Advancing analytical and graphical methods for binary and ternary mixtures: The toxic interactions of divalent metal ions in human lung cells

Abstract

Humans are exposed to various environmental chemicals, particles, and pathogens that can cause adverse health outcomes. These exposures are rarely homogenous but rather complex mixtures in which the components may interact, such as through synergism or antagonism. Toxicologists have conducted preliminary investigations into binary mixtures of two components, but little work has been done to understand mixtures of three or more components. We investigated mixtures of divalent metal ions, quantifying the toxic interactions in a human lung model. Eight metals were chosen: heavy metals cadmium, copper, lead, and tin, as well as transition metals iron, manganese, nickel, and zinc. Human alveolar epithelial cells (A549) were exposed to individual metals and sixteen binary and six ternary combinations. The dose-response was modeled using logistic regression in R to extract LC₅₀ values. Among the individual metals, the highest and lowest toxicity were observed with copper at an LC₅₀ of 102 μM and lead at an LC₅₀ of 5639 μM, respectively. First and second-order interaction coefficients were obtained using machine learning-based linear regression in Python. The resulting second-degree polynomial model formed either a hyperbolic or elliptical conic section, and the positive quadrant was used to produce isobolograms and contour plots. The strongest synergism and antagonism were observed in cadmium-copper and iron-zinc, respectively. A three-way interaction term was added to produce full ternary isobologram surfaces, which, to our knowledge, are a significant first in the toxicology literature.

1. Introduction

Humans are exposed to a variety of toxicants in the environment, including through inhalation. This exposure route is of particular concern in the occupational setting, where workers are repeatedly exposed to higher-than-normal concentrations of aerosolized substances such as minerals, metals, exhaust fumes, and organics [1,2]. The field of toxicology aims to achieve a thorough understanding of these exposures, including the mechanisms of toxic action and the overall risks involved. Because humans are often exposed to a complex mixture of toxicants rather than singular chemicals or particles, understanding the toxic interactions in mixtures remains a major enduring challenge for the field [3,4].

Traditionally, mixture interactions have been analyzed under concentration addition and independent action frameworks [5]. Concentration addition is the most common assumption, which presumes that toxicants have a similar mode of action such that any quantity of a toxicant could be replaced with the equivalent quantity of another. Independent action instead proposes that toxicants have dissimilar modes of action. Calculated values, such as toxic units and toxicity indices, have been applied to approximate the overall potency of a mixture or its broader characteristics, typically the presence of synergism or antagonism [6,7].

Several graphical methods have also been developed for binary mixtures, with isobolograms being the most used. In two dimensions, an isobologram contains axes corresponding to the two components and a curve representing all mixture ratios that produce a biological effect at a given threshold [[8], [9], [10]]. This is typically a set of EC₅₀ or LC₅₀ values equivalent to a contour of the three-dimensional response surface [[11], [12], [13]]. Isobolograms often include a line of additivity connecting the two intercepts (e.g., individual component EC₅₀ values). If the curve falls below or above the additive line, the behavior is considered synergistic or antagonistic, respectively.

Most of the theory behind mixtures toxicology is borrowed from or co-developed with concepts from pharmacology [14]. Most commonly, the pharmaceutical industry is concerned with undesired interference between two drugs a patient may take and the design of synergistic treatments [13,15]. The motivations impact toxicology, where there are few tools for analyzing mixtures of more than two components [16,17]. Because higher-order mixtures have exponentially more possible combinations, the potential resources required also serve as a deterrent [18]. We aim to show that it is possible to delve into ternary mixtures and expand our understanding through innovative techniques, both analytical and graphical.

For this study, we focused on metal ion mixture toxicity in the lungs [19,20]. Four heavy metals (cadmium, copper, lead, and tin) and four transition metals (iron, manganese, nickel, and zinc) were chosen to represent a range of substances where human occupational co-exposures might occur. All eight metals are mined from the earth's crust, with potential inhalation exposure during the mining and extraction. The metals are often present as impurities amongst each other; for example, cadmium may be obtained during the processing of zinc ores [21]. Copper, iron, lead, manganese, nickel, tin, and zinc are all commonly used in alloys, leading to workers' potential inhalation of mixed particles during their production or the subsequent manufacture of parts [22,23]. There are also other shared applications, such as batteries, ceramics, electronics, paints, and welding [24,25]. Once metal particles deposit in the lungs, ions are released, especially if they are compartmentalized into the acidic environment of macrophages [26]. Prior studies have shown the presence of divalent ions of cadmium, copper, iron, manganese, nickel, zinc, and other metals in the lungs, as well as potential mechanisms behind some of their interactions [[27], [28], [29], [30]]. While the speciation of individual metals may vary, the focus on divalent ions in this study is to increase the probability of observing interactions due to related mechanisms of action.

Using a medium-throughput approach, we quantified binary and ternary mixture interactions for toxicity of the eight metal ions toward human lung cells. These interactions were visualized using isobolograms and response surfaces. To our knowledge, the three-dimensional ternary isobolograms are the first in the toxicology literature [31].

2. Materials and methods

Materials. All chemicals and supplies - cadmium chloride, copper chloride, iron chloride, lead nitrate, manganese chloride, nickel chloride, tin chloride, zinc chloride, cell culture supplies, and CellTiter 96AQ One Solution Cell Proliferation (MTS) assay - were obtained from Fisher Scientific (Waltham, MA, USA). A549 human alveolar epithelial cells (Catalog number: CCL-185 and RRID: CVCL_0023) were obtained from ATCC (Manassas, VA, USA). Prior to initial use, cells were authenticated via STR analysis and tested for mycoplasma contamination via Hoechst DNA stain, agar culture, and PCR-based assay (none detected).

Metal ion cytotoxicity. A549 cells (passages 14–24) were cultured at 37 °C and 5 % CO₂ using RPMI 1640 media supplemented with 10 % fetal bovine serum and 1 % penicillin-streptomycin solution (5000 U/mL). After reaching 95 % confluency, cells were plated at 10,000 cells per well on 96-well plates and incubated 24 h before exposure. All chemicals were dissolved in fresh media and bath sonicated for 15 min. Each well was dosed by replacing the media with 200 μL of metal ion solution. A 0.1 % v/v Triton X-100 solution was used as a positive control for cell death, and an untreated control was used. Exposed cells were incubated for 24 h before adding 1:5 MTS:media. Cells were then incubated for 2 h before 490 nm absorbance reading on a Biotek H1 Synergy plate reader obtained from Agilent (Santa Clara, CA, USA). All data are an average of four replicates.

Mixtures interactions model. For a mixture of k components and a given response (e.g., a biological endpoint, such as live/dead), the response space has k+1 dimensions. A set response level results in k-dimensional iso-contours. For example, the response surface is three-dimensional in a binary mixture (k = 2), and the iso-contours at R = 0, R = 0.5, R = 1, etc., are two-dimensional. Assuming concentration addition yields purely additive behavior in a straight line, described by Equation (1) for binary mixtures and adding only a two-way interaction term results in hyperbolic behavior, as described by Equation (2). However, self-interactions are also possible for a 2nd-degree polynomial model, resulting in Equation (3).

$$b_1{x}_1+b_2{x}_2-1=0$$ (1)

$$b_{12}{x}_1{x}_2+b_1{x}_1+b_2{x}_2-1=0$$ (2)

$$b_{11}{x_1}^2+b_{12}{x}_1{x}_2+b_{22}{x_2}^2+b_1{x}_1+b_2{x}_2-1=0$$ (3)

Extending this model to ternary mixtures (k = 3), the additive behavior results in a plane instead of a line, as described by Equation (4). The iso-contour is now a surface with curvature. Among the ten possible third-order terms, we only included the three-way interaction term to avoid overfitting to limited data and overcomplicating the analyses. This model is described by Equation (5).

$$b_1{x}_1+b_2{x}_2+b_3{x}_3-1=0$$ (4)

$$b_{123}{x}_1{x}_2{x}_3+b_{11}{x_1}^2+b_{22}{x_2}^2+b_{33}{x_3}^2+b_{12}{x}_1{x}_2+b_{13}{x}_1{x}_3+b_{23}{x}_2{x}_3+b_1{x}_1+b_2{x}_2+b_3{x}_3-1=0$$ (5)

Modeling and statistics. Dose-response curves and LC₅₀ values for all exposures were modeled in R using logistic regression from a binomial generalized linear model (GLM). To obtain all first and second-order interaction coefficients, machine learning linear regression was conducted using numpy and scikit-learn packages in Python. Data were then processed to generate binary/ternary isobolograms and contour plots using the ggpubr, orca, plotly, tidyverse, and viridis packages in R [[32], [33], [34], [35], [36]].

3. Results and Discussion

Exposures were conducted for individual, binary, and ternary combinations in sequence, with concentrations and mixing ratios chosen to maximize information gain at each step. Although this study did not explore every possible mixture combination, each evaluated combination represents a plausible co-exposure scenario due to shared commercial and industrial applications for the component metals.

The dose-response curves for the eight individual metal ions were presented in Fig. 1. Each point averages four replicates, while simple logistic regression was used to model the curve. The LC₅₀ values in Table 1, representing the concentration of a toxicant that causes 50 % cell death, were calculated from the model coefficients. The dose-response curves and LC₅₀ values for mixtures were calculated similarly and can be found in the supplemental information (Fig. S1 and Table S1). Among the individual divalent metal ions, copper was the most toxic, with an LC₅₀ of 102 μM, while lead was the least toxic, with an LC₅₀ of 5639 μM. The ranked individual toxicities align strongly with a prior study for Cu, Fe, Mn, and Zn [37]. The only exception is Pb, a discrepancy that may warrant future investigation.

Fig. 1.

Dose-response curves for the cytotoxicity of individual metal ions to A549 cells. The dose in μM is reported on a log axis (3.125 μM–204800 μM), while the response ranges from 0 to 1 (0 %–100 % viable cells). Metals include cadmium (Cd), copper (Cu), iron (Fe), manganese (Mn), nickel (Ni), lead (Pb), tin (Sn), and zinc (Zn). The coordinates for all plots are identical.

Table 1.

LC₅₀ values in μM for all individual metal ion exposures as determined by logistic regression.

Metal Ion	LC50 (μM)
Cd2+	236
Cu2+	102
Fe2+	4850
Mn2+	192
Ni2+	138
Pb2+	5639
Sn2+	4764
Zn2+	701

Sixteen binary combinations of metal ions were tested for cytotoxicity to determine both self-interactions (e.g., Cd is the basic linear effect of cadmium while Cd:Cd is cadmium interacting with itself) and mixed interactions (e.g., Fe:Mn is the interaction between Fe and Mn). These interactions can be caused by parallel action, competition for receptors, or other mechanisms along the toxicity pathway. The resulting isobolograms and contour plots for binary mixtures are shown in Fig. 2, Fig. 3, respectively. The solid gray lines represent strictly additive behavior (Equation (1)) with no 2nd order interaction terms. The dashed colored lines show the isobologram as obtained from modeling experimental data. Equation (2) represents a simple hyperbolic model without self-interaction terms for each component, while Equation (3) represents the full second-degree polynomial model used in this study, which is also the general form for a conic section.

Fig. 2.

Binary mixture isobolograms for all tested combinations. Solid gray lines represent a theoretical additive relationship, while dashed colored lines show the observed behavior. Panels are labeled as [Component 1: Component 2]. The model used is described by Equation (3).

Fig. 3.

Binary mixture contour plots showing an estimated response gradient. The darker region represents low cytotoxicity, while the brighter region represents high cytotoxicity. Panels are labeled as [Component 1: Component 2]. The model used is described by Equation (3).

Compared to the simple hyperbolic model, the full second-degree model allows for both hyperbolic and elliptical behavior [[38], [39], [40]]. The full model can better capture strongly antagonistic behavior, as seen with Cu:Ni, Fe:Zn, and Zn:Cd. When using the simple model, the extreme points for Cu:Ni and Zn:Cd cause the hyperbolic to become unstable and inverted, while the extreme point for Fe:Zn causes a false reflection to the lower branch (Fig. S2). The full model does not encounter these difficulties, and additional self-interaction terms present constraints that result in higher consistency across the data.

A summary of the binary mixture results is reported in Table 2. Amongst the 16 binary combinations, seven were hyperbolic, and nine were elliptical. Regarding overall interaction, five were synergistic, five were additive, and six were antagonistic. The strongest synergism was observed in Cd:Cu, while the strongest antagonism was observed in Fe:Zn. With a set of only eight similar toxicants, the full range of interaction modes was observed. Previous studies on metal ion mixtures toxicity in lung cells are rare, though there are adjacent studies that may use related toxicants (e.g., metal oxide nanoparticles) or animal models [37,41,42]. Within this narrow range of potentially comparable results, there is mixed agreement (e.g., the Fe:Mn interaction is found to be synergistic in Yuan et al. but antagonistic in Wang et al., compared to the synergism reported here). The richness of mixture interactions found here emphasizes the continued need to expand our understanding of how toxicants can combine and produce novel toxic effects.

Table 2.

Summary of overall results for binary mixtures. The shape of the conic section is either hyperbolic (H) or elliptical (E) as determined by the discriminant. The resulting interaction is labeled as synergistic (S), additive (−), or antagonistic (A).

Mixture	Shape	Result
Cd: Cu	H	S
Cd: Mn	H	S
Cd: Ni	H	–
Cu: Ni	E	A
Fe: Mn	H	S
Fe: Ni	E	–
Fe: Zn	E	A
Mn: Cu	E	–
Mn: Ni	H	S
Pb: Cu	H	–
Pb: Sn	E	–
Sn: Cu	E	A
Zn: Cd	E	A
Zn: Cu	E	A
Zn: Mn	E	A
Zn: Ni	H	S

Six ternary combinations of metal ions were tested to capture different behaviors and potential co-exposure scenarios. The resulting LC₅₀ values and previously determined 1st and 2nd order terms were used to calculate three-way interaction coefficients. These were then used to plot ternary isobolograms, shown in Fig. 4. The grayscale triangles are planes of additivity (Equation (4)), while the colored surfaces are the ternary isobolograms (Equation (5)). While there are ten possible 3rd order interaction terms (e.g., x₁²x₂ and x₁³), we chose only to include the three-way term (x₁:x₂:x₃) to avoid overfitting the limited data and overcomplicating the analyses.

Fig. 4.

Ternary mixture isobolograms for all tested combinations. Grayscale triangular planes represent theoretical additive relationships. Colored surfaces represent the observed behavior. Axes were re-labeled for visibility. The model used is described by Equation (5).

The ternary Zn:Mn:Cu was found to be synergistic, though this only had the effect of dampening the overall antagonism from the binary interactions. The ternary Pb:Sn:Cu was additive, and the remainder were antagonistic. The overall interaction behavior of the mixture (account for both binary and ternary interactions) can be identified as synergistic or antagonistic when a given point on the surface falls below or above the plane of additivity, respectively.

We have not seen any ternary isobolograms in the toxicology literature, and only four examples in the pharmacology literature. Of those, three present a scatter of points, while only one shows a mesh isobologram surface [[43], [44], [45], [46]]. Here, we not only generated fully-resolved surfaces but could do so for multiple combinations effectively. We also quantified the underlying interactions using our model, whereas the existing studies only present averaged or single-point index values. Visualizing the entire surface is important because the isobologram can cross the plane of additivity in three dimensions, indicating an inflection in overall behavior that is important to consider when estimating risk [4].

4. Conclusions

Advancing our understanding of mixtures has been a persistent challenge in toxicology and human health risk assessment. In this study, we investigated binary and ternary combinations of divalent metal ions that have a wide range of commercial and industrial applications, leading to a variety of mixture exposure scenarios. We used an efficient design to collect data on their toxicity in an in vitro human lung model. Using a combination of logistic regression, linear regression, and a mixture interactions model, we generated a large set of fully-resolved 2D and 3D isobolograms. To our knowledge, this is the first such study in the toxicology literature [31].

There are many options to continue along the trajectory of the methods presented here. Expanding the ternary data with additional points would help improve estimates for the interaction terms. This would also allow the computation of ternary isobolograms using all 3rd order terms, possibly even with nonlinear terms [47]. Such an implementation might utilize the marching cubes algorithm or similar methods for estimating 3D contours from 4D data [48]. Other future work could involve expanding to more diverse combinations of toxicants that are still suspected to act along similar pathways, exploring new biological models or endpoints, or introducing heterogenous factors such as environmental conditions. As high-throughput experimental designs continue to progress, it may even be feasible to develop sophisticated models for quaternary mixtures or beyond. Sufficiently thorough screening could also pave the way for predicting interactions across classes of toxicants.

CRediT authorship contribution statement

James Y. Liu: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. Jonathan M. Beard: Writing – original draft, Methodology, Investigation, Formal analysis, Data curation. Saber Hussain: Resources, Project administration, Funding acquisition. Christie M. Sayes: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Data availability statement

Data will be made available on request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

The following is the supplementary data to this article.