FDTD simulation of optical phase contrast microscope imaging

Stoyan Tanev,1 James Pond,2 Paul Paddon,2 and Valery V. Tuchin3,4

1Department of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive,

Ottawa, ON, Canada K1S 5B6, e-mail: tanev@sce.carleton.ca 2Lumerical Solutions, Inc., Vancouver, BC, Canada

3Institute of Optics and Biophotonics, Saratov State University, Saratov, 410012 Russia 4Institute of Precise Mechanics and Control of RAS, Saratov 410028, Russia

ABSTRACT

The Finite-Difference Time-Domain (FDTD) approach is applied to model optical phase contrast microscope (OPCM) imaging of Gold nanoparticles (NPs) in singe biological cells. We first demonstrate the effect of optical clearing on the optical phase contrast microscope image of a realistic size biological cell containing a cytoplasm, a nucleus and a membrane. The FDTD-based OPCM model is then applied to visualize the presence of a cluster of Gold NPs in the cytoplasm at optical conditions. To the best of our knowledge, this is the first study using the FDTD approach in combination with Fourier optics techniques to construct OPCM images of realistic size cells containing NPs. The results demonstrate the potential of the FDTD modelling approach and extend its area of applicability into a new biomedical research area. Keywords: Biological cell, finite-difference time-domain method, optical phase contrast microscope, optical clearing effect, Gold nanoparticle

1. INTRODUCTION

One of the growing areas in biomedical optics and photonics1,2 is the application of optical software simulation and modeling tools to provide deeper understanding of newly developed optical diagnostics and imaging techniques. The numerical modeling of light interaction with biological cells and tissues within the context of a particular optical imaging technique are of particular relevance to the biomedical imaging community. There are a number of numerical modeling approaches that can be used for the modeling of the light scattering from biological cells3 including the finite-difference time domain (FDTD) method.4-9 The FDTD approach is one of the powerful tools in studying the nature of the light scattering mechanisms from both normal and pathological cells and tissues. Recently we demonstrated the applicability of the FDTD approach to numerically study how the optical immersion technique (OIT) enhances the effect of cell membrane thickness and Gold nanoparticles (NPs) on the phase contrast in the optical microscopic imaging of cancerous and non-cancerous cells.7,8 The OIT is connected with the so called “optical clearing” (OC) effect consisting in the increased light transmission through cells or tissues due to the matching of the refractive indices of some of their morphological components to that of the extra-cellular or interstitial media.10-12 In particular, we have studied the changes in the forward scattered light phase and intensity distributions due to different cell membrane thicknesses and different configurations of the NPs with and without the OIT. Here we present the main results of a pilot study applying the FDTD modeling technique to model optical phase contrast microscope (OPCM) imaging of Gold nanoparticles (NPs) in singe biological cells.13-15 The FDTD-based OPCM model provides a valuable tool for further studies on nanobioimaging and nanotherapetutics.

Biophotonics: Photonic Solutions for Better Health Care, edited byJürgen Popp, Wolfgang Drexler, Valery V. Tuchin, Dennis L. Matthews,

Proc. of SPIE Vol. 6991, 69912D, (2008) · 1605-7422/08/$18 · doi: 10.1117/12.781514

Proc. of SPIE Vol. 6991 69912D-12008 SPIE Digital Library -- Subscriber Archive Copy

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2. THE FDTD APPROACH

The FDTD technique is a numerical solution of Maxwell’s equations.16 In most light scattering simulation experiments one usually uses the so-called Total Field/Scattered Field (TFSF) formulation8,16 to excite the magnetic and electric fields and simulate a linearly polarized plane wave propagating in a finite region containing the simulation equivalent of the object under investigation, i.e. the biological cell. TFSF sources are used to separate the FDTD computation domain into two distinct regions. The first one is internal to the computational domain and contains the total near fields Etot = Einc+ Escat and Htot = Hinc+ Hscat, i.e. the sum of the incident field and the scattered near field. The second one occupies the rest of the computational domain and contains only the scattered near fields Escat = Etot - Einc and Hscat = Htot - Hinc.

16 The final FDTD simulation results require additional post-processing numerical procedures to transform the real values of the near fields into their complex far field counterparts. The far fields include the transverse distribution of the forward scattered light from the cell which is of high relevance for the modeling of optical phase contrast imaging. At the edges, the entire FDTD computational domain is truncated by the so called perfectly matched layer (PML) boundary conditions.16

Figure 1. Schematic representation of the FDTD computational domain and the Total Field/Reflected Field formulation. (NPs denotes nanoparticles)

The 3D simulation results provided here are based on a modified 3D TFSF formulation that could be more appropriately called “TFRF” (Total Field/Reflected Field). The 3D “TFRF” formulation uses a TFSF region which contains the biological cell and extends beyond the limits of the simulation domain (Fig. 1b). The extension of the transverse dimension of the input field beyond the limits of the computational domain through the PML boundaries would lead to distortions of its ideal plane wave shape and eventually distort the simulation results. To avoid these distortions one must use Bloch periodic boundary conditions (Fig. 1b) in the lateral x- and y-directions which are perpendicular to the direction of propagation - z. Bloch boundary conditions are periodic boundary conditions which take into account the phase effects due to the tilting of the input plane waves incoming at periodic structures, i.e. what we are actually modeling is a periodic row of biological cells. The near scattered fields, however, are calculated in transverse planes located in close proximity to the cell where the coupling effect due to waves scattered from adjacent cells is negligible. This effect can be further minimized or completely removed by controlling the lateral dimension of computational domain by using a large enough period of the periodic cell structure. The larger is this period, the smaller is the coupling

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effect. In the 3D TFRF formulation the location in the computational domain corresponding to the forward scattered light is positioned within the total field region (Fig. 1). The OPCM simulation model requires the explicit availability of the forward scattered transverse distribution of the fields. The phase of the scattered field accumulated by a plane wave propagating through a biological cell will be used in the FDTD model of the OPCM described in the next section.

3. OPTICAL PHASE CONTRAST MICROSCOPE MODEL

3.1. OPCM principle

Phase contrast microscopy is utilized to produce high-contrast images of transparent specimens such as microorganisms, thin tissue slices, living cells and sub-cellular components such as nuclei and organelles. It translates small phase variations into corresponding changes in amplitude visualized as differences in image contrast.

Figure 2. a) Schematic representation of an OPCM (a) and visualization of the OPCM design methodology. b) 2D visual representation of the FDTD OPCM model using incoherent illumination by two planes waves at a polar angle of 30 degrees. For each of the two plane waves light propagation is modeled as a combination of two parallel wave phenomena: a) propagation of the reference beam without the cell, and b) propagation of the diffracted beam due to the cell. The phase contrast image is created by adding up in intensity (incoherently) the two final interference images corresponding to the two input planes waves.

A standard phase contrast microscope design is shown in Fig. 2a, where an image with a strong contrast ratio is created by coherently interfering a reference (R) with a diffracted beam (D) from the specimen.a Relative to the reference beam, the diffracted beam has lower amplitude and is retarded in phase by approximately π/2 through interaction with the specimen. The main concept underlying the design of the phase contrast microscope is the spatial separation of the R and D wave fronts emerging from the specimen, which are projected onto different locations in the objective rear focal plane. In addition, the amplitude of the R beam light must be reduced and the phase advanced or retarded by another ±π/2 in order to maximize differences in intensity between the specimen and background in the image plane. The mechanism for generating relative phase retardation has two-steps: i) the D beam is being retarded in phase by a quarter wavelength (i.e., π/2) at the specimen, and ii) the R beam is advanced (or retarded) in phase by a phase plate positioned in or very near the objective rear focal plane in a way affecting only the R beam (Fig. 2a). This two steps process is enabled by a specially designed annular diaphragm – the condenser annulus – which is placed in the condenser front focal plane,

______ a http://www.microscopyu.com/articles/phasecontrast/phasemicroscopy.html

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matched in diameter, and optically conjugate to an internal phase plate residing in the objective rear focal plane (Fig. 2a). The resulting image, where the total phase difference is translated by interference at the image plane into an amplitude variation, can have a high contrast ratio, particularly if both beams have the same amplitude. Fig. 2a illustrates the part of the microscope that will become the subject of FDTD modeling combined with Fourier optics.

3.2.FDTD excitation and simulation of near fields

Figure 2b provides a visual representation illustrating the major steps in the FDTD OPCM model. The phase contrast microscope uses incoherent annular illumination that could be approximately modeled by eight different simulations using input plane waves incident at a given polar angle (30 degrees), an azimuthal angle (0, 90, 180 or 270 degrees), a specific polarization (in or perpendicular to the plane of the graph). Every single FDTD simulation provides the near field components in a transverse monitoring plane located right after the cell (Fig. 1).

3.3.Post-processing procedures

The near fields are recorded at a plane a few hundred nanometers after the cell. This near fields are projected to the far field using a near to far field projection. 16 In this analysis, the near to far field projection, like the FDTD simulation itself, takes into account the Bloch periodic boundary conditions in the transverse plane and is calculated only at angles that correspond to diffracted orders of the periodic structure such as defined by the Bragg conditions. The near to far field projection therefore provides the electric field amplitude and phase corresponding to each diffracted order. The zeroth order, ie. the light travelling through the scattering object without any deviation in angle, is the reference beam and the phase contrast microscope is designed to provide a phase delay to this order. The magnification factor is applied on the far fields before the interference of the scattered (S) and reference (D) beams. A lens system that provides magnification is only modifying the angle of light propagation. The numerical implementation of the magnification is equivalent to multiplying the direction cosines, ux and uy by the inverse value of the desired magnification factor M: Ux = ux/M and Uy = uy/M. Working in spherical coordinates provides the advantage that the vectorial components do not change when ux and uy are modified because they are part of a local coordinate system that is tied to the values of ux and uy. In our particular case the image is reconstructed assuming a 10X magnification. We have also applied the effect of a numerical aperture which clips any light that would not be collected by the lens system, i.e. all parts of the beams with Ux

2 + Uy2 > NA2 have been discarded. In our case NA = 0.8.

The OPCM images at the image plane are calculated by adding up the scattered and the reference beam at any desired phase offset Ψ:

I = abs(Ex-D + a Ex-R exp(iΨ))2 + abs(Ey-D + a Ey-R exp(iΨ))2 +-abs(Ez-D + a Ez-R exp(iΨ))2, (1) where Ex-D, Ey-D and Ez-D are the three components of the electric field of the diffracted beam and Ex-R, Ey-R and Ez-R are the three components of the electric field of the reference beam. The coefficient a and the phase Ψ correspond to the ability of the OPCM to adjust the relative amplitudes and the phase difference between the two beams. In all simulation results presented in this article a = 1 and Ψ is used as parameter.

4. CELL STRUCTURE AND FDTD OPCM IMAGE SIMULATION RESULTS

4.1. Cell structure

The cell is modeled as a dielectric sphere with realistic dimensions: radius Rc = 5 µm and membrane thickness d = 20 nm. The cell nucleus is also spherical, has a radius Rn = 1.5 µm and its center is located at a position which is 2.0 µm

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shifted from the cell center in a direction perpendicular to the principal direction of propagation. The refractive index of the cytoplasm is ncyto = 1.36, of the nucleus nnuc = 1.4, of the membrane nmem = 1.47 and of the extra-cellular material next = 1.33 (no refractive index matching) or 1.36 (refractive index matching).

4.2. Optical properties of Gold nanoparticles

Gold NPs exhibit the ability to resonantly scatter visible and near infrared light due to the excitation of Surface Plasmon

Resonances (SPR). The ability of Gold NPs to resonantly scatter visible and near infrared light is extremely sensitive to their size, shape, and aggregation state offering a great potential for optical cellular imaging and detection labeling studies. 18-22 Our FDTD approach uses the dispersion model for Gold derived from the experimental data provided by Johnson and Christy23 where the total, complex-valued permittivity is given by the following formula:

)()()( ωεωεεωε PLREAL ++= . (2)

Each of the three contributions to the permittivity arises from a different material model. The first term represents the contribution due to the basic, background permittivity. The second and third terms represent Lorentz and plasma contributions:

)2/()( 20

20

20 ωωδωωεωε −−= iLORENTZL , )/()( 22 ωωνωωε += CPP i , (3)

where all material constants are summarized in the table below.

Background permittivity Lorentz dispersion Plasma dispersion

REALε = 7.076500096493071

LORENTZε = 2.323056371040549

0ω = 4.634629996925336 × 10+015

0δ = 9.267097992234386 × 10+014

Pω = 1.390688491193091 × 10+016

Cν = 1.410836620968578 × 10+007

Table 1. Optical material constants of Gold [Ref. 23]

It is worth noting that the Gold optical properties described above are the bulk material ones and do not take into account confinement effects due to the nano-scale size of the NPs. The accounting of the confinement effects will lead to a slight shift of the resonant frequencies but will not change the nature of the optical phenomena under investigation. The optical resonant properties will be also slightly modified due to the fact that we are using a simulation model with staircase approximation of the ideal spherical shape of the nanoparticles as well as due to the imperfection of the boundary conditions. This, however, is not a limitation of our model since it will lead to a mere shift of the optical resonance. What is of importance here is to be able to model two different cases – on and off SPR. The ability to model both these cases together with the effect of optical clearing effect provides the opportunity to numerically study the possibility for imaging of the uptake of single a NP as well as of clusters of NPs – scenarios which, to the best of our knowledge, were not studied before. We used the FDTD technique to calculate the scattering and absorption cross-sections over a 400-900 nm wavelength range for a single 50 nm diameter Gold NP immersed in a material having the properties of the cytoplasm (n=1.36) and resolution dx=dy=dz=10nm. The scattering cross-section is defined as

)(/)( ωωσ incscatscat IP= , (4)

where Pscat is the total scattered power and Iinc

nc is the intensity of the incident light in [W/m

2]. It was calculated by

applying the Total Field/Scattered Field FDTD formulation16 and by creating 12 field power monitoring planes around the nanoparticle in the form of a box: 6 in the total field region and 6 in the scattered field region. The total scattered power is calculated by summing up the power flowing outward through 6 scattered field power monitors located in the scattered field region.

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3.7

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Wavelength [nm] Figure 3. Extinction cross-section of a 50 nm Gold nanoparticle immersed in material having the optical properties of the cytoplasm: n=1.36. The optical properties of Gold are described by Eqs. 2 and 3 and the parameters given in Table 1.

The absorption cross-section is similarly defined as

)(/)( ωωσ incabsabs IP= , (5)

where Pabss is the total power absorbed by the particle. The power absorbed by the particle is calculated by calculating the

net power flowing inward through the 6 total field power monitors located in the total field region. The extinction cross-section is the sum of the absorption and scattering cross-sections

absscatext σσσ += . (6)

Fig. 3 shows that the extinction ratio has a maximum at around 543.0 nm corresponding to one of the radiation wavelengths of He-Ne lasers. In this article we also present results for λ = 676.4 nm (a Krypton laser wavelength) which corresponds to the off SPR case. The FDTD results are compared with the theoretical curve calculated by Mie theory. The main reason for the slight discrepancy between the theoretical and FDTD results for the extinction cross-section is the larger mesh size. To improve the results, the mesh size should be decreased. We, however, have intentionally kept it as given above for consistency with the simulations of the biological cell with Gold NPs located in cytoplasm.

4.3. OPCM image simulation results

The simulation results presented here are based on the 3D FDTD OPCM model. The simulations use non-uniform meshing where the number of mesh points in space is automatically calculated to ensure a higher number of mesh points in materials with higher values of the refractive index.17 The realistic cell dimensions require a very fine numerical resolution making the simulations extremely computationally intensive (~100Gbs RAM) and requiring high performance computing facilities. The time step used during the simulation was defined by means of a factor of 0.99 in the Courant stability limit: ( ) 1/ 22 2 20.99 1/ 1/ 1/c t x y z

−∆ = × ∆ + ∆ + ∆ . Figure 4 shows a cross-section of OPCM images from a previous

work15 based on FDTD simulations of the same cell at λ = 632.8 nm with a phase difference Ψ = 180o between the reference beam and the scattered beam (assuming a = 1) for two different cases: Refractive Index Matching (RIM) and no RIM. The results demonstrate that the RIM leads to the optical clearing of the image and a significantly better OPCM image of the nucleus and of the membrane – the only sources of the phase contrast at optical immersion conditions. The cell center is located in the middle (x=y=z=0) of the computational domain with dimensions 25µm×25µm×15µm. The nucleus’ center is located at x=-2 µm, y=z=0 µm.

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-20 0 20 40 60 80 100 120X direction (tm]

Image Cross-Section at yO tm, Phase Delay = 18O3.0

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Figure 4. FDTD calculated cross-section of OPCM images of a biological cell with a nucleus and a membrane for λ = 632.8 nm and Ψ = 180o. The cross section goes through the centers of the nucleus and of the cell itself. It is shown that at optical immersion (next = ncyto

= 1.36) the nucleus is made sharply visible as compared to the rest of the cell. This is an illustration of the optical clearing effect. The numerical values of the X axe are in the far field and do not directly correspond to the exact cell dimensions.

RIM, no NPs, λ = 676.4 nm RIM, NPs cluster, λ = 676.4 nm RIM, NPs cluster, λ = 543.0 nm

Figure 5. FDTD simulation calculated OPCM images of a biological cell with a nucleus, a membrane and a cluster of 42 Gold NPs located in a position symmetrically opposite to the nucleus. The numerical values of the X axe are in the far field and do not directly correspond to the exact cell dimensions. The arrows indicate the position of the Gold NP cluster.

Figure 6. Comparison of the cross-sections of the three images shown in Fig. 5 at y=0 µm. The effect of the Gold NP SPR is clearly visible at λ = 543.0 nm. The curves are intentionally shown on the same background intensity level approximately equal to 0.88.

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Based on the fact that RIM enhances significantly the imaging of the cell components, we have used the FDTD OPCM model to create the OPCM images of the same cell at optical immersion conditions including a cluster of 42 Gold NPs located in a position which symmetrically opposite to that of the nucleus (Fig. 5, see also Fig. 1). The phase difference between the reference beam and the scattered beam (assuming a = 1) is again Ψ = 180o. The NPs are randomly distributed within a spherical space with a radius of 1 µm and center coordinate (x=2 µm, y=0 µm, z=0 µm). The coordinate of nucleus is again x=-2 µm, y=0 µm, z=0 µm. The insertion of the nanoparticles increases significantly the CPU and memory requirements. This is the reason why we needed to decrease the dimensions of the computational domain to 15 µm×15 µm×12 µm. The OPCM images in Fig. 5 show that the Gold NP cluster can be clearly identified at both the non-resonant (λ = 676.4 nm) and resonant (λ = 543.0 nm) wavelengths. The graph in Fig. 6 compares the cross-sections of the three images shown in Fig. 5 at y=0 µm showing the relevant half of the images (from zero to 80 µm) where the Gold NP cluster is located. At resonance the intensity of the Gold NP peak is 1.377 times higher than the one at non resonance. The enhanced imaging of the Gold NP cluster at resonant conditions is of no surprise. It however needs to be further studied as a function of particular phase delay Ψ between the reference beam and the scattered beam. The correct modeling of the Gold NPs required making sure that the FDTD resolution is the same as in the calculation of their optical scattering/absorption properties, i.e. dx=dy=dz=10nm. This was done by using the built in features of our FDTD simulation software17 enabling the possibility for using both non-uniform meshing in the entire computational domain and local hard coding of the resolution of the NPs. The results presented in Figures 5 and 6 have a preliminary character but clearly demonstrate the capability of our FDTD OPCM model to include the effect of Gold NPs.

5. CONCLUSIONS

In this article we provided pilot results on the application of the FDTD approach to model OPCM imaging of Gold NPs in singe biological cells. We have first reproducing the effect of optical immersion on the OPCM images of a simple biological cell containing a cytoplasm, a nucleus and a membrane. The model was then applied to include the presence of a cluster of Gold NPs in the cytoplasm at optical immersion conditions as well as the enhancing imaging effect of the SPR. The validation of the model provides a basis for future studies of OPCM imaging involving Gold NP effects. To the best of our knowledge this is the first time to process FDTD simulation results in a way providing OPCM images of cell with realistic cells sizes and NPs. The results demonstrate the potential of the FDTD approach to model optical nanobioimaging instruments.

ACKNOWLEDGMENTS ST, JP and PP acknowledge the use of the computing resources of WestGrid (Western Canada Research Grid) - a $50-million project to operate grid-enabled high performance computing and collaboration infrastructure at institutions across Canada. VVT was supported by grants of Federal Agency of Education of RF № 1.4.06, RNP.2.1.1.4473 and RFBR No. 06-02-16740.

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REFERENCES

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[16] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method, Artech House Publishers (3rd edition), Boston (2005).

[17] The simulations were performed by the FDTD Solutions™ software developed by Lumerical Solutions Inc., Vancouver, BC, Canada: www.lumerical.com

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