Advancing drug delivery: Neural network perspectives on nanoparticle-mediated treatments for cancerous tissues

Nazrul Islam; Yasmeen Akhtar; Shabbir Ahmad; Moin-ud-Din Junjua; Ahmed S. Hendy; Tmader Alballa; Hamiden Abd El-Wahed Khalifa

doi:10.1515/ntrev-2024-0129

Article Open Access

Advancing drug delivery: Neural network perspectives on nanoparticle-mediated treatments for cancerous tissues

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Published/Copyright: December 10, 2024

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From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

The article introduces a machine learning-based approach to enhance drug delivery to cancerous tissues via the human cardiovascular system. It addresses the need for improved drug transport in the presence of cardiovascular obstacles, such as foamy structures, which are implicated in cardiovascular diseases. By examining the impact of nanoparticles on drug transport and biomarkers like hydrogen peroxide, the study refines drug delivery strategies. The motivation is to understand how nanoparticles not only facilitate drug delivery to cancer cells but also mitigate hydrogen peroxide concentration in the blood. This study explores the interaction between nanoparticle behavior, hydrogen peroxide concentration, and drug delivery using machine learning techniques. The integration of modern-day approaches, mainly the Levenberg–Marquardt neural network (LM-NN), offers a healthy assessment of drug delivery systems. Blood flow is exhibited numerically as pulsatile flow in a parallel plate channel, incorporating the properties of foamy structures modeled as porous media. Nanostructures are treated as drug carriers by a concentration equation that considers diffusion, convection, and reaction dynamics in the blood flow. The investigation reveals that nanostructures serve a dual function by augmenting drug delivery to cancer cells and reducing hydrogen peroxide levels in the blood. Machine learning techniques, particularly the LM-NN, identify vital factors affecting drug delivery efficiency, offering insights into optimizing physiological parameters, drug properties, and patient-specific variables. This research presents a novel approach by integrating machine learning, specifically LM-NN, to optimize nanoparticle-mediated drug delivery. It exclusively combines modeling blood flow as pulsatile within a parallel plate channel with the contemplation of foamy structures as porous media. This dual-focus approach advances up-to-date methodologies by providing an inclusive understanding of the interplay between drug carriers and biomarkers, leading to potential enhancements in cancer treatment strategies.

Keywords: drug delivery; cardiovascular system; nanofluidics; machine learning; cancer tissues

1 Introduction

Pulsatile flow, characterized by rhythmic fluctuations in pressure and velocity, plays a critical role in various biological and industrial processes, including cardiovascular dynamics and fluid transport systems. In bioconversion, pulsatile flow can enhance mass transfer and reaction rates, making it valuable in processes such as enzyme-catalyzed reactions and fermentation. Numerical simulations, using methods like finite element or finite difference, are commonly employed to model and analyze pulsatile flow’s complex behavior, providing insights into optimizing parameters for improved efficiency in both medical applications and biotechnological processes. Manchi and Ponalagusamy [1] examined pulsatile flow in porous arterial stenosis, accounting for magnetohydrodynamic (MHD) and electro-osmotic with slip effects. They analyzed various gold nanoparticle shapes and solved the Poisson–Boltzmann equation for low zeta potential on walls. Ali et al. [2] modeled overlapping stenosed arteries using the Sisko non-Newtonian model and Buongiorno’s nanoscale effects formulation. They solved the nonlinear equations for mild stenosis numerically with an explicit forward time central space scheme and validated the results with the finite-element method. Ali et al. [3] explored Casson fluid flow in a porous channel, applying Darcy’s law for pulsatile flows. Their findings revealed that increasing the Hartman number reduces flow separation and streamwise velocity. Eldabe and Abou-zeid [4] analyzed the MHD pulsatile heat transfer of a non-Newtonian nanofluid between permeable plates, considering non-Darcy porous media, radiation, and viscous dissipation. The homotopy perturbation method provided solutions consistent with velocity, temperature, and nanoparticle distributions. Karim et al. [5] studied the Couette flow of Ag-water-based nanofluid between the parallel plates, incorporating Brownian motion, thermophoresis, and temperature-dependent viscosity. They generalized the flow equations using boundary layer approximations. Priyadharshini and Ponalagusamy [6] investigated the MHD effects on blood flow through a stenosed artery. Blood, modeled as a Casson fluid, was analyzed using finite-difference schemes to solve the nonlinear equations. Akdag et al. [7] numerically investigated heat transfer in Al₂O₃-water nanofluid within a wavy mini-channel under pulsating flow. They demonstrated improved thermal performance and nanoparticle dispersion, suggesting pulsating flow as a novel method for heat transfer enhancement.

Delivering drugs in synchronize with the body’s rhythm or the disease itself is a revolutionary approach that customs pulsatile flow. Pulsatile flow means the flow rate, velocity, and pressure of the drug change over time, affecting how it impacts the noticeable area. Scholars have planned various types of pulsatile drug delivery systems for numerous purposes, such as chronopharmacology, intrathecal delivery, and implantable devices. These systems can issue the drug after a certain time or in response to particular environments, such as enzymes, pH, or temperature. Hewlin et al. [8] have reported computational results on implant-assisted magnetic drug targeting (IA-MDT) using induced magnetism to deliver patient-specific therapeutic doses to targeted areas within the cardiovascular system. The multi-physics model applied in this study is based on their earlier work, with slight modifications to suit the current case scenario. The simulations indicate that the highest particle capture efficiency occurs for particle diameters between 0.7 and 4 µm, particularly in regions with minimal flow separation and vortex formation. Tripathi et al. [9] established a mathematical model for the Dynamic performance of a non-Newtonian, hydro-magnetically prejudiced nanoscale blood flow within an inflexible cylindrical artery with irregular stenoses and composite. The fluid model that designates how blood’s viscosity fluctuates with the rate of shear is named the Ostwald de-Waele power-law. This model can be employed to design biocompatible nanoparticles that can transport drugs to specific targets in the body. The study considered permeability effects in the arterial vessel, functional an external radial magnetic field, and used a combination of Buongiorno and Tiwari-Das nanoscale models. Hewlin and Tindall [10] have outlined the computational results using induced magnetism to administer intravenous within a Circle of Willis model. The simulations reveal that particle capture efficiency is highest for micron-sized particles, particularly in areas with minimal flow separation and vortex formation. Moreover, the study shows a significant decrease in capture efficiency for smaller particles, especially in the superparamagnetic range. Ponalagusamy and Priyadharshini [11] considered blood flow in a stenosed artery having magnetic particles, inspecting the influence of a Lorentz Force with recurring body acceleration. The model yields insights into the impact of different parameters regarding circulation disorders, drug delivery, and especially in cancer treatment on resistive impedance and stress caused by shear forces at the wall of the artery with potential implications. Hewlin et al. [12] developed a computational model to assess magnetic drug targeting in a patient-specific diseased carotid artery, showing promising results for localized cancer and cardiovascular treatments using magnetite particles. The study highlights efficient particle capture and suggests potential improvements with superparamagnetic particles or magnetized implants like stents. The researchers [13] investigated the behavior of very small particles that have magnetic properties in a thin tube. They applied an electric current and a magnetic field to the tube and observed how the particles reacted. The results display the extraordinary impact of electro-osmotic parameters on flow rate and carrier velocity. They described which factors are significant for apprehending magnetic nanoparticles in a small tube that can deliver drugs to the bloodstream. The factors are grounded on the numeral particles and their weight. Varmazyar et al. [14] evaluated the influence of stenosis on magneto nanoparticle distribution in pulsatile blood flow under a nonuniform magnetic field, revealing that proximal stenosis decreases drug availability near tumor tissue, highlighting the worth of considering plaque presence in drug delivery strategies.

A chemical reaction in blood on pulsatile flow is a process that includes the alliance of blood components with substances that are offered into the bloodstream, such as nanoparticles, drugs, or oxygen carriers. The chemical reaction can mark the behavior and attributes of the blood, such as its viscosity, density, concentration, temperature, and flow rate. Numerous circumstances, such as cardiovascular systems, implantable devices, and oral drug delivery, have been exposed by numerous researchers using chemical reactions with nanostructures in blood for drug delivery on pulsatile flow. Murugan et al. [15] calculated how a fluid with a certain viscosity acts when it flows in pulses under the action of magnetic and electric fields inside a tube filled with porous material. They simplified the mathematical model and found exact solutions for how the fluid spreads, how much of a substance it carries, and how it transfers heat. Hewlin and Kizito [16] conducted computational fluid dynamics simulations to compare three-dimensional pulsatile blood flow through a simplified stenosed carotid bifurcation model and two patient-specific stenosed carotid bifurcation geometries. Kumar et al. [17] analyzed the effects of Joules and viscous dissipations on the pulsatile flow of Casson-based nanofluid in a standing channel and thermal radiation. They utilized a perturbation scheme to decrease the governing partial differential equations to ordinary differential equations. Ali et al. [18] examined the interaction between an external Lorentz force and biofluid flow via a Darcy–Forchheimer tube, considering wall transpiration, pressure gradients, and chemical reactions. Additionally, the Forchheimer magnetic parameter and drag parameter showed contrasting effects. Hewlin and Kizito [19] investigated whether flow patterns and mechanical factors, like wall shear stress, help assess local cardiovascular risks and if simplified geometric models are reliable for flow modeling. Their study emphasizes the link between atherosclerosis and arterial geometry, highlighting the need for improved disease management. Mazumder and Das [20] studied the streamwise spreading of submissive contaminant fragments in a time-varying flow inside a channel. Utilizing an implicit finite-difference tactic, the scientists cracked the unstable convective–diffusion equation, showing the impact of pressure pulsation frequency and heterogeneous reactions.

Magnetohydrodynamics (MHD) is the study of magnetic fields with electrically conducting fluids such as blood. The phenomena of MHD in blood on pulsatile flow happen when blood moves through a magnetic field, such as in an MRI scanner. The magnetic field exercises energy on the charged particles in the blood, varying blood flow properties such as temperature, velocity, and pressure. MHD can be used to evaluate the volume and velocity of blood flowing over the arteries, which can assist in the diagnosis of cardiovascular disorders. MHD can also be employed to transport medications and control blood flow or nanoparticles to precise areas of the body, such as infections or tumors. Hussain and Shabbir [21] numerically investigated the effects of Brownian motion and thermophoresis on pulsating nanofluid (blood) flow in a curved artery with stenosis and post-stenotic dilatation using the Herschel–Bulkley model for fluid rheology. The study employed explicit finite-difference methods to solve the governing equations for momentum, energy, and mass concentration. Govindarajulu et al. [22] studied how a fluid with a certain viscosity and electric and magnetic properties flows in pulses between two walls with holes and different temperatures. Kumar and Suripeddi [23] studied how a mixture of blood and tiny particles of copper, alumina, gold, and silver behaves when it flows in pulses through a vertical channel filled with small holes under the influence of a magnetic field. They used a mathematical model that accounts for the fact that blood does not have a constant viscosity (flow rate), unlike water or oil. The study incorporated velocity slip and Joule’s heating at the walls, employing numerical solutions through the shooting and Runge–Kutta fourth-order methods. The researchers examined the impacts of flow-governing parameters, including cross-flow Reynolds number (Re), frequency parameter, temperature, and velocity profiles. Abbas et al. [24] employed numerical methods to simulate blood flow through an overlapping stenosed arterial vessel with periodic pressure gradient and external body acceleration. The Sutterby fluid model was employed to describe blood rheology, with the integration of a constant, uniform radial magnetic field within the blood vessel. In their final exploration, Ardahaie et al. [25] examined blood flow having nanoparticles considering the impact of a Lorentz force through analytical and mathematical approaches. The relevant works are available in the following sources [26,27,28,29].

1.1 Novelty of the current study

The literature has reputable that nanostructures are active transporters for drug delivery, particularly in targeting cancerous tissues due to their capability to navigate through intricate biological environments. Studies have revealed that these nanoparticles can be caused to enhance drug release at precise spots, enhancing therapeutic outcomes. Furthermore, the role of hydrogen peroxide as a biomarker of oxidative stress and its impact on many physiological processes is well documented. Outdated drug delivery methods frequently encounter challenges such as limited precision in obstacles and targeting within the cardiovascular system, which can hinder the effective delivery of therapeutic agents. In this study, a machine learning-based tactic was used to optimize the procedure of nanoparticle-mediated drug delivery to cancerous tissues. The methodology involved modeling blood flow as pulsatile flow within a parallel plate channel, taking into account the existence of a foamy structure (which has been modeled as a porous media). Nanostructures were exhibited as drug carriers, and their conduct was administrated by a concentration equation that reported for convection, diffusion, and reaction in the blood flow. The research employed unconventional machine learning techniques, specifically the LM-NN, to analyze how several drug-related and physiological parameters influence drug delivery efficacy. This technique aimed to classify key factors that could optimize therapeutic outcomes. This tactic allowable for a comprehensive examination of the dual role of nanoparticles in both drug delivery and the reduction of hydrogen peroxide concentration in the blood, contributing insights into potential developments in treatment strategies for cancer patients.

2 Problem formulation

We study the dynamics of nanofluid flow within a parallel channel embedded in a porous medium, as shown in Figure 1. The flow is two-dimensional, incompressible, laminar, and electrically conductive. Wall transpiration is modeled by injecting fluid through the bottom plate and extracting it through the top plate, which is located at y = ±c. The momentum equation has been developed by the utilization of Forchheimer’s law, with its terms modifying the Navier–Stokes equation. We also investigate the mass transportation of chemically reactive species in the channel, where the species, denoted by (C) in mol/m^³, forms a homogeneous mixture within the fluid. A first-order, irreversible reaction is presumed, with the concentration of species (C ₀) held constant at both the lower and upper walls of the channel. Initially, in the absence of transpiration, the temperature distribution is linear throughout a channel. However, the introduction of injection and suction alters the temperature significantly.

Figure 1

A simple diagram of human cardiovascular function.

The equations leading the system are expressed in dimensional form as outlined in reference:

Momentum equation:

(1) ∂ u ∂ τ + v 0 ∂ u ∂ x = − 1 ρ ∂ p ∂ x + v B ∂ 2 u ∂ y 2 − σ B 0 2 u ρ − v B k p u − b 0 u 2 ,

Energy equation:

(2) ∂ T ∂ τ + v 0 ∂ T ∂ y = α ∂ 2 T ∂ y 2 + ( ρ c p ) p ( ρ c p ) f D B ∂ C ∂ y ∂ T ∂ y + D T T c ∂ T ∂ y 2 ,

Equation for the nanoparticle concentration:

(3) ∂ C ∂ τ + v 0 ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T 0 ∂ 2 T ∂ y 2 − K c b 2 C ,

Equation for the concentration of the chemically reactive species (H _₂O_₂):

(4) ∂ b ∂ τ + v 0 ∂ b ∂ y = D 0 ∂ 2 b ∂ y 2 + K c b 2 C ,

where

α: thermal diffusivity
p: pressure
τ: dimensional time
σ: electrical conductivity
v 0: wall transpiration velocity
k p: porous medium permeability
C: concentration of the species
k: reaction rate constant
T: temperature
v B: kinematic viscosity
u: horizontal velocity
B 0: magnetic field strength
ρ: density
b 0: inertial drag coefficients
D B: mass diffusivity
C 0: characteristic concentration

Boundary conditions (for >0):

(5) b = b 0 , u = N ∂ u ∂ y , T = T 0 , C = C 0 at y = − a b = b 0 , u = N ∂ u ∂ y , T = T 0 , C = C 0 at y = a .

To streamline the analysis and minimize the number of variables, we define the subsequent dimensionless quantities:

(6) U = u v 0 , ξ = x a , η = y a , t = v 0 a τ , P = p ρ v 0 2 , θ = T − T 0 Δ T , ϕ = C − C 0 Δ C , ψ = b − b 0 Δ b .

By applying these dimensionless quantities, the governing equations can be reformulated into a more simplified form:

(7) ∂ U ∂ t + ∂ U ∂ η = − ∂ P ∂ ξ + 1 R ∂ 2 U ∂ η 2 − Ha . U − 1 λ U − N f U 2 ,

(8) ∂ θ ∂ t + ∂ θ ∂ η = 1 R Pr ∂ 2 θ ∂ η 2 + Nb ∂ θ ∂ η ∂ ϕ ∂ η + Nt ∂ θ ∂ η 2 + Ec R ∂ u ∂ η 2 + Ec Ha u 2 ,

(9) ∂ ϕ ∂ t + ∂ ϕ ∂ η = 1 Sc ∂ 2 ϕ ∂ η 2 − K l ϕ ψ 2 + Nt Nb 1 Sc ∂ 2 θ ∂ η 2 ,

(10) ∂ ψ ∂ t + ∂ ψ ∂ η = δ Sc ∂ 2 ψ ∂ η 2 + K l ϕ ψ 2 .

Below are the dimensionless parameters appearing in the equations:

(11) R = a v 0 v B ; Reynolds number λ = k p v 0 a v B ; Darcy parameter, H = σ B 0 α ρ v 0 ; Magnetic parameter, Nf = a b ; Forchheimer quadratic drag parameter, Sc = a v 0 D ; Schmidt number, γ = k a 2 v B ; Chemical reaction parameter, Pr = v B a ; Prandtl number .

Initially, we assume that the fluid is at rest, with both the concentration and the temperature distributions remaining unchanged across the regime. This supposition results in the subsequent initial conditions:

(12) U = 0 , θ = ϕ = ψ = 1 ∀ − 1 ≤ η ≤ 1 at t = 0 .

In this context, P s indicates oscillatory components, P 0 signifies the static component, and ω represents the occurrence of pressure gradient. Consequently, the boundary conditions are expressed subsequently:

(13) η = − 1 , U = s 0 ∂ U ∂ Y , θ = ϕ = ψ = 1 η = 1 , U = s 0 ∂ U ∂ Y , θ = ϕ = ψ = 1 .

The pressure gradient is defined as follows:

(14) − ∂ P ∂ ξ = P s + P 0 cos ( ω t ) .

3 Numerical solution

For the computational solution, we utilize a fully implicit method described above as follows:

(15) U m ( N + 1 ) − U m ( N ) Δ t + U m + 1 − U m − 1 2 Δ η ( N + 1 ) = { P s + P 0 cos ( t ( N + 1 ) ω ) } + 1 R U m + 1 − 2 U m + U m − 1 Δ η 2 ( N + 1 ) − Ha . U m ( N + 1 ) − 1 λ U m ( N + 1 ) − N f U m ( N + 1 ) 2 ,

(16) θ m ( N + 1 ) − θ m ( N ) Δ t + θ m + 1 − θ m − 1 2 Δ η ( N + 1 ) = 1 R Pr θ m + 1 − 2 θ m + θ m − 1 Δ η 2 ( N + 1 ) + Nb θ m + 1 − θ m − 1 2 Δ η ( N + 1 ) ϕ m + 1 − ϕ m − 1 2 Δ η ( N ) + Nt θ m + 1 − θ m − 1 2 Δ η ( N + 1 ) θ m + 1 − θ m − 1 2 Δ η ( N ) + Ec R U m + 1 − U m − 1 2 Δ η ( N + 1 ) 2 + Ec Ha U m ( N + 1 ) 2 ,

(17) ϕ m ( N + 1 ) − ϕ m ( N ) Δ t + ϕ m + 1 − ϕ m − 1 2 Δ η ( N + 1 ) = 1 Sc ϕ m + 1 − 2 ϕ m + ϕ m − 1 Δ η 2 ( N + 1 ) − K l ϕ m ( N + 1 ) ψ m ( N ) 2 + Nt Nb Sc θ m + 1 − 2 θ m + θ m − 1 Δ η 2 ( N + 1 ) ,

(18) ψ m ( N + 1 ) − ψ m ( N ) Δ t + ψ m + 1 − ψ m − 1 2 Δ η ( N + 1 ) = δ Sc ψ m + 1 − 2 ψ m + ψ m − 1 Δ η 2 ( N + 1 ) + K l ϕ m ( N + 1 ) ψ m ( N ) ψ m ( N + 1 ) .

It is important to note that the subscript and superscript indicate the position of the functional value and the time level on a homogenously spaced grid:

(19) η m = − 1 + 2 ( m − 1 ) N m − 1 .

We have observed the velocity distribution at several points through the computational domain to weigh grid independence. This encompasses how the velocity profiles converge as the grid is refined, confirming that the outcomes are not reliant on the grid resolution. By comparing the velocity distributions at different grid densities, we can check that the numerical results are independent and reliable of grid size. Figure 2 demonstrates the computational grid used in the current investigation, showcasing the grid structure and prominence of the zones where velocity measurements were taken to authenticate grid independence.

Figure 2

Computational grid used in the present work.

The detailed results associated with computational outcomes can be found in Table 1.

Table 1

Convergence of computational outcomes with decreasing step size (h):

η	U ( η )
	h = 0.05	h = 0.025	h = 0.0125
−1	0	0	0
−0.8	0.0574	0.0582	0.0594
−0.6	0.0955	0.1093	0.1080
−0.4	0.1301	0.1384	0.1382
−0.2	0.1566	0.1578	0.1567
0	0.1624	0.1692	0.1681
0.2	0.1631	0.1679	0.1680
0.4	0.1523	0.1461	0.1420
0.6	0.1189	0.1139	0.1140
0.8	0.0672	0.0692	0.0610
1	0	0	0

In the framework of the present research, a grid independence analysis was conducted to guarantee that the computational results were not prejudiced by the discretization of the computational domain. This is systematically purifying the computational mesh and noticing the changes in key flow variables such as pressure, velocity, concentration, and temperature profiles. The modification process started with a coarse grid and gradually increased the number of grid points. The grid was measured independently once extra refinement caused tiny changes in these variables. The final grid, which balanced accuracy and computational efficiency, was designated grounded on these criteria, certifying that the simulation outcomes were robust and not dependent on the exact grid size. This method agrees with the consistency of the numerical results existing in the study.

4 Machine learning procedure

The Neural-Fitting app offers a robust framework for building and evaluating neural networks, featuring tools for data selection, performance assessment, and training evaluation. It incorporates mean square error metrics and regression analysis for precise performance monitoring. Utilizing a two-layer feedforward network with sigmoid-hidden neurons and a linear output neuron, the app is capable of handling complex multi-dimensional plotting tasks. The optimal number of neurons in the hidden layer is determined by the task’s complexity, enhancing the model’s generalization ability. The app supports three key training methods: the Bayesian regularization algorithm, which excels in generalizing to complex or noisy datasets despite being time-consuming; the Levenberg–Marquardt algorithm, which is faster but requires more memory and stops training when validation sample errors increase; and the scaled-conjugate-gradient algorithm, which uses less memory and halts training when no further generalization improvements are observed. In neural networks, outputs are generated by summing weighted inputs (including a bias term) and applying a nonlinear activation function. The app relies on well-prepared training data, such as papers and images, to ensure accurate predictions and insights.

After determining the centers, the weights are computed by minimizing the output error, usually employing a linear-pseudo-inverse method, as described in equation (19)

(20) ω ̲ = A + b ̲ ,

where matrix A is defined as

(21) A i , j = ρ ( ∥ x i − c j ∥ ) .

At each time step, a gradient-based optimization algorithm adjusts the weights to minimize error. This process involves calculating the cost gradient, with the magnitude and direction of weight changes determined by moving opposite to the gradient. The new weights are then recalculated using a specific mathematical formula, as shown in equation (18)

(22) ω ̲ new = ω ̲ old + Δ ω ̲ .

The steps involved in calculating the gradient for updating weights are as follows:

(23) Δ ω j = − η ∂ J ∂ ω j ,

(24) Δ ω j = − η ∑ i ( target ( i ) − output i ) ( − x j ( i ) ) ,

(25) Δ ω j = η ∑ i ( target ( i ) − output i ) ( x j ( i ) ) .

The learning rate, denoted by η , is crucial in the gradient descent optimization process. Imagine a hiker, symbolizing the weight factor, trying to reach the lowest point in a valley, which represents the cost function. The hiker’s steps are influenced by the slope’s steepness (gradient) and the learning rate, which together determine the direction and distance of each step to minimize the cost.

4.1 The ANN controller

Recently, the ANN controller has become increasingly popular in energy management strategies [30]. The neural network’s architecture consists of three layers: input, hidden, and output, each populated with neurons (or nodes) that interconnect within the multi-layer network. The ANN controller operates in two main phases: the training phase, where the model learns from data, and the operational phase, where it is utilized for real-time applications. Figure 3 illustrates the design and structure of the ANN controller model.

Figure 3

Illustration of artificial neural network, analogous to neurons in a brain [31].

The artificial neural network (ANN) model is created using the MATLAB/Simulink environment. As depicted in Figure 4, the architecture employs a feedforward neural network design featuring two neurons in the input layer, ten neurons in the hidden layer, and one neuron in the output layer. The training process utilizes the Levenberg-Marquardt algorithm, optimizing the network’s performance with a dataset of duty cycle values obtained from simulations.

Figure 4

Layer assembly of the ANN model using the MATLAB NNET toolbox [32].

4.2 Flowchart illustrating our computational methodology

Figure 5 illustrates the numerical model employed for the alternating direction implicit (ADI) scheme.

Figure 5

Flow chart of our numerical approach.

5 Benchmarking the numerical scheme

The study’s predictions of temperature distributions and heat transfer coefficients were compared with the results from Ali et al. [18] for a certain limiting case. The comparison, as illustrated in Figure 6, revealed that the thermal gradients near the channel walls and the overall heat transfer rates were consistent with those reported in the literature, reinforcing the validity of the numerical approach. This comprehensive validation process reinforces the robustness of the numerical results, thereby lending greater confidence to the conclusions drawn in the study.

$Figure 6 Comparison of our computational outcomes for g ( η ) g(\eta ) and f ( η ) f(\eta ) , where C 1 = 0.1 , C 2 = 1 , {C}_{1}=0.1,{C}_{2}=1, C 3 = 0.1 , {C}_{3}=0.1,\hspace{.25em} Re = 0.1 , and M = 0 . \text{Re}=0.1,\hspace{.25em}\text{and}\hspace{.25em}M=0.$

Figure 6

Comparison of our computational outcomes for g ( η ) and f ( η ) , where C 1 = 0.1 , C 2 = 1 , C 3 = 0.1 , Re = 0.1 , and M = 0 .

The Re represents the balance between inertial and viscous forces in a fluid. In practical terms, a high Re indicates turbulent flow, characterized by chaotic motion, while a low Re corresponds to laminar flow with smooth, parallel fluid layers. In blood flow, a higher Re can result in turbulence, potentially disrupting uniform drug distribution. Turbulent flow augments the heat transfer rate due to the mixing of fluid layers; however, laminar flow may result in lower heat transmission, which is vital for keeping the constancy of thermally sensitive drugs during delivery. The Darcy parameter measures the penetrability of a porous media, such as blood vessels with porous walls. An advanced Darcy parameter designates a smaller amount of confrontation in flow, affecting pressure drop and flow velocity, which in turn affects nanoparticle distribution. Permeability also influences thermal conductivity, by lower permeability leading to condensed heat transfer, which can impact temperature regulation in drug delivery procedures. The Forchheimer parameter is responsible for nonlinear drag forces in porous media, with higher values representing enlarged resistance to fluid flow. This resistance can lessen the velocity of blood-carrying nanostructures, which is vital when considering flow through intricate vascular structures. Augmented drag forces can also decrease convective heat transport, affecting the thermal atmosphere everywhere drug delivery sites. The magnetic parameter replicates the impact of a magnetic field on electrically conducting fluids like blood. A higher magnetic field intensity can sluggish the flow due to the Lorentz force, permitting for control or direction of blood flow in medical trials. The magnetic field can also modify temperature distribution by affecting flow velocity, thereby impacting convective heat transport and empowering temperature management during therapy.

The Schmidt number is the ratio of momentum diffusivity to mass diffusivity, with higher values indicating slower momentum diffusion compared to mass diffusion. This can create sharp concentration gradients in the fluid, leading to localized regions of high or low drug concentration in drug delivery systems. Although primarily affecting mass transfer, the Schmidt number’s influence on fluid flow can indirectly impact heat transfer, particularly in multi-component flows. This parameter measures the rate of chemical reactions relative to the flow rate. A higher chemical reaction parameter indicates rapid reactions, significantly altering the concentration of reactive species like drugs in the bloodstream. A low Prandtl number (Pr) suggests rapid heat diffusion compared to momentum, leading to quicker temperature equilibration in the fluid. This impacts the thermal boundary layer thickness, with higher Pr leading to thinner layers and more efficient heat transfer, which is important for temperature-sensitive drug delivery. The slip parameter accounts for the relative motion between fluid and solid boundaries, such as blood vessel walls. A higher slip parameter indicates less friction, facilitating easier flow and reducing resistance, particularly in diseased vessels or engineered drug delivery systems. Each of these parameters significantly influences fluid flow and heat transfer, which are crucial for optimizing nanoparticle-mediated drug delivery systems, ensuring efficient and effective drug delivery to targeted tissues.

6 Results and discussion

This section examines the impact of numerous parameters on the thermal, concentration, and momentum profiles of blood-based nanofluid flow. The flow is motivated through the periodic force within a channel, as illustrated in Figure 1, and involves nanofluids comprising two types of nanostructures. Unless otherwise specified, the central parameters which are used in the current study are Sc = 0.5 , Re = 1 , λ = 0.5 , Pr = 21 , N f = 0.75 , Nt = 0.5 , Nb = 0.5 , p o = 1 , γ = 5 , p s = 1 , K l = 5 , Ec = 0.02 , ω = 2 , Ha = 0.3 , and Δ = 1 . To establish the reliability of our computational technique, we analyze variations in computational results, like temperature, concentration, and velocity, across different step sizes at a fixed time period. We compute the differences in computational outcomes between the next two successive grids, with one grid’s step size being half that of the previous grid. Figure 7(a) and (b) depicts the evolution of velocity parameters in a horizontal channel over time. We notice a repetition of a cycle in a flow field, which is determined by a parameter ω, which sets a cycle frequency. The stagnation renders the pressure gradient less effective until the flow has sufficient momentum, resulting in an initial cycle that is less symmetrical than later cycles. At the beginning, the stationary fluid among the walls of the channel leads to the in-line temperature distribution. This trend in Figure 7(c) and (d) suggests that the fluid’s highest temperature matches that of the bottom channel wall. Furthermore, Figure 7(e)–(h) reveals a notable similarity between the concentration of impurities, and the nanoparticle concentration surface, although a distinct gradient is observed near the top channel wall, which is less pronounced.

$Figure 7 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the cross-section of the channel.$

Figure 7

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the cross-section of the channel.

Before discussing the profiles, it is important to note that Figure 7 shows significant variations in the velocity and temperature profiles over time. This variation in the velocity profile is because of the term − ∂ p ∂ ξ = P s + P 0 cos ( ω t ) in the flow velocity equation. Similarly, the temperature field changes over time due to the term Ha⸱υ ² in the heat equation, which is responsible for heat generation. Simple profiles for both quantities ϕ and χ indicate that they tend to approach a steady-state solution regardless of governing parameters. In the following discussion, temperature and the velocity profiles at the time level (t = 10) have been considered, while ϕ ( η ) and χ ( η ) will represent the steady-state solution.

Turning our attention to Figure 8(a)–(d), it is evident that Darcy–Forchheimer parameter significantly influences the velocity of flow. This phenomenon is attributed to the diminishing impact of the Darcian force, which acts as a frictional or resistive force that increases with mounting (λ). Consequently, this leads to an enhanced impact of a pressure gradient. The inclusion of this term in governing equations arises from the application of Forchheimer’s law to model porous media in the given geometry. An increased significance of such a term results in greater flow resistance, eventually contributing to deceleration of the flow throughout the channel. From Figure 9(a) to (d), it is clear that the Pr exclusively impacts the heat profile due to the decoupling of the governing equations. Furthermore, an examination of the heat equation reveals that higher values of Pr reduce heat diffusion, which is influenced by the presence of the term 1 Pr R ∂ 2 θ ∂ ξ 2 in the corresponding equation. Further, Figure 10(a)–(d) shows a notable decrease in concentration profiles affected by both the Schmidt number (Sc) and the Re.

$Figure 8 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel cross-section with governing parameter (Nf).$

Figure 8

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel cross-section with governing parameter (Nf).

$Figure 9 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel cross-section with governing parameter (Pr).$

Figure 9

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel cross-section with governing parameter (Pr).

$Figure 10 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel cross-section with governing parameter (Sc).$

Figure 10

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel cross-section with governing parameter (Sc).

The chemical reactions (Kl) in the model primarily focus on the interaction between the nanoparticles and hydrogen peroxide within the bloodstream. Specifically, we utilized reaction rates derived from studies on similar nanoparticle systems, ensuring that the parameters align with realistic biological scenarios. In the model, the reaction kinetics were assumed to follow a first-order reaction with respect to the concentration of hydrogen peroxide. This assumption is based on the commonly observed behavior of nanoparticles in catalytic reactions, where the rate of reaction is directly proportional to the concentration of the reactant. Figure 11(a)–(d) illustrates that, without chemical reaction, there is no increase in the profiles of ϕ or χ . The temperature profile coincides with a line η = 1 , which explains why only three profiles are observed instead of the four profiles. We observe that as the values of K1 increase, ϕ ( ξ ) decrease. Thus, it has been concluded that an enhancement of chemical reaction leads to an increase in ϕ profiles while reducing nanoparticle concentration. Figure 12(a)–(d) suggests that an increase in Δ (the ratio of nanoparticle diffusivity to that of chemically reactive species) leads to a decrease in the measurable concentration of the species.

$Figure 11 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel cross-section with governing parameter (Kl).$

Figure 11

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel cross-section with governing parameter (Kl).

$Figure 12 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel cross-section with governing parameters Δ \Delta .$

Figure 12

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel cross-section with governing parameters Δ .

The intensity of the magnetic field in the current study is represented by a parameter H. An analysis of governing equations indicates that H significantly influences both heat transfer rate and shear stress as it directly appears in energy and the momentum equations. It was observed that by increasing H reduces the shear stress at walls, which could be beneficial in applications where relaxing artery walls by lowering blood pressure is desirable (Figure 13).

Figure 13

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter H.

As illustrated in Figure 14(a)–(d), the magnetic field has a stronger impact on flow velocity than on temperature variation. This weaker effect on the temperature profile is due to the term Ec⸱Ha⸱υ ² in the heat equation, coupled with the low Eckert number for this case. Additionally, we observe a steep decline in the velocity profile with a modest increase in the Hartmann number, suggesting that magnetic fields could serve as an effective means of controlling blood flow in targeted sections of human arteries.

$Figure 14 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel’s cross-section influenced by the governing parameter H.$

Figure 14

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel’s cross-section influenced by the governing parameter H.

Analyzing the effect of the Re on the flow regime, Figure 15(a) clearly shows an increase in flow velocity. The velocity profiles notably display a discernible tilt towards the upper wall. As the Re grows, there is a corresponding elevation in suction velocity, a condition assuming that fluid characteristics remain constant. This heightened suction contributes to the observed tilt in the velocity profile. A more detailed examination of Figure 15(b) uncovers that, upon beginning of flow, a confined region near the top wall experiences an intensified temperature gradient. This phenomenon effectively eradicates the preliminary linear thermal distribution.

$Figure 15 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel’s cross-section governed by the parameter R.$

Figure 15

Temporal variation in temperature ( θ ) , velocity (U), chemically reactive species ( χ ) , and nanoparticle concentration ( ϕ ) through the channel’s cross-section governed by the parameter R.

Examining Figure 16(a)–(d) demonstrates that the Darcy parameter λ fosters the velocity of the flow. This arises from the weakening Darcian force, acting by way of frictional and resistive force, with growing λ in the mathematical framework, translating into an increased impact of pressure gradient. An increased value of this parameter equates to reduced flow confrontation, eventually accelerating the flow across the channel, as shown in Figure 16(a).

$Figure 16 Temporal variation in temperature ( θ ) (\theta ) , velocity (U), chemically reactive species ( χ ) (\chi ) , and nanoparticle concentration ( ϕ ) (\phi ) through the channel’s cross-section influenced by the governing parameter λ.$

Figure 16

The study’s findings on magnetic fields and their impact on shear stress at artery walls have notable clinical implications. By optimizing shear stress through magnetic fields, it is possible to improve blood flow dynamics and develop novel non-invasive therapies for cardiovascular diseases. These magnetic fields may also help reduce plaque formation and stabilize existing plaques, potentially preventing serious cardiovascular events. Additionally, they could enhance magnetic drug delivery systems, allowing for more targeted and effective treatments. The research paves the way for innovations in medical devices that utilize magnetic fields, contributing to better vascular health and personalized medicine.

The influence of Re on the physical quantities of interest is illustrated in Figure 17. At the upper wall of the channel, the velocity gradient is the most significantly affected quantity. Additionally, there is a noticeable decrease in the heat transfer rate, while changes in ϕ ′ and ψ ′ are minimal. A similar trend is observed at the lower wall of the channel. However, it can be concluded that the Re does not have a major impact on the physical quantities at the lower wall.

Figure 17

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter Re.

Figure 18 is dedicated to understanding the influence of the parameter ( λ ) on physical quantities. We observe an increase in the velocity gradient, which is proportional to the corresponding shear stress. However, the rate of increase diminishes as λ increases. For instance, when λ varies from 1 to 2, the change v ′ ( ± 1 ) is noticeable in the first decimal place. Conversely, when λ changes from 4 to 5, the change is observed in the third decimal place despite the same rate of change. This trend is attributed to the factor ( 1 / λ ) in the governing equation. The remaining data shows minimal variation with changes in λ .

Figure 18

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter λ.

Figure 19 clearly illustrates the numerical data showing the influence of the parameter (Nf) on the physical quantities of interest. It is important to note that both Nf and λ the drag force on the flow, due to the presence of a porous medium, represent the same phenomenon. Consequently, it is expected that these two parameters will impact the flow similarly, at least quantitatively.

Figure 19

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter Nf.

Based on Figure 20, it is evident that the chemical reaction parameter Kl significantly influences the profiles of ϕ and ψ due to the nature of the governing equation. However, the table shows only a slight change in the heat transfer rate, attributed to the weaker coupling between the energy equation and the equation governing nanoparticle concentration. Additionally, this parameter greatly impacts the nanoparticle concentration, with a rapid increase in the concentration profile observed as Kl increases.

Figure 20

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter KL.

Now, Figure 21 illustrates that the Sc significantly increases the magnitudes of the two quantities ϕ ′ ( ± ) and ψ ′ ( ± ) . However, u ′ ( ± 1 ) remains constant because Sc does not appear in the momentum equation. Notably, the heat transfer rate at the upper wall shows a significant decrease with the increase in Sc. Additionally, the data reveals that the heat transfer rate at the lower channel wall is almost zero, indicating a flat thermal profile in that region.

Figure 21

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter Sc.

Finally, the influence of the parameter Δ on the current problem can be understood from Figure 22. Since Δ appears in the governing equation derived from ψ ( ξ ) , we observe no change in shear stress due to the decoupled governing equations. This indicates that “A” does not directly affect the shear stress in the system. However, the quantities ϕ ′ ( ± ) and ψ ′ ( ± ) play a more significant role. These parameters are the most influential factors in the system, impacting the overall behavior and outcomes. Their influence is evident in the variations observed in the results, suggesting that any changes in ϕ ′ ( ± ) and ψ ′ ( ± ) can lead to noticeable differences in the system’s performance. Understanding the roles of these parameters is crucial for accurately modeling and predicting the behavior of the system under various conditions.

$Figure 22 Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter Δ \Delta .$

Figure 22

Nanoparticle concentration, shear stress, the concentration of chemically reactive species, and heat transfer rate at the artery walls for the governing parameter Δ .

In our investigation of vortex generation through lid-driven cavity flow enhanced with nanostructures, we employed sophisticated methodologies, including cutting-edge machine-learning techniques. The implementation of the neural fitting application was crucial in refining our analytical framework, enabling the development of tailored neural networks and the effective selection of data through comprehensive training. We rigorously evaluated the performance of our optimized network using established metrics such as mean squared error and regression analysis, as illustrated in Figure 23(a)–(c). This evaluation highlights the importance of our state-of-the-art approach in achieving precision and improving the quality of our study findings. We recognize that the quality and diversity of the training dataset are fundamental to the success of machine learning models. Variations in factors such as seasonal changes, solar radiation levels, weather patterns, and solar energy exposure can have a significant influence on the model’s effectiveness. Moreover, the intrinsic complexity of a real-world system often necessitates an introduction of certain assumptions in the machine learning models that might not adequately replicate the complexities of phenomena being investigated. Therefore, it is essential to stay attentive regarding these encounters to guarantee the validity and applicability of our conclusions within the realm of the vortex dynamics. By addressing these issues, we can enhance the robustness and relevance of our research outcomes.

Figure 23

(a)–(c) Outputs generated by widely-used neural fitting application following training on the dataset produced by our self-developed code.

7 Conclusion

This research explores the effects of several parameters on thermal and concentration profiles in a channel where blood, mixed with nanostructures and impurities like H₂O₂, assists as a base fluid. It comprises the stability analysis of our computational scheme and studies parameters such as the strength of the magnetic field, Darcy parameter, Re, chemical reaction parameter, and Pr.

7.1 Key findings

The study reveals that a temperature profile flattens, with a complex thermal gradient neighboring the top wall. The magnetic field intensity (H) meaningfully influences shear stress and flow velocity, highlighting vital applications for regulating blood flow. The Re shifts velocity profiles (U) toward the top wall, increasing the suction velocity and the temperature gradients. The Darcy (λ) and Forchheimer (Nf) parameters are key in controlling the flow velocity and resistance, affecting the pressure gradient in a porous media. Moreover, a chemical reaction parameter (Kl) markedly increases concentration at higher values, while the Sc boosts concentration and velocity. The slip parameter (ω) influences shear stress and heat transfer, with implications for arterial wall relaxation.

7.2 Improvements over existing work

This study advances existing research by integrating a machine learning-based approach, specifically the LM-NN to optimize drug delivery systems. It addresses the limitations of traditional methods by providing a detailed examination of how nanoparticles enhance drug delivery while simultaneously reducing hydrogen peroxide concentration in the blood. This dual-focus approach offers a more comprehensive understanding of the interplay between drug carriers and biomarkers, which is a significant improvement over previous models that may not have accounted for these interactions.

7.3 Future work

Future research will focus on refining the model to include more complex biological scenarios and testing the effects of additional parameters on drug delivery efficiency. Further studies will explore experimental validation of the model and its application in personalized medicine to enhance therapeutic outcomes for patients with specific cardiovascular conditions.

Acknowledgments

The authors are grateful to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R404), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding information: This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R404), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-05-16

Revised: 2024-10-18

Accepted: 2024-11-14

Published Online: 2024-12-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ntrev-2024-0129

Keywords for this article

drug delivery; cardiovascular system; nanofluidics; machine learning; cancer tissues

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