Computers in Biology and Medicine 41 (2011) 771–779

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Computers in Biology and Medicine

0010-48

doi:10.1

n Corr

Institut

E-m

journal homepage: www.elsevier.com/locate/cbm

Predicting DNA-mediated drug delivery in interior carcinoma usingelectromagnetically excited nanoparticles

Soham Ghosh a, Tamal Das b, Suman Chakraborty c, Sarit K. Das a,n

a Department of Mechanical Engineering, IIT Madras, Chennai 600036, Indiab Department of Biotechnology, IIT Kharagpur, Kharagpur 721302, Indiac Department of Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India

a r t i c l e i n f o

Article history:

Received 7 June 2010

Accepted 17 June 2011

Keywords:

Magnetic nanoparticle

Drug delivery

DNA denaturation

Modeling

Simulation

25/$ - see front matter & 2011 Elsevier Ltd. A

016/j.compbiomed.2011.06.013

espondence to: HTTP Lab, Department of Me

e of Technology Madras, Chennai 600036, Ind

ail address: skdas@iitm.ac.in (S.K. Das).

a b s t r a c t

Tumor-site-specific delivery of anti-cancer drugs remains one of the most prevailing problems in

cancer treatment. While conventional means of chemo-delivery invariably leave different degrees of

side-effects on healthy tissues, in recent times, intelligent chemical designs have been exploited to

reduce the cross-consequences. In particular, the strategies involving superparamaganetic nanoparti-

cles with surface assembled oligonucleotides as therapeutic carrier have raised affirmative promises.

Process is designed in such a way that the therapeutic molecules are released preferentially at target

site as the complementary oligonucleotide chains dissociate over the heat generated by the nanopar-

ticles under the excitation of low frequency electromagnetic energy. In spite of the preliminary

demonstrations, analytical comprehension of the entire process especially on the purview of non-trivial

interactions between stochastic phase-transition phenomena of oligonucleotide chains and hierarchical

organization of in vivo transport processes remains unknown. Here, we propose an integrated

computational predictive model to interpret the efficacy of drug delivery in the aforementioned

process. The basic physics of heat generation by superparamagnetic nanoparticles in presence of

external electromagnetic field has been coupled with transient biological heat transfer model and the

statistical mechanics based oligonucleotide denaturation dynamics. Conjunctionally, we have intro-

duced a set of hierarchically appropriate transport processes to mimic the in vivo drug delivery system.

The subsequent interstitial diffusion and convection of the various species involved in the process over

time was simulated assuming a porous media model of the carcinoma. As a result, the model

predictions exhibit excellent congruence with available experimental results. To delineate a broader

spectrum of a priori speculations, we have investigated the effects of different tunable parameters such

as magnetizing field strength, nanoparticle size, diffusion coefficients, porous media parameters and

different oligonucleotide sequences on temperature rise and site-specific drug release. The proposed

model, thus, provides a generic framework for the betterment of nanoparticle mediated drug delivery,

which is expected to impart significant impact on cancer therapy.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Targeting specific tumor site is a pivotal theme in modernmedicine towards designing improved diagnosis and treatment ofcancer. Pertinently, nanotechnology based drug delivery systemspossess the ability of localized, controlled and targeted release oftherapeutic chemicals with minimal perturbation to the neigh-boring normal cells [1,2]. To this purpose, of diverse types ofmaterials being experimented over the years, magnetic elementsand alloys have been most effective, predominantly due to their

ll rights reserved.

chanical Engineering, Indian

ia.

easily modifiable surface chemistry, visibility to non-invasivemethods such as magnetic resonance imaging (MRI) and mostnotably, exclusive efficacy in converting weak electromagneticwaves to extremely concentrated heat energy. The heating ofmagnetic nanoparticles arises due to different processes ofmagnetization reversal in the particle system [3,4]. If the charac-teristics length scale of particle is reduced to few nanometers,intrinsic M–H curve becomes sigmoidal and, importantly, non-hysteretic. In this state, the magnetic moment of the particle is,as a whole, free to fluctuate in response to radiative energy,maintaining the individual atomic moments in their orderedstate relative to each other. Simultaneously, the energy barrierfor magnetization reversal also diminishes to such extent thatthermal fluctuations can lead to relaxation phenomena. Theseeffects, in juxtaposition, induce the heat generation process. It is

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779772

pertinent to mention that the frequency range used (350–400 kHz)to activate the magnetic nanoparticles, produces insignificant heat-ing in healthy tissue [5].

Previously, researchers have extensively utilized magneticneodymium-iron-bor capsules [6], magnetic microspheres [7,8],activated carbonated iron [9,10] and magnetoliposomes [11,12]as potential drug carriers. However, a recently invented strategy[13], as described in Fig. 1(a) exploiting superparamaganeticnanoparticles with surface assembled oligonucleotides (a shortchain of nucleotides; no100), has been perceived to possess theability of localized release possibly to most superior extent. In thismethod, instead of coupling the drug directly to nanoparticle, anintermediate heat-labile linker in the form of double strandedshort nucleic acid chain is introduced, which under a rise intemperature field facilitates the release process in a very confinedway. While one strand of the oligonucleotide chain is covalentlylinked with a multivalent superparamagnetic nanoparticle, thecomplementary strand is coupled with the therapeutic medicine.As imaged non-invasively by MRI, these multivalent nanoparticlesinvade the fenestrations of the angiogenic vasculature to extra-vasate into the tumor stroma [13]. The antibodies carried by thenanoparticles target the epitopes either exclusive or overex-pressed in cancer cells. Once the target is reached, the nanopar-ticles are excited with low frequency electromagnetic energy,which raises the local temperature (42–45 1C) enough to disrupthydrogen bonding between complementary strands and facilitate

Fig. 1. (a) Electromagnetic field triggered drug release from a ‘parent’ DNA strand attac

tumor through 1. Transvascular transport, 2. interstitial transport and 3. intracellular tr

gradient.

site-specific release of the therapeutic agent. After the drug getsreleased, it is transported through the tumor interstitial space,diffuses inside the cancer cells and culminates into gross necrosisof malignant tissues.

Assimilating the aforementioned facts, it becomes readilyevident that the entire process is composed of three criticalthermofluidic steps namely the heat generation by nanoparticles,dissociation of oligonucleotide strands and transport of drugmolecules inside the tumor site, culminating into tissue necrosis.Though these processes have been subjected to extensive experi-mental investigations either in isolation or in combination[14–17] there has not been much effort towards theoreticalelucidation of entangled and intricate thermo-fluidic-biologicalprocesses. On the ground of largely varying experimental results,we intent to emphasize the need for extensive analysis of thegoverning processes, which should endow us with more rationalview and optimized parametric control of this specific cancertreatment method. In the present work, we have proposed anintegrated computational predictive model to interpret the effi-cacy of drug delivery in the process. The basic physics of heatgeneration by superparamagnetic nanoparticles in presence ofexternal electromagnetic field has been coupled with transientbiological heat transfer model and the statistical thermodynamicsbased generalized Langevin formalism [18] of complete oligonu-cleotide denaturation to delineate a simulation tool in order tofind out the amount of drug released. With respect to the

hed to dextran coated iron oxide nanoparticle. (b) The drug molecule entering the

ansport and 4. the back flow of extravasated drug due to interstitial fluid pressure

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779 773

transport of drug molecules, it is important to conceive that nouniform dogma of fluid dynamics exists as different forcesprevails at different strata of physiological architecture rangingfrom the whole tissue to single cell. Hence, we resort to ahierarchically adaptive model approach. In this purview, first,the interstitial diffusion and convection of the various speciesinvolved in the process over time has been simulated assuming aporous media model of the tumor [19]. The transvascular migra-tion of nanoparticle-oligo-drug assembly dictated by bothhydraulic and osmotic pressure along with the drag force on thediffusing DNA strands has also been accounted to incorporate thein vivo complexities of the integrated process. We have alsoincluded the temperature dependence of diffusion coefficients,blood perfusion rate and other physical parameters. Importantly,our model retains numerous process parameters unchanged tothe values quantified by previous researchers. Subsequently, themodel has been validated with existing experimental results andgood congruencies have been obtained even in the presence ofutmost uncertainties in complex biological processes.

2. Mathematical model

As mentioned above, the complete model is segregated intoseveral thermo-fluidic modules, broadly including the process ofbiological heat transfer in the combined purview of blood perfu-sion and nanoparticle mediated heat generation, the kinetics ofoligonucleotide denaturation and the transport of either nano-particle-conjugated or free oligo-drug complexes within theinterstitial space of tumor.

2.1. Bioheat transfer

We have solved the 3-dimensional transient Pennes [20]bioheat transfer equation in its classical form

rc@T

@tþrUð�krTÞ ¼ QbloodþQmetabolicþQsource ð1Þ

Here, Qmetabolic is the volumetric rate of metabolic heat generation,which has been fixed to constant and physiologically relevantmagnitude. Qblood ¼ rbwbcbðTb�TÞ is the heat removal rate due toblood perfusion, where wb represents the blood perfusion rateand cb is the specific heat of blood. We introduce the temperatureand location dependence of the blood perfusion by considering,wb¼F(T, a,b), where the forms of the function F over differenttemperature ranges, as well as values of the parameter a and b aregiven in Table S-1 (see supplementary material).

The remaining magnetic source term Qsource has been deducedas follows. Firstly, the power generated per unit volume ofnanoparticle is given by

PSPM ¼ Pðf ,HÞ ¼ m0pfw00ðf ÞH2 ð2Þ

where f is the frequency, H is the strength of magnetizing field, m0

is the permeability of free space and w00 is the imaginary part ofthe complex susceptibility w¼ w0 þ iw00, where both w0 and w00arefrequency (f) dependent. The out-of-phase imaginary componentresults in the heat generation PSPM. From physical perspective, asM lags behind H, a positive conversion of magnetic energy intointernal energy becomes possible. Subsequently, imaginary com-ponent has been modeled as

w00ðf Þ ¼ w0F1þF2

, where F¼ ftN,B ð3Þ

Here, tN,Bis the effective relaxation time and is formulated astN,B ¼ tNtB=tNþtB. Further, the Neel relaxation time ðtNÞ is givenby, tN ¼ t0 exp½KVn=kBT�, where K is the anisotropic energydensity, Vn is the nanoparticle volume, kB is the Boltzmann

constant and t0is the attempt period, which is only weaklydependent on temperature [21] and has a typical value of 10�11 sfor non interacting particles. In the context of present scenario,due to the restricted Brownian motion of molecularly large nano-particle-oligonucletide-drug assembly, the Brownian relaxation timetB ¼ 4pZr3

n=kBT becomes relatively infinite, resulting tN,B ¼ tN

for Fc1, w00ðf Þ ¼ w0

F, where w0 ¼

m0MS2Vn

kBT ð4Þ

Here, Ms is the saturation magnetization in A/m. CombiningEqs. (2)–(4), we obtain

PðHÞ ¼m0

2Ms2H2Vn

t0 exp½KVn=kBT�kBTð5Þ

From this expression, it is evident that the power generatedper unit volume is frequency independent. Of specific signifi-cance, this phenomenon has been validated experimentally [22].For sake of further elaboration, we express K as 25kBTB=Vn, whereTB is the blocking temperature depending on the type and radiusof the nanoparticle. Now if c is the density of the nanoparticlethen combining all the previous equations we get in the final form

Qsource ¼Q ðH, rn, T , type of nanoparticleÞ ¼c3Nm0

2Vn2H2sðrnÞ

2pMnkBTt0 exp½25TB=T�

ð6Þ

s is the saturation magnetization in emu/g. In Eq. (5), both Mn

(molecular weight) and s are dependent on type of nanoparticles.We have shown this source term varies with nanoparticlediameter, field of magnetization and temperature. The depen-dence of these parameters on the source term is shown inFigure S-2(a) and (b) (see supplementary material) for constantmagnetic material density of 80 mg/cc.

2.2. Species transport

The drug vehicle travels intruding into the tumor through tumormicrocirculation, and then it enters the interstitial space by trans-vascular transport. Subsequently, on finding the epitopes on thecancer cell, it undergoes the intracellular transport. The path of itsmovement is explained in the Fig. 1(b). As it is conceived, the masstransfer equation should incorporate the transient 3-dimensionaldiffusion equation with generation or consumption of species andconvection through porous media. It is difficult to define a proper‘pore scale’ in soft tissue and biological systems are more like afibrilar structure consisting of proteins and proteoglycans notresembling classical porous media structure, which comprises ofsolid matrix with interconnected voids. For the last few decadesnumerous attempts have been made to evaluate the quantities likeporosity, tortuosity and void fraction. Though they were notintended to model biological system as porous media but eventuallyporous media model have been applied to biological systems fromextracellular to cellular level [23]. Though this modeling approach isnot even close to complete, they give excellent congruence withexperimental result and are able to explain the phenomena like cellblebbing and cytokinesis. Relevantly, approximating biological tis-sue as a porous media can lead to significant simplifications inimplementation and analysis in vivo transport [19]. In the process,for sake of a generalized representation, we designate nanoparticle-oligo-medicine assembly as species 1, nanoparticle-oligo as species2 and medicine molecule-oligo as species 3. The resulting genericdynamics thus yields to

@ci

@tþrUð�Deff

i rciÞ ¼ Ri�V!

ci ð7Þ

where Dieff is the effective diffusion coefficient of the species i (i¼1,

2 or 3) in the porous medium, given by Deffi ¼ ðe=tÞDi. Di is the

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779774

diffusivity of ith species in free condition, e is the void fraction and tis the tortuosity. The velocity vector is given by V

!¼�ðk=mÞrUP.

Where, k is the specific hydraulic permeability, m is the viscosityof interstitial fluid and k=m¼ K is the hydraulic conductivity.Empirically, the value of K is related to the tissue concentration ofglycosaminoglycans (GAG), in weight ratio (gm/100 gm) of tissueand is obtained [24] as

K ¼ 4:6� 10�13½GAG��1:202ðm37=mÞ ð8Þ

where m37 is the viscosity of interstitial fluid at 37 1C. Here it ispertinent to mention that the interstitial transport of drug is difficultto achieve both in systemic and local delivery protocols due to thelow intrinsic convective transport in the regime of elevated inter-stitial fluid pressure (IFP), the lack of functional lymphatic vesselsand the outward gradient of IFP. Importantly, the last componentenforces an adverse convective transport of extravasated drug fromthe interior to the periphery. Hence, diffusion is the predominantmode of transport throughout the tumor system except at theperiphery where convection becomes proportionally commensurate.Given that the variation of IFP (Fig. 2(b)) along the tumor depth is an

Fig. 2. (a) Tumor core temperature vs. time. The dots show the experimental results. T

pressure with tumor depth; (c) diffusion flux at t¼1300 s for all the species. (d) Diffusio

for t¼1300 s. (For interpretation of the references to color in this figure legend, the re

essential part of accurate determination of mass transport dynamics,we obtain this parameter by curve fitting with existing experimentalresults [25] and the result is given in Table S-3 (see supplementarymaterial).

2.3. Species generation and consumption through DNA denaturation

Next, we attempt to deduce the source or sink term Ri in thespecies equation from the prevailing biological phase transition inan increasing temperature field. It is trivial to note that the amountof species 2 or 3 generated is one mole each due to the dissociationof one mole of species 1. In order to obtain the converted fraction ofspecies 1, exclusively through the dissociation of oligonucleotidechains, we compute the ensemble average of fraction of dissociatedmolecules at a given temperature (op4), which is perceived to bea function of temperature G(T). The dissociation or ‘‘melting’’ of adouble stranded oligonucleotide chain into two mutually comple-mentary single strands sequences is essentially is temperaturesensitive phase-transition process. The intrinsic stochasticitiesof melting dynamics have recently been delineated utilizing

he black line shows the computational result; (b) the variation of interstitial fluid

n flux profiles for each species. The scale has the unit mole/m2 s. All the figures are

ader is referred to the web version of this article.)

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779 775

a generalized Langevin formalism. Due to randomly fluctuatingthermal energy, comparatively weak A–T couplings between twocomplementary strands break, which, due to the local constraints innucleotide motions, results in long-range cooperative effects thateventually govern the abrupt configurational entropy gradientdriven chemical transition. For this reason, the oligonucleotidesequences with a stretch of A/T nucleotides in the middle formspre-dissociation melting ‘‘bubbles’’, which facilitates completedecoupling of strands. As for representative intermediating oligonu-cleotides, we have used three well-characterized oligonucleotidesequences namely L60B36, L42B18 and L33B9, whose denaturationkinetics and thermodynamics have been investigated extensivelyfrom both theoretical and experimental perspectives. Thesesequences vary on the relative length of pre-melting bubble andthus, possess different dissociation probabilities and steepness indynamic transition kinetics at same temperature. In each stepof simulation, the magnitude of op4 [26] is obtained fromsimulation based on the modified Langevin dynamics of Peyrard–Bishop–Dauxois model, as introduced by Das and Chakraborty. It isworthwhile to note that oligonucleotide melting kinetics followsapproximately a generic relationship op4 ¼ 0:5tanh ððT�TcÞ=dÞþ

0:5, where Tc is the critical temperature at which op4¼0.5 and d

is sequence specific fitting parameter (tabulated in supplementarymaterial S-4). Once, op4 is known, it is integrated to the masstransport equation by R1¼�C1op4 and R2,3¼C1op4 .

2.4. Transvascular generation of species

The term R1 comprises one more term apart from �C1

G(T) due to transvascular transport. We define FB�FL as thetransvascular transport term. The first term is the rate of fluidextravasation rate from blood vessels per unit tissue volume,while the second term is the rate of lymphatic drainage per unittissue volume. There is no functional lymphatic in solid tumors.Thus, FL¼0. FB is given by the modified Kedem–Katchalskyequation [27]

FB ¼PS

VCP�

C

KAV

� �ð9Þ

P is the microvascular permeability coefficient, S/V is the vascularsurface area per unit tissue volume, Cp is the solute concentrationin plasma, C is the solute concentration in the interstitial fluid andKAV is the available volume fraction of solute in tissues.

2.5. Diffusion coefficients

Oligonucleotides differ structurally from globular proteins andare often treated as random coils [28]. For DNA in plasma, there issignificant interaction between the solvent and the molecule, andconsiderable solvent is associated with the macromolecule. In thiscase, we describe the root mean square end to end distance givenby, od2

041=2 ¼ Cnn1=2l, where Cn is known as the characteristic

ratio, n is the number of base pairs and l is the length of eachsegment. In general, Cn¼ lk/l, where lk is known as the Kuhnstatistical segment length [29]. For DNA, lk equals 150 nm and l

equals 0.34 nm. Assuming the random coil behaves as a spherewith a radius equal to the radius of gyration, RG, we find thetemperature dependent diffusion coefficient as [30]

D¼0:196kBTffiffiffi

6p

mRG

ð10Þ

For open ended coils

RG ¼/d2

0S1=2ffiffiffi

6p ð11Þ

For finding the diffusion coefficient of drug molecule, we use thetemperature dependent Wilkie–Chang correlation [31] given by

D¼ 7:4� 10�10 TðfMÞ0:5

mV0:6

!ð12Þ

where V is the molar volume of the solute (cm3/mole) at itsnormal boiling point, m is the viscosity of solution, M is themolecular weight of the solvent and f is an association parameterfor the solvent. The diffusion coefficients of these species are usedto find the modified diffusion coefficient in bound condition.A size averaged harmonic mean was used to find the effectivediffusion coefficient of the assembly as diffusion coefficientinversely varies with size of species.

V1þ2

D1þ2¼

V1

D1þ

V2

D2ð13Þ

where V1þ2, V1 and V2 are the volume of species 1 linked with 2,volume of species 1 and volume of species 2, respectively. D1þ2, D1

and D2 are the diffusion coefficient of species 1 linked with 2,diffusion coefficient of species 1 and diffusion coefficient of species2, respectively. From this equation it is clear that if the size of onespecies is very small resulting in high diffusion coefficient, the termcorresponding to that species equals zero and the resultant diffu-sion of the combined species is determined by the other species,which is expected from physical consideration.

We used Doxorubicin as the test drug molecule as the data for thisdrug is abundant in medical literature. Doxorubicin is a common anti-cancer agent used in the treatment of a number of neoplasia.

3. Computation and validation

We approximated the 3D tumor model for this simulation byconsidering tumor as a sphere resting over the cylindrical musclecovered by skin as shown in the Figure S-5(a) (see supplementarymaterial). The metabolic heat generation has been taken to be400 W/m3. The other thermophysical properties are given inTable S-2 (see supplementary material). COMSOL multiphysicssoftware based on the finite element method has been used tosolve the model by a direct matrix inversion technique withtemperature tolerance of 0.0001. After the creation of the3D model, it has been meshed, as shown in Figure S-5(b)(see supplementary material). The mesh structure contains 157elements, 330 mesh points. To confirm the robustness of oursolution and its non-dependence on mesh elements, we haveperformed several simulations with identical system parametersbut with variable number of mesh elements and points. Conse-quently maximum 0.07% change has been noted. Thus themeshed model has been solved transiently by conjugate gradientslinear system solver with algebraic multigrid preconditioner.

Next, the simulation results have been compared with theexperimental trends obtained by the Hilger group [32]. In arepresentative set of experiments on tumor-bearing mice,the animals have been subjected to alternating magnetic field(frequency 400 kHz, amplitude 8.8 kA/m) for 242 s after homo-geneous intratumoral injection of 21 mg79 of magnetite sampleof diameter 10 nm per 299 mm3 (70 mg/cc) of tissue. Due to theheat by magnetic nanoparticles, within the treatment time, theintratumoral temperature has been found to increase from 2671to 7178 1C. Including the aforementioned experimental para-meters into our model, an excellent congruency with experimen-tal results has been obtained (Fig. 2a). The temperature of thesimulated tumor is shown for different instants, which reveals theheating history (Figure S-5c) (See supplementary material). Onceour model has been validated, we further endeavor to proposean optimized therapeutic protocol, which should augment the

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779776

efficacy of nanoparticle mediated drug delivery system. In theactual model for the drug delivery simulation, we took a sphericaltumor of radius 4 mm implanted deep inside the healthy tissue.As the magnetic therapy is applied to tumors well embedded intissue, for further computations, the tumor has been modeled assituated inside tissue.

Table 2Summary of sensitivity analysis.

Input parameters Percentage change in the output due to

percentage change in input parameters

Maximum drug

concentration (mol/

m3)

Total drug delivered

(mol)

71% 75% 71% 75%

Magnetizing field (H) þ3.48 þ26.82 þ2.77 þ20.55

�3.71 �24.98 �2.93 �19.27

Number of DNA base pairs (n) �0.46 �1.62 �0.35 �1.37

þ0.69 þ1.84 þ0.58 þ1.69

Void fraction (e) þ0.24 þ0.69 þ0.26 þ0.52

�0.23 �0.64 �0.26 �0.44

Diffusion coefficient of drug �0.464 �0.696 �0.41 �0.464

�0.464 �0.465 �0.39 �0.334

4. Results and discussions:

4.1. Effect of nanoparticle diameter

The values for the relevant parameters used in this work havebeen tabulated in Tables 1, S-1, S-2, S-3 and S-4. Of severalprocess parameters on which the current therapeutic protocolrelies, the pattern of rise in temperature field has been noted to bethe most influencing. It is evident that with the decrease ofnanoparticle diameter, the heat generation per unit volumeincreases drastically due to the strong size dependence of s andthe relaxation times incorporated in blocking temperature. Forexample, with small decrease in nanoparticle size from 10 to9 nm, the power generation significantly increases causing extre-mely heated tissue (T4500 1C). On reverse side, with 12 nmdiameter nanoparticles, the power generation is seemingly notadequate for effective temperature (To40 1C) rise. Interestingly,most of the experiments conducted till date, presumably by trialand error, deals with 10 nm nanoparticles which, according to thepresent simulation results, is nearly the optimum for hyperther-mia and drug delivery applications.

4.2. Effect of temperature on the quantum of drug delivered

Here, it is relevant to narrate that temperature itself impartsboth increasing and decreasing effects on heat generation asshown from denominator of Eq. (6). However, in mutuallyopposing influence of terms involving T and exp[25TB/T], thepower generation increases considerably with temperature,which leads to further increase of the system temperature toattain a favorable condition for drug delivery or ablation. Again,attributing to the specific localization of nanopaticles on tumorsite, the temperature increase has been found to be extremelyconfined and on the healthy tissue away from the tumor site, theeffective rise in temperature field has been observed to benegligible (Figure S-5c; see supplementary material). The onlymeans by which temperature rise is possible in this regime isthrough heat diffusion, which is anticipated to lack adequatetransmission efficiency to implore a deleterious side-damage. Aswe can see from the sensitivity study (Table 2) the magnetizing

Table 1List of input parameters.

Parameter Valu

Avogadro constant (N) 6.02

Boltzmann constant (kB) 1.38

Magnetic constant (m0) 1.25

Saturation magnetization (s) 62.2

Molecular weight of nanoparticle (Mn) 232

Attempt period (t0) 10�

Blocking temperature (TB) 115

Void fraction (e) 0.6

Microvascular permeability coefficient (P) 2.5�

Vascular surface area per unit volume (S/V) 120

Solute concentration in plasma (Cp) 30

Available volume fraction of solute in tissues (KAV) 0.1

GAG 0.05

Diffusion coefficient of Doxorubicin 5.01

field has a significant effect on temperature generation and this inturn changes the amount of drug released considerably.

4.3. Space specificity of drug delivery

Another factor that perceptibly dictates the success of ther-apeutic treatment is the localized concentration of oligonucleo-tide conjugated drug molecules, which is, in turn, governed by theaccumulation of nanoparticle-oligo-drug assembly and oligonu-cleotide dissociation kinetics in the rising system temperature. Inthe representative theoretical scheme where the magnetic fieldwas set on for t¼1300 s, temperature rise up to 42 1C has beenrecorded (Fig. 3). Subsequently, this has been probed to beadequate for melting of significant fraction of double strandedoligonucleotides and release of drug molecules (Figure S-3; seesupplementary material). In this scenario, there exists a differencebetween concentration profiles of liberated chemical species thatessentially possess identical rate of generation. This is attributedto the difference in diffusivities, which arises from the intrinsicdisparity in molecular sizes. One must note that, within thepurview of coupled thermofluidic events, the nature and theforces involved in the transport of each type of molecularassembly can also be of immense importance. As it becomesevident, on application of magnetic field, oligo-drug compounddiffuses faster into the tumor-site than any other molecularassembly. Here, within the interstitial spaces of tumor, transportprocess is strongly diffusion dependent (Fig. 2c and d), which is incongruence of the experiment [25]. As already discussed, due to

e Unit Reference

2�1023 mole�1

06503�10�23 m2 kg s�2 K�1

663706�10�6 m kg s�2 A�2

emu/g [33]

11 s [21]

K [33]

[35]

10�8 m s�1 [27]

0 m�1 [27]

mole m�3 [27]

[36]

[24]

�10�11 m2 s�1 [34]

Fig. 3. At t¼1300 s after the application of magnetic field: (a) top left: temperature profile (unit K), (b) top right: concentration profile of species 1(unit mole/m3),

(c) bottom left: concentration profile of species 2 (unit mole/m3), (d) bottom right: concentration profile of species 3(unit mole/m3). Species 2 and 3 have similar profiles.

But the scale shows their different values of concentration. This occurs due to different diffusion coefficient.

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779 777

back flow induced by interstitial fluid pressure gradient, theconvective transport is effectively restricted only within theperipheral domains of tumor-site. This divergence in mode oftransport across the depth of interstitial space, reserves the intra-tumor penetration only to free drug molecules while nanoparti-cle-oligo-drug or nanoparticle-oligo compounds can accumulateon encircling region with the aid of both convective and diffusiveflux. Appreciating the specificity of delivery process, the biocom-patibility and the intracellular permeability of drug molecules,the afore-illustrated phenomenon can be of utter significance andshould be subjected to further experimental investigationstowards the desired betterment of the entire process.

4.4. Choice of proper DNA sequence

Finally, the protocol is anticipated to rely upon the oligosequence that forms chemical connection between the nanopar-ticle and the drug molecules. On the process of drug delivery, itimparts two way influences. While the size of oligonucleotidedecides the net diffusivity of the released oligo-drug complexthrough tumor site, both of its size and sequence should deter-mine the temperature induced dissociation kinetics. As delineatedin Fig. 4(b), the drug diffusivity increases with decreasing lengthof oligo-chain, implying short oligo-sequnces to be suitable forthis kind of therapeutic applications. However, the accessibledegree of reduction in oligo size is very much limited by therequired dissociation kinetics. Oligos of very short length (no20)have very low melting temperature and high probability ofdissociation even at ambient temperature, which would implyan undesirable non-specificity and ineffectuality of the deliverysystem. Hence, we propose oligonucleotides of 30–40 bases to be

optimum for this purpose. For sequence specification, it is con-jectured that oligonucleotides with bubble-in-the-middle (i.e. ATrich middle region flanked by GC rich sequences on either side)configuration should be the most suitable for the present ther-apeutic purpose. From the very nature of oligo-melting kinetics,as demonstrated by Das and Chakraborty [26], flanking GCregions prevent strand dissociation at ambient temperature,instead of formation of transient ‘‘melting bubble’’ in the AT richmiddle region. However, with increasing temperature beyond40 1C, the bubble becomes stabilized and generates sufficientconformational strain through inter-nucleotide phosphate linkingto decouple two complementary strands within a very shorttemperature window. This event, therefore, concurrently inhibitsnon-specific release in ambient temperature and yet, acceleratesthe drug release kinetics within physiologically tolerable range oftemperature increase, justifying our proposition. Interestingly, forsame oligo sequence, the concentration of drug is approximatelysame for different nanoparticle diameters within relevant rangeas the process is essentially dissociation kinetics limited andtemperature increase occurs at reasonable short time.

4.5. Sensitivity to various parameters

As the complete model consists of numerous process andparameters we conducted a sensitivity analysis (Table 2) tospecify the effect of the important parameters and to test therobustness of this predictive model. As we can see the error inporosity has minor effect on the total drug delivered as well onthe maximum drug concentration at a point. As the pore scale isnot well defined as pointed out earlier, it was worth studying tomeasure the robustness of this work to the parameter porosity.

Fig. 4. (a) Drug profile with varying magnetizing fields for nanoparticle diameter 10 nm and DNA sequence L60B36. (b) Drug profiles with different DNA sequences for

magnetizing field 350 kA and nanoparticle diameter 10 nm.

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779778

The diffusion coefficients and the parameters affecting them werealso used as input parameters for sensitivity study. The DNAlength and in turn the number of base pairs is a deciding factor forthe evaluation of diffusion coefficient and this was one of theinput parameters. The diffusion coefficient of drug, DNA andnanoparticles all entangle to give the effective diffusion coeffi-cients and it is clear from the Table 2 that the model is almoststable to the variation of these parameters. It is noteworthy thatmagnetizing field has quite high sensitivity on the temperaturegeneration and subsequent drug delivered. But as it is a para-meter that can be precisely controlled from outside during thetherapy this sensitivity should not be an impediment for theapplication of the modeling strategy.

4.6. Limitations

To discuss the limitations it is worth mentioning that thiswork is an attempt to computationally predict a truly multi-physics phenomena and as the drug delivery process encom-passes many more intricate phenomena, several effects should bestudied coupled with this generalized framework like double-layer effects, osmosis, hydration forces and steric effects on thediffusion parameters to name a few. Moreover we wish to pointout regarding hydraulic permeability that small structural changein tissue can lead to large variation in hydraulic parameters [36].So, proper characterization of tissue and quantification of this

parameter is required to make the model better. An application ofporoelastic model can be a good approach along with thischaracterization [23].

5. Summary

We have developed a fundamental mathematical model forsuperparamagnetic nanoparticle assisted drug delivery under exter-nal low frequency magnetizing field applicable for DNA-mediatedtherapy. The model has been rigorously tested to match experi-mental results. Most of the magnetic nanoparticle heating and drugdelivery studies till date are experiment based and not suitable fordevelopment of quantitative therapeutic protocol. However, wehave attempted here to understand the superparamagnetic nano-particle assisted heating mechanism from a fundamental viewpoint.We have successfully modeled the denaturation phenomena andextremely complicated transport process in biological systemthough leaving several intricate details. Our present studies revealthat with the advents in high-speed computing, enormous oppor-tunities and potentials do exist for developing a truly fundamental-based approach towards analyzing superparamagnetic nanoparticlemediated treatment instead of employing tunable parameters likespecific absorption rate (SAR), diffusion coefficients determinedmostly from experiments. This effort is expected to be contributorytowards the medical treatment under critical biophysical conditions,

S. Ghosh et al. / Computers in Biology and Medicine 41 (2011) 771–779 779

in which there is likely to be a natural emphasis on the optimizationand control of various therapeutic parameters, along with theminimization of drug quantity purely experience based treatmentprotocols. Towards that, the present study is expected to unveil adeeper understanding of physico-thermal processes involved withmagnetic nanoparticle-electromagnetic field interaction, by quanti-fying the role of type, radius and concentration of nanoparticle,magnetizing field, oligonucleotide type, drug type and exposuretime on the treated volume, thereby enabling one to control thetreatment by suitably regulating these parameters as well as pre-observing the medical treatment through simulations in real timewith an unprecedented accuracy, as a complementary to purelyexperience driven clinical trials.

Conflict of interest statement

The authors do not have any conflict of interest with anyone orany institution.

Appendix A. Supplementary material

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.compbiomed.2011.06.013.

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