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In this paper, we present a multi-source nanonetwork model for biomedical diagnosis applications, based on the Localized Surface Plasmon Resonance by different shape gold nanoparticles (i.e., cylinder, cube, and rod). We present the process of multi-source emission, diffusion, and reception of nanoparticles, based on the ligand/receptor binding. Then, a multi-detection process of DNA alterations is accomplished when nanoparticles are captured at the receiver. The colloidal particles are selectively functionalized with specific splice junctions of gene sequences to reveal simultaneously different alteration that could be associated to an early disease condition. Particularly, full-wave simulations have been carried out for the multi-detection of alternative splice junctions of breast cancer susceptibility gene 1. The proposed application is verified through numerical results and expressed in terms of Extinction-Cross Section, in the case of synchronous and asynchronous nanoparticles detection. We show that the proposed approach is able to detect DNA alterations, based on a selective nanoparticle reception process.

In the past few decades, nanotechnology has emerged as a novel evolution of technology enabling the design of miniaturized devices at nanoscale level (i.e., nanorobots and nanoparticles). At this scale, the behaviors and characteristics of nanodevices need more comprehension and a deep knowledge with respect to well-known features of devices at the macroscale level [

Nanoparticles can be designed accordingly to prolong circulation, enhance drug localization [

New materials are becoming available for nanoscale design and fabrication of nanodevices. Several physical, chemical and biological nanosensors have been developed by using novel nanomaterials, such as crystal materials. Gallium, phosphate, quartz, and ceramic are chosen for their durability and piezoelectric properties of developing and retaining an electric potential (charge), when subjected to mechanical stress. Graphene and its derivatives, namely Graphene NanoRibbons (GNRs) and Carbon NanoTubes (CNT), are used for the design of nanoantennas [10-12].

Several projects consider the use of nanodevices for medical applications as diagnosis and drug delivery. Among them we cite 1) NAD (Nanoparticles for Therapy and Diagnosis of Alzheimer Disease) whose aim is to utilize nanoparticles specifically engineered for the combined diagnosis and therapy (theranostics) of Alzheimer’s disease [

A set of nanodevices, sharing the same medium (e.g., the biological tissue or the bloody flow) and collaborating for the same task, forms a nanonetwork [

Communication and signal transmission techniques occurring in nanonetworks are challenging topics, due to the particular environment (i.e., the human body and biological tissues) and to the limited computation skill of nanodevices. Molecular Communication (MC) represents a promising communication paradigm for nanonetworks [

Recently, the modeling of diffusion-based nanoparticles communications has been investigated in [

However, in real scenarios (e.g., for extra-vivo sensing applications), the emission, diffusion and reception processes can show different behaviours. For example, many biomedical applications require a multi-source emission of nanoparticles, where each source (i.e., nanomachine) can emit a particular type of nanoparticles (i.e., with a given shape and size). As a consequence, synchronous and asynchronous nanoparticle emission can affect the diffusion process and degrade network performance (i.e., with an increase of interference and nanoparticles’ collisions). The same consideration exists at the receiver side, where a selective reception of nanoparticles occurs (i.e., a given nanoparticle can form a complex only with the “corresponding” receptor).

Other authors have investigated novel schemes of molecular communications, starting from classical concepts of the information theory and adapting into nanonetworks [18,19]. In [

Once the nanoparticles arrive at the receiver, they are bound and the detection process can occur. Electromagnetic fields and heat are largely used for sensing applications. For example, magnetic nanoparticles can be used to selectively damage or kill cancer cells by heating them, since intracellular hyperthermia has the potential to achieve localized tumor heating without any side effects [

BRCA1 and BRCA2 are human genes belonging to a class of genes known as tumor suppressors. Mutation of these genes has been linked to hereditary breast and ovarian cancer. For the BRCA1, this gene causes a genetic susceptibility to breast cancer and changes in its alternative splicing profile have been associated with malignant transformation [21,22]. A woman’s risk of developing breast and/or ovarian cancer is greatly increased if she inherits a harmful BRCA1/BRCA2 mutation.

Genetic tests are available to check for gene mutations. However, due to the vast number of alternative splice variants, tissue and cell type specific pattern of the relative expression levels of these variants, it is difficult to use alternative splicing profiling as a diagnostic tool for breast cancer.

Techniques based on the use of metallic nanoparticles illuminated by an impinging electromagnetic wave (visible and near infrared region) are largely exploited, as alternative approaches for gene alterations detection. The resonant frequency related to the collective electron motion has a strongly dependence on the shape, size, composition of the nanoparticles, as well as on the dielectric properties of the background environment. This technique is known as Localized Surface Plasmon Resonance (LSPR). In this paper, we apply the LSPR phenomenon for the detection of BRCA DNA alteration, when biological gold nanoparticles are captured at the receiver.

The remainder of this paper is organized as follows. In Section 2, we discuss the physical end-to-end model for flows of nanoparticles sent by a transmitter toward a receiver. How nanoparticles are emitted, diffused and received is the objective of this paper. Particularly, we first address the emission process for a multinanomachine source scenario in synchronous and asynchronous mode. Then, we present the diffusion and reception processes. Once the nanoparticles are captured by the receiver, the detection of BRCA DNA alterations occurs. The sensing platform is then illustrated in Section 3, while simulation results are described in Section 4. Finally, conclusions are drawn at the end of the paper.

In this section, we will present the physical model of the emission, diffusion and reception of nanoparticles. We will investigate how nanoparticles (1) are transmitted from a set of three nanomachines, (2) diffuse along the gap that separates the transmitter from the receiver, and then (3) arrive at the receiver. All the processes are assumed taking place inside the space, which contains a fluidic medium, initially filled with a homogeneous concentration of particles equal to zero.

In our model, we consider that a single nanoparticle is an indivisible object, which is released to (during the emission process), or collected from (during the reception process), a position in the space, by means of chemical reactions. Moreover, when a nanoparticle is not emitted or received, it is subjected to the diffusion process and moves into the space following the laws of diffusion of particles in a flow. The space is assumed as having infinite extent in any direction, and nanoparticle are free to move everywhere in the space, following a Random Mobility Model (RMM).

Nanoparticles are emitted by three nanomachines (sources), each of them transmitting nanoparticles of different shape (, cylinder, cube and rod) and same material (i.e., gold). The nanomachines have an independent behavior, such as the -th nanomachine can emit a given flow of nanoparticles in the time instant, with. For sensing applications, we assume that the nanomachines can emit nanoparticles both in synchronous (i.e., with), and asynchronous (i.e.,) mode.

A group of nanomachines forms what we define transmitter i.e., an entity emitting different kinds of nanoparticles at different rates and for positive concentrations. We assume the transmitter is placed in the space in the position along axis (i.e., at [nm]); the nanoparticles diffuse in the space and reach the receiver laying in the position, with [nm]. In our model, the receiver is comprised of a set of unit cells, each of them has three square gold patches (receptors) deposited on a silica substrate, and functionalized for the capture of target BRCA DNA sequences i.e., Target Sequences. Each patch is functionalized with the structure of the BRCA splice with the corresponding sandwich assays. The chemical receptors are BRCA alternative splice variants i.e., , , and [

The physical end-to-end model can be well explained through the scheme in

Each module is represented as a black box (i.e., , , and). Our aim is the modeling of the delay that occurs in 1) the transmitter module i.e., [s] the time necessary to transmit a flow of nanoparticles, 2) the diffusion process i.e., [s] the time for a nanoparticle to move along the gap from

the transmitter to the receiver, and 3) at the receiver i.e., [s] the time necessary to receive the nanoparticles. Equations (9), (34) and (39) will be introduced to describe the delay in the transmission, diffusion and reception process, respectively.

The emission process aims to modulate the particle concentration rate at the transmitter. This is comprised of three nanomachines, each of them is modeled as a box containing a nanoparticle concentration value, with, and provided with an aperture from where the nanoparticles can exit, when emitted.

Each nanomachine receives the same input signal, which represents the input of the end-to-end model. The particle flux is defined as the net particle concentration leaving/entering the transmitter per unit time. The particle concentration at the output of the -th nanomachine, with, is the average concentration in the proximity of the -th nanomachine, assumed equal to zero initially. This variation in the particle concentration at the output of the transmitter is expressed as:

where we assumed the superposition principle occurs.

The particle concentration flux emitted by the -th nanomachine is stimulated by a concentration gradient between and. Basically, the particle concentration inside each nanomachine is trigged according to the input signal: for increases (decreases) of, the transmitter increments (reduces) each, and then the -th particle concentration at the output is increased (decreased) too; as a result, the expression in (1) increases (decreases), as well. It is worth noticing that a negative rate modulation can occur when, and then is negative. However, this case is not assumed in our model i.e., the nanomachines emit flows of nanoparticles.

This behavior can be well depicted through an electrical parallel RC circuit, as that one shown in

The resistor and the capacitor are obtained as follows:

From the electrical circuit theory, we compute the Time Fast Fourier Transform (TFFT) of the equivalent

RC circuit of

where and are the Fourier transforms of the input and output voltage, respectively.

Notice that the nanoparticle concentration rate can be identified with the particle concentration flux at the output of the transmitter i.e., for. The particle concentration flux is dependent on the particle concentration gradient at time and position through the Fick’s first law, as follows

where [cm^{2}/s] is the diffusion coefficient, assumed as a constant value for a given fluidic medium. The diffusion coefficient depends on size and shape of nanoparticles, as well as the interaction with the solvent and viscosity of solvent. In (4), the nanoparticle concentration flux can be related to the output current, flowing along the resistor of the circuit in

From the circuit theory, we have

and then it follows

This shows that the diffusion coefficient takes into account the contributions of three nanoparticle flows. Now, we can consider the TFFT of the transmitter module be equivalent to (3), that is the TFFT of the RC circuit in

from which we can evaluate the delay for the transmitter module, as:

where is the phase of the TFFT of, whose expression is easily computed as

The diffusion process of nanoparticles in the space is strictly related to 1) the concentration rate, that is the output from the emission process, and 2) the way the particle diffuse in a medium i.e., in our case it is a fluid medium.

In the proposed model, we assume gold particles with different geometries that are similar to each other in terms of volume occupancy. This assumption makes easier the analysis of the diffusion process. It is well known that diffusion is always along a chemical potential gradient and stops when chemical potentials are constant along the space [

In order to understand how nanoparticles move in a three-dimensional space (lattice), we consider, and, as the concentration and the chemical potential at position in the space, respectively. Then, in position, where represents a space deviation, that is the step, we have:

Let [kJ/mol] be the activation barrier i.e., the minimum energy required to start a chemical reaction; this also means that for a chemical reaction to proceed at a reasonable rate, there should exist an appreciable number of molecules with energy equal to or greater than the activation barrier. Then, we can define the forward rate as

where is the universal gas constant, is a constant, and is the temperature. We assume that in our system there is no meaningful change in terms of temperature; in fact, as a specific case, we can consider an extra-vivo application (e.g., a hair sample), and then such assumption is plausible. Moreover, (12) is usually true with the assumption that, where the free energy change is small within short distances. This allows writing the following:

and also

Let us introduce as the vibrational frequency. Then, a nanoparticle will have enough energy per second to overcome the activation barrier times. In practice, we can define as the number of jumps of step per second, such as:

Since we are considering a three-dimensional space, modeled as a lattice, the nanoparticles are free to move along six equivalent directions of vectors, , and (i.e., , , and, respectively). This means that the number of jumps along the -th direction in the time unit [s] is

where. We can now calculate as the net forward rate along the direction:

where.

Let us consider the nanoparticles move along the direction of. Considering as the number of nanoparticles per unit area in position, we can compute the number of nanoparticles passing through the unit area in the unit time along as:

Since, (18) can be written as:

Now, let us consider the expression of chemical potential as

where is the activity of the particle in position, and is a constant, named activity coefficient. Then, the derivative of (20) is:

and by substituting (21) into (19), we will have:

and given the similarity between (4) and (22), we can calculate the intrinsic diffusion coefficient [cm/s] as:

By considering the diffusion process in a threedimensional space, where a nanoparticle can move following a Random Mobility Model, along six equivalent directions, we obtain:

from which we can recompute (15) as:

Unfortunately, the Fick's first law works when applied to steady state systems, that is the concentration will keep constant in the time and along the space. In the specific application we are considering, the concentration can change in the time, since our nanoparticle diffuse along the space by determining different levels of concentration. For this reason, we need to consider the Fick’s second law.

Let us define and as, respectively, the local concentration and the diffusion flux through an unit area in position. Then we can write:

and

from which it follows:

and by substituting with, (28) is equivalent to

Thus, from the first Fick’s law, we can obtain:

and for a three-dimensional space we can rewrite (30) as:

It is worth to notice that at the equilibrium state (i.e., when the concentration does not change), we obtain the first Fick’s law, that can be considered as a specific case of the second law, when applied to a steady state. However, as outlined in [

where the term is the relaxation time and is related with the heat diffusion process. In fact, the Telegraph Equation was originally formulated for the case of heat transfer, but it can also be applied to the diffusion of particles, as in our case.

For a random walk in a three-dimension space, after steps a nanoparticle travels an average distance of, where a step is the time necessary for one jump [s], and the jump length is [m]. Thus, after a time [s], the nanoparticle jumps times and moves a distance of

where [nm] is defined as the diffusion length. This represents the average space a nanoparticle moves during the diffusion process, depicted as a black box in

From (33) we obtain [s] that is the TFFT of the delay that a nanoparticle experiences to move along the gap from the transmitter to the receiver:

where is the TFFT of, is the TFFT of [mm/s], that is the velocity of the fluid where the nanoparticles diffuse, and is the convolutional operator. In accordance to Equation in [

In our specific scenario i.e., an extra-vivo application, the nanoparticles move into a fluid solution that is not constrained in a vessel (e.g., the bloody flow). Then, we can neglect the additional term, and consider the nanoparticles velocity approximates that of the fluid.

In this subsection we address the nanoparticle reception process. However, the structure of the receiver is presented in Section 3, where the geometry and electromagnetic properties are described.

The reception process has the task to sense the particle concentration at the receiver and to produce, accordingly, the output signal, which represents the output of the end-to-end model. The nanoparticle reception is realized by means of chemical receptors that homogeneously occupy the reception space, with, where the receiver lays. Receptors take place only in correspondence of BRCA alternative splice variants and the output signal is related to the number of obtained complexes, over the total number of available chemical receptors.

Similar to the transmission process, the behavior of the reception process can be well depicted through the electrical RC circuit shown in

The resistor and the capacitor are related, respectively, to the real and imaginary part of the following impedance:

From the electrical circuit theory, we compute the TFFT of the equivalent RC circuit of

where and are the Fourier transforms of the output current and the input voltage, respectively. It follows that we can consider the TFFT of the receiver module be equivalent to (37):

from which we can evaluate the delay for the receiver module, as:

where is the phase of the TFFT of, whose expression is easily computed as:

In this section we analyze the sensing platform able to reveal DNA multi-alterations of the BRCA gene based on the LSPR phenomenon. LSPR occurs when an electromagnetic plane wave impinges on metallic nanoparticles that are electrically small. In this condition the free electrons of the nanoparticle follow collectively the electromagnetic oscillations. Therefore, in order to study the electromagnetic behavior of nanoparticle, a quasistatic approximation can be carried out.

It is well known that in case of an arbitrary shaped nanoparticle, the dyadic polarizability, can be expressed as:

where is the particle volume, is the dielectric permittivity of the surrounding environment, is the complex dielectric permittivity of the metallic nanoparticle, are unit vectors in the direction of the principal axes of the particle, are the three components of the corresponding depolarization dyadic, that is

Therefore, the resonant frequency of the electron motion strongly depends on the nanoparticle size, shape, composition, and surrounding dielectric environment [

By using the same procedure conducted in [

where is the wavenumber, is the wavelength, and is the refractive index of the surrounding dielectric environment.

To design the sensing platform different nanoparticles are considered. The use of different nanoparticles shapes ensure the possibility to tune the electromagnetic response of each particle in a different frequencies range. According to [

where is the Complete Elliptic Integral of the second kind, [nm] is the cube side length, [nm] and [nm] are the elliptical cylinder base semi-axes lengths, [nm] is the rod thickness, and [nm] is the height length of the whole.

In order to provide the multi-detection of DNA alterations, we have designed the geometrical parameters of such particles in order to resonate at different frequencies. By using the aforementioned expressions we have assumed the following geometrical parameters for the selected particles: (1) the nanocube with nm, (2) the nanorod with nm and with nm, and (3) the elliptical cylinder nanoparticle with nm and nm and with nm. In this condition the cube, rod and elliptical cylinder nanoparticles resonate at nm, nm and nm respectively. For such nanoparticles configurations, we will show how the multi-detection condition for DNA alterations can be established.

It is well know that various shapes of nanoparticles can be functionalized with single DNA sequences [30,31]. In this paper we consider the nanoparticles functionalized with the corresponding DNA Probe Sequence (PS) of alternative splicing junctions of BRCA1, as shown in

The three corresponding DNA Capture Sequences (CS) of alternative splicing junctions of BRCA1 are allocated on three square gold patches deposited on silica substrate as shown in

In the next section, we will show the electromagnetic response when different ligand-receptor bindings are considered.

In this section we discuss and analyze the simulation results to verify the ability to detect the multi DNA alterations, particularly for BRCA. In order to test the proposed platform, extensive numerical analysis have been conducted through full-wave simulation [

The sensing platform is excited by an impinging plane wave, having the electric field parallel to the principal axis of the particles. The excitation is employed to analyze the electromagnetic properties, in terms of extinction cross-section. In the simulations we have assumed (1) for gold the experimental values of the complex permittivity reported in [

We considered two cases of nanoparticle detection i.e., the synchronous and asynchronous cases. In the first case, just one nanomachine has transmitted a flow of nanoparticles, and so we have just one kind of ligandreceptor bind (e.g., the nanomachine has transmitted a flow of nanocubes), while in the synchronous case, all

the nanomachines have transmitted the own nanoparticles.

As aforementioned, the electromagnetic response obtained from each nanoparticle must be different and independent in terms of resonant wavelength, amplitude and magnitude width, respectively.

The synchronous transmission can present other three binding combinations, such as the cases of 1) TS TS, 2) TS TS, and 3) TS TS. Due to the geometry of the unit cell, the coupled effects among near nanoparticles are negligible and also each nanoparticle shows near electric field enhancement at the proper resonant wavelength.

In this paper, we presented the end-to-end model from the transmittion up to the reception of flows of nanoparticles for biomedical sensing applications. How nanoparticles diffuse is also investigated. The proposed model has been verified for the multi-detection of DNA alterations, specifically for the BRCA1 gene.

In our model, we considered three kinds of metallic nanoparticles (i.e., cube, rod, and cylinder), which can form a complex with chemical receptors. Based on a ligand-receptor binding, it is possible to detect DNA

alterations and then obtain the extinction-cross section spectra, exploiting the LSPR phenomenon. Synchronous and asynchronous nanoparticle transmissions are allowed, while a selective reception process occurs.

Future work will address the use of the proposed end-to-end model in other scenarios, such as drug delivery for innovative therapeutic and diagnostic applications.