Exact and solitary wave structure of the tumor cell proliferation with LQ model of three dimensional PDE by newly extended direct algebraic method

Ghazanfar, Sidra; Ahmed, Nauman; Ali, Syed Mansoor; Iqbal, Muhammad Sajid; Akgül, Ali; Shar, Muhammad Ali; Bariq, Abdul

doi:10.1063/5.0153529

An essential stage in the spread of cancer is the entry of malignant cells into the bloodstream. The fundamental mechanism of cancer cell intravasation is still completely unclear, despite substantial advancements in observing tumor cell mobility in vivo. By creating therapeutic methods in conjunction with control engineering or by using the models for simulations and treatment process evaluation, tumor growth models have established themselves as a crucial instrument for producing an engineering backdrop for cancer therapy. Because tumor growth is a highly complex process, mathematical modeling has been essential for describing it because a carefully crafted tumor growth model constantly describes the measurements and the physiological processes of the tumors. This article discusses the exact and solitary wave behavior of a tumor cell with a three-dimensional linear-quadratic model. Exact solutions have been discussed in detail using the newly extended direct algebraic method, which presents a variety of answers to this issue based on the conditions applied. This article also illustrates its graphical behavior with surface and contour plots of several solitons.

I. INTRODUCTION

Nonlinear partial differential equations (NPDEs) are one of the key instruments for thoroughly analyzing the characteristics of nonlinear physical phenomena. In the fundamental sciences, they are being modeled in biology, physics, chemistry, etc.; in technology,¹ they can be seen in image creation, control of non-linear systems, material sciences, etc.; in the theoretical sciences,² they are playing a crucial role in many applied areas like meteorology, oceanography, and the aerospace industry, among others; and in social sciences,³ finance, economics, and many other fields. It is important to note that different classes of solutions may exist for the same NPDE, depending on the settings of the equation’s parameters.

Many analytical and numerical techniques, including the Bernoulli functional methodology,⁴ the F-expansion technique,⁵ the auxiliary equation technique,⁶ the simplest extended equation technique,⁷ the (G′/G)-expansion technique,⁸ the sub-ODE technique,⁹ the generalized Kudryashov technique (GKM),¹⁰ the Ricatti Bernoulli sub ODE method,¹¹ the newly extended direct algebraic method,¹² and many others, have been proposed in recent years to obtain solutions for NPDEs.^13–15 Further relevant literature that can also be studied here is.^16,17

The newly extended direct algebraic method (NEDAM) was chosen because it approaches the NPDE’s most consistent solutions. The suggested approach enables us to leverage the tedious algebraic calculations that are currently used to introduce solitary wave solutions.¹⁸ Our approach simply transforms the provided NPDEs into straightforward algebraic equations once we apply them. It is a very potent and successful strategy. This method is being used to replicate the precise resolution of NPDE, which occurs in many radio-biological situations.¹⁹

It is a common practice to use mathematical modeling of medical processes to establish quantitative discernment of bio-medical events. This quantitative strategy can be advantageous for both clinical and experimental research, and the field of cancer biology uses this mathematical modeling most frequently.²⁰ To simulate particular aspects of cancer, such as tumor growth, many models have been used for this purpose.²¹ Tumors affect human cells, and the body has a large number of cells that grow and divide, as well as die and replace themselves in a controlled way. Some genes in this system are in charge of regulating. However, if this process is left unchecked, cells can grow much more quickly and can go in any direction, leading to the development of swelling or inflammation, which is known as a tumor. Consequently, a tumor is an abnormal mass of tissues that can either be solid or liquid-filled. Any tumor’s carcinogenic activity is unpredictable; it could be benign (not malignant), precancerous (pre-malignant), or fully malignant (progressive and uncontrollable). A premalignant tumor²² is one that is not yet malignant but is likely to become so. A benign tumor²³ cannot metastasize (cannot spread, remains dormant). The cancerous malignant²⁴ ones can cause mortality because of their advancing nature. They spread rapidly, are progressive, and are unchecked as a result of their rapid growth.

The classification of genes that promote enhanced proliferation and survival and the quantification of physical stresses like the pressure that spatially regulates the direction and scale of tumor growth are just a few examples of experimental studies that span all physiological scales in the quest to understand the underlying mechanisms of tumor proliferation. Although a great deal of knowledge has been accumulated regarding tumor genesis, development, progression, and response to therapy, reliable methods to forecast tumor growth and response to certain therapeutic regimens for a given patient are still lacking. Researchers have created a multitude of mathematical models and methodologies to forecast the growth of cancer and how it will respond to treatment, most of which are largely independent of breakthroughs in cancer biology.

By investigating dosing regimens with cytotoxicity models that describe the effect on proliferation, cell signaling models that identify cellular transition rates for drug targeting, and tissue scale models that predict tumor response to therapy using patient-specific imaging data, these models have the potential to be used to optimize therapy.

Combination therapy with carboplatin and small molecule inhibitors that target an anti-apoptotic protein has been developed using a mathematical model of drug-resistant ovarian tumor growth to improve medication dose. A recent model that describes doxorubicin treatment for triple-negative breast cancer and doxorubicin treatment offers a framework for studies that quantitatively examine treatment response in vitro as well as a scalable method for forecasting tumor response in vivo. Therefore, there is some support in the literature for mathematical modeling of tumor cell responses at the population level in both in vitro and in vivo situations.

The development of various empirical models to represent radiation-induced cell death is similar to how chemotherapeutic models have evolved. The below-mentioned “linear quadratic” (LQ) model represents the survival fraction of cells as a function of radiation dose and radio sensitivity,²⁵

S (a, b, D o s e) = \exp^{(- a D o s e - b D o s e^{2})},

(1)

where S is the surviving proportion, the dosage is the given radiation therapy dose in Gray (Gy), and a and b are radiosensitive parameters. In contrast to dose, which is reported in units of Gy, a and b have units of Gy⁻¹ and Gy⁻², respectively. The linear-quadratic (LQ) model determines a factor that lowers the tumor volume. This model is also crucial to a three-dimensional partial differential equation (PDE) description of glioma cell proliferation and radiation therapy response that is calibrated by patient-specific magnetic resonance imaging (MRI) data, as illustrated below:

\begin{matrix} \frac{\partial N (x, t)}{\partial t} & = & \overset{diffusion}{\overset{︷}{\nabla \cdot (D \nabla N (x, t))}} + \overset{proliferation}{\overset{︷}{r (1 - \frac{N (x, t)}{N_{\max}}) \cdot N (x, t)}} \\ - \overset{death}{\overset{︷}{R \cdot N (x, t) \cdot (1 - \frac{N (x, t)}{N_{\max}})}}, \end{matrix}

(2)

where N(x, t) denotes the number of tumor cells present at position x and time t, and R(x, t, Dose(x, t)) denotes the cell death term as follows:

R = \{\begin{cases} 0 f o r t \notin therapy, \\ 1 - S (a, b, D o s e) f o r t \in therapy . \end{cases}

(3)

II. PROBLEM STATEMENT

Considering the second order LQ-model of tumor cells described earlier, written in the simplest form as

N_{t} = D N_{x x} + (r - R) [N - \frac{N^{2}}{N_{\max}}] .

(4)

Here, N is the tumor cell population growth, D is the diffusion coefficient of tumor cells, and R is the cell death term. As an alternative, the population can be depicted using logistic growth, which caps population growth depending on the ratio of carrying capacity to population density, N_max. Analytical resolution of the aforementioned nonlinear problem is the primary goal of this paper. This will help us arrive at the exact solutions to the problems presented by the tumor cell model.

III. STEPS TO USE NEWLY EXTENDED DIRECT ALGEBRAIC METHOD

This thorough and straightforward method is used by many experts to discover solitons and other wave solutions to the given challenge. A class of NPDEs can receive precise answers using this method. The given system of equations can be included by following the procedures below.

Step I: Considering the NPDE as follows:
$Ω (N, N_{t}, N_{x}, N_{x x}, N_{t t}, N_{x t}, N_{xxx} \dots) = 0,$
(5)
where N = N(x, t) is the unknown function, and Ω is a polynomial of N(x, t) and its derivatives with respect to x and t.
Step II: Applying the traveling wave transformation
$N (x, t) = Ω (η), η = x + c t .$
(6)
The aforementioned form is obtained by adding the real variables x, t, and a compound variable η. Here, the wave speed is taken to be c. Our NPDE (5) can be transformed into the following ODE using the aforementioned transformation (ordinary differential equation):
$Θ (Ω, Ω^{'}, Ω^{″}, Ω^{‴}, \dots) = 0,$
(7)
where Θ is the polynomial of Ω and its derivatives, and the superscripts identify regular derivatives with respect to η.
Step III: Assume that the initial solution of Eq. (7) can be stated as a polynomial Ψ(η) in the following form:
$N (η) = Σ_{i = 0}^{P} ϒ_{i} Ψ^{i} (η), ϒ_{P} \neq 0,$
(8)
where ϒ_i(0 ≤ i ≤ P) are constants concluded and Ψ(η) satisfies the following ODE:
$Ψ^{'} (η) = \ln (a) (ρ + ϱ Ψ (η) + σ Ψ^{2} (η)), a \neq 0, 1 .$
(9)
We currently have the following options for resolving the stated issue:
1. For ϱ² − 4ρσ < 0 and σ ≠ 0, there are five solutions
$\begin{gathered} Ψ_{1} (η) = - \frac{ϱ}{2 σ} + \frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{2 σ} \tan_{a} (\frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{2} η), \\ Ψ_{2} (η) = - \frac{ϱ}{2 σ} - \frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{2 σ} \cot_{a} (\frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{2} η), \\ Ψ_{3} (η) = - \frac{ϱ}{2 σ} + \frac{\sqrt{- (σ^{2} - 4 ρ σ)}}{2 σ} (\tan_{a} (\sqrt{- (ϱ^{2} - 4 ρ σ)} η) \pm \sqrt{p q} \sec_{a} (\sqrt{- (ϱ^{2} - 4 ρ σ)} η)), \\ Ψ_{4} (η) = - \frac{ϱ}{2 σ} + \frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{2 σ} (- \cot_{a} (\sqrt{- (ϱ^{2} - 4 ρ σ)} η) \pm \sqrt{p q} \csc_{a} (\sqrt{- (ϱ^{2} - 4 ρ σ)} η)), \\ Ψ_{5} (η) = - \frac{ϱ}{2 σ} + \frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{4 σ} (\tan_{a} (\frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{4} η) - \cot_{a} (\frac{\sqrt{- (ϱ^{2} - 4 ρ σ)}}{4} η)) . \end{gathered}$
(10)
1. For ϱ² − 4ρσ > 0 and σ ≠ 0, there are five solutions
$\begin{gathered} Ψ_{6} (η) = - \frac{ϱ}{2 σ} - \frac{\sqrt{ϱ^{2} - 4 ρ σ}}{2 σ} \tanh_{a} (\frac{\sqrt{ϱ^{2} - 4 ρ σ}}{2} η), \\ Ψ_{7} (η) = - \frac{ϱ}{2 σ} - \frac{\sqrt{ϱ^{2} - 4 ρ σ}}{2 σ} \coth_{a} (\frac{\sqrt{ϱ^{2} - 4 ρ σ}}{2} η), \\ Ψ_{8} (η) = - \frac{ϱ}{2 σ} + \frac{\sqrt{ϱ^{2} - 4 ρ σ}}{2 σ} (- \tanh_{a} (\sqrt{ϱ^{2} - 4 ρ σ} η) \pm ι \sqrt{p q} {sech}_{a} (\sqrt{ϱ^{2} - 4 ρ σ} η)), \\ Ψ_{9} (η) = - \frac{ϱ}{2 σ} + \frac{\sqrt{ϱ^{2} - 4 ρ σ}}{2 σ} (- \coth_{a} (\sqrt{ϱ^{2} - 4 ρ σ} η) \pm \sqrt{p q} {csch}_{a} (\sqrt{ϱ^{2} - 4 ρ σ} η)), \\ Ψ_{10} (η) = - \frac{ϱ}{2 σ} - \frac{\sqrt{ϱ^{2} - 4 ρ σ}}{4 σ} (\tanh_{a} (\frac{\sqrt{ϱ^{2} - 4 ρ σ}}{4} η) + \coth_{a} (\frac{\sqrt{ϱ^{2} - 4 ρ σ}}{4} η)) . \end{gathered}$
(11)
3. For ρσ > 0 and ϱ = 0, there are five solutions
$\begin{gathered} Ψ_{11} (η) = \sqrt{\frac{ρ}{σ}} \tan_{a} (\sqrt{ρ σ} η), \\ Ψ_{12} (η) = - \sqrt{\frac{ρ}{σ}} \cot_{a} (\sqrt{ρ σ} η), \\ Ψ_{13} (η) = \sqrt{\frac{ρ}{σ}} (\tan_{a} (2 \sqrt{ρ σ} η) \pm \sqrt{p q} \sec_{a} (2 \sqrt{ρ σ}) η), \\ Ψ_{14} (η) = \sqrt{\frac{ρ}{σ}} (- \cot_{a} (2 \sqrt{ρ σ} η) \pm \sqrt{p q} \csc_{a} (2 \sqrt{ρ σ}) η), \\ Ψ_{15} (η) = \frac{1}{2} \sqrt{\frac{ρ}{σ}} (\tan_{a} (\frac{\sqrt{ρ σ}}{2} η) - \cot_{a} (\frac{\sqrt{ρ σ}}{2}) η) . \end{gathered}$
(12)
4. For ρσ < 0 and ϱ = 0, there are five solutions
$\begin{gathered} Ψ_{16} (η) = - \sqrt{- \frac{ρ}{σ}} \tanh_{a} (\sqrt{- ρ σ} η), \\ Ψ_{17} (η) = - \sqrt{- \frac{ρ}{σ}} \coth_{a} (\sqrt{- ρ σ} η), \\ Ψ_{18} (η) = \sqrt{- \frac{ρ}{σ}} (- \tanh_{a} (2 \sqrt{- ρ σ} η) \pm ι \sqrt{p q} {sech}_{a} (2 \sqrt{- ρ σ}) η), \\ Ψ_{19} (η) = \sqrt{- \frac{ρ}{σ}} (- \coth_{a} (2 \sqrt{- ρ σ} η) \pm \sqrt{p q} {csch}_{a} (2 \sqrt{- ρ σ}) η), \\ Ψ_{20} (η) = - \frac{1}{2} \sqrt{\frac{- ρ}{σ}} (\tanh_{a} (\frac{\sqrt{- ρ σ}}{2} η) + \coth_{a} (\frac{\sqrt{- ρ σ}}{2}) η) . \end{gathered}$
(13)
5. For σ = ρ and ϱ = 0, there are five solutions
$\begin{gathered} Ψ_{21} (η) = \tan_{a} (ρ η), \\ Ψ_{22} (η) = - \cot_{a} (ρ η), \\ Ψ_{23} (η) = \tan_{a} (2 ρ η) \pm \sqrt{p q} \sec_{a} (2 ρ η), \\ Ψ_{24} (η) = - \cot_{a} (2 ρ η) \pm \sqrt{p q} \csc_{a} (2 ρ η), \\ Ψ_{25} (η) = \frac{1}{2} (\tan_{a} (\frac{ρ}{2} η) - \cot_{a} (\frac{ρ}{2} η)) . \end{gathered}$
(14)
6. For σ = −ρ and ϱ = 0, there are five solutions
$\begin{gathered} Ψ_{26} (η) = - \tanh_{a} (ρ η), \\ Ψ_{27} (η) = - \coth_{a} (ρ η), \\ Ψ_{28} (η) = - \tanh_{a} (2 ρ η) \pm ι \sqrt{p q} {sech}_{a} (2 ρ η), \\ Ψ_{29} (η) = - \coth_{a} (2 ρ η) \pm \sqrt{p q} {csch}_{a} (2 ρ η), \\ Ψ_{30} (η) = - \frac{1}{2} (\tanh_{a} (\frac{ρ}{2} η) + \coth_{a} (\frac{ρ}{2} η)) . \end{gathered}$
(15)
7. For ϱ² = 4ρσ, there is only one solution
$Ψ_{31} (η) = \frac{- 2 ρ (ϱ η \ln a + 2)}{ϱ^{2} η \ln a} .$
(16)
8. For ϱ = χ, ρ = κχ(κ ≠ 0) and σ = 0, there is only one solution
$Ψ_{32} (η) = a^{χ η} - κ .$
(17)
9. For ϱ = σ = 0, there is only one solution
$Ψ_{33} (η) = ρ η \ln a .$
(18)
10. For ρ = ϱ = 0, there is only one solution
$Ψ_{34} (η) = \frac{- 1}{σ η \ln a} .$
(19)
11. For ρ = 0 and ϱ ≠ 0. there are two solutions
$\begin{gathered} Ψ_{35} (η) = \frac{- p ϱ}{σ (\cosh_{a} (σ η) - \sinh_{a} (σ η) + p)}, \\ Ψ_{36} (η) = \frac{- (ϱ (\sinh_{a} (σ η) + \cosh_{a} (σ η)))}{σ (\sinh_{a} (σ η) + \cosh_{a} (σ η) + q)} . \end{gathered}$
(20)
12. For ϱ = χ, σ = κχ(κ ≠ 0) and ρ = 0, there is only one solution
$Ψ_{37} (η) = \frac{p a^{χ η}}{p - κ q a^{χ η}} .$
(21)
The generalized hyperbolic and trigonometric functions are defined as follows for the aforementioned solutions:
$\begin{gathered} \sin_{a} (η) = \frac{p a^{ι η} - q a^{- ι η}}{2}, \cos_{a} (η) = \frac{p a^{ι η} + q a^{- ι η}}{2}, \\ \tan_{a} (η) = - ι \frac{p a^{ι η} - q a^{- ι η}}{p a^{ι η} + q a^{- ι η}}, \cot_{a} (η) = ι \frac{p a^{ι η} + q a^{- ι η}}{p a^{ι η} - q a^{- ι η}}, \\ \sec_{a} (η) = \frac{2}{p a^{ι η} + q a^{- ι η}}, \csc_{a} (η) = \frac{2 ι}{p a^{ι η} - q a^{- ι η}}, \\ \sinh_{a} (η) = \frac{p a^{η} - q a^{- η}}{2}, \cosh_{a} (η) = \frac{p a^{η} + q a^{- η}}{2}, \\ \tanh_{a} (η) = \frac{p a^{η} - q a^{- η}}{p a^{η} + q a^{- η}}, \coth_{a} (η) = \frac{p a^{η} + q a^{η}}{p a^{η} - q a^{- η}}, \\ {sech}_{a} (η) = \frac{2}{p a^{η} + q a^{- η}}, {csch}_{a} (η) = \frac{2}{p a^{η} - q a^{- η}}, \end{gathered}$
where η is taken as an independent variable, p, q > 0, and are considered to be arbitrary constants.
Step IV: Now that we have the highest order derivative with the highest order nonlinear term, we can utilize the homogeneous balancing principle in Eq. (7) to determine the value of the positive integer P in Eq. (8).
Step V: The set of algebraic equations is produced by replacing Eq. (8) and its necessary derivatives in Eq. (7) and by equating the coefficients of a polynomial of the same power to zero. These equations can be analytically solved to obtain the necessary unknowns that were previously described in order to produce the final result of the solitary wave solution.

IV. APPLICATION OF THE PROPOSED METHOD ON LQ MODEL OF TUMOR CELL

Using the traveling wave transformation N(x, t) = N(η), η = x + ct, which can be reduced to Eq. to the following ODE:

D N^{″} - N^{'} + (r - R) (N - \frac{N^{2}}{N_{\max}}) = 0 .

(22)

Now, balancing the terms N″ and N² to get the value of P with the help of the homogeneous balance principle, we found this value to be P = 2. Using Eq. , the solution of Eq. takes the form as

N = a_{0} + a_{1} Ψ (η) + a_{2} {(Ψ (η))}^{2},

(23)

where a₀, a₁, and a₂ are the constants to be determined and Ψ(η) satisfies the following ODE:

Ψ^{'} (η) = \ln (a) (ρ + ϱ Ψ (η) + σ Ψ {(η)}^{2}), a \neq 0, 1 .

(24)

Using Eqs. and in Eq. and equating each of its coefficients equal to zero, we get the system of equations as

\begin{matrix} N^{4} : & 6 D {(\ln (a))}^{2} N_{\max} σ^{2} a_{2} + R {a_{2}}^{2} - r {a_{2}}^{2} = 0, \\ N^{3} : & 2 D {(\ln (a))}^{2} N_{\max} σ^{2} a_{1} - 2 \ln (a) N_{\max} σ a_{2} - 2 r a_{1} a_{2} + 2 R a_{1} a_{2} \\ + 10 D {(\ln (a))}^{2} N_{\max} ϱ σ a_{2} = 0, \\ N^{2} : & 3 D {(\ln (a))}^{2} N_{\max} ϱ σ a_{1} + 8 D {(\ln (a))}^{2} N_{\max} σ ρ a_{2} - \ln (a) N_{\max} σ a_{1} - r {a_{1}}^{2} \\ - 2 r a_{0} a_{2} + r a_{2} N_{\max} - 2 \ln (a) N_{\max} ϱ a_{2} + 4 D {(\ln (a))}^{2} N_{\max} ϱ^{2} a_{2} \\ + R {a_{1}}^{2} - R a_{2} N_{\max} + 2 R a_{0} a_{2} = 0, \\ N : & - 2 \ln (a) N_{\max} ρ a_{2} + 2 R a_{0} a_{1} + 6 D {(\ln (a))}^{2} N_{\max} ϱ ρ a_{2} - R a_{1} N_{\max} + r a_{1} N_{\max} \\ - \ln (a) N_{\max} ϱ a_{1} + 2 D {(\ln (a))}^{2} N_{\max} σ ρ a_{1} + D {(\ln (a))}^{2} N_{\max} ϱ^{2} a_{1} - 2 r a_{0} a_{1} = 0, \\ 1 : & - r {a_{0}}^{2} + R {a_{0}}^{2} + r a_{0} N_{\max} - R a_{0} N_{\max} - \ln (a) N_{\max} ρ a_{1} + D {(\ln (a))}^{2} N_{\max} ϱ ρ a_{1} \\ + 2 D {(\ln (a))}^{2} N_{\max} a_{2} ρ^{2} = 0 . \end{matrix}

By dissecting the aforementioned set of equations, we discovered the results listed below:

\begin{gathered} a_{0} = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D^{2} {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 D (R - r)}, \\ a_{1} = \frac{- 6 \ln (a) N_{\max} σ (- 1 + 5 D \ln (b) ϱ)}{5 (R - r)}, \\ a_{2} = \frac{- 6 D {(\ln (a))}^{2} N_{\max} σ^{2}}{R - r} . \end{gathered}

(25)

Substituting the values in Eq. (23), we get the summarized results as follows:

For ϱ² − 4ρσ < 0 and σ ≠ 0, the five solutions below were created using mixed-trigonometric and trigonometric formulas (ϱ² − 4ρσ = E):
$\begin{gathered} N_{1} (x, t) = - N_{\max} (- 25 D R - 1 + 25 r D + 25 D {(\ln (a))}^{2} ϱ^{2} - 30 \ln (a) D \sqrt{- E} \tan (\frac{\sqrt{- E} η}{2})) \\ - ({(\ln (a))}^{2} D^{2} (75 ϱ^{2} + 200 ρ σ + (75 ϱ^{2} - 300 ρ σ) {(\tan (\frac{\sqrt{- E} η}{2}))}^{2})) / ((R - r) 50 D), \\ N_{2} (x, t) = - N_{\max} (- 25 D R - 1 + 25 r D + 25 D {(\ln (a))}^{2} ϱ^{2} + 30 \ln (a) D \sqrt{- E} \cot (\frac{\sqrt{- E} η}{2})) \\ - ({(\ln (a))}^{2} D^{2} (75 ϱ^{2} + 200 ρ σ + (75 ϱ^{2} - 300 ρ σ) {(\cot (\frac{\sqrt{- E} η}{2}))}^{2})) / (R - r) 50 D, \\ N_{3} (x, t) = \frac{- (- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 D (R - r)} \\ - \frac{3 \ln (a) N_{\max} σ (- 1 + 5 D \ln (a) ϱ) (- ϱ + \sqrt{- E} (\tan (\sqrt{- E} η) \pm \sqrt{p q}) \sec (\sqrt{- E} η))}{5 σ (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- ϱ + \sqrt{- ϱ^{2} + 4 ρ σ} (\tan (\sqrt{- ϱ^{2} + 4 ρ σ} η)) \pm \sqrt{p q} \sec (\sqrt{- E} η))}^{2}}{2 (R - r)}, \end{gathered}$
(26)
$\begin{gathered} N_{4} (x, t) = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D \ln {(a)}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} \\ - \frac{3 \ln (a) N_{\max} (- 1 + 5 D \ln (a) ϱ) (- ϱ - \sqrt{- E} (\cot (\sqrt{- E} η)) \pm \sqrt{p q} \csc (\sqrt{- E} η))}{5 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- ϱ - \sqrt{- E} (\cot (\sqrt{- E} η)) \pm \sqrt{p q} \csc (\sqrt{- E} η))}^{2}}{2 (R - r)}, \\ N_{5} (x, t) = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 D (R - r)} \\ - \frac{3 \ln (a) N_{\max} σ (- 1 + 5 D \ln (a) ϱ) (- 2 ϱ + \sqrt{- E} (\tan (1 / 4 \sqrt{E} η) - \cot (\frac{\sqrt{- E} η}{4})))}{10 σ (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- 2 ϱ + \sqrt{- E} (\tan (\frac{\sqrt{- E} η}{4}) - \cot (\frac{\sqrt{- E} η}{4})))}^{2}}{8 (R - r)} . \end{gathered}$
(27)
For ϱ² − 4ρσ > 0 and σ ≠ 0 (considering ρ to be negative), there are five exact solutions derived below:
$\begin{align} N_{6} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D \ln {(a)}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} \\ - \frac{3 \ln (a) N_{\max} (- 1 + 5 D \ln (a) ϱ) (- ϱ - \sqrt{E} \tanh (\frac{\sqrt{E} η}{2}))}{5 (R - r)} - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- ϱ - \sqrt{E} \tanh (\frac{\sqrt{E} η}{2}))}^{2}}{2 (R - r)}, \\ N_{7} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} \\ - \frac{3 \ln (a) N_{\max} (- 1 + 5 D \ln (a) ϱ) (- ϱ - \sqrt{E} \coth (\frac{\sqrt{E} η}{2}))}{5 (R - r)} - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- ϱ - \sqrt{ϱ^{2} - 4 ρ σ} \coth (\frac{\sqrt{E} η}{2}))}^{2}}{2 (R - r)}, \end{align}$
$\begin{align} N_{8} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} \\ - \frac{3 \ln (a) N_{\max} (- 1 + 5 D \ln (a) ϱ) (- ϱ - \sqrt{E} (\tanh (\sqrt{E} η)) \pm ι \sqrt{p q} sech (\sqrt{E} η))}{5 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- ϱ - \sqrt{E} (\tanh (\sqrt{E} η)) \pm ι \sqrt{p q} sech (\sqrt{E} η))}^{2}}{2 (R - r)} . \\ N_{9} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} \\ - \frac{3 \ln (a) N_{\max} (- 1 + 5 D \ln (a) ϱ) (- ϱ - \sqrt{E} (\coth (\sqrt{E} η)) \pm \sqrt{p q} csch (\sqrt{E} η))}{5 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- ϱ - \sqrt{E} (\coth (\sqrt{E} η)) \pm \sqrt{p q} csch (\sqrt{E} η))}^{2}}{2 (R - r)}, \end{align}$
(28)
$\begin{matrix} N_{10} (x, t) & = & - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2} + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} \\ - \frac{3 \ln (a) N_{\max} (- 1 + 5 D \ln (a) ϱ) (- 2 ϱ - \sqrt{E} (\tanh (\frac{\sqrt{E} η}{4}) + \coth (\frac{\sqrt{E} η}{4})))}{10 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} {(- 2 ϱ - \sqrt{E} (\tanh (\frac{\sqrt{E} η}{4}) + \coth (\frac{\sqrt{E} η}{4})))}^{2}}{8 (R - r)} . \end{matrix}$
(29)
For ρσ > 0 and ϱ = 0, there are five trigonometric and mixed-trigonometric solutions stated below:
$\begin{align} N_{11} (x, t) & = - N_{m a x} (- 25 D R - 1 + 25 r D - 60 \ln (a) σ \sqrt{\frac{ρ}{σ}} \tan (\sqrt{ρ σ} η) D + D^{2} {(\ln (a))}^{2} σ ρ (300 {(\tan (\sqrt{ρ σ} η))}^{2} + 200)) / (50 (R - r) D), \\ N_{12} (x, t) & = - N_{m a x} (- 25 D R - 1 + 25 r D - 60 \ln (a) σ \sqrt{\frac{ρ}{σ}} \cot (\sqrt{ρ σ} η) D + D^{2} {(\ln (a))}^{2} σ ρ (300 {(\cot (\sqrt{ρ σ} η))}^{2} + 200)) / ((R - r) 50 D), \\ N_{13} (x, t) & = - \frac{(- 25 (D) R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{(R - r) 50 D} + \frac{6 \ln (a) N_{\max} σ \sqrt{\frac{ρ}{σ}} (\tan (2 \sqrt{ρ σ}) η \pm \sqrt{p q} \sec (2 \sqrt{ρ σ}) η)}{5 (R - r)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} σ ρ {(\tan (2 \sqrt{ρ σ}) η \pm \sqrt{p q} \sec (2 \sqrt{ρ σ}) η)}^{2}}{R - r}, \\ N_{14} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{(R - r) 50 D} + \frac{6 \ln (a) N_{\max} σ \sqrt{\frac{ρ}{σ}} (- \cot (2 \sqrt{ρ σ}) η \pm \sqrt{p q} \csc (2 \sqrt{ρ σ}) η)}{5 (R - r)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} σ ρ {(- \cot (2 \sqrt{ρ σ}) η \pm \sqrt{p q} \csc (2 \sqrt{ρ σ}) η)}^{2}}{R - r}, \\ N_{15} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{(R - r) 50 D} + \frac{3 \ln (a) N_{\max} σ \sqrt{\frac{ρ}{σ}} (\tan (\frac{\sqrt{ρ σ}}{2}) η - \cot (\frac{\sqrt{ρ σ}}{2}) η)}{5 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} σ ρ {(\tan (\frac{\sqrt{ρ σ}}{2}) η - \cot (\frac{\sqrt{ρ σ}}{2}) η)}^{2}}{2 (R - r)} . \end{align}$
(30)
For ρσ < 0 and ϱ = 0 (assuming the value of ρ is negative), there are five solutions constructed below:
$\begin{align} N_{16} (x, t) & = N_{\max} (25 D R + 1 - 25 r D - 60 \ln (a) σ \sqrt{- \frac{ρ}{σ}} \tanh (\sqrt{- ρ σ}) η D \\ + D^{2} {(\ln (a))}^{2} (300 ρ σ {(\tanh (\sqrt{- ρ σ}))}^{2} η^{2} - 200 ρ σ)) / ((R - r) 50 D), \\ N_{17} (x, t) & = N_{\max} (25 D R + 1 - 25 r D - 60 \ln (a) σ \sqrt{- \frac{ρ}{σ}} \coth (\sqrt{- ρ σ}) η D \\ + D^{2} {(\ln (a))}^{2} (300 ρ σ {(\coth (\sqrt{- ρ σ}))}^{2} η^{2} - 200 ρ σ)) / (50 (R - r) 50 D), \end{align}$
(31)
$\begin{align} N_{18} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} - \frac{6 \ln (a) N_{\max} σ \sqrt{- \frac{ρ}{σ}} (\tanh (2 \sqrt{- ρ σ}) η \pm ι \sqrt{p q} sech (2 \sqrt{- ρ σ}) η)}{5 (R - r)} \\ + \frac{6 D {(\ln (a))}^{2} N_{\max} σ ρ {(\tanh (2 \sqrt{- ρ σ}) η \pm ι \sqrt{p q} (sech (2 \sqrt{- ρ σ})) η)}^{2}}{R - r}, \\ N_{19} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{(R - r) 50 D} - \frac{6 \ln (a) N_{\max} σ \sqrt{- \frac{ρ}{σ}} \coth (2 \sqrt{- ρ σ}) η \pm \sqrt{p q} csch (2 \sqrt{- ρ σ}) η}{5 (R - r)} \\ + \frac{6 D (\ln (a)) N_{\max} σ ρ {(\coth (2 \sqrt{- ρ σ}) η \pm \sqrt{p q} {(csch (2 \sqrt{- ρ σ}))}^{2} η)}^{2}}{R - r}, \\ N_{20} (x, t) & = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max}}{50 (R - r) D} - \frac{3 \ln (a) N_{\max} σ \sqrt{- \frac{ρ}{σ}} (\tanh (\frac{\sqrt{- ρ σ}}{2}) η + \coth (\frac{\sqrt{- ρ σ}}{2}) η)}{5 (R - r)} \\ + \frac{3 D {(\ln (a))}^{2} N_{\max} σ ρ {(\tanh (\frac{\sqrt{- ρ σ}}{2}) η + \coth (\frac{\sqrt{- ρ σ}}{2}) η)}^{2}}{2 (R - r)} . \end{align}$
(32)
For σ = ρ and ϱ = 0, there are five solutions
$\begin{gathered} N_{21} (x, t) = - N_{\max} (- 25 D R - 1 + 25 r D - 60 \ln (a) ρ \tan (η ρ) D + D^{2} {(\ln (a) (300 ρ^{2} {(\tan (η ρ))}^{2} + 200 ρ^{2}))}^{2}) / (R - r) 50 D, \\ N_{22} (x, t) = - N_{\max} (- 25 D R - 1 + 25 r D - 60 \ln (a) ρ \cot (η ρ) D + D^{2} {(\ln (a) (300 ρ^{2} {(\cot (η ρ))}^{2} + 200 ρ^{2}))}^{2}) / (R - r) 50 D, \\ N_{23} (x, t) = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} ρ^{2}) N_{\max}}{(R - r) 50 D} + \frac{6 \ln (a) N_{\max} ρ \tan (2 η ρ) \pm \sqrt{p q} \sec (2 η ρ)}{5 (R - r)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} ρ^{2} {(\tan (2 η ρ))}^{2} \pm p q {(\sec (2 η ρ))}^{2}}{R - r}, \\ N_{24} (x, t) = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} ρ^{2}) N_{\max}}{(R - r) 50 D} + \frac{6 \ln (a) N_{\max} ρ (\cot (2 η ρ)) \pm \sqrt{p q} \csc (2 η ρ)}{5 (R - r)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} ρ^{2} {((\cot (2 η ρ)))}^{2} \pm p q {(\csc (2 η ρ))}^{2}}{R - r}, \\ N_{25} (x, t) = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} ρ^{2}) N_{\max}}{(R - r) 50 D} + \frac{3 \ln (a) N_{\max} ρ (\tan (\frac{η ρ}{2}) - \cot (\frac{η ρ}{2}))}{5 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} ρ^{2} {(\tan (\frac{η ρ}{2}) - \cot (\frac{η ρ}{2}))}^{2}}{2 (R - r)} . \end{gathered}$
(33)

For σ = −ρ and ϱ = 0, five solutions can be extracted below:
$\begin{gathered} N_{26} (x, t) = - N_{\max} (- 25 D R - 1 + 25 r D + 60 \ln (a) ρ \tanh (ρ η) D + D^{2} {(\ln (a))}^{2} (300 ρ^{2} {(\tanh (ρ η))}^{2} + 200 ρ^{2})) / (R - r) 50 D, \\ N_{27} (x, t) = - N_{\max} (- 25 D R - 1 + 25 r D + 60 \ln (a) ρ \coth (ρ η) D + D^{2} {(\ln (a))}^{2} (300 ρ^{2} {(\coth (ρ η))}^{2} + 200 ρ^{2})) / 50 (R - r) D, \\ N_{28} (x, t) = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} ρ^{2}) N_{\max}}{(R - r) 50 D} - \frac{6 \ln (a) N_{\max} ρ (\tanh (2 ρ η) \pm ι) \sqrt{p q} sech (2 ρ η)}{5 (R - r)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} ρ^{2} \tanh (2 ρ η) \pm ι^{2} p q {(sech (2 ρ η))}^{2}}{R - r}, \\ N_{29} (x, t) = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} ρ^{2}) N_{\max}}{(R - r) 50 D} - \frac{6 \ln (a) N_{\max} ρ (\coth (2 ρ η) \pm \sqrt{p q} csch (2 ρ η))}{5 (R - r)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} ρ^{2} \coth (2 ρ η) \pm p q {(csch (2 ρ η))}^{2}}{R - r}, \\ N_{30} (x, t) = - \frac{(- 25 D R - 1 + 25 r D + 200 D^{2} {(\ln (a))}^{2} ρ^{2}) N_{\max}}{(R - r) 50 D} - \frac{3 \ln (a) N_{\max} ρ (\tanh (\frac{ρ η}{2}) + \coth (\frac{ρ η}{2}))}{5 (R - r)} \\ - \frac{3 D {(\ln (a))}^{2} N_{\max} ρ^{2} {(\tanh (\frac{ρ η}{2}) + \coth (\frac{ρ η}{2}))}^{2}}{2 (R - r)} . \end{gathered}$
(34)
For ϱ² = 4ρσ, there exist only one solution stated below:
$\begin{matrix} N_{31} (x, t) & = & - (- 25 D R - 1 + 25 r D - 60 D \ln (a) \sqrt{ρ σ} + 100 D {(\ln (a))}^{2} ρ σ + 200 D^{2} {(\ln (a))}^{2} σ ρ) N_{\max} / (R - r) 50 D \\ + \frac{3 N_{\max} (- 1 + 10 D \ln (a) \sqrt{ρ σ}) ρ (2 \sqrt{ρ σ} η \ln (a) + 2)}{5 (R - r) ρ η} - \frac{3 D N_{\max} {(ρ (2 \sqrt{ρ σ} η \ln (a) + 2))}^{2}}{2 (R - r) ρ^{2} η^{2}} . \end{matrix}$
(35)
For ϱ = Ψ and ρ = κΨ, (κ ≠ 0) and σ = 0. the only solution is
$N_{32} (x, t) = - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) (Ψ) + 25 D {(\ln (a))}^{2} Ψ^{2}) N_{\max}}{(R - r) 50 D} .$
(36)
For ϱ = σ = 0, there is only one solution
$N_{33} (x, t) = - \frac{(- 25 D R - 1 + 25 r D) N_{\max}}{50 D (R - r)} .$
(37)
For ρ = ϱ = 0, the only solution we get is
$N_{34} (x, t) = - \frac{N_{\max} (- 25 η^{2} D R - η^{2} + 25 η^{2} r D + 60 D η + 300 D^{2})}{(R - r) 50 D η^{2}} .$
(38)
For ρ = 0 and ϱ ≠ 0, the two solutions are
$\begin{matrix} N_{35} (x, t) & = & - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2}) N_{\max}}{(R - r) 50 D} + \frac{6 \ln (a) N_{\max} σ (- 1 + 5 D \ln (a) ϱ) p ϱ}{5 (R - r) σ (\cosh (σ η) - \sinh (σ η) + p)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} σ^{2} p^{2} ϱ^{2}}{(R - r) {(σ (\cosh (σ η) - \sinh (σ η) + p))}^{2}}, \\ N_{36} (x, t) & = & - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) ϱ + 25 D {(\ln (a))}^{2} ϱ^{2}) N_{\max}}{(R - r) 50 D} + \frac{6 \ln (a) N_{\max} σ (- 1 + 5 D \ln (a) ϱ) ϱ (\sinh (σ η) + \cosh (σ η))}{5 (R - r) σ (\sinh (σ η) + \cosh (σ η) + q)} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} σ^{2} ϱ^{2} {(\sinh (σ η) + \cosh (σ η))}^{2}}{(R - r) {(σ (\sinh (σ η) + \cosh (σ η) + q))}^{2}} . \end{matrix}$
(39)
For ϱ = Ψ, σ = κΨ(κ ≠ 0) and ρ = 0, there is only one solution
$\begin{matrix} Φ_{37} (x, t) & = & - \frac{(- 25 D R - 1 + 25 r D - 30 D \ln (a) (Ψ) + 25 D {(\ln (a))}^{2} Ψ^{2}) N_{\max}}{50 (R - r) D} + \frac{6 \ln (a) N_{\max} κ Ψ (- 1 + 5 D \ln (a) Ψ) p a^{Ψ η}}{5 (R - r) (p - κ q a^{Ψ η})} \\ - \frac{6 D {(\ln (a))}^{2} N_{\max} κ^{2} Ψ^{2} p^{2} {(a^{Ψ η})}^{2}}{(R - r) {(p - κ q a^{Ψ η})}^{2}} . \end{matrix}$
(40)

V. GRAPHICAL BEHAVIOR

Below are various types of solitons that display the graphical behavior of the above-mentioned problem’s solution.

VI. RESULTS AND DISCUSSION

Since the majority of the solutions are trigonometric, including both simple and hyperbolic trigonometric solutions, the classification of the characteristics of different solutions is derived under these conditions. The surface and contour plots of each solution employing each specific condition listed in the approach are shown in the aforementioned graphs where Fig. 1 shows the periodic solitary waved behavior of the number of tumor cells N present at position x and time t represented in Eq. (27) under negative discriminant, Fig. 2 shows the surface and contour plots of the number of tumor cells N like a breather waved solitons present at position x and time t represented in Eq. (29) under positive discriminant, Fig. 3 represents the surface and contour plots showing periodic waved optical solitons of the number of tumor cells N present at position x and time t represented in Eq. (30) when ρσ > 0 and ϱ = 0, Fig. 4 signifies the surface and contour plots representing smooth solitons of the number of tumor cells N present at position x and time t represented in Eq. (32) when ρσ < 0 and ϱ = 0, Fig. 5 represents the surface and contour plots representing multi-peak singular solitons of the number of tumor cells N present at position x and time t represented in Eq. (33) when ρ = σ and ϱ = 0, Fig. 6 displays the surface and contour plots signifying dark solitons in the form of the number of tumor cells N present at position x and time t represented in Eq. (34) when ρ = −σ and ϱ = 0, Figs. 7 and 8 signify the surface and contour plots of pulse-like multi-peak singular solitons of the number of tumor cells N represented in Eq. (35) when ϱ² = 4ρσ and Eq. (38) when ϱ = ρ = 0, Fig. 9 presents the surface and contour plots of kink solitons of the number of tumor cells N present at position x and time t represented in Eq. (39) when ρ = 0 and ϱ ≠ 0, and Fig. 10 shows the surface and contour plots of dark solitons of the number of tumor cells N present at position x and time t represented in Eq. (40) when ϱ = Ψ, σ = κΨ(κ ≠ 0) and ρ = 0.

FIG. 1.

View large Download slide

The above surface and contour plots have been constructed by using the parametric values r = 1; R = 2; N_max = 1; b = −0.1; c = 10; G = 1; p = 1; q = 1; ρ = 2; ϱ = 1, σ = 3.

FIG. 2.

View large Download slide

The above graphs have been assembled with the help of the parameters r = 1; R = 2; N_max = 1; b = −0.1; c = 10; G = 1; p = 1; q = 1; ρ = 2; ϱ = 6; σ = 3.

FIG. 3.

View large Download slide

The above graphs have been made by using the parameters r = 1; R = 2; N_max = 1; b = −0.1; c = 10; G = 1; p = 1; q = 1; ρ = 2; ϱ = 0; σ = 3.

FIG. 4.

View large Download slide

The above graphs are made with the help of the parametric values r = −1; R = 1; N_max = −1; b = 0.1; c = 0.1; G = 1; p = 1; q = 1; ρ = 0.1; ϱ = 0; σ = −5.

FIG. 5.

View large Download slide

The above graphs are assembled by using the parameters r = 2; R = 1; N_max = 1; b = 0.1; c = 0.1; G = 2; p = 1; q = 1; ρ = 2; ϱ = 0; σ = 2.

FIG. 6.

View large Download slide

The above graphs have been constructed by using the parameters r = 2; R = 1; N_max = 1; b = 0.1; c = 0.1; G = 2; p = 1; q = 1; ρ = −2; ϱ = 0.

FIG. 7.

View large Download slide

The above graphs are made with the help of the parametric values r = 1.5; R = 1; N_max = 1; b = 1; c = 1.2; G = 2; p = 1; q = 1; ρ = 2; ϱ = 1; σ = 2.

FIG. 8.

View large Download slide

The above graphs have been assembled by using the parametric values r = 1.5; R = 1; N_max = 1; b = 1; c = 1.2; G = 2; p = 1; q = 1; ρ = 0; ϱ = 0; σ = 2.

FIG. 9.

View large Download slide

The above graphs have been developed by using the parameters r = 1.5; R = 1; N_max = 1; b = 0.01; c = 1.2; G = 5; p = 0.01; q = 0.01; ρ = 0; ϱ = 1; σ = −2.

FIG. 10.

View large Download slide

The above graphs have been constructed with the help of the parametric values r = 1.1; R = 1; N_max = 1; b = −0.45; c = 1.2; G = 5; p = 0.01; q = 0.01; ρ = 0; Ψ = 0.5; κ = 1.5.

VII. CONCLUSION

In cancer, mathematical modeling has evolved concurrently with experimental methods. Our understanding of tumor formation, specifically related to cancer cell proliferation at various scales, has improved as a result of numerous in silico studies that have demonstrated the effectiveness of this technique for developing mathematical frameworks. We discussed a number of solutions and strategies in this contribution to characterizing the dynamics of tumor cell proliferation. Future mathematical and experimental research on these theories is merited, especially for those defining therapeutic theories. In addition to the current analysis of mathematical models of tumor cells, we have discovered some precise soliton solutions in this article, utilizing the direct algebraic method, which is novel and unusual. The graphical behavior of solitons has successfully generated and displayed these solutions. The most significant accomplishment we have made with this strategy is the number of solutions we have produced by considering every potential result.

ACKNOWLEDGMENTS

The authors would like to extend their sincere appreciation to the researcher supporting program at King Saud University, Riyadh, for funding this work under Project No. (RSPD2023R699).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Sidra Ghazanfar: Conceptualization (equal). Nauman Ahmed: Formal analysis (equal). Syed Mansoor Ali: Investigation (equal). Muhammad Sajid Iqbal: Resources (equal). Ali Akgül: Supervision (equal). Muhammad Ali Shar: Writing – original draft (equal). Abdul Bariq: Investigation (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

REFERENCES

1.

M. Z.

Chishti

and

A.

Sinha

, “

Do the shocks in technological and financial innovation influence the environmental quality? Evidence from brics economies

,”

Technol. Soc.

68

,

101828

(

2022

).

https://doi.org/10.1016/j.techsoc.2021.101828

Google Scholar

Crossref

2.

V.

Quan-Hoang

,

Mindsponge Theory

(

De Gruyter

,

2023

).

Google Scholar

3.

T. G.

Jakobsen

and

M.

Mehmetoglu

,

Applied Statistics Using Stata: A Guide for the Social Sciences

(

Applied Statistics Using Stata

,

2022

), pp.

1

–

100

.

Google Scholar

4.

D. G.

Cacuci

, “

Fifth-order comprehensive adjoint sensitivity analysis methodology for nonlinear systems (5th-CASAM-N): II. Paradigm application to a Bernoulli model comprising uncertain parameters

,”

Am. J. Comput. Math.

12

(

01

),

119

–

161

(

2022

).

https://doi.org/10.4236/ajcm.2022.121008

Google Scholar

Crossref

5.

M. D. H.

Bashar

and

M.

Roshİd

, “

Exact travelling wave solutions of the nonlinear evolution equations by improved f-expansion in mathematical physics

,”

Commun. Adv. Math. Sci.

3

(

3

),

115

–

123

(

2020

).

https://doi.org/10.33434/cams.659225

Google Scholar

Crossref

6.

M.

Ozisik

,

A.

Secer

, and

M.

Bayram

, “

On the examination of optical soliton pulses of Manakov system with auxiliary equation technique

,”

Optik

268

,

169800

(

2022

).

https://doi.org/10.1016/j.ijleo.2022.169800

Google Scholar

Crossref

7.

E. M.

Zayed

and

R. M. A.

Shohib

, “

Optical solitons and other solutions to Biswas–Arshed equation using the extended simplest equation method

,”

Optik

185

,

626

–

635

(

2019

).

https://doi.org/10.1016/j.ijleo.2019.03.112

Google Scholar

Crossref

8.

M.

Mirzazadeh

,

M.

Eslami

,

D.

Milovic

, and

A.

Biswas

, “

Topological solitons of resonant nonlinear Schödinger’s equation with dual-power law nonlinearity by G′/G-expansion technique

,”

Optik

125

(

19

),

5480

–

5489

(

2014

).

https://doi.org/10.1016/j.ijleo.2014.03.042

Google Scholar

Crossref

9.

M. A. E.

Abdelrahman

and

M. A.

Sohaly

, “

The Riccati–Bernoulli sub-ode technique for solving the deterministic (stochastic) generalized-Zakharov system

,”

Int. J. Math. Syst. Sci.

1

(

3

),

1067

(

2018

).

https://doi.org/10.24294/ijmss.v1i3.810

Google Scholar

Crossref

10.

S.

Ghazanfar

,

N.

Ahmed

,

M. S.

Iqbal

,

A.

Akgül

,

M.

Bayram

, and

M.

De la Sen

, “

Imaging ultrasound propagation using the Westervelt equation by the generalized Kudryashov and modified Kudryashov methods

,”

Appl. Sci.

12

(

22

),

11813

(

2022

).

https://doi.org/10.3390/app122211813

Google Scholar

Crossref

11.

M. S.

Iqbal

,

M.

Inc

,

S.

Ghazanfar

, and

N.

Ahmed

, “

Existence and regularity of shock-effected solitons for nonlinear problems in electro-cardiac-physiology

,”

Int. J. Wavelets, Multiresolution Inf. Process.

21

(

02

),

2250053

(

2023

).

https://doi.org/10.1142/s0219691322500539

Google Scholar

Crossref

12.

W.

Gao

,

H.

Rezazadeh

,

Z.

Pinar

,

H. M.

Baskonus

,

S.

Sarwar

, and

G.

Yel

, “

Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique

,”

Opt. Quantum Electron.

52

(

1

),

52

(

2020

).

https://doi.org/10.1007/s11082-019-2162-8

Google Scholar

Crossref

13.

H.

Rehman

,

A. R.

Seadawy

,

M.

Younis

,

S. T. R.

Rizvi

,

I.

Anwar

,

M. Z.

Baber

, and

A.

Althobaiti

, “

Weakly nonlinear electron-acoustic waves in the fluid ions propagated via a (3 + 1)-dimensional generalized Korteweg–de-Vries–Zakharov–Kuznetsov equation in plasma physics

,”

Results Phys.

33

,

105069

(

2022

).

https://doi.org/10.1016/j.rinp.2021.105069

Google Scholar

Crossref

14.

A. R.

Seadawy

,

M.

Younis

,

M. Z.

Baber

,

S. T. R.

Rizvi

, and

M. S.

Iqbal

, “

Diverse acoustic wave propagation to confirmable time–space fractional KP equation arising in dusty plasma

,”

Commun. Theoretical Phys.

73

(

11

),

115004

(

2021

).

https://doi.org/10.1088/1572-9494/ac18bb

Google Scholar

Crossref

15.

W.

Islam

,

M. Z.

Baber

,

N.

Ahmed

,

A.

Akgül

,

M.

Rafiq

,

A.

Raza

,

I. S.

Yahia

,

H.

Algarni

, and

W.

Weera

, “

Investigation the soliton solutions of Mussel and Algae model leading to concentration

,”

Alexandria Eng. J.

70

,

133

–

143

(

2023

).

https://doi.org/10.1016/j.aej.2023.01.025

Google Scholar

Crossref

16.

M. Z.

Baber

,

A. R.

Seadway

,

N.

Ahmed

,

M. S.

Iqbal

, and

M. W.

Yasin

, “

Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross–Pitaevskii equation in dispersive media

,”

Int. J. Mod. Phys. B

37

,

2350191

(

2022

).

https://doi.org/10.1142/s0217979223501916

Google Scholar

Crossref

17.

M. Z.

Baber

,

N.

Ahmed

,

M. W.

Yasin

,

M. S.

Iqbal

,

A.

Akgül

,

M. B.

Riaz

,

M.

Rafiq

, and

A.

Raza

, “

Comparative analysis of numerical with optical soliton solutions of stochastic Gross–Pitaevskii equation in dispersive media

,”

Results Phys.

44

,

106175

(

2023

).

https://doi.org/10.1016/j.rinp.2022.106175

Google Scholar

Crossref

18.

Q.

Lu

and

Y.

Wang

, “

Nonlinear solitary waves in particle metamaterials with local resonators

,”

J. Acoust. Soc. Am.

151

(

3

),

1449

–

1463

(

2022

).

https://doi.org/10.1121/10.0009631

Google Scholar

Crossref

PubMed

19.

C. E.

Nurnberger

, “

Ionization theory and radio-biological reactions

,”

Proc. Natl. Acad. Sci. U. S. A.

23

(

4

),

189

–

193

(

1937

).

https://doi.org/10.1073/pnas.23.4.189

Google Scholar

Crossref

PubMed

20.

H.

Enderling

and

M. A. J.

Chaplain

, “

Mathematical modeling of tumor growth and treatment

,”

Curr. Pharm. Des.

20

(

30

),

4934

–

4940

(

2014

).

https://doi.org/10.2174/1381612819666131125150434

Google Scholar

Crossref

PubMed

21.

K. J.

Mahasa

,

R.

Ouifki

,

A.

Eladdadi

, and

L. D.

Pillis

, “

Mathematical model of tumor–immune surveillance

,”

J. Theor. Biol.

404

,

312

–

330

(

2016

).

https://doi.org/10.1016/j.jtbi.2016.06.012

Google Scholar

Crossref

PubMed

22.

K. K.

Unni

and

D. C.

Dahlin

, “

Premalignant tumors and conditions of bone

,”

Am. J. Surg. Pathol.

3

(

1

),

47

–

60

(

1979

).

https://doi.org/10.1097/00000478-197902000-00006

Google Scholar

Crossref

PubMed

23.

S. S.

Lazarus

and

L. D.

Trombetta

, “

Ultrastructural identification of a benign perineurial cell tumor

,”

Cancer

41

(

5

),

1823

–

1829

(

1978

).

https://doi.org/10.1002/1097-0142(197805)41:5<1823::aid-cncr2820410525>3.0.co;2-j

Google Scholar

Crossref

PubMed

24.

Y.

Kanai

and

S.

Hirohashi

, “

Alterations of DNA methylation associated with abnormalities of DNA methyltransferases in human cancers during transition from a precancerous to a malignant state

,”

Carcinogenesis

28

(

12

),

2434

–

2442

(

2007

).

https://doi.org/10.1093/carcin/bgm206

Google Scholar

Crossref

PubMed

25.

A. M.

Jarrett

,

E. A. B. F.

Lima

,

D. A.

Hormuth

,

M. T.

McKenna

,

X.

Feng

,

D. A.

Ekrut

,

A. C. M.

Resende

,

A.

Brock

, and

T. E.

Yankeelov

, “

Mathematical models of tumor cell proliferation: A review of the literature

,”

Expert Rev. Anticancer Ther.

18

(

12

),

1271

–

1286

(

2018

).

https://doi.org/10.1080/14737140.2018.1527689

Google Scholar

Crossref

PubMed

Exact and solitary wave structure of the tumor cell proliferation with LQ model of three dimensional PDE by newly extended direct algebraic method

I. INTRODUCTION

II. PROBLEM STATEMENT

III. STEPS TO USE NEWLY EXTENDED DIRECT ALGEBRAIC METHOD

IV. APPLICATION OF THE PROPOSED METHOD ON LQ MODEL OF TUMOR CELL

V. GRAPHICAL BEHAVIOR

VI. RESULTS AND DISCUSSION

VII. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Exact and solitary wave structure of the tumor cell proliferation with LQ model of three dimensional PDE by newly extended direct algebraic method

I. INTRODUCTION

II. PROBLEM STATEMENT

III. STEPS TO USE NEWLY EXTENDED DIRECT ALGEBRAIC METHOD

IV. APPLICATION OF THE PROPOSED METHOD ON LQ MODEL OF TUMOR CELL

V. GRAPHICAL BEHAVIOR

VI. RESULTS AND DISCUSSION

VII. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only