Developing and evaluation of a novel machine learning-derived formula for LDL-C estimation

Ethics committee approval

This study was conducted in accordance with the principles of the Declaration of Helsinki. Ethical approval was obtained from the Scientific Research Ethics Committee of the University of Health Sciences, Trabzon Faculty of Medicine, Turkey (Approval Date: February 27, 2024; Approval Number: 2024/59).

Lipid measurements and study variables

To develop accurate machine learning models for predicting low-density lipoprotein cholesterol (LDL-C) levels, a range of clinical variables – including age, sex, total cholesterol (TC), triglycerides (TG), high-density lipoprotein cholesterol (HDL-C), and directly measured LDL-C (d-LDL-C) – were incorporated as input features (Table 1). Retrospective data were retrieved from 2,270 patients who received care at Akcaabat Hackali Baba State Hospital (Trabzon, Turkey) between January 1 and October 31, 2023, via the hospital’s laboratory information system.

Venous blood samples were collected from the antecubital vein of participants following an overnight fast. Serum samples were analyzed for lipid profiles using enzymatic methods on an Architect c16000 analyzer (Abbott Laboratories, Abbott Park, IL, USA). TC was measured through a series of enzymatic reactions involving cholesteryl ester hydrolase, cholesterol oxidase, and peroxidase. TG was quantified through hydrolysis to glycerol and subsequent oxidation, with hydrogen peroxide measured colorimetrically. HDL-C and LDL-C were quantified using homogeneous enzymatic colorimetric assays after selective removal of interfering lipoproteins.

These records constituted the training dataset. Three independent test datasets were also obtained from hospitals within the province and included lipid profiles measured between 2023 and 2024 using the homogeneous enzymatic method. Test Dataset one consisted of 40,941 samples measured on the Architect c16000 analyzer, Test Dataset two included 42,520 samples from the AU5800 analyzer (Beckman Coulter Inc., Brea, CA, USA), and Test Dataset three comprised 5,813 samples from the Cobas analyzer (Roche Diagnostics, Mannheim, Germany). No data were excluded based on extreme values; all available results were included in the analysis. Basic demographic and lipid profile characteristics of patients across datasets are summarized in Table 1.

Analytical performance evaluation

The analytical performance of lipid measurements was assessed by calculating the total analytical error (%TEa), incorporating both systematic error (%Bias) and random error (%CV), using the formula: %TEa=%Bias + 1.65 × %CV.

Internal quality control was performed daily using two levels of Technopath Multichem S control materials (lot numbers: 11811210 and 11811212) on the Abbott Architect c16000 analyzer during the training phase. The observed coefficients of variation for the first six months were as follows:

TC: 2.31 % at 109.29 mg/dL, 3.03 % at 170.24 mg/dL

HDL-C: 2.94 % at 33.6 mg/dL, 2.88 % at 48.35 mg/dL

LDL-C: 2.22 % at 66.45 mg/dL, 2.55 % at 106.5 mg/dL

TG: 3.33 % at 82.3 mg/dL, 2.78 % at 132 mg/dL

External quality assurance was ensured via participation in the Bio-Rad EQAS Monthly Program 21. The calculated %TEa values were as follows:

Training dataset: TC: 3.87 %, HDL-C: 1.72 %, LDL-C: 6.84 %, TG: 2.23%

Abbott test dataset: TC: 4.68 %, HDL-C: 1.62 %, LDL-C: 9.03 %, TG: 2.29%

Beckman test dataset: TC: 5.36 %, HDL-C: 7.00 %, LDL-C: 6.63 %, TG: 1.20%

Roche test dataset: TC: 4.84 %, HDL-C: 1.68 %, LDL-C: 8.37 %, TG: 1.18%

All values were within the acceptable limits defined by the 2019 Clinical Laboratory Improvement Amendments (CLIA) guidelines [13].

Algorithm development

Symbolic regression, an evolutionary computation-based machine learning technique, was employed to derive mathematical expressions that best describe the relationship between input variables and LDL-C levels. The algorithm was implemented using the gplearn library (version 0.4.2). Unlike traditional regression methods, symbolic regression evolves equation structures dynamically via genetic programming.

The model was trained on the 2,270-patient dataset with the following hyperparameters: population size=1,000, generations=20, tournament size=20. The function set included addition, subtraction, multiplication, division, and square root operations. To enhance interpretability and clinical feasibility, model parsimony was prioritized using a high parsimony coefficient. Model fitness was optimized using mean absolute error (MAE) as the cost function. Genetic operators included a crossover probability of 0.7 and a mutation probability of 0.1.

The modelling pipeline utilized scikit-learn (v1.2.2) for performance evaluation, NumPy (v1.24.3) and pandas (v2.0.2) for data handling, and matplotlib (v3.7.2) and seaborn (v0.12.2) for visualization. The final formula derived from the training set was directly validated on all three independent external datasets without further tuning.

Comparative evaluation of LDL-C estimation formulas

LDL-C estimation methods differ primarily in how they calculate VLDL-C. The Friedewald equation uses a fixed TG:VLDL-C ratio by dividing TG by 5 [14]. Some modified formulas retain the Friedewald structure but use different fixed divisors, which still fail to address individual TG:VLDL-C variability. The Martin equation introduces a dynamic divisor based on stratified TG and non-HDL-C levels, improving precision [15]. The Sampson equation uses a bivariate quadratic regression derived from β-quantification data, calculating VLDL-C as:

The Sampson equation: VLDL = T G 8.59 + T G x   n o n H D L C 2250 − T G 2 16500 [16].

Our proposed formula introduces the term T G x T C 100 to better reflect the heterogeneity of TG:VLDL ratios.

Regression and bias analysis

LDL-C estimates were analyzed by entering the Friedewald, Sampson, Martin (180 cells), extended Martin (additional 240 cells), and Y-LDL-C formulas into the Python program. The data obtained as a result of the estimated formulas were compared with the measured LDL-C values. The Friedewald and Sampson equations have not been validated for triglyceride levels exceeding 400 mg/dL and 800 mg/dL, respectively; however, their estimates at these levels were included for comparative analysis.

We assessed the performance of our newly developed formula alongside the Friedewald, Sampson, and Extended Martin equations using three independent test datasets. To evaluate the agreement between the formulas and the homogeneous assay results, we performed Passing-Bablok regression analysis (Supplement 1) and Bland-Altman analysis (Figure 1).

Figure 1:

Bland–Altman plots showing the agreement between calculated LDL-C values using the Friedewald (A, E, I), Sampson (B, F, J), extended Martin (C, G, K), and Y-LDL-C (D, H, L) formulas and directly measured LDL-C concentrations on Abbott Architect (A–D, n=40,941), Beckman Coulter AU5800 (E–H, n=42,520), and Roche Cobas (I–L, n=5,813) platforms. Each panel shows the mean difference (solid green line) and the limits of agreement (±1.96 SD; dashed blue lines).

In the Bland-Altman analysis, F-LDL-C exhibited negative bias across all platforms, with mean differences of −10.48 mg/dL on Abbott Architect (Figure 1A), −5.12 mg/dL on Beckman Coulter (Figure 1E), and −6.01 mg/dL on Roche Cobas (Figure 1I). The 95 % limits of agreement ranged from 16.17 to −37.14 mg/dL, 24.48 to −34.73 mg/dL, and 8.97 to −40.98 mg/dL, respectively. S-LDL-C demonstrated similar negative bias patterns with mean differences of −7.08 mg/dL (Abbott), −2.23 mg/dL (Beckman), and −2.75 mg/dL (Roche). The 95 % limits of agreement were 14.74 to −29.89 mg/dL, 23.09 to −27.55 mg/dL, and 8.95 to −34.44 mg/dL, respectively (Figure 1B,F,J). EM-LDL-C showed mean differences of −7.37 mg/dL (Abbott), −2.23 mg/dL (Beckman), and −3.43 mg/dL (Roche), with 95 % limits of agreement ranging from 10.23 to −24.98 mg/dL, 17.24 to −21.70 mg/dL, and 7.57 to −34.43 mg/dL (Figure 1C, G, K). Y-LDL-C formula presented the lowest overall bias with mean differences of −4.92 mg/dL (Abbott), −0.34 mg/dL (Beckman), and −1.27 mg/dL (Roche). The 95 % limits of agreement were 11.84 to −21.69 mg/dL, 15.95 to −16.59 mg/dL, and 12.01 to −23.86 mg/dL, indicating the narrowest ranges among all formulas (Figure 1D, H, L).

The performance of the four formulas was compared across all three datasets using the following statistical metrics: Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), R-squared (R²), Concordance Correlation Coefficient (CCC), Quadratic Weighted Kappa (QWK), Bias, Concordance Rate (CR), Overestimation Rate (OER), and Underestimation Rate (UER) (Table 2).

To further evaluate formula performance across clinically relevant LDL-C ranges, we conducted a stratified analysis by LDL-C category. Patients were classified into six groups based on their LDL-C levels: <70 mg/dL, 70–99 mg/dL, 100–129 mg/dL, 130–159 mg/dL, 160–189 mg/dL, and ≥190 mg/dL. The distribution of patients in each group is provided in Table 1 and Table 3. To assess the agreement between direct and estimated LDL-C values within each group, a series of statistical measures were applied, including RMSE, MAE, MAPE, CCC, Mean Difference (MD), Mean Percentage Difference (MPD), CR, UER, and OER. The results of these statistical analyses are presented in Table 3.

To evaluate the performance of different LDL-C estimation methods across various TG levels, patients were categorized into five groups: <100 mg/dL, 100–149 mg/dL, 150–199 mg/dL, 200–399 mg/dL, and ≥400 mg/dL. To assess the agreement between direct and estimated LDL-C values within each group, a series of statistical measures were applied, including RMSE, MAE, MAPE, CCC, QW-Kappa, MD, MPD, CR, UER, and OER. The results of these statistical analyses are presented in Table 4.

The performance of LDL-C estimation formulas varies depending on TG levels. At TG<100 mg/dL, all formulas perform relatively well, with CR of 78–86 % on the Abbott and Beckman platforms. However, at TG≥400 mg/dL, performance drops significantly, with Friedewald demonstrating particularly poor results (CR% as low as 5.9 %). The Y-LDL-C formula consistently exhibits superior performance across most TG categories and platforms, especially at higher TG levels (≥200 mg/dL). The Friedewald formula continues to show the poorest performance, particularly at elevated TG levels. The Martin and Sampson formulas generally perform better than Friedewald, but not as well as Y-LDL-C. The RMSE increases significantly with TG levels for all formulas. The MAPE% surpasses 10 % for most formulas at TG≥150 mg/dL. At TG≥400 mg/dL, the Friedewald formula displays very high error metrics (MAPE% 39–45 %).

To visualize how each formula classifies patients across LDL-C categories, we created confusion matrices for each of the three datasets (Figure 2). The confusion matrices provide a detailed comparison between the predicted and homogeneous LDL-C classifications, highlighting the accuracy of each formula in correctly categorizing patients. The diagonal cells represent correctly classified samples, where the predicted LDL-C category matches the measured category, while off-diagonal cells indicate misclassifications. They also enable the identification of misclassifications, including false positives and false negatives, which are crucial for assessing the clinical reliability of each method.

Figure 2:

Confusion matrices comparing the clinical categorization agreement between calculated LDL-C values using the Friedewald (A, E, I), Sampson (B, F, J), extended Martin (C, G, K), and Y-LDL-C (D, H, L) formulas and directly measured LDL-C concentrations on Abbott Architect (A–D, n=40,941), Beckman Coulter (E–H, n=42,520), and Roche Cobas (I–L, n=3,813) platforms. Darker blue diagonal cells indicate higher concordance between calculated and measured LDL-C clinical categories. Overall accuracy percentages are shown at the top of each matrix.

Error Grid Analysis, a crucial tool for assessing the accuracy of LDL-C estimation formulas, plays a key role in understanding error levels in clinical decision-making processes. The National Cholesterol Education Program Adult Treatment Panel III (NCEP ATP III) have established a TEa threshold of 12 % as an acceptable limit. In our study, we adopted the same 12 % TEa value. For Error Grid Analysis, we selected clinical decision thresholds of 70 mg/dL and 190 mg/dL. To enhance the analysis, we stratified the dataset based on triglyceride (TG) levels, dividing it into two groups: TG<400 mg/dL and, TG≥400 mg/dL (Figure 3). In the TG<400 mg/dL group, when analyzing 40,098 LDL-C results obtained from the Abbott Architect c16000 analyzer (Test Dataset 1), the proportion of calculated LDL-C values falling within the 12 % TEa (a + b) threshold was as follows: 75.7 % for F-LDL-C, 83.7 % for S-LDL-C, 89.2 % for EM-LDL-C, and 96.3 % for Y-LDL-C. Similarly, analysis of 41,544 LDL-C results from the Beckman Coulter AU5800 analyzer (Test Dataset 2) revealed that 79.4 % of F-LDL-C, 84.0 % of S-LDL-C, 89.9 % of EM-LDL-C, and 95.4 % of Y- LDL-C values were within the 12 % TEa threshold. In contrast, when analyzing 5,709 LDL-C results from the Roche Cobas c501 analyzer (Test Dataset 3), the proportions were somewhat lower, with 52.5 % for F-LDL-C, 63.5 % for S-LDL-C, 59.8 % for EM-LDL-C, and 71.5 % for Y-LDL-C falling within the 12 % TEa threshold. Error Grid Analysis results further demonstrated clinically misclassifications at low and high LDL-C cut-points (zones e + f + g + h) In Test Dataset 1, the proportion of values falling within these clinically relevant error zones was 4.2 % for F-LDL-C, 2.6 % for S-LDL-C, 1.6 % for EM-LDL-C, and 0.6 % for Y-LDL-C. Similarly, in Test Dataset 2, the respective error rates were 3.5 % for F-LDL-C, 2.6 % for S-LDL-C, 1.6 % for EM-LDL-C, and 0.6 % for Y-LDL-C. In contrast, Test Dataset three exhibited higher clinically significant error rates, with 6.8 % for F-LDL-C, 5.2 % for S-LDL-C, 4.8 % for EM-LDL-C, and 3.5 % for Y-LDL-C falling within these critical zones (Figure 3) (Supplement 2).

Figure 3:

Error grid analysis for LDL-C measurements in samples with TG<400 mg/dL. Panels (A–L) show scatter plots comparing different LDL-C estimation methods against homogenous enzymatic assay across three analytical platforms. The error grid zones categorize results based on analytical accuracy and clinical impact: Zones a and b: Within ±12 % proportional error, considered analytically accurate. Zones c and d: Errors exceeding 12 % but without clinical impact. Zones e and f: Clinically significant misclassifications at high LDL-C thresholds. Zones g and h: Clinically significant misclassifications at low LDL-C thresholds. Panel (M) presents TE% accuracy for each method (zones a + b). Panel (N) quantifies analytical errors (c + d), while Panel (O) shows clinically significant errors (e–h). Panel (P) illustrates the LDL-C error grid structure. Numbers in shaded regions indicate misclassification counts. acceptable (zones c and d), and clinically significant errors (zones e to h) are presented in Panels N, O and P respectively.

For TG≥400 mg/dL, the analysis of LDL-C results included a smaller number of patients compared to the TG<400 mg/dL group (Test Dataset 1: n=843, Test Dataset 2: n=976, Test Dataset 3: n=104). When evaluating LDL-C results, the proportion of calculated LDL-C values falling within the 12 % TEa threshold (zones a + b) decreased across all four formulas in all three datasets. However, Y-LDL-C consistently demonstrated higher agreement with the TEa threshold compared to other estimation methods. Similarly, analysis revealed an increase in clinically significant misclassifications at both low and high LDL-C cut-points (zones e + f + g + h) across all three datasets (Figure 4) (Supplement 3).

Figure 4:

Error grid analysis for LDL-C measurements in samples with TG≥400 mg/dL. Panels (A–L) show scatter plots comparing different LDL-C estimation methods against homogenous enzymatic assay across three analytical platforms. The error grid zones categorize results based on analytical accuracy and clinical impact: Zones a and b: Within ±12 % proportional error, considered analytically accurate. Zones c and d: Errors exceeding 12 % but without clinical impact. Zones e and f: Clinically significant misclassifications at high LDL-C thresholds. Zones g and h: Clinically significant misclassifications at low LDL-C thresholds. Panel (M) presents TE% accuracy for each method (zones a + b). Panel (N) quantifies analytical errors (c + d), while Panel (O) shows clinically significant errors (E–H). Panel (P) illustrates the LDL-C error grid structure. Numbers in shaded regions indicate misclassification counts. acceptable (zones c and d), and clinically significant errors (zones e to h) are presented in Panels N, O and P respectively.

Study limitations

This study has several limitations. First, although beta quantification is the reference method for LDL-C measurement, we relied on homogeneous enzymatic d-LDL-C measurements as our comparator. While this reflects real-world laboratory practices, further validation of the novel formula using beta quantification or vertical ultracentrifugation in multicenter studies is warranted.

Second, our high-TG patient sample was relatively small – 843 cases in Test Dataset 1, 976 in Test Dataset 2, and only 104 in Test Dataset three – potentially limiting the generalizability of our findings in this subgroup.

Moreover, although the formula was derived from a dataset of 2,270 patients, its performance was validated only on three external datasets. Variables such as lipoprotein (a) and remnant lipoproteins were not separately accounted for in this study, which may influence LDL-C estimation. Additionally, treatment data for the included patients were not available, restricting our ability to control for medication effects. Finally, further studies exploring adaptive divisors (instead of the fixed 100 in the nonlinear term) based on LDL-C or TG stratification may yield even higher accuracy in future iterations of the formula.

To address these limitations, we are planning larger, multicentre prospective studies to further evaluate and refine the novel equation.