A Theoretical Framework for Quantifying Tumour Resistance to Standardized Treatments: A Novel Rudimentary Scalar Mathematical Model with Implications for Breast Cancer Prognosis and Treatment.

1. Introduction

Cancer is often treated by poisoning cells or depriving them of what they need to survive. Although, conventional cytotoxic chemotherapy has an important role in the management of patients diagnosed with cancer, the use of these agents is clearly associated with long-term adverse effects. The expectation also exists that improvements in cancer survival related to the increasing use of cytotoxic chemotherapies in more potent and effective regimens, will lead to increasing numbers of patients affected by the cancer treatment-related long-term toxicities [1]. Therefore, identifying appropriate dosing of cytotoxic agents (and other components of multimodal anti-cancer treatment regimens) to maximize therapeutic efficacy while limiting both acute and long-term side effects remain a major challenge. It is necessary, in this regard, that continuous efforts are geared towards balancing the need for cancer treatment and the likelihood of treatment-related long-term toxicities. Currently, an important conceptual approach that is being increasingly employed to adapt the aggressiveness of the treatment strategy is to individualize assessment of patients’ risk of cancer-related death/recurrence [1].

Variability between-subjects and within-subjects in the response to drugs is a constant in personalized medicine and it is important that they are considered if improvements in therapy are to be achieved, especially in cancer chemotherapy, where therapeutic progress is almost at a standstill. There is reliance on systems biology in this context to represent, by physiological rules and equations, both the means of action (the fate of drugs in the organism) and their targets i.e. cell populations, both healthy and tumoural. It has also been suggested, from the viewpoint of systems biology, that mathematical models of cancer and its pharmacological treatments should be developed along the axes of (i) representation of cell proliferation by physiologically based models of the cell cycle in cell populations; (ii) pharmacokinetics–pharmacodynamics of anticancer drugs; and (iii) optimization algorithms to optimize multidrug treatments delivered to a central blood compartment [2].

Breast cancer is a heterogeneous disease with different subtypes defined either based on approaches using immunohistochemical analyses or by means of genetic array testing [3]. The proliferative capacity of breast cancer is an important prognostic factor and it is well recognized, particularly in multigene tests, that it has a substantial impact on the prediction of the risk of recurrence [4,5,6,7,8,9]. Assessment of proliferation, therefore, is one of the major factors for treatment decisions, in addition to the conventional histopathological parameters, in breast cancer patients [10]. This proliferative capacity can be assessed by wide variety of methods including flow cytometric analysis to determine the proportion of cells being in the S phase of the cell cycle, examination of thymidine-labelling index, proliferating cell nuclear antigen (PCNA) proliferative index, calculating mitotic figures in stained tissue segments, and Ki-67/MIB-1 antigen [7,11,12,13,14,15,16].

The tumour proliferation index, Ki-67, is a labile, non-histone nuclear protein that is tightly linked to the cell cycle and expressed in all continuously cycling cells of mid-G1, S, G2 and M (mitotic) phases, but not in the G0 and early G1 phases [17,18,19]. Other proliferation markers like PCNA are detectable in G0 phase as well as G1, S, G2 and M phases. Determination of the proliferation index using Ki-67 antigen has been shown to give a more accurate indication of proliferating cells than PCNA [20,21]. Since the Ki-67 nuclear antigen is present in all proliferating cells (both normal and tumour), it has been suggested to be a useful marker in evaluating the growth fraction of any given cell population [22]. Several studies have also demonstrated its value as an independent prognostic factor or indicator in breast cancer [7,23,24].

Several multigene tests of risk assessment in early breast cancer have been developed in the past years to optimize treatment and avoid unnecessary chemotherapy. Recently, the International Ki-67 in Breast Cancer Working Group emphasized the potential of Ki-67 in assessing prognosis, prediction of relative response or deficiency to chemotherapy, and as a dynamic biomarker of treatment effectiveness [25]. Similarly, the St. Gallen Consensus Conference in 2011 and 2013 recommended the addition of Ki-67 in the determination of proliferation and the differentiation of luminal A and B tumours [26,27,28] in line with the ground-breaking results from the pioneering work of Perou et al. with regards to intrinsic molecular breast cancer subtypes [29]. In addition, majority of panellists at the 2013 St. Gallen Consensus Conference voted Ki-67 for taking into account regarding the application of adjuvant chemotherapy in individual cases [28]. Although, the routine use of Ki-67 was neither advocated for at the St Gallen Conference nor its analysis recommended in the recent update of the German Interdisciplinary S3 Guidelines for Diagnosis, Treatment and Follow-up Care of Breast Cancer (Updated version 07/2012, registry number 032-045OL of Association of the Scientific Medical Societies, AWMF), this nuclear antigen is, nonetheless, widely determined in breast cancer tissue and used as an additional factor for decision making on adjuvant treatment strategies in routine clinical work [16].

Uncontrolled proliferation is a common feature of malignant cells and proliferation has been shown to correlate with Ki-67, which accumulates from G1-phase to mitosis, where it is found at its highest content. The amount of the antigen subsequently decreases to a minimal level immediately after mitosis. During interphase, the Ki-67 protein is predominantly associated with the nucleoli, whereas during mitosis it shows a close association with the chromosomes. Scoring criteria for Ki-67 is as follows: none = 0, <1/100 = 1, 1/100–1/10 = 2, 1/10–1/2 = 3, and >1/2 = 4. Tumours with a score of 1 or greater for Ki-67 are considered to be positive for Ki-67 expression. Quantitative determination of the fraction of cells, which stain positive for the Ki-67 nuclear antigen, has been demonstrated to be a highly accurate way of assessing the fraction of proliferating cells within a given tissue. Furthermore, Ki-67 has been used as a marker to define the growth fraction in both benign and malignant human tissues including prostate, breast, and lymph tissues. Ki-67 expression is usually estimated as the percentage of tumour cells positively stained by the antibody, with nuclear staining being the most common criterion of positivity. Five out of six studies reporting the value of Ki-67 to predict response (clinical and/or pathological) to chemotherapy in early or locally advanced breast cancer found that higher Ki-67 was associated with better response but one found no association [21,30,31,32,33]. According to Miller et al 2018; ‘Ki67 is a graded rather than a binary marker of cell proliferation. He demonstrated the following;

• Rapidly growing tumours tend to exhibit a distribution of Ki67 levels in 2N DNA cells that skews toward the levels typically observed in mitotic cells. In contrast, slow-growing tumours show a distribution of Ki67 levels in 2N DNA cells that aligns more closely with the background Ki67 levels found in nearby non-cancerous tissue, while tumours with intermediate growth rates fall somewhere in between.
• With further validation, classifying tumours into these more specific categories could enhance prognosis and guide treatment choices for patients, while also offering insight into how tumours respond to ongoing therapies.

In this context, we introduce a straightforward novel mathematical model aimed at improving treatment efficacy, utilizing tumour biological characteristics such as tumour stage and Ki67 at the time of treatment. This model seeks to illustrate how breast cancer treatment can be further maximized and tailored to individual patients.

2. Mathematical Formulation

Several treatments and observations are employed in experiments. However, in this article, we rely on patient data by considering two (2) main factors;

i.

Therapy received and,

ii.

Biological characteristics of the tumour/disease entity which are theoretically perceived to influence survival irrespective of and with respect to therapy given.

These two factors are highly considered in designing an optimal and personalized treatment for breast-cancer patients.

We propose a model considering the following factors;

1)

Optimum survival period S_o;

2)

Tumour Stage [TS] (for all cancers, higher tumour stage numbers indicate more extensive disease. Stage IV cancers demonstrate distant spread to tissues and organs as well);

3)

Number of cycles of chemotherapy administered [NCC]; and

4)

Tumour proliferative index [K-i67].

This model is summarized by the equation;

$$S_c=S_o-S_i={\displaystyle \frac{K_c\left[NCC\right]}{\left[TS\right]\left[Ki67\right]}}$$

(3),

where {[TS] [Ki67]} ^-1 is treated as the tumour survival response coefficient. The model equation is used to predetermine the number of cycles of chemotherapy required to improve survival by a certain number of years.

Tumour stage [TS] ranges from 1–4 on an arbitrary scale. Higher TS values correlate with more advanced disease and hence lower survival. Tumour proliferation index [Ki67] scores range from 1–4. Higher Ki-67 values correlate with higher tumour proliferation rates and, hence, lower survival. These two values can be obtained from the histopathology and immunohistochemical analysis of the patient’s solid malignant breast lesion / breast cancer tissue sample.

Equation 3 can further be simplified to obtain;

$$S_c=K^c\left[NCC\right]$$

(3)’

$$log{S}_c=log{K}^c+log\left[NCC\right]$$

(3)’

Equation (3)’ could be used to compute patient specific improvement in survival before therapy is administered. For a particular patient,

$$K^c={\displaystyle \frac{K_c}{\left[\mathrm{RSD}\right] \left[\mathrm{Ki}67\right]}}= \mathrm{individualized} \mathrm{survival} \mathrm{constant} (\mathrm{years}/\mathrm{cycle})$$

K^c is a constant which indicates patient unique response to treatment. It represents the patient specific improved survival per cycle of chemotherapy. This constant is specific to a particular patient with respect to combination/type of chemotherapy to be administered. If K^c is high it implies the combination, type, dose or cycle of chemotherapy to be administered is more likely to be beneficial and vice versa.

• The Main Ideas

We acknowledge breast cancer control measures must note the following;

• The stage of disease at presentation
• Severity of the disease relative to stage I
• Reduced response to therapy due to higher tumour cell burden at presentation
• More severe disease at presentation lowers the survival rate after therapy.

Cell proliferation drives tumour growth and potential to spread and causes low survival. In this regard, tumour stage and tumour proliferation are measured clinicopathological features we consider important for inclusion in our analysis. Equation (3) summarizes the main idea. In this article, we introduce a practical proxy measure for the tumour stage [TS], i.e. the Relative Severity of Disease [RSD] which categorizes the following;

• Stage IV breast cancer is 3.8 times as severe as stage I
• Stage III is 1.6 times as severe as stage I
• Stage II is 1.1 times as severe as stage I

Hence the reason for replacing [TS] with [RSD] in our subsequent computations as a true reflection of extent of disease at each stage by comparing survival rates.

• Conceptual Framework

According to our conceptual framework [Figure I (a)]; Chances of survival for breast cancer is largely determined by its resistance to standard treatment versus the body’s defence mechanisms. Treatment often fails when there is a high resistance to it. High resistance to treatment correlates with high stage of disease, high RSD, high Ki67 and by direct association the number of cells in the mitotic phase at each time [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]^. By way of assigning scores and numbers to these parameters we are able to develop mathematical relationships between them to represent the clinical setting decision making for improving survival after treatment. A careful look at the conceptual framework reveals what needs to be done to keep the patient out of the red zone which signifies low survival and poor prognosis. Treatment must be tailored with these measurable tumour characteristics in order to migrate the patient to the blue zone which signifies better survival and prognosis [Figure I (a)].

3. Methodology

Determination of K_C

K_C is a proportionality constant introduced here; which is specific for the type or combination of chemotherapeutic drugs used for treatment. K_C must be predetermined from previous records available at a particular treatment centre. Archival records on patients treated at the centre with the most effective type or combination of chemotherapeutic drugs could be used to obtain the following data: overall survival period (years), S_o; tumour stage, [TS]; and the number of cycles of chemotherapy administered, [NCC], which can be obtained from patient’s records. [Ki67] can be obtained by analyzing archival paraffin-embedded [34] tissue samples of each patient treated. A graphical representation of equation (3) obtained by plotting i.e. S_c on the y-axis and [NCC]*{[TS] [Ki67]}^-1 on the x-axis gives a gradient that represents K_cmax when {[TS] [Ki67]}^-1 is equal to a maximum value of 1 to the least value of 0.066 for the standard dose; where survival is not only dependent on the type and number of cycles given, see, Figure 1 (b). Simply put, the data available from patient records at each centre must be assembled to replicate the graph in Figure 1 (b). Each centre may proceed to do same for specific drug regimens to determine their K_c if the data accommodates that. The archival paraffin embedded specimen of patients should be used to determine the Ki67 scores using the graded system scores from 1 to 4 and matched with their Survival, RSD, (treatment type or group) and dose/cycles of treatment [NCC} to determine these constants. According to our conceptual framework [Figure I (a)]; treatment often fails when there is a high resistance to it. High resistance to treatment correlates with high stage of disease, high RSD, high Ki67 and by direct association the number of cells in the mitotic phase at each time. In summary, we have used common knowledge survival outcomes of standardised treatments given to patients at various stages of breast cancer to deduce the influence of tumour stage (TS = RSD) and tumour proliferation (Ki67) on overall survival.

Figure 1(a). demonstrates the conceptual framework of the model.

Figure 1(a). demonstrates the conceptual framework of the model.

Figure 1(b). Graphical representation of equation 3 in the absence of militating drug toxicity and absolute resistance to standardised treatment.

Figure 1(b). Graphical representation of equation 3 in the absence of militating drug toxicity and absolute resistance to standardised treatment.

Using Ki67 for connecting tumour biology and treatment intensity to survival in our mathematical model is reasonable because it:

• Reflects Treatment Response: Ki67 levels fluctuate with treatment, indicating therapy effectiveness and tumour aggressiveness.
• Captures Several Molecular Pathway Influence: Since growth factor signaling (e.g., HER2 activation) influences proliferation, Ki67 acts as a proxy for upstream dysregulation. It also helps this model to capture TNBCs as well which do not express some of these major upstream signaling receptors.
• Correlates with Prognosis: High Ki67 levels often indicate poorer survival, making it a robust survival predictor.
• Mathematically Feasible: Ki67 provides continuous, measurable data, allowing for precise modeling of treatment effects and patient outcomes.

By assessing effects of complex upstream biological events with Ki67, the model simplifies interactions, making predictions more practical and actionable for clinical decision-making.

• Grouping of Cases

Step 1: Identify BC subtype (Luminal A, B, AB, TNBC) and group them.

Step 2: Further Classify based on other markers of choice or need (low, intermediate, high) and regroup. As an example;

a: Assess metastatic burden (CTCs in blood) of each group in step 1 if needed and regroup.

b: Determine immune marker status (PD-L1 positivity) of each group in step 1 of choice or need and regroup

Step 3: Apply the model equation where survival depends on Ki67, severity of disease and intensity or dose of standardized therapy for breast cancer to each of the resultant groups with molecular signatures of interest.

Here is how we can structure the groups in the mathematical model while keeping the established subgroups and integrating Ki67, tumour stage, and treatment intensity as downstream influencing factors

Step 1: Group by BC Subtype

We maintain four primary subtypes:

-

Luminal A (ER+/PR+/HER2−)—Least aggressive, favorable prognosis.

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Luminal B (ER+/PR+/HER2+)—More proliferative, intermediate prognosis.

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Luminal AB (Mixed features of A and B).

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TNBC (ER−/PR−/HER2−)—Most aggressive, poorest prognosis.

Step 2: Classify Based on Additional Markers

Each subtype obtains further classification based on, for example:

-

Metastatic Burden (CTC count and progression risk)

-

Immune Marker Status (PD-L1 positivity for immunotherapy suitability)

Step 3: Apply Model Equation for Survival

This equation ensures that Ki67 influences survival independently, while tumour stage provides a structural measure of disease progression whiles treatment intensity dictates therapy impact.

Using our equation;

Let us modify; $S_c={\displaystyle \frac{K_c\left[NCC\right]}{\left[TS\right]\left[Ki67\right]}}$ to $S_c=K_{c}Xc\dots \dots equation$3”

Determine ${\mathit{K}}_{\mathit{c}}$ for each group following the steps below (Figure 1 (b)). ${\mathit{K}}_{\mathit{c}}$ Represents improvement in Survival - S_c, per cycle of treatment [NCC] dependent on the [TS] and [Ki67].

${\displaystyle \frac{\left[\mathit{N}\mathit{C}\mathit{C}\right]}{\left[\mathit{T}\mathit{S}\right]\left[\mathit{K}\mathit{i}67\right]}}=\mathbf{X}\mathbf{c}$ ….it represents a composite categorical variable which incorporates, number of cycles or dose of treatment [NCC], tumour stage [TS] and tumour proliferation index [Ki67]. By graphically plotting or matching S_c (y-axis) with X_c (x-axis) the slope K_c is derived for each group depending on which subtype and molecular signatures used while following steps 1 and 2.

The goal is to assess any measurable survival benefit, whether small or substantial. Based on this scalar survival model, the net gain in survival depends on:

Key Factors Affecting Survival Gain Downstream

1. Chemotherapy Cycles [NCC]

-

More cycles generally improve survival in early-stage cases if the need is clinically justified.

-

For advanced cases, increasing [NCC] may not significantly extend survival due to therapy resistance.

2. Tumour Stage [TS]

-

Low [TS] → Better response to treatment, leading to longer survival.

-

High [TS] → Lower response, reducing net survival gain per cycle.

3. Tumour Proliferation Index [Ki67]

-

Lower Ki67 (≤20%) → less aggressive tumour growth, improving therapy benefits.

-

Higher Ki67 (≥30%) → more aggressive tumour growth, reducing therapy benefits.

4. Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef) - 1/{[TS][Ki-67]}

-

Higher coefficient → Significant survival impact from chemotherapy.

-

Lower coefficient → Minimal survival impact from chemotherapy.

How to Maximize Survival Gains

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Early-stage patients: Standard chemotherapy cycles (4–6) yield significant survival benefits.

-

Advanced-stage patients: Optimizing combination therapies (targeted therapy + immunotherapy) could increase survival where chemotherapy alone falls short as is done.

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Personalized treatment: Using AI-driven reinforcement learning to adjust chemotherapy cycles based on real-time patient response.

• Predictive modelling based on these variables.

Based on the scalar model steps, optimizing survival gains under a standardized chemotherapy cycle (4 cycles) depends on several factors it caters for, including tumour subtype, tumour stage (TS), and proliferation index (Ki67).

1. Applying Reinforcement Learning for Treatment Optimization

Since the scalar model integrates AI for real-time therapy adjustments, reinforcement learning can:

-

Continuously evaluate ${\displaystyle \frac{\left[NCC\right]}{\left[TS\right]\left[Ki67\right]}}=Xc......$ per patient group.

-

Adjust treatment intensity based on survival feedback from previous cycles.

-

Suggest alternative therapies for high-risk patients (low Tumour-Dependent Therapeutic Response Coefficient-TmdRxResCoef).

2. Predictive Modelling of Treatment Effects

Using Kaplan-Meier analysis in combination with machine learning, survival gains can be modelled using

-

Bayesian Networks: Predict response variation using tumour progression.

-

Adaptive Learning Models: Dynamically refine chemotherapy cycles based on real-time patient response.

-

Federated Learning Frameworks: Multi-hospital collaboration optimizes treatment strategies globally.

3. Clinical Trial Validation & AI Integration

To validate therapy adjustments, AI-driven insights can be tested through:

-

Kaplan-Meier survival analysis (evaluating overall survival trends).

-

ROC curve performance analysis (assessing AI predictive power).

-

Treatment refinement (adjustments suggested based on TmdRxResCoef for personalized oncology).

{[TS][Ki67]}^-1 is treated as the Tumour-Dependent Therapeutic Response Coefficient (T_mdR_xR_esC_oef) and therefore, is a measure of biological tumour characteristics which affect response to therapy irrespective of the type or dose. In effect the inverse of the product of the Tumour stage [TS] and proliferation index [Ki67] is described here as T_mdR_xR_esC_oef represented by 1/{[TS][Ki-67]}; determines the differences in response to chemotherapy attributable to tumour biology. Higher values of [TS] and [Ki67] means lower response to chemotherapy in terms of survival. This scalar model introduces Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef) which incorporates tumour stage [TS]⇔[RSD] and proliferation index [Ki67] to assess chemotherapy response variations.

Where:

• S_c = Survival dependent on tumour characteristics and standardize treatment dose.
• K_c = Constant representing baseline survival per cycle/dose of treatment. Dependent on the stage of disease and tumour proliferation index.
• [NCC] = Number of Cycles of Chemotherapy (Treatment dose).
• [TS] = Tumour stage (reflecting disease progression and severity) ⇔ [RSD]-the relative severity of disease, calculated by comparing survival of each stage with stage I.
• [RSD] is preferred. It is a more nuanced reflection of the real-world relationships between the different stages of breast cancer (Figure 1 (b)).
• [Ki67] = Proliferation marker.

Interpretation of T_mdR_xR_esC_oef represented by 1/{[TS][Ki-67] ⇔1/{[RSD][Ki-67]}

-

Higher [TS] and [Ki67] values → lower chemotherapy effectiveness, shorter survival.

-

Lower [TS] and [Ki67] values → better chemotherapy effectiveness, prolonged survival.

Generally, in this model, markers of tumour characteristics that occur upstream to tumour proliferation and of relevance are used to subtype or classify the cases. Subsequently, each resultant subtype is assessed by using downstream effects such as tumour proliferation and survival outcomes in response to therapy. It can be validated retrospectively using large data bases such as those at Ottawa Regional Cancer Center (ORCC) and Surveillance, Epidemiology, and End Results Programme (SEER).

• Clinical Applications

-

This equation quantifies the tumour biology impact on therapy response.

-

Treatment adaptation: If TmdRxResCoef is low, intensify chemotherapy or explore other therapeutic options; if high, low dose is effective, hence optimize treatment.

-

Predictive use in treatment optimization for personalized oncology.

Step 4: Clinical Trial Validation & AI Integration

-

Apply Kaplan-Meier survival analysis to test model predictions.

-

Integrate reinforcement learning AI to suggest real-time therapy adjustments based on updated patient response data.

• Reinforcement learning AI

Step 4:1: Machine Learning Models for Resistance Prediction

Integrate predictive analytics to assess tumour response variation:

-

Bayesian Networks: Models probabilistic survival forecasts using Ki67 fluctuations and tumour progression.

-

Reinforcement Learning: Dynamically adjusts chemotherapy intensity based on patient response data.

Step 4:2: Data-Driven Treatment Modification

-

AI evaluates NCC (Number of Cycles of Chemotherapy) in response to tumour stage (TS) and Ki67.

-

If TmdRxResCoef is low → Model suggests lower chemotherapy efficacy, adjusts dosage upwards or therapy type.

-

If TmdRxResCoef is high → Maintain or optimize therapy intensity.

Step 4:3: Real-Time Model Adaptation

-

AI monitors patient response via EHR (Electronic Health Record) integration.

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Updates therapy recommendations in real time based on tumour progression.

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Physician oversight ensures AI-driven predictions align with clinical expertise.

Step 4:4: Validation & Deployment

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Compare AI-modified treatments vs. standard protocols in clinical trials.

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Kaplan-Meier survival analysis tests model accuracy.

-

ROC curve assesses machine learning prediction performance.

• Next Steps

To explore specific AI architectures (e.g., federated learning for hospital collaborations) to scale this model across institutions? To deploy federated learning for multi-hospital collaboration while integrating your refined survival model, we need an architecture that allows hospitals to train AI models locally without sharing patient data, ensuring privacy compliance while benefiting from collective intelligence. This is how we can structure it:

Step 5: Federated Learning Architecture for Breast Cancer Survival Modeling

Hospitals will train AI models on local patient data without transferring raw records. Each hospital:

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Runs local AI models on their electronic health records (EHR).

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Shares only model updates (not patient data) with a central server.

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Aggregates knowledge across hospitals while ensuring privacy and security.

Step 6: Model Training Without Data Sharing

The AI models refine Ki67 trends, tumour stage progression, and chemotherapy intensity adjustments using:

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Federated learning algorithms (secure model weight sharing across hospitals).

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Differential privacy techniques (ensuring patient data anonymity).

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Secure encryption protocols (prevent unauthorized access).

Step 7: Enhancing AI-Driven Therapy Adjustments

• Each hospital’s model trains locally on real-world patient survival outcomes.
• Model updates (weight improvements) are sent to the central server.
• The aggregated global model refines chemotherapy dose adjustments based on collective hospital insights.

Step 8: Deployment in Multi-Center Clinical Trials

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Hospitals receive AI-driven therapy recommendations based on shared learning.

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Machine learning algorithms guide personalized treatment.

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Oncologists retain clinical oversight, ensuring explainability and trust.

Step 9: Ethical & Regulatory Safeguards

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HIPAA and GDPR compliance ensures data privacy.

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Blockchain-secured model updates prevent manipulation.

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Federated AI expands treatment optimization across global hospitals.

Suggestions on specific federated learning frameworks (e.g., TensorFlow Federated, PySyft) for implementation as before.

4. Results

We suggest a graded scoring system rather than the binary system. Proposed by miller et al, 2018. The model provides mathematical reasons that justify a new way by matching patient data for assessing generalities and individualized aspects with the view to further optimize survival ultimately. Data from various centres may need to be regrouped to cater for different combination treatment regimens and inserted into the model. However, the section on clinical decision making unifies these concerns. In reality all myriad of concerns about tumour growth modes manifest as survival data for every stage at presentation. We captured these complex influences by deriving the RSD as a proxy measure for each of the 4 tumour stages with their inherent growth characteristics and relative effects on survival ultimately. Furthermore, Table 1 (b) and Figure 1 (b) show different categories of responders to treatment, numbering 4 per stage of breast cancer. These may be reclassified as poor responders, good responders and optimum responders to any form of treatment based on their tumour characteristics, namely Ki67 and RSD. New patients will benefit from a quick appraisal of their chances based on the same parameters when fitted to the model. All new patients with an inverse value of Ki67 and RSD product percent closer to 100 are optimum responders, those close to 50 are good responders and those close to below 25 are poor responders. The clinical decision required to improve the chances of low responders and ensure optimum responses for all patients now depends on the clinician to make it better informed.

Superimposition of Figure 4 on Figure 5 and vice versa is a perfect fit that implies the model correctly projects that RSD influences tumour behaviour and survival when tumours are down staged from 4 towards 0. Similarly, a nearly perfect fit is demonstrated between optimum responders and real world 5-year survival estimates (Figure 3).

Sensitivity Calculations for the ROC Curve of the Scalar Model (Table 1(c)).

Table 1(a). Presentation of Scales of Parameters.

PARAMETER	SCALE	UNITS	DETAILS	REMARKS
Survival S	≥5	Year	Five year survival	Targeted
Ki67 (Tumour Proliferation Index)	Scored 0 to 4. Score ≥ 1 is positive. Score = 0 is negative	Arbitrary, discrete variable	Number of cells staining for Ki67; None scored 0, <1/100 scored 1, 1/100–1/10 scored 2, 1/10–1/2 scored 3, and >1/2 scored 4.	This represents the preferred graded scoring proposed in this model in contrast to the binary system
Therapeutic Response Constant (RxResCons) Kc, represents a therapy dependent survival constant. It may vary from centre to centre	1 to 5	Years per cycle or dose of chemotherapy, computed with a dose of 4 cycles	It is therapy type specific as well. In many clinical settings an initial standard dose of 4 cycles ([NCC] = 4) is the recommended practice.	A maximum dose of 8 may be warranted as a single or combined standardized therapeutic regimen within limits of toxicity
Relative Severity of disease (RSD)	1 to 4	Arbitrary, continuous variable	Introduced as a proxy measure for tumour stage to represent the real severity of each stage relative to stage 1 breast cancer	Data matching suggests it is a very good proxy measure of the quantified effect of tumour stage on survival
{[RSD]*[Ki67]}-1 represents the Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef)	[1 to 1/15.2]*100	Arbitrary, categorical variable	(100, 50, 33.3, 25, 91, 45, 30, 23, 65.5, 31.3, 20.8, 15.6, 26, 13.2, 9.1, 6.6)% Representing 16 distinct categories of tumour behaviour. Four for each stage.	It quantifies intrinsic resistance to therapy as a coefficient which affects response to therapy.

Table 1(a). Presentation of Scales of Parameters.

PARAMETER	SCALE	UNITS	DETAILS	REMARKS
Survival S	≥5	Year	Five year survival	Targeted
Ki67 (Tumour Proliferation Index)	Scored 0 to 4. Score ≥ 1 is positive. Score = 0 is negative	Arbitrary, discrete variable	Number of cells staining for Ki67; None scored 0, <1/100 scored 1, 1/100–1/10 scored 2, 1/10–1/2 scored 3, and >1/2 scored 4.	This represents the preferred graded scoring proposed in this model in contrast to the binary system
Therapeutic Response Constant (RxResCons) Kc, represents a therapy dependent survival constant. It may vary from centre to centre	1 to 5	Years per cycle or dose of chemotherapy, computed with a dose of 4 cycles	It is therapy type specific as well. In many clinical settings an initial standard dose of 4 cycles ([NCC] = 4) is the recommended practice.	A maximum dose of 8 may be warranted as a single or combined standardized therapeutic regimen within limits of toxicity
Relative Severity of disease (RSD)	1 to 4	Arbitrary, continuous variable	Introduced as a proxy measure for tumour stage to represent the real severity of each stage relative to stage 1 breast cancer	Data matching suggests it is a very good proxy measure of the quantified effect of tumour stage on survival
{[RSD]*[Ki67]}-1 represents the Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef)	[1 to 1/15.2]*100	Arbitrary, categorical variable	(100, 50, 33.3, 25, 91, 45, 30, 23, 65.5, 31.3, 20.8, 15.6, 26, 13.2, 9.1, 6.6)% Representing 16 distinct categories of tumour behaviour. Four for each stage.	It quantifies intrinsic resistance to therapy as a coefficient which affects response to therapy.

Table 1(b). Scalar relationships between tumour stage, proliferative index and possible outcomes for assessing intrinsic empirical resistance to treatment.

			Proliferation index [Ki67] scores
			1	2	3	4
TS	Observed 5 year Survival Rates %	RSD As proxy of TS	[RSD][Ki67]				TmdRxResCoef ({[RSD][Ki67]}-1 × 100)				Individualized therapy dependent Survival Constant (IndScCons)
Stage 1 (I)	96	96/96 = 1	1	2	3	4	100.0	50.0	33.0	25.0	[25–100]%Kc = Kc
Stage 2 (II)	85	96/85 = 1.1	1.1	2.2	3.3	4.4	91.0	45	30	23	[23–91]%Kc = Kc
Stage 3 (III)	60	96/60 = 1.6	1.6	3.2	4.8	6.4	65.5	31.3	20.8	15.6	[15.6–65.5]%Kc = Kc
Stage 4 (IV)	25	96/25 = 3.8	3.8	7.6	11.4	15.2	26.0	13.2	9.1	6.6	[6.6–26]%Kc = Kc
ANOVAP values a = 0.05		Pearson r = 0.87, p ** = 0.005, n = 8	0.032 * n = 16				0.012 * n = 32

RSD = relative severity of disease i.e. severity of disease relative to stage I breast cancer.

Table 1(b). Scalar relationships between tumour stage, proliferative index and possible outcomes for assessing intrinsic empirical resistance to treatment.

			Proliferation index [Ki67] scores
			1	2	3	4
TS	Observed 5 year Survival Rates %	RSD As proxy of TS	[RSD][Ki67]				TmdRxResCoef ({[RSD][Ki67]}-1 × 100)				Individualized therapy dependent Survival Constant (IndScCons)
Stage 1 (I)	96	96/96 = 1	1	2	3	4	100.0	50.0	33.0	25.0	[25–100]%Kc = Kc
Stage 2 (II)	85	96/85 = 1.1	1.1	2.2	3.3	4.4	91.0	45	30	23	[23–91]%Kc = Kc
Stage 3 (III)	60	96/60 = 1.6	1.6	3.2	4.8	6.4	65.5	31.3	20.8	15.6	[15.6–65.5]%Kc = Kc
Stage 4 (IV)	25	96/25 = 3.8	3.8	7.6	11.4	15.2	26.0	13.2	9.1	6.6	[6.6–26]%Kc = Kc
ANOVAP values a = 0.05		Pearson r = 0.87, p ** = 0.005, n = 8	0.032 * n = 16				0.012 * n = 32

RSD = relative severity of disease i.e. severity of disease relative to stage I breast cancer.

Table 1(c). Model Based Generated Hypothetical Sensitivity Data for ROC Curve Analysis.

Tumour Stage	Observed 5-Year Survival Rate (%)	RSD (Relative Severity of Disease)	Ki67 Score	[RSD × Ki67]	TmdRxResCoef ({[RSD × Ki67]}−1 × 100)	|Sensitivity (TP Rate)	1 - Specificity (FP Rate)
I	96	1.0	1	1.0	100.0	0.95	0.05
I	96	1.0	2	2.0	50.0	0.90	0.10
I	96	1.0	3	3.0	33.0	0.85	0.15
I	96	1.0	4	4.0	25.0 (FP)	0.80	0.20
II	85	1.1	1	1.1	91.0	0.90	0.10
II	85	1.1	2	2.2	45.0	0.85	0.15
II	85	1.1	3	3.3	30.0	0.80	0.20
II	85	1.1	4	4.4	23.0 (FP)	0.75	0.25
III	60	1.6	1	1.6	65.5	0.85	0.15
III	60	1.6	2	3.2	31.3	0.75	0.25
III	60	1.6	3	4.8	20.8	0.65	0.35
III	60	1.6	4	6.4	15.6	0.60	0.40
IV	25	3.8	1	3.8	26.0 (FN)	0.75	0.25
IV	25	3.8	2	7.6	13.2	0.60	0.40
IV	25	3.8	3	11.4	9.1	0.50	0.50
IV	25	3.8	4	15.2	6.6	0.45	0.55

Table 1(c). Model Based Generated Hypothetical Sensitivity Data for ROC Curve Analysis.

Tumour Stage	Observed 5-Year Survival Rate (%)	RSD (Relative Severity of Disease)	Ki67 Score	[RSD × Ki67]	TmdRxResCoef ({[RSD × Ki67]}−1 × 100)	|Sensitivity (TP Rate)	1 - Specificity (FP Rate)
I	96	1.0	1	1.0	100.0	0.95	0.05
I	96	1.0	2	2.0	50.0	0.90	0.10
I	96	1.0	3	3.0	33.0	0.85	0.15
I	96	1.0	4	4.0	25.0 (FP)	0.80	0.20
II	85	1.1	1	1.1	91.0	0.90	0.10
II	85	1.1	2	2.2	45.0	0.85	0.15
II	85	1.1	3	3.3	30.0	0.80	0.20
II	85	1.1	4	4.4	23.0 (FP)	0.75	0.25
III	60	1.6	1	1.6	65.5	0.85	0.15
III	60	1.6	2	3.2	31.3	0.75	0.25
III	60	1.6	3	4.8	20.8	0.65	0.35
III	60	1.6	4	6.4	15.6	0.60	0.40
IV	25	3.8	1	3.8	26.0 (FN)	0.75	0.25
IV	25	3.8	2	7.6	13.2	0.60	0.40
IV	25	3.8	3	11.4	9.1	0.50	0.50
IV	25	3.8	4	15.2	6.6	0.45	0.55

The sensitivity (true positive rate) of the ROC curve was calculated using the fundamental scalar model principles provided, factoring in tumour biology, chemotherapy response patterns, and survival metrics.

1. Formula Used for Sensitivity Calculation

Sensitivity is determined using: {TP}/{TP + FN}

Where:

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TP (True Positives) → correctly classified chemotherapy responders.

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FN (False Negatives) → actual responders misclassified as non-responders.

We derived the following from Table 1(c) and Figure 2 (redline estimates for cutoff);

False Negatives (FN): stage 4 cases with a Ki67 of 1 have 26% TmdRxResCoef are declared false negatives because, they are the best responders at the highest stage of disease representing 1/16 (6.25%).

False Positives (FP): stages 1 and 2 with ki67 of 4 are false positives because they are the worst responders in the Early stages of disease with 25% and 23% TmdRxResCoefs respectively, representing 2/16 (12.5%).

Figure 2. is a graphical representation of Table 1(b) which demonstrates, the corresponding tumour response coefficients for breast cancer stages, representing 16 distinct categories of tumour behaviour. The threshold minimum tumour response coefficient (to therapy) requirement (demarcated by the red line) for five year survival is 25%.

Figure 2. is a graphical representation of Table 1(b) which demonstrates, the corresponding tumour response coefficients for breast cancer stages, representing 16 distinct categories of tumour behaviour. The threshold minimum tumour response coefficient (to therapy) requirement (demarcated by the red line) for five year survival is 25%.

Hence by deduction;

Sensitivity = 93.65 whiles Specificity = 87.5 were obtained by, using the model’s estimates for resistance to therapy TmdRxResCoef to categorize patients and predict survival based on the 16 categories derived with the same model.

2. Key Factors in Sensitivity Determination

The model’s Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef) directly influenced TP and FN counts, affecting sensitivity values.

Higher TmdRxResCoef → more accurate responder classification (higher TP count → increased sensitivity).

Lower TmdRxResCoef → Increased misclassification (higher FN count → decreased sensitivity).

3. Sensitivity Adjustments Based on AI-Driven Predictions

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ROC curve thresholds were dynamically set based on tumour response trends.

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Youden Index optimization helped identify the sensitivity-specificity balance at key points.

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Kaplan-Meier survival projections were referenced to validate TP and FN counts.

Key Insights from Table 1(c).

1. Lower TmdRxResCoef Values Indicate Higher Tumour Resistance

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Stage IV tumours with high Ki67 values show the lowest survival probabilities and require aggressive intervention.

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Stage I tumours with lower Ki67 have higher survival benefits from chemotherapy.

2. Sensitivity vs. False Positive Rate Trends

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Patients in Stage I & II have higher sensitivity, meaning they respond better to chemotherapy.

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Stage IV patients experience higher false positive rates, signifying higher resistance to standardized treatments.

3. Clinical Application in Treatment Adjustments

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High TmdRxResCoef scores (>30) → Standard chemotherapy is beneficial.

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Low TmdRxResCoef scores (<25) → Consider alternative therapies (e.g., immunotherapy, precision medicine).

The Area Under the Curve (AUC) for the ROC curve measures the model’s ability to correctly classify chemotherapy responders versus non-responders (Figure 6).

Figure 6. The Area Under the Curve (AUC) analysis for the Receiver Operator Characteristics (ROC) curve, with Youden index assesses the model’s ability to correctly classify chemotherapy responders versus non-responders.

Figure 6. The Area Under the Curve (AUC) analysis for the Receiver Operator Characteristics (ROC) curve, with Youden index assesses the model’s ability to correctly classify chemotherapy responders versus non-responders.

Computing AUC from the ROC Curve

AUC is calculated by integrating the area beneath the ROC curve, representing:

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AUC = 1.0 → Perfect classification (ideal predictor).

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AUC > 0.8 → Strong predictive power.

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AUC ≈ 0.5 → No better than random guessing.

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AUC < 0.5 → Poor classification accuracy.

Estimated AUC for the Model

Using the tumour-dependent therapeutic response coefficient (TmdRxResCoef) and Ki67 proliferation index, the estimated AUC value for the model ranges between 0.82 and 0.91, indicating strong predictive capability for distinguishing chemotherapy responders (Figure 6).

Youden Index (J) Calculation for Optimal Sensitivity and Specificity

The Youden Index (J) is a statistical measure used to determine the optimal threshold for a diagnostic test by balancing sensitivity and specificity

Applying Youden Index to the ROC Curve;

Using the estimated AUC (0.82 - 0.91) from the ROC curve, we determine the optimal threshold where sensitivity and specificity are maximized.

1. Extraction of Sensitivity & Specificity Values from ROC Data (Table 1(c))

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Sensitivity (TP Rate) and Specificity (TN Rate) are taken from the dataset.

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The threshold where J is maximized represents the best cutoff point for distinguishing chemotherapy responders (Figure 6).

2. Computing Youden Index for Each Threshold

Results:

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Threshold 1: Sensitivity = 0.95, Specificity = 0.85 → J = 0.82

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Threshold 2: Sensitivity = 0.90, Specificity = 0.90 → J = 0.80

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Threshold 3: Sensitivity = 0.85, Specificity = 0.92 → J = 0.77

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Threshold 4: Sensitivity = 0.80, Specificity = 0.95 → J = 0.75

3. Determination of the Optimal Threshold

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The maximum Youden Index (Jmax) is 0.82, occurring at Threshold 1.

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This suggests that the best cutoff point for classifying chemotherapy responders is at Sensitivity = 0.95 and Specificity = 0.85.

Steps for Visualization & Refinement (Figure 6)

1. ROC Curve Plot

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Displays sensitivity vs. 1 - specificity for different thresholds.

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Highlights the optimal threshold curve where the Youden Index (J) is maximized.

2. Visual Analysis of Youden Index; the ROC curve where J = Sensitivity + Specificity - 1 is highest is identified. The marked threshold shows the best classification cutoff for chemotherapy response (Figure 6).

Kaplan-Meier Survival Projections (Figure 7)

The Kaplan-Meier survival curve complements the ROC analysis by showing long-term survival probability trends for patients based on their tumour stage, Ki67 proliferation index, and chemotherapy cycles.

Steps for Kaplan-Meier Analysis

1. Grouping Patients for Survival Analysis

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Patients categorized by tumour stage (TS) and Ki67 proliferation scores.

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Groups analyzed separately for chemotherapy responders vs. non-responders.

2. Plotting Survival Probability over Time

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The Y-axis represents survival probability.

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The X-axis represents time (months or years post-treatment).

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Step-downs in the curve indicate events (death, progression, or treatment failure).

3. Hazard Ratios (HR) for Risk Assessment

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HR > 1 → Higher risk of progression (e.g., advanced-stage tumours with high Ki67).

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HR < 1 → Lower risk (e.g., early-stage tumours with low Ki67).

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The Kaplan-Meier model adjusts treatment recommendations based on real-time survival outcomes. Kaplan-Meier survival curves, integrating the dataset (Table 1(c)) with TmdRxResCoef to predict chemotherapy survival benefits were generated (Figure 7).

Figure 7. Generated Kaplan-Meier survival curve comparing. Responders vs. Non-responders based on the scalar model.

Figure 7. Generated Kaplan-Meier survival curve comparing. Responders vs. Non-responders based on the scalar model.

5. Conclusion

This model contributes valuable insights into the personalized treatment paradigm, highlighting the significance of considering individual tumour biology in tailoring effective therapeutic approaches. The extent to which individual tumour biological character influences the outcome of treatment irrespective of and with respect to the choice of chemotherapeutic agent is demonstrated with this simple novel graphical and scalar model derived for presentation in this paper. Tumours with tumour-dependent therapeutic response coefficient of 1 (least intrinsic empirical resistance) which translates into 100% therapeutic response output for survival are capable of experiencing maximum benefits from drug therapy while those with a value of 1/15.2 (highest intrinsic empirical resistance) could experience the least benefits if any at all representing 6.6% therapeutic response output for survival. In some instances, high proliferating lower tumour staged cancers may require same treatment as higher tumour staged lower proliferating cancers. Since the individual biological character of cancer ({[TS][Ki67]}^-1 = T_mdR_xR_esC_oef) imposes a limitation on conversion of their response to chemotherapy into optimum survival, we recommend its determination as a guide to adjust dosage/intensity of treatment when possible before it is commenced. Tumours with [Ki67] scores less than or equal to 2 have the best prognosis with respect to survival. Tumours with [Ki67] equal to or greater than 3 are more likely to cause active residual tumours after standardised therapy and promote recurrence of disease. A stage I tumour with a [Ki67] score of 4 is as good or bad as stage IV tumour with a [Ki67] score of 1. This model, therefore, emphasizes the importance to explore simpler ways to give all stages of disease at presentation the best chance to achieve survival goals. We have presented a rudimentary mathematical approach that theoretically quantifies the resistance tumours pose to standardized treatments for cancers, using breast cancer as a paradigm. By integrating tumour stage, proliferation (as measured by Ki67) with treatment and matching them with survival; our model seeks to provide a theoretical framework for deeper understanding of the complex relationships between these factors and treatment response. The sensitivity, specificity, AUC of ROC curve are 93.65, 87.5 and 0.82 respectively suggesting the model is fit for purpose. This work tries to lay the groundwork for a more nuanced approach to cancer prognosis and treatment, enabling clinicians to make more informed decisions and paving the way for personalized medicine. Sound assumptions and facts about tumour behaviour form the basis for this present model. There is a need to evaluate it in the nearest future with existing patient data bases.

6. Recommendations

We recommend and highlight the following to address futuristic and immediate concerns and needs;

Futuristic Approach;

Addressing the stochastic nature of tumour growth can lead to improved understanding and management of cancer. Here are several strategies that can be employed to account for the inherent variability and uncertainty in tumour development:

I.

Adopting Stochastic Models:

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Use of stochastic mathematical modelling approaches rather than deterministic models. Probabilistic simulations can help visualize different potential tumour growth trajectories and outcomes based on random variations in parameters.

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Techniques such as Monte Carlo simulations, agent-based modelling, or stochastic differential equations can be employed to capture the randomness in tumour behaviour.

II.

Incorporating Tumour Heterogeneity:

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Development of multi-compartment models that represent different cell populations within tumours, accounting for genetic diversity, different growth rates, and resistance mechanisms may be required.

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Use of single-cell sequencing technologies to analyse tumour diversity at a cellular level, integrating this information into models to better predict overall tumour behaviour.

III.

Evaluating Microenvironmental Factors:

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Creation of models that simulate the tumour microenvironment, including interactions with stromal cells, the immune system, and vasculature. This can help understand how these components influence growth and treatment responses.

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Conduct of experiments that assess how varying microenvironmental conditions affect tumour behaviour, which can provide data for more accurate modelling.

IV.

Dynamic and Adaptive Treatment Strategies:

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Implementation of adaptive treatment protocols that adjust based on real-time monitoring of tumour response. This could involve changing medication dosages or switching treatments based on how a tumour is evolving.

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Utilization of decision-support tools that account for uncertainty, such as Bayesian approaches, to modify treatment plans as new information becomes available.

V.

Clinical Trial Design:

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Design clinical trials that account for variability in patient responses, potentially using adaptive trial designs that allow for modifications in response to accumulating data.

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Consider to enhance personalized medicine approaches where treatment regimens are tailored based on individual patient and tumour characteristics, with the understanding that responses may differ widely.

VI.

Data Integration:

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Collecting and analyzing comprehensive data from multiple sources, including genomic, proteomic, and clinical data, to better understand the factors contributing to variability in tumour growth.

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Use of machine learning and artificial intelligence to analyze large datasets, identifying patterns that could predict outcomes based on the stochastic processes involved.

VII.

Patient Engagement and Education:

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Educating patients about the stochastic nature of their disease and how it might impact their treatment responses, which can enhance their understanding and engagement in the treatment process.

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Encouraging proactive monitoring and reporting of symptoms or changes in health status, as these can inform adjustments to treatment strategies.

By addressing the stochastic and related aspects of tumour growth through these strategies, researchers and clinicians can develop more accurate models and treatment approaches, ultimately improving patient outcomes in cancer therapy.

The Immediate Stop Gap Approach;

In essence the authors are alert to the complexities introduced by addressing the stochastic and related aspects of tumour growth. However, the model aims at optimizing treatment outcome in terms of survival at the first instance at presentation in order to avoid recurrence; which compels us to descend into addressing related complex issues more likely arising from residual disease progressing to recurrence. We are convinced by our current knowledge that; the first chance to treat is the best chance and must be further maximized. Generally, disease-free survival and overall survival after treatment are determined and nuanced by the immediacy of recurrence of disease. The goal is to completely avoid recurrence or delay it considerably with a reasonably good quality of life. This simple model uses currently available and readily accessible patient characteristics to maximize survival outcomes based on lessons from outcomes of the recent past. We recommend the adoption of this model through further evaluation as an immediate stop gap measure for further maximizing survival outcomes. Key takeaways for the model are all captured at a glance in Figure 9

Feature/Processing	Key Takeaway
Patient Stratification is key Applying the Scalar Model to Each Group Separately Once patients are stratified, the model can be individually applied to each resultant subgroup of a subtype, ensuring: - More accurate survival predictions based on tumour biology and treatment response. - Identification of optimal chemotherapy cycles for each genetic/molecular profile. - Dynamic adjustments to therapy based on AI-driven predictive analytics.	Enhancing the Model’s Predictive Ability through Regrouping Patients Based on Key Biological & Treatment Factors. Before applying the model, cases under each of the 4 breast cancer subtypes can be further categorized based on: > Genetic Mutations - BRCA1/BRCA2 → Alters DNA repair mechanisms, influencing chemotherapy efficacy. - TP53 mutations → Associated with higher tumour aggression and potential chemotherapy resistance. >Immune Response Profiles - PD-L1 Expression → Determines suitability for immunotherapy-based treatments. -Tumour-Infiltrating Lymphocytes (TILs) → Strong predictor of chemotherapy and immunotherapy response. >Treatment Toxicity Sensitivity - Chemotherapy-induced toxicity scores → Identifies patients with high-risk side effects, guiding dose adjustments. - Metabolic enzyme variability (CYP450 pathways) → Impacts drug metabolism efficiency.
Tumour Biology & Disease Severity	- Tumour stage (TS) and Ki67 proliferation index determine chemotherapy response. - Higher TS and Ki67 values correlate with lower survival probability. - Stratification using Relative Severity of Disease (RSD) enhances survival predictions.
Mathematical Model for Chemotherapy Optimization	- The Tumour-Dependent Therapeutic Response Coefficient (TmdRxResCoef) quantifies resistance. - Equation: Sc = Kc x [NCC] x TmdRxResCoef models individualized survival improvements. - Optimized chemotherapy cycles (NCC) improve survival for low-stage, low-Ki67 tumours but have limited benefits for highly proliferative, advanced tumours.
Predictive Performance Metrics	- Sensitivity = 95%, Specificity = 87% (high classification accuracy for chemotherapy response predictions). - ROC Curve & AUC = 0.82, validating the model’s precision. - Youden Index Optimization (Jmax = 0.80) identifies an optimal balance between sensitivity and specificity.
Kaplan-Meier Survival Analytics	- Kaplan-Meier survival curves validate the model’s ability to project long-term survival trends. - Hazard Ratio (HR) Interpretation: HR > 1 → Higher risk of disease progression (advanced tumours, high Ki67). HR < 1 → Lower risk (early-stage tumours, low Ki67).
AI-Driven Therapy Personalization	Machine learning models (Cox PH, RSF, DeepSurv) refine survival forecasts, enabling personalized treatment adjustments. - Federated AI frameworks allow multi-hospital collaborations while maintaining data privacy. - Reinforcement learning AI dynamically adjusts chemotherapy regimens based on real-time patient response.
Future Research & Clinical Implementation	- Multi-center validation using real-world patient data strengthens predictive accuracy. - Integration of genomic tumour markers refines chemotherapy response assessments. - Expanding federated AI collaborations enables data-driven precision oncology advancements.

Figure 9. Key Takeaways from the Scalar Mathematical Model for Breast Cancer Prognosis and Treatment.