Analysis of heavy metal concentrations in breast milk by neutrosophic method in the locate of Lahore, Pakistan

Environmental contamination by heavy metals (e.g., As, Cu, Pb, Hg, Cd) poses serious global health risks. Long-term exposure can lead to kidney damage (Cd), neurological impairment (Pb), and systemic toxicity (Hg). Infants, whose diets rely primarily on breast milk during the first months of life, are especially vulnerable, as heavy metals can transfer from maternal blood and body stores into milk, and readily bind to milk proteins, potentially increasing infant exposure. In Pakistan, breastfeeding is common up to 24 months, with WHO recommending exclusive breastfeeding for the first six months, emphasizing the importance of monitoring contaminants in milk. The study addresses limitations of classical statistical methods that collapse interval measurements into single values, losing information about uncertainty and variation. Neutrosophic statistics, which explicitly model indeterminacy and interval-valued data, have been advanced as a more adaptable framework. Research question: Can a neutrosophic statistical approach more accurately characterize and interpret the variation and indeterminacy in heavy metal (Hg, Pb) concentrations measured in breast milk of lactating mothers around industrial areas of Lahore, thereby informing exposure assessment and health risk understanding? Purpose: To apply and demonstrate neutrosophic statistics on interval data from FAAS measurements of breast milk to capture uncertainty and extremes that classical statistics may obscure, and to assess patterns of Hg and Pb by maternal age in a polluted urban context.

Prior work has documented heavy metal pollution and toxicological effects on humans and links between maternal exposure and birth outcomes. In Pakistan, heavy metal contamination has been assessed in soils near industrial activities using innovative statistical approaches, and neutrosophic methods have been applied to physiological data (e.g., respiration, blood pressure changes during pregnancy and COVID-19). Neutrosophic approaches have also been used across domains such as signal quality enhancement, medical diagnosis, materials and sensor characterization, and resistance-temperature analyses, demonstrating utility for imprecise and interval data. Previous local studies evaluated heavy metal levels in foods and animal products and assessed lead, cadmium, and mercury levels in breast milk in Pakistani women. This study builds on that literature by explicitly using neutrosophic statistics to analyze interval FAAS measurements of Hg and Pb in human milk, contrasting results with classical averaging methods and highlighting how indeterminacy improves interpretability.

Study design and sampling: Approximately 70 breast milk samples were collected from lactating mothers residing near industrial areas of Lahore, Pakistan. Participants were grouped into two age groups covering ages 25–40 years, and samples were collected at the end of the 3rd month of lactation. Collection and preparation: Breast milk was collected manually into tubes cleaned with distilled water and dried on a hot plate. For each sample, 2 mL of milk underwent removal of organic content, drying at 50 °C, and dilution with 5 mL distilled water. Precipitates were filtered. Analytical measurement: Processed samples were analyzed for mercury (Hg) and lead (Pb) concentrations using a Flame Atomic Absorption Spectrometer (FAAS). Calibration was performed using reference standards, with concentration curves obtained for both metals across the age groups. Data structure: Results were recorded as interval values [minimum measured value, maximum measured value] for each age. Example intervals (ppb): Pb at age 25: [1.876, 2.043]; Hg at age 25: [5.876, 15.863]; … ; Pb at age 40: [1.264, 2.313]; Hg at age 40: [3.861, 14.334]. Neutrosophic statistical framework: The neutrosophic variable X_N ∈ [X_L, X_U] includes an indeterminacy interval I_N ∈ [I_L, I_U], typically with I_L = 0 and I_U = (X_U − X_L)/X_U. For heavy metals, HM_N = HM_L + HM_U I_N, with HM_L and HM_U the lower and upper bounds of the measured interval, respectively. This representation preserves indeterminacy and allows analysis across intervals without collapsing to a single average. Computational algorithm: (1) Read interval data; (2) compute indeterminacy bounds I_N; (3) compute HM_N via HM_N = HM_L + HM_U I_N; (4) iterate for all intervals; (5) visualize outputs via neutrosophic and classical comparison graphs. Visualization and comparison: Combined plots contrasted neutrosophic vs classical (average±error) representations for Hg and Pb across ages, illustrating how classical methods collapse intervals and may misrepresent extremes, whereas neutrosophic graphs retain variability and uncertainty.

- Neutrosophic statistics effectively preserved and quantified indeterminacy in interval FAAS measurements of Hg and Pb in breast milk, whereas classical statistics reduced intervals to single average values, potentially obscuring extremes. - Across ages 25–40 years, measured interval ranges (ppb) included: Pb overall approximately 0.845–2.865; Hg approximately 2.873–15.876. Examples by age (ppb): - Hg: 25 [5.876, 15.863]; 31 [10.634, 15.865]; 36 [2.873, 15.876] (highest upper bound observed); 40 [3.861, 14.334]. - Pb: 29 [1.009, 2.890] (near the highest upper bound); 36 [2.375, 2.865]; 35 [2.213, 2.475]. - Neutrosophic analysis indicated elevated levels and wider uncertainty for Hg and Pb among mothers from industrial localities, aligning with environmental contamination concerns. - Comparative graphs (neutrosophic vs classical) demonstrated that classical averages can mislead decision-making in the presence of wide intervals, while neutrosophic outputs provided a range-sensitive, adaptable depiction of variation. - The study concludes that neutrosophic statistics are more accurate, adaptable, and suitable for interval data than classical methods when assessing heavy metal concentrations in breast milk.

The research question centered on whether neutrosophic statistics offer a superior framework for analyzing imprecise, interval-based measurements of heavy metals in breast milk. Findings support that neutrosophic modeling captures indeterminacy and extremes, providing richer information than classical averages. This is critical for exposure assessment in infants, where upper-tail concentrations may drive risk. Elevated Hg and Pb intervals in mothers from industrial zones underscore environmental exposure pathways and potential infant vulnerability due to exclusive breastfeeding in early life. The preserved interval information informs public health decision-making, enabling more cautious interpretations (e.g., considering upper bounds) and better communication of uncertainty. By demonstrating the method on FAAS-derived intervals, the study highlights practical applicability in environmental health monitoring and suggests that policies or interventions should account for measurement uncertainty rather than relying on point estimates that may understate risk.

This study applies neutrosophic statistics to interval FAAS measurements of Hg and Pb in breast milk from lactating mothers near industrial areas in Lahore, Pakistan. The approach preserves indeterminacy and variation, outperforming classical averaging that can mask extremes. Results reveal elevated and variable Hg and Pb concentrations across maternal ages, highlighting concerns for infant exposure. Contributions include: (1) a practical neutrosophic formulation for heavy metal interval data; (2) an algorithmic workflow for computation and visualization; and (3) empirical evidence that neutrosophic analysis is more informative for environmental health assessments. Future research should extend the approach to larger and geographically diverse cohorts, additional contaminants (e.g., Cd, As), temporal sampling across lactation stages, diet and environmental covariates, and integrate health outcome data. The proposed future plan suggests leveraging single-interval neutrosophic graphs to simplify analysis, assist in exploring potential links to breast cancer risk factors, and improve accuracy in heavy metal detection analyses.

The neutrosophic approach, as presented, is tailored to datasets where measurements are available as interval values; its applicability depends on having interval data. The method was demonstrated on Hg and Pb only, within a single urban-industrial context, and sample grouping details (two age groups) were not deeply stratified beyond ages 25–40. As noted by the authors, the approach was designed for heavy metal detection datasets, and authenticity relies on the availability and quality of interval measurements.