Solvent-free reactions show a number of features that meet several of the principles of Green Chemistry, such as preventing formation of waste, increasing atom economy and savings in the use of auxiliary compounds [1]. In addition, these techniques can reduce the amount of hazardous products formed in chemical processes and increase the selectivity and yield of many organic reactions [2].

In this paper, we have focused on reactions employing microwave radiation. Microwave dielectric heating uses the capability of some solids or liquids to transform the electromagnetic energy into heat. Moreover, its magnitude depends on the properties of the molecules, so microwave radiation can be used to introduce a certain degree of selectivity [3] in the chemical process/es under study. Main applications of this technique are, among others, microwave assisted extraction [4], desorption and recovery of solids [5], SO₂ emissions reduction and synthesis of organometallic [6], organic and inorganic compounds [7,8].

Furthermore, there are many approaches that have been proposed to ease the prediction of molecular properties. Equations linking quantitative structure–property (QSPR) relationships are particularly relevant and can be applied to large libraries of compounds for virtual computational screening [9,10]. However, these models require good structural descriptors that reliably represent the molecular features responsible for the property of interest.

Molecular Topology (MT) has largely demonstrated its efficacy in depicting molecular structures and predicting their properties. It follows a two-dimensional approach only considering the internal arrangement, including atoms. The structure of each molecule is represented by specific subsets of topological indices (TIs). These indices, when well chosen, provide a unique way of characterizing a molecular structure [11]. TIs are able to characterize the most important features of molecular structure: molecular size, binding and branching. The computation of TIs is very swift and they also have the advantage of behaving as true structural invariants. This means that TIs are independent of the spatial position of the atoms in a particular moment, although extensions of the TIs, taking account of the three-dimensional structure, have been also devised [12,13].

MT has demonstrated to be an excellent tool in the prediction of physicochemical [14] and biological properties [15] of structurally heterogeneous groups of compounds.

Most pain likely to be suffered in a lifetime is sensitive to anti-inflammatory (AI) drugs, for instance myalgia, artralgia, cephalalgia, neuralgia, dysmenorrheal and acute or chronic inflammatory processes. Furthermore, they are often useful in the unrest linked to viral and bacterial processes. They constitute the first level treatment of pain in the World Health Organization (WHO) strategy. All this accounts for their selection as our object of study in this paper.

Amidine derivatives are well known for their broad range of pharmaceutically relevant properties (anticancer [16–18], antimicrobial [19], antifungal [20], antibacterial [21] to mention just a few). However, they also exhibit anti-inflammatory activity (AA) [22,23], and this is why we have focused our attention on them.

The purpose of this work is to build up some predictive models for the reaction yield, and the anti-inflammatory activity, of a set of heterocyclic amidine derivatives synthesized under environmental friendly conditions using microwave irradiation. Later on, the models were applied to virtual screening libraries in order to search for new amidine derivatives with higher values of reaction yields and anti-inflammatory activity.

Searching for equations capable to predict reaction yields (logYield) and anti-inflammatory activity (logAA) of the analyzed amidine derivatives, was the first objective. The best linear equations obtained, and their statistical parameters, were: (1) logYield = 3.927 + 0.029 Pol − 0.316 ATS 8 v − 0.534 EEig 01 d with   N = 16 ,   r = 0.884 ,   r 2 = 0.782 ,   Q 2 = 0.667 ,   SEE = 0.0117 ,   F = 14.3 ,   p = 0.0002 ; (2) and ,   logAA = 10.384 − 1.324 EEig 09 x − 4.193 EEig 06 r + 2.404 EEig 10 r with   N = 16 ,   r = 0.906 ,   r 2 = 0.820 ,   Q 2 = 0.629 ,   SEE = 0.0671 ,   F = 18.2 ,   p = 0.00009.

The above values of 0.75 and 0.5 of r² and Q², respectively, in addition to the low values of SEE in both cases (less than 12% of the average values of the property) confirm the validity of the models from a predictive standpoint.

Table 3 and Figure 2 show the yield and the anti-inflammatory activity predicted for each compound analyzed.

The EEig indices, topological descriptors derived from the eigenvalue of the adjacency matrix of edges weighed with different properties appear in both equations [29]. So, EEig01d takes into account the dipole moments of atoms, EEig09x the bond order of the various edges and EEig06r and EEig10r the resonance integral. Other indices present in Equation 1 are Pol, the number of polarity calculated as the number of pairs of vertexes at topological distance equal to 3 [26,27] and the Moreau-Broto autocorrelation index, ATS8v, weighed by Van der Waals volumes [28].

The predictive ability of the selected mathematical topological models was evaluated through cross-validation, using the leave-one-out test. Table 3 (columns 4 and 7) and Figure 3 show the obtained results. The values of Q² = 0.667 for reaction yield and Q² = 0.629 for anti-inflammatory activity are accepted as satisfactory [32].

In order to prevent the possible existence of fortuitous regressions, a randomization test was carried out. Thus, the values of the property of each compound are randomly permuted and linearly correlated with the aforementioned descriptors. This process is repeated as many times as needed. The usual way to represent the results of a randomization test is plotting the correlation coefficients versus the predicted ones, r² and Q², respectively. The results of the randomness tests, shown in Figure 4, suggest a high stability of both models (all regressions were rather poor except for the selected equation (black point) with the real values for each compound).

Once predictive equations were established, it was possible to carry out a search for new compounds showing anti-inflammatory activity, that could be reliably obtained from a highly efficient synthetic reaction. Based on the selected topological models, a virtual molecular screening, using the reaction scheme II and different structural fragments, was carried out. The results are exposed in Table 4. All proposed compounds, except 7a and 7c, show an expected yield exceeding 80%. With respect to the anti-inflammatory activity, compounds 7d–g exceeds the value of 50% in its pharmacological activity. In conclusion, it can be said that the proposed group of compounds is interesting from the anti-inflammatory activity standpoint.

Of course, these indicative results need to be confirmed by experimental tests. Should the test prove positive, the models proposed would be validated and could serve as a useful tool for the search of novel compounds synthesized under environmental friendly conditions and displaying anti-inflammatory activity.

Molecular topology was successfully used to arrange QSPR models for predicting the reaction yield and anti-inflammatory activity, in a group of 16 heterocyclic amidine derivatives, synthesized under environmental friendly conditions, using microwave irradiation. All the molecular descriptors used in this study were topological indices. The mathematical models achieved and described herein retain the main structural features of the correlatable properties, and hence can be applied to the search of new analogous compounds with an improved environmental profile.