A neural network computational structure for the fractional order breast cancer model

The current study provides the numerical performances of the fractional kind of breast cancer (FKBC) model, which are based on five different classes including cancer stem cells, healthy cells, tumor cells, excess estrogen, and immune cells. The motive to introduce the fractional order derivatives is to present more precise solutions as compared to integer order. A stochastic computing reliable scheme based on the Levenberg Marquardt backpropagation neural networks (LMBNNS) is proposed to solve three different cases of the fractional order values of the FKBC model. A designed dataset is constructed by using the Adam solver in order to reduce the mean square error by taking the data performances as 9% for both testing and validation, while 82% is used for training. The correctness of the solver is approved through the negligible absolute error and matching of the solutions for each model’s case. To validates the accuracy, and consistency of the solver, the performances based on the error histogram, transition state, and regression for solving the FKBC model.


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There are various studies that highlights the importance of genetic profiling based on BC prevention and its therapy by illuminating the role of genes using a predisposing individual of the illness 1,14,15 .Most pathogenic mutations with a high propensity to cause cancer are areas of the gene.Genes play a key role in the development of BC using the increase of estrogen level, which causes a natural body response toward growing the tumors in the BC dynamics 16 .Enderling et al. 17 presented the mathematical cancerous cell model using the regulated healthy pre-cancerous cells formed in the post-pubertal BC structure.BC development with higher estrogen levels is studied with the impacts of healthy breast epithelial cells, interactions among cancer cells, the body's natural immune response, and non -proliferative tumor cells 18 .
Mathematical systems are important tools in numerous fields of engineering, science, and economics.There are several mathematical models that have been studied in current decades and the solution of these models have been presented by using different schemes.To mention a few of them, Ganji et al. 19 presented the solutions of the brain tumor model using the fractional operator.Sánchez et al. 20 provided the solutions of the coronavirus model by applying a reliable numerical scheme.Sabir et al. 21provided the results of prediction differential system by using the explicit Runge-Kutta and Adams numerical techniques.Hart et al. 22 discussed the mathematical modelling of the functionally dependent bone remodeling and provided its numerical solutions.Fogelson et al. 23 designed and studied numerically by using a mathematical system based on the platelet adhesion and aggregation during blood clotting.Sana et al. 24 discussed a mathematical model based on the supply chain systems.The coronavirus time series data by applying the spectral analysis and deep learning methods have been proposed by Oshinubi et al. 25 .Few recent mathematical models that have been used to solve with different numerical schemes are provided in these references [26][27][28] .
The purpose of this research is to obtain the solutions of the fractional kind of breast cancer (FKBC) system in order to get more precise solutions as compared to integer order.Hence, a reliable stochastic scheme based on the Levenberg Marquardt backpropagation neural networks (LMBNNS) is proposed for the FKBC model.The stochastic process has been derived before to solve various kind of the mathematical model, but the solutions of the FKBC model have been presented first time with the implementation of this scheme.Currently, the neural network-based procedures have been used to solve functional form of singular models 29 , hepatitis virus models 30 , bone remodeling model 31 , fractional vector-host diseased model 32 , SIRC epidemic model 33 , Zika virus model 34 , breathing transmission system 35 , thermal explosion model 36 , Rabinovich-Fabrikant system 37 , Layla and Majnun model 38 and food chain model 39 .
The fractional order models are considered more challenging and provide more reliable results to solve the differential model.Fractional types of derivatives are implemented to test the effectiveness of the real-world applications 16,17 .Over the past 3 decades, the implementation of fractional calculus has been observed widely by applying the powerful operators of Weyl-Riesz 18 , Caputo 19 , Riemann-Liouville 20 , Erdlyi-Kober 21 , and Grnwald-Letnikov 22 .The Caputo derivative can be applied to solve both conditions of the model homogeneous and nonhomogeneous.All these operators have their own drawbacks; however, the Caputo derivatives are considered simple as compared to other operators.Some novel motivations of this study are given as: • The design of the FKBC system have been provided and the solutions have been performed by using the LMBNNs.• Three different cases based on the fractional order have been presented to solve the FKBC system.
• The correctness of the scheme is observed by using the overlapping of the outputs.
• The absolute error (AE) results in good performances enhance the reliability of the proposed solver.
The other paper's parts are organized as: Sect."Mathematical FKBC system" describes the construction of the FKBC system, Sect."Designed LMBNNs procedure" provides the proposed technique based LMBNNs, Sect."Results and discussions" is constructed based on the calculated outcomes, while conclusions are proposed in Sect."Conclusions".

Mathematical FKBC system
This section provides the FKBC system, which is divided into five different categories named as cancer stem cells C(t) , tumor cells T(t) , healthy cells H(t) , immune cells I(t) , and excess estrogen E(t) .The mathematical form of the integer order BC model is given as 16 : where k 1 , k 2 and q denote the normal rate of cell separation for first three dynamics, and M 1 indicates the carrying size based on the cells of first three dynamics.The rates at which estrogen promotes the growth of cancer stem (1) In addition, the rate at which healthy cells decease, s also shows the basis rate of resistant cells.u is the rate of immunological suppression caused by estrogen; v is the immune cell threshold; The letters τ stands the continuous estrogen infusion, the body's estrogen washout rate, and the letters d 1 , d 2 and d 3 are the rates at which cancer stem cells, tumor cells, and healthy cells absorb estrogen, respectively.The fractional form of the above system becomes as Ref. 16 : where α stands the fractional order Caputo derivative.To investigate the complicated features, such as super slow evolution and superfast transients are considered challenging as compared to integer order given in system (1).
There are various recent applications, where the fractional kind of the derivatives have been used.Some of them are inconsistent heat transmission 40 , pine wilt disease model with convex rate 41 , spatiotemporal outlines 42 , and soil animal approximation 43 , and soil animal constituent content 44 .

Designed LMBNNs procedure
The stochastic computational performances based on the LMBNNs scheme to solve the mathematical FKBC system as stated in the set of systems ( 2) is provided in this section.Three components of model using the developed technique, and outcomes are provided in Fig. 1a and b.These Figs depict the workflow diagram for the FKBC system.The design presentations are described in two different metrics including LMBNNs-based processes and the mathematical process.The design of the dataset is presented by using the Adam scheme and the division of the data contains 82% (training) and 9% for both verification and testing by taking 15 hidden neurons.Log-sigmoid transfer function in the hidden layers is selected and single input and output layer structure based on the neurons is obtained.The current neural network study is operated with untimely conjunction, overfitting, and concealed scenarios.Therefore, the parameters of the networks have been used carefully after considerable tests, experience, and knowledge.A fully concentration is required to adjust the parameter setting and a small modification in the setting can change the whole scenario and can impact the performance of the investigations.The "MATLAB" software (NFTOOL command) is used to implement the stochastic process based on the LMBNNs, which include the correct hidden neuron sections, testing statistics, learning strategies, and verification statics.Table 1 shows the parameter setting to perform the LMBNNs for solving the FKBC model.

Results and discussions
In this section, three different cases have been taken by using the values of α = 0.5, 0.7 and 0.9 for solving the mathematical model.

Analysis of results for case 1
Consider the values α = 0.5 , 2), while the values of the initial conditions have been selected 1.2 for each class.The mean square error (MSE) and state evolution (SE) results of the FKBC model's performance are shown in Fig. 2a,b for case 1.The first two parts of the Fig. 2 indicate the decrement of MSE for solving the FKBC model.The best validation performances are reported as 3.4006 ×10 −09 , while the gradient is reported as 1.4903 ×10 −06 , and the epochs have been calculated for this case are 220.Figure 3 shows the function fit and error histogram (EH) performances for case 1.The first half is designed based on the func fit, while the second half provides the EH.The values of the EH are presented as 5.95 ×10 −05 .Figure 4 provides the regression performances for case 1, which is calculated as 1 that shows the perfect model.
Table 2 shows the MSE performances based on the neural networks along with the counted epochs and complexity measures.

Conclusions
In this study, the numerical solution of the fractional breast cancer system have been presented, which are based on five different classes including cancer stem cells, healthy cells, tumor cells, excess estrogen, and immune cells.Some of the concluding remarks of this work are given as: • The fractional derivatives have been introduced to solve the breast cancer mathematical model.
• The fractional kind of derivatives have been provided to get more precise solutions of the model as compared to integer order.• A stochastic computing Levenberg Marquardt backpropagation neural networks scheme has been proposed for three fractional order cases of the FKBC model.• The constructions of the designed dataset based on the Adam solver has been presented to reduce the MSE by taking the data performances as 9% for both testing and validation, while 82% is used for training.• The correctness of the solver has been approved through the negligible absolute error along with the matching of the solutions for each model's case.• To authenticate the accuracy of the solver, the performances based on the error histogram, transition state, and regression for solving the FKBC model has been provided.

Figure 1 .
Figure 1.(a) A general structure of a single neuron.(b) A layer structure of the model.

Figure 2 .
Figure 2. MSE and STs for the FKBC system based case 1.

Figure 3 .
Figure 3. Func fit and EHs for the mathematical system for case 1.

Figure 5 .
Figure 5. MSE and STs for the FKBC system for case 2.

Figure 6 .
Figure 6.Func fit and EHs for the mathematical system based case 2.

Figure 12 .
Figure 12. (a-e) Result assessment for the FKBC model.
and tumor cells, as well as the rates at which healthy cells are lost because of estrogen-induced DNA mutation, are shown in p 1 , p 2 , and p 3 , respectively.It is also presented the rates at which immune cells react to tumor cells and cancer stem cells; a 1 , a 2 , and a 3 designate the number of first three dynamics.The values of n 1 and n 2 are the usual tumor.

Table 1 .
Parameters adjustment to perform the LMBNNs.