Construction of reliable QSPR models for predicting the impact sensitivity of nitroenergetic compounds using correlation weights of the fragments of molecular structures

Abstract

Impact sensitivity is a critical property of energetic molecules, indicating their tendency to react when subjected to mechanical stimuli such as impact. Nitro compounds are widely used as explosives across industrial, military, and civilian applications, making their safe handling a significant concern for engineers and scientists working with these materials. Predicting whether a molecule has the potential to pose safety risks is therefore of great importance. This study aimed to develop a QSPR model for predicting the impact sensitivity of 404 nitro compounds using the Monte Carlo algorithm implemented in CORAL-2023 software. The Simplified Molecular Input Line Entry System (SMILES) was employed to represent the molecular structures, while correlation weight descriptors were computed. Four target functions (TF0, TF1, TF2, and TF3) were used to generate robust models. The first model applied Monte Carlo optimization without the inclusion of IIC (information index of correlation) or CII (correlation index of information); the second incorporated IIC; the third incorporated CII; and the fourth applied both IIC and CII. Comparative statistical analyses indicated that the model integrating both IIC and CII demonstrated superior predictive performance, with the best results observed in split 2 (R²_Validation = 0.7821, IIC_Validation = 0.6529, CII_Validation=0.8766, Q²_Validation = 0.7715, and $\:{r}_{m}^{2}$ = 0.7464).

Introduction

The sensitivity of energetic molecules is a critical parameter for the explosives industry. These materials may detonate under various stimuli, including shock, impact, electric sparks, friction, and heat, which can occur during their synthesis, storage, handling, or application. Among the different types of sensitivity, impact sensitivity is particularly significant, as it has been implicated in numerous workplace accidents^1,2,3,4. Therefore, determining the impact sensitivity of energetic materials is essential for ensuring safety.

Experimentally, impact sensitivity is commonly assessed using the drop-weight impact test. In this method, milligram quantities of the energetic material are placed between a flat steel anvil and a steel striker. A 2.5 kg weight is dropped from a specified height (in centimeters) onto the striker plate, and the occurrence or absence of an explosion is recorded. The height at which there is a 50% probability of an explosion is denoted as H₅₀^5;6. However, the reproducibility of these experimental measurements is often affected by factors such as sample size, morphology, variations in test apparatus, and differing test conditions^7,8,9,10. Moreover, experimental determination of H₅₀ is time-consuming, and it is not feasible for compounds that have not yet been synthesized. Consequently, there is growing interest in theoretical methods to establish quantitative relationships between molecular structure and the impact sensitivity of energetic molecules¹¹.

Quantitative structure–property relationships (QSPR), along with related approaches such as QSAR (quantitative structure–activity relationships) and QSTR (quantitative structure–toxicity relationships), use mathematical equations to predict a desired property (e.g., property, activity, or toxicity) based on molecular descriptors derived from structural features^2;6,11;12. These models allow chemists to estimate properties of newly designed or undeveloped compounds^13,14. In recent years, the CORAL (CORrelation And Logic: http://www.insilico.eu/coral/). software, which leverages the Monte Carlo algorithm, has emerged as a powerful tool for building QSPR models using Simplified Molecular Input Line Entry System (SMILES) notations^15,16,17. Through Monte Carlo optimization, the software extracts optimal descriptors of correlation weight (DCW) from the SMILES representation of chemical structures, and these descriptors are used to calculate the target function^18,19,20. Literature indicates that the predictive performance of QSPR and QSAR models can be significantly enhanced through two statistical benchmarks: the index of ideality of correlation (IIC) and the correlation intensity index (CII). These metrics improve the models’ ability to account for both the correlation coefficient and the residual values of the test molecules’ endpoints^21,22,23.

The primary objective of this study is to develop robust QSPR models to estimate the impact sensitivity of 404 nitro energetic molecules using the Monte Carlo algorithm implemented in the CORAL program. Four different approaches are employed: the balance of correlation method without IIC or CII, the balance of correlation method with IIC, the balance of correlation method with CII, and the balance of correlation method incorporating both IIC and CII. The performance of these models is compared using four randomly generated dataset subsets: active training, passive training, calibration, and validation sets. To enhance the mechanical interpretation, correlation weights are used to identify structural features associated with increased or decreased impact sensitivity.

Methods

Data

To establish predictive QSPR models, a dataset comprising energetic compounds with known impact sensitivity (H₅₀) values was compiled from previously published literature sources^10,24,25. After excluding duplicate compounds and those with ambiguous molecular structures, a total of 404 unique compounds were retained. The molecular structures of these compounds were drawn using the free software BIOVIA Draw and converted into Simplified Molecular Input Line Entry System (SMILES) notations for computational analysis.

The impact sensitivity values, originally expressed as H₅₀ (in cm), were transformed into their corresponding logarithmic scale (log H₅₀), which served as the endpoint for QSPR modeling. The log H₅₀ values ranged from 0.699 to 2.690, encompassing a broad spectrum of sensitivity characteristics. Comprehensive details, including compound IDs, SMILES notations, observed and predicted log H₅₀ values, and information regarding the applicability domain of the constructed model (using the TF3 target function for four splits), are provided in the supplementary information (Table S1).

The dataset was divided into four splits to facilitate robust model development and evaluation. Each split was randomly partitioned into four subsets: active training, passive training, calibration, and validation sets. Table 1 demonstrates that the four splits employed in this study are non-identical, and the percent identity among the splits was calculated as follows:

Table 1 Percentage identity of splits 1–4 within the active training, passive training, calibration, and validation sets.

$$\:Identity\:\left(\%\right)=\:\frac{{N}_{i,\:j}}{0.5\times\:{(N}_{i}+{N}_{i})}\times\:100$$

(1)

where $\:{N}_{i,\:j}$ denote the number of elements that are assigned to the same set in both the $\:{i}^{th}$and $\:{j}^{th}$ splits (i.e., the set can be active training, passive training, calibration, or validation). Similarly, $\:{N}_{i}$ represents the number of elements assigned to a particular set in the $\:{i}^{th}$ split, while $\:{N}_{\:j}$ represents the number of elements assigned to the same set in the $\:{j}^{th}$ split.

The roles and tasks of each set in QSPR modeling have been thoroughly described in previous studies^26,27,28.

Optimal descriptor

The hybrid optimal descriptors, DCW (T, N), were utilized to construct the QSPR model for energetic compounds. These descriptors are derived by integrating molecular attributes obtained from SMILES notations and molecular graphs. By combining these two types of molecular representations, this hybrid approach improves the statistical quality of the model compared to those based exclusively on SMILES or molecular graphs²⁹.

The DCW of the hybrid optimal descriptors is calculated using the following mathematical function:

$$\:{}^{\text{H}\text{y}\text{b}\text{r}\text{i}\text{d}}\text{D}\text{C}\text{W}\left({\text{T}}^{\text{*}},\:{\text{N}}^{\text{*}}\right)={DCW}_{SMILES}\left({\text{T}}^{\text{*}},\:{\text{N}}^{\text{*}}\right)+{DCW}_{HSG}\left({\text{T}}^{\text{*}},\:{\text{N}}^{\text{*}}\right)$$

(2)

The following mathematical equations are used to calculate the Descriptors of Correlation Weight (DCW) for both SMILES and Hierarchical Structural Graph (HSG) representations.

$$\begin{aligned} ^{{{\text{SMILES}}}} {\text{DCW}}({\text{T}}^{{\text{*}}} ,\:{\text{N}}^{{\text{*}}} ) = & \sum {\text{CW}}\left( {{\text{S}}_{{\text{K}}} } \right) + \sum {\text{CW}}\left( {{\text{SS}}_{{\text{K}}} } \right) + \sum {\text{CW}}\left( {{\text{SSS}}_{{\text{K}}} } \right) \\ & + {\text{CW}}\left( {{\text{BOND}}} \right) + {\text{CW}}\left( {{\text{NOSP}}} \right) + {\text{CW}}\left( {{\text{HALO}}} \right) + {\text{CW}}\left( {{\text{HARD}}} \right) \\ \end{aligned}$$

(3)

$$\begin{aligned} ^{{HSG}} DCW\:({\text{T}}^{{\text{*}}} ,\:{\text{N}}^{{\text{*}}} ) = & \sum {\text{CW}}\left( {EC0_{k} } \right) + \sum {\text{CW}}\left( {EC1_{k} } \right) + \sum {\text{CW}}\left( {EC2_{k} } \right) \\ & + \sum {\text{CW}}\left( {EC3_{k} } \right) + \sum {\text{CW}}\left( {pt2_{k} } \right) + \sum {\text{CW}}\left( {pt3_{k} } \right) + \sum {\text{CW}}\left( {pt4_{k} } \right) \\ & + \sum CW\left( {VS2_{k} } \right) + \sum CW\left( {VS3_{k} } \right) + \sum {\text{CW}}\left( {nn_{k} } \right) \\ \end{aligned}$$

(4)

The notations employed in Eqs. (3) and (4) are explained in detail in Table 2. Based on the numerical values of the correlation weights for the descriptors, the predictive model for logH₅₀ can be formulated as follows:

Table 2 Detailed explanation of the SMILES and HSG features used in the established models for logH₅₀.

$$\:{LogH}_{50}\:={\text{C}}_{0}+{\text{C}}_{1}\times\:{}^{Hybrid}DCW\left({\text{T}}^{\text{*}},\:{\text{N}}^{\text{*}}\right)$$

(5)

where C₀ and C₁ are regression coefficients, and T and N represent parameters of the Monte Carlo optimization procedure. In this optimization process, the numerical values of T and N are determined to achieve the optimal statistical criteria for the calibration set, denoted as T^∗ and N^{∗ 16,30,31}.

Monte Carlo optimization

Equations (3) and (4) require the numerical values of the correlation weights (CW). Monte Carlo optimization is employed to calculate these numerical values using the CORAL software. In this study, to develop robust QSPR models, four target functions for the Monte Carlo optimization are evaluated: TF0 (correlation balance without CII or IIC), TF1 (correlation balance modified with IIC), TF2 (correlation balance modified with CII), and TF3 (correlation balance modified with both IIC and CII). The statistical quality of each target function is then compared. The mathematical expressions for the four target functions are defined as follows:

$$\:{\text{T}\text{F}}_{0}={\text{r}}_{\text{A}\text{T}\text{R}\text{N}}+{\text{r}}_{\text{P}\text{T}\text{R}\text{N}}-\left|{\text{r}}_{\text{A}\text{T}\text{R}\text{N}}-{\text{r}}_{\text{P}\text{T}\text{R}\text{N}}\right|\times\:Const$$

(6)

$$\:{\text{T}\text{F}}_{1}={\text{T}\text{F}}_{0}+{\text{I}\text{I}\text{C}}_{\text{C}\text{A}\text{L}}\times{\text{W}}_{\text{I}\text{I}\text{C}}$$

(7)

$$\:{\text{T}\text{F}}_{2}={\text{T}\text{F}}_{0}+{\text{C}\text{I}\text{I}}_{\text{C}\text{A}\text{L}}\times{\text{W}}_{\text{C}\text{I}\text{I}}$$

(8)

$$\:{\text{T}\text{F}}_{3}={\text{T}\text{F}}_{0}+{\text{I}\text{I}\text{C}}_{\text{C}\text{A}\text{L}}\times{\text{W}}_{\text{I}\text{I}\text{C}}+{\text{C}\text{I}\text{I}}_{\text{C}\text{A}\text{L}}\times\:{\text{W}}_{\text{C}\text{I}\text{I}}$$

(9)

Where r_ATRN represents the correlation coefficient between the predicted and observed logH₅₀ values for the active training set, r_PTRN represents the correlation coefficient for the passive training set. The constants (empirical constant), W_IIC, and W_CII generally have fixed values; specifically, these values are 0.1, 0.5, and 0.3, respectively. The flow chart of the optimization of the correlation weight with target function (TF) is cited in the literature³².

A literature survey indicates that the inclusion of IIC and CII enhances the statistical quality of the validation set^33,34,35. Data from the calibration set are used to calculate IIC, and the mathematical equations for calculating IIC and CII are well-established in the literature^21,33,36.

A flow diagram representation of the QSPR model established and methodology cited in the literature³⁷ and depicted in Fig. 1.

Fig. 1

Flow diagram representation of the QSPR model established methodology.

Results and discussion

In the present manuscript, the CORAL program was used to apply various methods for generating 16 QSPR models. These methods include: the balance of correlation approach without IIC or CII (TF0, Eq. 6), the balance of correlation approach with IIC (TF1, Eq. 7), the balance of correlation approach with CII (TF2, Eq. 8), and the balance of correlation approach with both IIC and CII (TF3, Eq. 9). The statistical outcomes of each method are then compared.

Comparison of target methods

In this study, to achieve reliable statistical results, four models were generated for each method using hybrid optimal descriptors derived from SMILES and HSG features. The statistical characteristics of the various QSPR models, constructed with different target functions, are presented in Table 3. Although all the established QSPR models meet the standard metrics and show acceptable statistical results, the models generated using the target function TF3 outperform those based on TF0, TF1, and TF2. The superior predictive performance of the models using TF3 can be attributed to the combined effect of IIC and CII on the system^38,39,40. Thus, TF3, computed with Eq. (9), is recognized as the best target function. The following QSPR model equations were obtained using Monte Carlo optimization with TF3 for logH₅₀:

Table 3 Summary of the statistical quality and predictability criteria for the QSPR models of logH₅₀ in nitro compounds, obtained for splits 1–4 using TF0, TF1, TF2, and TF3.

$${\text{Split}}\;1\;{\text{ LogH}}50=1.26767{\text{ }}\left( {0.00201} \right)+0.01327{\text{ }}\left( {0.00006} \right) * {\text{DCW}}\left( {7,35} \right)$$

(10)

$${\text{Split}}\;2\;{\text{LogH}}50=1.12604{\text{ }}\left( {0.00245} \right)+0.01700{\text{ }}\left( {0.00006} \right) * {\text{DCW}}\left( {7,35} \right)$$

(11)

$${\text{Split}}\;3\;{\text{LogH}}50=0.92802\;\left( {0.00294} \right)+0.01429\;\left( {0.00005} \right) * DCW\;\left( {7,35} \right)$$

(12)

$${\text{Split}}\;4\;{\text{LogH}}50=1.08749\;\left( {0.00300} \right)+0.01238\;\left( {0.00006} \right) * {\text{DCW}}\left( {7,35} \right)$$

(13)

The robustness of a QSPR model is assessed using various validation metrics. Among these, classical metrics such as Q² and R² are commonly employed to evaluate the predictive potential of the model. A model is considered acceptable when the numerical values of R² and Q² exceed 0.5 and 0.7, respectively^41,42. Roy and colleagues^43,44,45 proposed that $\:{r}_{m}^{2}$ may serve as a stricter metric than Q² and R². If the value of $\:{r}_{m}^{2}$ exceeds 0.5, the model can be interpreted as reliable. Additionally, other crucial statistical criteria for assessing various QSPR models include the average ($\:\stackrel{-}{{r}_{m}^{2}}$) and ∆r_m² for the validation set, where ($\:\stackrel{-}{{r}_{m}^{2}})\:$should be greater than 0.5 and $\:\varDelta\:{r}_{m}^{2}$ should be smaller than 0.2. In this study, these metrics, along with others, were utilized for model validation.

Table 3 shows that all four splits created using the target function TF3 meet the statistical quality criteria and fulfill the standards reported in the literature^42,43. Among the four models formulated with TF3, Split 2 was identified as the best, as it exhibited the highest R² and $\:{r}_{m}^{2}$ values for the validation set (R² = 0.7821 and $\:{r}_{m}^{2}$ = 0.7464).

Also, to compare these four models, the Akaike information criteria (AIC) values were calculated and compared. This parameter assessment the prediction error of the models generated from a single dataset and consequently, judges the relative efficiency of the models^46,47. It has been reported that among several models built from a data set, the model with the lowest AIC value has higher relative efficiency. Herein, to confirm superiority Split 2, the AIC criterion was computed for all four models formulated with TF3. It has been observed in each of the splits that the AICs of models are as follows: split 1= -325.95, split 2=-326.073, split 3= -323.029, split 4= -308.386, which demonstrates that in terms of performance, split 2 has a relatively better efficiency.

Figure 2 compares the numerical values of R² and $\:{r}_{m}^{2}$ for the validation set across the four splits designed using the four target functions. Figure 3 shows the plot of observed logH₅₀ versus calculated logH₅₀ for the six models developed with TF3. The calculated logH₅₀ values exhibit a strong correlation with their observed counterparts for all models constructed using TF3. Additionally, Fig. 4 illustrates the residuals of logH₅₀ versus experimental logH₅₀ for the four models. As shown in Fig. 4, the residuals of logH₅₀ are well-distributed around the horizontal line centered at zero. All these results indicate that the generated QSPR models are robust and well-fitted.

Fig. 2

Comparative numerical values of R² and r²m calculated using four target functions across all four splits.

Fig. 3

Plot of the correlation between predicted logH₅₀ and experimental logH₅₀ for the QSPR models developed using TF3.

Fig. 4

Plot of the residuals versus experimental logH₅₀ values for the four splits generated using TF3.

Model validation

To ensure the reliability, predictability, and quality of the constructed models, three validation strategies were employed: (1) internal validation or cross-validation using the training set data, (2) external validation using the test set data, and (3) Y-scrambling or data randomization. The training set was evaluated using various statistical criteria, such as R², Q²_LOO, MAE_train, and others. The robustness of the validation set was assessed using external statistical parameters, including R²_pred or Q²F₁, Q²F₂, Q²F₃, r²mtest (i.e. $\:\stackrel{-}{{r}_{m}^{2}}$ and $\:\varDelta\:{r}_{m}^{2}$ ), MAE_pred, and the concordance correlation coefficient (CCC), among others. To ascertain statistical significance and eliminate the possibility of random chance influencing the results, a Y-randomization test was performed on the generated models. Additionally, IIC and CII were applied to further validate the reliability and robustness of the QSPR models⁴⁸. The mathematical equations for these statistical metrics are well-documented in the literature^49,50,51.

Applicability domain

The applicability domain (AD) of a QSAR/QSPR model refers to the physicochemical, structural, or biological space within which the training set of the model has been developed. The resulting model is considered reliably applicable only to those molecules that fall within this domain⁵². It is essential to verify the AD of a QSAR/QSPR model to ensure its predictive reliability. The AD represents a theoretical region within the chemical space, defined using the model’s descriptors and responses, where reliable predictions can be made.

In this study, the AD of both the training and test sets was assessed to determine whether any compounds lie outside the defined AD.

For the Monte Carlo-based QSPR/QSAR models constructed using the CORAL program, the distribution of SMILES or HSG attributes in the active training, passive training, and calibration sets is used to define the AD. A substance is considered an outlier if it falls outside the scope of the AD.

In CORAL, the AD of a constructed model is characterized using the statistical defect of the SMILES attribute (Sk). The defects of the SMILES attribute (DSk) can be computed as follows:

$$\:{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}_{{\text{S}}_{\text{K}}}=\frac{\left|{\text{P}}_{\text{A}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})-{\text{P}}_{\text{P}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})\right|}{{\text{N}}_{\text{A}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})+{\text{N}}_{\text{P}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})}+\frac{\left|{\text{P}}_{\text{A}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})-{\text{P}}_{\text{C}\text{A}\text{L}}{(\text{S}}_{\text{K}})\right|}{{\text{N}}_{\text{A}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})+{\text{N}}_{\text{C}\text{A}\text{L}}{(\text{S}}_{\text{K}})}+\frac{\left|{\text{P}}_{\text{P}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})-{\text{P}}_{\text{C}\text{A}\text{L}}{(\text{S}}_{\text{K}})\right|}{{\text{N}}_{\text{P}\text{T}\text{R}\text{N}}{(\text{S}}_{\text{K}})+{\text{N}}_{\text{C}\text{A}\text{L}}{(\text{S}}_{\text{K}})\:\:}\:\:\:\:\:\:\:\:\:\:\:\text{I}\text{f}\:{\text{A}}_{\text{K}}>0$$

(14)

$$\:{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}_{{\text{S}}_{\text{K}}}=1\:\text{I}\text{f}\:{\text{S}}_{\text{K}}=0$$

P_ATRN(Sk), P_PTRN(Sk) and P_CAL(Sk) represent the probabilities of attributes in the active training set, passive training set, and calibration set, respectively. N_ATRN(Sk), N_PTRN(Sk), and N_CAL(Sk), are the frequencies of attributes in the active training set, passive training set, and calibration set, respectively.

The SMILES statistical defect (D) is calculated as the sum of the statistical defects of all attributes.

$$\:{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}_{\text{M}\text{o}\text{l}\text{e}\text{c}\text{u}\text{l}\text{e}}=\sum\:_{\text{k}=1}^{N{\text{A}}_{}}{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}_{{\text{S}}_{\text{K}}}$$

(15)

NA is the number of active SMILES attributes for the given compounds.

In CORAL, a SMILES is an outlier if:

$$\:{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}_{\text{m}\text{o}\text{l}\text{e}\text{c}\text{u}\text{l}\text{e}}\:>2\times\:{\stackrel{-}{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}}_{ATRN}$$

(16)

$\:{\stackrel{-}{\text{D}\text{e}\text{f}\text{e}\text{c}\text{t}}}_{\text{A}\text{T}\text{R}\text{N}}$ D represents the average statistical defect for the dataset of the active training set. While a statistical defect does not necessarily indicate that a compound is an outlier, it highlights concerns regarding the representativeness of the structural elements of the compound. Specifically, a large statistical defect suggests that a significant portion of the compound’s structure is not adequately supported by the correlation weights.

However, various logH₅₀ models constructed with TF3 exhibited different degrees of applicability domain (AD), with coverage ranging from 91 to 93% of the hypothetical space of compounds. Split 1 demonstrated the largest domain, covering 372 compounds out of 404, while Split 3 showed the smallest domain, covering 362 out of 404 compounds. In each model, AD varies with changes in the descriptors, making it model-specific. Nevertheless, if a compound lies outside the AD range of one split, it will typically fall within the AD range of another split (see supporting information). Consequently, all compounds were included in the model development process.

Mechanistic interpretation

In CORAL, the correlation weights (CWs) of structural attributes (Sk) are calculated across three Monte Carlo runs, followed by a mechanistic interpretation based on the numerical statistics of these CWs. Descriptors that either increase or decrease the endpoint (logH₅₀) are identified consistently across all runs. If the CW(Sk) is positive in every Monte Carlo optimization run, the attribute is classified as an enhancing promoter, whereas a consistently negative CW(Sk) indicates a reducing promoter. Attributes with CWs that alternate between positive and negative across runs are considered inconsistent and are excluded as valuable descriptors. The common descriptors that have consistently exhibited either an increasing or decreasing trend across multiple splits are summarized in Table 4.

Table 4 List of promoters for the increase and decrease of logH₅₀, along with their correlation weights in runs 1–4, the frequency of each attribute in each set, and examples of interpretative attributes.

The graph-based descriptors associated with increased logH₅₀ include the presence of fluorine and chlorine [(F.Cl).Y.Y.], the presence of nitrogen, oxygen, and absence of sulfur, and phosphorus [NOSP11000000], the absence of symmetry (xyzyx) ([xyzyx0]….), and the presence of aromatic carbon, with or without branching [c…1……., c…c…(…, c………., 1…c…(…, c…(…….].

Compounds 261, 264, and 280, which incorporate chlorine atoms (Table 5), demonstrate elevated logH₅₀ values. This observation indicates that the presence of chlorine atoms contributes to reduced impact sensitivity. Similarly, the high logH₅₀ value of compound 136 (Table 5), containing fluorine, further suggests that the incorporation of fluorine atoms also leads to decreased impact sensitivity. These four compounds were the only ones in the dataset that contained fluorine and chlorine. The impact of these descriptors on logH50 values will be further examined in detail, drawing on evidence from relevant literature and documented studies. This comprehensive analysis will help elucidate the relationship between molecular structure and the sensitivity characteristics of explosive compounds, contributing to a deeper understanding of their behavior under mechanical impact.

Table 5 Some examples of compounds from dataset for model interpretation.

The influence of halogens on the structure of explosive compounds and their effect on logH₅₀ values is a critical area of study. Halogens, due to their electronegativity and the ability to form strong bonds with carbon, significantly impact the thermal stability and sensitivity of energetic materials. For instance, the incorporation of halogens such as chlorine or fluorine into the molecular structure can reduce impact sensitivity, as reflected by higher logH₅₀ values. This effect arises from the halogen atoms’ ability to enhance the intermolecular interactions and increase the overall bond dissociation energy of the compound, thereby stabilizing the molecular framework. Moreover, halogens may alter the decomposition pathways of explosives, influencing the release of energy during detonation⁵³.

The presence of nitrogen (N) and oxygen (O) in the molecular structure of explosives can have significant effects on impact sensitivity (logH₅₀), stability, and explosive performance. These elements are commonly found in explosive compounds in the form of various groups such as nitro groups (-NO₂) and nitrogen groups (e.g., -NH₂), which can impart different characteristics to the explosive materials. Below are the primary effects of N and O presence in explosives:

Nitro groups are well-known and widely used in explosives, and they have a strong impact on explosive properties and impact sensitivity. Nitro groups generally increase impact sensitivity due to their high electronegativity. These groups weaken the bonds in the molecular structure, requiring less energy for detonation. As a result, materials containing nitro groups tend to have lower H50 values and are more easily affected by impact. Additionally, the presence of nitrogen and oxygen in nitro groups reduces the overall stability of the explosive, making it more prone to decomposition when subjected to impact or heat⁵⁴.

Nitrogen is often found in combination with other groups, especially in aromatic compounds such as 2,4,6-trinitrobenzene-1,3,5-triamine (TATB) or HMX (high-melting explosive). The presence of nitrogen in these compounds can have varying effects: In some materials, like TATB, the presence of amino groups (NH₂) forms hydrogen bonds that help reduce impact sensitivity, resulting in an increase in logH₅₀. These hydrogen bonds contribute to structural stability, making the material less sensitive to external forces. Amino-based explosives may demonstrate lower impact sensitivity because nitrogen-hydrogen and nitrogen-oxygen bonds enhance stability⁵⁵. Table 5 presents examples of aromatic compounds containing an amino group that exhibit high logH₅₀ values (compounds no. 223, 341, and 332).

The presence of aromatic carbon and the characteristics of aromaticity in the molecular structure can have significant impacts on the sensitivity of explosives, including logH₅₀. Generally, the aromatic structure, due to its unique features, can increase stability and reduce impact sensitivity. Aromatic structures, containing π bonds and six-membered carbon rings, are typically more stable. These π bonds in aromatic molecules can help distribute and stabilize energy across the molecule, which may lead to reduced sensitivity to impact and an increase in logH₅₀, as the explosive material would require more energy to break its stable structure. Furthermore, aromatic molecules are more resistant to impact due to the high stability of the carbon-carbon bonds in the aromatic rings. These bonds are usually stronger than non-aromatic intramolecular bonds, requiring more energy (i.e., a stronger impact) to initiate an explosion. This directly contributes to the increase in logH₅₀ ⁵³.

The SMILES-based descriptors associated with decreased logH₅₀ include presence of double bond with branching [=…(…….], presence of double bond [=……….], presence of double bond and oxygen with branching [ =…O…(…], presence of oxygen and double bond with branching [O…=…(…], and presence of oxygen with double bond [O…=…….]. The graph-based descriptors associated with decreased logH₅₀ include the Nearest Neighbor Codes for Oxygen (NNC-O) equal to 101 [NNC-O…101.], and the presence of the path of length 2 equal to 2 for oxygen atom (PT2-O…2…).

The presence of double bonds can significantly impact the impact sensitivity (H₅₀) of explosive materials, often leading to an increase in sensitivity and a decrease in H₅₀. Double bonds are typically more reactive under mechanical impact due to their relatively weaker bond strength compared to single bonds, making them more susceptible to rupture under pressure or impact. As a result, explosives with double bonds may more readily undergo reactions or detonation with minimal external force. Furthermore, the electronic structure of double bonds can create regions of high electron density or free electron areas, which may serve as vulnerable points in the molecular structure, increasing the likelihood of detonation under impact. Additionally, double bonds often alter the electronic density distribution within a molecule, influencing electrostatic interactions both within and between molecules. These changes can reduce stability and resistance to mechanical stimuli, further contributing to lower logH₅₀ values and higher impact sensitivity⁸.

The simultaneous presence of oxygen and double bonds in explosive materials can have a complex and multifaceted effect on impact sensitivity (logH₅₀), as each of these features independently influences the molecular structure and reactivity of the material. Oxygen, particularly in nitro groups, can enhance explosive potential by increasing the production of hot gases and high-pressure during detonation reactions. Double bonds, due to their electronic structure, create vulnerable sites in the molecule, such as π bonds, which are more susceptible to rupture under impact or pressure. This combination can decrease the overall stability of the explosive and increase its sensitivity to impact, thereby lowering logH₅₀ values. Furthermore, the presence of both oxygen and double bonds can alter the electronic density distribution within the molecule, potentially weakening the molecular structure and increasing the likelihood of detonation with less external force. As a result, the synergistic effect of oxygen and double bonds can increase explosive reactivity, reduce molecular stability, and lead to a decrease in logH₅₀, thereby enhancing impact sensitivity⁵⁶. Table 5 presents examples of aliphatic compounds containing an oxygen double bond that exhibit low logH50 values (compounds no. 110, 111, 109, and 33).

Comparison with earlier reported models

In previous studies, several QSPR models have been proposed to predict impact sensitivity. The predictive capabilities of the QSPR models discussed here have been compared with those of other reported models, and some statistical properties of the studied models are summarized in Table 6. Kamlet^7,57 developed models based on linear correlations between log50 (H50%) and oxygen balance (OB) for 78 nitro compounds. While his models showed strong correlations, they lacked external validation. Nefati et al.⁵⁸, Cho et al.²⁵, and Jun et al.⁵⁹ used artificial neural networks (ANNs) to create prediction models for impact sensitivity for 204, 234, and 33 energetic molecules, respectively. Keshavarz and colleagues^4,10,60,61 published multiple multilinear regression (MLR) and ANN models to predict the impact sensitivity of 291, 58, 69, and 46 nitro compounds across independent studies. Although most of the mentioned prediction models exhibited significant correlations, their predictive performance was not validated on an external dataset.

Table 6 Comparison with previously published models.

However, there are only a few models that have been validated on an external dataset. In two independent studies, Wang et al.^3,62 developed various models using different approaches, including multiple linear regression (MLR), partial least squares (PLS), and artificial neural networks (ANN), for 186 compounds, and backpropagation neural network (BPNN), MLR, and PLS for 156 compounds. For QSPR modeling of 186 non-heterocyclic nitroenergetic compounds, they employed a genetic algorithm-PLS method to select nine optimal descriptors. They then applied a nonlinear ANN method to model the relationship between the selected descriptors and impact sensitivity. The results obtained from the ANN method were superior to those from the PLS and MLR methods (Table 4).

For QSPR modeling of 156 nitroenergetic compounds, the entire dataset was randomly divided into a training set (64 compounds), a validation set (63 compounds), and a prediction set (29 compounds). Electrotopological-state indices (ETSI) were used as molecular structure descriptors to predict the impact sensitivity of the compounds. Although their BPNN models demonstrated high predictive potential, a lack of robustness was observed, likely due to over-parameterization, as the models relied on up to 16 descriptors. Additionally, the applicability domains of the models were not considered, which limits their practical use.

Nefati et al.⁵⁸ developed a model using thirteen molecular descriptors to predict the impact sensitivity of 204 explosive molecules via ANN. However, their model was not validated using an external dataset, which leaves unclear the prediction capability of the model for new molecules not included in the training process. Furthermore, it was not determined which specific property correlates most strongly with impact sensitivity.

Xu et al.¹¹, using 3D descriptors, developed a QSPR model to predict the impact sensitivity of 156 nitro compounds previously introduced by Wang et al.³. The entire dataset of 156 compounds was divided into a training set (127 compounds) and a test set (29 compounds). Both MLR and ANN were employed to select descriptors and build the models. The MLR-based model achieved an R² of 0.7222 and an s of 0.177 for the test set, while the ANN-based model showed an R² of 0.8656 and an s of 0.130. Based on these results, they concluded that the ANN model was more reliable than the linear MLR model.

Morrill and Byrd⁶³ developed a QSPR model based on eight quantum chemical descriptors for 227 nitroenergetic compounds using the best multilinear regression (BMLR) method. After removing seven outliers, the model’s R² value was 0.8141. However, despite having a large number of compounds, only eight compounds were used as a test set in their model, which raises concerns about the model’s generalizability and reliability.

In 2012, Fayet et al.² proposed a global QSPR model to predict the impact sensitivity of a dataset consisting of 161 nitro compounds. They used multilinear regression to establish correlations between chemical quantum descriptors and experimental impact sensitivities. The dataset was divided into a training set of 108 nitro compounds, with an R² of 0.750 and RMS of 0.19, and a test set of 53 nitro compounds, with an R² of 0.759 and RMS of 0.230. The model was developed using the multilinear regression (BMLR) method based on 15 quantum chemical descriptors.

In 2014, Fayet et al.¹² expanded their work by developing a series of QSPR models for 60 nitramines using topological and constitutional descriptors to predict their impact sensitivity. To predict the log h50% endpoint, they applied 17 different training and validation set partitions from the dataset, using two different ratios of compounds (40/20 or 45/15) and two division approaches (based on property ranking or random division). They focused on models derived from partitions 2, 6, 9, and 13, as these exhibited the best performance among the 17 partitions studied. The BMLR model from partition 9 was selected as the best model due to its highest predictive capabilities (R² _Ext = 0.90 and RMS = 0.14). All of Fayet’s models were constructed in accordance with the five OECD principles for the validation of QSPR models intended for regulatory use.

In 2021, Deng Q et al.⁶⁴ constructed six Machine learning (ML) models to probe the correlation between the impact sensitivity and the molecular structure of 240 nitroaromatic compounds using the ANN and sure independence screening and sparsifying operator (SISSO). The R2 and RMSE their ANN model for the test set were 0.95 and 0.92 respectively while the R2 and RMSE of SISSO model were 0.85 and 1.07 respectively, which is indicate that ANN significantly outperforms SISSO for the impact sensitivity model.

In 2013, Mathieu⁶ basis on the of single descriptor, built a QSPR model to predict the impact sensitivity of 156 nitro molecules which is used in Wang’s et al. models. The R2 and RMS for an external test set of the model were 0.721 and 0.265.

In 2014, Mathieu and Alaime⁶⁵ using the same database of 156 nitro compounds suggested a physically based quantitative model for impact sensitivities with a satisfactory reliability (R2 ≃ 0.8). Later, in 2015 ⁶⁶, they proposed a model to estimate impact sensitivities of 308 energetic compounds and it yields values of 0.6934 and 0.248 for the R² and SE computed for a test set of 74 compounds. In 2016, they followed their research with suggestion a semi empirical model based on physical assumptions and compared it with Fayet’s et al.¹² and Prana’s et al.⁶⁷ models. For construction their model, 129 compounds consist of 50 C-NO2 compounds studied by Prana et al.⁶⁷ 60 N-NO2 compounds studied by Fayet et al.¹². and 19 compounds studied by Mathieu and Alaime⁶⁶ considered. These data splitted into training and test sets and used 74 compounds in the training set (34 C − NO2 and 40 N − NO2 compounds). The R² and RMS for test set of the model were 0.71 and 0.22. Finally, in 2017, Mathieu²⁴ investigated the correlation between detonation performance and the sensitivity using the regression analysis, and explained there is a significant correlation analysis with R2 close to 0.8 for 156 aliphatic compounds.

In 2021, Deng Q. et al.⁶⁴ developed six machine learning (ML) models to explore the relationship between impact sensitivity and the molecular structure of 240 nitroaromatic compounds, employing ANN and the Sure Independence Screening and Sparsifying Operator (SISSO). The ANN model showed excellent predictive performance, achieving an R² of 0.95 and an RMSE of 0.92 for the test set, while the SISSO model had an R² of 0.85 and an RMSE of 1.07. These results suggest that the ANN model significantly outperforms the SISSO model for predicting impact sensitivity.

In 2013, Mathieu⁶ developed a QSPR model based on a single descriptor to predict the impact sensitivity of 156 nitro molecules, which were also included in Wang et al.‘s study. The model yielded an R² of 0.721 and an RMS of 0.265 for an external test set.

In 2014, Mathieu and Alaime⁶⁵ used the same dataset of 156 nitro compounds to propose a physically-based quantitative model for predicting impact sensitivities, which demonstrated satisfactory reliability (R² ≈ 0.8). The following year, in 2015 ⁶⁶, they extended their research by developing a model to estimate the impact sensitivities of 308 energetic compounds, which produced an R² of 0.6934 and a standard error (SE) of 0.248 for a test set of 74 compounds.

In 2016, they furthered their work by proposing a semi-empirical model based on physical principles and comparing it with the models of Fayet et al.¹² and Prana et al.⁶⁷. For their model, they utilized a dataset of 129 compounds, including 50 C-NO₂ compounds studied by Prana et al.⁶⁷, 60 N-NO₂ compounds studied by Fayet et al.¹², and 19 compounds from their own previous study⁶⁶. The data were divided into training and test sets, with 74 compounds in the training set (34 C − NO₂ and 40 N − NO₂ compounds). This model achieved an R² of 0.71 and an RMS of 0.22 for the test set.

Finally, in 2017, Mathieu²⁴ conducted a study to explore the correlation between detonation performance and sensitivity using regression analysis. His findings revealed a significant correlation, with an R² close to 0.8, for 156 aliphatic compounds.

To the best of our knowledge, only two studies^12,67 have reported OECD-compliant predictive models for impact sensitivity of nitro compounds. In contrast, all the models developed in the present study were constructed in accordance with all OECD principles for QSPR models intended for regulatory use. These principles include a defined endpoint, an unambiguous algorithm, a measure of predictive potential, a defined applicability domain, and a mechanistic interpretation.

The previously reported models were developed using smaller datasets (n < 308), whereas the current study uses the largest dataset to date, comprising 404 nitro compounds. Utilizing a larger dataset helps mitigate the occurrence of high R² values, making R² less reliable as a comparison metric. In earlier studies, QSPR models were typically built using two sets (training and test sets), except for the models by Wang et al., where the dataset was randomly divided into a training set, validation set, and prediction set. In contrast, the QSPR models in the present study were constructed using four sets: training, invisible training, calibration, and validation sets.

Another notable distinction is that none of the previously reported models provided mechanistic interpretations, whereas the models presented here offer mechanistic insights. Specifically, we have identified the key structural attributes responsible for the increase or decrease of logH50, based on the built models.

By utilizing a larger training dataset, the study has achieved acceptable statistical quality for the test set. The robustness and reliability of the constructed QSPR models have been validated using a comprehensive set of robust criteria, including R², Q², Q²F1, Q²F2, Q²F3, CCC, CII, IIC, CR2p, RMSE, MAE, $\:{\varDelta\:r}_{m}^{2}$, $\:\stackrel{-}{{r}_{m}^{2}}$, F, and Y-test. Consequently, based on these validation criteria, it can be concluded that the models generated in this work are statistically more significant and reliable when compared to the previously reported models.

Conclusion

This study presents the development of global and reliable QSPR models to predict the impact sensitivity of 404 nitro compounds using the Monte Carlo optimization procedure within the CORAL program. The models were established and evaluated according to the OECD principles for QSPR/QSAR acceptance in regulatory settings, demonstrating high internal and external predictivity and a broad applicability domain. A total of four splits were created, resulting in the establishment of 16 QSPR models. For each split, four target functions, TF0 (without IIC or CII), TF1 (with IIC), TF2 (with CII), and TF3 (with both IIC and CII), were employed. Among all the models, those generated using TF3 showed the most robustness and superior predictability. Based on the statistical quality of the model established for split 2 using TF3, it was selected as the best-performing model.

The results highlighted that the simultaneous use of IIC and CII significantly enhances the reliability of the models. Additionally, the structural attributes responsible for the increase or decrease of log H₅₀​ in nitro compounds were extracted. In conclusion, this study develops a simple, time- and cost-efficient, and highly effective in silico approach for predicting the impact sensitivity of energetic compounds.