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 tumor^{22} is one that is not yet malignant but is likely to become so. A benign tumor^{23} cannot metastasize (cannot spread, remains dormant). The cancerous malignant^{24} 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.

^{25}

*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:

*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:

## II. PROBLEM STATEMENT

*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 (4) 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:where(5)$\Omega (N,Nt,Nx,Nxx,Ntt,Nxt,Nxxx\u2026)=0,$*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 transformationThe aforementioned form is obtained by adding the real variables(6)$N(x,t)=\Omega (\eta ),\eta =x+ct.$*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):where Θ is the polynomial of Ω and its derivatives, and the superscripts identify regular derivatives with respect to(7)$\Theta (\Omega ,\Omega \u2032,\Omega \u2033,\Omega \u2034,\u2026)=0,$*η*.**Step III:**Assume that the initial solution of Eq. (7) can be stated as a polynomial Ψ(*η*) in the following form:where ϒ(8)$N(\eta )=\Sigma i=0P\u03d2i\Psi i(\eta ),\u03d2P\u22600,$_{i}(0 ≤*i*≤*P*) are constants concluded and Ψ(*η*) satisfies the following ODE:We currently have the following options for resolving the stated issue:(9)$\Psi \u2032(\eta )=ln(a)(\rho +\u03f1\Psi (\eta )+\sigma \Psi 2(\eta )),a\u22600,1.$For

*ϱ*^{2}− 4*ρσ*< 0 and*σ*≠ 0, there are five solutions

(10)$\Psi 1(\eta )=\u2212\u03f12\sigma +\u2212(\u03f12\u22124\rho \sigma )2\sigma tana\u2212(\u03f12\u22124\rho \sigma )2\eta ,\Psi 2(\eta )=\u2212\u03f12\sigma \u2212\u2212(\u03f12\u22124\rho \sigma )2\sigma cota\u2212(\u03f12\u22124\rho \sigma )2\eta ,\Psi 3(\eta )=\u2212\u03f12\sigma +\u2212(\sigma 2\u22124\rho \sigma )2\sigma tana\u2212(\u03f12\u22124\rho \sigma )\eta \xb1pqseca\u2212(\u03f12\u22124\rho \sigma )\eta ,\Psi 4(\eta )=\u2212\u03f12\sigma +\u2212(\u03f12\u22124\rho \sigma )2\sigma \u2212cota\u2212(\u03f12\u22124\rho \sigma )\eta \xb1pqcsca\u2212(\u03f12\u22124\rho \sigma )\eta ,\Psi 5(\eta )=\u2212\u03f12\sigma +\u2212(\u03f12\u22124\rho \sigma )4\sigma tana\u2212(\u03f12\u22124\rho \sigma )4\eta \u2212cota\u2212(\u03f12\u22124\rho \sigma )4\eta .$For

*ϱ*^{2}− 4*ρσ*> 0 and*σ*≠ 0, there are five solutions

(11)$\Psi 6(\eta )=\u2212\u03f12\sigma \u2212\u03f12\u22124\rho \sigma 2\sigma tanha\u03f12\u22124\rho \sigma 2\eta ,\Psi 7(\eta )=\u2212\u03f12\sigma \u2212\u03f12\u22124\rho \sigma 2\sigma cotha\u03f12\u22124\rho \sigma 2\eta ,\Psi 8(\eta )=\u2212\u03f12\sigma +\u03f12\u22124\rho \sigma 2\sigma \u2212tanha\u03f12\u22124\rho \sigma \eta \xb1\iota pqsecha\u03f12\u22124\rho \sigma \eta ,\Psi 9(\eta )=\u2212\u03f12\sigma +\u03f12\u22124\rho \sigma 2\sigma \u2212cotha\u03f12\u22124\rho \sigma \eta \xb1pqcscha\u03f12\u22124\rho \sigma \eta ,\Psi 10(\eta )=\u2212\u03f12\sigma \u2212\u03f12\u22124\rho \sigma 4\sigma tanha\u03f12\u22124\rho \sigma 4\eta +cotha\u03f12\u22124\rho \sigma 4\eta .$3. For*ρσ*> 0 and*ϱ*= 0, there are five solutions(12)$\Psi 11(\eta )=\rho \sigma tana(\rho \sigma \eta ),\Psi 12(\eta )=\u2212\rho \sigma cota(\rho \sigma \eta ),\Psi 13(\eta )=\rho \sigma tana2\rho \sigma \eta \xb1pqseca2\rho \sigma \eta ,\Psi 14(\eta )=\rho \sigma \u2212cota2\rho \sigma \eta \xb1pqcsca2\rho \sigma \eta ,\Psi 15(\eta )=12\rho \sigma tana\rho \sigma 2\eta \u2212cota\rho \sigma 2\eta .$4. For*ρσ*< 0 and*ϱ*= 0, there are five solutions(13)$\Psi 16(\eta )=\u2212\u2212\rho \sigma tanha(\u2212\rho \sigma \eta ),\Psi 17(\eta )=\u2212\u2212\rho \sigma cotha(\u2212\rho \sigma \eta ),\Psi 18(\eta )=\u2212\rho \sigma \u2212tanha2\u2212\rho \sigma \eta \xb1\iota pqsecha2\u2212\rho \sigma \eta ,\Psi 19(\eta )=\u2212\rho \sigma \u2212cotha2\u2212\rho \sigma \eta \xb1pqcscha2\u2212\rho \sigma \eta ,\Psi 20(\eta )=\u221212\u2212\rho \sigma tanha\u2212\rho \sigma 2\eta +cotha\u2212\rho \sigma 2\eta .$5. For*σ*=*ρ*and*ϱ*= 0, there are five solutions(14)$\Psi 21(\eta )=tana(\rho \eta ),\Psi 22(\eta )=\u2212cota(\rho \eta ),\Psi 23(\eta )=tana(2\rho \eta )\xb1pqseca(2\rho \eta ),\Psi 24(\eta )=\u2212cota(2\rho \eta )\xb1pqcsca(2\rho \eta ),\Psi 25(\eta )=12tana\rho 2\eta \u2212cota\rho 2\eta .$6. For*σ*= −*ρ*and*ϱ*= 0, there are five solutions(15)$\Psi 26(\eta )=\u2212tanha(\rho \eta ),\Psi 27(\eta )=\u2212cotha(\rho \eta ),\Psi 28(\eta )=\u2212tanha(2\rho \eta )\xb1\iota pqsecha(2\rho \eta ),\Psi 29(\eta )=\u2212cotha(2\rho \eta )\xb1pqcscha(2\rho \eta ),\Psi 30(\eta )=\u221212tanha\rho 2\eta +cotha\rho 2\eta .$7. For*ϱ*^{2}= 4*ρσ*, there is only one solution(16)$\Psi 31(\eta )=\u22122\rho (\u03f1\eta ln\u2061a+2)\u03f12\u2061\eta ln\u2061a.$8. For*ϱ*=*χ*,*ρ*=*κχ*(*κ*≠ 0) and*σ*= 0, there is only one solution(17)$\Psi 32(\eta )=a\chi \eta \u2212\kappa .$9. For*ϱ*=*σ*= 0, there is only one solution(18)$\Psi 33(\eta )=\rho \eta ln\u2061a.$10. For*ρ*=*ϱ*= 0, there is only one solution(19)$\Psi 34(\eta )=\u22121\sigma \eta ln\u2061a.$11. For*ρ*= 0 and*ϱ*≠ 0. there are two solutions(20)$\Psi 35(\eta )=\u2212p\u03f1\sigma (cosha(\sigma \eta )\u2212sinha(\sigma \eta )+p),\Psi 36(\eta )=\u2212(\u03f1(sinha(\sigma \eta )+cosha(\sigma \eta )))\sigma (sinha(\sigma \eta )+cosha(\sigma \eta )+q).$12. For*ϱ*=*χ*,*σ*=*κχ*(*κ*≠ 0) and*ρ*= 0, there is only one solution(21)$\Psi 37(\eta )=pa\chi \eta p\u2212\kappa qa\chi \eta .$The generalized hyperbolic and trigonometric functions are defined as follows for the aforementioned solutions:$sina(\eta )=pa\iota \eta \u2212qa\u2212\iota \eta 2,cosa(\eta )=pa\iota \eta +qa\u2212\iota \eta 2,tana(\eta )=\u2212\iota pa\iota \eta \u2212qa\u2212\iota \eta pa\iota \eta +qa\u2212\iota \eta ,cota(\eta )=\iota pa\iota \eta +qa\u2212\iota \eta pa\iota \eta \u2212qa\u2212\iota \eta ,seca(\eta )=2pa\iota \eta +qa\u2212\iota \eta ,csca(\eta )=2\iota pa\iota \eta \u2212qa\u2212\iota \eta ,sinha(\eta )=pa\eta \u2212qa\u2212\eta 2,cosha(\eta )=pa\eta +qa\u2212\eta 2,tanha(\eta )=pa\eta \u2212qa\u2212\eta pa\eta +qa\u2212\eta ,cotha(\eta )=pa\eta +qa\eta pa\eta \u2212qa\u2212\eta ,secha(\eta )=2pa\eta +qa\u2212\eta ,cscha(\eta )=2pa\eta \u2212qa\u2212\eta ,$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

*N*(

*x*,

*t*) =

*N*(

*η*),

*η*=

*x*+

*ct*, which can be reduced to Eq. (4) to the following ODE:

*N*″ and

*N*

^{2}to get the value of

*P*with the help of the homogeneous balance principle, we found this value to be

*P*= 2. Using Eq. (8), the solution of Eq. (22) takes the form as

*a*

_{0},

*a*

_{1}, and

*a*

_{2}are the constants to be determined and Ψ(

*η*) satisfies the following ODE:

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

- For
*ϱ*^{2}− 4*ρσ*< 0 and*σ*≠ 0, the five solutions below were created using mixed-trigonometric and trigonometric formulas (*ϱ*^{2}− 4*ρσ*=*E*):(26)$N1(x,t)=\u2212Nmax\u221225DR\u22121+25rD+25Dlna2\u03f12\u221230lnaD\u2212Etan\u2212E\eta 2\u2212lna2D275\u03f12+200\rho \sigma +75\u03f12\u2212300\rho \sigma tan\u2212E\eta 22/(R\u2212r50D),N2(x,t)=\u2212Nmax\u221225DR\u22121+25rD+25Dlna2\u03f12+30lnaD\u2212Ecot\u2212E\eta 2\u2212lna2D275\u03f12+200\rho \sigma +75\u03f12\u2212300\rho \sigma cot\u2212E\eta 22/R\u2212r50D,N3(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25Dlna2\u03f12+200D2lna2\sigma \rho Nmax50DR\u2212r\u22123\u2061lnaNmax\sigma \u22121+5D\u2061lna\u03f1\u2212\u03f1+\u2212Etan\u2212E\eta \xb1pqsec\u2212E\eta 5\sigma R\u2212r\u22123Dlna2Nmax\u2212\u03f1+\u2212\u03f12+4\rho \sigma tan\u2212\u03f12+4\rho \sigma \eta \xb1pqsec\u2212E\eta 22(R\u2212r),$(27)$N4(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25D\u2061lna2\u03f12+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\u22121+5D\u2061lna\u03f1\u2212\u03f1\u2212\u2212Ecot\u2212E\eta \xb1pqcsc\u2212E\eta 5(R\u2212r)\u22123Dlna2Nmax\u2212\u03f1\u2212\u2212Ecot\u2212E\eta \xb1pqcsc\u2212E\eta 22(R\u2212r),N5(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25Dlna2\u03f12+200D2lna2\sigma \rho Nmax50D(R\u2212r)\u22123\u2061lnaNmax\sigma \u22121+5D\u2061lna\u03f1\u22122\u03f1+\u2212Etan1/4E\eta \u2212cot\u2212E\eta 410\sigma R\u2212r\u22123Dlna2Nmax\u22122\u03f1+\u2212Etan\u2212E\eta 4\u2212cot\u2212E\eta 428(R\u2212r).$ - For
*ϱ*^{2}− 4*ρσ*> 0 and*σ*≠ 0 (considering*ρ*to be negative), there are five exact solutions derived below:$N6(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25D\u2061lna2\u03f12+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\u22121+5D\u2061lna\u03f1\u2212\u03f1\u2212EtanhE\eta 25(R\u2212r)\u22123Dlna2Nmax\u2212\u03f1\u2212EtanhE\eta 222(R\u2212r),N7(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25Dlna2\u03f12+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\u22121+5D\u2061lna\u03f1\u2212\u03f1\u2212EcothE\eta 25(R\u2212r)\u22123Dlna2Nmax\u2212\u03f1\u2212\u03f12\u22124\rho \sigma cothE\eta 222(R\u2212r),$(28)$N8(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25Dlna2\u03f12+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\u22121+5D\u2061lna\u03f1\u2212\u03f1\u2212EtanhE\eta \xb1\iota pqsechE\eta 5(R\u2212r)\u22123Dlna2Nmax\u2212\u03f1\u2212EtanhE\eta \xb1\iota pqsechE\eta 22(R\u2212r).N9(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25Dlna2\u03f12+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\u22121+5D\u2061lna\u03f1\u2212\u03f1\u2212EcothE\eta \xb1pqcschE\eta 5(R\u2212r)\u22123Dlna2Nmax\u2212\u03f1\u2212EcothE\eta \xb1pqcschE\eta 22(R\u2212r),$(29)$N10(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\u03f1+25Dlna2\u03f12+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\u22121+5D\u2061lna\u03f1\u22122\u03f1\u2212EtanhE\eta 4+cothE\eta 410(R\u2212r)\u22123Dlna2Nmax\u22122\u03f1\u2212EtanhE\eta 4+cothE\eta 428(R\u2212r).$ - For
*ρσ*> 0 and*ϱ*= 0, there are five trigonometric and mixed-trigonometric solutions stated below:(30)$N11(x,t)=\u2212Nmax\u221225DR\u22121+25rD\u221260lna\sigma \rho \sigma tan\rho \sigma \eta D+D2lna2\sigma \rho (300tan\rho \sigma \eta 2+200)/(50R\u2212rD),N12(x,t)=\u2212Nmax\u221225DR\u22121+25rD\u221260lna\sigma \rho \sigma cot\rho \sigma \eta D+D2lna2\sigma \rho (300cot\rho \sigma \eta 2+200)/(R\u2212r50D),N13(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\sigma \rho NmaxR\u2212r50D+6\u2061lnaNmax\sigma \rho \sigma tan2\rho \sigma \eta \xb1pqsec2\rho \sigma \eta 5R\u2212r\u22126Dlna2Nmax\sigma \rho tan2\rho \sigma \eta \xb1pqsec2\rho \sigma \eta 2R\u2212r,N14(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\sigma \rho NmaxR\u2212r50D+6\u2061lnaNmax\sigma \rho \sigma \u2212cot2\rho \sigma \eta \xb1pqcsc2\rho \sigma \eta 5R\u2212r\u22126Dlna2Nmax\sigma \rho \u2212cot2\rho \sigma \eta \xb1pqcsc2\rho \sigma \eta 2R\u2212r,N15(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\sigma \rho NmaxR\u2212r50D+3\u2061lnaNmax\sigma \rho \sigma tan\rho \sigma 2\eta \u2212cot\rho \sigma 2\eta 5R\u2212r\u22123Dlna2Nmax\sigma \rho tan\rho \sigma 2\eta \u2212cot\rho \sigma 2\eta 22(R\u2212r).$ - For
*ρσ*< 0 and*ϱ*= 0 (assuming the value of*ρ*is negative), there are five solutions constructed below:(31)$N16(x,t)=Nmax25DR+1\u221225rD\u221260lna\sigma \u2212\rho \sigma tanh\u2212\rho \sigma \eta D+D2lna2(300\rho \sigma tanh\u2212\rho \sigma 2\eta 2\u2212200\rho \sigma )/(R\u2212r50D),N17(x,t)=Nmax25DR+1\u221225rD\u221260lna\sigma \u2212\rho \sigma coth\u2212\rho \sigma \eta D+D2lna2(300\rho \sigma coth\u2212\rho \sigma 2\eta 2\u2212200\rho \sigma )/(50R\u2212r50D),$(32)$N18(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\sigma \rho Nmax50R\u2212rD\u22126\u2061lnaNmax\sigma \u2212\rho \sigma tanh2\u2212\rho \sigma \eta \xb1\iota pq\u2009sech2\u2212\rho \sigma \eta 5R\u2212r+6Dlna2Nmax\sigma \rho tanh2\u2212\rho \sigma \eta \xb1\iota pqsech2\u2212\rho \sigma \eta 2R\u2212r,N19(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\sigma \rho NmaxR\u2212r50D\u22126\u2061lnaNmax\sigma \u2212\rho \sigma coth2\u2212\rho \sigma \eta \xb1pqcsch2\u2212\rho \sigma \eta 5R\u2212r+6DlnaNmax\sigma \rho coth2\u2212\rho \sigma \eta \xb1pqcsch2\u2212\rho \sigma 2\eta 2R\u2212r,N20(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\sigma \rho Nmax50R\u2212rD\u22123\u2061lnaNmax\sigma \u2212\rho \sigma tanh\u2212\rho \sigma 2\eta +coth\u2212\rho \sigma 2\eta 5R\u2212r+3Dlna2Nmax\sigma \rho tanh\u2212\rho \sigma 2\eta +coth\u2212\rho \sigma 2\eta 22(R\u2212r).$ - For
*σ*=*ρ*and*ϱ*= 0, there are five solutions(33)$N21(x,t)=\u2212Nmax(\u221225DR\u22121+25rD\u221260lna\rho tan(\eta \rho )D+D2(lna300\rho 2(tan\eta \rho )2+200\rho 2)2)/R\u2212r50D,N22(x,t)=\u2212Nmax(\u221225DR\u22121+25rD\u221260lna\rho cot\eta \rho D+D2(lna(300\rho 2cot\eta \rho 2+200\rho 2))2)/R\u2212r50D,N23(x,t)=\u2212(\u221225DR\u22121+25rD+200D2lna2\rho 2)NmaxR\u2212r50D+6lnaNmax\rho tan2\eta \rho \xb1pqsec2\eta \rho 5(R\u2212r)\u22126Dlna2Nmax\rho 2tan2\eta \rho 2\xb1pqsec2\eta \rho 2R\u2212r,N24(x,t)=\u2212(\u221225DR\u22121+25rD+200D2lna2\rho 2)NmaxR\u2212r50D+6lnaNmax\rho (cot2\eta \rho )\xb1pqcsc2\eta \rho 5(R\u2212r)\u22126Dlna2Nmax\rho 2cot2\eta \rho 2\xb1pqcsc2\eta \rho 2R\u2212r,N25(x,t)=\u2212(\u221225DR\u22121+25rD+200D2lna2\rho 2)NmaxR\u2212r50D+3lnaNmax\rho tan\eta \rho 2\u2212cot\eta \rho 25(R\u2212r)\u22123Dlna2Nmax\rho 2tan\eta \rho 2\u2212cot\eta \rho 222(R\u2212r).$

- For
*σ*= −*ρ*and*ϱ*= 0, five solutions can be extracted below:(34)$N26(x,t)=\u2212Nmax(\u221225DR\u22121+25rD+60lna\rho tanh\rho \eta D+D2lna2(300\rho 2tanh\rho \eta 2+200\rho 2))/R\u2212r50D,N27(x,t)=\u2212Nmax(\u221225DR\u22121+25rD+60lna\rho coth\rho \eta D+D2lna2300\rho 2coth\rho \eta 2+200\rho 2)/50R\u2212rD,N28(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\rho 2NmaxR\u2212r50D\u22126\u2061lnaNmax\rho tanh2\rho \eta \xb1\iota pqsech2\rho \eta 5(R\u2212r)\u22126Dlna2Nmax\rho 2\u2061tanh2\rho \eta \xb1\iota 2pqsech2\rho \eta 2R\u2212r,N29(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\rho 2NmaxR\u2212r50D\u22126\u2061lnaNmax\rho coth2\rho \eta \xb1pqcsch2\rho \eta 5(R\u2212r)\u22126Dlna2Nmax\rho 2\u2061coth2\rho \eta \xb1pqcsch2\rho \eta 2R\u2212r,N30(x,t)=\u2212\u221225DR\u22121+25rD+200D2lna2\rho 2NmaxR\u2212r50D\u22123\u2061lnaNmax\rho tanh\rho \eta 2+coth\rho \eta 25(R\u2212r)\u22123Dlna2Nmax\rho 2tanh\rho \eta 2+coth\rho \eta 222(R\u2212r).$ - For
*ϱ*^{2}= 4*ρσ*, there exist only one solution stated below:(35)$N31(x,t)=\u2212(\u221225DR\u22121+25rD\u221260Dlna\rho \sigma +100Dlna2\rho \sigma +200D2lna2\sigma \rho )Nmax/R\u2212r50D+3Nmax\u22121+10Dlna\rho \sigma \rho 2\rho \sigma \eta lna+25R\u2212r\rho \eta \u22123DNmax\rho 2\rho \sigma \eta lna+222R\u2212r\rho 2\eta 2.$ - For
*ϱ*= Ψ and*ρ*=*κ*Ψ, (*κ*≠ 0) and*σ*= 0. the only solution is(36)$N32(x,t)=\u2212\u221225DR\u22121+25rD\u221230Dlna\Psi +25Dlna2\Psi 2NmaxR\u2212r50D.$ - For
*ϱ*=*σ*= 0, there is only one solution(37)$N33(x,t)=\u2212\u221225DR\u22121+25rDNmax50DR\u2212r.$ - For
*ρ*=*ϱ*= 0, the only solution we get is(38)$N34(x,t)=\u2212Nmax\u221225\eta 2DR\u2212\eta 2+25\eta 2rD+60D\eta +300D2R\u2212r50D\eta 2.$ - For
*ρ*= 0 and*ϱ*≠ 0, the two solutions are(39)$N35(x,t)=\u2212\u221225DR\u22121+25rD\u221230Dlna\u03f1+25Dlna2\u03f12NmaxR\u2212r50D+6lnaNmax\sigma \u22121+5Dlna\u03f1p\u03f15R\u2212r\sigma cosh\sigma \eta \u2212sinh\sigma \eta +p\u22126Dlna2Nmax\sigma 2p2\u03f12R\u2212r\sigma cosh\sigma \eta \u2212sinh\sigma \eta +p2,N36(x,t)=\u2212\u221225DR\u22121+25rD\u221230Dlna\u03f1+25Dlna2\u03f12NmaxR\u2212r50D+6lnaNmax\sigma \u22121+5Dlna\u03f1\u03f1sinh\sigma \eta +cosh\sigma \eta 5R\u2212r\sigma sinh\sigma \eta +cosh\sigma \eta +q\u22126Dlna2Nmax\sigma 2\u03f12sinh\sigma \eta +cosh\sigma \eta 2R\u2212r\sigma sinh\sigma \eta +cosh\sigma \eta +q2.$ - For
*ϱ*= Ψ,*σ*=*κ*Ψ(*κ*≠ 0) and*ρ*= 0, there is only one solution(40)$\Phi 37(x,t)=\u2212\u221225DR\u22121+25rD\u221230D\u2061lna\Psi +25Dlna2\Psi 2Nmax50R\u2212rD+6lnaNmax\kappa \Psi \u22121+5Dlna\Psi pa\Psi \eta 5R\u2212rp\u2212\kappa qa\Psi \eta \u22126Dlna2Nmax\kappa 2\Psi 2p2a\Psi \eta 2R\u2212rp\u2212\kappa qa\Psi \eta 2.$

## 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 *ϱ*^{2} = 4*ρσ* 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* = *κ*Ψ(*κ* ≠ 0) and *ρ* = 0.

## 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

*Applied Statistics Using Stata: A Guide for the Social Sciences*

*G*′/

*G*-expansion technique