Framingham Risk Tool Overview

• Framingham Risk Tool is a probabilistic algorithm that quantifies 10-year cardiovascular risk using clinical and biometric inputs such as age, cholesterol, and blood pressure.
• It employs logic-based explainers, including abductive and counterfactual methods, to provide transparent rationales and actionable prescriptive scenarios.
• Empirical evaluations on synthetic profiles reveal that targeted interventions on modifiable factors like SBP and cholesterol can effectively lower risk categories.

The Framingham Risk Tool (FRT), commonly referred to as the Framingham Risk Score (FRS), is a probabilistic point-based algorithm for estimating the 10-year risk of major cardiovascular disease (CVD) events in adults without established CVD. Utilizing demographic, biometric, and clinical variables, the FRT operationalizes empirical stratification of CVD risk and informs primary prevention strategies. While the FRT is widely recommended globally, recent work highlights both transparency limitations and the promise of logic-based explainers for actionable risk reduction (Bezerra et al., 25 Feb 2026).

1. Formal Definition and Scoring Algorithm

The FRS is defined by a set of sex-specific mappings from clinical and biometric features to an integer-valued score, which is subsequently transformed into an estimated 10-year CVD risk percentage (Bezerra et al., 25 Feb 2026).

Input Variables

• Age (years)
• Sex (male/female)
• Total cholesterol (TC, mg/dL)
• HDL cholesterol (HDL-C, mg/dL)
• Systolic blood pressure (SBP, mm Hg)
• Antihypertensive treatment for SBP (yes/no)
• Smoking status (yes/no)
• Diabetes status (yes/no)

Point Assignment (Male Example)

Key mappings for males (structurally analogous for females with differing thresholds):

$$\text{age\_points}_{\rm M} = \begin{cases} -1 & 30 \le \mathrm{age}<35 \ 0 & 35 \le \mathrm{age}<40 \ 1 & 40 \le \mathrm{age}<45 \ 2 & 45 \le \mathrm{age}<50 \ 3 & 50 \le \mathrm{age}<55 \ 4 & 55 \le \mathrm{age}<60 \ 5 & 60 \le \mathrm{age}<65 \ 6 & 65 \le \mathrm{age}<70 \ 7 & 70 \le \mathrm{age}<75 \ 8 & 75 \le \mathrm{age} \ \end{cases}$$

$$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$

$$\text{hdl\_points}_{\rm M} = \begin{cases} -2 & \mathrm{HDL}\ge60 \ -1 & 50\le\mathrm{HDL}<60 \ 0 & 45\le\mathrm{HDL}<50 \ 1 & 35\le\mathrm{HDL}<45 \ 2 & \mathrm{HDL}<35 \ \end{cases}$$

$$\text{sbp\_points}_{\rm M} = \begin{cases} \text{If untreated:} \ \quad 0 & 120\!-\!129 \ \quad 1 & 130\!-\!139 \ \quad 2 & 140\!-\!159 \ \quad 3 & \ge160 \ \text{If treated:} \ \quad 0 & <120 \ \quad 1 & 120\!-\!129 \ \quad 2 & 130\!-\!139 \ \quad 3 & 140\!-\!159 \ \quad 4 & \ge160 \ \end{cases}$$

$$\text{smoker\_points} = \begin{cases} 4 & \text{if smoker} \ 0 & \text{otherwise} \end{cases} \qquad \text{diabetes\_points} = \begin{cases} 3 & \text{if diabetic} \ 0 & \text{otherwise} \end{cases}$$

The sum $P_{\rm M}$ is mapped to risk via a table, e.g., $P_{\rm M} \ge 18 \implies \mathrm{Risk} > 30\%$.

Risk Category Thresholds

• Low: $0\% \leq \mathrm{Risk} < 6\%$
• Intermediate: $6\% \leq \mathrm{Risk} < 20\%$
• High: $\mathrm{Risk} \geq 20\%$

2. Logic-Based Formalization

The entire FRS can be encoded as a set $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$0 of quantifier-free formulas over linear real arithmetic, mapping feature intervals to point assignments and subsequently to risk categories via logical predicates (Bezerra et al., 25 Feb 2026). For example:

$$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$1

$$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$2

with aggregation constraints enforcing $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$3, $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$4, $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$5.

3. Explainable Artificial Intelligence Extensions

Lack of transparency in the native FRS motivates formal explainers that yield both abductive and counterfactual rationales for individual risk assignments (Bezerra et al., 25 Feb 2026).

Abductive Explanations

An abductive explanation is a minimal sufficient set of patient features $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$6 such that:

• $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$7
• For any proper subset $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$8, $$\text{tc\_points}_{\rm M} = \begin{cases} 0 & \mathrm{TC}<160 \ 1 & 160\le \mathrm{TC}<200 \ 2 & 200\le \mathrm{TC}<240 \ 3 & 240\le \mathrm{TC}<280 \ 4 & \mathrm{TC}\ge280 \ \end{cases}$$9

The extraction process iteratively removes features from the full set, retaining only those essential for the risk category. Empirical distributions (22,000+ cases) show that most explanations are sparse:

Features in Abductive Explanation	Percentage
3	4.0%
4	18.1%
5	25.2%
6	36.0%
7	16.1%
8	0.7%

Feature occurrence frequencies:

Feature	Presence in Explanations
Age	98.2%
SBP	92.4%
Smoker	71.2%
HDL-C	66.3%
TotalChol	59.5%
TreatedSBP	51.2%
Male sex	29.9%

This indicates age and SBP dominate explanations, while modifiable factors appear in ~50–75% of minimal sufficient sets.

Counterfactual Explanations

A counterfactual explanation identifies the smallest modifiable feature set $$\text{hdl\_points}_{\rm M} = \begin{cases} -2 & \mathrm{HDL}\ge60 \ -1 & 50\le\mathrm{HDL}<60 \ 0 & 45\le\mathrm{HDL}<50 \ 1 & 35\le\mathrm{HDL}<45 \ 2 & \mathrm{HDL}<35 \ \end{cases}$$0 whose alteration suffices to attain a lower risk category $$\text{hdl\_points}_{\rm M} = \begin{cases} -2 & \mathrm{HDL}\ge60 \ -1 & 50\le\mathrm{HDL}<60 \ 0 & 45\le\mathrm{HDL}<50 \ 1 & 35\le\mathrm{HDL}<45 \ 2 & \mathrm{HDL}<35 \ \end{cases}$$1. The logic-based solver identifies minimal feature-interval constraints (e.g., reduction of SBP, cholesterol, or cessation of smoking) that change the assignment. Frequencies for 22,000+ synthetic cases:

Features Changed	Percentage
1	47.2%
2	35.1%
3	13.1%
4	3.3%

SBP and TC were the most actionable modifiable targets.

4. Actionable Scenario Generation

The logical encoding enables automated generation of patient-specific prescriptive scenarios:

• Given a full profile, the solver determines which subset of modifiable features (SBP, TC, HDL-C, smoking, diabetes) must change, and to what extent, to reach a lower risk threshold.
• For example, for a 55-year-old male smoker with TC = 240 mg/dL, HDL = 40 mg/dL, untreated SBP = 150 mm Hg, the minimal scenario to drop from HighRisk (>30%) to ModerateRisk (6–20%) may require SBP < 140 mm Hg, TC < 200 mg/dL, and smoking cessation.

This formalism produces transparent, auditable, and actionable guidance for risk modification.

5. Empirical Evaluation and Clinical Interpretation

Evaluation on 22,000+ synthetic patient profiles revealed that abductive explanations are typically sparse, with most involving four to six features. Counterfactual interventions are parsimonious; over 80% of cases require modification of only one or two factors to achieve a lower risk category. SBP and TC are consistently identified as the most impactful levers for CVD risk reduction. A plausible implication is that resource allocation in preventive cardiology may benefit from prioritizing aggressive SBP and lipid management supported by cessation efforts in smoking (Bezerra et al., 25 Feb 2026).

Abductive explanations systematically emphasize the non-modifiable contributors (age, sex), while counterfactual explanations focus exclusively on modifiable factors. This dichotomy sharpens both risk communication and prevention strategy design.

6. Extensions, Limitations, and Comparative Approaches

The FRS, while robust and extensively validated, lacks granularity regarding the contribution of individual predictors to the final risk—a limitation directly addressed by logic-based explainers. Alternative AI-driven models, such as the retinal fundus imaging platform ORAiCLE, have demonstrated improved predictive accuracy (up to 12% over Framingham-based scoring) and introduce the prospect of decomposing composite risk into component factors, further individualizing treatment (Vaghefi et al., 2022). However, precise model architectures and equations for such alternatives were not disclosed.

A plausible implication is that the integration of transparent, explainable models with advanced machine learning could supersede black-box or point-sum methods, facilitating not only risk stratification but also precise targeting of interventions.

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References:

(Bezerra et al., 25 Feb 2026) Enhancing Framingham Cardiovascular Risk Score Transparency through Logic-Based XAI (Vaghefi et al., 2022) Replacing the Framingham-based equation for prediction of cardiovascular disease risk and adverse outcome by using artificial intelligence and retinal imaging