1. Introduction
Real time continuous monitoring of micron and submicron bioparticulates for biomedical and industrial applications requires both suitable instrumentation and sophisticated models that relate the measurements to the desired properties of biological particles. A rapid tool for submicron particle characterization is light spectroscopy [
11. C. E. Alupoaei, J. A. Olivares, and L. H. GarciaRubio, “Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores,” Biosens. Bioelectron. 19, 893–903 (2003) [CrossRef]
,
22. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron. 9, 277–287 (2003). [CrossRef]
], which yields information on their joint property distribution: chemical composition, size, shape, and orientation. These properties are inferred from the spectroscopy measurements through mathematical models derived from the theory of electromagnetic radiation (Maxwell’s equations), in particular from the theory of light scattering. To date rigorous computational techniques have been developed that enable the estimation of properties relevant to biological systems such as particle size and particle shape [
33. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “TMatrix Computations of Light Scattering by Nonspherical Particles: A Review,” J. Quantum Spectrosc. Radiat.55, No. 5 (1996).
]. Unfortunately, these methods (Tmatrix and PurcellPennypacker methods) are computationally intensive and do not yet lend themselves to the evaluation of particle ensembles for realtime continuous monitoring applications. Therefore there is a need for approximations that can provide adequate particle characterization results in relatively short computational times. Mie theory and the RayleighDebyeGans (RDG) approximation are special solutions and limiting cases of the electromagnetic theory that can be effectively used in the context of realtime applications using spectrophotometric measurements. In particular, RDG enables the estimation of the particle shape, an important feature when characterizing biological systems.
RayleighDebyeGans is an approximate theory for particles of any shape and size having a relative refractive index near unity and has found extensive use in direct light scattering applications [
22. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron. 9, 277–287 (2003). [CrossRef]
] and other techniques such as flow cytometry [
44. A. L. Koch, B. R. Robertson, and D. K. Button, “Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry,” J. Microbio. Meth. 2, 40–61 (1996).
]. Mie theory is the exact solution to the boundary value problem for light scattering by a sphere and is used as a reference [
55. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (Wiley Science Paper Series, New York, 1998). [CrossRef]
,
66. M. Kerker, The Scattering of Light and other Electromagnetic Radiation. (Academic Press, New York, 1969).
]. There are inherent limitations to using RayleighDebyeGans theory as a characterization tool. The relative complex refractive index
m must be close to one and the size of the particle must be much smaller than
λ/
m1, where
λ is the wavelength. There exists a trade off for the limits of RDG theory; first, if
m is close to one and no absorption is present then the size of the particle can be the same order of magnitude as the wavelength. Conversely, if absorption is present and
m is greater than one, the particle size must be smaller than the wavelength. It is noteworthy that RDG has been used for the characterization of blood [
77. M. Hammer, D. Schweitzerk, B. Michel, E. Thamm, and A. Kolb, “Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418 (1998). [CrossRef]
] and bacterialike particles [
22. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron. 9, 277–287 (2003). [CrossRef]
,
88. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, “In situ identification of bacteria size by light scattering,” Proc. SPIE 4965, 7376 (2003)
] which clearly have sizes and optical properties beyond the expected range of application of the theory.
The importance of the RDG approximation for biological particles lies on the flexibility of this approximation for generating the scattering structures and shapes of complex particles through the formulation of the form factors [
44. A. L. Koch, B. R. Robertson, and D. K. Button, “Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry,” J. Microbio. Meth. 2, 40–61 (1996).
,
99. B. P. Latimer, “Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods,” J. Colloid Interface Science 63, 310–316 (1977). [CrossRef]
,
1313. P. J. Wyatt and D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493–501 (1972). [CrossRef]
]. An important motivation for the study of RDG in the spectral window of 200–1100 nm has to do with the fact that most of the key components of biological particles (proteins, DNA, Chlorophyll, etc.) absorb and scatter light in UV portion of the spectrum. Therefore, if biological particles are to be characterized in terms of their absorption and scattering properties both, the spectral regions where absorption and scattering are significant have to be considered. A complication brought about by the use of a broad wavelength spectral region (200–1100 nm) is that particles may not adhere to the constraints required by RDG across the complete spectral range and it is therefore desirable, if not completely necessary, to increase the range of application of the RDG approximations.
In this paper, RDG and Mie theory are explored in the context of multiwavelength spectroscopy over a wavelength range where strict adherence to the conditions implicit in the approximations could not be met, but where important compositional information can be acquired [
11. C. E. Alupoaei, J. A. Olivares, and L. H. GarciaRubio, “Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores,” Biosens. Bioelectron. 19, 893–903 (2003) [CrossRef]
]. As a result from this study, the differences between RDG and Mie theory are generalized and their range of application more accurately defined as a function of both wavelength and magnitude of the optical properties. Reported corrections to RDG proposed to obtain a better agreement with Mie theory have also been evaluated. These corrections include the use of effective optical properties [
99. B. P. Latimer, “Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods,” J. Colloid Interface Science 63, 310–316 (1977). [CrossRef]
] and the inclusion of hypochromic effects [
1010. N. L. Veshkin, “Screening Hypochromism of Biological Macromolecules and Suspensions,” J. Photochem. Photobiol. B: Biology , 3, 625–630 (1989). [CrossRef]
–
1212. N. L. Veshkin, “Screening Hypochromism of Chromophores in Macromolecular Biostructures,” Biophys. 44, 41–51 (1999)
]. The results from this study indicate that the range of application of RDG is considerably more limited than suggested by the single wavelength evaluations reported in the literature [
22. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, “Bacteria size determination by elastic light scattering,” IEEE J. Sel. Top. Quantum Electron. 9, 277–287 (2003). [CrossRef]
,
66. M. Kerker, The Scattering of Light and other Electromagnetic Radiation. (Academic Press, New York, 1969).
,
1313. P. J. Wyatt and D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493–501 (1972). [CrossRef]
], and that neither the correction of the optical properties for hypochromicity, nor the use of effective optical properties result in better agreement between RDG and Mie theories. It is evident that to have a theoretically based spectroscopy interpretation model suitable for realtime characterization of complex bioparticles the formulation of RDG must be revisited.
2. Methods
2.1 Simulations
The programs for Mie theory, RayleighDebyeGans theory, and hypochromicity were developed in Matlab v6.5.1. Computations for these programs were conducted using a Dell Inspiron 4100 with 1GHz Pentium III processor and 512 MB RAM. The experimentally determined optical properties (refractive indices) utilized were provided by Dr. GarciaRubio and the SAPD laboratory through the College of Marine Science at the University of South Florida^{14}.
The ranges of particle volumes were chosen between 12700 nm
^{3} and 87000e6 nm
^{3}. The spherical diameter equivalents to the volume range are between 25 nm–5500 nm.
Table 1 gives the wavelength, concentration and density values used to define the suspensions for the analyses conducted in this manuscript. The simulation spectra are graphed by optical density (a standard unit for intensity ratio per unit pathlength * concentration) and wavelength.
Table 1. Simulation Parameters for Turbidity using Mie and RayleighDebyeGans Theory. 

2.2 Materials
The computer codes developed for the analysis of RayleighDebyeGans and Mie particles were tested against published values of the scattering functions [
55. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (Wiley Science Paper Series, New York, 1998). [CrossRef]
,
1515. W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, VectorSpeed Computer Codes,”NCAR/TN140+STR. National Center for Atmospheric Research, Boulder Colorado (1979).
]. In testing and exploring the algorithms for RayleighDebyeGans the refractive indices selected were those of soft bodies and hemoglobin, where soft bodies are defined here as particles whose relative refractive index is close to one with no absorption component. The values of the index of refraction
n+iκ for biological particles commonly used are soft bodies (1.04≤
n≤1.45) and hemoglobin (1.48≤
n≤1.6, 0.01≤
κ≤0.15) [
77. M. Hammer, D. Schweitzerk, B. Michel, E. Thamm, and A. Kolb, “Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418 (1998). [CrossRef]
]. Polystyrene (1.5≤
n≤2.2, 0.01≤
κ≤0.82), silver bromide (2.6≤
n≤3.5, 0.001≤
κ≤1.6) and silver chloride (2≤
n≤2.7, 0.001<
κ≤0.85) are materials found in industrial applications whose properties are used as standards for the calibration and evaluation of optical instruments [
77. M. Hammer, D. Schweitzerk, B. Michel, E. Thamm, and A. Kolb, “Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418 (1998). [CrossRef]
]. Water (1.3≤
n≤1.4) was used as the suspending medium.
The optical properties are graphed as function of wavelength in the Appendix, numerical values are available upon request from Prof. GarciaRubio.
3. Theory
3.1 Turbidity
The turbidity equation is an energy balance equation applicable to small acceptance angle transmission measurements. Turbidity has been traditionally defined as an attenuation coefficient due to scattering (only) for the transmission of the incident beam. Herein, turbidity is described as the total attenuation observed due to scattering and absorption. The expression for the turbidity of a monodisperse system is
where
N
_{p} is the number of particles,
l is the pathlength of the sample,
Q
_{ext} is the extinction efficiency,
Q
_{sca} is the scattering efficiency, and
Q
_{abs} is the absorption efficiency. The efficiencies are determined using RayleighDebyeGans theory, from which the turbidity equation can be expressed explicitly by the following equation [
1616. A. C. GarciaLopezHybrid Model for Characterization of Submicron Particles using Mulitwavelength Spectroscopy, (University of South Florida, 2005).
].
where P(θ) is the form factor (a function of wavelength, λ) for spheres as defined by Kerker6
Note that the turbidity is a function of m, the relative complex refractive index (a function of wavelength, λ) and θ the angle of observation.
3.2 Complex Refractive Index.
The connection between light scattering and absorption phenomena and particle’s joint property distribution (size, shape, orientation, chemical composition, and internal structure) is made through the optical properties that are characteristic of the materials contained in the particle. The complex refractive index is given by
where n and κ are non negative values, n is the refractive index (real), and κ is the absorption coefficient (imaginary). The scattering of light is due to differences in refractive indices between the medium and the particle. The refractive index of the particle (N
_{1}) relative to the suspending medium (N
_{0}) is
3.3 Optical Property Corrections
Differences in the scattering behavior predicted by Mie and by RDG have been previously investigated within the context of the effect of shape and monochromatic angular scattering measurements [
66. M. Kerker, The Scattering of Light and other Electromagnetic Radiation. (Academic Press, New York, 1969).
,
99. B. P. Latimer, “Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods,” J. Colloid Interface Science 63, 310–316 (1977). [CrossRef]
,
1717. P. Latimer, A. Brunsting, B. E. Pyle, and C. Moor, “Effects of asphericity on single particle scattering,” Appl. Opt. 17, 3152–3158 (1978). [CrossRef] [PubMed]
], and it has been suggested that it is possible to compensate RDG through the use of
effective refractive indices; in other words, the refractive indices become adjustable parameters. Latimer et al [
99. B. P. Latimer, “Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods,” J. Colloid Interface Science 63, 310–316 (1977). [CrossRef]
] have reported formulas to calculate the
effective refractive indices for a variety of shapes. The differences in behavior observed between the spectra calculated with Mie and with RDG theory suggests that a similar approach may be used to correct RDG to obtain closer approximations to Mie theory. The corrections made to the optical properties can be approached empirically and/or theoretically. The theoretical models developed by Veshkin [
1010. N. L. Veshkin, “Screening Hypochromism of Biological Macromolecules and Suspensions,” J. Photochem. Photobiol. B: Biology , 3, 625–630 (1989). [CrossRef]
–
1212. N. L. Veshkin, “Screening Hypochromism of Chromophores in Macromolecular Biostructures,” Biophys. 44, 41–51 (1999)
] to account for hypochromic effects offer a distinct possibility since these models compensate the absorption coefficient in the direction of decreasing intensity, as it would be required by RDG to approximate Mie. Both the empirical and the theoretical approaches are explored in this paper.
3.4 Hypochromicity as a Correction for RDG
The observable light scattering phenomena depends on the configuration of the instrumentation and on the optical properties of the material investigated. The optical properties (real and imaginary parts of the refractive index) are wavelengthdependent and intrinsic properties of matter. It is known that the optical properties depend on the state of aggregation [
55. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (Wiley Science Paper Series, New York, 1998). [CrossRef]
]. However, under certain conditions (i.e. infinite dilution) the optical properties are additive and independent of concentration. The presence of absorbing groups (chromophores) in high concentration within particles gives rise to a concentration dependence of the observed optical phenomena. This phenomenon generally results in a decrease of the observed imaginary component of the refractive index–relative to its value in solution (hypochromicity) [
1818. A. Nonoyama, Using multiwavelength UVVisible spectroscopy for the characterization of red blood cells: an investigation of hypochromism. (University of South Florida, 2004).
]. Note that any change in value of the imaginary component of the refractive index (
–) will be reflected in the absorption component of the turbidity spectra.
A consequence of Mie theory is that the absorption component of the turbidity spectrum decreases relative to the scattering efficiency as the particle size increases [
55. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (Wiley Science Paper Series, New York, 1998). [CrossRef]
]. Similarly, in the context of hypochromicity, as the concentration of chromophores increases, there is a decrease in the observed imaginary component (absorption) of the complex refractive index. Therefore, it seems plausible that the mathematical structure of successful models developed to account for hypochromic effects may be able to account for the decrease in the absorption component present in Mie theory, but absent in the RDG approximation. The attenuation of light due to absorption through a particle is shown schematically in
Fig. 1 for Mie theory and RDG approximation.
The most recent models for hypochromicity are those developed by Veshkin [
1010. N. L. Veshkin, “Screening Hypochromism of Biological Macromolecules and Suspensions,” J. Photochem. Photobiol. B: Biology , 3, 625–630 (1989). [CrossRef]
–
1212. N. L. Veshkin, “Screening Hypochromism of Chromophores in Macromolecular Biostructures,” Biophys. 44, 41–51 (1999)
] and take into consideration the molecular structure, the molecular orientation, and the number of chromophoric groups per unit volume of particle. Vekshin’s model describes screening of chromophores when stacked along the molecular chain axis. In this paper the chromophores were considered to be stacked within the particle, by doing so the model developed by Veshkin has been extended to include the effect of the wavelength and implemented for multiwavelength spectroscopy. The concentration of chromophoric groups per particle size per wavelength can be related as depicted in
Fig. 2 [see
Eq. (12)].
Fig. 1. The attenuation of light due to absorption through a particle shown schematically for Mie theory and RDG approximation.
Fig. 2. Schematic of chromophoric groups per particle size per wavelength within a particle.
3.5 Wavelength Dependent Hypochromicity
Experimentally the hypochromism value h at a given wavelength is defined by:
where
ε is the extinction coefficient for the situation of single chromophore absorption in units of 1/
M
cm and
ε̃ is the average extinction coefficient per chromophore. From the screening model [
1010. N. L. Veshkin, “Screening Hypochromism of Biological Macromolecules and Suspensions,” J. Photochem. Photobiol. B: Biology , 3, 625–630 (1989). [CrossRef]
,
1111. N. L. Veshkin, “Screening Hypochromism of Molecular Aggregates and Biopolymers,” J. Biol. Phys. 25, 339–354 (1999). [CrossRef]
]
This equation predicts the hypochromic extinction coefficient in a solution of stack chromophores (cluster) if the values E, s, q, and k are known. E is in units of molecular extinction coefficients (Å/molecule) and is a function of wavelength, Ẽ is the average extinction coefficient, s is the effective geometric area of a chromophore (Å
^{2}), q is the orientation factor, and k (not to be confused with the wave number) is the quantity of chromophores.
Transforming Vekshin’s model from units 1/M
cm to Å results in rewriting the equation above as
where M
_{w} is the molecular weight of the chromophore and V is the unit volume transformation of 1000 cm^{3}/L. The number of chromophores k calculated from the volume fraction or concentration of the sample
where v
_{f} is the volume fraction, λ is the wavelength, d is the diameter of the sample. The probability of absorption of a photon by a molecule P can be presented as
where the extinction coefficient for one chromophore is calculated by
From this, the hypochromicity can be calculated using
Eq. (8). The corrected imaginary part of the refractive index
κ
_{c} can be computed using the following equation
Notice that the hypochromicity models given by
Eqs. (15)–
(17) can be used directly within their theoretical context, or empirically as a calibration model to correct the absorption component.
4. Results
4.1 Exploration of theoretical limits.
Three approaches were taken to explore, through simulation, the constraints of RayleighDebyeGans theory for spheres. First, the sizes of the spherical particles were kept small compared to the wavelengths, but the wavelengthdependent relative refractive index was allowed to significantly exceed the value of one. Refractive index ratios greater than one are typical of actual materials. Second, the relative refractive index was kept close to one while the absorption was held at zero, and particle sizes comparable to the wavelengths were considered. Third, the contribution of absorption in the relative refractive index, kept close to 1, was investigated for particle sizes comparable to the wavelengths. The following subsections describe in more detail the parameters used and the conclusions and observations drawn.
4.2 Particle diameter ≪ wavelength.
The first of the sensitivity studies conducted tested the limits of RayleighDebyeGans for relative refractive indices greater than 1 and the absorption greater than zero, while keeping small sized spherical particles, compared to the wavelengths (200 nm–900 nm). The multiwavelength transmission spectra were calculated for Mie and RayleighDebyeGans using spheres of silver bromide (1.1≤n/n
_{o}≤2.4, 0.0001≤κ≤0.85) and spheres of silver chloride (1.1≤n/n
_{o}≤2.4, 0.0001≤κ≤ 0.6). The spherical diameter sizes chosen were 25 nm and 50 nm. Particle concentration, particle density, and wavelength range were kept constant
Figures 3–
4 show that RayleighDebyeGans gives an adequate approximation to Mie for particle sizes much smaller than the wavelength; notice that across the UVVis spectral region and shape and the amplitude of the spectral features are quite similar.
Figures 5–
6 reveal that for slightly larger particles RayleighDebyeGans no longer closely follows Mie Theory. Notice that in the spectral region where absorption is small (300 nm–900 nm) both theories coincide even though
n/n
_{o}>1. However, where strong absorption is present, the theories rapidly diverge, clearly suggesting that absorption plays an important role in the disparity between the theories, and that RDG is not a good approximation whenever absorption is present.
Fig. 3. Comparison of the calculated transmission of Mie and RayleighDebyeGans for 25 nm AgBr spheres suspended in water.
Fig. 4. Comparison of the calculated transmission of Mie and RayleighDebyeGans for 25 nm AgCl spheres suspended in water.
Fig. 5. Comparison of the calculated transmission of Mie and RayleighDebyeGans for 50 nm AgBr spheres suspended in water. Spectral differences in RDG are visible below 400 nm wavelength compared to those is
Fig. 1.
Fig. 6. Comparison of the calculated transmission of Mie and RayleighDebyeGans for a 50 nm AgCl spheres suspended in water. The spectral differences in RDG are visible below 400 nm wavelength compared to those in
Fig. 4.
4.3 Particle diameter≈wavelength, no absorption.
The restriction of RayleighDebyeGans theory with respect to size was tested through the calculation of transmission spectra at wavelength range between 200 nm and 900 nm for nonabsorbing spherical particles with relative refractive index close to one. The refractive indices chosen were soft bodies (n=1.04) and hemoglobin (1.01≤n/n
_{o}≤1.2); only the real part of the refractive index was used for hemoglobin. Particle diameters used were 500 nm, 1 µm, and 5.5 µm. Note that the refractive index spectrum of hemoglobin changes as function of wavelength and that it is not possible to meet the conditions for the application of RDG across the spectrum.
Figures 7 and
8 show that RayleighDebyeGans theory produces similar spectral patterns as Mie theory for relatively small particle sizes. However, RDG begins to diverge at the lower wavelengths (200–300 nm) for the particle size of 1 µm.
Figure 9 shows that for a particle size of 5.5 µm RayleighDebyeGans theory simply does not approximate Mie theory. The combination of zero absorption and relative refractive index ratio close to 1 does not increase the particle size ranges for which RDG is said to be applicable; this is in disagreement with the results reported in Kerker [
66. M. Kerker, The Scattering of Light and other Electromagnetic Radiation. (Academic Press, New York, 1969).
].
Fig. 7. Comparison of Mie and RayleighDebyeGans calculated spectra of 500 nm Soft Body spheres suspended in water. Notice that the spectrumcalculated with RayleighDebyeGans theory does not approximate Mie theory.
Fig. 8. Calculated transmission of Mie and RayleighDebyeGans for 1 µm Soft Body spheres suspended in water. Shape of the spectra by both theories remains similar.
Fig. 9. Transmission spectra calculated using Mie and RayleighDebyeGans for 5.5 µm Soft Body spheres suspended in water. RayleighDebyeGans spectra does not coincide with Mie theory at such a large particle size.
The multiwavelength transmission calculations conducted with only the real part of the refractive index of hemoglobin show that for 500 nm diameter particles (
Fig. 10), the theories follow one another closely in spectral shape but there are quantifiable differences in amplitude. If turbidity is used for analysis, the spectral differences between the two theories would result in considerable variation in the estimate of particle size and concentration. With increasing of the particle diameter to 1 µm (
Fig. 11), the spectral shape for Mie theory relative to RayleighDebyeGans flattens considerably at the shorter wavelengths.
Figure 12 shows a semilogarithmic turbidity plot of 5.5 µm particles to show the differences in shape and amplitude for the two theories. The effect of a relative refractive index greater than one with no absorption results in a limited particle size range for the application of RDG theory, in contrast to particles with a refractive index close to one with no absorption. It is noteworthy that RDG is being used for the characterization of microorganisms and red blood cells at wavelengths where the discrepancies are considerable.
Fig. 10. Calculated spectra of Mie and RayleighDebyeGans for 500 nm Hemoglobin spheres with no absorption (κ=0) suspended in water. Note: the shape of the spectra are similar.
Fig. 11. Spectral differences between Mie and RayleighDebyeGans theory for 1 µm Hemoglobin spheres with no absorption (κ=0) suspended in water. At shorter wavelengths the spectra calculated by Mie theory flattens considerably.
Fig. 12. Semi log plot spectra for 5.5 µm Hemoglobin spheres with no absorption (κ=0) calculated using Mie and RayleighDebyeGans theory. The limits of RDG have been met for particle size.
4.4 Particle diameter ~ wavelength, absorption κ>0.
The limits of validity of RayleighDebyeGans theory with a relative refractive index close to 1 and an absorption coefficient greater than zero were tested through the calculation of the transmission spectra or spherical particles whose sizes are comparable to those of the wavelengths. The refractive indices of whole hemoglobin corresponding to wavelengths between 200 nm and 900 nm (1.01≤
n/
n
_{o}≤1.2, 0.001≤
κ≤0.1), were considered. The particle diameter sizes used were 100 nm, 500 nm, and 1 µm.
Figure 13 show that RayleighDebyeGans and Mie closely follow one another for a 100 nm sphere. As the particle size was increased to 500 nm and 1 µm the calculated turbidity from Rayleigh DebyeGans slowly deviates from Mie (see
Fig. 14 and
Fig. 15). As the size increases, the features of the spectra calculated with Mie theory flatten yielding completely different spectra. These differences are due primarily to the fact that absorption efficiency
Q
_{abs} which is directly proportional to the volume in the case of RDG theory.
Fig. 13. Close approximation of the calculated transmission of RayleighDebyeGans to Mie theory for 100 nm Hemoglobin spheres suspended in water.
Fig. 14. Comparison of calculated transmission for RayleighDebyeGans to that of Mie theory shows the divergence of theories for 500 nm Hemoglobin spheres suspended in water. Note the difference of RDG compared to
Fig. 10 where no absorption is present.
Fig. 15. The calculated spectra for 1 µm Hemoglobin spheres suspended in water shows that Mie theory appears to flatten when compared to RayleighDebyeGans.
4.5 Compensation of RDG through the modification of the Optical Properties
The practical range of application of RayleighDebyeGans has been considered limited to small deviations from the relative refractive index (
n(
λ)/
n
_{0}(
λ)~1) and small values of the absorption coefficient (
κ(
λ)~0) [
66. M. Kerker, The Scattering of Light and other Electromagnetic Radiation. (Academic Press, New York, 1969).
]. The derivation of RayleighDebyeGans theory assumes each dipole absorbs and scatters independently and only considers the interference of the scattered wave. As a result, the angular scattering intensity is shape and orientation dependent, whereas the absorption efficiency is independent of the particle shape; in other words, the total absorption is only dependent on the particle volume (the total number of chromophoric groups in the particle) [
55. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. (Wiley Science Paper Series, New York, 1998). [CrossRef]
,
66. M. Kerker, The Scattering of Light and other Electromagnetic Radiation. (Academic Press, New York, 1969).
]. When comparing the total absorption calculated from RDG and Mie theories for
κ(
λ)>0, a large discrepancy can be observed; the absorption efficiencies calculated with Mie theory always being smaller (hypochromic) than the values calculated with RDG. This holds particularly true for large absorption coefficients (i.e., Hemoglobin, DNA). The apparent hypochromicity observed for Mie theory suggests that the theoretical models developed to account for hypochromic or “screening” effects may be able to compensate RDG and bring it into a better agreement with Mie theory.
To explore the effect of hypochromicity corrections, the volume fraction of chromophoric groups is treated as an adjustable parameter in Veshkin’s model. Two cases are considered:
v
_{f}=0 which corresponds to 100% hypochromicity and translates to the corrected
κ
_{c}(
λ) being equal to zero; and
v
_{f}=1 which corresponds to using the value of
κ
_{c}(
λ) equal to
κ(
λ). Spherical hemoglobin particles with a diameter of 1 µm were used as test cases where Veshkin’s correction was applied only to
κ(
λ). The volume fraction values used in this study were 0.15, 0.20, 0.33, and 0.50. The molecular parameters for hemoglobin are: characteristic length=68
Å, with a cross sectional area of 20
Å, and a molecular weight of 16100 [
1919. S. Narayanan, Aggregation and Structural Changes in Biological Systems: An Ultraviolet Visible Spectroscopic Approach for Analysis of Blood Cell Aggregation and Protein Conformation. (University of South Florida, 1999). [PubMed]
]. The orientation value
q was set to one, meaning the molecules are randomly oriented [
1111. N. L. Veshkin, “Screening Hypochromism of Molecular Aggregates and Biopolymers,” J. Biol. Phys. 25, 339–354 (1999). [CrossRef]
].
Fig. 16. Intermediate levels hypochromicity using Veshkin correction for RayleighDebyeGans compared to no hypochromicity correction in Mie and RayleighDebyeGans for 1 µm Hemoglobin spheres suspended in water.
Fig. 17. Hypochromicity of 0% and 100% calculated using Mie and RayleighDebyeGans theory for a 1 µm Hemoglobin spheres suspended in water. Note that at 100% hypochromicity RayleighDebyeGans does not achieve reduced difference to Mie theory.
Fig. 18. Calculated transmission of Mie and RayleighDebyeGans for 1 µm Hemoglobin spheres with Veshkin Correction to k
_{c} and an n
_{eff} calculated through KramersKronig Transform.
Using the effective n calculated from corrected κ
_{c}, one would expect the transmission by RDG to more closely match the transmission calculated by Mie; however, the contrary is observed. At different volume fractions of the chromophore relating to κ
_{c} and n
_{eff} values, there are distinct differences in the shape of the spectra. The differences in the spectra are rooted in determining the values for n
_{eff} from the κ
_{c} using the KramersKronig transform.
An alternate approach for bringing together the spectra calculated from Mie and RDG theories is the mathematical adjustment of the refractive indices at each wavelength [
1717. P. Latimer, A. Brunsting, B. E. Pyle, and C. Moor, “Effects of asphericity on single particle scattering,” Appl. Opt. 17, 3152–3158 (1978). [CrossRef] [PubMed]
]. This approach explored in the next section.
4.6 Correction to RDG through the estimation of Effective Refractive Indices
The concept behind calculating effective refractive indices is that, given the extinction efficiency
Q
_{ext} (
λ) calculated from Mie theory there may be a set of effective n(
λ) and
κ(
λ) that can compensate for the differences between the extinction efficiencies calculated from RDG and Mie theories. This approach has been successfully applied by Latimer et al [
1717. P. Latimer, A. Brunsting, B. E. Pyle, and C. Moor, “Effects of asphericity on single particle scattering,” Appl. Opt. 17, 3152–3158 (1978). [CrossRef] [PubMed]
] to compensate for the effect of particle shape in monochromatic angular scattering measurements. The absorption and scattering efficiencies for RDG can be expressed as:
and
Introducing
Λ=∫0πf2(θ)(1−cos2θ)sinθdθ, and realizing that Λ is independent of
n and
κ, for
m≈1
Eq. (19) can be expressed as:
Notice that
Eq. (21) is a quadratic in terms of
n(
λ) and
κ(
λ). Also notice that there is only one degree of freedom at each wavelength, and therefore only one parameter, either
n
_{eff} (
λ) or
κ
_{eff} (
λ), can be estimated from the observed
Q
_{ext}. Replacing the extinction efficiency in
Eq. (21) with the extinction efficiency evaluated from Mie theory (
Q
_{Mie}), explicit expressions for both
n
_{eff} (
λ) and
κ
_{eff}(
λ) can be obtained.
If Q
_{ext}(λ)=Q
_{Mie}(λ) and κ(λ) is estimated, then n
_{eff} (λ) can be obtained directly from:
Alternatively, if Q
_{ext}(λ)=Q
_{Mie}(λ) and n(λ) is estimated then κ
_{eff}(λ) can be obtained directly from:
The terms Q
_{ko} and Q
_{no} are given by
where q1=49(ka)2Λ,q2=83(ka),andq3=49(ka)4Λ
Previous reports on the range of validity of RDG
^{6} and the simulation results using
Eq. (18) and
Eq. (19) 125
[
1616. A. C. GarciaLopezHybrid Model for Characterization of Submicron Particles using Mulitwavelength Spectroscopy, (University of South Florida, 2005).
] have shown that the difference
Q
_{ext}(RDG)
Q
_{ext}(Mie) is always positive for the particle size range of interest; on this basis it can be readily shown that
Therefore, there exists a value for 0≤n
_{eff}≤n that will equate the Mie and RDG extinction efficiencies. Similarly, it can be shown that a positive κ
_{eff} can be obtained from
subject to the following constraints
and
It is evident that, although effective values for the optical properties can be obtained, the approach is not satisfactory; the effective properties are also functions of the particle size, they have to be evaluated at every wavelength, and the optical property values calculated through
Eqs. (26)–
(29) do not have any physical meaning. In addition, the estimation of either
n
_{eff} (
λ) or
κ
_{eff} (
λ) conditional upon
κ(
λ) or
n (
λ) will not necessarily satisfy the KramersKronig transforms.
5. Discussion
Prior to this study, a comparative evaluation of the RayleighDebyeGans approximation relative to Mie theory had not been studied for particle sizes relevant to biological systems over the UVVis spectral region (200–900nm). The importance of this study lies on the potential use of multiwavelength spectrophotometric measurements for the characterization of biological particles. Of particular interest is the region where relevant biological chromophores absorb (190–400nm); it is primarily in this region where strict adherence to the RDG constraints cannot be ensured. The simulation studies reported herein lead to the conclusion that there are significant differences in the spectroscopy behavior of homogeneous spherical particles when their transmission spectra are predicted with RayleighDebyeGans and with Mie theory. In these studies, the optical properties relevant to the study of industrial and biological particles have been used. The disagreement between RDG and Mie theories is most severe when absorption is present and when the particle size becomes larger than the wavelength. The effect of changes in the refractive indices has been explored as a means to extend the range of validity of RDG over the complete UVVis wavelength range. In studying the effects of changes in the refractive indices the expectation was to bring RayleighDebyeGans into better agreement with Mie theory for larger particles and for particles containing strong chromophoric groups. It has been demonstrated that it is not possible to adequately, or realistically, compensate for the differences between Mie and RDG through the use of hypochromicity models and/or effective refractive indices. The simplicity of the RayleighDebyeGans approximation for the prediction of shape and orientation, and its potential application to the characterization of complex biological particles, continues to justify the efforts to improve its accuracy and precision. Although the compensation approaches reported herein have shown not to be effective over a broad wavelength range, the reformulation of RDG in terms of a hybrid theory shows considerable promise. The results from such formulation will be reported shortly.