Quantifying the impact of non-adherence to drug therapy: a technical note concerning an application of a branch and bound algorithm

Abstract

Pharmacokinetic models typically rely on a key assumption that patients take their medication as prescribed, whereas this is often not the case. We present a branch and bound algorithm that can be used to estimate the time-varying probability that, given a specified pattern of non-adherence to a prescribed regimen, a patient receives no therapeutic benefit from treatment. Use of this algorithm is a much faster method for obtaining this probability than exhaustive computation of the relevant probability distribution. The use of this algorithm to assess, in quantitative terms, the impact of non-adherence on the effectiveness of treatment provides a rational basis for evaluating the potential harm to patients.

Appendix: description of the branch and bound algorithm

We use the vertex set V = {v _i } to denote the set of nodes of the directed tree, which are arbitrarily indexed except that the top node is denoted v ₁. For 1 ≤ i ≤ 2^{N  + 1} − 1 we denote the probability that the outcomes of Bernoulli trials would result in a path passing through the node v _i by p(v _i ).

In the summary of the branch and bound algorithm below, the symbol j is used to denote the index of the node within U that is selected for branching. The algorithm is summarised as follows:

• Step 1 Set up Set A and B as empty and U as containing just one node v ₁, the top node in the tree. Set p(v ₁) = 1, since all possible paths emanate from v ₁. Compute the maximum value of drug concentration at time T associated with this node if all the scheduled drug administrations occurred and assign v ₁ to B if possible. END if U is now empty.

• Step 2 Branch Update U, replacing the node selected for branching by the two nodes it leads to in the next level of the tree. Compute the probabilities p() and the minimum and maximum values of drug concentration at time T associated with these nodes.

• Step 3 Bound If possible, reduce U, by re-assigning the two new nodes to the sets A or B if possible.

• Step 4 Select IF the set U is empty, sum p() for nodes in B to give the solution THEN END.

This is a stopping condition indicating that the search for the probability that the drug concentration at time T is in the sub-therapeutic range is complete.

ELSE select as the branching node that v _j in U which leads to the highest value of Z(T).

GO TO Step 2

To illustrate the operation of the algorithm, consider a sequence of six scheduled administrations. For one such sequence, Fig. 2 shows the nodes of the tree that belonged to the set U at some point during the operation of the algorithm.

Fig. 2

Example nodes evaluated during a run of the algorithm for six scheduled administrations. Nodes shown as black are those in the set A at the final step of the algorithm. Those shown in white are in set B at the final step of the algorithm and are included in the calculation of the cumulative probability that the drug concentration is in the sub-therapeutic range

The nodes which are shown as black represent nodes of the tree assigned to the set A by the time the algorithm finished. Those shown as white are those included in the set B used to calculate the probability of the final drug concentration being in the sub-therapeutic range. This is the total probability associated with all paths passing through one of the white nodes. The nodes shaded grey are those that were members of U, were branched on and removed from U.