Optimizing Combinatory Drugs using Markov Chain-Based Models

13 Background: Combinatory drug therapy for complex diseases, such as HSV infection and 14 cancers, has a more significant efficacy than single-drug treatment. However, one key 15 challenge is how to effectively and efficiently determine the optimal concentrations of 16 combinatory drugs because the number of drug combinations increases exponentially with the 17 types of drugs. 18 Results: In this study, a searching method based on Markov chain is presented to optimize the 19 combinatory drug concentrations. Its performance is compared with four stochastic 20 optimization algorithms as benchmark methods by simulation and biological experiements. 21 Both simulation results and experimental data demonstrate that the Markov Chain-based 22 approach is more reliable and efficient than the benchmark algorithms. 23 Conclusion: This article provides a versatile method for combinatory drug screening, which is 24 of great significance for clinical drug combination therapy. 25


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In the practice of clinical treatment, a single drug often fails to achieve the desired efficacy  The infection of parasitic nematodes (or roundworms) poses a serious safety hazard to humans 39 and livestock [8], and the anthelmintics (or antinematode drugs) are highly susceptible to drug 40 3 resistance. It has been proved that a variety of combinations of multiple anthelmintic drugs, 41 rather than a single medicine, can enhance the deworming effect [9]. In the case of the 42 eradication of wild-type Caenorhabditis elegans worms, it is more effective to use four 43 combinatory drugs (levamisole, pyrantel, tribendimidine, and methyridine) than single drugs 44 [10]. Traditional treatments of HSV-I, one of the most common sexually transmitted infections, 45 often include virus-specific drugs, which are effective at the beginning but exhibit limited long-46 term efficacy as drug-resistant strains develop. However, a combination of six drugs (IFN-α, 47 acyclovir, IFN-γ ,ribavirin, IFN-β and TNF-α ) was demonstrated to be the most promising 48 therapy for the reason that the drugs in the combinatory treatment can act simultaneously on 49 the multiple pathways and cellular protein complexes, and, therefore, regulate all relevant 50 pathways, potentially blocking HSV-I replication [11]. Combined use of multiple drugs is also 51 a common practice in the treatment of cancers to achieve higher efficacy and potency. For 52 example, in the treatment of non-Hodgkin's lymphoma, the drugs, pirarubicin, velet, cytarabine 53 and prednisone, are usually used in combination, which the chemotherapy effect is 54 remarkablely enhanced [12]. 55 However, owing to the inherent complexity of biological systems and internal structure of 56 cells and, particularly, to the huge searching space, it is extremely challenging to effectively 57 and efficiently to determine the optimal drug mixture from all possible drug combinations 58 through trial and error. For example, there are drugs and each drug has concentration 59 candidates, it is necessary to find the optimal drug mixture in the space of combinations. In this paper, an optimization method based on Markov chain models is proposed to search 87 for optimal combinatory drug concentrations with excellent performance. In this method, the   approach are obvious. As approaches infinity, the state transfers to the optimal state x*, 183 which means that the probability at the global maximum of the objective function is the greatest.

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Reconsidering the Markov chain model described above, from the state transition diagram 185 shown in Fig. 1, it is obvious that it is a random walk with any two adjacent states. Searching 186 for the optimal drug concentration is equivalent to seeking the state with the largest steady- is the corresponding optimal state sought, that is, the optimal combination of drug 195 concentration levels.

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It is noteworthy that the initialization of the transition probability matrix is not unique. The steady state that has the maximal distribution probability is referred to as the optimal drug 209 combination.

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The general procedure of the optimization algorithm for combinatory drugs based on the 211 Markov chain model is described as follows.

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Step 1: The Markov chain and the corresponding transition probability matrix are 213 initialized according to the numbers of drugs and concentration levels

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Step 2: Suitable adjacent combinations of experimental points are selected.

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Step 3: The transition probability matrix of the Markov chain is updated according to the 216 difference in the drug response functions at the corresponding suitable experimental points.

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Step 4: The corresponding stationary distribution is solved according to the updated 218 transition probability matrix using the balance equation.

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Step 5: It is determined whether the stationary distribution converges. If it converges, the 220 algorithm stops; otherwise, it returns to the second step, or, when the predetermined number of 221 iterations is reached, the algorithm stops.

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As shown in Fig. 2, the GG-based algorithm oscillates around some states (Fig. 2(d)) or stays 233 in a suboptimal state (Fig. 2(e) and Fig. 2(f)). The DE algorithm converges too early to the 234 local extremum (Fig. 2(i)). The CAPR algorithm oscillates in a relatively small range but does    In the Fig. 4A-(d) and Fig. 4B-(d), when the Markov-chain-based algorithm is used, the 317 global optimal combination can be found within only a few iterations. As the experiment and 318 calculation are parallel, the proposed algorithm is much more efficient.

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Shown in Table 2   As shown in Fig. 5, two drug response functions of the two combinatory drugs were used.  In this study, a novel Markov-chain-based approach was proposed to solve the problem of 342 the optimization of combinatory drugs. The basic principle of the proposed method was 343 introduced, and the steps of the algorithm used in the general case were illustrated in detail.

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Furthermore, the algorithm was promoted to cases of one-dimensional and two-dimensional