Enhancing Framingham Cardiovascular Risk Score Transparency through Logic-Based XAI

Enhancing F ramingham Cardio v ascular Risk Score T ransparency through Logic-Based XAI Emann uel L. de A. Bezerra 1 [0009 − 0004 − 0195 − 9683] , Luiz H. T. Viana 1 [0009 − 0005 − 1509 − 6038] , Vinícius P . Chagas 1 [0009 − 0003 − 3871 − 640 X ] , Diogo E. Rolim, Thiago Alv es Ro c ha 1 [0000 − 0001 − 7037 − 9683] , and Carlos H. L. Ca v alcante 1 [0000 − 0001 − 9395 − 8338] Instituto F ederal do Ceará (IF CE), Maracanaú, Ceará Brazil {emannuel.bezerra07, henrique.viana06, vinicius.peixoto.chagas61}@aluno.ifce.edu.br {thiago.alves, henriqueleitao}@ifce.edu.br Abstract. Cardio v ascular disease (CVD) remains one of the leading global health challenges, accoun ting for more than 19 million deaths w orldwide. T o address this, several to ols that aim to predict CVD risk and support clinical decision making hav e b een developed. In particular, the F ramingham Risk Score (FRS) is one of the most widely used and recommended w orldwide. Ho wev er, it do es not explain why a patien t w as assigned to a particular risk category nor how it can be reduced. Due to this lack of transparency , our approach introduces a logical explainer for the FRS. Based on first-order logic and explainable artificial intelligence (XAI) fundamen ts, our metho d is capable of identifying the minimal set of patien t attributes that are sufficien t to explain a giv en risk classi- fication (ab duction). The explainer pro duces actionable scenarios that illustrate which mo difiable v ariables would reduce a patien t’s risk cate- gory (counterfactual). W e ev aluated all p ossible input com binations of the FRS (ov er 22,000 samples) and tested them with our solution, suc- cessfully identifying imp ortan t risk factors and suggesting focused inter- v entions for each case. The results improv es clinician trust and facilitates the wider implemen tation of CVD risk assessment by con verting opaque scores into transparent, prescriptive insigh ts, particularly in areas with restricted access to sp ecialists. Keyw ords: F ramingham Risk Score · Explainable Artificial · Logic- Based · Explanation Intelligence · Cardio v ascular Disease. 1 In tro duction Cardio v ascular disease (CVD) is a critical public health issue, impacting millions of p eople worldwide and imp osing a significan t economic strain on healthcare systems [10]. In 2017, the global mortality rate w as appro ximately 55 million deaths, with cardiov ascular diseases (CVD) accounting for 32% of the total [10]. Predicting the risk of developing CVD for individuals pla ys a crucial role in 2 Bezerra et al. prev ention, allowing for the early identification of those who may need more fo cused in terven tions [2]. Man y risk prediction models ha v e been presen ted in the last few decades [6]. The F ramingham Risk Score (FRS) is still one of the most widely v alidated instrumen ts among them [9]; the Brazilian So ciet y of Cardiology’s (SBC) De- partmen t of Atherosclerosis uses it as well [13]. Originally designed to predict the 10-year risk of coronary heart disease (CHD), the F ramingham Heart Study (FRS) is based on a prospective study that was started in 1948 and has follow ed thousands of individuals [14]. Later on, it w as expanded to forecast a wider range of cardiov ascular conditions [7]. FRS employs sex-sp ecific equations that integrate key risk factors: age, total c holesterol (TC), high-density lip oprotein c holesterol (HDL-C), systolic blo od pressure (SBP), smoking status, and diab etes. Based on an individual’s score, FRS assigns them to low, mo derate, or high risk categories to develop CVD o ver the next decade [7]. Evidence from multiple cohorts demonstrates that FRS reliably iden tifies those at elev ated risk in v arious p opulations - including samples from the gen- eral communit y [7], multiethnic groups [3], patients with diab etes or metab olic syndrome [15], and women with p olycystic ov ary syndrome [1]. F ortunately , most of the risk factors are mo difiable, and timely in terven tion can reduce adv erse outcomes and deaths, making the CVD risk prediction of individuals critical in the precise and efficien t reduction of deaths. Although FRS effectively supp orts clinical decision-making, it only rep orts a n umerical risk score and category , lea ving unclear which factors dro v e the classification and what sp ecific changes a patient should mak e to low er their risk. This opacity can impact the confidence in the mo del’s result and limits actionable guidance. Therefore, it is essential to highligh t the individual risk factors that influ- enced the classification and to indicate whic h mo difiable characteristics the pa- tien t should address to reduce their future risk. W e address this gap b y us- ing first-order logic to build an ab ductiv e and counterfactual explainer for the F ramingham Risk Score, thereby ensuring the correctness of the in terpretations. F eatures present in an ab ductiv e explanation are those sufficient to imply the assigned risk category [8], without redundancy . With them, we can improv e the in terpretability of the FRS mo del and understand ho w it mak es its decisions. The features presen t in contrastiv e explanations are those that, if appropriately c hanged, would cause the patient’s score to fall in to a low er risk category (e.g., from mo derate to lo w or from high to moderate) [4]. This pap er is organized as follows. In Section 1, w e introduce the research problem and motiv ate the need for more transparen t CVD risk assessmen ts. Section 2 reviews the most relev ant related work in explainable AI and risk- prediction mo dels. Section 3 des cribes our metho dology in detail, outlining the logical framework, model formulation, and system architecture. Section 4 presents the exp erimen tal ev aluation, implementation of the explainer, and an analysis Enhancing FRS with Logic-Based XAI 3 of its p erformance. Finally , Section 5 summarizes our main findings, discusses their implications, and suggests directions for future work. 2 Bac kground 2.1 Cardio v ascular diseases (CVDs) Cardio v ascular disease (CVD) encompasses a broad range of conditions affecting the heart and bloo d v essels, and it remains one of the leading causes of death and disabilit y worldwide. [11] This group of disorders includes ailments such as coro- nary artery disease, heart failure, and arrhythmias. The onset and progression of CVD are influenced b y b oth non-mo difiable factors, suc h as age and genetics, and mo difiable factors, including lifest yle choices and environmen tal influences [5]. 2.2 F ramingham Risk Score The F ramingham Risk Score is a to ol that helps calculate the estimated risk of having cardiov ascular disease in 10 y ears. This to ol was created using the F ramingham Heart Study , where the comprehensive data collected from the par- ticipan ts was analyzed to identify k ey risk factors. The study results were then translated into practical risk assessment tables b y p erforming statistical analy- ses to determine how eac h risk factor contributes to the ov erall cardiov ascular risk. These analyses pro duced regression co efficien ts for v arious features (such as age, c holesterol lev els, blo od pressure, and smoking status), which were then con verted in to point v alues. These p oin t v alues are organized into tables that group the risk factors in to different ranges, allowing clinicians to quickly assess an individual’s risk level [7]. The F ramingham Risk Score is computed using dual-p oin t tables, one for eac h feature for males and another for females. F or a given individual, eac h feature is lo cated within a sp ecified range in the corresp onding table, and the asso ciated p oin t v alue is added to the ov erall score. According to T able 1, for instance, a w ould receiv e a total of 26 p oin ts. Once the total score is calculated, it is mapp ed to T able 2, which determines the corresponding risk percentage. In our example, the man’s risk w ould be >30%. There are three p ossible risk categories: low-risk (0% to 6%), mo derate- risk (6% to 20%), and high-risk ( ≥ 20% ) [7]. Therefore, the man w ould fall into the high-risk category . 2.3 First-order Logic ov er LRA In this w ork, we use first-order logic (FOL) to give explanations with guaran tees of correctness. W e use quantifier-free first-order form ulas o ver the theory of linear rational arithmetic (LRA). Then, first-order v ariables are allow ed to tak e v alues 4 Bezerra et al. T able 1. CVD Prediction Poin ts T able for Male adapted from [7] HDL: High-Density Lip oprotein, SBP: Systollic Blo od Pressure P oints Age (y ears) HDL (mg/dL) T otal Chol (mg/dL) SBP - Not T reated (mm Hg) SBP - T reated (mm Hg) Smok er Diab etic -2 60+ <120 -1 50–59 0 30–34 45–49 <160 120–129 <120 No No 1 35–44 160–199 130–139 2 35–39 <35 200–239 140–159 120–129 3 240–279 160+ 130–139 Y es 4 280+ 140–159 Y es 5 40–44 160+ 6 45+ 8 50–54 10 55–59 11 60–64 12 65–69 14 70–74 15 75+ from the real num b ers R . F or details, see [12]. Therefore, we consider formulas as defined b elo w: F , G := p | ( F ∧ G ) | ( F ∨ G ) | ( ¬ F ) | ( F → G ) , p := n X i =1 w i x i ≤ b, (1) suc h that F and G are quantifier-free first-order form ulas o ver the theory of linear real arithmetic. Moreo v er, p represents the atomic formulas suc h that n ≥ 1 , each w i and b are fixed real num b ers, and each x i is a first-order v ariable. F or example, (2 . 5 x 1 + 3 . 1 x 2 ≥ 6) ∧ ( x 1 = 1 ∨ x 1 = 2) ∧ ( x 1 = 2 → x 2 ≤ 1 . 1) is a form ula by this definition. Observ e that we allow standard abbreviations as ¬ (2 . 5 x 1 + 3 . 1 x 2 < 6) for 2 . 5 x 1 + 3 . 1 x 2 ≥ 6 . Since we are assuming the seman tics of form ulas o ver the domain of real n umbers, an assignment A for a formula F is a mapping from the first-order v ariables of F to elements in the domain of real num bers. F or instance, { x 1 7→ 2 . 3 , x 2 7→ 1 } is an assignment for (2 . 5 x 1 + 3 . 1 x 2 ≥ 6) ∧ ( x 1 = 1 ∨ x 1 = 2) ∧ ( x 1 = 2 → x 2 ≤ 1 . 1) . An assignment A satisfies a formula F if F is true under this assignment. F or example, { x 1 7→ 2 , x 2 7→ 1 . 05 } satisfies the formula in the Enhancing FRS with Logic-Based XAI 5 T able 2. Risk p ercen tage according to total p oin ts for male [7] P oints Risk (%) ≤ − 3 or less < 1 − 2 1.1 − 1 1.4 0 1.6 1 1.9 2 2.3 3 2.8 4 3.3 5 3.9 6 4.7 7 5.6 8 6.7 9 7.9 10 9.4 11 11.2 12 13.2 13 15.6 14 18.4 15 21.6 16 25.3 17 29.4 18+ > 30 ab o v e example, whereas { x 1 7→ 2 . 3 , x 2 7→ 1 } do es not satisfy it. Moreo ver, an assignmen t A satisfies a set Γ of formulas if all formulas in Γ are true under A . A set of form ulas Γ is satisfiable if there exists a satisfying assignment for Γ . T o give an example, the set { (2 . 5 x 1 + 3 . 1 x 2 ≥ 6) , ( x 1 = 1 ∨ x 1 = 2) , ( x 1 = 2 → x 2 ≤ 1 . 1) } is satisfiable since { x 1 7→ 2 , x 2 7→ 1 . 05 } satisfies it. As another example, the set { ( x 1 ≥ 2) , ( x 1 < 1) } is unsatisfiable since no assignment satisfies it. Giv en a set of formulas Γ and a formula G , the notation Γ | = G is used to denote lo gic al c onse quenc e or entailment , i.e., each assignment that satisfies Γ also satisfies G . As an illustrativ e example, let Γ = { x 1 = 2 , x 2 ≥ 1 } and G = (2 . 5 x 1 + x 2 ≥ 5) ∧ ( x 1 = 1 ∨ x 1 = 2) . Then, Γ | = G . The essence of en tailment lies in ensuring the correctness of the conclusion G based on the set of premises Γ . In the con text of computing explanations, as presented in [8], logical consequence serv es as a fundamen tal to ol for guaranteeing correctness. The relationship b et w een satisfiability and entailmen t is a fundamental as- p ect of logic. It is widely kno wn that, for all sets of form ulas Γ and all for- m ulas G , it holds that Γ | = G iff Γ ∪ {¬ G } is unsatisfiable. F or instance, 6 Bezerra et al. { x 1 = 2 , x 2 ≥ 1) , ¬ ((2 . 5 x 1 + x 2 ≥ 5) ∧ ( x 1 = 1 ∨ x 1 = 2)) } has no satisfy- ing assignment since an assignment that satisfies { x 1 = 2 , x 2 ≥ 1 } also sat- isfies (2 . 5 x 1 + x 2 ≥ 5) ∧ ( x 1 = 1 ∨ x 1 = 2) and, therefore, do es not satisfy ¬ ((2 . 5 x 1 + x 2 ≥ 5) ∧ ( x 1 = 1 ∨ x 1 = 2)) . Since our approach builds up on the concept of logical consequence, we can lev erage this connection in the con text of computing explanations. 2.4 Logic Based Explanations Due to the F ramingham risk score b eing defined by assigning v alues to tables according to the features of a user, the scoring mo del can b e readily adapted to a First-Order Logic (FOL) framework b y formalizing the rules as logical impli- cations. F or example, for contin uous v ariables such as age, one can express: 40 ≤ age < 45 → age_p oin ts = 5 . F or Bo olean v ariables, suc h as smoking status (denoted by is_smoker), the as- signmen t of points is defined as follows: if is_smok er is true, the patien t receives 4 p oin ts; if false, 0 p oin ts are added. This can b e represented in FOL as: ( is_smok er → smok er_p oin ts = 4) ∧ ( ¬ is_smoker → smoker_points = 0) . Once all features hav e b een assigned their p oin t v alues, the total score determines the patien t’s risk category (low, mo derate, or high). Explanations in this FOL framew ork then pro ceed by identifying tw o complementary types of feature-sets T o compute the ab ductiv e features, we iterate ov er each feature in the current in terpretation, remov e it, and c heck whether the remaining features still logically en tail the originally assi gned risk category . If en tailment still holds, that feature is deemed irrelev an t and is not included in the ab ductiv e explanation [8]. T o compute the coun terfactual features, w e start from an empt y set and incremen tally add each mutable feature (i.e., excluding immutable features such as age or sex) together with the desired tar get category (e.g. “low risk”). If the conjunction of this set w ith the target category is unsatisfiable, we remov e that feature from consideration. The resulting counterfactual explanation is the difference b et ween the original in terpretation and the final feature set [4]. Applying this approac h to the example presented previously , the 70-y ear- old man with diab etes, who is not on medication for systolic blo od pressure, curren tly do es not smok e and has a total cholesterol of 283 mg/dL, an HDL of 30 mg / dL and a systolic blo o d pressure of 170 mm Hg, we identify systolic bloo d pressure, diab etes, and age as ab ductiv e features. This o ccurs even though total c holesterol contributes more p oin ts than systolic blo o d pressure and diab etes, as sho wn in T able 1. The fact that the F ramingham Risk Score (FRS) is based on a p oin t system may lead to the assumption that the features contributing the highest num ber of p oints are necessarily the most imp ortan t. How ev er, as this example illustrates, that is not alw ays the case, highligh ting the relev ance of our prop osed approac h. Enhancing FRS with Logic-Based XAI 7 3 Metho dology 3.1 Data Collection T o comprehensively ev aluate our framew ork, w e constructed a syn thetic dataset that exhaustiv ely cov ers all p ossible input com binations used in the FRS calcula- tion. F or each contin uous feature, we selected representativ e v alues corresp ond- ing to the distinct ranges defined by the FRS guidelines. Since v alues within the same range yield equiv alent risk contributions, this discretization p ermits a finite and complete enumeration of all meaningful input configurations. T able 3. Number of possible v alues for eac h feature after contin uous-features quanti- zation for males and famale F eature #V alues (Male) #V alues (F emale) Age 10 10 HDL 5 5 T otal ChoL 5 5 SBP 5 6 T reatment for SBP 2 2 Smok er 2 2 Diab etic 2 2 Based on T able 3, the total num ber of distinct FRS inputs is 22,000. This quan tity is straightforw ard to pro cess exhaustively . All distinct samples (each reflecting a legitimate com bination of FRS input v alues) were automatically created using Python and the pandas pac k age. The relev an t risk score for each sample was then calculated and explained by feeding this dataset in to our logic- based explanation engine. 3.2 Explainer Using the z3py API, w e enco ded the en tire F ramingham Risk Score (FRS) cal- culation for the created explanation as a set of logical constraints in the Z3 SMT solv er, including threshold chec ks and risk table lo okups. Z3 uses logical infer- ence to calculate the risk score based on the patient’s input features. Without using heuristics or appro ximation techniques, w e w ere able to pro cess our whole syn thetic dataset of 22,000 samples and generate logically consistent explana- tions for eac h instance thanks to Z3’s effective constrain t-solving capabilities. W e pro duced tw o kinds of explanations using this formal enco ding: counterfactual explanations, which identify the smallest changes to input features necessary to c hange the risk score to a giv en target v alue, and ab ductiv e explanations, which iden tify the minimal subset of input features sufficient to justify the computed risk score. 8 Bezerra et al. 3.3 Exp erimen ts The experiments in this study w ere conducted under t w o distinct scenarios. The first scenario aimed to iden tify ab ductiv e features, that is, those whose presence is sufficient to justify the assigned risk category . The goal was to enhance the interpretabilit y of the FRS mo del and to gain a b etter understanding of the rationale b ehind its decisions. The second scenario fo cused on identifying coun terfactual features, meaning v ariables that, if appropriately mo dified, could lead to a reclassification of the patien t (e.g., from high to mo derate or from mo derate to low). 4 Results In the first scenario, our explainer achiev ed goo ds results using the synthetic dataset with 22,000 inputs. The ab ductiv e explanations are richer, with nearly 77% including five or more features, suggesting that justifying a giv en risk score t ypically requires citing a constellation of factors, where we first mensured the quan tified ho w concise eac h explanation metho d tends to b e. The T able 4 shows this results. T able 4. Distribution of ab ductiv e explanation sparsity ov er all FRS inputs Num b er of F eatures Ab ductiv e (%) 3 4.00 4 18.14 5 25.15 6 35.97 7 16.05 8 0.70 Subsequen tly , w e examined whic h features app ear most often in each explana- tion type. T o ab ductiv e explanations, age and systolic blo od pressure dominate, app earing in ov er 90% of justifications, as you can see in T able 5. Moreov er, the distinction b et w een mo difiable factors ( blo o d pr essur e, cholester ol, smoking status and me dic ation ) and non-mo difiable factors ( sex and age ) is clear: sex app ears in only 30% of cases, while all mo difiable factors app ear in 50–75% of cases, highlighting their central role in justifying risk. In the second scenario, displa yed in T able 6, w e observ e that coun terfac- tual explanations are sparse. Ov er 80% inv olve at most tw o features (1-feature: 47.17%, 2-features: 35.07%), indicating that only one or t wo features is enough to change the risk category . In counterfactual explanations, systolic blo o d pr essur e and total cholester ol w ere the ma jor factor to c hange the risk class, each one app earing in ov er 40% of cases. The smoking status and HDL cholester ol are less frequent, suggesting Enhancing FRS with Logic-Based XAI 9 T able 5. F eature presence in ab ductiv e explanations ov er all inputs F eature Coun t % of all samples Age 21 593 98.2% Systolic blo o d pressure 20 329 92.4% Smok er status 15 662 71.2% HDL cholesterol 14 588 66.3% T otal cholesterol 13 095 59.5% T reatment for SBP 11 257 51.2% Male sex 6 579 29.9% T able 6. Distribution of counterfactual explanation sparsity o ver Moderate-Risk and High-Risk FRS inputs Num b er of F eatures Counterfactual (%) 1 47.17 2 35.07 3 13.06 4 3.32 5 0.54 6 0.84 that change blo o d pressure or cholesterol alone is often sufficient to cross risk thresholds. T able 7 shows the results. Overall, these patterns align with clinical practice: ab ductiv e explanations highlight core driv ers such as age and sex, while coun terfactual explanations fo cus on modifiable factors for interv en tion. T able 7. F eature presence in counterfactual explanations (excluding lo w-risk) F eature Coun t % Systolic blo o d pressure 8 330 43.7% T otal cholesterol 8 019 42.1% T reatment for SBP 5 958 31.3% HDL cholesterol 4 983 26.2% Smok er status 2 450 12.9% Sex, age 0 0.0% Ov erall, thes e results align closely with established clinical practice: ab duc- tiv e explanations emphasize the foundational drivers of cardiov ascular risk, while coun terfactual explanations highlight the mo difiable factors that clinicians target for interv ention. 5 Conclusion T wo complementary explanation t yp es (ab ductiv e and counterfactual) for the FRS were successfully produced by our system. The ab ductiv e explanations 10 Bezerra et al. demonstrate that our approach accurately reflects the established clinical hi- erarc hy of risk factors. Coun terfactual explanations target sp ecific areas for in- terv ention to c hange the patien t’s risk category . F uture enhancements could include testing the explainer on real-w orld datasets and ev aluating its effective- ness through exp ert assessmen t. A dditionally , the approac h can impro ve the coun terfactual explanations to calculate sp ecific features v alues to alter the risk classification. A ckno wledgmen ts. The authors ackno wledge the supp ort of Instituto F ederal do Ceará (IFCE) through the research gran t calls PIBITI No. 11/2024 and PIBIC No. 07/2024, issued b y the PRPI/IF CE, as well as the supp ort of F undação Cearense de Ap oio ao Desenv olvimen to Científico e T ecnológico (FUNCAP) and Conselho Nacional de Desenv olvimen to Científico e T ecnológico (CNPq) in the dev elopment of this work. 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