## Evaluation of photon migration using a two speed model for characterization of packed powder beds and dense particulate suspensions

Optics Express, Vol. 13, Issue 10, pp. 3600-3618 (2005)

http://dx.doi.org/10.1364/OPEX.13.003600

Acrobat PDF (574 KB)

### Abstract

A two-speed photon diffusion equation is developed for light propagation in a powder bed of high volume fraction or dense particulate suspension, whereby the light speed is impacted by the refractive index difference between particles and the suspending medium. The equation is validated using Monte Carlo simulation of light propagation coupled with dynamic simulation of particle sedimentation for the non-uniform arrangement of powder particles. Frequency domain experiments at 650 nm for a 77-µm-diameter resin-powder and 50-µm-diameter lactose-powder beds as well as resin-water and lactose-ethanol suspensions confirm the scattering and absorption coefficients derived from the two-speed diffusion equation.

© 2005 Optical Society of America

## 1. Introduction

*in situ*. Recently, our group has focused upon employing frequency domain photon migration (FDPM) measurements and the optical diffusion equation for monitoring the blend homogeneity of active pharmaceutical ingredient (API) within pharmaceutical blending operations by relating the measured absorption coefficient of the powder bed to the API concentration [1

1. R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,” J. Pharm. Sci. **88**, 959–966 (1999). [CrossRef] [PubMed]

5. S. E. Torrance, Z. Sun, and E. M. Sevick-Muraca, “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 (2004). [CrossRef] [PubMed]

*C/n̄*

_{med}, where

*c*is the light speed in vacuum and

*n̄*

_{med}is the average refractive index of scattering medium. For dilute systems,

*n̄*

_{med}

*≈n*

_{med}, where

*n*

_{med}is the refractive index of the suspending medium. In complex media whereby the difference of the refractive index between the scattering particles and suspending fluid can be significant, (such as the case of metal oxides, lactose, or other powders), the relative refractive index and the structure resulting from the packing of scatters may dramatically impact time-dependent measurements of multiply scattered light. As illustrated in Fig. 1, propagating photons in dense powder beds have two velocities. Outside particles, photons travel at a velocity of

*c/n*

_{med}, whereas they travel within particles at velocity,

*c/n*

_{p}, where

*n*

_{p}is the refractive index within the particle of the powder or particulate suspension. While a one-speed diffusion equation may describe photon migration in tissue media, [6

6. V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing medium and small source-detector separations,” Phys. Rev. E **58**, 2395–2407 (1998). [CrossRef]

## 2. Methods and approach

### 2.1 Two-speed photon migration diffusion equation

*ϕ*

_{p}, propagating at speed

*c/n*

_{p}within the particle phase with particle absorption,

*µ*

_{a,p}, can be written as

*ϕ*

_{med}, propagating at speed

*c/n*

_{med}within the medium of absorption cross section, on,

*µ*

_{a,med}, can be expressed from the solution of

*r*

_{loss,p}

*·ϕ*

_{p}represents the loss of photons from the particle phase into the suspending medium phase; and

*r*

_{loss,med}

*·ϕ*

_{med}represents the loss of photons from the suspending medium phase into the particle phase. We assume that the photon loss comes only from the absorption processes and that photons travel between the particle and surrounding medium phases in a purely elastic scattering process. The terms

_{med}and

_{p}are current flux of photons within the suspending medium and particle phase, respectively.

*D*is a diffusion coefficient), there is a local equilibrium between photon densities within each phase such that:

*K*

_{e}is the equilibrium constant, which depends upon the available volume fraction for photon transit inside particles and their surrounding media.

*ϕ*

_{p}and

*ϕ*

_{med}and given

*Fick*’s law for multi-component diffusion,[8]

*D*

_{ij}

*(i, j=med, p*) is an element of the multi-component diffusion coefficient tensor, Equations (1)–(5) can be combined to obtain

### 2.2 Dynamic simulation of sedimentation process

*et al*. [9

9. R. Y. Yang, R. P. Zou, and A. B. Yu, “Effect of material properties on the packing of fine particles,” J. Appl. Phys. **94**, 3025–3034 (2003). [CrossRef]

*t*are:

_{i}is the velocity of mass center of particle

*i*and

_{i}is its rotational velocity. The total force on particle

*i,*F →

_{i}, arises from (i) gravity,

*m*

_{i}

_{i,vdW};(iii) the normal and frictional forces due to particle contacting,

_{i,normal}and

_{i,friction}; and (iv) drag forces,

_{i,drag}.

_{i}is the total torque, which arises from

_{ij}, friction and

_{ij,rolling}

*; I*

_{i}is the inertial moment; and

*N*

_{i,contact}is the number of neighboring particles that contact particle

*i*. Each force in Eqs. (9) and (10) and associated parameter values used in the simulation are described in Appendix I.

*h*, of 2×10

^{-8}seconds and computes particle position and translational and rotational velocities at each time step:[10]

*x*and

*y*directions are used to approximate semi-infinite settling of a bed of particles. The computation required 5×10

^{6}seconds for the simulation of 50,000 particles on a Pentium 4, 2.8 GHz machine with a Linux operating system. The calculation flowchart is given in Appendix II.

*D*

_{bed}and

*µ*

_{a,bed}, the magnitude of forces acting on the particles can be adjusted in order to generate various particle ensembles. However, it is difficult to experimentally manipulate properties which govern these forces. Consequently, in order to investigate the influence of volume fraction, we simulated fluid particulate suspension by uniformly expanding the final simulated resin and lactose powder beds at

*f*

_{v}=0.64 and conducting Monte Carlo simulation of photon trajectories (as described below) on particulate arrangements at lower volume fractions. This enables comparison of Monte Carlo simulation results (described below) to experimental measurements in dense resin and lactose suspensions.

### 2.3 Monte Carlo simulation of photon migration in particulate suspensions

*d*

_{p}, was simulated based upon the generated powder structure within a 10

*d*

_{p}×10

*d*

_{p}×32

*d*

_{p}cubic with periodic boundary conditions along

*x, y*, and

*z*directions. Photon trajectories were computed using Monte Carlo technique.

*Mie*scattering for characterizing dense colloidal suspensions, where

*d*

_{p}

*≤λ*(wavelength), [11

11. Z. Sun and E. M. Sevick-Muraca, “Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: bidisperse systems,” Langmuir , **18**, 1091–1097 (2002). [CrossRef]

12. Y. Huang, Z. Sun, and E. M. Sevick-Muraca, “Assessment of electrostatic interaction in dense colloidal suspensions with multiply scattered light,” Langmuir **18**, 2048–2053 (2002). [CrossRef]

*Fresnel*’s formula [14

14. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. **47**, 131–146 (1995). [CrossRef] [PubMed]

*w*, after each

*i*

^{th}scattering step is attenuated by

*j*denotes the medium in which the step has occurred (i.e., either within the suspending,

*med*, or particle,

*p*, phases) and

*t*

_{i}, represents the “time-of-flight” associated with the photon scattering step length.

*ϕ*

_{med}, if it is within the medium, and contributes to the photon density within particles,

*ϕ*

_{p}, if it is located within the particle.

^{-8}seconds. We simulate 10

^{6}photon trajectories with a computational time of 4×10

^{5}seconds on a Pentium 4, 2.8 GHz machine with a Linux operating system. The details of the Monte Carlo simulation of geometric optics are provided in Appendix III.

### 2.4 Determination of scattering and absorption coefficients from Monte Carlo simulation

*t*, <

*r*(

*t*)>, derived from Monte Carlo statistics, the effective diffusion coefficient,

*D*

_{bed}, in powder bed can be determined [15

15. S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Review of Modern Physics **15**, 1–89 (1943). [CrossRef]

*t*

_{total,p}> and <

*t*

_{total,med}> can be computed from the mean scattering length within particle phase, scat, <

*l*

_{scat,p}>, and the mean scattering length within the suspension medium, <

*l*

_{scat,med}>,

*N*

_{scat, j}is the number of total scattering events within suspending medium or particle phase.

*Fresnel*’s formula, [14

14. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. **47**, 131–146 (1995). [CrossRef] [PubMed]

*N*

_{scat,med}

*=N*

_{scat,p}(see Appendix IV for proof). Therefore, the photon equilibrium constant and associated absorption coefficient can be expressed as:

*l*

_{scat,med}>, which is a function of volume fraction,

*f*

_{v}, and particles diameter,

*d*

_{p}. In this contribution, we us Monte Carlo simulation to investigate the dependence of <

*l*

_{scat,med}> upon

*f*

_{v}, which varies from 0.05 to 0.64. For a system of monodisperse particles, we provide a mathematical expression for <

*l*

_{scat,med}> as a function of

*f*

_{v}and

*d*

_{p}.

### 2.5 Quantitative experimental time-dependent measurements

#### 2.5.1 Samples

*f*

_{v}, of lactose-ethanol suspensions was varied from 0.17 to 0.48, and

*f*

_{v}of resin-water suspension was varied from 0.20 to 0.64 by adding desired amount of resin and lactose particles into water and ethanol.

^{™}Lac 455; Leprino Food, Denver, CO), shown in Fig. 2(b), have irregular shapes and a broad particle size distribution with a mean value of around 50 µm and a standard deviation of 30 µm.

16. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,” Phys. Med. Biol. **48**, 4165–4172 (2003). [CrossRef]

^{P}-4; Fisher Scientific, Fair Lawn, NJ) are both 1.33 at 650 nm [18

18. H.-J. Moon, K. An, and J.-H. Lee, “Single spatial mode selection in a layered square microcavity laser,” Appl. Phys. Lett. **82**, 2963–2965 (2003). [CrossRef]

*f*

_{v}, of the powder bed was determined by the ratio of the mass density of powder bed,

*ρ*

_{bed}, to the particle mass density,

*ρ*

_{p}. In a densely packed state,

*ρ*

_{bed}of resin bed was determined to be 0.65 gram/cm

^{3}by measuring the weight of powder bed and the associated bed volume, while

*ρ*

_{p}of resin particle was determined to be 1.02 gram/cm

^{3}by measuring the weight of particle bed and the associated net volume occupied by particles, which was measured by settling the resin particles in water and reading the volume increment. We found

*f*

_{v}for the monodisperse resin powder bed in air to be 0.64. For polydisperse lactose powders, the same measurements gave

*ρ*

_{p}=1.46gram/cm

^{3}, 0.95

*ρ*

_{bed}=gram/cm

^{3}, and

*f*

_{v}=0.65.

#### 2.5.2 Frequency domain photon migration measurements

1. R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,” J. Pharm. Sci. **88**, 959–966 (1999). [CrossRef] [PubMed]

*PS*), amplitude (

*AC*), and mean intensity (

*DC*) of the signal from the sample PMT relative to the reference PMT were recovered for data analysis.

#### 2.5.3 Analysis of phase shift, amplitude, and mean-intensity of photon density wave

*PS*), amplitude (

*AC*), and mean intensity (

*DC*) for (i) the resin and lactose powder beds with a volume of 150 cm

^{3}and (ii) the resin-water and lactose-ethanol suspensions with volumes of more than 400 ml were evaluated from the solution of the standard diffusion equation to provide the absorption coefficient,

*µ*

_{a,bed}and scattering coefficient,

*µ′*

_{s,bed}[19

19. J. B. Fishkin, P. T. C. So, A. E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, “Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. **34**, 1143–1155 (1995). [CrossRef] [PubMed]

*µ*

_{a,bed}and

*µ′*

_{s,bed}.

*PS, AC*, and

*DC*, we compute a parameter,

*Y*:

*ω*is the modulation frequency,

*r*is the source-detector separation, and the subscript, 0, denotes the values at the smallest source-detector separation of (i) 5 mm in resin powder bed and (ii) 10 mm in lactose powder bed. Equation (21) was employed to extract the FDPM measured scattering coefficients,

*µ′*

_{s,bed}, from the measurements of

*Y*(

*PS,AC,DC,r*) when

*ω*was varied (i) from 65 to 125 MHz in the resin powder bed and (ii) from 40 to 120 MHz in lactose powder bed.

## 3. Results and discussion

### 3.1 Powder structure from dynamic simulation of sedimentation processes

*p*

_{d}=50

*µm*in a (5×10

^{-4}

*m*).(5×10

^{-4}

*m*).(2×10

^{-3}

*m*) cube with periodic boundary conditions along horizontal

*x*and

*y*directions, is shown in Fig. 3.

*f*

_{v}, are 0.21, 0.28, 0.32, 0.53, and 0.59, respectively. It is assumed that a mechanically balanced state is achieved when the volume fraction no longer changes with sedimentation time.

*f*

_{v}, grew to 0.61 and 0.64, respectively, due to the increase of vertical pressure caused by bed weight.

### 3.2 Photon Equilibrium

*K*

_{e}at each volume fraction can be computed by Eq. (18) based on the statistical quantities of <

*l*

_{scat,med}> and <

*l*

_{scat, p}>. Figure 5(a) illustrates the values of <

*l*

_{scat,med}>/

*d*

_{p}computed from Monte Carlo as a function of

*f*

_{v}, which varies from 0.05 to 0.64, the relative refractive index,

*n*

_{p}

*/n*

_{med}, which varies from 1.15 to 1.8, and particle diameter,

*d*

_{p}, which varies from 20 to 50

*µm*. As expected, the relative refractive index of the particle has little impact on <

*l*

_{scat,med}>. The reduction of <

*l*

_{scat,med}>/

*d*

_{p}is consistent with the decrease of the void space between particles when the volume fraction is increased. Quantitatively, in the ranges of

*f*

_{v}measured for the resin-water and lactose-ethanol suspensions, the dependence of <

*l*

_{scat,med}>upon fv in Fig. 5(a) can be described empirically by a power form fit:

*l*

_{scat,p}>/

*d*

_{p}increases with the increase of

*n*

_{p}

*/n*

_{med}, which varies from 1.1 to 1.8 and that the volume fraction, which varies from 0.1 to 0.64, has little impact upon <l

_{scat,p}>

*d*

_{p}. Therefore, <

*l*

_{scat,p}> becomes a constant when both

*d*

_{p}and

*n*

_{p}

*/n*

_{med}are given. From Fig. 5, the mean scattering lengths were determined to be, <

*l*

_{scat,p}>=0.895

*d*

_{p}and <

*l*

_{scat,med}>=0.575

*d*

_{p}for the powder bed of

*f*

_{v}=0.61 and

*n*

_{p}

*/n*

_{med}=1.6. For this powder bed, Eq. (18) predicts 2.49 Ke=2.49, which agrees well with

*K*

_{e}=2.44 obtained from the ratio of

*ϕ*

_{p}to

*ϕ*

_{med}computed from Monte Carlo.

### 3.3 Measurements in powder beds for verifications of two-speed diffusion equation and simulated scattering coefficient

*PS*), amplitude (

*AC*), and mean intensity (

*DC*) were made in the resin and lactose powder beds. Figure 6(a) shows the FDPM measurements for

*Y*(

*PS,AC,DC,r*) (i.e., Eq. (20)) versus the source-detector separation,

*r*, at 65 MHz (diamonds), 95 MHz (triangles), and 125 MHz (squares) for the monodisperse resin powder bed with

*d*

_{p}=77

*µm*and 0.64

*f*

_{v}=0.64. Fig. 6(b) shows

*Y*(

*PS,AC,DC,r*) versus

*r*at 40 MHz (diamonds), 80 MHz (triangles), and 120 MHz (squares) for the polydisperse lactose powder bed with a mean diameter of 50 µm and

*f*

_{v}=0.65. In both cases,

*Y*(

*PS,AC,DC,r*) changes linearly with

*r*at each modulation frequency, as predicted by Eq. (21) derived from the diffusion equation. From Eq. (21), we found the values of FDPM measured scattering coefficients to be

*µ′*

_{s,bed}=47.8 cm

^{-1}and

*µ′*

_{s,bed}=243cm

^{-1}for resin and lactose powder beds, respectively. The Monte Carlo predicted scattering coefficient,

*µ′*

_{s,bed}, for a simulated monodisperse resin powder bed with particle diameter of 77

*µm*, refractive index of 1.6, and volume fraction of 0.64 was found to be

*µ′*

_{s,bed}=48.9 cm

^{-1}. In the monodisperse resin powder bed, the scattering coefficient predicted by Monte Carlo simulation is close to the actual FDPM measured scattering coefficient and correspondingly, the predictions of

*Y*(

*PS,AC,DC,r*) with use of simulated

*µ′*

_{s,bed}in Eq. (21) are also close to the measured values, as shown in Fig. 6(a).

*d*

_{p}=50

*µm*and

*f*

_{v}=0.64, the scattering coefficient predicted by Monte Carlo simulation was found to be

*µ′*

_{s,bed}=75 cm

^{-1}. Contrary to the results for the resin powder bed, the scattering coefficient predicted by Monte Carlo simulation for the monodisperse lactose powder bed of

*d*

_{p}=50

*µm*differs significantly from the actual FDPM measured scattering coefficient for the polydisperse powder bed with a mean diameter of

*d̄*

_{p.poly}=50

*µm*. The prediction of scattering coefficient in the polydisperse powder bed may require more considerations of the particle size distribution and particle shapes.

### 3.4 Particulate suspension experiments for confirmation of the dependence of absorption coefficient upon volume fraction

*µ*

_{a,bed}, versus volume fraction,

*f*

_{v}. In the resin-water particulate suspension,

*µ*

_{a,bed}decreased from 0.0193 cm

^{-1}to 0.0098 cm

^{-1}when

*f*

_{v}was increased from 0.20 to 0.64. In the lactose-ethanol particulate suspension,

*µ*

_{a,bed}decreased from 0.0143 cm

^{-1}to 0.0082 cm

^{-1}when

*f*

_{v}was increased from 0.17 to 0.48.

*µ*

_{a,bed}upon

*f*

_{v}can be evaluated explicitly based on the combination of Eq. (19) and the empirical relationship between <

*l*

_{scat,med}> and

*f*

_{v}(Eq. (22)):

*a*

_{1}and

*a*

_{2}are constants defined as

*µ*

_{a,med},

*a*

_{1}and

*a*

_{2}. The values of

*µ*

_{a,med}at 650 nm were thus found to be 0.028 and 0.022 cm

^{-1}for suspending water and ethanol, respectively, and the fitting curves (solid lines) are also shown in Fig. 7. The determined

*µ*

_{a,med}for suspending water is higher than the corresponding absorption coefficient for pure water reported in literature[20

20. L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. **32**, 3513–3540 (1993). [CrossRef]

*d*

_{p}, has little impact on the absorption coefficient of powder bed,

*µ*

_{a,bed}, since both <

*l*

_{scat,med}> and <

*l*

_{scat, p}> are proportional to

*d*

_{p}(as shown in Fig. 5.), which is then cancelled as a common factor in the denominator and the numerator of Eq. (19). The conclusion of the independence of

*µ*

_{a,bed}upon

*d*

_{p}is consistent with our past experimental results.

^{[44. Z. Sun, S. Torrance, F. K. McNeil-Watson, and E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003). [CrossRef] [PubMed] ]}

## 4. Conclusions

## 5. Appendix

## 5.1 Calculation of forces on particles

_{i,vdW}[21

21. R. Y. Yang, R. P. Zou, and A. B. Yu, “Computer simulation of the packing of fine particles,” Phys. Rev. E **62**, 3900–3908 (2000). [CrossRef]

## 5.2 Calculation flow-chart for dynamic simulation

*See reference [29]*

29. L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometrics,” J. Opt. Soc. Am A **12**, 1772–1781 (1995). [CrossRef]

## 5.3 Algorithm for Monte Carlo of geometric optics

^{6}and the migration time,

*t*

_{end}, for a photon in Monte Carlo is 2.3×10

^{-8}second. The photon trajectories are simulated one by one. A photon package includes time,

*t*, propagating direction,

*w*, and refractive index for the surrounding phase,

*n*

_{med}or

*n*

_{p}.

*t*=0. The initial propagating direction,

_{0}, at the origin,

_{0}, is randomly chosen from the isotropic distribution and the photon weight,

*w*, is set to be 1. The photon package at the origin is [

*t,*v →
,r →
,w,n]⃖[

*0,*v →
0,r →
0,1,n

_{med}].

_{0}, and the particle positions, which are obtained from the dynamic simulation of sedimentation process for powder bed. The time for a photon propagating from the origin,

_{0}, to the encountering point,

_{ext}, on the external surface of the interacting particle with a speed,

*c/n*

_{med}, is Δ

*t*. The attenuation of photon weight,

*w*, is exp(-

*µ*

_{a,med}·|

_{ext}-

_{0}|). Correspondingly, the photon package at

_{ext}, is updated [

*t,*v →
,r →
,w,n]⃖[(

*t+Δt*),

_{ext},(

*w*.exp(-

*µ*

_{a,med}·|

_{ext}-

_{0}|)),

*n*].

_{ext}, the incident angle,

*α*

_{i}, between the photon propagating direction,

_{0}, and the normal,

*Fresnel*’s formulas[11

11. Z. Sun and E. M. Sevick-Muraca, “Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: bidisperse systems,” Langmuir , **18**, 1091–1097 (2002). [CrossRef]

*α*

_{t}is the transmission angle determined by the Snel’s law

^{[1010. J. P. Hansen and I. R. McDonald, “Theory of Simple Liquid,” (Academic Press, 1986), Chapter 3.]}

*ξ*, from the uniform distribution in the range of [0,1

1. R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,” J. Pharm. Sci. **88**, 959–966 (1999). [CrossRef] [PubMed]

*ξ<P*. Otherwise, it undergoes a refraction from outside to inside of the particle.

*Snel*’s law and the photon package becomes [

*t,*v →
,r →
,w,n]⃖[

*t,*

_{1},

*w,n*].

_{1}and the particle positions, the next interacting particle is determined and similar updating for propagating outside particles is conducted.

_{2}in terms of the

*Snel*’s law. The refractive index in the photon package is changed from

*n*

_{med}to

*n*

_{p}since the photon transmits inside the particle. The photon package is updated [

*t*,

*w,n*]⃖[

*t*,

_{2},

*,w,np*].

_{2}, the position of the interacting particle, and the particle radius, the ending point for the photon’s internal travel is determined as

_{int}. The propagating time, Δ

_{t}, is calculated from the distance between

_{ext}and int

_{int}and the particle refractive index,

*n*

_{p}. The attenuation of the photon weight is ext (-

*µ*

_{a.p}·|

_{int}-

_{ext}|). At position,

_{int}, the photon package is updated [

*t*,

*w,n*]⃖[(

*t*+Δ

*t*),

_{int}, (

*w*.exp(-

*µ*

_{a.p}·|

_{int}-

_{ext}|)),

*n*].

_{int}, the incident angle,

*α*

_{i}, between

_{2}and the normal of the particle surface,

*P*, is calculated in terms of

*Fresnel*’s formulas. Then a random number,

*ξ*, is generated from the uniform distribution in the range of [0,1

**88**, 959–966 (1999). [CrossRef] [PubMed]

*ξ*<

*P*. Otherwise, it undergoes a refraction from inside to outside of the particle.

_{3}in terms of

*Snel*’s law and the photon package is updated 3 [

*t*,

*w,n*]⃖[

*t*,

_{3},

*w,n*].

*ξ*≥

*P*, the photon undergoes an refraction from inside to outside of the particle. The propagating direction is changed to

_{4}in terms of the

*Snel*’s law and the associated refractive index is changed from

*n*

_{p}to

*n*

_{med}. Then the photon package is updated [

*t*,

*w,n*]⃖[

*t*,

_{4},

*w,n*

_{p}].

*t*and

*t*

_{end}is compared. When

*t*<

*t*

_{end}, the simulation for the photon continues. Otherwise, the simulation for the photon terminates and another photon is launched into the powder bed. When the total simulated photon trajectories reaches 10

^{6}, the simulation is finished.

_{reflection}is the propagating direction after reflection at an interface; and

_{transmission}is the propagating direction after refraction at an interface.

## 5.4 Monte Carlo simulation of scattering numbers

## References and links

1. | R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, “Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,” J. Pharm. Sci. |

2. | T. Pan and E. M. Sevick-Muraca, “Volume of pharmaceutical powders probed by frequency-domain photon migration measurements of multiply scattered light,” Anal. Chem. |

3. | T. Pan, D. Barber, D. Coffin-Beach, Z. Sun, and E. M. Sevick-Muraca, “Measurement of low dose active pharmaceutical ingredient in a pharmaceutical blend using frequency-domain photon migration,” J. Pharm. Sci. |

4. | Z. Sun, S. Torrance, F. K. McNeil-Watson, and E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. |

5. | S. E. Torrance, Z. Sun, and E. M. Sevick-Muraca, “Impact of excipient particle size on measurement of active pharmaceutical ingredient absorbance in mixtures using frequency domain photon migration,” J. Pharm. Sci. |

6. | V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing medium and small source-detector separations,” Phys. Rev. E |

7. | J. J. Duderstadt and L. J. Hamilton, “ |

8. | A. F. Henry, “ |

9. | R. Y. Yang, R. P. Zou, and A. B. Yu, “Effect of material properties on the packing of fine particles,” J. Appl. Phys. |

10. | J. P. Hansen and I. R. McDonald, “Theory of Simple Liquid,” (Academic Press, 1986), |

11. | Z. Sun and E. M. Sevick-Muraca, “Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: bidisperse systems,” Langmuir , |

12. | Y. Huang, Z. Sun, and E. M. Sevick-Muraca, “Assessment of electrostatic interaction in dense colloidal suspensions with multiply scattered light,” Langmuir |

13. | C. F. Bohren and D. R. Huffman, “ |

14. | L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. |

15. | S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Review of Modern Physics |

16. | X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,” Phys. Med. Biol. |

17. | A. N. Winchell, “ |

18. | H.-J. Moon, K. An, and J.-H. Lee, “Single spatial mode selection in a layered square microcavity laser,” Appl. Phys. Lett. |

19. | J. B. Fishkin, P. T. C. So, A. E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, “Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. |

20. | L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. |

21. | R. Y. Yang, R. P. Zou, and A. B. Yu, “Computer simulation of the packing of fine particles,” Phys. Rev. E |

22. | N. V. Brilliantov, F. Spahn, and J. M. Hertzsch, “Model for collisions in granular gases,” Phys. Rev. E |

23. | P. A. Thompson and G. S. Grest, “Granular flow: friction and the dilatancy transition,” Phys. Rev. Lett. |

24. | L. G. Leal, “ |

25. | F. Podczeck, J. M. Newton, and M. B. James, “The adhesion force of micronized salmeterol xinafoate particles to pharmaceutically relevant surface materials,” J. Phys. D: Appl. Phys. |

26. | R. J. Roberts, R. C. Rowe, and P. York, “The relationship between Young’s modulus of elasticity of organic solids and their molecular structure,” Powder Technol. |

27. | Y. Shimada, Y. Yonezawa, H. Sunada, R. Nonaka, K. Katou, and H. Morishita, “Development of an apparatus for measuring adhesive force between fine particles,” KONA, No 20, 223–230 (2002). |

28. | K. Z. Y. Yen and T. K. Chaki, “A dynamic simulation of particle rearrangement in powder packings with realistic interactions,” J. Appl. Phys. |

29. | L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometrics,” J. Opt. Soc. Am A |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(170.5280) Medical optics and biotechnology : Photon migration

(290.1990) Scattering : Diffusion

(290.4210) Scattering : Multiple scattering

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 14, 2005

Revised Manuscript: April 29, 2005

Published: May 16, 2005

**Citation**

Tianshu Pan, Sarabjyot Dali, and Eva Sevick-Muraca, "Evaluation of photon migration using a two speed model for characterization of packed powder beds and dense particulate suspensions," Opt. Express **13**, 3600-3618 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3600

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### References

- R. R. Shinde, G. V. Balgi, S. L. Nail, and E. M. Sevick, �??Frequency-domain photon migration measurements for quantitative assessment of powder absorbance: a novel sensor of blend homogeneity,�?? J. Pharm. Sci. 88, 959-966 (1999). [CrossRef]
- T. Pan and E. M. Sevick-Muraca, �??Volume of pharmaceutical powders probed by frequency-domain photon migration measurements of multiply scattered light,�?? Anal. Chem. 74, 4228-4234 (2002). [CrossRef]
- T. Pan, D. Barber, D. Coffin-Beach, Z. Sun, and E. M. Sevick-Muraca, �??Measurement of low dose active pharmaceutical ingredient in a pharmaceutical blend using frequency-domain photon migration,�?? J. Pharm. Sci. 93, 635-645 (2004). [CrossRef]
- Z. Sun, S. Torrance, F. K. McNeil-Watson, and E. M. Sevick-Muraca, �??Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,�?? Anal. Chem. 75, 1720-1725 (2003). [CrossRef]
- S. E. Torrance, Z. Sun, and E. M. Sevick-Muraca, �??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 (2004). [CrossRef]
- V. Venugopalan, J. S. You, and B. J. Tromberg, �??Radiative transport in the diffusion approximation: an extension for highly absorbing medium and small source-detector separations,�?? Phys. Rev. E 58, 2395-2407 (1998). [CrossRef]
- J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis, (John Wiley & Sons, 1976 ), Chapter 4.
- A. F. Henry, Nuclear-Reactor Analysis, (The MIT Press, 1975), Chapter 9.
- R. Y. Yang, R. P. Zou, and A. B. Yu, �??Effect of material properties on the packing of fine particles,�?? J. Appl. Phys. 94, 3025-3034 (2003). [CrossRef]
- J. P. Hansen and I. R. McDonald, Theory of Simple Liquid, (Academic Press, 1986), Chapter 3.
- Z. Sun and E. M. Sevick-Muraca, �??Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: bidisperse systems,�?? Langmuir 18, 1091-1097 (2002). [CrossRef]
- Y. Huang, Z. Sun, and E. M. Sevick-Muraca, �??Assessment of electrostatic interaction in dense colloidal suspensions with multiply scattered light,�?? Langmuir 18, 2048-2053 (2002). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, 1983), Chapter 7.
- L. Wang, S. L. Jacques, and L. Zheng, �??MCML �?? Monte Carlo modeling of light transport in multi-layered tissue,�?? Comput. Methods Programs Biomed. 47, 131-146 (1995). [CrossRef]
- S. Chandrasekhar, �??Stochastic problems in physics and astronomy,�?? Review of Modern Physics 15, 1-89 (1943).
- X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, �??Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,�?? Phys. Med. Biol. 48, 4165-4172 (2003). [CrossRef]
- A. N. Winchell, The Optical Properties of Organic Compounds, (Academic Press, 1954), Chapter 4.
- H.-J. Moon, K. An, and J.- H. Lee, �??Single spatial mode selection in a layered square microcavity laser,�?? Appl. Phys. Lett. 82, 2963-2965 (2003). [CrossRef]
- J. B. Fishkin, P. T. C. So, A. E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, �??Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,�?? Appl. Opt. 34, 1143-1155 (1995).
- L. Kou, D. Labrie, and P. Chylek, �??Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,�?? Appl. Opt. 32, 3513-3540 (1993).
- R. Y. Yang, R. P. Zou, and A. B. Yu, �??Computer simulation of the packing of fine particles,�?? Phys. Rev. E 62, 3900-3908 (2000). [CrossRef]
- N. V. Brilliantov, F. Spahn, and J. M. Hertzsch, �??Model for collisions in granular gases,�?? Phys. Rev. E 53, 5382-5392 (1996). [CrossRef]
- P. A. Thompson and G. S. Grest, �??Granular flow: friction and the dilatancy transition,�?? Phys. Rev. Lett. 67, 1751-1754 (1991). [CrossRef]
- L. G. Leal, Laminar Flow and Convective Transport Processes, (Butterworth-Heinemann, 1992), Chapter 4.
- F. Podczeck, J. M. Newton, and M. B. James, �??The adhesion force of micronized salmeterol xinafoate particles to pharmaceutically relevant surface materials,�?? J. Phys. D: Appl. Phys. 29, 1878-1884 (1996). [CrossRef]
- R. J. Roberts, R. C. Rowe, and P. York, �??The relationship between Young�??s modulus of elasticity of organic solids and their molecular structure,�?? Powder Technol. 65, 139-146 (1991).
- Y. Shimada, Y. Yonezawa, H. Sunada, R. Nonaka, K. Katou, and H. Morishita, �??Development of an apparatus for measuring adhesive force between fine particles,�?? KONA No. 20, 223-230 (2002).
- K. Z. Y. Yen and T. K. Chaki, �??A dynamic simulation of particle rearrangement in powder packings with realistic interactions,�?? J. Appl. Phys. 71, 3164-3173 (1992). [CrossRef]
- L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, �??Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometrics,�?? J. Opt. Soc. Am A 12, 1772-1781 (1995).

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