Pharmaceutical Applications of Separation of Absorption and Scattering in Near-Infrared Spectroscopy (NIRS)

REVIEW Pharmaceutical Applications of Separation of Absorption and Scattering in Near-Infrared Spectroscopy (NIRS) ZHENQI SHI, CARL A. ANDERSON Graduate School of Pharmaceutical Sciences, Duquesne University, Pittsburgh, Pennsylvania 15282 Received 14 December 2009; accepted 20 April 2010 Published online 2 June 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jps.22228 ABSTRACT: The number of near-infrared (NIR) spectroscopic applications in the pharmaceutical sciences has grown significantly in the last decade. Despite its widespread application, the fundamental interaction between NIR radiation and pharmaceutical materials is often not mechanistically well understood. Separation of absorption and scattering in near-infrared spectroscopy (NIRS) is intended to extract absorption and scattering spectra (i.e., absorption and reduced scattering coefficients) from reflectance/transmittance NIR measurements. The purpose of the separation is twofold: (1) to enhance the understanding of the individual roles played by absorption and scattering in NIRS and (2) to apply the separated absorption and scattering spectra for practical spectroscopic analyses. This review paper surveys the multiple techniques reported to date on the separation of NIR absorption and scattering within pharmaceutical applications, focusing on the instrumentations, mathematical approaches used to separate absorption and scattering and related pharmaceutical applications. This literature review is expected to enhance the understanding and thereby the utility of NIRS in pharmaceutical science. Further, the measurement and subsequent understanding of the separation of absorption and scattering is expected to increase not only the number of NIRS applications, but also their robustness. ß 2010 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 99:4766–4783, 2010

Keywords: near-infrared spectroscopy; absorption spectroscopy; light-scattering; Monte Carlo; radiative transport equation

INTRODUCTION The number of near-infrared spectroscopic applications in pharmaceutical science has grown significantly in the last decade. These applications have rapidly permeated the research and development activities in the pharmaceutical industry, including raw material characterization, powder blending monitoring, granulation process control, tablet manufacture and finished products characterization.1,2 Much of the appeal of this technique is due to the fact that a wealth of chemical and physical information can be obtained noninvasively within seconds, often without the need for any sample preparation. As NIR spectra contain information pertaining to both the chemical and physical properties of Correspondence to: Carl A. Anderson (Telephone: 412-396-1102; Fax: 412-396-4660; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 99, 4766–4783 (2010) ß 2010 Wiley-Liss, Inc. and the American Pharmacists Association

4766

the sample matrix, multivariate modeling is often required to correlate spectral information with the chemical or physical properties of the analyte of interest. In spite of the extensive application of NIRS, the underlying optical behavior of NIR radiation in pharmaceutical materials is often not mechanistically well understood. It is well known that two primary events occur when NIR light impinges on a turbid medium (e.g., biological tissues or pharmaceutical solids): absorption and scattering.2 Absorption reduces the intensity of photons of specific energy due to an alteration of the molecular dipole of a bond; therefore, chemical attributes such as concentration are expected to affect absorption events. Scattering, on the other hand, is caused by mismatched refractive indices at particle– air/particle–particle interstitial spaces within the sample; therefore, physical parameters such as sample density and porosity are dominant factors in determining scattering events. Considerable success has been achieved in the field of biomedical optics when separated absorption and scattering properties

JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

were used to describe underlying optical behaviors in tissue samples.3,4 The separated absorption and scattering properties are often used to understand the optical behaviors in tissues, correlating them directly with the chemical and physical features of the tissue medium. The major goal in tissue optics is to utilize the separated absorbing and scattering properties of tissue samples for diagnostic and therapeutic applications. Due to the common features shared by tissue samples and pharmaceutical solids (as both are turbid media), more and more attention has been paid to the separation of NIR absorption and scattering in pharmaceutical applications. The two main purposes of the separation are to improve the understanding of individual roles played by absorption and scattering in NIRS, and to utilize the separated absorption and scattering spectra for qualitative and quantitative applications. To date, five techniques to separate NIR absorption and scattering of pharmaceutical related materials have been reported: spatially resolved spectroscopy,5–8 frequency-resolved spectroscopy,9–21 time-resolved spectroscopy,22–27 the integrating sphere based reflectance and transmittance measurements,28–31 and measurements of remission, absorption and transmission fractions through layers of material of different thicknesses.32,33 A review of how these techniques are applied specifically to pharmaceutical application has yet to be reported. A clear understanding of these optical phenomena is expected to facilitate the application of NIRS to pharmaceutical science, particularly as efforts to engage process analytical technology (PAT)-based manufacturing strategies intensify. Therefore, a literature review detailing the separation of absorption and scattering in pharmaceutical materials is expected to aid in the implementation of NIRS through a fundamental understanding of the underlying optical phenomena. This fundamental understanding will greatly increase the potential success and longevity of NIRS methods in the pharmaceutical industry. The goal of this paper is to review the approaches that have been applied in pharmaceutical analyses, focusing on the following perspectives: (1) instrumentation, (2) mathematical approaches to separate absorption and scattering, and (3) related pharmaceutical applications for the individual techniques as mentioned above. The review begins with a theoretical background of light propagation in turbid media, followed by a literature survey and a comparison among these techniques from both theoretical and application standpoints, and concludes with a discussion of the authors’ perspective on future trends of these techniques in the pharmaceutical applications. DOI 10.1002/jps

4767

THEORY—LIGHT PROPAGATION IN TURBID MEDIA Turbid Medium In physics and chemistry, light propagation is often described by its wave-like and particle-like properties. Wave-like properties indicate light to be an oscillating electromagnetic (EM) field with a continuous range of energies. This is also referred to as the classical theory of light propagation. Particle-like properties indicate that light waves consist of packets of energy called photons, which are described by the quantum model. Quantum theory introduces the idea that light and matter exchange energy as photons. It is well known that light propagation in dilute, nonscattering solution systems is described by the Lambert–Beer’s law. Since a diluted sample does not scatter light, a fixed path-length is typically used in Lambert–Beer’s law. If the sample multiply scatters light, then a distribution of path-lengths will be observed. Multiply scattering, also called ‘‘multiple scattering,’’ is the photon behavior in which individual photons are scattered a large number of times before eventually escaping from or being absorbed by the medium. Thus, the description of light propagation in strongly scattering media becomes more complicated than is expressed by the Lambert–Beer’s law. Examples for multiply scattering media are concentrated solutions, colloids, semi-solids, and solids, etc. This is also the reason that models for light propagation in strongly scattering media have been used so widely in medicine (i.e., tissue optics), agriculture, the paper industry, and pharmaceuticals. The most common examples of strongly scattering media in the pharmaceutical field are solid materials, either in free powder or consolidated compact forms. A sampling medium exhibiting the multiply scattering property is normally referred to as a turbid medium. Radiative Transport Equation Considerable success has been achieved in describing the light propagation in turbid media by the application of radiative transfer theory (radiative transport equation, RTE).3,4 Radiative transfer theory is not specific to light and has other important applications in areas such as neutron transport and thermodynamics. In the RTE formalism, light propagation is considered equivalent to the flow of discrete photons, which are either absorbed or scattered by the medium. RTE only accounts for the transport of light energy in the medium. It ignores the wave amplitude and phases and does not itself include effects such as JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4768

SHI AND ANDERSON

diffraction, interference, or polarization. Further, no correlation between the radiation fields is considered in RTE (i.e., photons are independent of each other). In radiative transfer theory, a small packet of light energy (I) is considered by defining its position (r) directed in a cone of solid angle (dV) and oriented in the direction of propagation ð^ sÞ in a sample medium. As it propagates in the medium, the packet loses a fraction of its energy due to absorption and scattering out of s^ (the first term on the right side of Eq. 1), but also gains energy from light scattered into the s^ direction from other s^0 directions (the second term on the right side of Eq. 1). These processes are quantified by the integral-differential equation known as the RTE:3 ~ s^  rIðr; s^Þ ¼ ðma þ ms ÞIðr; s^Þ Z m þ ms pð^ s  s^0 ÞIðr; s^ÞdV þ a 4p

Diffusion Approximation to the RTE Due to the multiply scattering events that occur when NIR light interacts with turbid media, the diffusion approximation is commonly used to simplify the RTE. The diffusion approximation is applicable to situations where scattering dominates the light propagation process. In the diffusion process, the particles (in the current case, photons) move through a medium in a series of steps of random length and direction (i.e., random walk). Each step begins with a scattering event that is equally likely to be taken in any direction. In the diffusion approximation, the scattering event in RTE is described by the reduced scattering coefficient m0s , which is related to the previously defined parameters by m0s ¼ ð1  gÞms

ð1Þ

where pð^ s  s^0 Þ is the scattering phase function (SPF) between the s^ and s^0 directions. ma and ms are the absorption and scattering coefficients of the sample medium, respectively. In RTE, the ms and ma are defined as follows. A medium containing a uniform distribution of identical scattering and absorbing particles is characterized by the scattering and absorption coefficients, respectively ms ¼ rs s

(2)

ma ¼ rs a

(3)

where ss and sa are the cross sectional areas of the scattering and absorption particles, respectively, which describe the propensity to scatter and absorb. r is the number density of scattering and absorption particles in the sample medium. The standard units of for ms and ma are cm1. Different terminologies are often used to represent the ms and ma when individual techniques are used to separate absorption and scattering, despite the fact that they often share the same underlying optical definitions. The absorption coefficient is defined as the probability of photons being absorbed, while the scattering coefficient is defined as the probability of photons being scattered. The reciprocal of ms is the scattering mean free path, representing the average distance a photon travels between consecutive scattering events. The reciprocal of ma is the absorption mean free path, representing the average distance a photon travels between consecutive absorption events. Absorption and scattering coefficients together are often referred to as the optical coefficients or the optical properties of a sample medium. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

(4)

where g is the anisotropy factor, describing the angular distribution of SPF between s^0 and s^ in Eq. (1). The anisotropy factor is zero for isotropic scattering. A nonzero g value is representative of anisotropic scattering, in which total forward scattering is described by g ¼ 1, while total backward scattering is described by g ¼ 1. As it can be seen in Figure 1, a scatterer with a g in the range of 0 to 1 means it is more likely to forward-scatter the incident photons, while a scatter with g ranging from 1 to 0 indicates it is more likely to backward-scatter the incident photons. Pharmaceutical solids are reported to demonstrate substantial forward scattering within the NIR wavelength region.2 After introducing m0s , the light propagation behavior in a sample with optical properties ma, ms, and g 6¼ 0 can be alternatively described by the same sample with optical properties ma, m0s , and g ¼ 0. Alternatively stated, the reflectance and transmit-

Figure 1. The angular distribution of SPF for g ¼ 0.9, 0.5, 0.1, 0.2, 0.5, and 0.8, assuming the light impinges upon the center of the polar plot from the left. DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

tance for a sample with optical properties ma, ms, and g 6¼ 0 are the same as those for the same sample with optical properties ma, m0s , and g ¼ 0. This is referred to as the ‘‘Similarity Principle’’ in tissue optics.34 Therefore, the optical properties needed to describe light propagation behavior in turbid media are simplified to ma and m0s . Based on the first-order expansion in the unit vector s^, the diffusion equation is obtained as follows; the detailed derivation can be found elsewhere.3,35,36 @ fðr; tÞ ¼ Dr2 fðr; tÞ  ma cm fðr; tÞ þ Qðr; tÞ @t

(5)

In the diffusion approximation, the properties of photon movement in the medium are contained in the diffusion constant, D ¼ ð1=3½ma þ m0s Þ. fðr; tÞ is the fluence rate, which is the light intensity per unit area at position (r) at a given time (t). F(r, t) is the net intensity vector (i.e., diffuse photon flux), which is the photon energy per unit area in the direction of s^. cm symbolizes the speed of light in the sample medium. In order for the diffusion approximation to the RTE to describe the light propagation behaviors in turbid media, the following assumptions are typically considered. m0s =ðma þ m0s Þ, also known as the albedo (v0), is close to unity, that is, when the absorption of the medium is low. If scattering is not dominant, the photon migration behavior cannot be appropriately described by the random walk. Then the diffusion approximation to RTE also does not stand.37 The point of interest is far from sources or boundaries. A common criterion is the distance between the light source and detector is larger than 10 times the mean free path of the photons in the sample. This indicates that the diffusion approximation to RTE is only suitable for describing photon migration behaviors that experience large numbers of scattering events with small scattering angles, in comparison with photons experiencing a few scattering events with large scattering angles.22 The sample medium has to be of finite thickness. If the sample becomes thinner to the point where its thickness is comparable to theffi effective penepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tration depth 1= 3ma ðma þ m0s Þ, the diffusion approximation will break down as a result of the increased relative importance of potential photon–photon interactions.34 Monte Carlo simulation Many situations of practical interest involve a variety of light sources, multiple sample types, and DOI 10.1002/jps

4769

complex illumination geometries. These situations typically require assumptions regarding individual boundary conditions and parameters to generate a closed form analytical solution to the RTE (Eq. 1). Thus, analytical solutions for realistic scenarios are complicated at best (if an analytical solution even exists). For cases when the analytical solution is not available, the problem can be approached using numerical techniques. The most widely used approach for radiative transfer theory is Monte Carlo (MC) simulation method-based photon migration. Monte Carlo refers to the city in Monaco, where the primary attractions are casinos that offer games of chance. The random behavior in games of chance is similar to how MC simulation statistically samples the probability distribution for the photon migration parameters, such as step size, scattering direction, internal reflection/out of boundary, etc. These parameters are simulated using functions of random number generators. Because of these functions, individual photon movement can be traced step-by-step through the sample medium, and the distribution of light (i.e., reflectance and/or transmittance) in the system can be recorded from these individual photon trajectories. As the number of photons in the simulation grows toward infinity, the MC simulation for the light distribution approaches an analytical solution to the RTE. The actual number of photons necessary for a realistic result depends on the specifics of the simulation and which quantities one desires to determine. As few as 3000 photons may be adequate for determining diffuse reflectance from a sample, while more than 100000 may be required for a complex three-dimensional simulation.3 There are several advantages to using the MC method to describe photon migration behavior.3,4 First, the initial large-scale applications of MC methods were radiative transfer problems involving neutron transport. Thus, the MC approach is well suited for problems involving light transport in turbid media because a photon can be treated as a neutron particle whose propagation behaves according to the rules of radiative transport. Second, the algorithms for implementing the basic elements of an MC simulation are straightforward. Third, the technique possesses a great deal of flexibility and is widely applicable to practical transport problems. Through the use of corresponding mathematical relationships, MC simulation has the ability to simulate virtually any source, detector, and sample boundary condition, as well as any combination of sample optical properties. Finally, the MC approach can be used for any albedo and SPF. A sample MC protocol for simulation-based photon migration is illustrated in Figure 2.38 JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4770

SHI AND ANDERSON

Figure 3. Integrating sphere based reflectance and transmittance measurements. (A) Single sphere measurement. (B) Double sphere measurement. The figure was adapted from Wilson.34

Figure 2. Schematic diagram of Monte Carlo simulation. The figure was adapted from Wang et al.38

TECHNIQUES USED FOR SEPARATING ABSORPTION AND SCATTERING IN NIRS Integrating Sphere-Based Reflectance and Transmittance Measurement The integrating sphere-based approach, reported by Fricke and coworkers is the first documented method for the separation of NIR absorption and scattering in pharmaceutical samples.28–31 Measurement of diffuse reflectance (Rd) and diffuse transmittance (Td) provides access for deconvolution of analytical equations to determine ma and m0s . Instrumentation Integrating sphere-based approaches can be generally classified into two different instrument set-ups.34 The first set-up uses a single integrating sphere for both the reflectance and transmittance geometries (Fig. 3A). The incident light can be either a collimated beam or diffuse irradiation. The measurement via diffuse irradiation is less accurate than that made with collimated irradiation as a result of the high background signal of the diffuse source in the reflectance measurement.34 The other set-up consists of a double integrating sphere system (Fig. 3B), where the sample is placed in a common port between two spheres. One of the main JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

advantages of using a double sphere is to increase signal in both spheres over the single-sphere case, which is a result of the ‘‘cross talk’’ between the two spheres.34 Additionally, the double sphere provides the opportunity to determine Rd and Td simultaneously. This instrument configuration has not yet been applied to pharmaceutical samples. In the integrating sphere-based approach, the measurement is designed to obtain values of both Rd and Td. Therefore, the sample must have a finite thickness, which is dependent on the optical properties of the sample. If the sample is too thick, not enough signal will be detected in Td. If the sample is too thin, however, the assumption of the diffusion approximation will be violated as a result of individual photons interfering with each other.34 It has been suggested that performing a complete calibration of the experimental set-up using samples of known absorption and scattering properties over the range of those anticipated for samples and wavelengths of interest is essential to obtain accurate measurements of Td and Rd.34 Thus, both singlesphere and double-sphere systems require calibration using standard samples to characterize the method accuracy (i.e., percentage deviation from known values of the standards) in order to further correct the estimated ma and m0s . Mathematical Approaches to Determine la and ls0 Fricke and coworkers were the first to develop the closed-form equations for Rd and Td, and they used these equations to determine the optical properties of pharmaceutical samples.28–31 The detailed derivations can be found elsewhere.30 Briefly, a three-flux approximation was applied to derive the measured DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

quantities of transmission (Td) and reflectance (Rd). The three-flux approximation represents the three surfaces through which light propagates: transmittance, reflectance, and side scatter (as described in Mathematical Approaches to Determine ma and m0s Section).   Td v 0 ; t 0   cg aT bT kt 0 kt 0 t0 e e þ 3e TF þ ¼ F 1  ð2=3Þk 1 þ ð2=3Þk

þ et0 TF (6)   Rd v 0 ; t 0   cg aR bR 3

þ þ þ 3e2to RF ¼ F 1 þ ð2=3Þk 1  ð2=3Þk 5

þ e2t0 RF (7) The formulae for calculating constants c, g, k, aT, bT, aR, bR, A, and B in Eqs. (6) and (7) can be found elsewhere.30 In addition, F, TF, and RF are experimental constants, representative of the impinging flux of the light source, the transmission and reflectance coefficient of the supporting layer on which the Td and Rd measurements of the powder sample were measured. Except for F, TF, and RF, the rest of constants shown in the right side of Eqs. (6) and (7) are functions of the scaled albedo ðv 0 Þ and the scaled optical depth (t ).30 The v 0 and t are defined as v0 ð1  gÞ 1  v0 g

(8)

t ¼ tð1  v0 gÞ

(9)

v 0 ¼

where t ¼ m0t x, m0t ¼ m0s þ ma , and x represents the photon position along the depth-axis of the sample. Since Eqs. (6) and (7) express the diffuse reflectance and transmittance as a nonlinear function of v 0 and

t , nonlinear regression tools (e.g., Newton–Raphson method) are typically applied to fit the measured Rd

and Td profiles in order to estimate the v 0 and t . Subsequently, the reduced scattering coefficient m0s and absorption coefficient ma can be determined using Eqs. (8) and (9).29 Since the determined optical coefficients are typically normalized by the sample density, the resultant units for the optical coefficients determined by this approach are m2/kg. Related Pharmaceutical Applications Fricke and coworkers conducted a series of studies in which they utilized the integrating spherebased approach to determine optical coefficients of DOI 10.1002/jps

4771

pharmaceutical powder samples within the infrared (IR) and NIR ranges.28,29,31 The first paper addressed the general information of absorption and scattering properties of pharmaceutical powders in the IR range,29 which was later confirmed in the NIR range.31 The inter-relationship between the absorption and reduced scattering coefficients for specific materials was investigated. It was found that an inverse correlation between the reduced scattering and absorption coefficient existed for the sample materials (e.g., lactose and microcrystalline cellulose powder), in which a peak in the absorption coefficient always corresponded to a valley in the reduced scattering coefficient. The inverse relationship between the absorption and reduced scattering coefficients was also confirmed using a theoretical Mie calculation.29 The wavelength dependences of the optical coefficients were also explored. The reduced scattering coefficient gradually decreased with increasing wavelengths (and absorption). Further, the absorption coefficients showed wavelength dependence due to specific absorption bands within certain wavelength ranges. The wavelength dependence of the reduced scattering coefficient was also found to be material dependent. The reduced scattering coefficients of lactose and microcrystalline cellulose powder were found to vary over the observed wavelength range, while the reduced scattering coefficients of paracetamol and ascorbic acid powder were relatively constant. Given the wavelength dependency of scattering, Fricke and coworkers pointed out the potential invalidity in the commonly used scattering correction methods, such as multiplicative scattering correction (MSC) and standard normal variate (SNV), which assume constant scattering across the wavelength axis. Additionally, the authors investigated the effect of particle size on the optical coefficients. It was demonstrated that the reduced scattering coefficient was inversely related to particle size in both the IR and NIR ranges; smaller particle size led to a larger reduced scattering coefficient. Meanwhile, the absorption coefficient was also found to increase when particle size was reduced, especially within the IR range where strong absorption bands are located.29 The correlation between the absorption coefficient and particle size diminished in the NIR range, except for large particle size difference, such as that observed for 90 mm versus 490 mm ascorbic acid powder.31 Fricke and coworkers attributed the relationship between the absorption coefficient and particle size to the decreased hidden mass in the small particles, which results in a corresponding increase in absorption. The hidden mass was defined as the portion of material contributing to the total mass but not contributing to absorption. For example, JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4772

SHI AND ANDERSON

absorption is much stronger for several small particles having the same mass as a single large particle, which is a consequence of the decreased hidden mass of the former.29,31 Given the effects of particle size on the optical coefficients, especially for the absorption coefficient, Fricke and coworkers realized its potential impact on the quantitative analysis of powder mixtures.29 The authors cautioned that the development of a proper calibration for a two-component mixture, subsequent spectral analysis of unknown mixtures may contain significant errors. They postulated that this might be due to a change in the degree of agglomeration (owing to humidity), which alters the particle size distribution of the mixture and leads to differences in the absorption intensity and both optical coefficients.29 However, the authors did not address any potential solutions to the variability in the absorption coefficient as a result of physical variation (e.g., particle size) and its potential effect on prediction errors in future samples. The same group of researchers applied the extracted absorption and reduced scattering coefficients to enhance the prediction capability of a method for determining the amount of active ingredient in two-component mixtures containing paracetamol and lactose.28 Since MSC treats scattering as a wavelength-independent phenomenon, MSC was used only on wavelength regions that were independent of scattering (i.e., wavelength ranges for which constant reduced scattering coefficients of the powder mixture were observed). Partial least squares modeling resulted in a more accurate (lower RMSEC) model compared to MSC applied to the entire wavelength region. Additionally, an artificial neural network (ANN) was trained to build a calibration model between separated absorption coefficients and paracetamol concentrations. The ANN-derived model showed significantly lower RMSEC and RMSEP compared to the PLS calibration. Burger et al. attributed the enhanced performance of the ANN to its capacity to model nonlinear relations. The application of an ANN was justifiable as a certain degree of nonlinearity was found between absorption coefficient and paracetamol concentration. Measurements of Remission, Absorption, and Transmission Fractions through Layers of Material of Different Thicknesses Separation of absorption and scattering was reported for diffuse reflectance measurements of pharmaceutical samples of different thicknesses.32,33 An integrating sphere is typically used to acquire the measurement. The difference compared to the approach used in the section of Integrating SphereBased Reflectance and Transmittance Measurement is that here, only reflectance is used; no transmittance JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

component is involved. Instead of estimating the absorption and reduced scattering coefficients, a simplified solution of the Kubelka–Munk (K–M) function describing light flux into and from samples was used to calculate the K–M absorption (K) and scattering (S) coefficients. The following equations were used to describe the relation between S and K within diffuse reflectance measurements.   2:303 R1 R1 ð1  R1 R0 Þ log (10) S¼ d 1  R21 R1  R0   2:303 1  R1 R1 ð1  R1 R0 Þ K¼ log 2d 1 þ R1 R 1  R0

(11)

Here, R0 denotes the spectrum measured at a defined sample thickness (d) and R1 denotes the measurement of the same sample with an optically infinite thickness. As it can be seen from Eqs. (10) and (11), K and S can be directly calculated from the reflectance measurements. Although the coefficients K and S used in K–M theory are not directly comparable with ma and m0s , they are related through the relationship expressed in Eq. (12).17,21,29    K 8 ma ¼ (12) S 3 m0s The sole pharmaceutical application of the optical coefficients determined by this method was reported in terms of hard model constraints for multivariate curve resolution.33 Multivariate Curve Resolution is a group of chemometric algorithms that help resolve mixtures by determining the number of constituents, their spectral profiles and their estimated concentrations when no prior information is available about the nature and composition of these mixtures. The dataset contained NIR spectra of pharmaceutical tablets compressed at 31, 156, and 281 MPa. The spectra of the tablets compressed at 31 MPa were used for calibration, while the remaining spectra were reserved for validation. It was found that multivariate curve resolution-alternating least squares (MCR-ALS), using the background information of K–M scattering and absorption coefficients, was comparable in calibration, but superior in validation, when compared to PLS modeling without any spectral pretreatment. The study also showed slightly better validation performance for optical coefficients-based MCR-ALS, compared to pure component spectra based MCR-ALS and PLS modeling on extended multiplicative scattering correction (EMSC) preprocessed spectra. Moreover, only three samples were necessary for a reliable and robust calibration when optical coefficients-based MCR-ALS was applied, despite the fact that strong changes in the scattering behavior were expected. DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

Time-Resolved Spectroscopy (TRS) Absorption and scattering properties of a turbid medium can be estimated by measuring the temporal dispersion of a short light pulse as it propagates through the medium. Such measurements have long been of interest in atmospheric research. For example, Weinman and Shipley39 used the time dependence of a transmitted pulse to deduce the optical thickness of clouds. This method was first developed for medical applications, but it has since been extended to other fields, such as pharmaceutics and agriculture. It uses picoseconds laser pulses to irradiate a sample. The light signal diffusively remitted by the sample at a given distance from the irradiation point is then temporally recorded. The temporal shape of the pulse is altered by absorption and scattering events as it passes through the sample. By analyzing the modified temporal shape of the pulse, the optical properties of that sample can be deduced.

4773

fiber (ICF). Due to the optical behaviors of an ICF, light pulses with approximately the same temporal width as the laser are accessible with a spectral width spanning from 500 nm to at least 1200 nm. In contrast to a single photon counting system, this system uses a streak camera to provide a unique combination of a relatively short acquisition time with high spectral and temporal resolution. The system measures a 700 nm wavelength region with a spectral resolution of 5 nm. The system has a total temporal range of 2.1 ns with resolution of 4.5 ps. Since time resolution must be on the order of tens of picoseconds, the major limitation for the current TRS system is the expensive instrument set-up, including both the pulsed laser and the photon-counting detector.34 Also, the system is limited by its current wavelength range, which is relatively narrow compared to that of the NIR region.23 The primary reason for this limitation is the lack of commercially available efficient photon cathode materials for streak tubes that operate in the NIR range.

Instrumentation

Mathematical Approaches to Determine la and ls0

The basic instrument set-up of TRS requires a picoseconds laser pulse, a sample interface (either reflectance or transmittance), and a photon-counting system to record the temporal spreading. The most recent instrument set-up was developed by the Lund Institute of Technology, Sweden.24 This system utilizes an index-guided crystal fiber for light delivery and a streak camera, which is necessary to achieve the temporal resolution. The optic arrangement of this system is presented in Figure 4.24 The laser pulse is focused into a 100 cm long index-guiding crystal

The analytical equations for TRS were initially derived from tissue optics. It was developed by Patterson et al.40 for either reflectance measurements of a semi-infinite homogenous medium or reflectance/ transmittance measurements of a finite medium. To date, all reported pharmaceutical applications of TRS involved finite media. After using specific boundary conditions on the diffusion approximation to the RTE (Eq. 5), the expressions for the reflectance R(d, t) and transmittance T(d, t) at a specific time (t) for a sample with

Figure 4. Optical arrangement of TRS. The figure was reproduced, with permission, from Abrahamsson et al.24 DOI 10.1002/jps

JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4774

SHI AND ANDERSON

finite thickness (d), such as a pharmaceutical tablet, are Rðd; tÞ ¼ ð4pDcÞ1=2 t3=2 expðma ctÞ ( " #   z20 ð2d  z0 Þ2  z0 exp   ð2d  z0 Þexp  4Dct 4Dct " #) ð2d þ z0 Þ2 þ ð2d þ z0 Þ exp  ð13Þ 4Dct Tðd; tÞ ¼ ð4pDcÞ1=2 t3=2 expðma ctÞ ( " # ðd  z0 Þ2  ðd  z0 Þexp   ðd þ z0 Þ 4Dct " # " # ðd þ z0 Þ2 ð3d  z0 Þ2  exp  þ ð3d  z0 Þ exp  4Dct 4Dct " #) ð3d þ z0 Þ2  ð3d þ z0 Þexp  ð14Þ 4Dct where c represents the speed of light propagation in the sample medium, D stands for the diffusion constant, and z0 ¼ ½ð1  gÞms 1 . As can be seen in Eqs. (13) and (14), measurement of either reflectance or transmittance allows for a direct estimate of ma and m0s . In order to deconvolve these two equations and extract the optical coefficients, multiple methods have been reported, including nonlinear regression (Levenberg-Marquardt algorithm, LMA),22,24 least-square support vector machine (LSSVM)26 and other linear approaches, that is, MAximum Determination for Solving Time-REsolved Spectroscopy Signal (MADSTRESS).25 Nonlinear regression is the most commonly used method among pharmaceutical applications.22,24 In mathematics and computing, LMA generates a numerical solution to a problem by minimizing a function, which generally nonlinear, over a space of parameters specific to function. LMA can also be used as a (nonlinear) leastsquares curve-fitting algorithm. Abrahamsson et al. applied LMA to determine the optical coefficients of phantom samples and compared this method to the measurements acquired using integrating spherebased approaches. A phantom sample is a type of lipid emulsion (Intralipid, Sigma–Aldrich, St. Louis, MO) with known optical properties in the short NIR wavelength range. Phantom samples have been widely used as standards in tissue optics to characterize method accuracy. Abrahamsson et al.24 concluded that the two methods offered comparable results. Related Pharmaceutical Applications The first application of TRS to pharmaceutical samples was published by Johansson et al.23 Transmittance mode was used to acquire measurements of a 3.5 mm-thick tablet. Based on the temporal JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

spreading and an assumed refractive index, the degree of scattering within the tablet matrix, in terms of the total optical path length, was determined to be 20–25 cm. This indicated that very strong multiple scattering events took place within the sample. Monte Carlo simulation and a corresponding comparison with the experimental data estimated the reduced scattering coefficients of the tablet to be on the order of 500 cm1 at 790 nm. The same group of researchers published a second paper focusing on the application of extracted reduced scattering coefficients from TRS to enhance the quantitative analysis of pharmaceutical tablets through scatter correction.22 Pharmaceutical tablets produced at different compression forces and various granule sizes were used. Multiple comparisons were performed between the scattering-corrected spectra and raw NIR spectra using different calibration and validation datasets. When compared to raw NIR scans, the scatter-corrected spectra resulted in lower RMSEPs across all of the evaluated conditions. In addition, Abrahamsson et al.27 utilized the slope of the time dispersion curve from the time-resolved measurement to determine the chemical concentration of binary compacts containing iron oxide and microcrystalline cellulose. When compared to traditional transmission-based NIRS, the time-resolved measurement resulted in a fivefold increase in accuracy for the determination of iron oxide concentration. Further, due to the direct relationship between the slope of time dispersion curve and light absorption, the calibration model based on the time-resolved measurement reliably predicted the concentration of iron oxide in samples with physical properties outside those included in the calibration set. Frequency-Resolved Spectroscopy (Frequency Domain Photon Migration, FDPM) The principle of FDPM involves monitoring the timedependent propagation characteristics of multiply scattered light in turbid media. Briefly, this technique launches intensity-modulated light onto a multiply scattering medium via a single point source, and detects it at other discrete points of known distances from the incident light. Upon modulating the incident light at various modulation frequencies or varying the source-to-detector distance, the measurements of phase-shift and amplitude attenuation can be determined as functions of the optical properties of the sample medium. The propagation of such a photon density wave within a turbid medium is influenced by its absorption and scattering properties and can be modeled by the diffusion approximation to the RTE. By solving the diffusion equation with appropriate boundary conditions, the measurement data (i.e., the phase-shift and amplitude attenuation) can be used to determine the optical properties of the sample medium DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

4775

Figure 5. Schematic diagram of the FDPM setup with the enlarged insert denoting a powder configuration for multiple scattering of photons. The figure was reproduced, with permission, from Pan and Sevick-Muraca.12

(i.e., the absorption and reduced scattering coefficients). Instrumentation The latest instrument used in FDPM was developed by Sevick-Muraca et al. This group holds a number of patents on FDPM instrumentation and their related applications.13–15 As shown in Figure 5,12 modulated light of modulation frequency v (typically 30–200 MHz) is launched from a monochromatic laser diode, which is directed to a beam splitter to form reference and sampling beams. The reference beam is delivered to a reference photomultiplier tube (PMT) through a 1-mm diameter optical fiber. The sampling beam is introduced to the sampling medium through a second 1-mm diameter optical fiber whose end is placed within the sampling medium. A third 1-mm-diameter fiber is located a distance of r from the point of illumination to detect the propagated light. The relative distance between the source and detector fibers has to be at least 10 times that of the scattering mean free path to ensure multiple light scattering, typically 1–15 cm. The source and detector fibers are normally maintained in a coplanar geometry. Detection is accomplished with a second PMT. The two PMTs are modulated at the same frequency as the laser diode, with the exception of an additional offset frequency of Dv ¼ 100 Hz. Using the heterodyne technique, the mixed signals are created to contain the sum and difference between the signal at the laser modulation frequency and at 100 Hz higher. Then, DOI 10.1002/jps

the resulting mixed signals from the PMTs are passed through two transimpedance amplifiers to filter the high frequency components, leaving the 100 Hz signals intact. The remaining phase-shift and amplitude attenuation are then measured. Finally, data acquisition software is used to acquire the heterodyned signals and record the phase shift (PS), amplitude (AC), and mean intensity (DC) of the signal from the sample PMT relative to the reference PMT. The optical properties of the sampling medium can then be accurately extracted by solving the equations for PS, AC, and DC as functions of ma and m0s at the wavelength of the monochromatic laser. For the above instrument, Sevick-Muraca and coworkers18 developed two general methods to determine the optical properties, including multiple frequency and multiple distance methods; each method will be discussed in detail in the following section. Under different experimental conditions, individual qualification criteria for each method were developed to assess the accuracy and precision of FDPM measurements,18 including (1) whether abnormal measurement error exists during the FDPM experiment; (2) which ranges of modulation frequency and relative distance are suitable for FDPM experimentation for a given sample; and (3) which segments of the measurement can be used to generate accurate and reliable optical properties. Mathematical Approaches to Determine la and ls0 Fishkin and Gratton solved the diffusion approximation to the RTE (Eq. 5) for an infinite and macroscopically uniform medium. The outcome was JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4776

SHI AND ANDERSON

surements: the multiple frequency and multiple distance methods.

Figure 6. Time evolution of the intensity from a sinusoidally intensity-modulated source. The figure was adapted from Fishkin and Gratton.42

expressions for three experimentally determined quantities: (1) the steady-state photon density or the time invariant average intensity, the DC component, (2) the amplitude of the photon-density oscillation, the AC component, and (3) the phase shift of the photon-density wave, the PS component.42,43 These three quantities are illustrated in Figure 6.42 To eliminate measurement error at a given v, the properties of the photon density wave at two different source-detector separations, namely, r and r0, are normally measured and compared.43 Thus, the DC, AC, and PS are normally measured in their relative quantities, which are expressed as a function of the optical coefficients as  r

3m ðm þ m0s Þ 1=2 0 ln DCrel ¼ ðr  r0 Þ a a (15) 2 r   r

3ma ðma þ m0s Þ 1=2 0 ln ACrel ¼ ðr  r0 Þ 2 r 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2   v 2 4 þ 15  1þ yma

ð16Þ

PSrel ¼ ðr  r0 Þ 

3ma ðma þ m0s Þ  2

1=2

2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2  2 v 4 1þ  15 yma (17)

  3ma ðma þ m0s Þ 1=2 lnðModrel Þ ¼ ðr  r0 Þ 2 2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11=2 3  2 p ffiffiffi v 6 7 þ 1A 5 4 2@ 1þ yma

ð18Þ

Eqs. (15)–(18) express two general approaches for extracting the optical coefficients from FDPM meaJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

Multiple Frequency (MF) Method. Eqs. (15)–(18) show that at fixed distances between the source and detector, r and r0, the values of ln[(r/r0)ACrel], PSrel, and ln(Modrel) are nonlinear functions of the modulation frequency. Thus, measurements of DC, AC, and PS (and therefore Mod), which are functions of the modulation frequency at two fixed source-detector distances, can be used to estimate the optical properties via nonlinear regression. Since there are only two unknowns, multiple combinations of DC, AC, PS, and Mod can be used to determine ma and m0s . It was found that a regression approach based on only one type of measurement data was insufficient to obtain accurate results.18,44 It was also determined that simultaneous regression of DC þ PS, AC þ DC þ PS gave comparable results to simultaneous regression of AC þ PS, while simultaneous regression of DC þ AC or Mod þ PS failed to accurately estimate the optical properties.18,43 Error associated with the nonlinear regression was investigated via a Monte Carlo method of error analysis.18 The study used a polystyrene colloidal suspension to determine m0s and compared it with the value calculated by Mie theory. Based on the error analysis, the accuracy and precision of the determined reduced scattering coefficients obtained via the multiple frequency method were studied and compared at multiple source-detector distances. The comparison concluded that a minimum relative distance (r  r0) of about 2.5 mm was necessary for reasonable accuracy and precision. If the relative distance is smaller than this minimum distance threshold, unsatisfactory accuracy and precision of the extracted m0s will be induced due to the large uncertainty in the FDPM measurement. Multiple Distance (MD) Method. Eqs. (15)–(18) also show that at a fixed modulation frequency, v, the values of ln[(r/r0)DCrel], ln[(r/r0)ACrel], PSrel, and ln(Modrel) are linear functions of the relative distance between the detectors (r  r0). Thus, the slopes (k) can be determined from plots of ln[(r/r0)DCrel], ln[(r/ r0)ACrel], PSrel, and ln(Modrel) versus (r  r0). Subsequently, simultaneous regression via different combinations of kDC, kAC, kMod, and kPS can be used to obtain the optical properties of the sample medium. Comparisons were performed between multiple combinations of kDC, kAC, kMod, and kPS to determine the reduced scattering coefficients via the MD method and those calculated by Mie theory.18 Results indicated that the reduced scattering coefficients derived from simultaneous regression of DC þ PS, AC þ PS, and AC þ DC þ PS agreed well with the theoretical calculations, but simultaneous fitting of DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

DC þ AC or Mod þ PS failed to accurately estimate the reduced scattering coefficients. The results obtained via MD linear regression were similar to those obtained via MF nonlinear regression. This suggested that PS data combined with DC and/or AC data provide the most accurate information of the optical properties. The modulation data (i.e., AC/DC), however, are not suitable for deriving the optical parameters, even when they are combined with PS data. The uncertainties of determination via the MD method were derived from error analysis using Eqs. (15)–(18).45,46 Error analysis was performed to calculate the precision and accuracy associated with the of determination of the reduced scattering coefficient for a polystyrene colloidal suspension measured at multiple frequencies via the MD method. Both precision and accuracy indicated that a modulation frequency greater than 60 MHz was necessary to obtain reasonable results.18 If the modulation frequency is less than the threshold, unsatisfactory accuracy and precision will be induced as a result of the uncertainty associated with the FDPM measurement. In general, the MF and MD methods perform similarly. The most obvious advantage of the MD method is that the analytical solution for the optical parameters can be directly derived without nonlinear regression. Although a study showed that the accuracy of the MD method was better than that of the MF method, the precision was worse.18 The authors suggested combining the two methods to increase the signal-to-noise ratio and improve the accuracy and precision for the estimation of the optical properties. For a MF measurement, the experiment can be performed at several different relative distances (called the combined MF method), while for a MD measurement, the experiment can be performed at several different modulation frequencies (called the combined MD method). The combined approaches were found to improve the accuracy and precision for the estimation of the optical coefficients. Related Pharmaceutical Applications To date, FDPM has been predominately utilized to separate absorption and scattering in NIR spectral responses of pharmaceutical samples. A number of studies were performed by Dr. Sevick-Muraca and collaborators, where they applied FDPM for the analysis of particle size in suspensions and powder media, and for the determination of constituent concentration in powder mixtures.9–12,16,19–21,47 For particle size analysis, an inversion algorithm was developed to determine the particle size distribution (PSD) and volume fraction for noninteracting colloidal suspension samples (<2% volume DOI 10.1002/jps

4777

fraction) from FDPM measurements.19 When particle interaction was accounted for by the full polydisperse hard sphere Percus–Yevick (HSPY) model, FDPM was capable of determining PSD and volume fraction for polydisperse interacting suspensions with volume fractions up to 40%.20,47 The samples used in these studies were simple laboratory systems such as polydisperse polystyrene suspensions and titanium dioxide suspensions. Given these findings, the potential for on-line analyses of PSD and volume fraction of concentrated samples within dense process streams using FDPM is promising. FDPM was also reported to be a useful technique for determining the mean particle size of pharmaceutical powders.17 The MD method was used to determine the optical coefficients for lactose powders with mean particle sizes of 650, 785, and 828 nm. It was found that the reduced scattering coefficient was linearly related to the reciprocal of the mean particle size determined by different reference methods, including sieve analysis, laser diffraction, and image analysis. The absorption coefficients were insensitive to differences in particle size; however, they were wavelength-dependent. As a result of the linear relationship between the reduced scattering coefficient and the reciprocal of the mean particle size, it was proposed that FDPM can be used for particle sizing analysis without spectral pretreatment and chemometric calibration (traditionally used in NIRS). Because of its noninvasive nature of measurement without sample preparation, FDPM is a potential tool for on-line applications. Multiple studies have reported a linear relationship between the absorption coefficients determined by FDPM and the chemical concentrations.16,21 Even when API was mixed with different particle-sized excipients, the separated absorption and scattering properties by FDPM captured the linear relationship between absorption coefficient and API concentration without effect of particle size and without further data pretreatment.21 Comparatively, intensity attenuation-based measurement (e.g., NIRS) was unable to effectively differentiate between light absorption and scattering processes, which require spectral pretreatment to enhance the linearity between spectral intensity and API concentration. Moreover, due to the minimal noise associated with measurements in the frequency domain,18,48 the absorption coefficient determined by FDPM was sensitive to the API signal from both the low (<1%, w/w) and ultralow dose (<0.1%, w/w) formulations.16,21 Meanwhile, the reduced scattering coefficient showed little sensitivity to changes in API concentration within the same concentration range. The major limitation of FDPM for monitoring chemical composition is that the monochromatic laser source is not sufficient for determining the concenJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4778

SHI AND ANDERSON

tration of both excipients and API, especially for multi-component pharmaceutical matrices. Thus, multiple wavelengths measurements are necessary if FDPM is to be widely used for industrial applications. Given that FDPM can determine both absorption and scattering properties (shown by the above studies), Sevick-Muraca and coworkers16,17,21 proposed it as a tool for on-line monitoring of powder blending, including both the variation of API concentrations from changes in the absorption coefficient, and variation in the packing arrangement of the powder bed from changes in the reduced scattering coefficient. In order to apply a noninvasive measurement (e.g., FDPM) for monitoring powder blending, one of the key questions to be answered is the size of sampling volume. The determination of sampling volume provides a means to directly compare FDPM with other analytical measurements, for example, NIRS and HPLC. Mathematical expressions predicting the sampling volume of FDPM were developed for infinite and semi-infinite powder beds via probability distribution analysis to describe the propagation of multiply scattered light between a point source and point detector separated by a known distance.10 The predicted volume of interrogation was in agreement with that determined by empirical measurements of FDPM. Based on the derived equation, the sampling volume of FDPM is determined by the (1) separation distance between the incident point source and the point detector; (2) optical properties of the sample, and (3) modulation frequency. The first article using FDPM to monitor powder blending was published in 2004 by Pan et al.9 In this article, FDPM was compared to HPLC as an off-line method to trace the concentration variation of API in a terazosin powder blend (0.72%, w/w). Thieved samples were used for both FDPM and HPLC measurements. Although the off-line sampling protocol used to monitor powder blending was not ideal, the paper did present evidence demonstrating the relationship between sampling volume and blending variance. Based on the mathematical expression developed earlier,10 the sampling volume by FDPM was estimated to be 1.4 cm3, which was shown to be larger than that determined by both HPLC (0.65 cm3) and the reported value for optic fiber-based NIR spectroscopy (<0.3 cm3). A complete-random-mixture (CRM) model was used to calculate the blending variance in terms of relative standard deviation. After using FDPM, HPLC, and NIR to analyze the homogenous powder mixture, the relative standard deviations were found to be 4.8%, 7%, and 10%, respectively. Thus, the expected inverse relationship between blending variance and sampling volume was JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

clearly illustrated. The larger sampling volume of FDPM was attributed to the separation between the source and detector, which is larger when compared to the close proximity of source and detector commonly used in optic fiber-based NIRS. Additionally, the instrument error for FDPM was also determined using repetitive measurements of one location of a stationary powder bed. The instrument error was determined to be about 0.12%. When compared to a blending variance of 4.8%, instrument noise can be neglected in the calculation of variance. Given the large sampling volume and low instrument error, Pan et al. expected FDPM to provide highprecision determination of low dose API content in pharmaceutical processing streams. The most recent pharmaceutical application of FDPM combined a two-speed diffusion model with FDPM to allow both absorption and scattering information to contribute to monitoring concentration variation during a powder blending process.11,12 It is well known that the solid volume fraction changes irregularly over time during powder blending. Monte Carlo simulation and FDPM measurement were used to demonstrate that both the absorption and reduced scattering coefficients of the powder beds increased with the increment of solid-volume fraction.12 Since both the variations of API concentration and the solid-volume fraction in the powder blending process contribute to changes in the absorption coefficient of the powder bed, it may not be possible to simultaneously determine the variation of API concentrations and the variation of the solid-volume fraction when only the absorption coefficient of the powder bed is considered. This indicates that, besides the absorption coefficient, additional information is required to accurately determining the blend homogeneity. Thus, instead of using the absorption coefficient, the ratio of the absorption to the reduced scattering coefficient was proposed and confirmed to be insensitive to the solid volume fraction, while still maintaining sensitivity to API concentration.12 Pan et al. concluded that the ratio would be appropriate for monitoring powder blending homogeneity, regardless of the variation of solid volume fraction. Spatially-Resolved Spectroscopy (SRS) Spatially resolved spectroscopy involves the measurement of the spatial (either radial or depth) distribution of reflectance and transmittance of a turbid medium to determine its optical properties. SRS was first used to determine the effective extinction coefficient ðm0t Þ. This was accomplished by measuring the fluence-depth distribution in tissue with broad beam illumination using three interstitially placed optical fibers connected to photodetectors.49 DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

4779

where there is no derived analytical equation matching the experimental conditions, numerical simulation (i.e., Monte Carlo simulation-based photon migration) can be used as an alternative to estimate the optical coefficients. These techniques typically involve forward calculation of R(r) over a expected range of ma and m0s values, followed by either iterative interpolation of the measured R(r) inside the calculated range to find the optical properties that yield the smallest interpolation error,52 or prediction of the optical properties by certain regression algorithms, such as artificial neural network (ANN)54 and partial least square (PLS).5,51 Figure 7. Optical set-up of SRS. The figure was adapted from Wilson.34

Instrumentation The most common instrument set-up measures diffuse reflectance (R) as a function of the radial distance (r) on the sample surface (i.e., radially diffused reflectance). The initial set-up used in the field of tissue optics relied on optic-fibers, in which point or pencil-beam source was used to perpendicularly illuminate sample surface, and diffusively reflected signals were detected by optical-fibers at a specified distances from the illumination spot (Fig. 7).34 Detection could be accomplished using one single fiber moving radially and measuring reflectance one radial distance at a time, or using a linear fiber array picking up signal simultaneously from different radial distances. With the development of imaging technology, charge-coupled device (CCD) cameras have emerged as a means of detecting radially diffused reflectance.5,50–52 The advantage of using CCD cameras to capture the entire radial distribution of diffusively reflected signals is to enhance the signal-to-noise ratio of the reflectance measurements via spatial image processing (i.e., signal binning) of equivalent radial distances.50 Mathematical Approaches to Determine la and ls0 In the field of tissue optics, a closed form analytical equation was developed by Farrell and Patterson53 to correlate the radially diffused reflectance with ma and m0s . However, the derived equation was based on the assumptions and boundary conditions for a semiinfinite medium, which is not necessarily applicable for pharmaceutical samples. A semi-infinite homogenous medium has optical boundaries that are infinitely wide, which indicates that the boundaries are much wider than the spatial extent of the photon distribution. Therefore, an alternative approach is needed before SRS can be applied for pharmaceutical applications. Numerical methods have also been used to determine ma and m0s from SRS measurements. For the case DOI 10.1002/jps

Related Pharmaceutical Applications Shi and Anderson were the first to explore the potential applications of SRS in the pharmaceutical field. They, along with other researchers, published a series of reports that focused on SRS method development for pharmaceutical samples,5 the enhanced understanding that separated optical coefficients offer to practical uses of NIRS6,7 and the application of optical coefficients to spectroscopic analyses under practical conditions.5,8 The authors established a chemical imaging-based spatially resolved spectroscopic measurement.5 A chemical imaging system was used to capture both the spatial and spectral information from the radially diffused reflectance of pharmaceutical solid samples (either as powder or tablets). Subsequently, a Monte Carlo simulation-oriented PLS model was used to predict the optical coefficients from the measured radially diffused reflectance. Simulation and reference correction by Intralipid at 1064 nm normalized the simulated radially diffused reflectance such that it was comparable to the measured counterpart. This comparability indicated that the model based on simulated data could be applied to the radially diffused reflectance measurements to predict the optical coefficients. The optical coefficients extracted from SRS have been used to enhance the understanding of practical applications of NIRS.6 The samples used here were pharmaceutical powders of various particle sizes or compacts of various densities. An increase in either particle size or tablet density induced a proportional change in ma and an inversely proportional change in m0s . The separated ma and m0s were input into the Monte Carlo simulation-based photon migration program to trace the photon absorption behavior and record the depth of penetration. The consistency observed between the measured and simulated results indicated that the ma and m0s were the dominant factors in the NIR absorbance profile and the depth of penetration characteristics, respectively. The combination of optical coefficients determined by SRS and Monte Carlo simulation-based photon JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4780

SHI AND ANDERSON

migration provides a unique tool to understand the depth- and radially resolved profiles of NIR radiation on pharmaceutical samples.7 The depth-/radially resolved profile exploits the relationship between the cumulative percentage of reflected information and the depth/radial distance in a sample matrix. In silico studies revealed that both the depth- and radially resolved profiles are nonlinear, indicating that portions of the sample close to the point of interest, along either the depth/the radial distance, contribute more to the final reflectance than those further away. The nonlinearity of the profiles is expected to be dependent on the ma and m0s at their corresponding wavelength. Additionally, the simulated depth-/radially resolved profile was also applicable to chemical imaging systems. The depth and radial distance corresponding to 95% of the reflected information were determined to be approximately 150 and 300 mm, respectively, for the simulation conditions. These values were larger than the physical size of a single pixel in any commercially available chemical imaging system. Thus, the observed information from a single pixel was believed to be representative of the information within a specific 3-D volume. In other words, the reflected intensity captured in a given pixel of a chemical image is a weighted average across a specific depth and radial distance. These results underscore the precautions that must be taken when interpreting NIR chemical images. Based on the enhanced understanding by the above studies about individual roles of absorption and scattering in NIRS, the determined ma and m0s were subsequently applied to spectroscopic analyses under practical conditions. Due to the wavelength and absorption dependency of m0s , a m0s -based scattering correction method was proposed5 as an alternative to wavelength-independent scattering correction methods, such as SNV and MSC. When applied to model the chemical compositions of tablets, the m0s -based scattering correction method, termed scattering orthogonalization, resulted in superior calibration and prediction statistics compared to SNV. The enhanced performance of scattering orthogonalization was attributed to its ability to mitigate the physical interferences while preserving the chemical information. Therefore, this method is expected to be useful for routine model calibration and model update procedures as it minimizes changes to the calibration resulting from physical variations in the samples related to the m0s . Since pure component materials are typically available in the pharmaceutical industry, both ma and m0s of a pure component raw material can be used to represent interfering signals when predicting the chemical concentrations of other components within a powder or tablet mixture. In a recent paper, ma and m0s were integrated into specific chemometric algoJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

rithms.8 For example, net analyte signal (NAS) and generalized least squares (GLS) were used to simplify a NIRS multivariate calibration model using only pure component spectra and concentration values from one formulation mixture. It was found that the simplified model was conducive to parsimonious multivariate models and reached the same or even lower prediction error than traditional approaches. Thus, optical coefficient-based signal processing is expected to be beneficial to both calibration and update efforts during routine NIR spectroscopic analyses.

COMPARISON AMONG TECHNIQUES USED TO SEPARATE ABSORPTION AND SCATTERING IN NIRS Five categories of techniques have been applied to pharmaceutical samples to separate absorption and scattering in NIRS. To simplify the following discussion, these techniques can be reorganized into two major groups: time and intensity related measurements. TRS and FDPM are both time dependent measurements, while integrating sphere-based reflectance and transmittance measurements, measurements through layers of material of different thicknesses and SRS are intensity based techniques. The relationship between TRS and FDPM can be described as follows.48 A broadened pulse will be observed in the time domain h(t), if an infinitesimally short pulse is applied to a turbid scattering medium. Alternatively, if a sinusoidally modulated light source is applied to the same medium, the photon flux at the detector will also be sinusoidal in time, but the oscillation will be delayed in phase and amplitude relative to the source. In essence, the time domain signal h(t) can be linked to the phase and amplitude by the Fourier transform such that any information acquired in the time domain can also be, in principle, obtained in the frequency domain. However, certain practical differences between these two techniques do exist.48 First, typical FDPM measurements using frequencies of 300 MHz or less are considerably less expensive than the time-resolved techniques. Second, phase and amplitude measurements can be made in near-real time such that the influence of time-varying phenomena can be studied in a sample medium. Comparatively, the acquisition rate of time-resolved data collected using a time-correlated single photon counter is usually limited by electronic constraints imposed by the count rate. Therefore, while the fundamental observations are the same, the underlying details of the two techniques dictate their applicability. The integrating sphere-based approach, measurement through layers of material of different thicknesses and SRS are all intensity based measureDOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

ments. The first two techniques detect reflected signals exiting the sample surface and record these signals as a single magnitude at a given wavelength. This method is referred to as the total diffuse reflectance measurement.34 SRS, on the other hand, measures radiallydiffused reflectance, which is the individual reflectance signal at a specific radial distance for a given wavelength. This method is referred to as the local diffuse reflectance measurement.34 Limited studies have been reported to date to compare the practical performance of different techniques across the same sample platform (i.e., the same analyte of interest). Swartling et al.55 compared the practical performance of the integrating sphere-based approach, TRS and SRS for determining the optical coefficients of a set of tissue phantom samples. Their study was limited to a wavelength range of 660–785 nm. The integrating sphere-based method was shown to be the best approach to estimate the reduced scattering coefficients. The authors’ results were supported by previously reported values for phantom samples.56 The integrating sphere-based method, however, had a poor limit of detection for determining absorption coefficients. Comparatively, TRS demonstrated the capacity to determine low absorption coefficients. Overall, Swartling et al. concluded that the differences between the approaches for the determination of the optical coefficients were minimal. Sevick-Muraca et al.57 investigated the theoretical differences between the time dependent measurements (TRS or FDPM) and SRS. The authors concluded that SRS, unlike TRS and FDPM, does not provide direct measurements of photon path length, and it relies solely on the detection of light intensity attenuation to describe the absorption and scattering behaviors. Compared to SRS, TRS, and FDPM do not measure relative intensity, but rather absolute ‘‘time-of-flight’’ and phase delay. Thus, TRS and FDPM are essentially self-calibrating, and are not subjected to the measurement errors associated with the calibration with respect to an external standard. In summary, there are two main reasons for the limited number of studies comparing the practical performances of the various techniques for separating absorption and scattering. First, optical set-ups interrogate different sample volumes.55 For instance, because of the large source-detector distance, FDPM may interrogate a larger sample volume compared to the other techniques. A larger sampling volume minimizes the potential effects of sample heterogeneity on the resultant optical coefficients, leading to better precision and reduced measurement error. The effects of inhomogeneity are also mitigated when transmittance rather than reflectance is used in TRS as the former often interrogates larger volumes.23 DOI 10.1002/jps

4781

Second, the accuracy of the determined optical coefficients is dependent upon the mathematical approaches used. For instance, nonlinear regression performed on multiple, rather than two, radial distance points may enhance the robustness of SRS for extracting the optical coefficients.55

CONCLUSIONS AND PERSPECTIVES Overall, publications detailing the separation of absorption and scattering phenomena in NIRS have improved the current understanding of NIR diffuse reflectance, especially with regard to the individual roles of absorption and scattering. The enhanced understanding has provided and continuously will offer the basis for improved spectroscopic analyses under practical conditions. With the increasing awareness of the importance of NIRS in PAT, the spectroscopic application with mechanistic understanding of underlying optical phenomena is expected to save time and effort when integrating PAT into pharmaceutical processes, and enhance the robustness of multivariate models to provide effective process monitoring and control to ultimately improve end-product quality. For instance, a spectral library of ma(l) and m0s ðlÞ of pure component materials is expected to provide tremendous leverage for both multivariate calibration and routine calibration update in pharmaceutical applications of NIRS. Additionally, upcoming generations of NIRS instrumentation are expected to integrate the techniques used in the separation of absorption and scattering, such as those reviewed in this article, to delineate NIR absorbance spectra directly into absorption and scattering profiles, which will simplify subsequent qualitative and quantitative applications. In the meantime, improvements to the techniques used for the separation of absorption and scattering in NIRS are necessary. A standard with known ma(l) and m0s ðlÞ in NIR range should be developed to improve the accuracy of individual measurements and provide a platform to compare measurements across different techniques. The most common standard in the field of tissue optics is Intralipid. However, the wavelength range used in tissue optics is narrow (600–1100 nm) compared to that which is typically used in pharmaceutical applications (780–2500 nm). Therefore, the determination of ma and m0s for Intralipid over the NIR spectral range, or the design and measurement of some new standard, will be essential for continuous improvement of the techniques reviewed herein. Many of the individual techniques might also benefit from advances in instrumentation. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

4782

SHI AND ANDERSON

For example, the extended wavelength range covered by TRS or instrumentation offering simultaneous FDPM measurements over multiple wavelengths, or better yet, a continuous wavelength range, is expected to improve the functionally and generality of these techniques in pharmaceutical applications.

12.

13.

14.

Finally, the idea of applying the separation of absorption and scattering in NIRS to enhance the mechanistic understanding of the fundamental optical phenomena and improve the practical spectroscopic analyses is expected to facilitate future applications of NIRS.

ACKNOWLEDGMENTS The authors would like to acknowledge Dr. Steve Short for his skillful scientific and grammatical editing, which has been helpful in preparing this manuscript.

15.

16.

17.

18.

19.

REFERENCES 20. 1. Reich G. 2005. Near-infrared spectroscopy and imaging: Basic principles and pharmaceutical applications. Adv Drug Deliv Rev 57:1109–1143. 2. Cogdill RP, Drennen JK. 2006. Near-Infrared Spectroscopy. In: Brittain H, editor. Spectroscopy of pharmaceutical solids. New York: Taylor & Francis. pp. 313–412. 3. Mobley J, Vo-Dinh T. 2003. Optical properties of tissue. In: Vo-Dinh T, editor. Biomedical photonics handbook. Boca Raton: CRC Press. pp. 2.1–2.75. 4. Welch AJ, van Gemert MJC, Star WM, Wilson BC. 1995. Definitions and overview of tissue optics. In: Welch AJ, van Gemert MJC, editors. Optical-thermal response of laser-irradiated tissue. New York: Plenum Press. pp. 15–46. 5. Shi ZQ, Anderson CA. 2009. Scattering orthogonalization of near-infrared spectra for analysis of pharmaceutical tablets. Anal Chem 81:1389–1396. 6. Shi Z, Anderson CA. 2009. Application of Monte Carlo simulation-based photon migration for enhanced understanding of near-infrared (NIR) diffuse reflectance. Part I: Information depth in pharmaceutical materials. J Pharm Sci 99:2399–2412. 7. Shi Z, Anderson CA. 2009. Application of Monte Carlo simulation-based photon migration for enhanced understanding of near-infrared (NIR) diffuse reflectance. Part II: Photon radial diffusion in NIR chemical images. J Pharm Sci 10.1002/ jps.22125. 8. Shi Z, Cogdill RP, Martens H, Anderson CA. 2009. Optical coefficient-based multivariate calibration on near-infrared spectroscopy. J Chemometrics 24:288–299. 9. Pan T, Barber D, Coffin-Beach D, Sun Z, Sevick-Muraca EM. 2004. Measurement of low-dose active pharmaceutical ingredient in a pharmaceutical blend using frequency-domain photon migration. J Pharm Sci 93:635–645. 10. Pan T, Sevick-Muraca EM. 2002. Volume of pharmaceutical powders probed by frequency-domain photon migration measurements of multiply scattered light. Anal Chem 74:4228– 4234. 11. Pan TS, Dali SS, Sevick-Muraca EM. 2005. Evaluation of photon migration using a two speed model for characterization JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010

21.

22.

23.

24.

25.

26.

27.

28.

29.

of packed powder beds and dense particulate suspensions. Opt Express 13:3600–3618. Pan TS, Sevick-Muraca EM. 2006. Evaluation of ingredient concentration in powders using two-speed photon migration theory and measurements. J Pharm Sci 95:530–541. Sevick-Muraca EM, Sun ZG, Huang YQ. 2005. Method for characterizing particles in suspension from frequency domain photon migration measurements. US Patent 6,930,777. Sevick-Muraca EM, Sun ZG, Huang YQ. 2007. Method for characterising particles in suspension from frequency domain photon migration measurements. US Patent 7,268,873. Sevick-Muraca EM, Sun ZG, Pan TS. 2004. Characterizing powders using frequency-domain photon migration. US Patent 6,771,370. Shinde RR, Balgi GV, Nail SL, Sevick-Muraca EM. 1999. Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity. J Pharm Sci 88:959–966. Sun Z, Torrance S, McNeil-Watson FK, Sevick-Muraca EM. 2003. Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders. Anal Chem 75:1720–1725. Sun ZG, Huang YQ, Sevick-Muraca EM. 2002. Precise analysis of frequency domain photon migration measurement for characterization of concentrated colloidal suspensions. Rev Sci Instrum 73:383–393. Sun ZG, Sevick-Muraca EM. 2001. Inversion algorithms for particle sizing with photon migration measurement. AIChE J 47:1487–1498. Sun ZG, Tomlin CD, Sevick-Muraca EM. 2001. Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light. Langmuir 17:6142–6147. Torrance SE, Sun Z, Sevick-Muraca EM. 2004. Impact of excipient particle size on measurement of active pharmaceutical ingredient absorbance in mixtures using frequency domain photon migration. J Pharm Sci 93:1879–1889. Abrahamsson C, Lowgren A, Stromdahl B, Svesson T, Andersson-Engels S, Johansson J, Folestad S. 2005. Scatter correction of transmission near-infrared spectra by photon migration data: Quantitative analysis of solids. Appl Spectrosc 59: 1381–1387. Johansson J, Folestad S, Josefson M, Sparen A, Abrahamsson C, Andersson-Engels S, Svanberg S. 2002. Time-resolved NIR/ Vis spectroscopy for analysis for solids: Pharmaceutical tablets. Appl Spectrosc 56:725–731. Abrahamsson C, Svensson T, Svanberg S, Andersson-Engelsen S, Johansson J, Folestad S. 2004. Time and wavelength resolved spectroscopy of turbid media using light continuum generated in a crystal fiber. Opt Express 12:4103–4112. Chauchard F, Roger JM, Bellon-Maurel V, Abrahamsson C, Andersson-Engels S, Svanberg S. 2005. MADSTRESS: A linear approach for evaluating scattering and absorption coefficients of samples measured using time-resolved spectroscopy in reflection. Appl Spectrosc 59:53–59. Chauchard F, Roussel S, Roger JM, Bellon-Maurel V, Abrahamsson C, Svensson T, Andersson-Engels S, Svanberg S. 2005. Least-squares support vector machines modelization for time-resolved spectroscopy. Appl Opt 44:1–7. Abrahamsson C, Johansson J, Andersson-Engels S, Svanberg S, Folestad S. 2005. Time-resolved NIR spectroscopy for quantitative analysis of intact pharmaceutical tablets. Anal Chem 77:1055–1059. Burger T, Ploss HJ, Ebel S, Fricke J. 1997. Diffuse reflectance and transmittance spectroscopy for the quantitative determination of scattering and absorption coefficients in quantitative powder analysis. Appl Spectrosc 51:1323–1329. Burger T, Kuhn J, Caps RJF. 1997. Quantitative determination of the scattering and absorption coefficients from diffuse DOI 10.1002/jps

SEPARATION OF ABSORPTION AND SCATTERING IN NIRS

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

reflectance and transmittance measurements: Application to pharmaceutical powders. Appl Spectrosc 51:309–317. Kuhn J, Korder S, Arduini-Schuster MC, Caps R, Fricke J. 1993. Infrared-optical transmission and reflection measurements on loose powders. Rev Sci Instrum 64:2523–2530. Burger T, Fricke J, Kuhn J. 1998. NIR radiative transfer investigations to characterise pharmaceutical powders and their mixtures. J Near Infrared Spectrosc 6:33–40. Dahm DJ, Dahm KD. 1999. Spectrographic analysis instrument and method based on discontinuum theory United States Patent 5,912,730. Kessler W, Oelkrug D, Kessler RW. 2008. Using scattering and absorption spectra as MCR-hardmodel constraints for diffuse reflectance measurements of tablets. 11th International Conference on Chemometrics for Analytical Chemistry, Montpellier, France. Wilson BC. 1995. Measurement of tissue optical properties: methods and theories. In: Welch AJ, van Gemert MJC, editors. Optical-thermal response of laser-irradiated tissue. New York: Plenum Press. pp. 233–274. Abrahamsson C. 2005. Dissertation: Time-resolved spectroscopy for pharmaceutical applications. Department of Physics. Lund: Lund University. pp. 29–36. Nichols MG. 1996. Dissertation: Transport of oxygen and light in model tumor systems. Department of Physics and Astronomy. Rochester: University of Rochester. pp. 84–98. Nichols MG, Hull EL, Foster TH. 1997. Design and testing of a white-light, steady-state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems. Appl Opt 36:93–104. Wang LH, Jacques SL, Zheng LQ. 1995. MCML—Monte Carlo modeling of light transport in multi-layered tissues. Comput Methods Programs Biomed 47:131–146. Weinman JA, Shipley ST. 1972. Effects of multiple scattering on laser pulses transmitted through clouds. J Geophys Res 77:7123–7128. Patterson MS, Chance B, Wilson BC. 1989. Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties. Appl Opt 28:2331–2336. van Staveren HJ, Moes CJM, van Marle J, Prahl SA, van Gemert MJC. 1991. Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm. Appl Opt 30:4507–4514. Fishkin JB, Gratton E. 1993. Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge. J Opt Soc Am A 10:127–140. Fishkin JB, So PTC, Cerussi AE, Fantini S, Franceschini MA, Gratton E. 1995. Frequency-domain method for measuring spectral properties in multiple-scattering media: Methemoglobin absorption spectrum in a tissuelike phantom. Appl Opt 34:1143–1155. Pham TH, Coquoz O, Fishkin JB, Anderson E, Tromberg BJ. 2000. Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy. Rev Sci Instrum 71:2500– 2513.

DOI 10.1002/jps

4783

45. Fantini S, Franceschini MA, Fishkin JB, Barbieri B, Gratton E. 1994. Quantitative determination of the absorption spectra of chromophores in strongly scattering media: A light-emittingdiode based technique. Appl Opt 33:5204–5213. 46. Gerken M, Faris GW. 1999. High-precision frequency-domain measurements of the optical properties of turbid media. Opt Express 24:930–932. 47. Dali SS, Rasmussen JC, Huang YQ, Roy R, Sevick-Muraca EM. 2005. Particle sizing in dense suspensions with multiwavelength photon migration measurements. AIChE J 51:1116– 1124. 48. Patterson MS. 1995. Principles and applications of frequencydomain measurements of light propagation. In: Welch AJ, van Gemert MJC, editors. Optical-thermal response of laser-irradiated tissue. New York: Plenum Press. pp. 333–364. 49. Doiron DR, Svaasand LO, Profio AE. 1983. Light dosimetry in tissue: application to photoradiation therapy. In: Kessel D, Dougherty TJ, editors. Porphyrin photosensitization. New York: Plenum Press. pp. 63–76. 50. Kienle A, Lilge L, Patterson MS, Hibst R, Steiner R, Wilson BC. 1996. Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue. Appl Opt 35: 2304–2314. 51. Pham TH, Eker C, Durkin A, Tromberg BJ, Andersson-Engels S. 2001. Quantifying the optical properties and chromophore concentrations of turbid media by chemometric analysis of hyperspectral diffuse reflectance data collected using a Fourier interferometric imaging system. Appl Spectrosc 55:1035– 1045. 52. Pham TH, Bevilacqua F, Spott T, Dam JS, Tromberg BJ, Andersson-Engels S. 2000. Quantifying the absorption and reduced scattering coefficients of tissuelike turbid media over a broad spectral range with noncontact Fourier-transform hyperspectral imaging. Appl Opt 39:6487–6497. 53. Farrell TJ, Patterson MS. 1992. A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo. Med Phys 19:879–888. 54. Farrell TJ, Wilson BC, Patterson MS. 1992. The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements. Phys Med Biol 37: 2281–2286. 55. Swartling J, Dam JS, Andersson-Engels S. 2003. Comparison of spatially and temporally resolved diffuse-reflectance measurement systems for determination of biomedical optical properties. Appl Opt 42:4612–4620. 56. Dam JS, Dalgaard T, Fabricius PE, Andersson-Engels S. 2000. Multiple polynomial regression method for determination of biomedical optical properties from integrating sphere measurements. Appl Opt 39:1202–1209. 57. Sevick EM, Chance B, Leigh J, Nioka S, Maris M. 1991. Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation. Anal Biochem 195:330–351.

JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 12, DECEMBER 2010