Adaptive reference update (ARU) algorithm. A stochastic search algorithm for efficient optimization of multi-drug cocktails
- Mansuck Kim^{1} and
- Byung-Jun Yoon^{1}Email author
https://doi.org/10.1186/1471-2164-13-S6-S12
© Kim and Yoon; licensee BioMed Central Ltd. 2012
Published: 26 October 2012
Abstract
Background
Multi-target therapeutics has been shown to be effective for treating complex diseases, and currently, it is a common practice to combine multiple drugs to treat such diseases to optimize the therapeutic outcomes. However, considering the huge number of possible ways to mix multiple drugs at different concentrations, it is practically difficult to identify the optimal drug combination through exhaustive testing.
Results
In this paper, we propose a novel stochastic search algorithm, called the adaptive reference update (ARU) algorithm, that can provide an efficient and systematic way for optimizing multi-drug cocktails. The ARU algorithm iteratively updates the drug combination to improve its response, where the update is made by comparing the response of the current combination with that of a reference combination, based on which the beneficial update direction is predicted. The reference combination is continuously updated based on the drug response values observed in the past, thereby adapting to the underlying drug response function. To demonstrate the effectiveness of the proposed algorithm, we evaluated its performance based on various multi-dimensional drug functions and compared it with existing algorithms.
Conclusions
Simulation results show that the ARU algorithm significantly outperforms existing stochastic search algorithms, including the Gur Game algorithm. In fact, the ARU algorithm can more effectively identify potent drug combinations and it typically spends fewer iterations for finding effective combinations. Furthermore, the ARU algorithm is robust to random fluctuations and noise in the measured drug response, which makes the algorithm well-suited for practical drug optimization applications.
Keywords
Background
Biological networks are known to be redundant at various levels, which makes them robust to various types of perturbations. As a consequence, it is generally difficult to change their long-term dynamics by blocking a specific gene or intervening in a specific pathway. This is one of the reasons why monotherapy is often not very effective in treating complex diseases, such as cancer and diabetes. In fact, multi-target therapeutics based on combinatory drugs are known to be much more effective, and they are commonly used these days for treating various diseases [1–6]. However, considering the huge number of possible ways to mix multiple drugs, it is practically impossible to find the optimal "drug cocktail" simply by trial and error or by exhaustive testing. Clearly, we need a systematic way of identifying the most effective drug cocktail, and recently, several algorithms have been proposed to address the problem of combinatorial drug optimization [7–12].
For example, Calzolari et al. [7] developed a drug optimization algorithm based on sequential decoding algorithms that have been traditionally used in digital communications [13, 14]. In [7], it was shown that we can algorithmically identify the optimal drug combination by testing only a relatively small number of drug combinations, compared to exhaustive search. Unlike the approach proposed by Calzolari et al. [7], which was deterministic, Wong et al. [9] proposed a different approach based on a stochastic search algorithm, called the Gur Game algorithm [15, 16]. In this work [9], they formed a closed-loop optimization framework, in which the Gur Game algorithm was used to predict an updated drug combination that is likely to improve the current drug response, and the drug combination is iteratively updated until the response is maximized. It was shown that this closed-loop optimization method can quickly identify potent drug combinations. More recently, another stochastic search algorithm was proposed in [11] that addresses the limitations of the Gur Game algorithm, thereby further improving the performance of the closed-loop optimization approach originally proposed in [9].
In this paper, we propose a novel stochastic search algorithm, called the adaptive reference update (ARU) algorithm, that can significantly improve the performance of the existing stochastic search algorithms [9, 11]. The key idea of this algorithm is to adaptively update the reference drug combination to guide the search algorithm and help it to make better informed guesses without requiring extensive prior knowledge of the underlying biological network. We demonstrate that the proposed ARU algorithm outperforms existing stochastic drug optimization algorithms, in terms of both efficiency, success rate, and robustness.
Methods
Combinatorial drug optimization problem
As we can see, this is a combinatorial optimization problem, in which we have to find the optimal drug combination out of M_{1} M_{2} . . . M_{ N } possible combinations. The total number of distinct drug combinations quickly grows as the number of drugs increases. Considering the practical cost of experimentally measuring the normalized drug response function f (x), it is apparent that we cannot test all drug combinations to find the most effective one.
Stochastic search algorithms
Stochastic search algorithms [9, 11] aim to efficiently identify the potent drug combinations without exploring the entire combinatorial solution space. The basic idea is to randomly search through the solution space by iteratively updating the drug combination until an effective combination emerges. At each step, the current drug combination is incrementally updated towards the direction that is likely to improve the overall drug response. The updating decision is made in a stochastic manner, which allows the search to proceed towards directions that are deemed to be less likely to improve the response. This is an important characteristic of stochastic search algorithms, which is critical for keeping the search from being trapped in local maxima. Since a stochastic search algorithm tries to arrive at the optimal solution (i.e., the most effective drug combination) by performing iterative local searches, its overall performance depends on how it chooses the next solution state (i.e., an updated drug combination in $\mathcal{X}$, the set of all possible combinations) from a given state (i.e., the current drug combination). The two performance metrics of interest are: (i) the effectiveness of the predicted drug combination, in terms of how close its response is to the optimal response, and (ii) the number of search steps that the algorithm needs to take until an effective combination is found. Basically, we want to predict a potent drug cocktail by testing minimal number of drug combinations to minimize the actual experimental cost for measuring the cell response to combinatorial drugs. When choosing the next state, the search algorithm has to be as parsimonious as possible, in terms of the number of function evaluations, so that the overall experimental cost for identifying the optimal drug combination can be minimized. This has been one of the main design considerations of existing stochastic search algorithms that have been developed for combinatorial drug optimization [9, 11].
where α ∈ [0, 1] is a control parameter that adjusts the randomness of the algorithm [11]. This g(x, x') is compared with a uniformly distributed random number r_{ n } ∈ [0, 1]. If g(x, x') > r_{ n }, the n-th drug is rewarded, i.e., updated in such a way that appears to be more beneficial for enhancing the drug response according to the rules shown in (3). Otherwise, the n-th drug is penalized, i.e., updated in a way that appears to be less beneficial based on the past observations. It is not difficult to see that this algorithm is always more likely to reward, or beneficially update, a given drug. Since the algorithm proposed in [11] adaptively determines how to update the drug concentration based on previous observations, it can also effectively deal with drug response functions shown in Figures 1A and 1B, for which the Gur Game algorithm does not perform well. Despite its merits, this algorithm also has its own shortcomings. For example, as the update rule for a given drug is determined only based on the two observations that correspond to its latest update, not on a longer-range trend, the algorithm may be sensitive to small variations in the drug response. As a result, it may not show satisfactory search performance for drug response functions that are similar to the one in Figure 1C. Furthermore, considering that f(x) has to be experimentally estimated, where a certain level of measurement noise and small variations due to a number of practical factors may not be avoidable, such sensitivity may adversely affect the overall performance of the algorithm. Another weakness of the algorithm is that it only utilizes a very small part of the past observations without fully utilizing them. In the following section, we introduce a novel stochastic search algorithm that can effectively address the aforementioned issues.
The adaptive reference update (ARU) stochastic search algorithm
and comparing it with a random number r_{ n } ∈ [0, 1]. If g(x^{ c }, x^{ref}) >r_{ n }, the drug concentration x_{ n } is updated by a single level, following the beneficial update direction predicted by (5). Otherwise, the concentration is updated in the opposite direction. As briefly mentioned before, the parameter α ∈ [0, 1] controls the randomness of the algorithm. For example, α = 0 will make the search process completely random, regardless of the observed drug responses. Using a larger α means that we are giving a larger weight to the past observations when deciding how to update the drug concentrations. The value of this control parameter is limited to α ≤ 1 such that g(x^{ c }, x^{ref}) ≤ 1. Also note that we always have g(x^{ c }, x^{ref}) ≥ 0.5, which implies that at any drug update step, the update is always more likely to take place in accordance with the rules in (5), which have been derived based on past observations of the drug response. In other words, the ARU algorithm tries to effectively utilize the past response data to beneficially update the drug concentrations, and ultimately, to identify a potent drug combination, while keeping the search still stochastic. For illustration, let us again consider the drug response function in Figure 2, where the hypothetical search process proceeds from the lowest drug concentration to the highest concentration. The black solid arrows below the graph shows the drug update direction that gets higher probability according to the new algorithm, described above. For example, in region-A (c_{min} < × < x^{ref1}), the algorithm tends to increase the drug concentration x further, as the response f(x) is larger than f(c_{min}) of the initial reference concentration (i.e., c_{min}). As x continues to increase and passes the first local maximum point x^{ref1}, the reference is updated to x^{ref} ← x^{ref1}. In region-B (x^{ref1} < × < x^{ref2}), the search algorithm tends to drive the concentration towards x^{ref1} by decreasing the concentration. Suppose the search continues to increase the drug concentration x beyond x^{ref2}, the second local maximum point, despite the tendency of the algorithm to decrease x back to x^{ref1}. After passing x^{ref2}, the reference is updated to x^{ref} ← x^{ref2}. In region-C, the search algorithms assigns higher probability to the update rule that tries to bring the concentration down to x^{ref2}, since f(x) < f(x^{ref2}) in the given region. However, once x enters region-D, where f(x) > f(x^{ref2}), the algorithm begins to drive the drug concentration x further to the right until it passes the third local maximum point x^{ref3}. The reference concentration is updated to x^{ref} ← x^{ref3}, once the search continues to the right and the drug concentration x gets larger than x^{ref3}. Since f(x^{ref3}) is larger than f(x) in region-D (x^{ref3} < × < c_{max}), the search algorithm will tend to bring the concentration down to the current reference concentration, namely, x^{ref} = x^{ref3}.
Choosing a local maximum solution as a reference combination has a number of practical advantages. First of all, it allows the algorithm to adjust the drug update rules based on a long-range trend of the given drug response function, which makes the algorithm robust to small variations in the observed response. Another advantage of using a long-range trend is that the search process will become also less sensitive to random fluctuations that may exist in the observed drug response. Considering that the drug response function f(x) has to be experimentally estimated through actual biological experiments, where random factors (e.g., measurement noise) that affect the estimation results cannot be completely ruled out, such robustness is critical for the algorithm to be used in practical drug optimization applications. It is also beneficial to use the drug combination that corresponds to the most recent local maximum response, instead of the drug combination that has yielded the highest response among all past combinations, as the reference point. This prevents the search process from dwelling too much on past observations, while keeping it robust to variations and random fluctuations.
Drug response functions
In order to evaluate the overall performance of the ARU algorithm, we used the algorithm to search for the optimal drug cocktail for several different drug response functions.
where every drug concentration can take its value from one of the 11 discrete concentrations that evenly divide the range [-2.5, 2.5].
Results
Optimizing the combination of two drugs
Performance for optimizing the combination of two drugs.
Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | ||||||
---|---|---|---|---|---|---|---|---|---|
success rate | unique comb. | success rate | unique comb. | success rate | unique comb. | success rate | unique comb. | ||
f_{2a}(x) : HIV INHIBITION | (M = 80) | 97% | 13.2 | 95% | 17.2 | 100% | 13.4 | 100% | 12.1 |
f_{2b}(x) : DE JONG (2ND) | (M = 441) | 9% | 3.6 | 10% | 4.5 | 99% | 56.2 | 99% | 46.2 |
f_{2c}(x) : CANCER INHIBITION | (M = 100) | 58% | 11.3 | 53% | 13.0 | 98% | 13.2 | 98% | 12.4 |
f_{2d}(x): BACTERIA INHIBITION | (M = 81) | 96% | 5.9 | 91% | 6.8 | 100% | 4.8 | 100% | 4.5 |
Optimizing multi-drug cocktails
Performance for optimizing multi-drug cocktails.
Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | ||||||
---|---|---|---|---|---|---|---|---|---|
success rate | unique comb. | success rate | unique comb. | success rate | unique comb. | success rate | unique comb. | ||
f_{3a}(x) | (M = 11^{3}) | 1% | 4.3 | 2% | 5.5 | 100% | 105.3 | 100% | 74.0 |
f_{3b}(x) | (M = 11^{3}) | 83% | 229.4 | 58% | 204.8 | 100% | 88.5 | 100% | 79.4 |
f_{4a}(x) | (M = 11^{4}) | 20% | 823.9 | 11% | 666.6 | 100% | 177.9 | 100% | 136.8 |
f_{4b}(x) | (M = 11^{4}) | 52% | 706.7 | 24% | 520.9 | 100% | 117.9 | 100% | 91.6 |
f_{5a}(x) | (M = 11^{5}) | 8% | 2.1 | 2% | 4.8 | 100% | 138.1 | 100% | 80.6 |
f_{5b}(x) | (M = 11^{5}) | 89% | 1013.4 | 54% | 976.2 | 100% | 252.9 | 100% | 216.8 |
f_{6a}(x) | (M = 11^{6}) | 90% | 1269.1 | 44% | 1260.8 | 100% | 191.9 | 100% | 178.1 |
f_{6b}(x) | (M = 11^{6}) | 90% | 446.7 | 40% | 1033.2 | 100% | 238.1 | 100% | 190.1 |
Drug optimization in the presence of measurement noise
In order to use a drug optimization algorithm in practical applications, the algorithm has to be robust to random fluctuations in the estimated drug response. To evaluate the robustness of the proposed ARU algorithm, we evaluated its search performance in the presence of measurement noise and compared it with other existing stochastic search algorithms. In these experiments, we considered two different types of search strategies. In the first search strategy (referred as type-A), when the search algorithm happens to revisit a drug combination that was previously tested, it does not re-evaluate the drug response and simply uses the previously estimated value. On the other hand, according to the second strategy (referred as type-B), the search algorithm always re-evaluates the drug response, even if it revisits a previously evaluated drug combination, since the measured response may be different every time due to the random measurement noise. The first strategy may be useful when the noise level is relatively low, in which case this strategy may be able to reduce the total number of drug response evaluations, thereby reducing the overall experimental cost for identifying a potent drug combination. However, when the noise level is high, the search performance may be degraded as the search algorithm clings to the past (noisy) response, once it has been measured. In contrast, the second search strategy generally requires a relatively larger number of drug response evaluations, but it tends to be more robust to random fluctuations and noise in the measured drug response function.
Performance for optimizing the combination of two drugs in the presence of noise.
Noise level | Search type | Performance metric | Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | |
---|---|---|---|---|---|---|---|
f_{2a}(x) | (2%) | A | success rate unique comb. | 97% | 96% | 100% | 100% |
12.5 | 17.0 | 13.3 | 11.6 | ||||
B | success rate iterations | 97% | 95% | 100% | 100% | ||
37.8 | 45.2 | 25.8 | 20.0 | ||||
(5%) | A | success rate unique comb. | 97% | 96% | 100% | 100% | |
12.6 | 17.1 | 13.3 | 11.8 | ||||
B | success rate iterations | 97% | 95% | 100% | 100% | ||
38.0 | 45.4 | 26.6 | 20.2 | ||||
(8%) | A | success rate unique comb. | 97% | 96% | 100% | 100% | |
12.6 | 17.0 | 13.3 | 12.0 | ||||
B | success rate iterations | 97% | 95% | 100% | 100% | ||
38.2 | 45.4 | 26.8 | 20.4 | ||||
f_{2b}(x) | (2%) | A | success rate unique comb. | 10% | 10% | 99% | 99% |
3.9 | 4.2 | 57.0 | 45.1 | ||||
B | success rate iterations | 10% | 10% | 99% | 99% | ||
4.0 | 44 | 148.5 | 120.0 | ||||
(5%) | A | success rate unique comb. | 9% | 9% | 98% | 98% | |
4.1 | 4.5 | 62.9 | 52.2 | ||||
B | success rate iterations | 9% | 9% | 98% | 99% | ||
4.3 | 4.7 | 172.1 | 143.8 | ||||
(8%) | A | success rate unique comb. | 8% | 9% | 97% | 98% | |
4.1 | 4.7 | 66.2 | 55.6 | ||||
B | success rate iterations | 9% | 9% | 97% | 98% | ||
4.4 | 4.9 | 198.1 | 167.3 | ||||
f_{2c}(x) | (2%) | A | success rate unique comb. | 61% | 54% | 98% | 98% |
10.9 | 12.7 | 13.1 | 12.5 | ||||
B | success rate iterations | 60% | 54% | 98% | 98% | ||
33.1 | 36.0 | 35.9 | 35.2 | ||||
(5%) | A | success rate unique comb. | 71% | 66% | 98% | 98% | |
11.4 | 13.2 | 12.9 | 12.4 | ||||
B | success rate iterations | 60% | 54% | 98% | 98% | ||
33.2 | 35.4 | 36.7 | 36.0 | ||||
(8%) | A | success rate unique comb. | 88% | 83% | 98% | 98% | |
11.7 | 13.4 | 12.6 | 12.0 | ||||
B | success rate iterations | 6.% | 54% | 98% | 98% | ||
33.4% | 34.3 | 37.4 | 37.0 | ||||
f_{2d}(x) | (2%) | A | success rate unique comb. | 100% | 100% | 100% | 100% |
4.9 | 6.0 | 4.6 | 4.1 | ||||
B | success rate iterations | 96% | 9.% | 100% | 100% | ||
18.3 | 19.7 | 8.2 | 7.7 | ||||
(5%) | A | success rate unique comb. | 100% | 100% | 100% | 100% | |
4.9 | 5.7 | 4.3 | 4.1 | ||||
B | success rate iterations | 96.% | 91% | 100% | 100% | ||
18.4 | 19.6 | 8.1 | 7.5 | ||||
(8%) | A | success rate unique comb. | 100% | 100% | 100% | 100% | |
4.8 | 5.6 | 4.4 | 4.2 | ||||
B | success rate iterations | 96% | 91% | 100% | 100% | ||
18.6 | 19.6 | 8.1 | 7.1 |
Performance for optimizing the combination of three drugs in the presence of noise.
Noise level | Search type | Performance metric | Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | |
---|---|---|---|---|---|---|---|
f_{3a}(x) | (2%) | A | success rate unique comb. | 1% | 3% | 99% | 100% |
2.4 | 7.3 | 110.6 | 77.3 | ||||
B | success rate iterations | 1% | 3% | 99% | 100% | ||
2.8 | 10.9 | 201.1 | 139.6 | ||||
(5%) | A | success rate unique comb. | 1% | 3% | 99% | 100% | |
2.4 | 7.4 | 111.7 | 78.4 | ||||
B | success rate iterations | 1% | 3% | 99% | 100% | ||
2.5 | 10.1 | 201.6 | 144.0 | ||||
(8%) | A | success rate unique comb. | 1% | 3% | 99% | 100% | |
2.5 | 7.7 | 113.3 | 80.5 | ||||
B | success rate iterations | 1% | 3% | 99% | 100% | ||
2.3 | 9.4 | 210.9 | 151.7 | ||||
f_{3b}(x) | (2%) | A | success rate unique comb. | 86% | 69% | 99% | 99% |
224.3 | 201.6 | 110.8 | 93.3 | ||||
B | success rate iterations | 83% | 59% | 99% | 99% | ||
367.1 | 419.1 | 211.5 | 205.3 | ||||
(5%) | A | success rate unique comb. | 89% | 72% | 99% | 99% | |
222.3 | 201.6 | 116.6 | 106.2 | ||||
B | success rate iterations | 83% | 59% | 98% | 98% | ||
359.3 | 439.9 | 225.5 | 222.9 | ||||
(8%) | A | success rate unique comb. | 90% | 74% | 97% | 98% | |
225.9 | 200.9 | 126.4 | 114.3 | ||||
B | success rate iterations | 82% | 60% | 97% | 98% | ||
359.4 | 431.3 | 249.1 | 246.8 |
Performance for optimizing the combination of four drugs in the presence of noise.
Noise level | Search type | Performance metric | Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | |
---|---|---|---|---|---|---|---|
f_{4a}(x) | (2%) | A | success rate unique comb. | 21% | 13% | 96% | 98% |
798.9 | 711.7 | 393.9 | 327.4 | ||||
B | success rate iterations | 21% | 12% | 96% | 97% | ||
941.9 | 1032.1 | 558.3 | 452.3 | ||||
(5%) | A | success rate unique comb. | 21% | 14% | 90% | 95% | |
816.3 | 653.6 | 473.1 | 398.3 | ||||
B | success rate iterations | 21% | 13% | 90% | 95% | ||
895.6 | 1022.9 | 675.4 | 581.2 | ||||
(8%) | A | success rate unique comb. | 24% | 14% | 85% | 95% | |
858.7 | 681.6 | 505.7 | 433.6 | ||||
B | success rate iterations | 23% | 13% | 84% | 92% | ||
997.1 | 1008.9 | 720.8 | 648.5 | ||||
f_{4b}(x) | (2%) | A | success rate unique comb. | 62% | 41% | 100% | 100% |
634.5 | 523.1 | 138.0 | 103.1 | ||||
B | success rate iterations | 51% | 26% | 100% | 100% | ||
932.4 | 903.0 | 236.9 | 182.8 | ||||
(5%) | A | success rate unique comb. | 75% | 68% | 100% | 100% | |
610..2 | 468.0 | 231.1 | 150.9 | ||||
B | success rate iterations | 50% | 25% | 100% | 100% | ||
855.9 | 921.2 | 411.0 | 258.8 | ||||
(8%) | A | success rate unique comb. | 86% | 82% | 98% | 100% | |
525.8 | 430.1 | 314.2 | 215.9 | ||||
B | success rate iterations | 50% | 24% | 94% | 100% | ||
835.3 | 979.2 | 602.0 | 393.8 |
Performance for optimizing the combination of five drugs in the presence of noise.
Noise level | Search type | Performance metric | Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | |
---|---|---|---|---|---|---|---|
f_{5a}(x) | (2%) | A | success rate unique comb. | 8% | 9% | 100% | 100% |
2.1 | 4.4 | 139.3 | 122.5 | ||||
B | success rate iterations | 9% | 11% | 100% | 100% | ||
2.1 | 6.0 | 172.4 | 154.9 | ||||
(5%) | A | success rate unique comb. | 9% | 11% | 100% | 100% | |
3.9 | 7.9 | 142.1 | 129.1 | ||||
B | success rate iterations | 9% | 12% | 100% | 100% | ||
108.3 | 38.6 | 177.1 | 155.6 | ||||
(8%) | A | success rate unique comb. | 10% | 13% | 100% | 100% | |
7.1 | 20.2 | 144.5 | 131.4 | ||||
B | success rate iterations | 9% | 13% | 100% | 100% | ||
191.4 | 70.8 | 182.3 | 156.5 | ||||
f_{5b}(x) | (2%) | A | success rate unique comb. | 89% | 55% | 100% | 100% |
917.9 | 1026.3 | 407.1 | 343.9 | ||||
B | success rate iterations | 90% | 55% | 100% | 100% | ||
999.8 | 1325.1 | 516.5 | 444.3 | ||||
(5%) | A | success rate unique comb. | 90% | 59% | 97% | 98% | |
932.8 | 1002.7 | 507.7 | 463.6 | ||||
B | success rate iterations | 90% | 56% | 99% | 99% | ||
1004.7 | 1332.7 | 656.9 | 562.4 | ||||
(8%) | A | success rate unique comb. | 91% | 59% | 97% | 98% | |
959.5 | 971.2 | 578.8 | 534.8 | ||||
B | success rate iterations | 90% | 56% | 99% | 99% | ||
1015.2 | 1341.0 | 735.0 | 668.6 |
Performance for optimizing the combination of six drugs in the presence of noise.
Noise level | Search type | Performance metric | Gur Game (simultaneous) | Gur Game (sequential) | Previous search algorithm[11](α= 1) | ARU algorithm (proposed) (α= 1) | |
---|---|---|---|---|---|---|---|
f_{6a}(x) | (2%) | A | success rate unique comb. | 91% | 43% | 100% | 100% |
1280.0 | 1214.1 | 476.8 | 432.6 | ||||
B | success rate iterations | 90% | 43% | 100% | 100% | ||
1352.6 | 1662.6 | 531.4 | 503.6 | ||||
(5%) | A | success rate unique comb. | 90% | 44% | 99% | 99% | |
1262.3 | 1247.2 | 621.0 | 598.3 | ||||
B | success rate iterations | 90% | 45% | 100% | 100% | ||
1396.8 | 1675.9 | 763.7 | 736.1 | ||||
(8%) | A | success rate unique comb. | 90% | 46% | 98% | 98% | |
1204.7 | 1302.4 | 723.2 | 698.8 | ||||
B | success rate iterations | 90% | 46% | 98% | 98% | ||
1412.2 | 1681.9 | 875.2 | 834.3 | ||||
f_{6b}(x) | (2%) | A | success rate unique comb. | 91% | 43% | 100% | 100% |
509.9 | 971.0 | 341.4 | 237.7 | ||||
B | success rate iterations | 94% | 44% | 100% | 100% | ||
1240.7 | 1646.0 | 436.5 | 293.5 | ||||
(5%) | A | success rate unique comb. | 90% | 42% | 100% | 100% | |
473.8 | 970.7 | 349.9 | 279.8 | ||||
B | success rate iterations | 94% | 44% | 100% | 100% | ||
1278.6 | 1704.9 | 480.8 | 324.6 | ||||
(8%) | A | success rate unique comb. | 89% | 42% | 100% | 100% | |
454.4 | 969.9 | 457.4 | 353.2 | ||||
B | success rate iterations | 94% | 44% | 100% | 100% | ||
1333.1 | 1775.5 | 545.3 | 391.5 |
As we can see in these Tables, measurement noise certainly affects the overall performance of the ARU algorithm, where a higher noise tends to reduce the success rate and increase the number of iterations as well as that of the unique drug combinations to be tested. For many drug response functions considered in our simulations, the performance degradation is typically not too significant for the proposed algorithm, showing that the ARU algorithm is relatively robust to measurement noise. However, we can also observe that the extent of performance degradation will critically depend on the landscape of the underlying drug response. In most cases, the ARU algorithm continued to substantially outperform other stochastic search algorithms [9, 11], demonstrating that it is better suited for practical drug optimization applications.
One interesting observation is that the performance of the Gur Game algorithm is typically not very sensitive to measurement noise. In fact, in some cases, its performance even improves as the noise level goes up. The main reason for this phenomenon is as follows. As discussed earlier, the Gur Game algorithm does not adapt to the observed drug response function, and for this reason, its overall performance crucially depends on whether or not its predetermined FSA matches the drug response function at hand. As a result, if the FSA does not match the original drug response function well, ironically enough, the measurement noise may perturb the search process in such a way that improves the overall performance. In this sense, the fact that the Gur Game algorithm is not very sensitive to measurement noise reflects its inaptitude for handling various types of drug response functions, rather than its robustness to random fluctuations and noise in the measured drug response.
Conclusions
In this paper, we proposed a novel stochastic search algorithm, called the adaptive reference update (ARU) algorithm, which can be effectively used for optimizing the composition of combinatory drugs. The proposed algorithm intelligently utilizes the drug response values observed in the past to reliably predict how to beneficially update the drug concentrations to improve the drug response. As we demonstrated throughout this paper, the proposed algorithm addresses several shortcomings of previous drug optimization algorithms [9, 11], thereby improving the overall search performance. Numerical experiments based on various types of multi-drug response functions show that the ARU algorithm results in a higher success rate (i.e., higher probability of identifying a potent drug combination) while requiring significantly fewer drug response evaluations. Furthermore, the proposed algorithm is robust to random measurement noise, where its search performance is not substantially affected in the presence of noise and degrades gracefully as the noise level increases. Throughout our experiments, we used α = 1 for the ARU algorithm as well as the previous search algorithm [11]. As discussed earlier, the parameter α controls the randomness of the search, by determining how much weight we should give to the drug response values that we observed in the past. In general, unless the observations are very noisy or the underlying drug response function is assumed to be extremely nonlinear, it would be best to set the parameter to the largest allowed value (i.e., α = 1), so that we fully utilize the past observations for making our best informed guess about the beneficial drug update strategy. For comparison, we also repeated our simulations using α = 0.5 and α = 0.75, whose results are summarized in Table S1 - Table S7 (see Additional file 1). We can see from these results that α = 1 indeed leads to the best performance for the drug response functions and the noise levels we have considered in this paper.
Declarations
Acknowledgements
Based on “Efficient combinatorial drug optimization through stochastic search”, by Mansuck Kim and Byung-Jun Yoon which appeared in Genomic Signal Processing and Statistics (GENSIPS), 2011 IEEE International Workshop on. © 2011 IEEE [19].
This work was supported in part by the National Science Foundation, through NSF Award CCF-1149544.
This article has been published as part of BMC Genomics Volume 13 Supplement 6, 2012: Selected articles from the IEEE International Workshop on Genomic Signal Processing and Statistics (GENSIPS) 2011. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/13/S6.
Authors’ Affiliations
References
- Borisy AA, Elliott PJ, Hurst NW, Lee MS, Lehar J, Price ER, Serbedzija G, Zimmermann GR, Foley MA, Stockwell BR, Keith CT: Systematic discovery of multicomponent therapeutics. Proc Natl Acad Sci USA. 2003, 100: 7977-7982. 10.1073/pnas.1337088100.PubMed CentralView ArticlePubMedGoogle Scholar
- Chesney MA, Ickovics J, Hecht FM, Sikipa G, Rabkin J: Adherence: a necessity for successful HIV combination therapy. AIDS. 1999, 13 (Suppl A): S271-278.PubMedGoogle Scholar
- De Francesco R, Migliaccio G: Challenges and successes in developing new therapies for hepatitis C. Nature. 2005, 436: 953-960. 10.1038/nature04080.View ArticlePubMedGoogle Scholar
- Fitzgerald JB, Schoeberl B, Nielsen UB, Sorger PK: Systems biology and combination therapy in the quest for clinical efficacy. Nat Chem Biol. 2006, 2: 458-466. 10.1038/nchembio817.View ArticlePubMedGoogle Scholar
- Sawyers CL: Cancer: mixing cocktails. Nature. 2007, 449: 993-996. 10.1038/449993a.View ArticlePubMedGoogle Scholar
- Zimmermann GR, Lehar J, Keith CT: Multi-target therapeutics: when the whole is greater than the sum of the parts. Drug Discov Today. 2007, 12: 34-42. 10.1016/j.drudis.2006.11.008.View ArticlePubMedGoogle Scholar
- Calzolari D, Bruschi S, Coquin L, Schofield J, Feala JD, Reed JC, McCulloch AD, Paternostro G: Search algorithms as a framework for the optimization of drug combinations. PLoS Comput Biol. 2008, 4: e1000249-10.1371/journal.pcbi.1000249.PubMed CentralView ArticlePubMedGoogle Scholar
- Feala JD, Cortes J, Duxbury PM, Piermarocchi C, McCulloch AD, Paternostro G: Systems approaches and algorithms for discovery of combinatorial therapies. Wiley Interdiscip Rev Syst Biol Med. 2010, 2: 181-193. 10.1002/wsbm.51.View ArticlePubMedGoogle Scholar
- Wong PK, Yu F, Shahangian A, Cheng G, Sun R, Ho CM: Closed-loop control of cellular functions using combinatory drugs guided by a stochastic search algorithm. Proc Natl Acad Sci USA. 2008, 105: 5105-5110. 10.1073/pnas.0800823105.PubMed CentralView ArticlePubMedGoogle Scholar
- Wang Y, Yu L, Zhang L, Qu H, Cheng Y: A novel methodology for multicomponent drug design and its application in optimizing the combination of active components from Chinese medicinal formula Shenmai. Chem Biol Drug Des. 2010, 75: 318-324. 10.1111/j.1747-0285.2009.00934.x.View ArticlePubMedGoogle Scholar
- Yoon BJ: Enhanced stochastic optimization algorithm for finding effective multi-target therapeutics. BMC Bioinformatics. 2011, 12 (Suppl 1): S18-10.1186/1471-2105-12-S1-S18.PubMed CentralView ArticlePubMedGoogle Scholar
- Zinner RG, Barrett BL, Popova E, Damien P, Volgin AY, Gelovani JG, Lotan R, Tran HT, Pisano C, Mills GB, Mao L, Hong WK, Lippman SM, Miller JH: Algorithmic guided screening of drug combinations of arbitrary size for activity against cancer cells. Mol Cancer Ther. 2009, 8: 521-532. 10.1158/1535-7163.MCT-08-0937.View ArticlePubMedGoogle Scholar
- Jelinek F: Fast sequential decoding algorithm using a stack. IBM Journal of Research and Development. 1969, 13 (6): 675-685.View ArticleGoogle Scholar
- Viterbi A, Omura J: Principles of digital communication and coding. 1979, McGraw-Hill, Inc. New York, NY, USAGoogle Scholar
- Tsetlin M: Finite automata and modeling the simplest forms of behavior. Uspekhi Matem Nauk. 1963, 8: 1-26.Google Scholar
- Tung B, Kleinrock L: Using finite state automata to produce self-optimization and self-control. Parallel and Distributed Systems, IEEE Transactions on. 1996, 7 (4): 439-448. 10.1109/71.494637.View ArticleGoogle Scholar
- Ji C, Zhang J, Dioszegi M, Chiu S, Rao E, Derosier A, Cammack N, Brandt M, Sankuratri S: CCR5 small-molecule antagonists and monoclonal antibodies exert potent synergistic antiviral effects by cobinding to the receptor. Mol Pharmacol. 2007, 72: 18-28. 10.1124/mol.107.035055.View ArticlePubMedGoogle Scholar
- Storn R, Price K: Differential Evolution - A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. Journal of Global Optimization. 1997, 11: 341-359. 10.1023/A:1008202821328. [http://dx.doi.org/10.1023/A:1008202821328].[10.1023/A:1008202821328]View ArticleGoogle Scholar
- Kim M, Yoon B: Efficient combinatorial drug optimization through stochastic search. Genomic Signal Processing and Statistics (GENSIPS), 2011 IEEE International Workshop on: 4-6 December 2011. 2011, 33-35. 10.1109/GENSiPS.2011.6169434.View ArticleGoogle Scholar
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