All features are continuous, real-valued numbers

The MATLAB code performs the following preprocessing steps:

Conversion of diagnosis labels to binary (M = 1, B = 0)

The logistic regression model estimates malignancy probability through the following formulation: (1)P(y=1|x)=σ(wTx+b)=11+e−(wTx+b),where •

x∈R30 is the input feature vector containing: –

10 core features (radius, texture, perimeter, etc.)

The model is trained via maximum likelihood estimation with: •

L2 regularization (ridge penalty) to prevent overfitting: (2)ℒreg=λ‖w‖22

The L-BFGS optimization ensures convergence by approximating the Hessian matrix with a limited-memory update, guaranteeing descent directions [38,39].

The decision tree implements recursive partitioning through: (4)f(x)=∑m=1Mcm⋅I(x∈Rm)where •

Rm⊂R30 are hyperrectangular regions formed by axis-aligned splits with: –

Maximum depth of 30 (early stopped via validation)

Splits are determined by Gini impurity minimization: (6)G(Q)=∑k=12pk(1−pk),pk=1|Q|∑xi∈QI(yi=k)where

Q is the data subset at each node

Advanced features include:

Cost-complexity pruning with complexity parameter α=0.01

The tree provides interpretable decision paths: (9)Path(x)=⋂j=1d{xj≤θj or xj>θj}where θj are the learned split thresholds for each feature.

The Radial Basis Function (RBF) kernel SVM solves the dual optimization problem: (10)maxα∑i=1nαi−12∑i,jαiαjyiyjK(xi,xj)s.t.0≤αi≤C,∑iαiyi=0,where

αi∈R+ are Lagrange multipliers encoding support vector importance

with the RBF kernel: (11)K(xi,xj)=exp⁡(−γ‖xi−xj‖2),where

γ=12σ2 controls kernel width (inverse squared standard deviation)

The decision function is: (12)f(x)=sgn(∑i∈SVαiyiK(xi,x)+b),where

SV denotes the set of support vectors (αi>0)

An ensemble of B=100 decision trees with bagging: (13)f^(x)=majority vote{fb(x)}b=1B,where

B is the number of trees in the ensemble (set to 100 via cross-validation)

Each tree fb is trained on a bootstrap sample with:

Random feature selection at each split (typically 30≈5 features considered)

(14)G(Q)=∑k=12pk(1−pk),pk=1|Q|∑(xi,yi)∈QI(yi=k)where Q is the data subset at each node and I(⋅) is the indicator function

Maximum tree depth determined by early stopping (typically 10–15 levels)

Key properties:

Out-of-bag (OOB) samples used for unbiased error estimation

Here’s the enhanced XGBoost subsection with detailed explanations of all terms and mechanisms:

The gradient boosted trees model minimizes the following regularized objective function: (15)ℒ(ϕ)=∑i=1nl(yi,y^i)+∑k=1KΩ(fk),where

l(yi,y^i) is the loss function measuring prediction error: (16)l(yi,y^i)=−[yilog⁡(σ(y^i))+(1−yi)log⁡(1−σ(y^i))]with σ(⋅) as the logistic sigmoid function for binary classification

Key optimization features:

Tree Construction: Uses weighted quantile sketch for approximate greedy algorithm

Implementation details for breast cancer prediction:

Early stopping with 50-round patience on validation AUC

The shallow (2 hidden layers) and deep (4 hidden layers) networks implement: (22){h1=ReLU(BN(W1x+b1))h2=ReLU(BN(W2h1+b2))(shallow network)hl=ReLU(BN(Wlhl−1+bl))(for l=3 to 5 in deep network)P(y=1|x)=softmax(WLhL−1+bL)where

ReLU(z)=max(0,z) is the Rectified Linear Unit activation function

with layer dimensions: •

Shallow Network (2 hidden layers): (23)d→128→64→2

All models use the categorical cross-entropy loss with L2 regularization: (25)ℒ=−1n∑i=1n∑c=12yi,clog⁡pi,c+λ∑l=1L‖Wl‖F2where

n is the number of training samples

Key implementation details:

Batch normalization after each hidden layer

The fractional-order model in (26) is constructed and informed by the machine learning algorithm and results in Section 3. The optimized parameters are obtained from fitting the model to the dataset. The system describes interactions between three key tumor features identified in the feature importance analysis (Table 3):

(26){Dtα0cR(t)=a1R(t)(1−R(t)K)−b1R(t)T(t)1+T(t)+c1C(t)1+C(t)(Radius growth)Dtα0cT(t)=a2T(t)(1−T(t)K/2)−b2R(t)T(t)1+R(t)+c2C(t)1+C(t)(Texture dynamics)Dtα0cC(t)=a3C(t)R(t)1+R(t)−b3C2(t)1+C(t)−c3R(t)T(t)C(t)(1+R(t))(1+T(t))(1+C(t))(Concavity evolution)where Dtα0c denotes the Caputo fractional derivative of order α∈(0,1], R(t) represents tumor radius (mm), T(t) describes tumor texture (gray-scale variation), and C(t) quantifies concavity score (contour irregularity). The Dtα0c of order α∈(0,1] is given by [42,43]: Dtα0cg(t)=1Γ(1−α)∫0tg′(τ)(t−τ)αdτ,where g(t) is an absolutely continuous function, and Γ(⋅) is the gamma function.

The Riemann-Liouville integral and Caputo derivative satisfy the following fundamental relation [44,45]: Itα0RL(Dtα0cg(t))=g(t)−∑k=0n−1g(k)(0)tkk!,n−1<α≤n,n=⌈α⌉,where Itα0RL is the Riemann-Liouville fractional integral, defined as: (27)Itα0RLg(t)=1Γ(α)∫0tg(τ)(t−τ)1−αdτ,α∈(0,1].

The Caputo operator is chosen because it accommodates classical initial conditions and aligns with biological systems where memory effects depend on past states.

Definition 1. The solution space is 𝒳=C([0,T],R+3) equipped with the weighted norm: ‖(R,T,C)‖𝒳=supt∈[0,T](e−λt[|R(t)|+|T(t)|+|C(t)|]),λ>0

where R+3={(R,T,C)∈R3:R,T,C≥0}.

This weighted norm construction ensures biological feasibility by enforcing non-negativity of tumor characteristics (radius, texture, concavity), a critical constraint since negative values would lack physical meaning in cancer progression. The exponential weighting accounts for clinical observation that recent tumor states disproportionately influence current growth rates.

Lemma 1. For α∈(0,1) and f∈AC([0,T]), the Caputo derivative satisfies: 1.

Semi-group property: Dtα0c(Itα0RLf)(t)=f(t)

These operator properties mirror fundamental cancer behaviors: the semi-group property reflects self-consistent tumor evolution, the fundamental theorem ensures measurable changes from baseline pathology, and the comparison principle permits ranking of tumor aggression levels, analogous to histological grading systems.

Theorem 1 (Existence, Uniqueness, and Positivity). For the fractional tumor system in (26), with initial data (R0,T0,C0)∈R+3 and parameters

ai,bi,ci>0 for i=1,2,3,

there exists a unique, non-negative solution (R,T,C)∈𝒳 for all t≥0.

Proof. Define the nonlinearities fi as the right-hand sides of (26). For (R,T,C),(R′,T′,C′)∈Bρ⊂𝒳 such that Bρ={x∈𝒳|‖x‖≤ρ,ρ>0}. Since 1/(1+R)(1+R′),1/(1+T)(1+T′),1/(1+C)(1+C′)<1, we have |f1(R,T,C)−f1(R′,T′,C′)|≤(a1+2a1ρK+b1ρ(1+ρ))|R−R′|+b1ρ|T−T′|+c1|C−C′|≤η1(|R−R′|+|T−T′|+|C−C′|),where η1=max{a1+2a1ρK+b1ρ(1+ρ),b1ρ,c1}. The other components satisfy similar bounds: |f2(R,T,C)−f2(R′,T′,C′)|≤b2ρ|R−R′|+(a2+4a2ρK+b2ρ(1+ρ))|T−T|+c2|C−C′|≤η2(|R−R′|+|T−T′|+|C−C′|),where η2=max{b2ρ,a2+4ρa2K+b2ρ(1+ρ),c2}. |f3(R,T,C)−f3(R′,T′,C′)|≤(a3ρ+c3ρ2+2c3ρ3+c3ρ4)|R−R′|+(c3ρ2+3c3ρ3)|T−T′|+(a3ρ(1+ρ)+b3ρ(2+ρ)+c3ρ2+2c3ρ3+c3ρ4)|C−C′|≤η3(|R−R′|+|T−T′|+|C−C′|),where η3=max{a3ρ+c3ρ2+2c3ρ3+c3ρ4,c3ρ2+3c3ρ3,a3ρ(1+ρ)+b3ρ(2+ρ)+c3ρ2+2c3ρ3+c3ρ4}. Using Lemma 1, convert the given system to the following Volterra form: R(t)=R0+1Γ(α)∫0tf1(R,T,C)(t−s)1−αds,T(t)=T0+1Γ(α)∫0tf2(R,T,C)(t−s)1−αds,C(t)=C0+1Γ(α)∫0tf3(R,T,C)(t−s)1−αds.

Define the operator ℱ:𝒳→𝒳 componentwise via the integral formulation. For λ sufficiently large: ‖ℱ(R,T,C)−ℱ(R′,T′,C′)‖𝒳≤LλαΓ(α)‖(R,T,C)−(R′,T′,C′)‖𝒳where L is the Lipschitz constant such that L=max{η1,η2,η3}. Choose λ>(L/Γ(α))1/α to ensure contraction.

For non-negative initial data, the right-hand sides satisfy:

f1(0,T,C)=c1C1+C≥0,

By the fractional comparison principle (Lemma 1), solutions remain non-negative.

Using the barrier method with: Dtα0cR≤a1R(1−R/K) ⟹ R(t)≤max(R0,K),and similar bounds for T(t), C(t), solutions cannot blow up in finite time. □

The contraction mapping guarantees clinically observed tumor trajectory stability, where small measurement errors don’t drastically alter predictions. The non-negativity preservation aligns with radiologically measurable tumor features, while the barrier method’s carrying capacity bound (K) models physiological space constraints in breast tissue

Theorem 2 (Global Asymptotic Stability of Tumor Progression Model). For the fractional-order system (26) with parameters satisfying: 1.

ai,bi,ci>0 for i=1,2,3,

there exists a unique positive equilibrium E∗=(R∗,T∗,C∗) that is globally asymptotically stable in Ω for all α∈(0,1].

Proof. The equilibrium points satisfy the nonlinear algebraic system: (28)0=a1R∗(1−R∗K)−b1R∗T∗1+T∗+c1C∗1+C∗ (29)0=a2T∗(1−T∗K/2)−b2R∗T∗1+R∗+c2C∗1+C∗ (30)0=a3C∗R∗1+R∗−b3C∗21+C∗−c3R∗T∗C∗(1+R∗)(1+T∗)(1+C∗)

Eq. (30) can be rewritten as: C∗[a3R∗1+R∗−b3C∗1+C∗−c3R∗T∗(1+R∗)(1+T∗)(1+C∗)]=0

The nontrivial solution satisfies: a3R∗1+R∗=b3C∗1+C∗+c3R∗T∗(1+R∗)(1+T∗)(1+C∗)

This implicitly defines a function C∗=ϕ(R∗,T∗) that is continuously differentiable by the Implicit Function Theorem. Substitute C∗=ϕ(R∗,T∗) into (28) and (29): F1(R∗,T∗)=a1(1−R∗K)−b1T∗1+T∗+c1ϕ(R∗,T∗)R∗(1+ϕ(R∗,T∗))=0F2(R∗,T∗)=a2(1−T∗K/2)−b2R∗1+R∗+c2ϕ(R∗,T∗)T∗(1+ϕ(R∗,T∗))=0

The Jacobian determinant of (F1,F2) is non-zero in Ω when K>max(2a2/b2,a3/b3), guaranteeing a unique solution by the Inverse Function Theorem. Define the Volterra-type Lyapunov functional: (31)V(t)=Itα0RL[w1(R−R∗−R∗ln⁡RR∗)+w2(T−T∗−T∗ln⁡TT∗)+w3(C−C∗−C∗ln⁡CC∗)]where wi>0 are weighting parameters to be determined. Using the composition rule for Caputo derivatives: Dtα0cV(t)≤w1(1−R∗R)Dtα0cR+w2(1−T∗T)Dtα0cT+w3(1−C∗C)Dtα0cC=w1(1−R∗R)[a1R(1−RK)−b1RT1+T+c1C1+C]+w2(1−T∗T)[a2T(1−TK/2)−b2RT1+R+c2C1+C]+w3(1−C∗C)[a3CR1+R−b3C21+C−c3RTC(1+R)(1+T)(1+C)]

After extensive algebraic manipulation and application of Young’s inequalities, we obtain: Dtα0cV(t)≤−∑i=13λi(Xi−Xi∗)2−Ψ(R,T,C)where: λ1=w1(a1K−b1ϵ12−b2w2ϵ22w1)λ2=w2(a2K/2−b1w12w2ϵ1−c3w3ϵ32w2(1+C∗))λ3=w3(b3−c1w12w3ϵ4−c2w22w3ϵ5−c3R∗2w3ϵ3(1+R∗)(1+T∗))

Here ϵi>0 are small constants chosen such that all λi>0 under the theorem’s parameter conditions, and Ψ contains higher-order terms that are positive definite.

Applying the fractional comparison principle and the properties of Mittag-Leffler functions: V(t)≤V(0)Eα(−κtα)+μκ(1−Eα(−κtα))where κ=min(λi)>0 and μ bounds the residual terms. As t→∞, Eα(−κtα)→0 exponentially, proving global convergence to E∗. □

The equilibrium conditions (28)–(30) encode balanced tumor microenvironments. The exponential convergence rate (κ) via Mittag-Leffler decay provides a theoretical basis for clinical timelines, where higher κ values predict faster stabilization post-treatment, and lower value characterizes aggressive tumors requiring prolonged therapy, matching chemotherapy response patterns.

The numerical implementation of our fractional-order tumor model employs a predictor-corrector scheme adapted for Caputo fractional derivatives, implemented in MATLAB R2021a. The complete computational framework consists of the following components: Discretization Scheme:

Predictor Step: (32){yk+1P=y0+1Γ(α)∑j=0kbj,k+1f(tj,yj)withbj,k+1=hαα[(k+1−j)α−(k−j)α]

Corrector Step: (33){tk=t0+kh,k=0,1,…,Nwithh=tfinal−t0Nyk+1=y0+hαΓ(α+2)[f(tk+1,yk+1P)+∑j=0kaj,k+1f(tj,yj)]whereaj,k+1={kα+1−(k−α)(k+1)α,j=0(k−j+2)α+1+(k−j)α+1−2(k−j+1)α+1,1≤j≤k1,j=k+1