Organic contaminant sorption parameters should only be compared across a consistent system of linear functions.

Modeling contaminant sorption data using a linear model is very common; however, the rationale for whether the y-intercept should be constrained or not remains a subject of debate. This article justifies constraining the y-intercept in the linear model to zero. By doing so, one imposes consistency on the system of linear equations, allowing for direct comparison of the sorption coefficients.

Introduction

Over the years, numerous reports published in the scientific literature have focused on various aspects of contaminant sorption modeling, from the development of new mathematical formulas to modifications of the more common linear and nonlinear sorption models (e.g., Freundlich, Langmuir). Sorption represents the simplest and most experimentally accessible of contaminant interactions with soils, providing a quantitative parameter to describe the transfer of a solute in bulk solution across the soil-water interface to be “solubilized” in the solid-phase. For this reason, sorption parameters have been expanded to incorporate kinetic information [1], describe partitioning across different soil chemical domains [2], and discriminate the particularities regarding the contaminant of interest.

With few exceptions [3], conventional contaminant sorption models are strictly empirical in nature given that the thermodynamic state of the soil surface currently remains impossible to define. The simplest of the models, linear partitioning, is:where, CW = contaminant concentration in solution at equilibrium, CS = sorbed concentration of contaminant on the soil surface, and KD = the distribution coefficient of sorption referring to the affinity of the solid phase for the solute. Modeling sorption data with the linear sorption function may be conducted in two modes, either allowing the slope (representing the contaminant distribution coefficient or KD) and y-intercept (which has unresolved physical and chemical meaning) to “float” independently (i.e, unconstrained) or, forcing the y-intercept through zero. Which of these approaches is preferred or entirely justifiable, to our knowledge, is not satisfactorily answered in the scientific literature. For example, abundant discussion exists among the sample of published studies surveyed for this paper [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] regarding the appropriateness of the linear model and its mechanistic implications on contaminant sorption. However, most authors avoided any discussion regarding their treatment of the y-intercept although it is obvious from the sorption plots that this parameter was overwhelmingly forced through zero. Notable exceptions include Ruffino and Zanetti [14] and Dontsova et al. [15], where the y-intercept was allowed to float during the linear modeling of the sorption isotherms. The author has largely preferred forcing the y-intercept through zero to give KD estimates that are more readily comparable among different soils or treatments [16, 17]. Overall, it may be concluded that setting the y-intercept equal to zero represents a theoretically reasonable assumption that the soil was not been previously exposed to the contaminant. Thus, forcing the sorption model through zero strictly conforms to Eq. (1), where no y-intercept is depicted. However, this is a far from satisfying response, especially when it is apparent from the sorption data that the plot unexplainably deviates substantially from zero sorption. Aside from its practical value, this short article explains the mathematical justification for controlling the y-intercept, and its necessity for making meaningful comparisons among contaminant KD values.

In this paper, we propose that KD values can only be compared among the different soils if the system of linear functions (by which KD was extracted) describing sorption are consistent, which is defined has having at least one solution. For example, consider three linear functions with three unknowns [18]:

By simple elimination, a single solution exists for this system represented by a linear combination of three values (1,1,0). Graphically, this represents a single point in where the lines passing through three separate dimensions all intersect at (1,1,0). This is important as a consistent system of linear equations is considered independent, meaning that each function in the system is not proportional to any other of the functions, or in the same plane (such as for parallel functions). In practical terms, an independent equation provides unique information to the system that is not duplicated by any of the other equations. If a system of equations is consistent, then all equations in the system are independent. This means that all equations within the system are equally valuable for describing the system and directly comparable.

To our knowledge, a linear algebra-based justification for controlling the y-intercept has never been presented in linear sorption isotherm modeling.

Materials and methods

Here, this note draws on a recently published study [19] investigating the sorption of the insensitive munition compound, 2,4-dinitroanisole (DNAN), on different taxonomic soil “types”. All sorption isotherms were modeled with the linear sorption model (Eq. 1) using the R programming language [20] via the RStudio interface [21]. Modeling this data without any constraints on the fitting generated a system of linear equations, in the form of y = Ax + C, where A = the slope (representing the KD) and C = the y-intercept or offset. System consistency was tested using using the ‘matlib’ package [22] for R.

Results and discussion

As evident in Figure 1A, the DNAN sorption isotherms appeared strongly linear among the different soils, exhibiting small deviations in the y-intercept away from zero. Both the slope and y-intercept parameters (Table 1) were statistically significant (p < 0.05) for all soils. With the sorption curves parameterized, it was of interest to compare the KD values in order to gain a relative sense of how the different soils ranked in terms of their preference for the contaminant.

Figure 1

DNAN sorption isotherms and tests for consistency based on constraints on the y-intercept (A) DNAN sorption data where modeling of the linear sorption parameters were unrestrained, and (B) corresponding consistency test results (C) Graphical analysis of the consistency of the linear functions after forcing the y-intercept through zero. In B and C, the linear sorption functions were extrapolated past the origin for comparing the curves graphically. Black circles represent all intersection points for the linear sorption functions.

Table 1

Fit results and statistics for DNAN sorption modeling on different soil samples. Here, y-intercept was unconstrained in the linear regression modeling.

Individual parameters						Overall model
Sample name	term	estimate	SE	t-stat	p0.05	R2	F-stat	p0.05
Dismal1	y-int	21.706	4.710	4.608	4.902E-04	0.858	78.509	7.183E-07
Dismal1	slope (KD)	6.116	0.690	8.861	7.183E-07
Dismal5	y-int	6.008	1.540	3.900	1.825E-03	0.989	1126.665	5.133E-14
Dismal5	slope (KD)	6.120	0.182	33.566	5.133E-14
FtPolk2	y-int	4.578	0.802	5.710	7.173E-05	0.959	303.132	2.168E-10
FtPolk2	slope (KD)	1.079	0.062	17.411	2.168E-10
Holmes3	y-int	3.343	0.789	4.237	9.703E-04	0.860	79.952	6.482E-07
Holmes3	slope (KD)	0.521	0.058	8.942	6.482E-07
Huntsville1	y-int	6.285	2.268	2.771	1.588E-02	0.883	97.791	2.048E-07
Huntsville1	slope (KD)	1.813	0.183	9.889	2.048E-07
Laurel3	y-int	14.391	1.582	9.097	5.332E-07	0.994	2147.739	8.007E-16
Laurel3	slope (KD)	15.256	0.329	46.344	8.007E-16
Laurel4	y-int	12.372	1.253	9.878	2.075E-07	0.988	1068.427	7.220E-14
Laurel4	slope (KD)	4.677	0.143	32.687	7.220E-14
Morrow1	y-int	14.373	1.908	7.531	4.302E-06	0.981	683.179	1.268E-12
Morrow1	slope (KD)	6.464	0.247	26.138	1.268E-12
Morrow3	y-int	9.367	1.070	8.756	8.215E-07	0.984	776.227	5.605E-13
Morrow3	slope (KD)	2.798	0.100	27.861	5.605E-13
Morrow5	y-int	4.814	0.919	5.236	1.606E-04	0.983	758.667	6.489E-13
Morrow5	slope (KD)	2.111	0.077	27.544	6.489E-13
Ohiopyle3	y-int	16.284	1.436	11.342	4.098E-08	0.992	1701.271	3.607E-15
Ohiopyle3	slope (KD)	8.348	0.202	41.246	3.607E-15
Stewart1	y-int	8.140	0.417	19.507	5.202E-11	0.997	4896.307	3.856E-18
Stewart1	slope (KD)	2.572	0.037	69.974	3.856E-18
Susequehenna1	y-int	10.069	1.066	9.443	3.485E-07	0.985	836.766	3.465E-13
Susequehenna1	slope (KD)	2.795	0.097	28.927	3.465E-13
Susequehenna3	y-int	13.053	3.979	3.280	5.969E-03	0.078	1.095	3.144E-01
Susequehenna3	slope (KD)	0.346	0.330	1.046	3.144E-01
Toledo1	y-int	7.802	2.027	3.850	2.008E-03	0.945	224.753	1.390E-09
Toledo1	slope (KD)	2.705	0.180	14.992	1.390E-09
Toledo2	y-int	7.445	2.152	3.459	4.232E-03	0.881	96.243	2.246E-07
Toledo2	slope (KD)	1.711	0.174	9.810	2.246E-07
Twin2	y-int	5.625	0.578	9.725	2.484E-07	0.995	2733.787	1.682E-16
Twin2	slope (KD)	2.663	0.051	52.286	1.682E-16
Woodlake1	y-int	10.144	2.748	3.691	2.718E-03	0.871	87.947	3.770E-07
Woodlake1	slope (KD)	2.360	0.252	9.378	3.770E-07
Woodlake2	y-int	4.744	1.855	2.557	2.386E-02	0.922	153.060	1.448E-08
Woodlake2	slope (KD)	1.986	0.161	12.372	1.448E-08
Woodlake6	y-int	2.328	1.041	2.237	4.340E-02	0.917	144.487	2.048E-08
Woodlake6	slope (KD)	0.939	0.078	12.020	2.048E-08

Converting the functions into the standard form (Ax + By = C), the information was stored in a 20 × 3 matrix and tested for consistency. To be considered as consistent, there must be a unique solution for the entire system. This is represented graphically by a single point where all of the lines intersect in two-dimensional space (given the two unknowns in the equations). Analysis [20] showed that the linear sorption isotherms intersected at multiple points (indicated by the circles in Figure 1B), suggesting that this system of equations was inconsistent. Given that each sorption curve exhibited its own unique slope and y-intercept, it is reasonable to assume that the sorption curves were consistent at least on a pairwise basis. However, there was no mathematical basis for comparing KD values across the entire system without first imposing constraints on the fitted linear sorption model.

The problem of inconsistent linear functions can be circumvented by forcing the y-intercept to zero during the linear regression analysis. As a result, this gave a new 20 × 3 matrix where the last column (C) was populated by zeroes, representing what is known as a homogeneous system of linear equations. Under this scenario, all of the curves intersected (Figure 1C) at a single point at (0,0), making the origin the unique solution for the entire system. Thus, statistically comparing the KD values across the entire system became mathematically viable with this step. As a consequence of constraining the y-intercept = 0, the KD values were on average 24% higher (representing an average 16% increase in the estimates’ standard error), showing the “pull” of the y-intercept parameter on the overall linear regression (Table 2). However, all models remained highly significant, with the computed p-values for the F-statistic ranging from 10−9 – 10−17.

Table 2

Fit statistics for the DNAN sorption modeling when the y-intercept when the y-intercept = 0.

Individual parameters						Overall model
Sample name	term	estimate	SE	t-stat	p0.05	R2	F-stat	p0.05
Dismal1	slope (KD)	8.707	0.626	13.903	1.384E-09	0.932466	193.302	1.38E-09
Dismal5	slope (KD)	6.716	0.141	47.665	6.788E-17	0.993876	2271.945	6.79E-17
FtPolk2	slope (KD)	1.377	0.060	22.842	1.763E-12	0.973869	521.7696	1.76E-12
Holmes3	slope (KD)	0.730	0.046	15.924	2.301E-10	0.947681	253.5891	2.30E-10
Huntsville1	slope (KD)	2.239	0.122	18.382	3.363E-11	0.960217	337.9125	3.36E-11
Laurel3	slope (KD)	17.689	0.502	35.236	4.510E-15	0.98885	1241.593	4.51E-15
Laurel4	slope (KD)	5.835	0.231	25.283	4.392E-13	0.978568	639.2423	4.39E-13
Morrow1	slope (KD)	7.984	0.319	25.048	4.993E-13	0.978173	627.3933	4.99E-13
Morrow3	slope (KD)	3.528	0.142	24.916	5.366E-13	0.977947	620.8228	5.37E-13
Morrow5	slope (KD)	2.448	0.070	34.871	5.210E-15	0.988618	1215.995	5.21E-15
Ohiopyle3	slope (KD)	10.217	0.374	27.319	1.516E-13	0.981586	746.3041	1.52E-13
Stewart1	slope (KD)	3.171	0.107	29.570	5.091E-14	0.984241	874.3937	5.09E-14
Susequehenna1	slope (KD)	3.552	0.146	24.305	7.544E-13	0.976849	590.7205	7.54E-13
Susequehenna3	slope (KD)	1.255	0.234	5.366	9.952E-05	0.672858	28.79483	9.95E-05
Toledo1	slope (KD)	3.285	0.140	23.400	1.268E-12	0.97507	547.5729	1.27E-12
Toledo2	slope (KD)	2.214	0.128	17.275	7.755E-11	0.955188	298.4187	7.75E-11
Twin2	slope (KD)	3.079	0.077	40.205	7.237E-16	0.991413	1616.42	7.24E-16
Woodlake1	slope (KD)	3.129	0.195	16.074	2.032E-10	0.948601	258.3765	2.03E-10
Woodlake2	slope (KD)	2.333	0.101	23.036	1.570E-12	0.974297	530.6801	1.57E-12
Woodlake6	slope (KD)	1.087	0.047	23.110	1.503E-12	0.974457	534.0845	1.50E-12

Not only does this simple approach allow for comparing KD values across the system discussed here, but also allows for expanding the size of the system to include legacy data or, conversely, new sorption information. On the odd chance that the system of linear functions are consistent, then forcing the y-intercept through zero would not be necessary. Alternatively, the y-intercept could be forced through a positive, non-zero value, but the value chosen for the offset would need to be determined as well as applicable across the entire system.

Declarations

Author contribution statement

Mark A Chappell: Conceived and designed the experiments; Analyzed and interpreted the data; Wrote the paper.

Jennifer M Seiter, Haley M West, Lesley F Miller, Maria E Negrete, Beth E Porter, Matthew A. Middleton: Performed the experiments.

Joshua J LeMonte, Cynthia L Price: Analyzed and interpreted the data.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.