# A New 3D Simulation Method for the Construction of Optical Phase Contrast Images of Gold Nanoparticle Clusters in Biological Cells

Authors: Stoyan Tanev; James Pond; Paul Paddon; Valery V. Tuchin
Journal: Advances in Optical Technologies
 (2008)

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## Abstract

A new 3D simulation method based on the finite-difference time domain (FDTD) approach in combination with Fourier optic techniques is applied to the modeling of optical phase contrast microscope (OPCM) imaging of gold nanoparticles (NPs) in singe biological cells. We consider a realistic size 3D cell model at optical immersion conditions, that is, when the refractive index values of the cytoplasm and of the extracellular medium are equal. For the first time, an FDTD-based OPCM model is applied to visualize the presence of a cluster of gold NPs in the cytoplasm at both resonant and nonresonant conditions. The results demonstrate the capability to model OPCM image enhancement by optically controlling the resonant properties of the NPs. Our research study extends the applicability of the FDTD modeling approach into a new biomedical optics research area.


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## Body

## 1. Introduction

One of the  growing areas in biomedical optics and
photonics [1, 2] is the  application of optical software simulation and modeling
tools to provide deeper understanding of newly developed optical diagnostics
and imaging techniques. There are a number of numerical modeling approaches
that can be used for the modeling of the light scattering from biological cells
[3] including the finite-difference time domain (FDTD) method [4–9]. Recently, we
demonstrated the applicability of the 2D FDTD approach to numerically study the
changes in the forward scattered light phase and intensity distributions due to
different cell membrane thicknesses and different configurations of gold NPs in
the cytoplasm of small size cells using the optical immersion technique (OIT) [7, 8]. The OIT is associated 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 extracellular or interstitial media [10–12]. In this
article, we present the first results of a research study applying a 3D FDTD-based
optical phase contrast microscope (OPCM) imaging approach to the modeling of a
cluster of gold nanoparticles (NPs) in realistic size single cells. We
demonstrate a new simple way of using the FDTD approach for the construction of
OPCM images by moving its modeling power from the calculation of phase
distribution curves [7] to realistic cell images. We validate our approach by
modeling a gold nanoparticle cluster in the cell cytoplasm to provide a
valuable design methodology for further studies in optical nano-bioimaging and
nanotherapeutics.


## 2. The FDTD Approach

The FDTD technique is a numerical solution of
Maxwell’s equations [13]. In most light scattering simulation experiments, one usually
uses the so-called total field/scattered field (TFSF) formulation [8, 13] 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 [13]. A schematic representation of the 2D FDTD
computational domain is shown in Figure 1(a). 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, that is, 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. The final FDTD simulation results
require additional postprocessing computational procedures to transform the
real values of the calculated 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 [13].Schematic representation of the
FDTD computational domain: (a) total field/scattered field 2D formulation;
(b) total field/reflected
field 3D formulation.
(a)(b)The 3D case of the TFSF formulation described above
follows a similar logic. The 3D simulation results provided here are based on a
modified 3D TFSF formulation that could be more appropriately called total field/reflected
field (TFRF). The 3D “TFRF” formulation uses a TFSF region which contains the
biological cell and extends beyond the limits of the simulation domain (Figure1(b)).
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 (Figure 1(b)) in the lateral x- and
y-directions which are perpendicular to the direction of propagation-z [13]. 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, that is,
what we are actually modeling is a periodic row of biological cells. The near
scattered fields, however, are calculated in the transverse planes located in the
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 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 (Figure 1(b)). 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 that will be described in Section 3.


## 3. Optical Phase Contrast Microscope Model

### 3.1. FDTD OPCM Principle

Phase contrast microscopy is utilized to produce
high-contrast images of transparent specimens such as microorganisms, thin
tissue slices, living cells, and subcellular components such as nuclei and
organelles. It translates small-phase variations into
corresponding changes in amplitude visualized as differences in image contrast.
A standard phase
contrast microscope design is shown in Figure2(a), where an image with a
strong contrast ratio is created by coherently interfering a reference (R) with
a diffracted beam (D) from the specimen (http://www.microscopyu.com/articles/phasecontrast/phasemicroscopy.html). 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.(a) Schematic
representation of an OPCM. (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, the propagation of light is modeled
as a combination of two parallel wave phenomena: (i) propagation of the
reference (R) beam without the cell, and (ii) propagation of the diffracted (D) beam
due to the cell.
(a)(b)The main concept underlying the design of the phase contrast microscope is
the spatial separation of theR beam from D wave front emerging from the
specimen. In addition, the amplitude of the R beam light must be reduced and the phase should be  advanced or
retarded by another ±π/2 in order to maximize the differences
in the intensity between the specimen and the 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 (Figure 2(a)).
This two-step process is enabled by a specially designed annular diaphragm—the condenser annulus—which is placed
in the condenser front focal plane, is matched in diameter, and optically conjugates to the phase
plate residing in the objective rear focal plane. 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.


### 3.2. Excitation and Simulation of the FDTD near Fields

Figure2(a)
illustrates the part of the microscope that will become the subject of FDTD
modeling combined with Fourier optics. Figure 2(b) 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 adding up the results of eight different simulations using ideal input plane waves
incident at a given polar angle (30 degrees), an azimuthal angle (0, 90, 180,
or 270 degrees), and a specific light polarization (parallel or perpendicular
to the plane of the graph). Every single FDTD simulation provides the near
field components in a transverse monitoring plane located right behind the cell
(Figure 1(b)).


### 3.3. Far-Field Transformation

The far field transformations (Figure3) use the FDTD
calculated near fields right behind the cell and return the three complex
components of the electromagnetic fields at 1 meter distance (i.e., long
enough) from the location of the near fields, that is, in the far field [13, 14]: Er(ux,uy),Eθ(ux,uy), and EΦ(ux,uy), where (r,θ,Φ) refer to a spherical coordinate system and the
variables ux and uy are the x and y components of the unit
direction vector u. The unit
direction vector is related to the angular variables by ux=sin(θ)cos(Φ),uy=sin(θ)sin(Φ),uz=cos(θ), with ux2+uy2+uz2=1. The in-plane wave vectors for each
plane wave are given by kx=kux and ky=kuy, where k=2π/λ. To scale the far field to a
different distance, we can recognize that in 3D, the electric field intensity
decreases like 1/R2: |E(R)|2=|E0|2/R2, where E0 is the field at 1 m distance resulting from the far field transformation and
R  is measured in meters. The specific details of the far field transformation
can be found elsewhere [13].Figure 3
Visual
representation of the far field transformation procedure within the context of
the optical phase contrast microscope schematics presented in Figures2.It is important to note that the near-to-far
field projection must take into account the Bloch periodic boundary conditions
in the lateral dimension and is calculated only at angles that correspond to
diffracted orders of the periodic structure as defined by the Bragg conditions. This is done
by calculating the direction cosines of all the diffracted orders that meet the
Bragg condition, and interpolating the previously far-field distributions onto
those specific directions. The near-to-far field projection, therefore,
provides the electric field amplitude and phase corresponding to each
diffracted order. The zeroth order, that is, 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.


### 3.4. Optical Magnification

Now that we have the field componentsE(kx,ky), we can use their amplitudes and
phases to determine all the properties of the polarization as well as do
Fourier optics for both the scattered and reference beams. We assume an ideal
optical lens system that could be characterized by a given magnification factor.
This simple model could be easily extended to include the numerical equivalent
of the two lenses (Figure 2) together with an additional model to take into
account any aberrational effects. The magnification was implemented by modifying
the angle of light propagation, that is, by multiplying the direction cosines ux and uy by the inverse value of the desired
magnification factor M: Ux=ux/M and Uy=uy/M. In any other circumstances, the
modification of the direction cosines would lead to complications because of
the vectorial nature of the E field. In our case, however, working in
spherical coordinates (Er,Eθ, and EΦ) leads to 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. The magnification factor M=10 was applied to the far fields before the
interference of the diffracted (D) and reference (R) beams at the image plane.


### 3.5. Numerical Aperture

It was also possible to apply the effect of a
numerical apertureNA=0.8 which clips any light that has too steep angle  and would not
be collected by the lens system. This means that all the light with Ux2+Uy2>NA2 is being clipped. The effect of the aperture
is defined by the inequality Ux2+Uy2>NA2 applied to the corrected aperture angles θ′ and Φ′: (1)sin(Φ′)=UyUxy,cos(Φ′)=UxUxy,cos(θ′)=Uz,sin(θ′)=Uxy,where Uxy=sqrt(Ux2+Uy2),Uz=sqrt(1−Uxy2) and the “sqrt” labels a square root
mathematical operation. The magnified field components will then have the
following form.Diffracted(D) Beam:
(2)Ex−D(kx,ky)=−EΦsin(Φ′)+Eθcos(Φ′)cos(θ′),Ey−D(kx,ky)=EΦcos(Φ′)+Eθsin(Φ′)cos(θ′),Ez−D(kx,ky)=−Eθsin(θ′).Reference(R) Beam:
(3)Ex−R(kx,ky)=−EΦ−Rsin(Φ′)+Eθ−Rcos(Φ′)cos(θ′),Ey−R(kx,ky)=EΦ−Rcos(Φ′)+Eθ−Rsin(Φ′)cos(θ′),Ez−R(kx,ky)=−Eθ−Rsin(θ′),   where Eθ−R are EΦ−R are the far field components of the reference
beam.


### 3.6. OPCM Image Construction

The fields given above are then used to calculate back
the Fourier inverse transform of the far field transformed fields leading to
the distribution of the scattered and the references beams in the image plane.Diffracted(D) Beam:
(4)Ex−D=sum(Ex−D(kx,ky)exp(ikxx+ikyy+ikzz)),Ey−D=sum(Ey−D(kx,ky)exp(ikxx+ikyy+ikzz)),Ez−D=sum(Ez−D(kx,ky)exp(ikxx+ikyy+ikzz)).Reference(R) Beam:
(5)Ex−R=sum(Ex−R(kx,ky)exp(ikxx+ikyy+ikzz)),Ey−R=sum(Ey−R(kx,ky)exp(ikxx+ikyy+ikzz)),Ez−R=sum(Ez−R(kx,ky)exp(ikxx+ikyy+ikzz)),   where the summation is over all angles.The OPCM images at the image plane are calculated by
adding up the scattered and the reference beam at any desired phase offsetΨ:(6)I=abs(Ex−D+aEx−Rexp(iΨ))2+abs(Ey−D+aEy−Rexp(iΨ))2+abs(Ez−D+aEz−Rexp(iΨ))2. 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.


## 3.1. FDTD OPCM Principle

Phase contrast microscopy is utilized to produce
high-contrast images of transparent specimens such as microorganisms, thin
tissue slices, living cells, and subcellular components such as nuclei and
organelles. It translates small-phase variations into
corresponding changes in amplitude visualized as differences in image contrast.
A standard phase
contrast microscope design is shown in Figure2(a), where an image with a
strong contrast ratio is created by coherently interfering a reference (R) with
a diffracted beam (D) from the specimen (http://www.microscopyu.com/articles/phasecontrast/phasemicroscopy.html). 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.(a) Schematic
representation of an OPCM. (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, the propagation of light is modeled
as a combination of two parallel wave phenomena: (i) propagation of the
reference (R) beam without the cell, and (ii) propagation of the diffracted (D) beam
due to the cell.
(a)(b)The main concept underlying the design of the phase contrast microscope is
the spatial separation of theR beam from D wave front emerging from the
specimen. In addition, the amplitude of the R beam light must be reduced and the phase should be  advanced or
retarded by another ±π/2 in order to maximize the differences
in the intensity between the specimen and the 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 (Figure 2(a)).
This two-step process is enabled by a specially designed annular diaphragm—the condenser annulus—which is placed
in the condenser front focal plane, is matched in diameter, and optically conjugates to the phase
plate residing in the objective rear focal plane. 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.


## 3.2. Excitation and Simulation of the FDTD near Fields

Figure2(a)
illustrates the part of the microscope that will become the subject of FDTD
modeling combined with Fourier optics. Figure 2(b) 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 adding up the results of eight different simulations using ideal input plane waves
incident at a given polar angle (30 degrees), an azimuthal angle (0, 90, 180,
or 270 degrees), and a specific light polarization (parallel or perpendicular
to the plane of the graph). Every single FDTD simulation provides the near
field components in a transverse monitoring plane located right behind the cell
(Figure 1(b)).


## 3.3. Far-Field Transformation

The far field transformations (Figure3) use the FDTD
calculated near fields right behind the cell and return the three complex
components of the electromagnetic fields at 1 meter distance (i.e., long
enough) from the location of the near fields, that is, in the far field [13, 14]: Er(ux,uy),Eθ(ux,uy), and EΦ(ux,uy), where (r,θ,Φ) refer to a spherical coordinate system and the
variables ux and uy are the x and y components of the unit
direction vector u. The unit
direction vector is related to the angular variables by ux=sin(θ)cos(Φ),uy=sin(θ)sin(Φ),uz=cos(θ), with ux2+uy2+uz2=1. The in-plane wave vectors for each
plane wave are given by kx=kux and ky=kuy, where k=2π/λ. To scale the far field to a
different distance, we can recognize that in 3D, the electric field intensity
decreases like 1/R2: |E(R)|2=|E0|2/R2, where E0 is the field at 1 m distance resulting from the far field transformation and
R  is measured in meters. The specific details of the far field transformation
can be found elsewhere [13].Figure 3
Visual
representation of the far field transformation procedure within the context of
the optical phase contrast microscope schematics presented in Figures2.It is important to note that the near-to-far
field projection must take into account the Bloch periodic boundary conditions
in the lateral dimension and is calculated only at angles that correspond to
diffracted orders of the periodic structure as defined by the Bragg conditions. This is done
by calculating the direction cosines of all the diffracted orders that meet the
Bragg condition, and interpolating the previously far-field distributions onto
those specific directions. The near-to-far field projection, therefore,
provides the electric field amplitude and phase corresponding to each
diffracted order. The zeroth order, that is, 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.


## 3.4. Optical Magnification

Now that we have the field componentsE(kx,ky), we can use their amplitudes and
phases to determine all the properties of the polarization as well as do
Fourier optics for both the scattered and reference beams. We assume an ideal
optical lens system that could be characterized by a given magnification factor.
This simple model could be easily extended to include the numerical equivalent
of the two lenses (Figure 2) together with an additional model to take into
account any aberrational effects. The magnification was implemented by modifying
the angle of light propagation, that is, by multiplying the direction cosines ux and uy by the inverse value of the desired
magnification factor M: Ux=ux/M and Uy=uy/M. In any other circumstances, the
modification of the direction cosines would lead to complications because of
the vectorial nature of the E field. In our case, however, working in
spherical coordinates (Er,Eθ, and EΦ) leads to 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. The magnification factor M=10 was applied to the far fields before the
interference of the diffracted (D) and reference (R) beams at the image plane.


## 3.5. Numerical Aperture

It was also possible to apply the effect of a
numerical apertureNA=0.8 which clips any light that has too steep angle  and would not
be collected by the lens system. This means that all the light with Ux2+Uy2>NA2 is being clipped. The effect of the aperture
is defined by the inequality Ux2+Uy2>NA2 applied to the corrected aperture angles θ′ and Φ′: (1)sin(Φ′)=UyUxy,cos(Φ′)=UxUxy,cos(θ′)=Uz,sin(θ′)=Uxy,where Uxy=sqrt(Ux2+Uy2),Uz=sqrt(1−Uxy2) and the “sqrt” labels a square root
mathematical operation. The magnified field components will then have the
following form.Diffracted(D) Beam:
(2)Ex−D(kx,ky)=−EΦsin(Φ′)+Eθcos(Φ′)cos(θ′),Ey−D(kx,ky)=EΦcos(Φ′)+Eθsin(Φ′)cos(θ′),Ez−D(kx,ky)=−Eθsin(θ′).Reference(R) Beam:
(3)Ex−R(kx,ky)=−EΦ−Rsin(Φ′)+Eθ−Rcos(Φ′)cos(θ′),Ey−R(kx,ky)=EΦ−Rcos(Φ′)+Eθ−Rsin(Φ′)cos(θ′),Ez−R(kx,ky)=−Eθ−Rsin(θ′),   where Eθ−R are EΦ−R are the far field components of the reference
beam.


## 3.6. OPCM Image Construction

The fields given above are then used to calculate back
the Fourier inverse transform of the far field transformed fields leading to
the distribution of the scattered and the references beams in the image plane.Diffracted(D) Beam:
(4)Ex−D=sum(Ex−D(kx,ky)exp(ikxx+ikyy+ikzz)),Ey−D=sum(Ey−D(kx,ky)exp(ikxx+ikyy+ikzz)),Ez−D=sum(Ez−D(kx,ky)exp(ikxx+ikyy+ikzz)).Reference(R) Beam:
(5)Ex−R=sum(Ex−R(kx,ky)exp(ikxx+ikyy+ikzz)),Ey−R=sum(Ey−R(kx,ky)exp(ikxx+ikyy+ikzz)),Ez−R=sum(Ez−R(kx,ky)exp(ikxx+ikyy+ikzz)),   where the summation is over all angles.The OPCM images at the image plane are calculated by
adding up the scattered and the reference beam at any desired phase offsetΨ:(6)I=abs(Ex−D+aEx−Rexp(iΨ))2+abs(Ey−D+aEy−Rexp(iΨ))2+abs(Ez−D+aEz−Rexp(iΨ))2. 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, Gold Nanoparticle Model and Simulation Results

### 4.1. Cell Structure

The cell is modeled as a dielectric sphere with a realistic
radiusRc=5μm. It is critically important to pay
special attention to the membrane model since assigning the membrane the
precisely desired thickness value leads to inadequate modeling of optical
immersion experiments. This is due to the fact that the FDTD technique averages
out the refractive index values at the interface of two different materials
effectively reducing the membrane thickness value and eventually, due to the
staircase approximation error, destroying the continuity of the membrane. This
is the reason for us to select a membrane thickness d=20 nm which corresponds to effective thickness of
approximately 10 nm. The cell nucleus is also spherical with a radius Rn=1.5μm centered at a position which is 2.0μm shifted
from the cell center in a direction perpendicular to the direction of light 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 extracellular material next=1.33 (no refractive index matching) or 1.36
(refractive index matching).


### 4.2. Optical Properties of Gold Nanoparticles

Gold NPs exhibit theability to resonantly
scatter visible and near infrared light due to the excitation of surface plasmonresonances (SPRs). The scattering ability is extremely sensitive to their
size, shape, andaggregation state offering a great potential for optical
cellular imaging and detection labeling studies [15–19]. Our FDTD
approach uses the dispersion model for gold derived from the experimental data
provided by Johnson and Christy [7, 20] where the total, complex-valued
permittivity is given as (7)ε(ω)=εREAL+εL(ω)+εP(ω). 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: (8)εL(ω)=εLORENTZω02ω02−2iδ0ω−ω2,εP(ω)=ωP2iωνC+ω2, where all material
constants are summarized in Table 1.Table 1
Optical material constants of gold [20].
Background permittivityLorentz dispersionPlasma dispersionεREAL=7.077εLORENTZ=2.323ωP=1.391×1016 Hzω0=4.635×1015 HzνC=1.411×107 Hzδ0=9.267×1014 HzWe have modeled two different cases—resonant and
nonresonant. 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 clusters of NPs—a scenario which,
to the best of our knowledge, was 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=10 nm. The scattering cross-section is defined as (9)σscat=Pscat(ω)Iinc(ω), where Pscat is the total scattered power and Iinc is the intensity of the incident
light in (W/m2). It was calculated by applying the total field/scattered field FDTD formulation [13] 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.The absorption cross-section is similarly defined as(10)σabs=Pabs(ω)Iinc(ω), where
Pabs
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 (11)σext=σscat+σabs. Figure 4 shows that the extinction
cross-section has a maximum of 3.89 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 nonresonant case (extinction cross-section value 0.322, ~12
times smaller than 3.89). 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 finite
mesh size.Figure 4
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 (7) and (8) and the parameters given in Table 1.To consistency of the results could be improved by reducing
the mesh size. In
the FDTD simulations including both the cell and the nanoparticles, the
numerical resolution of the nanoparticles was hard-coded todx=dy=dz=10 nm to make sure that their numerically manifested optical resonant
properties will be the same as the ones shown in Figure 4. However, Figure 5
visualizes the positioning of a cluster of 42 nanoparticles in the cytoplasm
that was used to produce the FDTD simulation results presented in
Section 5. It was
randomly generated without controlling the possibility for spatial overlaps or
contacts between the nanoparticles. The closer visual inspection showed that
there were no such overlaps and contacts.Figure 5
A cluster of 42 gold nanoparticles randomly
distributed in a spatial sphere with a radius of 1μm at a cell
location opposite to the location of the nucleus (x=2μm, y=0,z=0). The nucleus
is located at x=−2μm. The gold
nanoparticle size (diameter  =  50 nm) on right-hand graph is slightly exaggerated.


### 4.3. OPCM Image Simulation Results

The simulation results presented here are based on the
3D FDTD OPCM model. The simulations use nonuniform 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 [14]. The
realistic cell dimensions (including both cell radius and membrane) require a
very fine numerical resolution making the simulations computationally
intensive. The locally refined resolution of the nanoparticles leads to
additional requirements for the CPU time and memory (~120 Gbs RAM) requiring
high performance computing resources. The time step used during the simulation
was defined by means of a factor of 0.99 in the Courant stability limit: cΔt=0.99×(1/Δx2+1/Δy2+1/Δz2)−1/2.
The cell center is located in the middle (x=y=z=0) of the computational domain with dimensions 15μm×12μm×15μm. The nucleus’ center is located at x=−2μm,y=z=0μm. The cluster of gold nanoparticles
is located at x=2μm,y=z=0μm (Figure 5).The refractive
index matching (RIM) between the cytoplasm and the extracellular medium 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 phase contrast at optical immersion conditions. 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 the cluster of 42 gold NPs (Figure5, see also Figure 1(b))
and for different phase offsets between the reference beam and the scattered
beam (assuming a=1). The insertion of the nanoparticles increases significantly the CPU and
memory requirements. The dimensions of the computational domain are 15μm×15μm×12μm. The OPCM images in Figure 6 show that
the gold NP cluster can be clearly identified at both the nonresonant (λ=676.4 nm) and resonant (λ=543.0 nm) wavelengths.OPCM images of a biological cell for different values  (a:−150°, b: −90°, c: 90°, d: 180°) of the phase offset between the reference and diffracted beam of the OPCM at optical immersion conditions (left) including a cluster of 42 gold NPs (middle: no resonance, right: at resonance) located in a position symmetrically opposite to the nucleus. The arrows indicate the position of the gold NP cluster.
(a)(b)(c)(d)The graphs in Figure7 compare the geometrical
cross-sections of the three OPCM images shown at the bottom of Figure 6 at y=0μm showing the relevant half of the images (right)
where the gold NP cluster is located. At resonance, the optical contrast of the gold NP peak is ~2.24
times larger than the one at no resonance and 24.79 times larger than the
background optical contrast corresponding to the case when there are no
nanoparticles. 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. Figure 8
visualizes the optical contrast due to the gold nanoparticle cluster as a
function of the phase offsets between the reference (R) and the diffracted (D)
beams of the optical phase microscope. It shows that the enhancement of the  optical contrast due to the nanoparticle resonance changes significantly from a
minimum of 0.0 (Ψ=0°) to a maximum of 3.60 (Ψ=−150°).Comparison
of the geometrical cross sections aty=0μm of the three OPCM images corresponding to 180° phase delay between the reference and the
diffracted beam (bottom row in Figure 6) in terms of optical contrast. The right-hand side graph
illustrates the optical contrast enhancement due to the effect of the gold NP
resonance at λ=543nm.
(a)(b)Figure 8
Optical contrast due to the gold nanoparticle cluster
as a function of the phase offsets between the reference (R) and the diffracted
(D) beams of the optical phase microscope.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, that is,dx=dy=dz=10 nm. This was done by using the built-in features of our FDTD simulation
software [14] enabling the possibility for using both nonuniform
meshing in the entire computational domain and local hard coding of the
resolution of the NPs. The results presented in Figures 6–8 clearly
demonstrate the capability of the proposed FDTD OPCM approach to model the
visual effect of gold NPs at nonresonant and resonant conditions. A future
extension of this research will be to study the capability of the model to provide
valuable insights for the application of gold NPs in optical
nanotherapeutics.


## 4.1. Cell Structure

The cell is modeled as a dielectric sphere with a realistic
radiusRc=5μm. It is critically important to pay
special attention to the membrane model since assigning the membrane the
precisely desired thickness value leads to inadequate modeling of optical
immersion experiments. This is due to the fact that the FDTD technique averages
out the refractive index values at the interface of two different materials
effectively reducing the membrane thickness value and eventually, due to the
staircase approximation error, destroying the continuity of the membrane. This
is the reason for us to select a membrane thickness d=20 nm which corresponds to effective thickness of
approximately 10 nm. The cell nucleus is also spherical with a radius Rn=1.5μm centered at a position which is 2.0μm shifted
from the cell center in a direction perpendicular to the direction of light 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 extracellular material next=1.33 (no refractive index matching) or 1.36
(refractive index matching).


## 4.2. Optical Properties of Gold Nanoparticles

Gold NPs exhibit theability to resonantly
scatter visible and near infrared light due to the excitation of surface plasmonresonances (SPRs). The scattering ability is extremely sensitive to their
size, shape, andaggregation state offering a great potential for optical
cellular imaging and detection labeling studies [15–19]. Our FDTD
approach uses the dispersion model for gold derived from the experimental data
provided by Johnson and Christy [7, 20] where the total, complex-valued
permittivity is given as (7)ε(ω)=εREAL+εL(ω)+εP(ω). 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: (8)εL(ω)=εLORENTZω02ω02−2iδ0ω−ω2,εP(ω)=ωP2iωνC+ω2, where all material
constants are summarized in Table 1.Table 1
Optical material constants of gold [20].
Background permittivityLorentz dispersionPlasma dispersionεREAL=7.077εLORENTZ=2.323ωP=1.391×1016 Hzω0=4.635×1015 HzνC=1.411×107 Hzδ0=9.267×1014 HzWe have modeled two different cases—resonant and
nonresonant. 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 clusters of NPs—a scenario which,
to the best of our knowledge, was 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=10 nm. The scattering cross-section is defined as (9)σscat=Pscat(ω)Iinc(ω), where Pscat is the total scattered power and Iinc is the intensity of the incident
light in (W/m2). It was calculated by applying the total field/scattered field FDTD formulation [13] 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.The absorption cross-section is similarly defined as(10)σabs=Pabs(ω)Iinc(ω), where
Pabs
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 (11)σext=σscat+σabs. Figure 4 shows that the extinction
cross-section has a maximum of 3.89 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 nonresonant case (extinction cross-section value 0.322, ~12
times smaller than 3.89). 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 finite
mesh size.Figure 4
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 (7) and (8) and the parameters given in Table 1.To consistency of the results could be improved by reducing
the mesh size. In
the FDTD simulations including both the cell and the nanoparticles, the
numerical resolution of the nanoparticles was hard-coded todx=dy=dz=10 nm to make sure that their numerically manifested optical resonant
properties will be the same as the ones shown in Figure 4. However, Figure 5
visualizes the positioning of a cluster of 42 nanoparticles in the cytoplasm
that was used to produce the FDTD simulation results presented in
Section 5. It was
randomly generated without controlling the possibility for spatial overlaps or
contacts between the nanoparticles. The closer visual inspection showed that
there were no such overlaps and contacts.Figure 5
A cluster of 42 gold nanoparticles randomly
distributed in a spatial sphere with a radius of 1μm at a cell
location opposite to the location of the nucleus (x=2μm, y=0,z=0). The nucleus
is located at x=−2μm. The gold
nanoparticle size (diameter  =  50 nm) on right-hand graph is slightly exaggerated.


## 4.3. OPCM Image Simulation Results

The simulation results presented here are based on the
3D FDTD OPCM model. The simulations use nonuniform 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 [14]. The
realistic cell dimensions (including both cell radius and membrane) require a
very fine numerical resolution making the simulations computationally
intensive. The locally refined resolution of the nanoparticles leads to
additional requirements for the CPU time and memory (~120 Gbs RAM) requiring
high performance computing resources. The time step used during the simulation
was defined by means of a factor of 0.99 in the Courant stability limit: cΔt=0.99×(1/Δx2+1/Δy2+1/Δz2)−1/2.
The cell center is located in the middle (x=y=z=0) of the computational domain with dimensions 15μm×12μm×15μm. The nucleus’ center is located at x=−2μm,y=z=0μm. The cluster of gold nanoparticles
is located at x=2μm,y=z=0μm (Figure 5).The refractive
index matching (RIM) between the cytoplasm and the extracellular medium 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 phase contrast at optical immersion conditions. 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 the cluster of 42 gold NPs (Figure5, see also Figure 1(b))
and for different phase offsets between the reference beam and the scattered
beam (assuming a=1). The insertion of the nanoparticles increases significantly the CPU and
memory requirements. The dimensions of the computational domain are 15μm×15μm×12μm. The OPCM images in Figure 6 show that
the gold NP cluster can be clearly identified at both the nonresonant (λ=676.4 nm) and resonant (λ=543.0 nm) wavelengths.OPCM images of a biological cell for different values  (a:−150°, b: −90°, c: 90°, d: 180°) of the phase offset between the reference and diffracted beam of the OPCM at optical immersion conditions (left) including a cluster of 42 gold NPs (middle: no resonance, right: at resonance) located in a position symmetrically opposite to the nucleus. The arrows indicate the position of the gold NP cluster.
(a)(b)(c)(d)The graphs in Figure7 compare the geometrical
cross-sections of the three OPCM images shown at the bottom of Figure 6 at y=0μm showing the relevant half of the images (right)
where the gold NP cluster is located. At resonance, the optical contrast of the gold NP peak is ~2.24
times larger than the one at no resonance and 24.79 times larger than the
background optical contrast corresponding to the case when there are no
nanoparticles. 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. Figure 8
visualizes the optical contrast due to the gold nanoparticle cluster as a
function of the phase offsets between the reference (R) and the diffracted (D)
beams of the optical phase microscope. It shows that the enhancement of the  optical contrast due to the nanoparticle resonance changes significantly from a
minimum of 0.0 (Ψ=0°) to a maximum of 3.60 (Ψ=−150°).Comparison
of the geometrical cross sections aty=0μm of the three OPCM images corresponding to 180° phase delay between the reference and the
diffracted beam (bottom row in Figure 6) in terms of optical contrast. The right-hand side graph
illustrates the optical contrast enhancement due to the effect of the gold NP
resonance at λ=543nm.
(a)(b)Figure 8
Optical contrast due to the gold nanoparticle cluster
as a function of the phase offsets between the reference (R) and the diffracted
(D) beams of the optical phase microscope.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, that is,dx=dy=dz=10 nm. This was done by using the built-in features of our FDTD simulation
software [14] enabling the possibility for using both nonuniform
meshing in the entire computational domain and local hard coding of the
resolution of the NPs. The results presented in Figures 6–8 clearly
demonstrate the capability of the proposed FDTD OPCM approach to model the
visual effect of gold NPs at nonresonant and resonant conditions. A future
extension of this research will be to study the capability of the model to provide
valuable insights for the application of gold NPs in optical
nanotherapeutics.


## 5. Conclusions

In this article, we provided simulation results on the application of the FDTD approach to the modeling of the OPCM imaging of gold NPs in singe biological cells. We have first reproduced the effect of optical immersion on the OPCM images of a realistic size cell containing a cytoplasm, a nucleus, and a membrane. The model was then extended 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 optical resonance of the nanoparticles. The results do not allow analyzing the scaling of the NP imaging effect as a function of the number of the NPs in the cluster. However, the validation of the model provides a basis for future studies of OPCM nanobioimaging including the effects of NP cluster size, NP size and number, as well as average dimension between the NPs. To the best of our knowledge, this is the first time that FDTD simulation results were post-processed in a way providing realistic OPCM images of biological cells including a cluster of gold NPs. The results demonstrate the potential of the FDTD approach to model advanced optical nanobioimaging instruments.



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*Source: 727418-2008-12-31.xml*