IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 25, NO. 2, MARCH/APRIL 2019 7101508

Superradiant Cancer Hyperthermia Using aBuckyball Assembly of Quantum Dot Emitters

Sudaraka Mallawaarachchi , Student Member, IEEE, Malin Premaratne , and Philip K. Maini

Abstract—With the emergence of nanomedicine, targeted hy-perthermia has a high potential of becoming a first-line cancertreatment modality. However, hyperthermia needs to be preciselycontrolled to avoid damaging adjacent healthy tissues. Due to theuncontrollable transfer of heat from the tumor boundary to healthytissues, it is extremely difficult to control the temperature increase.As a solution, in this paper, we propose using a superradiant emit-ter assembly to deliver an ephemeral and powerful thermal pulseto enhance cancer hyperthermia by reducing damage to healthytissues. Our assembly comprises quantum dot emitters arrangedin the shape of a buckyball. We obtain the criteria for our assemblyto be superradiant and prove that it is possible to control the su-perradiance using an external electric field. We analytically obtainexpressions for the assembly dynamics and conduct thermal studiesusing a simple breast cancer model constructed using experimentalparameters. Our results indicate that using a series of superradi-ant pulses can enhance cancer hyperthermia by minimizing thedamage to adjacent healthy tissues.

Index Terms—Coupled mode analysis, nanotechnology, photonradiation effects.

I. INTRODUCTION

HYPERTHERMIA refers to the cytotoxic increase of bodytemperature beyond its normal value. The effects of hy-

perthermic cytotoxicity are known to intensify in deoxygenatedand acidic cellular environments, establishing its potency as anideal cancer treatment modality. The clinical appeal of hyper-thermia is further enhanced by the fact that it does not involveusing harmful chemicals or radiation. However, high tempera-ture exposure causes protein denaturation in healthy and malig-nant cells alike. Therefore, caution must be exercised in clinicalhyperthermia to minimize potential harm to healthy cells [1].Fortunately, the advent of nanomedicine has paved the wayto targeted treatment modalities that facilitate the direct deliv-ery of thermal energy to malignant cells. This spares healthytissues from significant heat exposure [2]. However, even tar-geted heating of cancer cells will generate undesired results due

Manuscript received May 26, 2018; revised August 22, 2018; accepted August23, 2018. Date of publication August 27, 2018; date of current version October9, 2018. (Corresponding author: Sudaraka Mallawaarachchi.)

S. Mallawaarachchi and M. Premaratne are with the Department of Electricaland Computer Systems Engineering, Advanced Computing and Simulation Lab-oratory, Monash University, Melbourne, Vic 3000 Australia (e-mail:,sudaraka@ieee.org; malin.premaratne@monash.edu).

P. K. Maini is with the Wolfson Centre for Mathematical Biology, Math-ematical Institute, Oxford University, Oxford OX2 6GG, U.K. (e-mail:,maini@maths.ox.ac.uk).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTQE.2018.2867417

to heat propagation, which leads to induced hyperthermia ofneighboring healthy cells. The emergence of nanobiophotonics,which is the amalgamation between biomedical engineering andnanophotonics, presents us with a myriad of control techniquesfor heat propagation and delivery. The utilization of such meth-ods could provide us with the means to tackle these complicatedissues and aid in clinically establishing cancer hyperthermia asa first-line treatment modality.

In nanophotonics, spontaneous emission of photons from ex-cited emitter assemblies is generally regarded without involvingthe contributions from inter-emitter interactions. In such sce-narios, the emission linearly scales with the number of systememitters,N [3]. However, under certain stringent conditions, theinvolvement of inter-emitter interactions gives rise to various in-teresting quantum electrodynamic (QED) phenomena such asFano interferences, spasing, chemical interface damping and su-perradiance [4]–[8]. In particular, the effect of superradiance hasthe potential to scale the emission quadratically with the numberof system emitters (N 2) and curtail the emission time by a factorof (1/N ), metamorphosing the emission to an ephemeral pulsecomprising a large amount of energy. If controlled, the natureof these pulses should cause hyperthermia in adjacent malig-nant cells while sparing the heat propagation into neighboringhealthy cells.

Numerous efforts have been made at observing, understand-ing and controlling superradiance [9], [10]. Unfortunately, su-perradiance in its raw form is a single-occurring event, andtherefore is of minute practical significance. However, success-ful attempts have been made recently at sustaining superradi-ance in the steady-state [11], which has enabled its utilizationin fields such as energy harvesting, biomedical engineering andlasing [12]–[14]. Therefore, a superradiant emitter assembly forcancer hyperthermia is an intriguing concept that is worthwhileexploring.

Although a conceptual proposal of utilizing superradiance forcancer hyperthermia is unequivocally sound, validation of suchan assembly is highly challenging. For instance, although thecharacteristics of superradiance and its sustenance have beenstudied in great detail, few attempts have been made at vali-dating the structural requirements of an emitter assembly to besuperradiant. It is well known that the existence of superradi-ance is strictly contingent on fulfilling two superradiant crite-ria (SRC); namely size and symmetry [3]. These SRC requirestrict geometric distributions for the emitters and significantlylimit the number of emitters that can partake in superradiantemission. In order to propose using superradiance for cancer

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7101508 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 25, NO. 2, MARCH/APRIL 2019

Fig. 1. Proposed superradiant emitter assembly configuration where 60 QDsare placed at the vertices of a truncated icosahedron. The assembly is designednanoscopically to fulfill the size SRC, where the emitter dimensions (r) aremuch smaller than the wavelength with which it interacts (λ). Furthermore,each QD is assumed to comprise a permanent dipole and an external controlfield is used to manipulate the behavior of the resultant dipole of each QD.Under the correct conditions, this assembly fulfills the symmetry SRC as well.

hyperthermia, it is necessary to validate that the assembly ful-fills SRC. Furthermore, it should be possible to involve a largenumber of emitters in order to fulfill the power requirements forcancer hyperthermia.

In this paper, we propose delivering cancer hyperthermia us-ing an emitter assembly comprising quantum dots (QDs), placedat the vertices of a buckyball, as shown in Fig. 1. This shape,formally known as a truncated icosahedron, is the well-knowngeometry of the C60 buckminsterfullerene [15]. The radiativecoupling nature of QD ensembles is experimentally provenand the observed broad spectrum emission characteristics makethem ideal for cancer hyperthermia [16]. Furthermore, QD as-semblies have previously been used for in-vitro hyperthermiawith success [17]. Moreover, the fact that numerous QD typeshave proven biocompatible characteristics undoubtedly provestheir suitability for this application [18]. Furthermore, recentadvances in designing complex QD structures strongly bolstersthe possibility of realizing our proposed shape in an experi-mental setting [19], [20]. We propose using an external controlfield to alter the resultant dipole characteristics of the QDs as amechanism to control superradiant emission. Furthermore, weperform a complete spectral analysis of the assembly to assessits emission dynamics. Then we use realistic parameters forbreast cancer and healthy tissues to conduct thermal simula-tions [21]. Our results prove that the proposed emitter assemblyis capable of delivering superior cancer hyperthermia comparedto conventional models that do not incorporate inter emitterinteractions.

II. METHODS

A. Quantum Dot Emitter Assemblies

We begin our analysis by considering the emission of a gen-eral assembly of N QDs without adhering to any geometric

constraints. We approximate each QD by a two-level dipole andassume that at t = 0, the entire assembly is at the fully excitedstage. Now we model the decaying behavior of the assembly intwo configurations.

1) Non-Interacting Assemblies: Assuming that the QDs areplaced sufficiently far away from each other, there will be nointer-emitter interactions. This is the conventional arrangementand corresponds to the typical scenario of spontaneous emission.The emission of each photon is governed by the single QDspontaneous emission rate, denoted by [3]:

Λat =8π2 |q|23εb�λ3 , (1)

where |q| is the norm of the dipole vector (dipole matrix ele-ment), εb is the bath permittivity, � is the reduced Planck constantand λ is the emitted wavelength.

Let Pω denote the power emitted by the entire assembly atfrequency ω. Now, without loss of generality, it is possible toapproximate the emission characteristics for the assembly using[22]:

Pω,t ≈ N exp(−t/Λat)Pω . (2)

Equation (2) describes an exponentially decaying pulse of en-ergy, the frequency characteristics of which are completely gov-erned by the power spectrum of the emitter assembly.

2) Superradiant Assemblies: An emitter assembly needs toadhere with two SRC in order to exhibit superradiant character-istics. These SRC are discussed in detail in Section II-B. Withoutloss of generality, we now assume that our QD assembly ful-fills these criteria. Due to superradiance, the emitted pulse willscale quadratically and its duration will scale inversely with thenumber of emitters [22]:

Pω,t ≈ N 2 exp(−t/(NΛat))Pω . (3)

Once the spectral characterization is complete, equations (2)and (3) will allow us to simulate and compare the emissiondynamics of these two emitter assemblies.

B. Superradiant Criteria

Superradiance is a well-studied phenomenon that has at-tracted attention due to its unique temporal and spectral char-acteristics. There are strict requirements to which an emitterassembly must adhere, in order to depict superradiance.

1) Size SRC: This SRC is intuitive; when the linear size ofthe assembly (S) is much smaller than the wavelength withwhich it interacts (S � λ), the emitters become indiscernibleto the emitted energy. Therefore, rather than functioning asindividual emitters, the entire assembly functions as a globalradiating dipole. This phenomenon is fundamental to superra-diance. Therefore, any emitter assembly confined within sub-wavelength volumes fulfills the size criterion and is a potentialcandidate for superradiance.

2) Symmetry SRC: This SRC depends on the form of inter-emitter interactions. For QDs that are sufficiently proximal toeach other, but not close enough for Forster type coupling ortunneling, it is possible to assume that Van der Waal’s coupling

MALLAWAARACHCHI et al.: SUPERRADIANT CANCER HYPERTHERMIA USING A BUCKYBALL ASSEMBLY OF QUANTUM DOT EMITTERS 7101508

Fig. 2. (a) Each QD in the assembly (say p) is assumed to interact with 3nearest neighbors (a, b, c), each at a distance of 2r. Interactions beyond thislimit can be safely ignored for sufficiently large r. (b) The symmetry SRC ofthe assembly depends on the dot product quantity: qp .rp b , the value of whichrelies on the shown inter-vector angle β . Due to the symmetry of a truncatedicosahedron, this angle is constant for all QDs across the assembly. This impliesthat the model is capable of fulfilling the symmetry SRC.

is the main form of inter-emitter photon transfer. In order tounderstand the behavior of this coupling, it is necessary to beginwith fundamental QED theory to compose and solve the masterequation for the superradiant system. This lengthy derivation isdetailed elsewhere in the literature [3].

The imaginary part of the master equation presents the fol-lowing expression for inter-emitter Van der Waal’s interactionsbetween the QDs p and b as shown in Fig. 2(a):

Λint(p, b) =|qp |24πεb�

1r3pb

[1 − 3

(qp · rpb)2

|rpb |2], (4)

where qp denotes the resultant unit dipole vector of QD p.The terms rpb and |rpb | denote the interaction vector betweenthe QDs and its norm, respectively. This quantity is commonlyreferred to as the hopping-interaction strength because it definesthe virtual photon exchange rate among emitters [9].

The dot product term: qp · rpb , as shown in Fig. 2(b), is piv-otal to the symmetry SRC. An assembly of QDs must maintainthis coupling rate at a uniform value across all interacting QDs.If this criterion does not hold, then the emitters will couplewith each other at different rates and superradiance will ceaseto exist. An arbitrary geometric distribution of QDs will obvi-ously breach the symmetry SRC because Λint(p, b) will takearbitrary values for different interacting QDs, depending on theinter-emitter distances and their unique resultant dipoles. Thisclearly implies that the symmetry SRC cannot be guaranteedfor an arbitrary geometric distribution of interacting QDs. How-ever, a symmetric ring or an equidistant linear QD configura-tion clearly satisfies the SRC, since it is possible to maintainqp ⊥ rpq ⇒ qp · rpq = 0,∀p, q, which explains the popularityof these geometries in the literature [23], [24]. However, it iswell-known that such structures support superradiant emissiononly up to a few emitters. This is clearly a limitation and willhinder the practical applicability of superradiance [9].

Furthermore, it is hypothesized that an assembly maintainingsuperradiant criteria should also exhibit superabsorption char-acteristics due to quantum time reversal symmetry [9]. This im-plies that the proposed assembly should theoretically depict ananalogous enhancement in the initial heating process involvingthe absorption of electromagnetic radiation.

Next we show that our proposed emitter assembly can fulfillboth these SRC while maintaining superradiance for an assem-bly of 60 QDs.

C. Proposed Emitter Assembly

The emitter assembly proposed in Fig. 1 comprises 60 QDsthat are placed at the vertices of a truncated icosahedron. TheQDs are positioned according to Cartesian coordinates that arecomputed using the well-known even permutations method. Fur-thermore, the QDs are labeled using the numbering schemebased on permutations in the alternating group A5 [15].

1) Maintaining Size SRC: When constructed using a scalingdimension (say r), all vertices of the truncated icosahedron willlie on a sphere with radius: r

√9ψ + 10, where ψ is the golden

mean and the edge length will be 2r [15]. This implies thatrpb = 2r, where QD b is a nearest neighbor of p. As shown inFig. 2(a), we assume that each QD will have exactly 3 nearestneighbors, each at a distance of 2r. This is a fair assumption,given that the hopping-interaction strength is proportional to1/r3

pq and within nanoscopic dimensions, interactions beyondthe nearest neighbor limit are usually safely ignored [3], [9],[10]. It is noted that by choosing r appropriately, our modelcan be designed to fulfill the size SRC. Furthermore, by varyingthe QD dimensions while maintaining the size SRC, it will bepossible to tune the emission characteristics due to the resultingchanges to QD parameters such as absorption and emission.

2) Maintaining Symmetry SRC: Assuming that it is possibleto obtain a geometrical QD assembly where the resultant po-larization vector of each QD intersects with the origin of thetruncated icosahedron and the center of the respective QD, it ispossible to maintain the SRC. In this configuration, each QD willhave a unique unit polarization vector. As shown in Fig. 2(b), un-der this assumption, the angle β then determines the value of thehopping-interaction strength in equation (4). Simple trigonom-etry shows that Λint(p, b) = 0.8778616|qp |2/4πεb�r3 is a con-stant where b is a nearest neighbor of p. Maintaining a constantvalue for Λint(p, b),∀(p, b), where b is a nearest neighbor of p,is clearly in agreement with the symmetry SRC.

Therefore, our proposed emitter assembly should be superra-diant under these conditions. However, it is noted that by usingnon-spherical QDs, the symmetry SRC is potentially challengeddue to anisotropy and the existence of multiple resonant modes.

D. Superradiant Controllability

Now we propose a method to realistically achieve the re-sultant dipole arrangement to maintain the symmetry SRC. Itis known that QDs built using specific material and size cri-teria possess intrinsic, permanent dipoles of high potentials[25]–[27]. Furthermore, it is well-known that the applicationof an external electric field will induce a dipole moment dueto the polarizability of the exciton. Fig. 3(a) depicts this sce-nario. Here we have shown the field-induced dipoles using redarrows. Given the nanoscopic dimensions of the emitter assem-bly, we have assumed that all QDs uniformly interact with theincident electric field. The white arrows denote the direction ofthe permanent dipoles of each QD. When two dipole moments

7101508 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 25, NO. 2, MARCH/APRIL 2019

Fig. 3. (a) An assembly of QDs with large, intrinsic, permanent dipoles (whitearrows) placed at the vertices of a truncated icosahedron. When the assemblyis exposed to an electric field, a temporary dipole is induced (red arrows)within the QDs. Since the assembly is minuscule, it is assumed that each QDinteracts with the incident field in the same manner. Therefore, the induceddipoles have the same magnitude and direction. (b) The resultant dipole mo-ments for the two dipoles are shown by arrows. When the permanent and induceddipoles are perfectly matched, it is possible to have all resultant dipoles origi-nating from the center of the buckyball and traverse through the center of thecorresponding QD.

are present within the same QD, the resultant dipole momentis the pivotal parameter controlling inter-emitter couplings. Theresultant dipole moment for each QD is shown by the greenarrows in Fig. 3(b).

Under this configuration, superradiant emission of the pro-posed assembly can clearly be controlled by alternating thecontrol field. When the control field is OFF, the resultant dipolewill be the permanent dipole for each QD and the symmetrySRC is clearly violated. Emission will still exist but withoutsuperradiant enhancement. When the control field is ON, theinduced dipoles will change the resultant dipole to the configu-ration in Fig. 3(b). This will reinstate the superradiant behaviorto the assembly. When superradiant, the system will emit a highburst of energy within a very short time. Alternating the controlfield will result in pulsated emission of high energy, ephemeralpulses. It is noted that this controllable pulsating behavior is notpresent in conventional, non-interacting emitter assemblies.

However, in a purely practical perspective, arranging a QDassembly to perfectly meet the presented mathematical require-ments will be challenging. However, recent advancements inQD technology strongly suggests the possibility of realizing thestructure [19], [20]. Furthermore, it is noted that the value ofΛint(p, b) beyond the nearest neighbor condition is not a con-stant under our configuration. However, in a practical setting, thepresence of such trivial interactions will not hinder the effectsof superradiance [28].

Assuming all these criteria are maintained, the next step is tostudy the spectral characteristics of the QD emitter assembly.Once the spectra are known, it is possible to use equations (2)and (3) to conduct simulations and study the dynamics of theproposed assembly.

E. Spectral Characterization

Any emitter assembly should be benchmarked using spec-tral analysis in order to ascertain its power delivering capabil-ities. Without loss of generality, coupled mode theory presents

a method to directly express the fluctuations of electric energywithin the emitters of the assembly [29].

1) Coupled Mode Treatment: Let the normalized, time andfrequency varying, electric field amplitudes within the QD emit-ters be denoted by: a(ω, t) = [a1(ω, t) . . . aN (ω, t)]T . Then thesystem dynamics are captured using the simple expression:

∂

∂ta(ω, t) = [jΩ0 − Λ]a(ω, t) + �(ω, t), (5)

where Ω0 = diag[ω10 . . . ω

N0 ] contains the resonant frequencies

of each emitter. The N ×N matrix Λ = ΛA + ΛE + Λint ,encodes the rates of photon absorption: ΛA = diag[Λ1

A . . .ΛNA ]

and emission: ΛE = diag[Λ1E . . .Λ

NE ] of all emitters. The inter-

emitter coupling rates are encoded within Λint and takes theform of equation (4). Generally, the N × 1 matrix �(ω, t)expresses the photon generation rate within emitters [29].

For the case of QDs operating in the thermal regime, weassume that each emitter is modeled based on a gray-body radi-ator. Since each source emits in accordance with the fluctuationdissipation theorem, and assuming a stationary characteristicfor each source, �(ω, t) converts to the time invariant form:�(ω) =

√ΛAn(ω). Now, the N × 1 matrix n(ω) has the form

[30]:

〈n∗p(ω)nb(ω)〉 = Θ(ω, T )δ(ω − ω′)δpb , (6)

where Θ(ω, T ) = εω�ω/(2π exp(�ω/kBT )) is the well-knownPlanck energy of thermal photons radiating at temperatureT andangular frequency ω. Here, εω and kB denote the emissivity ofthe QDs and the Boltzman constant, respectively.

Equation (5) describes the temporal and spectral dynamicsof our system, the characteristics of which are fully definedusing photon couping rates. In order to conduct simulations, itis necessary to establish analytical relationships between them.

It is obvious that all generated photons of the assembly musteither be absorbed within the generated emitter, coupled to freespace or coupled to other emitters. Furthermore, due to energyconservation, all emitted photons from the system must eitherbe coupled to other emitters or free space. Considering only theemitted photons, equation (5) modifies to ∂

∂ t a(ω, t) = [jΩ0 −(ΛE + Λint)]a(ω, t), which yields:

∂

∂ta∗a = a∗[−2(ΛE + Λint)]a. (7)

The negative sign implies that the process is an emission.We continue our analysis by focusing on a finite region of

free space of length L around an arbitrary emitter p, as shownin Fig. 4. This region is quantified into 2m+ 1 vertical radi-ation channels, where m = L/λ0 . Since all these photons areemanating from the set of emitters, energy conservation yields:

2(ΛE + Λint) =[Λ∀p→∀φE

]T [Λ∀p→∀φE

], (8)

where Λ∀p→∀φE is the rate matrix for all emitters coupling to all

radiation channels.To obtain a relationship for Λ∀p→∀φ

E , we analyze couplingbetween the arbitrary pth emitter shown in Fig. 4 and free space.We assume that the pth QD is an isotropic radiator and the

MALLAWAARACHCHI et al.: SUPERRADIANT CANCER HYPERTHERMIA USING A BUCKYBALL ASSEMBLY OF QUANTUM DOT EMITTERS 7101508

Fig. 4. This diagram illustrates the coupling of emission from the pth QD withfree space. For analysis, the free space is first confined to a width of L and thevertical region is divided into 2m + 1 radiation channels where m = L/λ0 . Acircular geometry of radiusL/2 is selected around the pth emitter. Each channelcan now be uniquely identified using the vertical angle φ. Realistic free spaceconditions are later achieved by applying the limit L → ∞.

emission is distributed across the vertical radiation channels. Formathematical convenience, we begin by analyzing the radiationchannels confined within a circular geometry of radius L/2.Now, each channel can be uniquely characterized by the verticalangle (φ).

The photon emission rate from the pth QD can be expressedas: Λp

E = 2∑pi/2

φ=−π/2 Λp→φE . Here, the term Λp→φ

E denotes thephoton emission rate to channel φ. The rate to each channelcan be projected onto the perpendicular direction by using therelationship: Λp→⊥

E = Λp→φE cosφ. Hence,

ΛpE = 2Λp→⊥

E

∑−π

2 <φ<π2

1cosφ

. (9)

Note that φ can only assume discrete and realistic valuescorresponding to channels in Fig. 4. Applying trigonomet-ric relations and the limit L→ ∞ to equation (9) yields:ΛpE = 2Λp→⊥

E mπ. Based on this result, it is possible to ob-tain the following relationship for the emitter-channel coupling:Λp→φE = Λp

E secφ/2mπ. This is the general form of an ele-ment in matrix Λ∀p→∀φ

E . Substituting this result in equation (8)directly yields:

[Λint ](p, q) =12

√ΛpEΛq

E (1 − δp,q ), (10)

which is the sought after relationship to completely define theemitter assembly with measurable photon coupling rates. It isnoted that since only the nearest neighbor interactions are usedin our model, Λint is a sparse matrix as shown in Fig. 5.

2) Analytical Emission Spectrum: Fourier transformationsand the framework of the fluctuation dissipation theorem forthermal emission can be used to establish an analytical expres-sion for the spectrum. We have published a lengthy and descrip-tive derivation of the analytical emission spectrum previously

Fig. 5. In this grid arrangement, an interacting pair of QDs is highlightedusing a colored box. The presence of these interactions is critical in achievingsuperradiant emission. The assembly QDs are labeled using the same techniqueused in Fig 1. Only the neighboring QDs that are highlighted in colored boxesare considered to interact with each other. This implies that the interactionrate matrix Λin t is sparse and occupies the same rate value at each of thehighlighted places. Incorporating interactions beyond this limit will break thesymmetry SRC, thus rendering the system subradiant.

[31], [32] and summarize the following result for the sake ofcompleteness of this paper.

We begin our analysis using the free space channel modelshown in Fig. 4. For a single radiation channel, the total emittedpower from the pth QD across the spectrum can be expressedby considering the relationships in equations (7) and (8):

〈Pq(t,φ)〉 =

N∑p=1

N∑q=1

⟨ap(t)∗ [E ]T(p,φ) [E ](q ,φ) aq (t)

⟩. (11)

For convenience, we define: E = Λ∀p→∀φE , the matrix al-

ready known through equation (8). Now, solving equa-tion (11) depends on deriving the expected value of inter-nal emitter amplitudes at time t. Using Fourier relations,we can convert this quantity as follows: 〈ap(t)∗aq (t)〉 =∫ ∞

0 dω∫ ∞

0 dω′(exp [−j(ω − ω′)t]〈ap(ω)∗aq (ω′)〉). Now weneed to solve for 〈ap(ω)∗aq (ω′)〉, which can be done by mod-ifying equation (5). We avoid strenuous recalculations and usethe result directly for this derivation. Furthermore, according toFourier analysis: 〈Pq

(t,φ)〉 =∫ ∞

0 Pq(ω,φ)dω. Finally, the collec-

tive emission from the assembly can be expressed as a summa-tion. Based on these facts, we write:

P(ω,φ) =λ0Θ(ω, T )4mπ2 cosφ

N∑p=1

N∑q=1

N∑r=1

Q∗(p,r)Q(r,u)

√ΛpEΛq

E , (12)

where the matrix Q = [[jΩ + Λ][√

ΛA ]]−1 results from the ex-pression for 〈ap(ω)∗aq (ω′)〉. Finally, the channel dependenceof equation (12) can be omitted by normalizing using the chan-nel flux density [22], which yields the following for the final

7101508 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 25, NO. 2, MARCH/APRIL 2019

emission spectrum:

Pω = P0λ0

2π

N∑p=1

N∑q=1

N∑r=1

Q∗(p,r)Q(r,u)

√ΛpEΛq

E , (13)

where P0 = εω�ω2/(4π2c exp (�ω/kBT ) − 1) is the powerspectral density of a gray-body radiator [22].

This is the final analytical result obtained in this manuscriptand can now be substituted in equations (2) and (3) to conducttissue model simulations.

III. RESULTS AND DISCUSSION

A. Simulation Parameters

First we analyzed the non-interacting emitter assembly, thedynamics of which are described by equation (2). Here, weassumed that Λint(p, b) = 0, ∀p, b and solved equation (13).A similar approach was used to solve equation (3), where weassumed that Λint(p, b) takes the form of the sparse matrixshown in Fig. 5.

We assumed that the initial temperature of both emitterassemblies is at 323.15 K, which is well suited for cancerhyperthermia. This corresponds to a resonance frequency of7.84 × 1013 rads−1 for thermal photons. In the simulation, weassume that all emitters are engulfed within cancerous tissue.Therefore, we assumed the bath permittivity: εb = 7.17, whichis in agreement with published results [33]. For simplicity, weignored the temperature and frequency dependencies of εb forthis simulation. In order to perform cancer hyperthermia effec-tively, we assumed the resultant dipole matrix element of eachQD as 3.3336 × 10−28 Cm. This value is in agreement withmeasured values for CdSe QDs, each with a diameter of 5.6 nm[25]. The separation between two adjacent QDs was maintainedat 4.4 nm, which corresponds to r = 2.2 nm.

For the superradiant assembly, these parameters correspond toa total inter-emitter coupling rate of 1.309 × 1013 s−1 for all in-teracting QDs. Since we are assuming a 3-nearest neighbor sce-nario, this value has to be scaled appropriately to find the effec-tive coupling rate Λint . The value for ΛE = 2.618 × 1013s−1

can be directly calculated using equation (10). Due to the lackof experimental data for photon emissivity and absorptivity, wepicked a variety of photon generation rates for our simulations.Finally, the value for ΛA is found using the principle of energyconservation.

The heat transfer simulation for the tissue model is solvedfor both transient and steady-states. At t = 0, the emitters, tu-mor and healthy tissue regions are assumed to be at 323.15 K,311.15 K and 310.5 K, respectively. For both tissue types,thermal conductivity and mass density were maintained at0.48 Wm−1K−1 and 1080 kgm−3, respectively. Furthermore,specific heat capacities of 3000 and 3500 Jkg−1K−1 were main-tained for healthy and tumor tissue types, respectively. For thesimulation, we placed 8 emitter bundles close to the edge ofthe tissue region. Each emitter bundle was assumed to comprise1000 buckyball assemblies. The size of a single emitter assem-bly was assumed to be negligibly small compared to the size ofthe tumor and the specific heat capacity of each assembly wasassumed to be negligibly small. All emitter parameters were ob-

Fig. 6. The emission spectra are plotted using the derived analytical spectrumin equation (13). Continuous and dashed lines correspond to the superradiantand non-interacting assemblies, respectively. The curves correspond to differentQD configurations with varying photon generation rates. The non-interacting as-sembly always outperforms the superradiant counterpart, which is the expectedbehavior.

tained by solving equations (2) and (3) for the non-resonant andsuperradiant emitter assembly simulations, respectively [21].

However, it is noted that different QD parameters are re-quired for a realistic simulation because the biocompatibilityof TOPO-coated CdSe QDs is unestablished. Furthermore, itis necessary to functionalize the surface of each QD prior toin-vivo applications, which will result in further modified QDparameters. Furthermore, we have ignored the temperature de-pendent variations to these parameters. When simulating thetissue model, we have assumed that the effects of blood perfu-sion and metabolic heat generation are negligible. We have alsoassumed that the simulated region is free from blood vessels.Moreover, it is known that the application of hyperthermia willalter the thermal tissue characteristics. We have ignored suchchanges as well. In fact, the simulated tissue model is a crudeapproximation, that is presented to juxtapose and validate thesuperior cancer hyperthermia characteristics of the superradiantassembly in comparison with its conventional counterpart.

B. Emission Characteristics

As shown in Fig. 6, for the time independent emission, aLorentzian pattern is observed for both non-resonant and super-radiant assemblies. This is the expected result for emissions ofthis nature [22]. These results are obtained by solving equation(10) for four photon generation rates. For each instance, it isevident that the non-resonant assembly emits a larger numberof photons, compared to its superradiant counterpart. This is ex-pected, given that the resonant emitter assembly suffers lossesdue to inter-emitter interactions.

Fig. 7 shows the complete dynamics for the two assemblies.Color values depict the log of generated energy, at each fre-quency and time point. These results are for the highest photongeneration rate shown in Fig. 6. It is noted that, although thesuperradiant assembly generates a lower number of photons fora given frequency as shown in Fig. 6, the emission power atresonance at the beginning of the superradiant pulse is much

MALLAWAARACHCHI et al.: SUPERRADIANT CANCER HYPERTHERMIA USING A BUCKYBALL ASSEMBLY OF QUANTUM DOT EMITTERS 7101508

Fig. 7. These plots depict the emission dynamics for two analogous(a) superradiant, (b) non-interacting, emitter assemblies. The photon genera-tion rate is assumed to be 1.04014 × 1013 s−1. These results are obtained bysolving (a) equation (3) and (b) equation (2), respectively. Color values corre-spond to the log of generated energy at each time and frequency point. Thenegative values suggest that a single assembly only generates a minute amountof heat. The black contour lines highlight the contrasting emission dynamicpatterns of the two emitter assemblies.

Fig. 8. Artistically illustrated thermal characterization of the non-interactingand superradiant assemblies. (a) The simulation state at t = 0. Three clearlydisparate temperature regions are visualized. The emitters, tumor and healthytissue are at 323.15, 311.15, and 310.15 K, respectively. (b, d) During transience,the superradiant assembly has clearly obtained a sufficiently high temperaturewithin the tumor tissue for cancer hyperthermia. The non-resonant assemblyhas a lower power emission and the temperature increase is marginal. (c, e) Atthe steady-state, the superradiant assembly maintains the tumor temperature ataround 312 K while maintaining the healthy tissues at harmless temperatures.In contrast, the non-resonant assembly has increased the temperature of healthytissues adjacent to the tumor to dangerously high values.

higher (∼ × 104) than the power generated by the non-resonantcounterpart at any frequency and time point.

C. Tissue Model Temperature Characteristics

Initially, it is assumed that there are 3 clearly disparate temper-ature zones as shown in Fig. 8(a). The tumor region is confinedto a circle of radius 15 μm and initially maintains a temperatureof 311.15 K. The sub-figures throughout Fig. 8 maintain thesame scale and the human breast outline is overlaid as a purelyartistic illustration and its dimensions are ignored.

At t = 0, both emitter assemblies have the same conditionsas shown in Fig. 8(a). During transience, it is observed thatthe superradiant assembly is capable of effectively increasingthe temperature of the tumor region to around 317 K as shownin Fig. 8(b). This increase is sufficient for successful cancer

hyperthermia. Comparatively, the non-resonant assembly barelyincreases the tumor temperature beyond its starting value dur-ing transience as shown in Fig. 8(d). This result is clearly inagreement with Fig. 7. The high powered and ephemeral super-radiant pulse clearly increase the temperature of tumor tissue atthe commencement of emission, compared to the conventionalcounterpart. At steady-state, it is observed that the superradi-ant assembly barely increases the normal tissue temperaturebeyond its starting value as shown in Fig. 8(c). Conversely, atsteady-state, the conventional assembly has increased the tem-perature of the adjacent healthy tissue beyond 320 K as shown inFig. 8(e). Since the conventional assembly emits a higher num-ber of photons compared to the superradiant counterpart, thisresult is expected. It is noted that both these results are recordedfor the decay of a single energy pulse.

These observations imply that, when using a conventionalemitter assembly for cancer hyperthermia, inadvertent hyper-thermia of healthy tissues is unavoidable. This is attributed tothe slow release of a large number of thermal photons, whichleads to undesired temperature rises in adjacent healthy tissues.However, our results suggest that by utilizing a superradiantpulse instead, it is possible to significantly reduce the hyperther-mia of adjacent healthy tissues. Therefore, using our proposedassembly to repeatedly deliver a superradiant pulse of heat fromwithin the tumor environment will enhance cancer hyperthermiaover using its conventional counterpart.

IV. CONCLUSION

We have analyzed and illustrated that an assembly of QDthermal emitters, placed at the vertices of a truncated icosahe-dron, can achieve controlled superradiant emission under theinfluence of an external electric field. We simulated and provedthat under the right conditions, such an emitter assembly can beused to enhance cancer hyperthermia by minimizing the harmto adjacent healthy tissues.

For our analysis, we derived analytical expressions to char-acterize the dynamics of the emitter assembly. To validate ourclaims, we used experimental parameters to model a breast can-cer environment and conducted thermal simulations. Our resultsclearly indicate that the superradiant emitter assembly outper-forms the conventional, non-resonant counterpart in deliveringenhanced cancer hyperthermia by reducing harm to adjacenthealthy tissues.

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Sudaraka Mallawarachchi received the B.Sc. de-gree in electronic and telecommunication engineer-ing from the University of Moratuwa, Moratuwa, SriLanka, in 2014. Until 2016, he was a Lecturer with thesame institute. Since then, he has been working withProf. M. Premaratne for the Ph.D. degree in electricaland computer systems engineering, Monash Univer-sity, Melbourne, Vic, Australia. His current researchinterests include nanobiophotonics, spasers, and pho-tothermal cancer therapy with an emphasis on com-puter modeling and simulations.

Malin Premaratne received the B.Sc. degree inmathematics, the B.E. degree in electrical and elec-tronics engineering with first-class honors, and thePh.D. degree from the University of Melbourne(UoM), Melbourne, Australia, in 1995 and 1998,respectively. From 1998 to 2000, he was with thePhotonics Research Laboratory, UoM where he wasthe coproject leader of the CRC Optical Amplifierproject and was associated with Telstra, Australia,and Hewlett Packard, USA. From 2001 to 2003, hewas as a consultant to several companies including

Cisco, Lucent, Ericsson, Siemens, VPISystems, Telcordia, Ciena, and Tellium.Since 2004, he has guided the research program in high-performance computingapplications and complex systems simulations with the Advanced Computingand Simulation Laboratory, Monash University where he is currently the VicePresident of the academic board and a Full Professor. He is also a VisitingResearcher with the Jet Propulsion Laboratory with Caltech, the UoM, Aus-tralian National University, Canberra ACT, Australia, University of California,Los Angeles, Los Angeles, CA, USA, the University of Rochester New York,Rochester, NY, USA, and Oxford University, Oxford, OX, U.K. He has au-thored more than 200 journal papers and a book. He has given presentations onmodeling and simulation of optical devices at major international meetings andscientific institutions in the USA, Europe, Asia, and Australia. He is currentlyan Associate Editor for the IEEE PHOTONICS TECHNOLOGY LETTERS, the IEEEPHOTONICS JOURNAl, and The Optical Society Advances in Optics and Photon-ics Journal. He is a Fellow of the Optical Society of America, and The Instituteof Engineers Australia.

Philip K. Maini received the B.A. degree in math-ematics from Balliol College, Oxford, OX, U.K., in1982 and the D.Phil. degree, in 1985. In 1988 hewas appointed an Assistant Professor with the De-partment of Mathematics, University of Utah, SaltLake City, UT, USA. In 1990 he returned to Oxfordas a University Lecturer and in 1998 was appointedProfessor of Mathematical Biology by Recognitionof Distinction and Director of the WCMB. In 2005he was appointed Statutory Professor of Mathemati-cal Biology. He is on the editorial boards of a large

number of journals, including serving as the Editor-in-Chief of the Bulletin ofMathematical Biology from 2002 to 2015. He is a Fellow of the IMA, a SIAMFellow, an Inaugural SMB Fellow, a Fellow of the Royal Society of Biology, andMiembro Correspondiente (Foreign Fellow), La Academia Mexicana de Cien-cias. In 2015 he was elected Fellow of the Royal Society, and in 2017 he waselected Fellow of the Academy of Medical Sciences, and Foreign Fellow of theIndian National Science Academy. His research interests include the modelingof avascular and vascular tumours, normal and abnormal wound healing, and anumber of applications of mathematical modeling in pattern formation in earlydevelopment, as well as the theoretical analysis of the mathematical modelsthat arise in all these applications. He has more than 300 publications and hasheld visiting positions at a number of universities worldwide. He was a Distin-guished Foreign Visiting Fellow, Hokkaido University (2002). He coauthoredwith Jonathan Sherratt and Paul Dale a Bellman Prize winning paper (1997),was awarded a Royal Society Leverhulme Trust Senior Research Fellowshipfrom 2001 to 2002 and a Royal Society-Wolfson Research Merit Award from2006 to 2011. In 2009 he was awarded the LMS Naylor Prize and Lectureshipand in 2014 he was listed in “The World’s Most Influential Scientific Minds”(Thomson Reuters). In 2007 he was awarded the Arthur T. Winfree Prize fromthe Society of Mathematical Biology.