\section{Introduction}

Eye tumors require dedicated radiation techniques. These techniques are demanding owing to the close connection of the tumor to sensitive structures and also because of the small size of the target volume, which represents a challenge from the dosimetry point of view. Applicators using $^{106}$Ru/$^{106}$Rh \cite{Schueler06a} and external beam radiotherapy (EBRT) with either protons \cite{Mourtada,Woong}, electrons \cite{Brualla11b, Brualla09b} or photons \cite{Sauerwein} are among the established techniques. Common to all these modalities is the necessity of an accurate determination of the dose distribution inside the eye and in the surrounding structures and tissues \cite{Fluhs, ICRU, Schueler06b}. In this respect, Monte Carlo (MC) simulation has permitted a significant advance in the last years, providing an accurate description of the dose imparted to these small-sized irradiated volumes \cite{Chiu, Miras, Thomson08, Brualla12c}.

Retinoblastoma (Rb), the most common intraocular malignancy in the early childhood, responds very well to radiotherapy. Rb was treated with first line EBRT as the treatment of choice with a total dose from 40--50 Gy \cite{Messmer} resulting in a close to 90\% eye preservation in group I--IV (Reese-Ellsworth classification \cite{Reese}). There are two different types of Rb: a hereditary and a non-hereditary form. Both forms are induced by the biallelic inactivation of the RB1 tumor suppressor gene. The non-hereditary form results in a solitary tumor in one eye. For the hereditary variant the first mutation of the RB1 allele is inherited following the somatic inactivation of the second allele resulting in multiple tumor lesions in both eyes. The genotype in hereditary Rb patients predisposes the appearance of secondary malignancies, with radiotherapy further enhancing the risk of tumors occurring in the irradiated area. Treatments with EBRT have showed very good results in the control of primary tumors as well as in preserving the visual function \cite{Sauerwein}. However, during the last 15 years the trend has been to move away from radiotherapy mainly with the goal to reduce radiation induced secondary malignancies \cite{Abramson, Dommering, Turaka, Vasudevan, Rodjan}, but also in view of the devastating side effects originated by inappropriate irradiation techniques leading to deformation of the bones of the midface \cite{Imhof, Peylan}. This tendency has favored the use of other techniques in young children \cite{Gallie, Gobin, Munier, Schueler06a, Schueler06b, Shields96, Temming}. Most of them involve chemotherapy giving rise to damages in the retina, a reduction of the visual acuity, an increase of the toxicity on other organs \cite{Qaddoumi}, and an increase of the incidence of leukaemia and solid tumors \cite{Felix, Gombos}. In case of failure of these therapies, EBRT remains a salvage option, but adding further local damage and augmenting the risk of secondary malignancies.

In these circumstances EBRT can be reconsidered as first line treatment of Rb using dedicated treatment techniques that avoid asymmetric growth of the face while protecting sensitive structures of the eye. In this article we examine the possibility to further improve the lens sparing irradiation technique, which was initially proposed by Schipper \cite{Schipper83, Schipper97}, by reducing the irradiated volume and limiting further the penumbra.

In a previous work \cite{Brualla12a} we assessed by MC simulation the absorbed dose distribution in a water phantom obtained with the Shipper's technique \cite{Sauerwein09} as it is currently applied at the University Hospital of Essen. The technique consists of an EBRT irradiation that uses a dedicated `D'-shaped collimator whose purpose is to minimize the absorbed dose to normal tissues, hence reducing the incidence of radio-induced secondary tumors. This collimator can conform two field sizes depending whether an additional brass insert is employed or not. The large field (5.2~cm$^2$) is conformed without the brass insert and it is usually employed in children older than 6 months. For the small field (3.1~cm$^2$), employed in children younger than 6 months old, the brass insert is required. In the present article we investigate, by using the MC code {\sc penelope} \cite{PENELOPE}, the role played by the various pieces conforming the collimation system, paying attention to both the shape of its constructive elements and the materials conforming them. The ultimate goal is to propose a modified collimator that even reduces further the field size of the large field conformed without the brass insert, while maintaining the shape of the penumbrae and guaranteeing at least 95\% of the maximum dose along 2.4~cm of a centered lateral profile in the $x$ direction (corresponding to the inferior-superior direction of the eyeball) measured at a depth of 2~cm in a water phantom. The scope of reducing the field size is to pursue a lower incidence of radio-induced secondary tumors by reducing the volume of irradiated normal tissue in children older than 6 months. The aim of mantaining the characteristics of the dose distribution, while only reducing the field size, is to be able to continue applying the clinical experience gained in nearly 30 years of daily application of this irradiation technique at Essen. With respect to the collimator with the brass insert in use (small field), it was found that the irradiated volume is adequate for children younger than 6 months old and, therefore, no modification is proposed.

The studies conducting to the currently used version of the modified Shipper's technique were based on experiments and calculations done on water phantoms. Adequate computerized tomograpies of young children are rare and we do not have any. Thus, the actual patient's geometry is not available to MC treatment planning programs for its simulation. Treatment planning must resort to results obtained in water phantoms. For the design of the proposed collimator in this article we have compared our absorbed dose distributions in water phantoms, obtained with MC simulations, with results from the currently used collimator also obtained with MC simulations in water phantoms.

About 60\% of the Rb patients irradiated at Essen have one eye enucleated at the time of treatment with EBRT. These patients are irradiated with a single field. Non-hereditary Rb patients with a single tumor in one eye are also irradiated with a single field. Only patients with tumors in both eyes at the time of radiotherapy are irradiated with two opposed fields. The modified collimator proposed herein has been optimized for a single field irradiation, so to serve to the majority of the treated Rb patients.

\section{Material and methods}

\subsection{`D'-shaped collimation system currently in use}

In this work we have studied the dosimetry of a `D'-shaped collimation system used for Rb treatment when various changes in the shape of its constructive elements and in the material composition of the collimator are considered. The Rb collimator is designed to be inserted in the accessory tray holder of a Varian Clinac~2100~C/D operating at 6~MV. Recently \cite{Brualla12a}, the dosimetry of this system was analyzed by comparing results obtained in MC simulations with {\sc penelope} \cite{PENELOPE}, experimental measurements in a water phantom and the predictions of the analytical anisotropic algorithm \cite{Sievinen, Ulmer95, Ulmer03} implemented in the Eclipse (Varian Medical Systems, Palo Alto, California) treatment planning system.

Figure~\ref{fig:G0} shows the collimation system currently in use. It includes a primary and a secondary Rb collimator made of Cerrobend plus and optional brass insert that permits to reduce the radiation field maintaining its `D'-shape. Thus, patients can be treated with either $5.2$ cm$^2$ or $3.1$ cm$^2$ fields. Usually, patients younger than 6 months old are treated with the smaller field. The aluminum plate (figure~\ref{fig:G0}) permits to insert the collimator in the accessory tray holder of the linac. The beam central axis defines the direction of increasing $z$.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{fff1.pdf}
\caption{Geometry G$_0$ corresponding to the dedicated `D'-shaped collimator studied in the present work. Dashed lines indicate the cuts at which the insets have been taken. Detailed blueprints of this collimator can be found in \cite{Brualla12a}.
\label{fig:G0}}
\end{figure}

The coordinate system plotted in this figure is the one we have used for all calculations. The origin of coordinates is situated at $100$ cm from the isocenter upstream in the $z$ axis. The $x$-jaws of the linac define a symmetric field of $5.5$~cm width about the $y$ axis. The $y$-jaws in the negative and positive $y$-axis are displaced 0.7 and 3.5 cm, respectively, away from their closed position. A radiation field of $5.5\times4.2$~cm$^2$, centered at $x = 0$ and $y = 1.4$~cm is produced at the isocenter. The movable jaws are situated in the same place independently whether the brass insert is used or not. We called this geometry G$_0$ and we used it as reference to compare the results obtained with the modified geometries that we  studied. Results obtained with and without the brass insert were labeled as G$_0^{\rm w}$ and G$_0^{\rm wo}$, respectively. Depth doses at the center of the field, and $x$ and $y$ lateral profiles for $y = 1.3$~cm and $x = 0$, respectively, at three different depths ($z = 1.5$, $3.0$ and $5.0$~cm) were estimated in~\cite{Brualla12a}.

Figure~\ref{fig:Picture} shows a photograph of the Rb collimator mounted on a Varian Clinac~2100~C/D with its gantry and collimator angles set for irradiating a patient in head-first supine position (i.e., the left eye would be placed upstream). The scope of this figure and the following explanation is to relate the coordinate system used in the simulation with the anatomical directions of the eyeball. For clarity reasons the position of both eyes is related to the coordinate system, although, in the clinical practice most patients have one eye enucleated. The coordinate system defined in figures~\ref{fig:G0} and~\ref{fig:Picture}, relates to the anatomical directions of the eyeball as follows: i) the $x$-axis increases in the inferior-superior direction; ii) the $y$-axis increases in the anterior-posterior direction; iii) the $z$-axis increases in the lateral-nasal direction for the eye upstream and in the nasal-lateral direction for the eye downstream. During the treatment the eyes of the patient are fixed with the vacuum lenses appearing in the photograph mounted on the rail, so that the posterior pole of both eye lenses lie at $x=y=0$, that is, 1~mm away from the flat side of the `D' into the shielded part of the field. Thus, effectively protecting the eye lense and its surrounding structures from damaging radiation.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{fff-pic01.pdf}
\caption{Photograph of the Varian Clinac 2100~C/D set in position for irradiating a Rb patient. The patient is placed on the couch in head-first supine position, thus, his left eye is found upstream. The coordinate system used in the simulations presented herein is shown. The flat wall of the `D' is 1~mm away from the $z$-axis.
\label{fig:Picture}
}
\end{figure}

\subsection{Simulation code}

All simulations were run using the main steering code {\sc penEasy}~\cite{Sempau11} that runs on the general-purpose MC system {\sc penelope} \cite{Baro, PENELOPE, Sempau97}. The geometry of the Varian Clinac~2100~C/D was automatically generated with the code {\sc penEasyLinac}~\cite{Brualla09b, Sempau11,Rodriguez2013}. The calculations with this set of codes were benchmarked specifically for the linac considered in this work operating at 6~MV (which is the nominal energy used in all simulations presented herein) in previous works~\cite{Brualla09b, Brualla09a, Fernandez, Panettieri, Sempau11}

{\sc penelope} simulates the coupled transport of electrons, positrons and photons and makes use of a mixed simulation scheme that classifies electron and positron interactions either as hard or soft events. On the one hand, hard events correspond to interactions in which the angular deflections and/or the energy losses are larger than certain cutoffs; these events are simulated in a detailed way. On the other hand, the soft interactions occurring between two hard events are simulated as a single artificial event described within a multiple scattering theory. The particles are transported while their kinetic energies are larger than the absorption energies defined by the user. In our case, these energies were set to 100~keV for electrons and positrons and 20~keV for photons. Electron and positron transport is controlled by five user defined parameters whose role is described in detail in the {\sc penelope} user's manual \cite{PENELOPE}. In our simulations {\sc penEasyLinac} set C1 = C2 = 0.1, WCC = 100~keV, WCR = 20~keV and DSMAX equals to one tenth of the thickness of the different geometry bodies along the beam path.

The depth dose and lateral dose profiles were scored in a $50 \times 50 \times 50$~cm$^3$ water phantom situated at a source-to-surface distance of 100~cm. The simulations were performed in three steps. In the first one, a phase-space file was tallied at the exit of the linac gantry for each considered configuration of the jaws. In this step the source was simulated as a monoenergetic point-like pencil beam with zero divergence and emitting electrons with an initial kinetic energy of 6.26~MeV. In the second step, the phase-space files scored in the first one were used as sources and particles were transported downstream through the corresponding `D'-shaped dedicated collimator with or without the brass insert. In this step, new phase-space files were tallied at the surface of the water phantom. Finally, absorbed dose distributions in water were obtained by transporting the particles in the phase-space files tallied in the second step into the water phantom. Absorbed dose distributions were tallied using scoring bins of $0.05\times0.05\times0.1$ cm$^3$.

\subsection{Modified geometries}

We are interested in investigating the dosimetric impact of various elements of the dedicated `D'-shape collimation system described in figure~\ref{fig:G0}. To that end, we studied a set of new geometries obtained by modifying the original geometry G$_0$. Three of these modifications are sketched in figure~\ref{fig:Gmod}.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{fff2.pdf}
\caption{Modified geometries studied in the present work.
\label{fig:Gmod}
}
\end{figure}

With the first one, G$_1$, we have studied the role played by the primary collimator. As shown in figure~\ref{fig:Gmod}, in this geometry the primary collimator was eliminated, while the secondary collimator and the brass insert were kept inaltered. Simulations with and without the brass insert were performed and, as in the case of the G$_0$ geometry, we labeled the corresponding calculations as G$_1^{\rm w}$ and G$_1^{\rm wo}$, respectively.

The next group of geometries is related to the material and shape of the brass insert. First, we have assumed that this insert is made of Cerrobend (as the primary and the secondary Rb collimators are). The corresponding geometry was labeled G$_{21}$. In a second modification, the shape of the brass insert was changed to match exactly the length of the secondary collimator. We labeled G$_{22}$ this new geometry. The remaining elements of the collimator were maintained as in the G$_{0}$ geometry.

In these three modified geometries, the jaws were situated as described for the G$_{0}$ geometry. The radiation field defined by the jaws at the isocenter ($5.5\times 4.2$ cm$^2$, centered at $x = 0$ and $y = 1.4$ cm) is considerably larger than the field delimited by the Rb collimator which is, as said above, of 3.1 or $5.2$ cm$^2$ if the brass insert is included or not. In order to analyze how the jaw position can affect the final dose distribution, we considered a modified linac configuration in which the field delimited by the jaws was reduced to circumscribe the field defined by the Rb collimator, with and without the brass insert. Though the linac jaws are focalized to the target, the Rb collimator is not (i.e., its inner walls are parallel to the $z$-axis) and then two different positions of the jaws were analyzed where the circumscribed field corresponds to the fields delimited by the projections to the isocenter of the upstream and the downstream ends of the secondary Rb collimator. We labeled these geometries as G$_{\rm U}$ and G$_{\rm D}$, respectively. We simulated the geometries including or not the brass insert, without modifying the geometry of the Rb collimator.

The changes in the jaw positions produce a modification of the fluence and the dose delivered to the water phantom changes. To compare the dose distributions obtained with the modified geometries they were scaled so that all the maxima of the different dose distributions match that obtained with the G$_0$ geometry. The dose distributions obtained for the geometries G$_{\rm U}^{\rm wo}$, G$_{\rm U}^{\rm w}$, G$_{\rm D}^{\rm wo}$ and G$_{\rm D}^{\rm w}$ were multiplied by 1.010, 1.014, 1.018 and 1.023, respectively.

The MC standard statistical relative uncertainties for all bins with a dose larger than 50\% than the maximum dose was below 0.1\% in average. In the figures shown below, these uncertainties are smaller than the symbols used to represent the results obtained with the G$_0$ geometry.

\subsection{Profile analysis}

In order to analyze the various profiles calculated in the simulations, we have determined the field size and the penumbra of each one. Figure \ref{fig:defi} sketches the definitions of both quantities.

\begin{figure}[!b]
\centering
\includegraphics[width=8cm]{fff-defi.pdf}
\caption{Scheme indicating the definitions of the field size (at 95\% of the maximum dose) and the penumbra used in the present work.
\label{fig:defi}
}
\end{figure}

The field size is defined as the distance between the two positions having 95\% of the maximum dose of the profile. The level of 95\% is that required in clinics to fix the treatments. We require from the proposed collimator intended to substitute the current collimator without brass insert to maintain at least 95\% of the maximum dose along 2.4~cm of a centered lateral profile in the $x$ direction (corresponding to the inferior-superior direction of the eyeball) measured at a depth of 2~cm in a water phantom.

To obtain the penumbrae, the following procedure was carried out. First we fitted a straight line to the data around the 50\% level of the maximum dose. The penumbra was defined as the distance between the positions having 20\% and 80\% of the maximum dose, measured on the fitted line. We checked that the penumbra was the same on both sides of the $x$ profiles.

\section{Results and discussion}

For the various modified geometries analyzed, depth dose and lateral profiles are compared to those found with the G$_0$ geometry. The results obtained with this geometry are shown with solid circles while those corresponding to the modified geometries are plotted as colored curves. In each case, the differences between the doses obtained with the modified geometries and the original one are also shown as dotted curves. The depth dose distributions were obtained at $x = 0$ and $y = 1.3$~cm. The $x$ and $y$ lateral profiles have been obtained at $y = 1.3$~cm and $x = 0$, respectively, and for depths $z = 1.5$ (red), 3.0 (green) and 5.0 (blue)~cm.

In figure \ref{fig:G1} the results found with the geometry G$_{1}$ are shown. This geometry permits to investigate the role played by the primary Rb collimator (see figure~\ref{fig:Gmod}). The differences between the results obtained G$_{0}$ and those with G$_{1}$ are very small, independently of the fact that the brass insert is included or not. As it can be seen in the medium and lower right panels, if the brass insert is used, the elimination of the primary Rb collimator does not produce any effect. If the brass insert is absent, a slight modification of the lateral profiles occurs. The changes are limited to the penumbra regions where the dose found for G$_{1}$ is greater than that corresponding to G$_{0}$ . This increased dose is very small in the region where the eye lens is situated ($y < 0$) and a little bit larger in the posterior pole of the eye ($y\geq 3$ cm). In any case, these results indicate that the Rb primary collimator must be kept if the brass insert is not used, in order to maintain the dose distribution found with the G$_{0}$ geometry.

\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{fff-G1.pdf}
\caption{Depth doses (upper panels) and $x$ (medium panels) and $y$ (lower panels) lateral profiles obtained with the geometry G$_{1}$ without (left panels) and with (right panels) the brass insert, represented with curves, are compared to those found with the reference geometry G$_0$ (points). The dotted curves are the differences $D_{{\rm G}_1}-D_{{\rm G}_0}$ (right-ordinate axes).
\label{fig:G1}
}
\end{figure}

Figure \ref{fig:G2} shows the comparison of the G$_{21}$ (left panels) and G$_{22}$ (right panels) results to those found with the reference geometry. As indicated in figure \ref{fig:Gmod}, in the G$_{21}$ geometry, the material of the insert has been changed from brass to Cerrobend. This change produces a small variation of the penumbrae: the G$_{21}$ dose reduces with respect to the G$_0^{\rm w}$ one and this occurs only where the insert is present ($x \sim \pm1.4$~cm in the medium panel and $y \sim 2.2$~cm in the lower one). This means a dose reduction in the posterior pole of the eye without additional modifications of the dose absorbed at the eye lens. No relevant changes are observed in the depth dose distributions (upper panels). Also, as expected, there are no effects in the penumbrae of the $y$-lateral profiles (lower panels) at $y \sim 0$ (see figure \ref{fig:G0}) that remains as in G$_0^{\rm w}$. In geometry G$_{22}$, where the insert shape is modified to match the secondary Rb collimator (see figure \ref{fig:Gmod}), the effect seen in G$_{21}$ is largely compensated and the modifications in the penumbrae disappear to a large extent.

\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{fff-G2.pdf}
\caption{Depth doses (upper panels) and $x$ (medium panels) and $y$ (lower panels) lateral profiles obtained with the geometries G$_{21}$ (left panels) and G$_{22}$ (right panels), represented with curves, are compared to those found with the reference geometry G$_0^{\rm w}$ (points). The dotted curves are the differences $D_{{\rm G}_{2\alpha}}-D_{{\rm G}_0^{\rm w}}$ (right-ordinate axes).
\label{fig:G2}
}
\end{figure}

The modifications produced on the absorbed dose distribution due to changes in the position of the jaws are shown in figures \ref{fig:G3U} and \ref{fig:G3D}. In the first one we have plotted the results corresponding to the geometry G$_{\rm U}$ in which the jaws were moved to circumscribe the field delimited by the upstream end of the secondary Rb collimator. The modifications on the depth dose distributions (upper panels) owing to the change of the jaws position is rather reduced independently of the presence of the brass insert. The modifications on the lateral profiles due to the aforementioned displacement of the jaws when the brass insert is present are also small and restricted to the penumbrae (medium and lower right panels). A slightly larger modification, also restricted to the penumbrae, is found when the insert is not included in the geometry. Therein, the doses found for G$_0$ are smaller than those obtained with G$_{\rm U}$ and a roughly 10\% increase of the dose at the posterior pole of the eye occurs.

\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{fff-G3U.pdf}
\caption{Depth doses (upper panels), and $x$ (medium panels) and $y$ (lower panels) lateral profiles obtained with the geometry G$_{\rm U}$ without (left panels) and with (right panels) the brass insert, represented with curves, are compared to those found with the reference geometry G$_0$ (points).  The dotted curves are the differences $D_{{\rm G}_{\rm U}}-D_{{\rm G}_0}$ (right-ordinate axes). \label{fig:G3U}
}
\end{figure}

\begin{figure}[!th]
\centering
\includegraphics[width=8cm]{fff-G3D.pdf}
\caption{Depth doses (upper panels), and $x$ (medium panels) and $y$ (lower panels) lateral profiles obtained with the geometry G$_{\rm D}$ without (left panels) and with (right panels) the brass insert, represented with curves, are compared to those found with the reference geometry G$_0$ (points). The dotted curves are the differences $D_{{\rm G}_{\rm D}}-D_{{\rm G}_0}$ (right-ordinate axes). \label{fig:G3D}
}
\end{figure}

\begin{figure}[!th]
\centering
\includegraphics[width=8cm]{fff-GD-PDD.pdf}
\caption{Depth doses obtained at $x = 0$ and $y = 0.1$, $0.3$ and $1.3$~cm with the geometry G$_{\rm D}$ without (left panels) and with (right panels) the brass insert, represented with curves, are compared to those found with the reference geometry G$_0$ (points).
\label{fig:G3D-PDD}
}
\end{figure}

A different behavior is observed in the case of the G$_{\rm D}$ geometry (see figure \ref{fig:G3D}) in which the jaws were moved to circumscribe the field delimited by the downstream end of the secondary Rb collimator. The changes produced by the geometry modification are not relevant in the depth doses (upper panels). In G$_{\rm D}^{\rm wo}$, when the brass insert is not present, the effects of the modification in the jaws position on the lateral profiles (medium and lower left panel) change sign with respect to those found for G$_{\rm U}^{\rm wo}$ geometry (see figure \ref{fig:G3U}). These effects are of the same order for the $x$ profiles (medium panel) but appear to be largely increased in the case of the $y$ profiles. In fact, a reduction of $\sim 40$\% in the dose absorbed at the posterior pole is observed. Similar results are found when the brass insert is included (medium and lower right panels) but in this case the sign of the modifications does not change with respect to G$_{\rm U}^{\rm w}$.

In the case of the geometry G$_{\rm D}$, we have also investigated the effect that has on the depth dose profiles the fact of closening the jaws. In figure \ref{fig:G3D-PDD} we show these profiles, obtained for $x = 0$ and three different values of $y$: $1.3$~cm (which coincides with that shown in the upper panels of figure~\ref{fig:G3D}), $0.3$~cm and $0.1$~cm (this last very close to the end position of the field delimited by the jaws). We see that there is not a great effect due to the jaw situation. Only in the case in which the brass insert is present (right panel), for $y = 0.1$~cm, a small reduction in the maximum of the curves ($\sim 2$\%) is apparent.

Penumbrae are a relevant aspect of the dose profiles. Table~\ref{tab:penun} shows the values found for the different geometries studied at two depths in water, with and without the brass insert. In general, the increase of the depth produces slightly larger penumbrae. At $z = 3.0$~cm they are between 0.03 and 0.05~cm greater than those found at $z = 1.5$~cm. It is worth pointing out that the presence of the brass insert produces small modifications in the penumbra below 0.02~cm. The values obtained for the geometries including the brass insert are all statistically compatible with that obtained with the geometry G$_{0}^{\rm w}$. When the brass insert is absent greater differences occur at $z = 3.0$~cm among the values obtained for some of the geometries (see table~\ref{tab:penun}).

\begin{table}[!b]
\caption{Penumbrae (in cm) of the $x$ profiles for two depths in water ($z=1.5$ and $3$~cm) for all geometries studied in the present work, including (w) or not (wo) the brass insert. Standard statistical uncertainties are 0.01~cm.
\label{tab:penun}}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
&\multicolumn{2}{c}{wo} &~~~&\multicolumn{2}{c}{w} \\
\cline{2-3}\cline{5-6}
$z$ (cm) & 1.5 & 3.0 && 1.5 & 3.0 \\\hline
G$_0$ & 0.31 & 0.34 && 0.32 & 0.36 \\
G$_1$ & 0.32 & 0.37 && 0.32 & 0.37 \\
G$_{21}$ & & && 0.31 & 0.35 \\
G$_{22}$ & & && 0.31 & 0.35 \\
G$_{\rm U}$ & 0.32 & 0.36 && 0.32 & 0.37 \\
G$_{\rm D}$ & 0.30 & 0.33 && 0.31	 & 0.35 \\ \hline
G$_{\rm opt}$ & 0.30 & 0.33 && & \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[!bh]
\caption{Field sizes (in cm) of the $x$ profiles for two depths in water ($z=1.5$ and 3~cm) for all the geometries studied in the present work, including (w) or not (wo) the brass insert. Standard statistical uncertainties are 0.04~cm.
\label{tab:field}}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
&\multicolumn{2}{c}{wo} &~~~&\multicolumn{2}{c}{w} \\
\cline{2-3}\cline{5-6}
$z$ (cm) & 1.5 & 3.0 && 1.5 & 3.0 \\\hline
G$_0$ & 2.75 & 2.70 && 1.65 & 1.65 \\
G$_1$ & 2.75 & 2.75 && 1.65 & 1.65 \\
G$_{21}$ & & && 1.65 & 1.65 \\
G$_{22}$ & & && 1.65 & 1.65 \\
G$_{\rm U}$ & 2.75 & 2.75 && 1.65 & 1.65 \\
G$_{\rm D}$ & 2.70 & 2.70 && 1.65 & 1.65 \\ \hline
G$_{\rm opt}$ & 2.45 & 2.45 && & \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

An important point to be considered is the size of the radiation fields. Presently, the brass insert providing the smaller field is used to treat children younger than 6 months old. This field is, however, too small for older patients for whom a field of width 2.4~cm in the $x$-axis direction (inferior-superior direction in the eyeball) measured at 2~cm depth with 95\% of the maximum dose would suffice for an adequate treatment, even taking into account a safety margin of about 0.2~cm. Table~\ref{tab:field} summarizes the field widths obtained in our simulations at two of the depths analyzed, namely 1.5 and 3.0~cm. The field width along the $x$-axis does not change too much neither with the specific geometry considered nor with the depth in the water phantom. It is 1.65 cm when the brass insert is present and ranges from 2.70 to 2.75~cm without it. Our scope is to modify the geometry without the brass insert so to reduce its field width along the $x$-axis from 2.7~cm to 2.4~cm.

\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{fff3.pdf}
\caption{Optimal geometry G$_{\rm opt}$ proposed in the present work. The primary collimator has been eliminated and the radius of the cylindrical surface limiting the curved part of the `D' has been reduced from 1.485~cm in G$_0^{\rm wo}$ to 1.365~cm. In addition, the jaws are closed as in the G$_{\rm D}$ geometry, so that they circumscribe the field delimited by the projections to the isocenter of the downstream end of the G$_{\rm opt}$ collimator.
\label{fig:Geom-opt}
}
\end{figure}

\begin{figure}[!b]
\centering
\includegraphics[width=8cm]{fff-Gopt.pdf}
\caption{Depth doses (upper panels) and $x$ (medium panels) and $y$ (lower panels) lateral profiles obtained with the geometry G$_{\rm opt}$, represented with curves, are compared to those found with the reference geometry G$_0$ without (left panels) and with (right panels) the brass insert (points). The dotted curves, plotted together with the depth dose profiles, are the differences $D_{{\rm G}_{\rm opt}}-D_{{\rm G}_0}$ (right-ordinate axes).
\label{fig:Gopt}
}
\end{figure}

To ensure the therapeutic requirement about the field width we considered a new optimized geometry that we labeled G$_{\rm opt}$, shown in figure~\ref{fig:Geom-opt}. It includes the modifications of the geometries G$_1$ (in which the primary collimator was eliminated) and G$_{22}$ (in which the insert was modified to match the secondary Rb collimator). The radius of the cylindrical surface defining the curved part of the `D'-shaped collimator is 1.485~cm for the geometry G$_0^{\rm wo}$ and 1.085~cm for G$_0^{\rm w}$. For the proposed G$_{\rm opt}$ this radius is 1.365~cm. As in G$_{22}$ the collimator is made of Cerrobend (i.e., there is no brass). In addition, the jaws are positioned with the same scope followed in the G$_{\rm D}$ geometry, that is moving them in such a way that they circumscribe the field delimited by the projection to the isocenter of the downstream end of the secondary Rb collimator.

In figure~\ref{fig:Gopt}, the results found with G$_{\rm opt}$ are compared to those obtained with G$_0^{\rm wo}$ (left panels) and G$_0^{\rm w}$ (right panels). To adequately perform the comparison shown in figure~\ref{fig:Gopt} the dose distribution obtained for the G$_{\rm opt}$ was multiplied by 1.024 and 1.009 when comparing with the G$_0^{\rm wo}$ and the G$_0^{\rm w}$ results, respectively.

The upper panels of figure~\ref{fig:Gopt} show the depth dose distributions. The differences with the G$_0^{\rm wo}$ results are very small, except very close to the surface ($z\sim0$). However, these differences are more apparent in the case the brass insert is included. The expected reduction/increasing of the field size with respect to those defined by G$_0^{\rm wo}$ and G$_0^{\rm w}$ geometries is apparent in the medium panels. As we can see in table \ref{tab:field}, the field goes, respectively, from $\sim 2.7$ cm and $\sim 1.6$ cm to the $\sim 2.4$ cm in G$_{\rm opt}$. The large modification that this optimal geometry implies with respect to G$_0^{\rm w}$ is responsible for the differences observed in the case of the depth dose distributions. Also $y$-profiles (see lower panels) show the modification in the field size.

Penumbrae (see table \ref{tab:penun}) do not change with respect to the other geometries and the variations with the depth in water are moderate. The field size does not change with $z$ and it maintains the required value of $\sim 2.4$~cm in all the range of depths analyzed.

\section{Conclusion}

In this work, the dosimetric impact produced by different geometry modifications introduced in the original design of the `D'-shaped collimation system currently in use for Rb treatments have been investigated. The changes affect the primary and secondary collimators of this system as well as the situation of the linac jaws delimiting the radiation field at the entrance of the Rb collimation system.

The various geometries analyzed have a rather small impact on the depth dose distributions at the beam center. The changes produced in both $x$ and $y$ lateral profiles are limited to the penumbrae and result to be relatively important only in the case of the G$_{\rm D}$ geometry which corresponds to a modification in the jaw position in order to delimit the radiation field that they define to that conformed by the downstream end of the secondary Rb collimator.

An optimal geometry G$_{\rm opt}$ has been proposed; it produces a field of width $2.4$~cm along $x$-axis, corresponding to the inferior-superior direction of the eyeball, at $2$~cm depth with 95\% of the maximum dose. With this new design an important reduction of the irradiated volume is obtained, without reducing the dose to the target volume of the patients that are currently treated with the G$_0^{\rm wo}$ collimator.

\acknowledgments{The work of PAM and AML has been supported in part by the Junta de
Andaluc\'{\i}a (FQM0220) and FEDER and the Spanish Ministerio de Ciencia e
Innovaci\'{o}n and FEDER (project FPA2009-14091-C02-02) and Ministerio de Econom\'{\i}a y Competitividad (project FPA2012-31993). LB is grateful to the Deutsche Forschungsgemeinschaft (project BR~4043/1-1).}


\section{Introduction}

Since its original systematic study 190 years ago by Robert Brown \cite{Brown},
diffusion of molecular and (sub-)micron-sized entities has been identified as
the dominant form of thermally driven, passive transport in numerous biological
and inanimate systems. The two hallmark features of diffusion is the linear
growth $\langle\mathbf{r}^2(t)\rangle=2Ddt$ of the mean squared displacement
(MSD) with diffusion coefficient $D$ in $d$ spatial dimensions, and the Gaussian
distribution of displacements \cite{vankampen}. With increasing complexity of
the studied systems deviations from these two central properties have been
unveiled over the years. Thus, anomalous diffusion with an MSD of the form
$\langle\mathbf{r}^2(t)\rangle\simeq t^{\alpha}$ was observed in a large
range of systems \cite{hoefling,lene_rev}. Along with such observations a rich
variety of generalised stochastic processes has been developed \cite{bouchaud,
pccp}. The displacement distribution of anomalous diffusion processes may be
inherently Gaussian (such as for fractional Brownian motion \cite{fBM}) or
non-Gaussian (for instance, for processes characterised by scale-free trapping
time distributions \cite{scher} or space-dependent diffusivity models
\cite{andrey}).

Recently a lage variety of systems have been reported in which the MSD exhibits
the linear growth in time $\langle\mathbf{r}^2(t)\rangle\simeq Dt$ of Brownian
(Fickian) transport, however, the distribution of displacements $P(\mathbf{r},t)$
is pronouncedly non-Gaussian \cite{Wang:BYNG2}. Pertinent examples include the
motion of tracer beads along tubular or membrane structures or in gels and
colloidal suspensions \cite{Wang:BYNG2,Wang:BYNG3,Goldstein:BYNG9}, and
the motion of nematodes \cite{hapca}
and single cells on substrates \cite{beta}. As long as the displacement
distribution $P(\mathbf{r},t)$ has a fixed shape for any times $t$, one possible way
to model the non-Gaussianity is the concept of superstatistics \cite{Superst_1,
Superst_2} which introduces a distribution $p_D(D)$ of the diffusion coefficient
and then averages individual Gaussian distributions with one given $D$ value
over this $p_D(D)$. However, this approach does not work when eventually a
crossover to an effective Gaussian is observed \cite{Wang:BYNG2,Wang:BYNG3}. For
the latter case Chubynsky and Slater introduced the diffusing diffusivity (DD) model
\cite{Chubynsky:DD1}, see also \cite{Chubynsky_new:DD1,Sebastian:DD3,
Sebastian:DD3_bis,tyagi:DD5,Chechkin:DD2,Sposini:DD2,Grebenkov:DD4}: In this
popular approach the diffusion coefficient is assumed to be a stochastic variable
itself, described by a stationary process. Consequently the system is initially
described by a non-Gaussian displacement distribution. Beyond a characteristic
time scale a crossover occurs to a Gaussian behaviour characterised by an effective
value of the diffusivity.

We here study the first passage behaviour of the DD model.
The concept of first passage is ubiquitously used in statistical physics and
its applications, for instance, to quantify when a diffusing particle reaches
a reaction centre or a stochastic process exceeds a given threshold value
\cite{redner,ralf_gleb_sid_book}. Based on the minimal model for DD
\cite{Chechkin:DD2} we derive the first passage
behaviour in both semi-infinite and finite systems. We find that the DD
dynamics may outperform Brownian-Gaussian normal diffusion at
intermittent times in a semi-infinite domain while the long time behaviour
matches exactly the Brownian-Gaussian result with an effective diffusivity.
We also observe an interesting universal crossover point of the survival probability
which is independent of the initial particle position.
In finite domains the mean first passage time of the DD model is longer than
in the Brownian-Gaussian case. Concurrently, in the DD model
the divergence of the mean first passage
time observed in the superstatistical approach is rectified.

In section 2 we briefly recall the basic properties of the minimal diffusing
diffusivity model \cite{Chechkin:DD2}. The survival probabilities for the
semi-infinite and finite domains are then derived in section 3 along with
their short and long time asymptotes. Section 4 provides a detailed
discussion of the results including a relation to the standard Brownian-Gaussian
first passage behaviour. A short conclusion is presented in section 5.

\section{Minimal model for Brownian yet non-Gaussian diffusion}

The model we study is the so-called minimal diffusing diffusivity (DD) model which was
introduced to describe diffusion in heterogeneous environments \cite{Chechkin:DD2}.
In this model the diffusivity is defined as a stochastic process itself, in terms
of the squared Ornstein-Uhlenbeck process, guaranteeing the stationarity of $D(t)$.
In dimensionless units the minimal DD model is defined by the set of Langevin equations
\cite{Chechkin:DD2}
\begin{eqnarray}
\frac{d}{dt}\mathbf{r}(t)&=&\sqrt{2D(t)}{\bm \xi}(t)\nonumber\\
D(t)&=& \mathbf{Y}^2(t),\quad\frac{d}{dt}\mathbf{Y}(t)=-\mathbf{Y}+{\bm\eta}(t),
\label{eqn:XDD}
\end{eqnarray}
where the components of $\bm\xi(t)$ and $\bm\eta(t)$ are independent white Gaussian
noises and $\mathbf{Y}$ represents an $d$-dimensional Ornstein-Uhlenbeck process. The
dimensionless Ornstein-Uhlenbeck process here has a characteristic crossover time of
unity. We assume the diffusivity to start from equilibrium initial conditions (the
non-equilibrium case is discussed in \cite{Sposini:DD2}). This leads to the
superstatistical short time diffusivity distribution
\begin{equation}
p_D(D)=\left\{\begin{array}{lr}(\sqrt{\pi D})^{-1}\e^{-D}, & d=1\\
\e^{-D}, & d=2\\
(2\sqrt{D/\pi})\e^{-D}, & d=3
\end{array}\right.,
\label{eqn:PD}
\end{equation}
for 1, 2 and 3 dimensions. While the dominating exponential tail is common to all
$d$, there is a pole at $D\to0$ in $d=1$ \cite{Chechkin:DD2}. As we
showed previously, the minimal DD model can be written using the concepts of
subordination \cite{Subord} through the relation \cite{Chechkin:DD2}
\begin{equation}
P(\mathbf{r},t|\mathbf{r}_0)=\int_0^{\infty}G(\mathbf{r},\tau|\mathbf{r}_0,D=1)
T_d(\tau,t)d\tau,
\label{eqn:PDD}
\end{equation}
of the probability density function (PDF) $P(\mathbf{r},t|\mathbf{r}_0)$ of
displacement and the Gaussian
\begin{equation}
G(\mathbf{r},t|\mathbf{r}_0,D)=(4 \pi D t)^{-d/2}\exp\left((\mathbf{r}-\mathbf{r}_0)^2
/[4Dt]\right)
\end{equation}
with fixed diffusion coefficient $D$. The subordinator $T_d(\tau,t)$ represents the PDF
of the process $\tau(t)=\int_0^t\mathbf{Y}^2(t')dt'$ and is defined through its Laplace
transform \cite{Chechkin:DD2}
\begin{equation}
\fl\tilde{T}_d(s,t)=\exp(dt/2)\left[\frac{1}{2}(\sqrt{1+2s}+\frac{1}{\sqrt{1+2s}})
\sinh \left(t \sqrt{1+2s}\right)+\cosh \left(t \sqrt{1+2s}\right)\right]^{-d/2}
\label{eqn:T}
\end{equation}
with short and long times limits
\begin{eqnarray}
\fl\tilde{T}_d(s,t)&\sim t^{-d/2}\left(s+1/t\right)^{-d/2}, & t\ll 1,\label{eqn:TST}\\
\fl\tilde{T}_d(s,t)&\sim2^{d/2}\exp\left(\frac{dt}{2}(1-\sqrt{1+2s})\right)
\left(1+\frac{1}{2}\left(\sqrt{1+2s}+\frac{1}{\sqrt{1+2s}}\right)\right)^{-d/2}, &
t\gg 1.
\label{eqn:TLT}
\end{eqnarray}
At short times the diffusivity varies slowly and we can assume it to be almost
constant. In this limit the DD model thus reduces to the superstatistical
approximation of the DD model in which each particle has a constant random diffusion
coefficient with distribution $p_D(D)$ \cite{Superst_1,Superst_2}: on the ensemble
level this implies that the PDF can be written as $P_\mathrm{sup}(\mathbf{r},t|
\mathbf{r}_0)=\int_0^\infty G(\mathbf{r},t|\mathbf{r}_0,D)p_D(D)dD$, such that the
short time PDF explicitly reads
\begin{equation}
P_\mathrm{ST}(\mathbf{r},t|\mathbf{r}_0)=P_\mathrm{sup}(\mathbf{r},t|\mathbf{r}_0) =
\left\{\begin{array}{lr}(\pi t^{1/2})^{-1}K_0\left(|x-x_0|/t^{1/2}\right), & d=1\\
(2\pi t)^{-1}K_0\left(|r-r_0|/t^{1/2}\right), & d=2\\
(2\pi^2 t^{3/2})^{-1}K_0\left(|r-r_0|/t^{1/2}\right), & d=3
\end{array}
\right.
\label{eqn:PDDST}
\end{equation}
where $K_0(x)$ is a modified Bessel function of the second kind with exponential
asymptote $K_0(z)\sim\sqrt{\pi/(2z)}e^{-z}$ \cite{prudnikov2}. At long times
the DD process crosses over to a purely Gaussian process with
PDF $ P_\mathrm{LT}(\mathbf{r},t|\mathbf{r}_0)=G(\mathbf{r},t|\mathbf{r}_0,D=
\langle D\rangle_\mathrm{st})$ with the stationary diffusivity $\langle
D\rangle_\mathrm{st}=d/2$ \cite{Chechkin:DD2}. For all $t$ the MSD is given by
$\langle(\mathbf{r}(t)-\mathbf{r}_0)^2\rangle=2d\langle D \rangle_\mathrm{st}t$.

We showed in \cite{Sposini:DD2} that there is a stochastic counterpart to this
superstatistical approximation, defined through the generalised grey Brownian
motion (ggBM) formalism \cite{Mura:ggBM}, $\mathbf{r}(t)=\sqrt{2D}\times\mathbf{
W}(t),$ where $D$ is the random and constant diffusion coefficient and $\mathbf{
W}(t)$ is the $d$-dimensional Wiener process or standard Brownian motion. Note
that while the DD model represents the heterogeneity of the medium in some mean
field sense \cite{Chechkin:DD2} the ggBM model describes an heterogeneous
ensemble of particles \cite{spos-pagni:2} .

\section{Results for the survival probabilities}

The first passage time PDF of a stochastic process is the negative time
derivative of the survival probability, $\wp(t)=-d S(t)/dt$. We here obtain
the survival probability for semi-infinite and finite domains using the
above subordination relation.

\subsection{Survival of diffusing diffusivity model in semi-infinite domain}

We begin our study with the semi-infinite interval $d=1$. Following the approach
for standard diffusion \cite{redner} we use the method of images for the initial
particle position $x_0$. Combined with the subordination principle (\ref{eqn:PDD})
we get the image propagator
\begin{equation}
\fl P(x,t|x_0)=\int_0^{\infty}\left(G(x,\tau|x_0,D=1)-G(x,\tau|-x_0,D=1)\right)T_1(
\tau,t)d\tau.
\end{equation}
After Fourier transform we obtain
\begin{equation}
\fl\hat{P}(k,t|x_0)=\int_0^\infty T_1(\tau,t)e^{-k^2\tau}\left(e^{ikx_0}-e^{-ikx_0}
\right)d\tau=(e^{ikx_0}-e^{-ikx_0})\tilde{T}_1(s=k^2,t).
\end{equation}
Here $\hat{\cdot}$ and $\tilde{\cdot}$ indicate the Fourier and Laplace transforms
of the functions, respectively. We then calculate the survival probability in the
semi-infinite domain,
\begin{equation}
S(t|x_0)=\int_0^\infty P(x,t|x_0)dx=\int_0^\infty dx\int_{-\infty}^{+\infty}
\frac{dk}{2\pi}e^{-ikx}\hat{P}(k,t|x_0).
\label{eqn:SDDint}
\end{equation}
To check normalisation, we first see from expression (\ref{eqn:T}) that $\tilde{
T}_1(s,0)=1$. Then,
\begin{eqnarray}
S(0|x_0)=\int_0^\infty dx\int_{-\infty}^{+\infty}\frac{dk}{2\pi}(e^{-ik(x-x_0)}
-e^{-ik(x+x_0)})=1,
\end{eqnarray}
where we used that $\int_{-\infty}^{\infty}dk/(2\pi)\exp(-ikx)=\delta(x)$. Moreover,
plugging the long time limit for $\tilde{T}_1(s=k^2,t)$ in (\ref{eqn:TLT}) into the
expression for $S(t|x_0)$ one can readily show that $S(t\to\infty|x_0)=0$, as it
should.

The direct calculation of the integral (\ref{eqn:SDDint}) is not easy to perform,
we here focus on the short and long time regimes. At short times, $\tilde{T}_1(s
=k^2,t)$ is given by (\ref{eqn:TST}) and thus
\begin{eqnarray}
\fl S_\mathrm{ST}(t|x_0)&=&\int_0^{\infty}dx\int_{-\infty}^{\infty}\frac{dk}{2\pi}
e^{-ikx}(e^{ikx_0}-e^{-ikx_0})\frac{t^{-1/2}}{\sqrt{k^2 +1/t}}\nonumber\\
\fl &=&\frac{1}{2\pi\sqrt{t}}\left(\int_0^\infty dx\int_{-\infty}^{\infty}dk
\frac{e^{-ik(x-x_0)}}{\sqrt{k^2 +1/t}}-\int_0^\infty dx \int_{-\infty}^{\infty}dk
\frac{e^{-ik(x+x_0)}}{\sqrt{k^2 +1/t}}\right)\nonumber\\
\fl &=& \frac{1}{\pi \sqrt{t}} \int_0^\infty \left[ K_0\left(\frac{|x-x_0|}{\sqrt{
t}}\right)-K_0\left(\frac{|x+x_0|}{\sqrt{t}}\right)\right]dx,
\label{DD=ggBM}
\end{eqnarray}
Splitting the integral and changing variables we obtain
\begin{eqnarray}
\fl S_\mathrm{ST}(t|x_0)&=&\frac{1}{\pi\sqrt{t}}\left[\int_0^{x_0}K_0\left(\frac{
x_0-x}{\sqrt{t}}\right)dx+\int_{x_0}^\infty K_0\left(\frac{x-x_0}{\sqrt{t}}\right)dx
-\int_0^\infty K_0\left(\frac{x+x_0}{\sqrt{t}}\right)dx\right]\nonumber\\
\fl &=&\frac{2}{\pi}\int_0^{x_0/\sqrt{t}}K_0(z)dz.
\end{eqnarray}
Using $\int_0^a K_0(z)dz=a\pi/2\left(K_0(a)L_{-1}(a)+K_1(a)L_0(a)\right)$, with the
modified Struve function $L_\nu(z)$ \cite{prudnikov2},
\begin{equation}
S_\mathrm{ST}(t|x_0)=\frac{x_0}{\sqrt{t}}\left[K_0\left(\frac{x_0}{\sqrt{t}}\right)
L_{-1}\left(\frac{x_0}{\sqrt{t}}\right)+K_1\left(\frac{x_0}{\sqrt{t}}\right)L_0\left(
\frac{x_0}{\sqrt{t}}\right)\right].
\label{eqn:S_DD_ST}
\end{equation}
The same result can be obtained both inserting directly the short times approximation
(\ref{eqn:PDDST}) of the propagator in the result of the images method of images and
calculating directly the superstatistical integral valid for the survival probability.

At long times, when in equation (\ref{eqn:TLT}) we only consider the tails of the
distribution, $\tilde{T}_1(s=k^2,t)\sim\exp\left(-k^2t/2\right)$. This approximation
leads to
\begin{eqnarray}
\fl S_\mathrm{LT} (t|x_0)&=&\int_0^\infty dx\left[\int_{-\infty}^{\infty}\frac{dk}{2
\pi}\exp\left(-ik (x-x_0)-\frac{k^2t}{2} \right)-\exp\left(-ik (x+x_0)-\frac{k^2t}{2}
\right)\right]\nonumber\\
\fl &=& \frac{1}{\sqrt{2 \pi t}} \int_0^\infty dx \left[ \exp\left(-\frac{(x-x_0)^2}{2t}\right) -
\exp\left(-\frac{(x+x_0)^2}{2t}\right) \right]  \nonumber \\
\fl &=& \mathrm{erf}\left( \frac{x_0}{\sqrt{2 t}} \right)= \mathrm{erf}\left( \frac{x_0}
{\sqrt{4 \langle D \rangle_\mathrm{st} t}} \right) .
\label{S_DD_lt}
\end{eqnarray}
This result equals the one for Brownian diffusion in a semi-infinite domain, in
agreement with the fact that at long times the DD model shows a crossover to
Gaussian diffusion with effective diffusivity $ \langle D \rangle_\mathrm{st}$.
In analogy with Brownian diffusion, this particularly leads to the divergence of
the mean first passage time, $\langle t\rangle=\infty$.

\subsection{Survival of diffusing diffusivity model in finite domain}

We now turn to a finite domain $[0,L]$ with absorbing boundaries at $x=0$
and $x=L$. Drawing on the subordination approach again, we map the images result
for the finite domain to obtain the DD propagator,
\begin{equation}
P(x,t|x_0)=\frac{2}{L}\sum_{n=1}^{\infty}\sin\left(\frac{\pi n}{L}x_0\right)\sin
\left(\frac{\pi n}{L}x\right)\tilde{T}_1(\lambda_n^2,t).
\label{eqn:DDsubL}
\end{equation}
By integration we obtain the survival probability
\begin{equation}
S(t|x_0)=\frac{4}{\pi}\sum_{n=0}^{\infty}\sin\left(\frac{\pi(2n+1)}{L}x_0\right)
\frac{\tilde{T}_1(\lambda_{2n+1}^2,t)}{(2n+1)},
\end{equation}
from which we obtain the limiting behaviours for short times,
\begin{equation}
S_\mathrm{ST} (t|x_0) \sim \frac{4}{\pi} \sum_{n=0}^{\infty} \sin
\left( \frac{\pi (2n +1)}{L} x_0 \right) \frac{1}{(2n+1)\sqrt{\lambda_{2n+1}^2 t +1}},
\end{equation}
and for long times,
\begin{eqnarray}
\fl S_\mathrm{LT}(t|x_0)&\sim&\frac{4\sqrt{2}}{\pi} \sum_{n=0}^{\infty} \sin
\left( \frac{\pi (2n +1)}{L} x_0 \right)\exp\left(-\frac{t}{2}
\left[\sqrt{1+2\lambda_{2n+1}^2}-1\right]\right)\nonumber\\
\fl &&\times(2n+1)\left( 1+ \frac{1}{2}
\left(\sqrt{1+2\lambda_{2n+1}^2}+\frac{1}{\sqrt{1+2\lambda_{2n+1}^2}}
\right)\right)^{1/2}.
\label{eqn:SDDL_LT}
\end{eqnarray}
Note that, as in the previous case, the asymptotic behaviour at short times can also
be found through direct calculation of the superstatistical integral.

\section{Discussion of results}

Figures \ref{fig:S_x0_IFFT} and \ref{fig:S_DD_L} show a comparison of the results
obtained for the DD model with the classical ones for Brownian-Gaussian motion in
the semi-infinite and finite domains, respectively. In figure \ref{fig:S_x0_IFFT}
(left) and figure \ref{fig:S_DD_L} we include results from simulations, demonstrating
excellent agreement with our analytical results. As expected, we observe significant
dissimilarities between the two models mostly in the short time limit. At intermediate
time scales the DD model shows a crossover from short time superstatistical behaviour
to the limiting Brownian-Gaussian behaviour with effective diffusivity $\langle D
\rangle_{\mathrm{st}}$.

For the semi-infinite domain figure \ref{fig:S_x0_IFFT} demonstrates that in the
short time regime the DD process exhibits a faster decay of the survival probability
and thus a more efficient first passage dynamics. This effects is particularly
visible in the right panel, in which short times correspond to large values on the
abscissa $x_0/\sqrt{t}$. To clarify this effect we express result (\ref{eqn:S_DD_ST})
and the one for Brownian motion in terms of elementary functions,
\begin{eqnarray}
S_\mathrm{BM}(t|x_0)&\sim1-\frac{\sqrt{2}e^{-(x_0^2/2t)}}{\sqrt{\pi}x_0}t^{1/2},
& x_0/\sqrt{t}\to\infty, \label{eqn:series_BM}\\
S_\mathrm{ST}(t|x_0)&\sim1-\frac{\sqrt{2}e^{-(x_0/\sqrt{t})}}{\sqrt{\pi x_0}}t^{1/4}
+\frac{5e^{-(x_0/\sqrt{t})}}{4\sqrt{2\pi x_0^3}}t^{3/4}, \quad & x_0/\sqrt{t}\to\infty.
\label{eqn:series_DD}
\end{eqnarray}
Comparing the asymptotes (\ref{eqn:series_BM}) with (\ref{eqn:series_DD}) along with
the inset in figure \ref{fig:S_x0_IFFT} (right), we observe that for a fixed initial
position $x_0$ the DD survival probability initially indeed drops faster than the one
for Brownian-Gaussian motion. This behaviour is more visible for larger $x_0$ and
becomes less and less relevant when $x_0$ approaches the absorbing boundary. From a
physical point of view, this can be understood due to the fact that the closer to
the boundary we place the particle initially the more likely it is that the particle
is absorbed immediately, independently from the underlying diffusive model.

\begin{figure}
\includegraphics[width=0.5\textwidth]{Surv_IFFT_x0s.eps}
\includegraphics[width=0.5\textwidth]{Surv_IFFT_x0s_rescale_inset.eps}
\caption{Left: Comparison of numerical and analytical results for the survival
probability $S(t|x_0)$ in the semi-infinite interval. Different colours represent
the initial positions $x_0=0.5$, $1$, $1.5$, $2$, and $5$. Dashed lines in both
panels represent the results of the corresponding Brownian-Gaussian motion. The
numerical results (dots) obtained through Monte Carlo simulations are in full
agreement with the analytical trend (solid line) obtained from numerical
integration of the inverse Fourier transform (\ref{eqn:SDDint}). Right: analytical
results for the survival probability in rescaled units in the semi-infinite domain
as function of $x_0/\sqrt{t}$. In the inset the short time behaviour of $S(t|x_0)$
is reported, the universal crossover at $S\approx0.925$ is distinct.}
\label{fig:S_x0_IFFT}
\end{figure}

Figure \ref{fig:S_x0_IFFT} (right) demonstrates two universalities. First, we observe
that at intermediate times the survival probabilities for any initial position show
a universal convergence to a common crossover point at around $S(t|x_0)\approx0.925$,
including the Brownian-Gaussian survival probability. At times shorter than this
crossover point Brownian-Gaussian motion is outperformed by the DD model, which
assumes smaller values of $S(t|x_0)$. At times longer than the crossover time the
decay of Brownian-Gaussian motion is the fastest. Second, the initial advantage of
the DD first passage dynamics over Brownian-Gaussian motion which reverts after the
universal crossover point, appears to balance out: at long times the survival
probability in all cases converges to the exact result of Brownian-Gaussian motion
with effective diffusivity $\langle D\rangle_{\mathrm{st}}$. This can be seen directly
from result (\ref{S_DD_lt}), the associated first passage density of which is
exactly the well-known L{\'e}vy-Smirnov form $\wp(t)=(x_0/\sqrt{4\pi\langle D\rangle_
{\mathrm{st}}t^3})\exp(-x_0^2/[4\langle D\rangle_{\mathrm{st}}t]$).

Qualitatively a similar behaviour is observed for finite domains at short times. As
shown in figure \ref{fig:S_DD_L}, in contrast, the long time behaviour is dominated
by the exponential shoulder (\ref{eqn:SDDL_LT}) corresponding to the lowest non-zero
eigenvalue in the DD model. The corresponding characteristic time scale $\tau$ in
figure
\ref{fig:S_DD_L} is longer than for Brownian-Gaussian motion. This is due to the fact
that in the finite interval the particles will reach the boundary before experiencing
the entire diffusivity space, and so the effective Brownian limit is not recovered.
The larger the interval $L$ is the smaller the difference between the characteristic
times of DD and Brownian-Gaussian models will be. In the limit of $L\to\infty$ the
same long time behaviour is observed. Figure \ref{fig:S_DD_L} also demonstrates an
interesting behaviour of the superstatistical model. When the diffusivity distribution
(\ref{eqn:PD}) governs the particle motion at all times $t$, even in the finite domain
a power law scaling of the survival probability emerges, and thus a diverging mean
first passage time is produced. This behaviour is caused by appreciable fraction of
immobile particles manifested in the divergence or nonzero value of $p_D(D=0)$ in
$d=1$ and $d=2$, respectively. This behaviour is rectified in the DD model.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Surv_sin_x0s_L1.eps}
\caption{Survival probability for the finite interval $[0,L]$, showing a comparison
between Brownian-Gaussian diffusion, DD, and superstatistical dynamics. Different
colour shades represent different initial positions $x_0$, and the solid lines
represent the analytical trends. The normalisation is chosen considering that all 3
models present the dependence on the initial position through the sinusoidal function
reported in the $y$-axis and that at long times just the first eigenstate dominates.
Indeed we observe that at long times all lines approach a quasi-master curve which
is different for each model and in agreement with the analytical results: a power law
asymptotic for the superstatistical model and exponential tails for the DD and
Brownian-Gaussian models, the latter with two different $\tau$ values of the
dominant exponential tail.}
\label{fig:S_DD_L}
\end{figure}

\section{Conclusions}

We studied the first passage behaviour of the popular DD model used as a mean field
proxy for diffusion of test particles in heterogeneous environments, in which the
particle experiences varying diffusivities. Our analysis demonstrated that at
short times the DD dynamics leads to a faster decay of the survival probability and
thus to more efficient first passage. In a semi-infinite domain, fully independent
of the initial particle position a universal crossover occurs, beyond which the
DD dynamics becomes less efficient than pure Brownian-Gaussian motion, and the
ultimate decay is determined by the conventional L{\'e}vy-Smirnov behaviour for
initial particle position $x_0$ and effective diffusivity $\langle D\rangle_{\mathrm{
st}}$. The initial advantage of the DD dynamics may be particularly relevant in
cases of molecular regulation processes at very low concentrations (few-encounter
limit) \cite{aljaz}. At long times in finite domains the DD first passage
behaviour is dominated by an exponential shoulder with a characteristic time
(approximately the mean first passage time) that is longer than that for
Brownian-Gaussian motion.

These results are in agreement with the expectation that rare events, represented
by the exponential tails of the particles displacement distribution at short
times, may dominate triggered actions. Thus, even if in general heterogeneity
in the environment does not improve the mean first passage result (in fact
some of the particles are slowed down) it allows some other particles to
have a diffusion coefficient greater than the average, and this is enough
to increase the efficiency of the reaction activation.  Moreover, we proved
that the amount of fast particles is independent on the initial position,
representing the distance between particle and target. This suggests that
the obtained results may be qualitatively generalised to any distribution
of the initial particle position.

The study developed here is not limited to the one-dimensional case. First of
all, we know that in the semi-infinite domain the results of the survival
probability of Brownian-Gaussian motion in $d=2$ and $d=3$ are the same as
the one in $d=1$. Then, the same analysis of the first passage problem can be
performed by solely changing to the corresponding $d$-dimensional subordinator. For finite
domains the analysis is also similar since, for all $d$ we have an exponential
behaviour in time of the propagator which allows us to relate the DD survival
probability to the Laplace transform of the corresponding subordinator,
as we did for the one-dimensional case.

We finally note that similar non-Gaussian effects have been reported for systems,
in which the (subdiffusive) motion is dominated by viscoelastic effects. With a
fixed diffusivity this would be a Gaussian process, and the non-Gaussianity was
shown to stem from varying diffusivity values \cite{prx,Lampo:BYNG6,natcomm}.
It will be interesting to study the associated first passage behaviour in this
case, as well.

\ack

VS thanks the organisers of the 671st WE-Heraeus-Seminar on Search and Problem Solving
by Random Walks: Drunkards vs Quantum Computers at Bad Honnef, Germany, in May 2018
where the main results of this work were presented. RM and AVC acknowledge funding
from Deutsche Forschungsgemeinschaft, grants ME 1535/6-1 and ME 1535/7-1. RM
acknowledges the Foundation for Polish Science (Fundacja na rzecz Nauki Polski) for
funding within an Alexander von Humboldt Polish Honorary Research Scholarship.

\section*{References}

\bibliographystyle{iopart-num}


\section*{1 Introduction}

In the recent years, the meaning of  quantum discords has been attracted a lot of interest because it indicates that entangled states are not the only kind of quantum states exhibiting nonclassical features\cite{zurek, Zurek2, Henderson, Luo, Vedral, Datta}. Quantum discord is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information . Quantum discord captures the fundamental feature of the quantumness of nonclassical correlations, which is much like quantum entanglement, but it is beyond quantum entanglement because quantum discord is even present in separable quantum states. Quantum discords have strong potentials in studying dynamical processes\cite{zurek3, oppenheim, Horodecki, mahdian}, some quantum information processes such as the broadcasting of quantum states\cite{piani1, piani} quantum state merging \cite{cavalcanti, madhok}, quantum entanglement distillation\cite{cornelio, piani3} ,
entanglement of formation \cite{fanchini}.\\
Besides, there has been considerable interest in the quantum information properties of solid state environment particularly, semiconductor quantum dots (QDs) due to their well defined controllable atom-like and molecule-like properties\cite{filippo, filippo2,hawrylak}. To study the electro-optical properties of  QDs (specially coupled ones), one of the most interesting parameter is the exciton-exciton interaction. This kind of interaction in first neighbors dots will allow to implement a scheme for quantum information processing on QDs \cite{chen}. It was shown that the biexcitonic shift due to the dipolar interaction allows for subpicosecond quantum gate operations\cite{rinaldis02}.
It was proven that the optically driven QDs, that are carrying excitons are good candidates for  implementation of quantum gates and quantum computation\cite{xiaoqin,Herschbach,Friedrich}.

In order to study the amount of concurrence and discord in the array of excitons inside QDs, we employ the lower bound of concurrence for thermal density matrix of identical and equidistant coupled QDs at different bath temperatures. Furthermore, By the means of electric field, the manipulation of entanglement and discord will be discussed. We show that, quantum correlations may be enhanced upon increasing the temperature and decreased with incresing the electric field across the system. Our theoretical model detects nonvanishing quantum discord, ensuring the existence of nonclassical correlations as measured by the quantum discord

\section{2 Theory: quantum dot model and Hamiltonian}

The model sample to study the quantum correlation properties of  optically driven QDs is a series of InAs coupled QDs with small equal spacing between them along the axis. This model can be realized experimentally(see for example reference\cite{Nishibayashi}). In this model, F\"{o}rster mechanism\cite{Forster} is a valid model to explain the energy transfer between QDs through dipolar interaction between the excitons. Here, the qubits are the excitonic electric dipole moments located in each QD which can only orient along ($|0\rangle$) or against ($|1\rangle$) the external electric fields. For such a system, tuning and controlling the quantum correlation between the dipoles is of great importance. The governing Hamiltonian of dipoles in the presence of external electric field simply reads:

\begin{eqnarray}
H = \hbar\sum _{i=1}^{n}\omega_i[S_z^i+\frac{1}{2}]+\nonumber\\
+\hbar\sum_{i=1}^{n}\Omega_i\hat{S_i^z}+\hbar\sum_{i=1}^{n}J_z[S_+^i][S_-^j]+\nonumber\\
\frac{1}{2}\sum_{i,j=1}^{n}\lambda[S_+^iS_-^j+S_-^jS_+^i],
\end{eqnarray}

Where $S_+^i$=$(|0\rangle\langle1\rangle)$,$S_-^i$=$(|1\rangle\langle0\rangle)$, and $S_z^i$=$\frac{1}{2}(|0\rangle\langle0\rangle-|1\rangle\langle1\rangle)$. $\omega_i$ presents the frequency of the excitons in QDs, $\Omega_i$ is the frequency related to dipole moment(exciton) that is a function of dipole moment and the external electric field (E) at \textit{i}th QD:

\begin{equation}
\hbar\Omega_i=|\vec{d}.\vec{E}|,
\end{equation}

Where $\vec{d}$ is the electric dipole moment carried by the exciton that is assumed to be same for each QD. $\lambda$ presents the F\"{o}rster interaction which transfers an exciton from one QD to other ones. $J_z$ presents the exciton-exciton dipolar interaction energy, reads:

\begin{eqnarray}
\hbar J_z=\frac{\vec{d}^2(1-3\cos^2\theta)}{\vec{r}_{ij}^3},
\end{eqnarray}

where $r_{ij}$ is the distance between dipoles \textit{i} and \textit{j} that is assumed along the $z$ axis.

For a qualitative discussion on quantum correlations we assume that the dipole carried by exciton is the one order of magnitude in debyes, a typical
experimental electric field $10^6$ $V/m$, so the dipolar interaction parameter($\hbar J_z$) will be of the order of $meV$. F\"{o}rster interaction energy($\hbar \lambda$) and $\hbar \Omega$ are assumed to be in the order of $meV$. This values are consistent with experimental observations and calculations\cite{zhu}.

\subsection{2.1 Lower bound concurrence}

Since entanglement is conceived as a resource to perform various tasks of quantum information processing \cite{23,24,25,26}, knowledge about the amount of entanglement in a quantum state is so important. Indeed, awareness from the value of entanglement, means knowing how well a certain task can be accomplished. The quantification problem of entanglement only for bipartite systems in pure states \cite{27} and two-qubit system in mixed state \cite{28} is essentially solved. In multi-partite systems, even the pure state case, this problem is not exactly solved and just lower bounds for the entanglement have been proposed \cite{29,30,31,32}. Here, in order to determination the exact minimum of entanglement between three dipoles(excitons), we use the lower bound of concurrence for three-qubit state which is recently suggested by Li et al. \cite{33}
\begin{eqnarray}
\tau_{3}(\rho)=\frac{1}{\sqrt{3}}(\sum_{j=1}^{6}(C_{j}^{12|3})^{2}+(C_{j}^{13|2})^{2}+(C_{j}^{23|1})^{2})^{\frac{1}{2}},
\end{eqnarray}
where $C_{j}^{12|3}$ is terms of the bipartite concurrences for qubits $12$ and $3$ which is given by
\begin{eqnarray}
C_{j}^{12|3}=\max \{0,\lambda_{j}^{12|3}(1)-\lambda_{j}^{12|3}(2)-\lambda_{j}^{12|3}(3)-\lambda_{j}^{12|3}\}.
\end{eqnarray}
In this notation, $\lambda_{j}^{12|3}(\kappa)$, $(\kappa=1..4)$, are the square nonzero roots, in decreasing order, of the non-Hermitian matrix $\rho \tilde{\rho}_{j}^{12|3}$. The matrix $\tilde{\rho}_{j}^{12|3}$ are obtained from rotated the complex conjugate of density operator, $\rho^{*}$, by the operator $S_{j}^{12|3}$ as $\tilde{\rho}_{j}^{12|3}=S_{j}^{12|3}\, \rho^{*} \, S_{j}^{12|3} $. The rotation operators $S_{j}^{12|3}$ are given by tensor product of the six generators of the group SO(4), ($L_{j}^{12}$), and the single generator of the group SO(2), ($L_{0}^{3}$) that is $S_{j}^{12|3}=L_{j}^{12} \otimes L_{0}^{3}$. Since the matrix $S_{j}^{12|3}$ has four rows and columns which are identically zero, so the rank of non-Hermitian matrix $\rho \tilde{\rho}_{j}^{12|3}$ can not be larger than 4, i.e., $\lambda_{j}^{12|3}(\kappa)=0$ for $\kappa\geq 5$. The bipartite concurrences $C^{13|2}$ and $C^{23|1}$ are defined in a similar way to $C^{12|3}$.

In order to calculate thermal entanglement, we need the temperature dependent density matrix and the density matrix
for a system in equilibrium at a temperature T reads: $\rho=\exp(-\beta\hat{H}/Z)$ with $\beta=1/KT$ and $Z$ is partition function, $Z=Tr(\exp(-\beta\hat{H}))$. In this case, the partition function is

\begin{eqnarray}
Z(T)=\sum_{i=1}g_ie^{-\beta\lambda_i},
\end{eqnarray}

where $\lambda_i$ is the $i$th eigenvalue and $g_i$ is the degeneracy. and the corresponding density matrix can be written

\begin{eqnarray}
\rho(T)=\frac{1}{Z}\sum_i^N e^{-\beta\lambda_i}|\Phi_i\rangle\langle\Phi_i|,
\end{eqnarray}

here $|\Phi_i\rangle$ is the $i$th eigenfunction. The density matrix for our considered system has the form as:

\begin{equation}
 \rho(T) = \left( \begin{array}{cccccccc}
\rho_{11}&0&0&0&0&0&0&0 \\  0&\rho_{22}&\rho_{23}&0&\rho_{23}&0&0&0 \\ 0&\rho_{23}&\rho_{22}&0&\rho_{23}&0&0&0 \\ 0&0&0&\rho_{44}&0&\rho_{46}&\rho_{46}&0 \\ 0&\rho_{23}&\rho_{23}&0&\rho_{22}&0&0&0 \\ 0&0&0&\rho_{46}&0&\rho_{44}&\rho_{46}&0 \\ 0&0&0&\rho_{46}&0&\rho_{46}&\rho_{44}&0 \\ 0&0&0&0&0&0&0&\rho_{88}\end{array} \right),
\end{equation}

with
\begin{eqnarray}
\rho_{11}&=&\frac{e^{\beta\lambda}}{1+e^{\beta(2d+Fz+w)}}
\cr\cr&\times&[e^{\beta(2d+2Fz+w)}+e^{\beta\lambda}-e^{\beta(2d+Fz+w+\lambda)}\cr&&
+e^{\beta(4d+2Fz+2w+\lambda)}+2e^{\beta(2d+2Fz+w+\frac{3}{2}\lambda)}]^{-1},
\cr\cr\cr
\rho_{22}&=&\frac{1}{3}(1+2e^{\frac{3}{2}\beta\lambda})\cr&\times&
[(1+e^{\beta(2d+Fz+w)})(1+2e^{\frac{3}{2}\beta\lambda})\cr&&
+ e^{\beta(\lambda-2d-2Fz-w)}+e^{\beta(4d+Fz+2w+\lambda)}]^{-1},
\cr\cr\cr
\rho_{23}&=&\frac{e^{\beta(2d+2Fz+w)}(1-e^{\frac{3}{2}\beta\lambda})}{3(1+e^{\beta(2d+Fz+w)})}
\cr\cr&\times&
[e^{\beta(2d+2Fz+w)}+e^{\beta\lambda}-e^{\beta(2d+Fz+w+\lambda)}\cr&&
+e^{\beta(4d+2Fz+2w+\lambda)}+2e^{\beta(2d+2Fz+w+\frac{3}{2}\lambda)}]^{-1},
\cr\cr\cr
\rho_{44}&=&\frac{1}{3}(1+2e^{\frac{3}{2}\beta\lambda})\cr&\times&
[(1+e^{-\beta(2d+Fz+w)})(1+2e^{\frac{3}{2}\beta\lambda})\cr&&
+ e^{\beta(\lambda-4d-3Fz-2w)}+e^{\beta(2d+w+\lambda)}]^{-1},
\cr\cr\cr
\rho_{46}&=&\frac{e^{\beta(4d+3Fz+2w)}(1-e^{\frac{3}{2}\beta\lambda})}{3(1+e^{\beta(2d+Fz+w)})}
\cr\cr&\times&
[e^{\beta(2d+2Fz+w)}+e^{\beta\lambda}-e^{\beta(2d+Fz+w+\lambda)}\cr&&
+e^{\beta(4d+2Fz+2w+\lambda)}+2e^{\beta(2d+2Fz+w+\frac{3}{2}\lambda)}]^{-1},
\cr\cr\cr
\rho_{88}&=&e^{3\beta d}
\cr\cr&\times&[e^{3\beta d}+e^{-3\beta(d+Fz+w)}+2e^{\beta(d-w+\frac{1}{2}\lambda)}\cr&&
+e^{\beta(d-w-\lambda)}+e^{-\beta(d+Fz+2w+\lambda)}\cr
&&+2e^{-\beta(d+Fz+2w+\frac{1}{2}\lambda)}]^{-1}.
\end{eqnarray}

Since the explicit expressions of solutions of Eq. (7) are very complicated, here we skip the details and give our results in terms of figures. The discussion of results will be postponed to the next section.

\subsection{2.2 Quantum discord}

For a bipartite system AB quantum discord is defined by the discrepancy between quantum versions of two classically
equivalent expressions for mutual informationis \cite{20}
\begin{eqnarray} \label{2-2}
\textit{D}(\rho^{AB})=\textit{I}(\rho^{AB})- \textit{C}(\rho^{AB}),
\end{eqnarray}

where $\textit{I}(\rho^{AB})=S(\rho^{A})+S(\rho^{B})-S(\rho^{AB})$ and the classical correlation is the maximum information about one subsystem $\rho^{A}$ or $\rho^{B}$ which depends on the type of measurement performed on the other subsystem such as $\textit{C}(\rho^{AB})=max[S(\rho^{A})-S(\rho^{AB}|\{\Pi_{k}\})]$ with $S(\rho)=-Tr[\rho \log_{2}\rho]$ as the von-Neumann entropy. Notice that the maximum is taken over the set of projective measurements $\{\Pi_{k}\}$ \cite{4}.
\\By definition the conditional density operator $\rho^{AB}_{k}=\frac{1}{p_{k}}\{(I^{A}\otimes \Pi_{k}^{B})\rho^{AB}(I^{A}\otimes \Pi_{k}^{B})\}$ with $p_{k}=Tr[(I^{A}\otimes \Pi_{k}^{B})\rho^{AB}]$ as the probability of obtaining the outcome $k$, we can define the conditional entropy of $A$ as
$S(\rho^{AB}|\{\Pi_{k}\})=\sum_{k}p_{k}S(\rho_{k}^{A})$ with $\rho_{k}^{A}=Tr_{B}[\rho_{k}^{AB}]$ and $S(\rho_{k}^{A})=S(\rho_{k}^{AB})$.
It has been shown that $\textit{D}(\rho^{AB})\geq 0$ with the equal sign only for classical correlation \cite{5}.
\\Very recently, Rulli et al. \cite{3} have proposed a global measure of quantum discord based on a systematic extension of the bipartite quantum discord. Global quantum discord (GQD) which satisfy the basic requirements of a correlation function, for an arbitrary multipartite state $\rho^{A_{1}...A_{N}}$ under a set of local measurement $\{\Pi_{j}^{A_{1}}\otimes ... \otimes \Pi_{j}^{A_{N}}\}$ is defined as

\begin{eqnarray}
\textit{D}(\rho^{A_{1}...A_{N}})=\min_{\{\Pi_{k}\}}\,[S(\rho^{A_{1}...A_{N}}\|\Phi(\rho^{A_{1}...A_{N}}))\nonumber\\
-\sum_{j=1}^{N}S(\rho^{A_{j}}\|\Phi_{j}(\rho^{A_{j}}))].
\end{eqnarray}

Where $\Phi_{j}(\rho^{A_{j}})=\sum_{i}\Pi_{i}^{A_{j}}\rho^{A_{j}}\Pi_{i}^{A_{j}}$ and $\Phi(\rho^{A_{1}...A_{N}})=\sum_{k}\Pi_{k}\rho^{A_{1}...A_{N}}\Pi_{k}$ with $\Pi_{k}=\Pi_{j_{1}}^{A_{1}}\otimes ... \otimes\Pi_{j_{N}}^{A_{N}}$ and $k$ denoting the index string ($j_{1}...j_{N}$).
We could eliminate dependence on measurement by minimization the set of projectors $\{\Pi_{j_{1}}^{A_{1}}, ... ,\Pi_{j_{N}}^{A_{N}}\}$.
\\ By a set of von-Neumann measurements as

$$
\begin{array}{cc}
 \Pi_{1}^{A_{j}}=\left(\begin{array}{cc}
  \cos^{2}(\frac{\theta_{j}}{2})& e^{i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2}) \\
  e^{-i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2}) & \sin^{2}(\frac{\theta_{j}}{2}) \\
\end{array}
\right),
\end{array}
$$
$$
\begin{array}{cc}
 \Pi_{2}^{A_{j}}= \left(
\begin{array}{cc}
 \sin^{2}(\frac{\theta_{j}}{2})& -e^{-i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2}) \\
 -e^{i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2}) & \cos^{2}(\frac{\theta_{j}}{2})\\
\end{array}
\right),
\end{array}
$$

with $\theta_{j}\in[0,\pi)$ and $\varphi_{j}\in[0,2\pi)$ for $j=1,2,3$, the equation 10 reduces to

\begin{eqnarray}
\textit{D}(\rho(T))=\min_{\{\theta_{j},\varphi_{j}\}}\,[S(\rho(T)\|\Phi(\rho(T)))\nonumber\\
-\sum_{j=1}^{3}S(\rho^{A_{j}}\|\Phi_{j}(\rho^{A_{j}}))].
\end{eqnarray}

By tracing out two qubits, the one qubit density matrices representing the individual subsystems are

$$
\begin{array}{cc}
 \rho^{A_{j=1,2,3}}= \left(
 \begin{array}{cc}
 \rho_{11}+2\rho_{22}+\rho_{44}& 0 \\
 0 & \rho_{22}+2\rho_{44}+\rho_{88} \\
\end{array}
\right).
\end{array}
$$

To find the measurement bases that minimize quantum discord, after some algebraic calculation we have perceived that by adopting local
measurements in the $\sigma_{z}$ eigen basis for each particle, the value of quantum discord will be minimized. It leads to $S(\rho^{A_{j}}\|\Phi_{j}(\rho^{A_{j}}))=0$ and

\begin{eqnarray}
S(\Phi(\rho(T)))&=&-\{\rho_{11}\log_{2}(\rho_{11})+\rho_{88}\log_{2}(\rho_{88})\cr&+&
3\rho_{22}\log_{2}(\rho_{22})+3\rho_{44}\log_{2}(\rho_{44})\}.
\end{eqnarray}

The Entropy $S(\rho((T)))$ can be obtained as

\begin{eqnarray}
S(\rho(T))&=&-\{\rho_{11}\log_{2}(\rho_{11})+\rho_{88}\log_{2}(\rho_{88})\cr&+&2(\rho_{22}-\rho_{23})\log_{2}(\rho_{22}-\rho_{23})\cr
&+&(\rho_{22}+2\rho_{23})\log_{2}(\rho_{22}+2\rho_{23})\cr&+&
2(\rho_{44}-\rho_{46})\log_{2}(\rho_{44}-\rho_{46})\cr&+&(\rho_{44}+2\rho_{46})\log_{2}(\rho_{44}+2\rho_{46})\}.
\end{eqnarray}

In this case, the time evolution of quantum discord is explicitly obtained as

\begin{eqnarray}
\textit{D}(\rho(T))&=&-3\rho_{22}\log_{2}(\rho_{22})-3\rho_{44}\log_{2}(\rho_{44})\cr&+&2(\rho_{22}-\rho_{23})\log_{2}(\rho_{22}-\rho_{23})\cr
&+&(\rho_{22}+2\rho_{23})\log_{2}(\rho_{22}+2\rho_{23})\cr&+&
2(\rho_{44}-\rho_{46})\log_{2}(\rho_{44}-\rho_{46})\cr&+&(\rho_{44}+2\rho_{46})\log_{2}(\rho_{44}+2\rho_{46})\}.
\end{eqnarray}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=1.2\linewidth]{1.eps}
\caption{(Color online) Discord(top panel) and concurrence(bottom panel) measures versus temperature for different values of F\"{o}rster interaction. Electric filed is zero.}
\label{fig1}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=1.2\linewidth]{2.eps}
\caption{(Color online) Discord(top panel) and concurrence(bottom panel) measures versus temperature for different values of F\"{o}rster interaction. The amount of applied Electric filed is about $20\times10^6$ $V/m$.}
\label{fig2}
\end{center}
\end{figure}

\section{3 Results and discussion}

In what follows, we theoretically test for the existence of quantum discord in aforementioned system containing excitonic qbits and that how it behaves with external electric field. From figures 1, comparing the concurrence and the discord measures at different temperatures shows that, quantum discord survives at relatively high temperatures where concurrence is zero, also figures depict at very low temperatures both measures yield the same result. At  higher temperatures, concurrence suddenly diminishes while quantum discord is still finite, slowly decreasing to zero. This is in accord with the fact that discord quantifies nonclassical correlations beyond entanglement. Generally, both quantum correlations
decay with temperature due to thermal relaxation effects. Figures demonstrate that when electrical bias is off the quantum correlations increase monotonically with increasing F\"{o}rster interaction. This can be explained in terms of the increasing  excitonic interaction that leads to increase the correlations. At F\"{o}rster interactions higher than about $10meV$ and at temperatures smaller than $20K$, concurrence is slightly larger than discord. In contrast, at lower F\"{o}rster interactions the opposite trend is observed and discord overcomes the concurrence almost for all temperatures that is because of as mentioned fact that discord indicates nonclassical correlations beyond entanglement even when $\lambda$ is low. From figures it is clear that at lower $\lambda$, Temperature helps to make the correlations so that for very low temperatures, till about $20K$(for discord) and $10K$(for concurrence), discord and concurrence increase with increasing temperature due to the thermal entanglement effects\cite{Arnesen}.\\
Figures 2 displays discord and correlation measures when $\hbar\Omega$ is assumed to be 2.5 $meV$ that corresponds to the electric field($E$) of about $20\times10^6$ $V/m$( this small value for electric field can be realized by experiments very easily). It is clear that discord and concurrence are smaller than that of observed in figure1. This indicates that  turning electric field on, decreases the correlations. It is because electric field makes all the dipoles align in the same direction that results in increasing dipolar repulsive interaction that immediately leads to the reduction of coulomb induced correlations\cite{Saeid}. For all the values of $\lambda$ the discord is survived with increasing $E$ while the concurrence dies for smaller $\lambda$. Our results shows that increasing the electric field overcomes the effect of F\"{o}rster interaction and removes the concurrence for all values of $\lambda$, however, nonzero discord can still be observed for higher values of electric field(figure3). This manifests the importance of discord to measure the quantum correlations.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=1.2\linewidth]{3.eps}
\caption{(Color online) Discord(top panel) and concurrence(bottom panel) measures versus temperature for different values of F\"{o}rster interaction. The amount of applied Electric filed is about $40\times10^6$ $V/m$.}
\label{fig3}
\end{center}
\end{figure}

\section{4. Conclusions}

In summary, We studied the quantum discord and concurrence  measures in the array of optically driven coupled quantum dots. Qubits are excitons in each quantum dot that can be model by dipoles. We used the lower bound of concurrence for thermal density matrix of identical and equidistant coupled QDs at different temperatures. We found that the discord and concurrence are enhanced by increasing the parameter of  F\"{o}rster interaction that is resulted by increasing correlations with increasing this parameter. For very low temperatures and higher F\"{o}rster interaction,  concurrence is slightly larger than discord. In stark contrast, at low F\"{o}rster interactions the opposite trend is observed. Also in very low temperature region, because of thermal entanglement, both measures increase and then tend to zero for higher temperatures induced by thermal relaxation effects. We observed that switching electric field on, correlations diminish, however, discord is survived while concurrence dies for the all values of F\"{o}rster interaction. It is because electric field makes all the dipoles to be parallel that results in increasing dipolar repulsive interaction and finally decrease the entanglement.

\section*{References}