\r\n\t
",isbn:"978-1-83881-119-8",printIsbn:"978-1-83881-118-1",pdfIsbn:"978-1-83881-120-4",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"8bd4f03c89e63ef15984ee1b7f1485c4",bookSignature:"Prof. Andrew James Manning",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10407.jpg",keywords:"Hydrodynamics, Suspension/Saltation/Bedload, Numerical Modeling / CFD, Deposition, Flocculation, Sediment Types, Regional/Temporal Variability, Turbidity Currents, Dust Storms, Socio-Economic Effects, Contaminants, Storm / Severe Weather Effects",numberOfDownloads:205,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 27th 2020",dateEndSecondStepPublish:"September 11th 2020",dateEndThirdStepPublish:"November 10th 2020",dateEndFourthStepPublish:"January 29th 2021",dateEndFifthStepPublish:"March 30th 2021",remainingDaysToSecondStep:"6 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Dr. Manning is a highly published and world-renowned scientist in the field of depositional sedimentary flocculation processes.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"23008",title:"Prof.",name:"Andrew James",middleName:null,surname:"Manning",slug:"andrew-james-manning",fullName:"Andrew James Manning",profilePictureURL:"https://mts.intechopen.com/storage/users/23008/images/system/23008.jpeg",biography:"Professor Andrew J. Manning is a Principal Scientist (Rank Grade 9) in the Coasts & Oceans Group at HR Wallingford (UK) and has over 23 years of scientific research experience (in both industry and academia) examining natural turbulent flow dynamics, fine-grained sediment transport processes, and assessing how these interact, (including both field studies and controlled laboratory flume simulations). Andrew also lectures in Coastal & Shelf Physical Oceanography at the University of Plymouth (UK). Internationally, Andrew has been appointed Visiting / Guest / Adjunct Professor at five Universities (Hull, UK; Delaware, USA; Florida, USA; Stanford, USA; TU Delft, Netherlands), and is a highly published and world-renowned scientist in the field of depositional sedimentary flocculation processes. Andrew has contributed to more than 100 peer-reviewed publications in marine science, of which more than 60 have been published in international scientific journals, plus over 180 articles in refereed international conference proceedings, and currently has an H-index of 24. He supervises graduates, postgraduates and doctoral students focusing on a range of research topics in marine science. 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Therefore the ability to generate and manipulate single photon determines how far we can go in QIST. Generally, there are two distinct methods for generating single photons: one is based on excitation-reemission of photon in a semiconductor quantum dot [4], a single defect in NV center [5], or a single atom [6]; another convenient method is based on spontaneous emission based on a second- [7] or a third-order nonlinear process [8]. In this chapter, we will focus on single photon generation by using nonlinear processes. Usually, there are two nonlinear processes for generating nonclassical photon pairs: (1) spontaneous parametric down-conversion (SPDC), which is a second-order nonlinear process; (2) spontaneous four-wave mixing (SFWM), which is a third-order nonlinear process. In both SPDC and SFWM, energy, linear momentum, and angular momentum conservations should be fulfilled. Due to these conservation laws and the technology of quantum interference used, two photons in each pair generated in SPDC and SFWM can be correlated in various degrees of freedoms, for example, polarization, energy-time, orbital angular momentum, position-linear momentum, angular momentum, and photon number and path [1]; we can utilize these freedoms in a specific application scenario in QIST.
\nIn the subsequent section, we will first introduce the basic principle of SPDC and SFWM for generating photon pairs and then the various materials used for SPDC and SFWM. In the key part of this chapter, we will review the developments of various entangled photon pair sources and methods for charactering these sources. After that we will introduce a nonlinear method for transducing the wavelength of the photon from one to another while keeping its quantum properties unchanged, which is suitable for building up a quantum interface to connect different quantum systems. Finally, we will give a brief summary in which some future perspectives for nonclassical photon pair generation and potential applications are discussed.
\nSPDC is realized in a second-order nonlinear process (see Figure 1 left image), in which a pump photon at higher frequency (ωp\n) is split into two daughter photons at lower frequencies with certain probability in a nonlinear crystal; these two daughter photons are usually called signal (ωs\n) and idler (ωi\n) photons. The conservation laws of energy, linear momentum, and angular momentum require that the frequency, linear momentum (k), and angular momentum (l) of the pump, signal, and idler photon fulfill the following conditions: ωp = ωs + ωi\n; kp = ks + ki\n; lp = ls + li\n. These conservation laws are responsible for the generation of various entangled sources.
\nA simple diagram for SPDC and SFWM. The conservation of energy, linear momentum, and angular momentum holds in both nonlinear processes.
In correspondence to SPDC, SFWM is a third-order nonlinear process; a big difference is that there are two pump beams in SFWM (see Figure 1 right image), in comparison to SPDC in which only one pump beam is used. The conservation laws in SFWM require that the corresponding parameters of the pump, signal, and idler photons have the following relationships: ω\n\np1 + ω\n\np2 = ωs + ωi\n; k\n\np1\n+ k\n\np2 = ks + ki\n; l\n\np1\n+ l\n\np2 = ls + li\n.
\nFor quantum optical description of SPDC and SFWM, the Hamiltonian of the two processes can be expressed as [9]:
\nwhere ξ depends on the pump intensity, the nonlinear coefficient of the crystal, crystal length, and focusing parameters. Therefore the photon states generated in SPDC and SFWM can be expressed in Fock state basis as [9]:
\nIt can be seen from Eq. (2) that we obtain a vacuum state with a high probability if the pump is weak. The second term is the photon pair state we need, and the other terms are multiphoton states which should be avoided. It is clear from Eq. (2) that the pump beam should be at a moderate intensity level in order to eliminate the effects of higher photon number states. The photon pair generated in SPDC and SFWM is of probability and is undetermined, which is a disadvantage for photon sources generated from nonlinear processes. For this reason, one needs to choose a proper pump intensity level in order to balance between different experimental parameters.
\nAll materials have third-order nonlinearity, but only those materials that are central asymmetric have second-order nonlinearity. The commonly used materials for SPDC can be divided into two kinds according to different phase matching: one is birefringent angle phase-matching materials, such as LBO, BBO, KTP, and LN [10]; another is quasi-phase-matching (QPM) crystals such as PPKTP and PPLN [11]. QPM crystals have the advantages of high generation rate and narrow bandwidth, which are frequently used in photon pair generation in modern quantum optics experiments [12, 13, 14, 15, 16, 17, 18, 19]. For SFWM, the commonly used materials are hot or cold atomic ensembles [20, 21, 22, 23, 24] and guided-wave materials such as dispersion-shifted fibers (DSF) [25, 26, 27, 28, 29], photonic crystal fibers (PCF), [30, 31, 32], and silicon-on-insulator (SOI) waveguide [33, 34, 35, 36, 37, 38, 39, 40, 41]. To look for new kinds of nonlinear materials for generating high-quality photon sources is still a very hot topic in QIST.
\nBecause of the conservation of energy, linear momentum, and angular momentum in SPDC and SFWM, various kinds of nonclassical sources can be generated; in this section we will review the recent development and key points in generating various kinds of nonclassical photon sources.
\nA polarization-entangled photon source (PEPS) is one of the most important entangled photon sources that have been studied for decades of years. In the literatures, people generate PEPS using different materials with different experimental configurations [12, 13, 14, 15, 16, 17, 18, 19, 42, 43]. For SPDC, in the early times, PEPSs are created using birefringence phase-matching (BPM) crystals, for example, a type-II phase-matched BBO crystal is used to create a PEPS in the first practical and effective experiment, in which orthogonal polarized photons are emitted at the intersection cones [42]. Later on, a beam-like design is used for high-brightness photon pair generation, which is widely used in multiphoton quantum experiments [44]. The significant progress in nonlinear crystal fabrication makes a QPM crystal a better choice for researchers in many nonlinear optics applications [11]. The most important merit of using QPM crystals in generation photon pairs is its high spectral brightness in contrast to BPM crystals, due to its large effective nonlinear coefficient and long allowable interaction length.
\nRecently, to generate PEPS by placing a QPM crystal inside a Sagnac interferometer configuration has been demonstrated to be superior to other configurations (see Figure 2). The basic idea for a Sagnac loop-based PEPS is as the following: a pump beam is split into two beams by a double polarized beam splitter (DPBS) and counter-propagates in the Sagnac loop; each beam generates a pair of photon with orthogonal polarization, in one circulation direction; the photon pair is rotated by a double half wave plate; then two pairs of photons are recombined in the DPBS; and a PEPS with a form of \n
The experimental setup for a typical polarization-entangled source based on Sagnac interferometer (figure cited from [13]).
In SFWM, PEPS is generated using an atomic ensemble with different configurations. The PEPS generated with the atomic ensemble has narrow bandwidth; the wavelength is fixed to specific atomic transition lines [50, 51]. Many works report PEPS generation based on guided-wave materials such as DSF [8, 25, 27, 28], PCF [30], and SOI waveguide [37, 52], the advantages of using guided-wave materials are free of free-space coupling, low loss, low cost, and easy to integrate. The guided-wave platform is very promising in large-scale applications which require hundreds of optical components. It is also convenient for building up a compact, versatile photonic source platform for various kinds of applications in QIST.
\nBecause of conservation of energy in nonlinear processes, the two photons in each pair generated are correlated in frequency and are also generated simultaneously. Although the uncertainty in time and frequency domain for individual particle should meet the requirement of uncertainty principle, the sum of the frequency of signal and idler multiplies the difference between arrive times of the two photon should have a very small value, and violates an inequality for two photons existed classical correlations [53]. A two-photon Franson-type interference is used to characterize the correlations between the two photons; the phases between the two unbalanced Michelson interferometers (UMI) are correlated [54, 55]. To generate a time-energy entangled photon pair, a laser with long coherent time is needed (see Figure 3(a)); the time difference between two paths in UMI should be much larger than the coherence time of the single photon but much shorter than the coherent time of the pump laser [53]. A similar kind of entangled photon source is a time-bin entangled photon source [56], in which a pulse pump is split into two pulses in an UMI, and then these two pulses have a certain probability to generate a pair of photon separately; the photon pairs generated by these two pulses are indistinguishable after passing through two UMIs (the time difference of the UMI in measurement part is the same as the UMI in pump part, see Figure 3(b)). The quantum states for a time-energy or a time-bin entangled photon source can be expressed as \n
Simplified diagrams for (a) time-energy; (b) time-bin entangled photon generation. A narrow bandwidth CW laser is used for generating of time-energy entangled photon pair, while a pulse laser is used for generating time-bin entangled photon pair.
Another important degree of freedom of photon is orbital angular momentum (OAM), which has been widely investigated since 1992 [59]. OAM has unbounded dimensions, which is very promising for high-capacity communication task in both classical and quantum optical communications [60, 61, 62]. OAM entangled photon pairs can be generated in SPDC and SFWM based on crystals [48, 63, 64, 65, 66, 67, 68, 69] and atomic vapors [70, 71]. The quantum state for an OAM entangled photon pair generated directly by pumping a nonlinear crystal (Figure 4, left image) can be expressed as \n
Generate HD OAM entangled state directly from SPDC (left image); 2D OAM entangled state generation by converting a polarized entangled state into 2D subspace of OAM entangled state (right image, cited from [48]).
Nonclassical photon sources can be characterized from different aspects. For characterizing the properties of a heralded single photon, the heralded efficiency [72], the coincidence to accidental coincidence ratio (CAR) [73], and the single photon Glauber function [74, 75] are important parameters. The heralded efficiency is the probability of detecting the second photon when the first photon is detected. It is a measurement of the photon collection efficiency, filter and transmission losses, and the single photon detector efficiency. The heralded efficiency is the ratio of the coincidence count to the single count rate of the first detected photon. CAR is a measurement of the signal to background noise ratio for a two-photon experiment; high CAR can ensure the quantum nature existed between the two photons. CAR depends on pump power and detector performance. Usually, CAR will increase when the pump power is increased in the low pump power regime. After reaching the maximum value, CAR will decrease with the increase of the pump power [76]. The single photon Glauber function can be measured as shown in [74]. The measured photon is firstly split by a beam splitter, and then by measuring the three party coincidence, single count and two-photon coincidence, we can calculate the single photon g(2) function (see Figure 5). A near zero g(2)(0) indicates the high quality of single photon nature. For a pulse pumped photon source, the single photon purity is also an important parameter [17, 77]. The purity of photon is a measurement of spectral correlations between two photons; the purity is determined by the Schmitt number in the Schmitt decomposition. The unity single photon purity indicates that the two-photon spectral can be factorized into product of two separate functions of the signal and idler photons. The high single photon purity is very important for realizing high visibility HOM interference between two independent single photon sources, which is the key technique for realizing high photon number entangled states.
\nExperimental setup for measuring heralded single photon autocorrelation function for single photon generated from SPDC (image cited from [74]).
There are various available and faithful methods to characterize the quality of entanglement of an entangled two-photon source, including two-photon interference fringes [65, 78], Bell CHSH inequality [79, 80], and quantum state tomography (QST) [81]. Two-photon interference fringe is much easier to measure; through calculating the interference visibility from the measured data, we can evaluate the quality of an entangled source. A high visibility indicates a high quality of the generated state by comparing the ideal maximum Bell states. When the visibility is greater than a threshold value, the two photons have Bell nonlocality; the threshold value is different for two-photon states in different dimension. For two photons existing in classical correlation, the Bell CHSH parameter S is not greater than a certain value. The violation of this value indicates a nonclassical correlation between the two-photon states. Bell CHSH parameter S is an indicator of whether the two-photon state has Bell nonlocality and how strong this kind of nonlocal correlation is. The violation of Bell inequality has been widely studied in literatures for a 2D and a HD entangled state. To fully know the content of a generated quantum state, QST can be used to reconstruct the density matrix of a certain quantum state. By the density matrix of a quantum state, all the properties of the quantum state can be predicted.
\nThere are many quantum systems for QIST based on different materials, including atomic ensembles, trapped ions, solid-state materials, and fibers for transmission [82, 83, 84, 85, 86]. Each quantum system has some advantages in QST, and these systems usually work at different frequencies, which may have a big frequency mismatching. To build up a quantum network consisting of various quantum systems for information encoding, storage, transmission, and processing, a quantum frequency converter (QFC) to link different quantum systems is indispensable. Such a frequency transducer can be realized by utilizing nonlinear processes such as sum frequency generation (SFG) and Bragg reflection in four-wave mixing. The theory of quantum frequency conversion for SFG was first proposed by P. Kumar in 1990 [87]. In SFG, a strong pump laser can convert a weak signal beam with high quantum efficiency; the unity quantum efficiency can be reached when the pump beam is strong enough (see Figure 6 left image), and the quantum correlations are unchanged after frequency conversion. Since it has been proposed, some significant progresses have been made in this field; researchers have realized that frequency up-conversion and down-conversion for a single photon generated from quantum dot, and various qubit states or entangled states such as polarization, time-energy, and OAM entangled state [39, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101] have been up- or down-converted. A typical setup for QFC of an OAM qubit, an OAM-polarization hybrid entangled state, and an OAM-OAM entangled state is shown in Figure 6 (right image). For frequency down-conversion, the visible single photons generated from atomic ensemble, trapped irons, or NV centers have been converted to telecom band successfully [102, 103, 104, 105]. In all these demonstrations, the single photon properties and entanglement are preserved in the conversion processes, which ensures that quantum information can be coupled to different quantum systems by using a quantum frequency interface.
\nLeft image: simple diagram for sum frequency generation in QPM crystal. Right image: experimental setup of QFC for an OAM qubit, an OAM-polarization hybrid entangled state, and an OAM-OAM entangled state (cited from [99]).
In QFC, there are four parameters to evaluate the quality of a converter: quantum conversion efficiency, noise level, spectral bandwidth, and spatial bandwidth. These parameters are not independent; therefore in practical applications, one needs to balance between different parameters [98]. A longer nonlinear crystal is preferred to reach maximum conversion efficiency when the pump power is limited, but a longer nonlinear crystal would lead to a smaller spectral bandwidth and spatial bandwidth; a proper crystal length should be chosen to balance conversion efficiency and spectral bandwidth. The noise in QFC comes from SPDC and SRS of the intense pump beam; therefore a longer pump wavelength is preferred to reduce the noise photon in QFC [106]. The noise photon can also be dramatically reduced by using a narrow bandwidth filters to filter out the converted photon.
\nNonclassical photon sources are used in almost all fields of QIST; the ability to generate and control its properties is at the heart for applications in QIST. Though many progresses have been made in single photon generation and manipulation in nonlinear processes, lots of further techniques should be developed to harness the quality of single photon generated in SPDC or SFWM. The detailed techniques for optimizing the parameter of the photon source depend on the specific applications. For a pulsed heralded single photon source, the heralded efficiency and total photon count are important parameters, but the probability of single photon generation per pulse is very low, which limits the flux of photon pair generation. These defects can be overcome by using time, frequency, and OAM multiplexing to enhance the photon generation probability per pulse and total count rate [41, 107, 108, 109, 110, 111, 112]. When the optical elements for multiplexing have low losses, the heralded efficiency and rate can be increased substantially [112]. For applications taking advantages of the sharp time correlations in SPDC, a broadband spectrum of the photon pair is needed. Such a broadband photon pair can be realized with an ultrathin nonlinear crystal or using a chirp quasi-phase-matching crystal; the bandwidth of the photon pair generated can be greater than 100 nm, which has a time correlation of sub-femtosecond [113]. For quantum information applications, a multiplexed time-energy and polarized entangled photon pair is preferred for high-capacity quantum communication by using dense-wave division multiplexing technique. The multiplexed entangled sources are easier to be realized using a waveguide platform such as a PPLN waveguide, a DSF, or SOI waveguide. A SOI ring cavity is also preferred in generating frequency comb entangled states [114]. For generating HD entangled states, by shaping the profile of a pump beam, a much greater Hilbert space can be reached [115, 116]. For QFC of OAM entangled states, the mode-dependent conversion efficiency has not been solved yet. We recently proposed and demonstrated that if we use a flat-top beam to pump the SFG crystal, then we can solve the problem of mode-dependent conversion efficiency by using a Gaussian pump beam [117].
\nFor a compact application, integrated optics will offer a great advantage over free-space implementation; the trends of modern optics are to convert a free-space module to an equivalent integrated optical device, which will be of high compact, robust to environment fluctuations and much easier for larger amounts of fabrication [118, 119].
\nIn conclusion, most of the advances and progresses for generation and manipulation of single photon sources in nonlinear processes are briefly reviewed in this chapter; this review will provide a glance at the current status, and challenges remain to be solved in this field. The general properties for single photon generation in nonlinear processes are introduced firstly; then we introduce the development of various entangled states and the methods to characterize nonclassical photonic states. Next, we review the progresses for frequency conversion of a photonic state in nonlinear processes. Finally, we give comprehensive discussions about the remaining challenges in generating high-quality and HD entangled states, the unsolved problems for QFC of HD OAM photonic states, and the development of integrated optics for small footprint optical devices and large-scale quantum information processing on chip. This book chapter should be helpful for new researchers working in this field.
\nThe discussion in the succeeding paragraphs is not limited to only iron oxide nanoparticles, it deals with all nanoparticles which are magnetic in nature; hence, there is a reoccurrence of the phrase “magnetic nanoparticles.”
\nThe technique or the procedure which I am going to discuss is widely known as hyperthermia. It is the phenomenon which involves selective heating of magnetic particles using high-frequency magnetic field. The case presented here uses the fact that the tumor in the affected area can be removed by heating it up to certain temperatures depending on the different parameters of the nanoparticles. The whole idea started with the introduction of the magic bullet, a concept given by Nobel laureate Paul Ehrlich (1854–1915), 1908, in the field of medicine in the field of immunity. The idea of Magic Bullet projected by selective targeting of disease causing organism in addition to delivery of toxin for the affected area. The procedure suggested was to first identify the cancerous cell/tissue and then target the magic bullet of nanoparticles at the site and the blast the cell for destroying the un-repairable cell or delivering dose with the help of nanoparticles’ magic bullet, which at the site will open to deliver the drug.
\nLater, after the invention of the magic bullet, a number of hyperthermia techniques were suggested since 1970. Scientists Zimmermann and Pilwat in 1976 proposed the use of magnetic erythrocytes for delivery of drug at the affected site. Research group led by Freeman et al. in 1960 came up with the idea wherein magnetic nanoparticles could be transported through the vascular system and grouped in a specific part of the body using magnetic field. As recently as 2009, Boris Polyak and Gary Friedman explored clinical potential and applications of magnetic targeting for site-specific drug delivery.
\nThe treatment of hyperthermia involves heating of injected cancer-specific biomolecules coated with magnetic nanoparticles at the affected area. It involves selective heating of magnetic particles, which are positioned at the affected site, using high-frequency magnetic field. Removal of tumor (different diameter sizes) located in the liver is studied by varying power applied for different exposure times theoretically using heating model given by Tsafnat et al.
\nI again present the research by my group wherein we optimized the power requirements for the destruction of diseased cell at the location. The entire work was around the concept of the treatment of tumor using hyperthermia, which involves heating of injected cancer-specific biomolecules coated with magnetic nanoparticles at the affected area. The procedure involved selective heating of magnetic nanoparticles, which are positioned at the affected site of the diseased cell, using a very high-frequency magnetic field. The hypothesis used is that the tumor in the affected area can be removed by heating it up to temperatures, in the range of 41–46°C based on earlier research in the area. It was proposed that a tumor with a diameter size of 5 cm can be efficiently removed, if magnetic nanoparticles (present at the tumor site) were exposed for 10 s, with a power range of 2.75–6.5 W.
\nBased on the different research models given, we used the theory according to which the tumor in the affected area can be removed by heating it up to temperatures [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], in the range of 41–46°C. It was proposed that if the power ranges 2.75–6.5 W, when applied to the magnetic nanoparticles (present at the tumor site), for duration of (up to) 10 s, a tumor having diameter of size 5 cm can be successfully/efficiently removed. Dependency of temperature in the affected area on the diameter of the magnetic nanoparticle in addition to exposure time of magnetic nanoparticles by alternating magnetic field and power was studied [20, 21].
\nThe work tries to carry forward the results presented in orphan drugs [22], which proved that, if heat is applied for a duration of 10 s for an applied power to magnetic nanoparticles of 6.5 W, it leads to removal of tumor (up to radius size of 10 cm). We explore the variation in applied power within which the desired results can be obtained, which in turn leads to minimizing the running cost and undue heating of healthy tissue/cell in the vicinity of the affected area. The study presented is based on the model suggested by Tsafnat et al. [19] for heating of liver tumors. According to their study, the affected area demonstrates lower levels of blood perfusion than a healthy one. This results in partial safety for the healthy liver tissue during localized hyperthermia treatment. The model (mathematical) reproduced here simulates the practical heat diffusion from the affected area (tumor) to its surrounding (unaffected) tissue. It is assumed that temperatures in the two respective areas have an effect on each other at the boundary interface.
\nIt is assumed that the shape of the tumor is a spherical tissue with radius a which is surrounded by a normal tissue again assumed to be a bigger concentric sphere with radius b (see Figure 1).
\nTumor tissue, spherical in shape, surrounded by normal tissue concentric sphere [20].
The model presented also works on a hypothesis that the heat source with constant power density P is concentrated within the small sphere of radius a surrounded by a medium of homogenous heat conductivity.
\nAs the model under study is having the spherical symmetry and the homogenous time-independent power density P inside the sphere, the temperature distribution is a function of distance r from the center of the sphere and on time t.
\nThe differential equations of heat conduction [11, 22, 23] are used for defining the required mathematical model given by [11, 22, 23]
\nThe subscript “1” is used to represent tumor tissue, while the subscript “2” refers to normal tissue, and the various parameters in these equations are defined as follows:
\n\nT is for the temperature.
\n\nc represents specific heat capacity.
\n\nρ represents the density.
\n\nk is the heat conductivity.
\nBased on the hypothesis that the temperature and flux at the boundary are continuous and fine, the boundary conditions can be written as
\nTo solve the above system of differential equations, we discretize it using the Euler method, which is reproduced here [20].
\nLet h > 0 and k > 0 be the step lengths in the space and time directions, respectively.
\nAlso let N1 and N2 be integers such that
\nWe replace the region Ω = {(r,t) l 0 ≤ r ≤ b, t ≥ 0} by a set of grid points (r1, tj), denoted by (l, j),
\nwhere
\nwhere J is a positive integer.
\nLet
\ndenote the solution of (1) and (2), respectively, at the grid point (l, j).
\nWe approximate the solution of (1), (2) at the grid point (l, j) by the scheme
\nFrom the boundary condition at the tumor-healthy tissue edge, we can write the approximation at the grid point (N1, j) as
\nSolving Eqs. (8)–(11), using values of the different constants from Table 1 [11, 24, 25, 26], the graphs can be plotted for studying dependency of temperature in and around tumor on radius (of tumor), time (of heat exposure), and power applied (on magnetic nanoparticles).
\nParameter | \nConstant | \nValue | \n
---|---|---|
Radius of liver tumor | \n\nA\n | \n2.50 cm | \n
Tumorous liver tissue specific heat | \n\nc1\n\n | \n3.758 kJ/kg K | \n
Healthy liver tissue specific heat | \n\nc2\n\n | \n3.617 kJ/kg K | \n
Liver tissue heat conductance | \n\nk1\n=k2\n\n | \n0.5122 W/m K | \n
Liver density | \n\nρ1\n = ρ2\n\n | \n1.0492 g/mL | \n
From Table 1, it can be observed that varying r from center (0 cm) to the boundary of the affected area (=a), time of exposure up to 10 s with power ranging from 2.75 to 6.5 W, MATLAB software is used to obtain Figures 2–6. In the given model, it was assumed that magnetic nanoparticles used were of the size up to 10 nm in radius. The surface plot (Figure 2) shows dependency of temperature in the affected area on hyperthermia time (t in s) and radius of the tumor (r in cm). It can be concluded from it that on application of 5 W power, temperature in tumor increases to 46°C at the middle region of the tumor, gradually reducing to body temperature at the interface (of affected and healthy area), thus causing least effect to the unaffected area. Similar conclusion can be drawn from Figure 3. The temperature variation at the center of the tumor on varying power of magnetic nanoparticles and time of heat exposure can be studied from Figures 4 and 5. It is observed that the temperature at the middle of the tumor cell gradually increases from body temperature at the interface (r=a) to 48°C for an applied power range of 2.75–6.5 W. It can be further observed that for a standard time of exposure (t = 7 s), if the power is varied from 2.75 to 6.5 W over a radius of 2.5 cm, it leads to annihilation of the tumor, from Figure 6.
\nFor a constant magnetic nanoparticle power of 5 W, temperature in the tumorous cell/tissue plotted, as a surface plot and as a function of hyperthermia time (t, in s) and distance from the center of the tumor (r, in cm) [20].
For a constant magnetic nanoparticle power of 5 W, plot of temperature inside the tumorous cell/tissue as a function of distance from the center of the tumor (r in cm) and hyperthermia time (t, in s) [20].
Temperature at the center of the tumorous cell/tissue as a function of hyperthermia time (t in s) and a magnetic nanoparticle power (p, in Wts) [20].
Temperature at the center of the tumorous cell/tissue plotted as a function of magnetic nanoparticle power (p, in W) and hyperthermia time (t, in s) [20].
Temperature in the tumorous cell/tissue plotted as a function of distance from the center of the tumor (r, in cm) and magnetic nanoparticle power (p, in watts) for a constant exposure time (t = 7 s) [20].
From the discussion, it can be concluded that hyperthermia treatment involving magnetic nanoparticles can be efficiently and effectively used for the removal of tumorous cell/tissue with not much collateral damage. Reduced damage to the neighboring healthy cells makes the technique more successful, clinical results are also in tandem with results if we use nanoparticles with power in the range of 2.75–6.5 W with a heat exposure time up to 10 s, this futuristic approach will make treatment more effective with fewer side effects and less cost, leading to widespread use and finally conquering the disease, which even in this robotic age is considered a taboo.
\nAs recently engineers from MIT are working on designing of tiny robots, in nano range, which can assist in drug delivery, the engineered robots called the “microbots” are based on bacterial propulsion. The scientists have proposed that the procedure can help in overcoming the hindrances to drug delivery loaded with nanoparticles, enabling them to exit blood vessels and hit the right place [27].
\nAs recently as March 2019, Angl Apostolov et al. reported the study based on similar concept of destroying hyperthermia. The group suggests, theoretically, the use of mixed ferrite nanoparticles with structure formula Me1−x\nZn\nx\nFe2O4 (Me = Co, Ni, Cu, Mn) appropriated for self-controlled magnetic hyperthermia (SMHT) for both in vivo and in vitro applications.
\nThus, there is a lot of scope in the area which can be reinvented and researched to make the world if not free at least less cancer deaths or sufferings.
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