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## Graph theoretic analysis of neighbourhood relations with respect to simultaneously applied drugs

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Published: | December 9, 2015 |
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**Background: **Multimedication in connection with multimorbidity offers many challenges in medical care. We will consider which drugs and group of drugs are most important in relation to other prescribed drugs and drug groups with respect to simultaneous application for patients. Thereby we get additional aspects of drug efficiency, compliance and safety. We will use methods of network topology analysis to examine drug interactions.

**Materials and Methods: **We examined drug prescription data with respect to § 301 SGB V of regional statutory health insurances of a quarter in 2014. We use the international ATC (anatomical therapeutic chemical) classification in order to get groups of drugs. We specify simultaneous application by prescibing both in the same quarter. Different application dates are thereby no problem in long term application. The drugs and groups of drugs determined by ATC level 4, 5 and 7 digits are used as the vertices of a graph. The (directional) edges are given by most important (resp. the two most important) drug/ drug group with respect to the considered drug/ drug group. If we don’t use the direction, we get an undirected graph. For the analysis of standard graph properties and for the visualization we use the Mathematica program system of Wolfram Research.

**Results: **At ATC 4 level the graph with one outgoing edge in the directed version has 3 connected components with 166, 4 and 3 vertices, cf. Figure 1 [Fig. 1] for the undirected version. If we use a decreasing order of the number of adjacent edges we get in the top positions 51 edges for the vertex A02A (drugs for peptic ulcer and gastro-oesophageal reflux disease), 41 edges for C07A (beta blocking agents), 22 edges for M01A (antiinflammatory and anirheumatic producs, non-steroids) and 18 edges for R01A (decongestants and other nasal preparations for topical use). The large component has diameter 9 and radius 5.

In the same context at ATC 5 level we get connected components with 326, 9, 6, 4 and 3 vertices, cf. Figure 2 [Fig. 2]. The vertex A02BC (antacids, aluminium compounds) has 117 edges and C07AB (beta blocking agents, selective) has 60 edges. The largest connected component with diameter 8 and a tree structure has a week connectivity: if we exclude 4 edges, it splits in 5 components, c.f. Figure 3 [Fig. 3] and Figure 4 [Fig. 4]. This is related to a small isoperimetric constant of the graph. In other words: Each of 326 ATC 5 groups is mainly related to one of the groups A02BC (proton pump inhibitors), C07AB (beta blocking agents, selective), H02AB (glucocorticoids), R01AA (sympathomimetics, plain) and M01AE (propionic acid derivatives).

If we consider ATC 7 level, still with one outgoing edge, we get 8 connected components, the largest with 418 and 196 vertices, cf. Figure 5 [Fig. 5]. The top positions with respect to the number of vertices are A02BC02 (pantoprazole), C07AB02 (metoprolol), M01AE01 (ibuprofen), R01AA07 (xylometazoline) and N02BB02 (metamizole sodium). The diameter of the largest component is 8.

If we use two outgoing edges at ATC 7 level we get 3 connected components with the same top positions pantoprazole, metoprolol, ibuprofen, xylometazoline and metamizole sodium with respect to the number of vertices, cf. Figure 6 [Fig. 6].

**Conclusion: **We have identified drugs and group of drugs, which are interfere with a large number of other drugs and are therefore of special interest with respect to drug safety. At ATC 4 level these are drug groups for peptic ulcer and gastro-oesophageal reflux disease, beta blocking agents, antiinflammatory and anirheumatic producs, non-steroids and decongestants and other nasal preparations for topical use. Within these groups the drugs pantoprazole, metoprolol, ibuprofen and xylometazoline are of special importance with respect to interactions.

Each drug or drug group can be reached from every other drug or drug group by at least a path length of 8 or 9. The considered path length is similar to the social psychological small-world theses by Stanley Milgram and other researchers examining path length for social networks.

It suggested that human society is a small-world-type network characterized by short path-lengths, often associated with the phrase "six degrees of separation". Modern methods of network topology can be applied to examine drug interactions.

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