Magneto-Hemodynamic of Blood Flow Having Impact of Radiative Flux Due to Infrared Magnetic Hyperthermia: Spectral Relaxation Approach

The science of blood fluid dynamics in the presence of static high magnetic fields is known as magneto-hemodynamics. The importance of this study stems from the fact that hemodynamic dysfunction causes the majority of cardiovascular ailments such as stenosis and abnormalities, leading to cancer. Hyperthermia is a treatment that involves exposing bodily tissue to high temperatures in order to harm and kill cancer cells or make them simpler to kill with radiation therapy or chemotherapy. Hyperthermia treatment can take several forms, including putting the full body or a portion of it in a heated chamber or machine, as pictured.

According to the World Health Organization, cancer-related death is projected to surge by 15% across the globe for the period 2010-2020. According to the Surveillance, Epidemiology, and End Results (SEER) initiative of the National Cancer Institute (NCI), cancer incidence affects roughly 9% of the US population. In Europe, Ferlay et al. [1] reported approximate death tool of 1.9 million and nearly 4 million new cases for the 2018. While nations in the developed world are achieving substantial progress in cancer management, Africa is currently experiencing rapid surge in cancer incidence. For instance, experts in the field of oncology in Africa reported that Nigeria has the highest mortality in age-standardized study [2]. However, the improvement on thermal and non-thermal therapies in diagnosis and management of cases of tumor-related illnesses is suggested to hold the potential to douse the mortality burden associated with patients inhibited with cancerous infections. Present hope in combating cancer plague is triggered by capabilities of several strategies such as hyperthermic perfusion, conductive hyperthermia, electromagnetic and ultrasonic hyperthermia and electrothermic procedures.

Hyperthermia (HT) has been an important strategy to treat cancer-related illness over the years. Innovations around applications of thermal biology for tumor in several anatomical sites of the body have been reported. High temperature inducement of malignant region is described as an adjunctive procedure where elevated temperature field exposure on portions of tumor growth is administered to provide curative effect on the tissue [3]. To improve the potency of clinical interventions with far-reaching therapeutics effects such as chemotherapy and radiotherapy [4], used finite element method for the computation of magneto-hemodynamic flow and heat transfer in a bifurcated artery with saccular aneurysm using the Carreau-Yasuda biorheological model. Ige et al. [5] examined nanoparticle-embedded blood flow control with magnetic field influence using spectra homotopy analysis. Using implantable anticancer drug delivery with thermal ablation, Al-Zu’bi and Mohan [6] studied Modelling of combination therapy in solid tumor. Shorif et al. [7] investigated magnetic nanoparticles mediated cancer hyperthermia while the Study of heat and mass transfer to magnetohydrodynamic (MHD) pulsatile couple stress fluid between two parallel porous plates was considered by Oyelami et al. [8]. In the treatment planning of radiofrequency hyperthermia therapy in clinics [9] studied the role of simulations. Subject to a transient laser irradiation, Zhang et al. [10] examined modelling and simulation on heat transfer in blood vessels. In order to improve the accumulation and penetration of magnetic nanocarriers into solid tumors, Liu et al. [11] used oppositely polarized external magnets in his analysis. Magneto-Hemodynamics Mixed Convection with radiative heat transfer in a stenosed artery was investigated by Ngufor and Makinde [12]. Nguyen et al. [13] reported a non-invasive HT strategy for mammalian cancer using microwave HT in a three-dimensional antenna-beam approach to describe electromagnetic wave penetration in breast cancer model. Nguyen et al. [14] experimentally showed that 65-W microwave HT penetration for breast cancer treatment by focusing 3D-antenna in the vicinity of tumor embedded in glands at elevated temperature 42℃.

During hyperthermia-type treatment in cancer issues, perfusion of blood across the vasculature of the diseased region is regarded because of its impact for circulation and removal of thermal energy in the vicinity of neighboring tissue. Numerical procedures have the capacity to predict and provide guidance to clinicians during administration of hyperthermia therapy in cancer patients.

In this study, investigation on the impact of magnetic field mediated radiative flux on infrared-type hyperthermia on a model spongy tissue under perfusion is presented and implemented by Chebyshev and pseudo-Spectral relaxation scheme. The elucidation of transport of biogenic fluid across the microvasculature of the perfused lesion is captured in our investigation. For the first time, we report field-based augmentation infrared-mediated hyperthermia with externally imposed magnetic field in infrared-mediated. While the perfusion in a near-native vascular spongy tissue is considered with predicated degree of porosity to analyzed the impact of controlled infrared over selected permeability of the substrate lesion.

The SRM is utilized to provide answer to the transformed non - dimensional Eqns. (9) and (10). This is an iterative technique for linearizing and decoupling sets of nonlinear differential equations by employing Gauss-Siedel relaxation procedure. The linear system of equations are then discretized using [16], a Chebyshev pseudo-spectral approach. The linear functions are calculated at the recent iteration step (r+1), but the non-linear functions are presumed to exist at the previous step (r).

The fact that this method (SRM) can solve both ordinary and partial differential equations makes it unique. Apply the SRM to the modified momentum and energy Eqns. (9) and (10) to get Eqns. (11) and (12) as shown below:

$\frac{\partial u_{r+1}}{\partial t}=\frac{\partial^2 u_{r+1}}{\partial y^2}-\frac{1}{K_p} u_{r+1}-m u_{r+1}$          (11)

$\frac{\partial \theta_{r+1}}{\partial t}=\alpha\left(1+\frac{4}{3} R\right) \frac{\partial^2 \theta_{r+1}}{\partial y^2}+(r+\beta)-(r-\lambda) \theta_{r+1}+E_c\left(\frac{\partial u}{\partial y}\right)^2$              (12)

Subject to:

$u_{r+1}(y, 0)=37^{\circ} \mathrm{C}, u_{r+1}(0, t)=37^{\circ} \mathrm{C}$               (13)

$\theta_{r+1}(y, 0)=37^{\circ} \mathrm{C}, \theta_{x+1}(0, t)=37^{\circ} \mathrm{C}$               (14)

Setting:

$a_{0, r}=\frac{1}{K_p}, a_{1, r}=-m, b_{0, r}=\alpha\left(1+\frac{4}{3} R\right), b_{1, r}=-(r-\lambda)$

$b_{2, r}=(\beta+r), b_{3, r}=E_c\left(\frac{\partial u}{\partial y}\right)^2$

Substituting the above coefficient parameters into (11) and (12) to obtain:

$\frac{\partial u_{r+1}}{\partial t}=\frac{\partial^2 u_{r+1}}{\partial y^2}+a_{0, r} u_{r+1}+a_{1, r} u_{r+1}$                (15)

$\frac{\partial \theta_{r+1}}{\partial t}=b_{0, r} \frac{\partial^2 \theta_{r+1}}{\partial y^2}+b_{1, r} \theta_{r+1}+b_{2, r}+b_{3, r}$               (16)

Subject to (13) and (14).

We proceed to define the Gauss-Lobatto points as given by Idowu and Falodun [18] as:

$\Sigma_j=\cos \frac{\pi j}{N}, j=0,1,2, \ldots, N ; 1 \leq \xi \leq-1$                    (17)

The realm of the physical neighborhood is converted into [-1, 1] from [0, 1], wherein N stand for number of collocation points.

In reference to the boundary conditions, an initial guess is chosen as:

$u_0(y, t)=\theta_0(y, t)=37^{\circ} \mathrm{C}+e^{-\eta}-1$                 (18)

As a result, starting with the initial approximation defined in Eq. (8), the dimensionless Eqns. (9) and (10) were solved iteratively for the unknown functions (18). Whenever r=0, 1, 2, the iterative schemes (15) and (16) are solved iteratively for u_r+1(y,t) and θ_r+1(y,t) respectively. The Chebyshev spectral collocation method will be used to discretize the equations in the y-path, as well as the implicit difference approach for centering mid-point within tⁿ⁺¹ and tⁿ.

The midpoint is conveyed as:

 $t^{n+\frac{1}{2}}=\frac{t^{n+1}+t^n}{2}$               (19)

Thus, using the centering point about $t^{n+\frac{1}{2}}$ to the unidentified functions, say u(y,t) and θ(y,t) with its associated derivative as noted by Alao et al. [16] and defined in this study as:

$u\left(y_j, t^{n+\frac{1}{2}}\right)=u_j^{n+\frac{1}{2}}=\frac{u_j^{n+1}}{2},\left(\frac{\partial u}{\partial t}\right)^{n+\frac{1}{2}}=\frac{u_j^{n+1}-u_j^n}{\Delta t}$                 (20)

$\theta\left(y_j, t^{n+\frac{1}{2}}\right)=\theta_j^{n+\frac{1}{2}}=\frac{\theta_j^{n+1}}{2},\left(\frac{\partial \theta}{\partial t}\right)^{n+\frac{1}{2}}=\frac{\theta_j^{n+1}-\theta_j^n}{\Delta t}$                     (21)

We proceed to apply the concept of spectral collocation method to estimate derivatives of unidentified variables given in (22) and (23) using differential matrix D:

$\frac{d^r u}{d v^r}=\sum_{K=0}^N D_{i K}^r u\left(\varepsilon_K\right)=D^r u, i=0,1, \ldots, N$                (22)

$\frac{d^r \theta}{d y^r}=\sum_{K=0}^N D_{i K}^r \theta\left(\varepsilon_K\right)=D^r \theta, i=0,1, \ldots, N$             (23)

Applying Eqns. (22) and (23) on (15) and (16) before applying the finite difference scheme to obtain:

$\frac{d u_{r+1}}{d t}=u_{r+1} D^2+a_{0, r} u_{r+1}+a_{1, r} u_{r+1}$                        (24)

$\frac{d \theta_{r+1}}{d t}=b_{0, r} D^2 \theta_{r+1}+b_{1, r} \theta_{r+1}+b_{2, r}+b_{3, r}$                       (25)

Subject to:

$u_{r+1}\left(x_0, t\right)=37^{\circ} \mathrm{C}, u_{r+1}\left(x N_r, t\right)=37^{\circ} \mathrm{C}$                    (26)

$\theta_{r+1}\left(x_0, t\right)=37^{\circ} \mathrm{C}, \theta_{r+1}\left(x N_x, t\right)=37^{\circ} \mathrm{C}$              (27)

Simplifying Eqns. (24) and (24) to obtain:

$\frac{d u_{r+1}}{d t}=\left(D^2+a_{0, r}+a_{1, r}\right) u_{r+1}$         (28)

$\frac{d \theta_{r+1}}{d t}=\left(b_{0, r} D^2+b_{1, r}\right) \theta_{r+1}+b_{2, r}+b_{3, r}$                 (29)

We proceed to apply the finite difference scheme to obtain:

$\left(\frac{u_{r+1}^{n+1}-u_{r+1}^n}{\Delta t}\right)=\left(D^2+a_{0, r}+a_{1, r}\right)\left(\frac{u_{r+1}^{n+1}+u_{r+1}^n}{2}\right)$                (30)

$\left(\frac{\theta_{r+1}^{n+1}-\theta_{r+1}^n}{\Delta t}\right)=\left(b_{o, r} D^2+b_{1, r}\right)\left(\frac{\theta_{r+1}^{n+1}+\theta_{r+1}^n}{2}\right)+b_{2, r}+b_{3, r}$               (31)

Simplifying further to obtain:

$\left(\frac{1}{\Delta t}\right) u_{r+1}^{n+1}-\left(\frac{1}{\Delta t}\right) u_{r+1}^n=\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right) u_{r+1}^{n+1}+\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right) u_{r+1}^n$               (32)

$\left(\frac{1}{\Delta t}\right) \theta_{r+1}^{n+1}-\left(\frac{1}{\Delta t}\right) \theta_{r+1}^n=\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right) \theta_{r+1}^{n+1}+\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right) \theta_{r+1}^n+b_{2, r}+b_{3, r}$            (33)

Upon further simplification,

$\left(\frac{1}{\Delta t}\right) u_{r+1}^{n+1}-\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right) u_{r+1}^{n+1}=\left(\frac{1}{\Delta t}\right) u_{r+1}^n+\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right) u_{r+1}^n$              (34)

$\left(\frac{1}{\Delta t}\right) \theta_{r+1}^{n+1}-\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right) \theta_{r+1}^{n+1}=\left(\frac{1}{\Delta t}\right) \theta_{r+1}^n+\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right) \theta_{r+1}^n+b_{2, r}+b_{3, r}$                 (35)

Simplifying further to obtain:

$\left[\left(\frac{1}{\Delta t}\right)-\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right)\right] u_{r+1}^{n+1}=\left[\left(\frac{1}{\Delta t}\right)+\right.\left.\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right)\right] u_{r+1}^n+K_1$               (36)

$\left[\left(\frac{1}{\Delta t}\right)-\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right)\right] \theta_{r+1}^{n+1}=\left[\left(\frac{1}{\Delta t}\right)+\right.\left.\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right)\right] \theta_{r+1}^n+K_2$                     (37)

Hence, the iterative scheme becomes:

$A_1 u_{r+1}^{n+1}=B_1 u_{r+1}^n+K_1$                 (38)

$A_2 \theta_{r+1}^{n+1}=B_2 \theta_{r+1}^n+K_2$           (39)

where,

$A_1=\left[\left(\frac{1}{\Delta t}\right)-\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right)\right], B_1=\left[\left(\frac{1}{\Delta t}\right)+\right.\left.\left(\frac{D^2+a_{o, r}+a_{1, r}}{2}\right)\right], K_1=0$

$A_2=\left[\left(\frac{1}{\Delta t}\right)-\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right)\right], B_2=\left[\left(\frac{1}{\Delta t}\right)+\right.\left.\left(\frac{b_{o, r} D^2+b_{1, r}}{2}\right)\right], K_2=b_{2, r}+b_{3, r}$

Investigation on the impact of radiative flux due to magnetic and infrared hyperthermia has been investigated under the influence of porosity term numerically. We considered the blood tissue at a temperature of 37℃ while heat source is studied to serve as an external device to intensify the temperature.

After critical examination of the problem, the following are the key findings:

(1) An incremental value of the heat source reduces the temperature;

(2) The blood viscosity reduces when there is high temperature, this signifies elevation in the temperature plot;

(3) A higher magnetic parameter was found to degenerate the velocity plot but has no significance on the temperature profile;

(4) The permeable tissue spurs the movement of the blood cell to accelerate;

(5) A higher thermal radiation was found to hike the fluid temperature profile but has no effect on the velocity profile.