Gail Continuous Risk Model Visualization

Model Flow Diagram

flowchart TD %% Input Variables A["`**Inputs**
• Age: current_age
• Menarche: age_menarche
• Biopsies: num_biopsies
• First Birth: age_first_birth
• Relatives: num_relatives
• Follow-up: followup_years`"] %% Variable Transformations B["`**Variable Categories**
AM_Cat = cases
NB_Cat = min(biopsies, 2)
NR_Cat = min(relatives, 2)`"] %% First Birth Category C["`**First Birth Category**
AF_Cat = age-based categories
(0=<20, 1=20-24, 2=25-29, 3=30+)
Nulliparous = 2`"] %% Relative Risk Components D["`**Menarche RR**
1.0 (≥14) → 1.21 (12-13) → 1.47 (<12)`"] E["`**Biopsy RR (Age < 50)**
1.0 → 1.11 → 1.23
(0, 1, 2+ biopsies)`"] F["`**Biopsy RR (Age ≥ 50)**
1.0 → 1.34 → 1.8
(0, 1, 2+ biopsies)`"] G["`**Family History RR**
Complex interaction matrix
AF_Cat × NR_Cat
Range: 1.0 - 3.82`"] %% Combined Risk H["`**Combined Relative Risk**
RR_raw = RR_menarche × RR_biopsy × RR_family

RR_adj = RR_raw × AR_factor
AR_factor = 0.5788413`"] %% Survival Functions I["`**Survival from Competing Risks**
S₂(age) = exp(-∫h₂(u)du)

Using NCI competing hazard rates`"] %% Integration J["`**Absolute Risk Integration**
R = ∫ h₁(t) · RR_adj · S₁(t) · S₂(t)/S₂(age) dt

S₁(t) = exp(-∫[h₁(v)·RR_adj + h₂(v)]dv)`"] %% Final Result K["`**5-Year Risk Percentage**
Risk% = R × 100`"] %% Age decision AGE_DECISION{"`Age < 50?`"} %% Flow connections A --> B A --> C B --> D B --> AGE_DECISION B --> G C --> G AGE_DECISION -->|Yes| E AGE_DECISION -->|No| F D --> H E --> H F --> H G --> H H --> I H --> J I --> J J --> K

Mathematical Details

1. Variable Categorization

Age at Menarche Category:

$$AM_{Cat} = \begin{cases} 0 & \text{if menarche} \geq 14 \\ 1 & \text{if } 12 \leq \text{menarche} < 14 \\ 2 & \text{if menarche} < 12 \end{cases}$$

First Birth Category:

$$AF_{Cat} = \begin{cases} 2 & \text{nulliparous} \\ 0 & \text{if } < 20 \\ 1 & \text{if } 20-24 \\ 2 & \text{if } 25-29 \\ 3 & \text{if } \geq 30 \end{cases}$$

2. Relative Risk Calculations

Age-dependent Biopsy Relative Risk:

For age < 50:

$$RR_{biopsy} = \begin{cases} 1.0 & \text{if } NB_{Cat} = 0 \\ 1.11 & \text{if } NB_{Cat} = 1 \\ 1.23 & \text{if } NB_{Cat} = 2 \end{cases}$$

For age ≥ 50:

$$RR_{biopsy} = \begin{cases} 1.0 & \text{if } NB_{Cat} = 0 \\ 1.34 & \text{if } NB_{Cat} = 1 \\ 1.8 & \text{if } NB_{Cat} = 2 \end{cases}$$

3. Survival Analysis

Survival from Competing Risks:

$$S_2(age) = \exp\left(-\int_{20}^{age} h_2(u) \, du\right)$$

Absolute Risk Integration:

$$R = \int_{age}^{age+5} h_1(t) \cdot RR_{adj} \cdot S_1(t) \cdot \frac{S_2(t)}{S_2(age)} \, dt$$

where the survival function is:

$$S_1(t) = \exp\left(-\int_{age}^{t} [h_1(v) \cdot RR_{adj} + h_2(v)] \, dv\right)$$

Key Components

1. Input Processing

Variable categorization and capping for clinical risk factors
Age thresholds that determine different calculation paths

2. Relative Risk Calculation

Age-dependent lookup tables for different risk factor combinations
Interaction terms between first birth age and family history
Multiplicative risk model combining all factors

3. Survival Analysis

Competing risks from other causes of death
Baseline hazard rates from NCI epidemiological data
Attributable risk adjustment factor (0.5788413)

4. Numerical Integration

Nested trapezoidal integrals for survival probability calculations
Age-interval interpolation using NCI hazard rate tables
Continuous integration over 5-year follow-up period

Model Characteristics

This visualization demonstrates how simplified clinical risk factors (menarche age, number of biopsies, family history, etc.) are transformed through complex mathematical operations including:

Categorical encoding of continuous variables
Age-stratified relative risk calculations
Competing risks survival analysis
Numerical integration for absolute risk estimation

The model successfully bridges clinical simplicity (5 input variables) with mathematical sophistication (nested integrations and survival analysis) to produce accurate breast cancer risk estimates.