absorption, corresponding to µa values ranging from 0 to 0.47 mm-1. A total of 20 unique samples having a variety of particle sizes and dye concentrations were analyzed. Methods of Analysis. Power law analysis examines selfsimilarities of features in data at different scales. This self-similarity exhibits scaling with magnification of regions of the data and leads to a specific rate of change of detail exhibited at each magnification level. These values can be used to describe the dimension of the data. Higher rates of change suggest higher complexity of the measured signal. Simple shapes such as lines, squares, and cubes result in rates of change of 1, 2, and 3, respectively. Signals having substantial self-similarity, such as a snowflake (Koch curve), can have noninteger dimensions, which has led to the definition of the fractal dimension (D). Various methods exist that allow the determination of fractal dimension such as box counting and correlation algorithms.16 These methods have been used extensively to characterize self-similarity in a wide variety of systems. For example, power law analysis has been used in a wide range of areas as a means for fingerprinting chromatograms obtained (16) Theiler, J. J. Opt. Soc. Am. A 1990, 7, 1055-1073.
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exhibit a linear portion in the CDP that should have a slope equal to the Dc. Calculation of Dc is thus simply the first derivative of the CDP:16
Dc )
δ log(Cr) δ log(r)
(2)
passed over the data is varied, and the procedure is repeated for a variety of radii. A plot of log(Cr) versus log(r) is constructed, which is known as the correlation dimension plot (CDP). Data analyzed in this fashion, which contains fractal characteristics, will
Equation 2 is valid when a single fractal relationship exists in the data. When multiple fractal processes occur, a complex convolution of the processes is obtained in the output. As an example, original fractal signals may be further modified by other fractal processes, leading to a more complicated signal. Trends in the CDP will be apparent in different regions, depending on the scale at which the process is occurring.20 Thus, CDPs resulting from multifractal processes will contain multiple regions. For most signals, noise is present and will dominate at very small time scales. This leads to a Dc of 1 for random noise contribution, whereas another region may be dominated by a process that results in the major portion of the signal of interest. Multifractal processes have even been proposed to exist in simulations of enzymatic reactions where reactant crowding is present.21 Complex systems leading to multifractal processes require more complex analysis routines than simple processes, since multiple regions of a CDP need to be investigated. Estimation of Particle Size and Absorption Coefficients. For absorption coefficient and particle size estimation, two multilinear regression approaches were used on information contained in the CDPs. In the first technique, a stagewise multilinear regression was used to identify regions that were highly correlated to sample optical properties. It has been shown that information can be collapsed into simple slopes and intercepts from selected regions in CDPs from TOF data.10 As a result, the second approach employed a multilinear regression on the slope (fractal dimension) and intercept information from regions identified as having high correlation to sample optical properties. Photon TOF peak area was also used as a parameter since it has been shown previously to be effective in estimation of µa.8 For the computation of Dc, the linear portions of the correlation dimension plots were determined. This was done by taking the first point of the line at a large log(r) value and successively adding points to it moving toward the lower log(r) portion of the plot. The initial line contained only 10 points and was constructed to coincide with the regions of highest correlation changes in sample properties. An f-test with a 99% confidence level was used to evaluate when the addition of a point to the line significantly increased the error about the regression (sy).22 From the optimal regression line, the slope and x- and y-intercepts were obtained for subsequent calculations. These multilinear processing techniques allowed the determination of particle sizes and µa values for each sample with coefficients of variance (CV) and R2 statistics being calculated. In both approaches, a similar multilinear model was used, where the dependent vector Y (1 × p), p being the number of
(17) Yiyu, C.; Minjun, C.; Welsh, W. J. J. Chem. Inf. Comput. Sci. 2003, 43, 1959-1965. (18) Spillman, W. B., Jr.; Robertson, J. L.; Huckle, W. R.; Govindan, B. S.; Meissner, K. E. Phys. Rev. E 2004, 70, 061911, 1-12. (19) Martinis, M.; Knezˇevic´, A.; Krstacˇic´, G.; Vargovic´, E. Phys. Rev. E 2004, 70, 012903, 1-4.
(20) Falconer, K. Fractal Geometry: Mathematical foundations and applications; John Wiley & Sons: New York, 1990. (21) Aranda, J. S.; Salgado, E.; Mun ˜oz-Diosdado, A. J. Theor. Biol. 2006, 240, 209-217. (22) Christian, G. D. Analytical Chemistry, 6th ed.; John Wiley & Sons: Hoboken, NJ, 2004.
Figure 2. Schematic representation of the correlation dimension calculation process.
from separations of medicinal herbs,17 characterizing malignant cancer tumors,18 and for diagnosing heart conditions.19 Correlation Dimension. Factors determining which method to use for calculating dimension are linked to the type of data that is being investigated, time and computational constraints, and the information required from the data. Since the TOF data consist of vectors containing equally spaced data points, a correlation dimension approach was chosen. The correlation dimension method is an average pointwise mass algorithm and is a straightforward way of measuring the dimension of this type of data. The correlation dimension (Dc) obtained is one of many different types of dimensions used to characterize shapes, often those containing fractal characteristics. In practice, what it represents for a time series data plot is a measure of interpoint separation. Implementation of the correlation dimension method is based on a circle counting procedure, shown in Figure 2. As depicted, the process begins by taking a circle having radius r1 and passing it along the data curve beginning with it being centered at the first point in the data set, point x1. The circle is then moved along the entire plot one point at a time with the number of data points that fall within the circle’s radius along the curve being summed up. For data having a total of N points, and with xi representing the ith point within the data set, the correlation sum (Cr) value for a given circle radius is given in eq 1. The radius of the circle N
∑ (total number of points within r of point x ) i
Cr )
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i)1
N(N - 1)
(1)
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Figure 3. (a, b) Photon time-of-flight profiles for source/detector separation of 5 mm for (a) constant particle size of 1.61 mm and a dye concentration of 0, 26 ppm, 74, and 258 ppm as well as (b) constant dye concentration of 48 ppm and a particle size of 1.61, 0.93, 0.29, and 0.16 mm.
sample properties being estimated, was related to the independent variable xm by using the weighing coefficients bm. The subscript m denotes which parameter is specified by the equation, namely, from 1 to m. In essence, the weighting coefficients (b1-bm) are applied to individual data points from the CDPs (x1 xm). The general form of the multilinear equation is then given by
Y ) b0 + b1x1 + ... + bmxm
(3)
A different number of independent parameters can be included in the model to best describe the data. However, an f-test at the 95% confidence interval is used to determine the most parsimonious model, to avoid overfitting the data. In addition, a separate estimation of an independent data set is used to determine the robustness of the model with the CV and R2 of the models being calculated. When there are few variables to select from, all possible combinations can be used to determine the most parsimonious calibration. This applies to the MLR model involving the use of slope and intercept information. However, when there are many variables, this cannot be readily done and some search method is required. This situation presents itself when examining the whole CDP, and as a result, stagewise multilinear regression is used to select variables with respect to their correlation to sample properties. Details of this are given in Draper and Smith.23 In general, what is done is models with different numbers of parameters are evaluated to see whether there is a significant difference between their standard errors. When the addition of a parameter does not significantly reduce the error at the 95% confidence level, the previous model is kept as the most parsimonious. RESULTS AND DISCUSSION Examples of photon TOF curves obtained for samples with constant particle size and constant dye concentration, but varying dye concentration and particle size, are shown in Figures 3a and b, respectively. When one of the sample properties is changed, there are characteristic shifts in the profiles. If particle size is kept (23) Draper, N.; Smith, H. Applied Regression Analysis, 2nd ed.; Wiley: New York, 1981.
constant and dye concentration is increased, there is a notable attenuation in signal intensity, as well as a slight shift to shorter times. For samples with constant dye concentration, when the particle size is decreased, this leads to an attenuated signal as well as a shift to longer times. As there are no models that effectively predict sample properties when both particle size and absorption coefficients vary simultaneously in granular samples, power law analysis facilitates analysis because it aids in noting subtle features within data. Properties of Correlation Dimension Plots. When the data are transformed by a 10∧(time axis), this provides greater separation of the effects from concurrently changing absorption and scattering coefficients within samples. Generation of CDPs shows that they appear to be different from colloidal samples,10 which exhibit monotonically changing curves, due to the presence of different regions. Characteristic correlation dimension plots for samples with constant particle size and varying dye concentration, as well as constant dye concentration and varying particle size, are shown in Figures 4a and b, respectively. Below the log(r) value of 0, both sample properties show similar profiles, with slopes close to 1, which is consistent with random noise fluctuations in measurements.10 Above this region, there are characteristic changes describing variations in both sample absorption and particle size, which is also consistent with the work in the aforementioned reference. When looking at just absorbance variations, the region that exhibits changes is in the upper region (above log(r) of 0, especially above log(r) of 1.5). When looking at just particle size variations, the log(r) region between about 0 and 1.5 is most correlated to sample properties. This is an interesting finding in itself for two main reasons. First, this means that different regions of CDPs can be used to effectively estimate individual sample properties without significant influence from the other concurrently varying sample properties. For example, particle size could be estimated while variations in absorption coefficients were present with little influence on the estimation. Second, this provides information on the fundamental changes in the photon TOF profiles arising from variations in sample properties. For absorption coefficients, relatively large circles are used to probe variations in these sample properties (large log(r) region of CDP), whereas smaller circles are more effective in probing variations in particle size. This means that changing dye Analytical Chemistry, Vol. 79, No. 17, September 1, 2007
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Figure 4. (a, b) Characteristic correlation dimension plots at a source/detector separation of 5 mm for samples with (a) constant particle size of 1.61 mm and a dye concentration of 0, 26, 74, and 258 ppm as well as (b) constant dye concentration of 48 ppm and a particle size of 1.61, 0.93, 0.29, and 0.16 mm. Note that regions are labeled corresponding to the highest correlation with estimation of µa and particle size.
concentration leads to larger variations in TOF profiles, as opposed to smaller variations in the profile exhibited by particle size variation (refer to Figure 2a and b). Thus, this shows that distinct differences in the shape of the TOF profiles occur when sample properties vary and that the two confounding variations can be separated from one another without significant difficulty by using a power law analysis approach. Absorption Coefficient Estimation. Conventional methods for the estimation of absorption coefficients are based on taking steady-state absorbance measurements, which would correspond to using only the peak area from a photon TOF profile. This is because the area under the curve represents the integrated intensity, and the time axis for the TOF profile is related to the path length that the photon travels. However, using only the integrated area for these highly scattering samples does not account for the multiple scattering events, which results in poor estimations. The best model using only peak area was obtained at a source/detector separation of 15 mm and only had an R2 of 0.60 and a CV of 60.1%. This multiple scattering bias can be largely avoided when power law analysis and multiple parameters are used for the estimation of sample absorption coefficients. SMLR was used to objectively select areas of the CDPs that had highest correlation with analyte concentration, regardless of particle size. The particular regions of the CDPs that were effective in robustly estimating µa are shown in Figure 4a, with the regions numbered in order of highest to lowest correlation. The first region of interest is associated with the upper portion of the CDP, while the next area, having lower correlation, is located lower in the CDP. The more highly correlated upper log(r) regions on the CDPs are related to large-scale variations of the photon TOF profiles. It is noted that, for a given increase in dye concentration, there is a shift of the CDP in the upper region to a lower log(r) value. Similar results were observed for all of the path lengths for the samples. It was found that highest correlations are generally obtained when the source/detector separation is shortest. Results obtained show that the estimation of absorption coefficients can be effectively achieved by using only two parameters (at a separation of 5 mm), with an R2 of 0.95 and CV of 20.7%. This model was found to be the most parsimonious; however, improved estimation can be achieved with the use of additional parameters. These results are consistent with those presented in the literature for solid granular samples by Long and Burns.15 6796 Analytical Chemistry, Vol. 79, No. 17, September 1, 2007
Table 1. Results from MLR Analysis for the Estimation of Absorption Coefficients separation (mm)
parameters used
R2
CV (%)
5 15 15 15
m, by, bx, A m, by, A m, by bx
0.93 0.92 0.92 0.88
25.5 26.5 27.3 33.0
Since the SMLR was able to objectively suggest a region that was most correlated with sample properties, MLR models were built by using slope and intercept information from the most highly correlated regions. It was found that the best models were obtained by probing the CDP within approximately log(r) 1.5 and 2.5. Thus, all possible combinations of the linear regression parameters obtained from the linear portion of the CDPs, as well as the peak area from the original TOF profiles, were used to build models. The parameters used are denoted m (slope), by (yintercept), bx (x-intercept), and A (peak area of original TOF profile). R2 and the CVs were calculated for each model, and the best models using 1, 2, 3, and 4 parameters for estimation are shown in Table 1. The best estimation of µa was obtained at a 5-mm source/detector separation using m, by, bx, and A and had an R2 of 0.93 and a CV of 25.5%. Interestingly, the most parsimonious model obtained used only a single parameter, namely, bx, at a source/detector separation of 15 mm with an R2 of 0.88 and a CV of 33.0%. It is important to note that the MLR and SMLR approaches discussed, involving the analysis of CDPs, lead to a significant reduction in the error of estimation compared to conventional methodologies. Compared to conventional integrated peak intensity measurements, approximately a 3-fold improvement in the error of estimation of µa can be achieved by using these power law analysis techniques. Particle Size Estimation. As with absorption coefficient estimation, the use of integrated peak area for correlation to particle size is shown to provide poor estimations. This is because the effects due to absorption coefficient variation within samples cannot be separated from those of particle size using only one parameter in a linear equation, for samples that have concomitant variations in sample properties. Using only the integrated peak area of photon TOF profiles, the best model (at 10-mm source/ detector separation) only has an R2 of 0.23 and a CV of 65.5%,
Table 2. Regression Results for the Estimation of Particle Size Using MLR separation (mm)
parameters used
R2
CV (%)
5 5 5 5
m, by, bx, A m, by, A m, by m
0.97 0.97 0.95 0.95
13.4 13.1 16.4 16.8
which would be quite unacceptable for sample analysis. Similar to absorption coefficient estimation, power law analysis and multiple parameters can be used in linear equations to account for the affects of absorption and particle size changes to greatly improve these results. SMLR was used to independently estimate particle sizes for samples with varying analyte concentrations. Different regions of the CDPs were found to be most highly correlated with changes in particle size, compared to absorption coefficients, which are illustrated in Figure 4b. Unlike estimation of absorption coefficients, which had the highest correlated region in the upper section of the CDP, the most highly correlated region for particle sizes is at smaller regions on the log(r) axis. It is noted that the slope of the region between log(r) of 0 and 1.5 decreases when the particle size decreases (scattering increases). At the same time, there is also a visible increase in the y-intercept (by) for a linear regression within this region. Better estimations are obtained when the source/detector separations are smallest, which is again consistent with the estimation of absorption coefficients using the SMLR approach. Results show that accurate particle size estimations can be made using few parameters. The most parsimonious model obtained, using three parameters, was at a 5-mm source/detector separation and had an R2 of 0.96 and a CV of 13.7%. MLR models were built using the regions specified by the results of SMLR in order to estimate particle sizes independently from absorption coefficients. All possible combinations of the linear regressions parameters from CDPs, as well as the peak area from the original TOF profiles were used as parameters. R2 and CVs are shown in Table 2 for the best models using 1, 2, 3, and 4 parameters. In general, estimations were better when the separation distance was shorter. The best model was obtained at a source/detector separation of 5 mm using only three parameters (m, by, and A) and had an R2 and CV of 0.97 and 13.1%, respectively. However, the most parsimonious model employed only the slope of the line (m) at 5-mm separation and had an R2 of 0.95 and CV of 16.8%. A plot of the known versus estimated particle sizes is shown in Figure 5 for the best model. It is important to note that the error bars denote one standard deviation for the known particles and was included since it is significant compared to the actual particle sizes being estimated. Knowing that there is considerable spread in particle size uniformity, this makes estimation of sample properties more difficult, and as a result, both of the multilinear regression approaches are quite effective in estimating sample particle sizes. It is quite clear that correlation dimension analysis leads to significantly improved estimation of particle size, with up to a 5-fold reduction in errors of estimation over conventional techniques.
Figure 5. Results from MLR estimation of absorption coefficient with a source/detector separation of 5 mm and using m, by, and A parameters (R2 ) 0.97, CV ) 13.1%). Error bars denote ( 1 standard deviation for particle sizes.
CONCLUSION Power law analysis has been shown to reveal clear trends relating photon TOF profiles to absorption and scattering processes from solid granular samples. Changes in dye concentration lead to visible shifts in CDPs in regions mostly associated with large-scale variation of photon TOF signals. This is different than particle size variation, which leads to changes in CDPs corresponding predominantly to smaller photon TOF signal variations. Information from these CDPs can then be used to simultaneously estimate analyte concentration, as well as particle size of samples. CDPs generated by power law analysis exhibited two distinct regions affected by scattering and absorption processes. These results are different from colloidal sample analysis, since multiple regions are observed in CDPs and may be related to the dependent absorption and scattering processes of the samples investigated. This is significant since estimations can be made effectively by selecting the region being analyzed, to minimize effects due to concurrent variation in other sample properties. More accurate estimations can be obtained since the two processes are readily separated. The temporal scale at which individual processes dominate can then be identified. It was found that particle size variations are characteristic of integrations over small scales, whereas absorption processes are related to long time scales. Both multilinear regression approaches lead to accurate estimation of sample properties. The SMLR approach specified regions of highest correlation within CDPs to sample properties. Compared to conventional integration of a signal, results from SMLR analysis lead to an approximate 3-fold and more than a 5-fold reduction in error of estimation for µa and particle size estimation, respectively. The second method related slope and intercept information from CDPs and photon TOF profiles to sample properties. When applying the second approach, some similarities were noted compared to separate analyses of colloidal samples. For example, when estimating µa, the single best parameter for both analyses was bx. Errors of estimation were generally lower in the colloidal sample analysis, yet it is important to note that the parameters used for the estimation of µa were normalized to µs. This may be attributed to the inherent differences in sample composition, mainly because scattering and absorption processes are linked for granular samples. Analytical Chemistry, Vol. 79, No. 17, September 1, 2007
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In general, the CDP signal was similar for different source/ detector separations. However, results were clearly better at shorter source/detector separations. This suggests that both scattering and absorption processes are present with close source/ detector separation and that this information may be attenuated at longer separations. Thus, optimizing instrumental parameters is seen as being crucial in order to obtain accurate estimations of sample properties. Power law analysis of photon TOF experiments has been shown to lead to accurate determination of both absorption coefficients and particle sizes of highly scattering granular samples. Experiments are straightforward and data processing does not require significant computational time. This technique has the potential for use in a variety of research areas. One such
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area is the on-line analysis of solid samples where information about analyte concentration and particle size is required, especially when both parameters vary concurrently. This would prove useful for pharmaceutical or polymer industries. ACKNOWLEDGMENT We thank Dr. William F. Long for his work with the photon TOF measurements. Funding for the research was provided by the National Science and Engineering Research Council of Canada (NSERC). Received for June 25, 2007. AC070961X
review
May
11,
2007.
Accepted
