Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128, USA

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.

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

For example, Calzolari et al.

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

Methods

Combinatorial drug optimization problem

Suppose we have _{n }_{n }_{n }**x **= (_{1}, _{2}, . . . , _{N}**x**) be the normalized drug response that quantitatively measures the effectiveness of a given drug combination **x**. We assume the response has been normalized such that **x **is completely ineffective, and larger **x**) implies higher efficacy. In practical applications, **x**) may be obtained by monitoring the cell response to a drug intervention using fluorescent imaging, microarrays, or sequencing techniques. Under this setting, we aim to find the optimal drug combination **x*** that maximizes the normalized drug response:

As we can see, this is a combinatorial optimization problem, in which we have to find the optimal drug combination out of _{1 }_{2 }. . . _{N }**x**), it is apparent that we cannot test all drug combinations to find the most effective one.

Stochastic search algorithms

Stochastic search algorithms

For example, the Gur Game algorithm adopted in **x**^{c}_{n }_{n }**x**^{c}_{n}**x**^{c}_{n}**x**^{c}_{n}

The reference concentration **x**^{c}_{n}

Note that penalization moves the current drug concentration closer to the reference concentration **x**) is properly normalized and the reference concentration _{min}, _{max}]. Since _{ref }although it is clearly not optimal. Figure _{ref}. This will push the concentration either towards _{min }or _{max}, both of which are suboptimal.

Drug response functions

**Drug response functions**. (A) A normalized drug response

The enhanced stochastic algorithm proposed in **x**^{c}

where **x **and _{n }**x**) and

The above rules allow the algorithm to adaptively determine

where **x**, **x'**) is compared with a uniformly distributed random number _{n }**x**, **x'**) _{n}**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

In order to make the search algorithm robust to small variations in the observed drug response, the update rules have to be decided based on the general trend of the drug response change over a wide range of drug concentration, not just based on the response change resulting from a **adaptive reference update (ARU) algorithm**. In this algorithm, we compare the current drug response **x**^{c}**x**^{ref}) of a **x**^{ref}, which is adaptively updated based on past observations. In the beginning, **x**^{ref }is set to the initial drug concentration, where we start the search process. During the search, when the algorithm encounters a local maximum, the current reference combination is replaced by the corresponding drug combination. As an example, let us consider the one-dimensional drug response function _{min }to the highest concentration _{max}. Suppose the search begins at the concentration _{min}. Initially the reference concentration is also set to this initial drug concentration ^{ref }← _{min}. As the search reaches the first local maximum, the reference concentration is updated to this local maximum solution ^{ref }← ^{ref1}. As the search continues to the right, this reference concentration is used until we reach the next local maximum. After passing the second local maximum solution ^{ref2}, the reference point is updated to ^{ref }← ^{ref2}. In a similar manner, as the search continues further to the right, the reference point is updated to ^{ref }← ^{ref3 }after passing the third local maximum point.

Updating the reference point

**Updating the reference point**. As the search for the optimal drug concentration continues from left to right (from the lowest concentration to the highest one), the reference concentration is updated from the initial drug concentration _{min }to the local maximum points ^{ref1}, ^{ref2}, and ^{ref3}, according to this order.

At each iteration, the current drug response **x**^{c}**x**^{ref}) of the reference drug combination, based on which the drug update rule is determined. For example, let **x**^{c}**x**^{ref}). As before, we have the following four possible cases.

Conceptually, we can view the above as estimating the "virtual" slope between two points

based on which we determine how to update the concentration _{n }

and comparing it with a random number _{n }**x**^{c}**x**^{ref}) >_{n}_{n }**x**^{c}**x**^{ref}) ≤ 1. Also note that we always have **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 _{min }^{ref1}), the algorithm tends to increase the drug concentration _{min}) of the initial reference concentration (i.e., _{min}). As ^{ref1}, the reference is updated to ^{ref }← ^{ref1}. In region-B (^{ref1 }^{ref2}), the search algorithm tends to drive the concentration towards ^{ref1 }by decreasing the concentration. Suppose the search continues to increase the drug concentration ^{ref2}, the second local maximum point, despite the tendency of the algorithm to decrease ^{ref1}. After passing ^{ref2}, the reference is updated to ^{ref }← ^{ref2}. In region-C, the search algorithms assigns higher probability to the update rule that tries to bring the concentration down to ^{ref2}, since ^{ref2}) in the given region. However, once ^{ref2}), the algorithm begins to drive the drug concentration ^{ref3}. The reference concentration is updated to ^{ref }← ^{ref3}, once the search continues to the right and the drug concentration ^{ref3}. Since ^{ref3}) is larger than ^{ref3 }_{max}), the search algorithm will tend to bring the concentration down to the current reference concentration, namely, ^{ref }= ^{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 **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.

**Two-dimensional drug response functions **For performance assessment, we first used the four two-dimensional drug response functions that are shown in Figure _{2a}(_{1}, _{2}) shown in Figure _{1 }∈ {0, 0.01, 0.03, 0.09, 0.27, 0.82, 2.47, 7.41, 22.22, 66.67}(_{2 }∈ {0, 0.09, 0.27, 0.8, 2.41, 7.22, 21.67, 65}(_{2b}(_{1}, _{2}) shown in Figure _{1}, _{2 }∈ {_{0}, _{1}, ... , _{20}}, where _{k }_{2c}(_{1}, _{2}) in Figure _{1 }∈ {0, 1, 2, 4, 6, 8, 12, 16, 20, 22}(_{2 }∈ {0, 0.25, 0.4, 0.6, 0.8, 1, 1.5, 2, 4, 6.8}(_{2d}(_{1}, _{2}), the normalized bacterial (_{1 }∈ {0, 0.08, 0.16, 0.32, 0.63, 1.25, 2.5, 5, 10} was considered for Trimethoprim and _{2 }∈ {0, 0.31, 0.62, 1.25, 2.5, 5, 10, 20, 40} was considered for Sulfamethoxazole. All four drug response functions were normalized to span the entire range [0, 1], such that the minimum response is 0 and the maximum response is 1.

Two-dimensional drug response functions

**Two-dimensional drug response functions**. (A) Inhibition of HIV. (B) Second De Jong function (Rosenbrock's saddle). (C) Inhibition of A549 lung carcinoma cell proliferation. (D) Inhibition of bacteria (

**Multi-dimensional drug response functions **To evaluate the performance for optimizing multi-drug cocktails, we defined several hypothetical drug response functions with up to six drugs. First, we defined two 3-dimensional drug functions _{3a}(_{1}, _{2}, _{3}) and _{3b}(_{1}, _{2}, _{3}). The first function is defined as

where each of _{1}, _{2}, _{3 }can take one of the 11 discrete concentrations that evenly divide the range [-2.5, 2.5]. The second function is defined as follows

using the Matlab peaks(_{1}, _{2}) function. We assume that each drug can take one of the 11 discrete values that evenly divide [-3, 3]. Next, we defined two 4-dimensional drug functions

and

We assume that _{1 }and _{2 }in the first function _{4a}(_{1}, _{2}, _{3}, _{4}) can take one of the 11 discrete values that evenly divide the range [-2, 2], while _{3 }and _{4 }can take one of the 11 discrete values that evenly divide the range [-3, 3]. For the second drug response function _{4b}(_{1}, _{2}, _{3}, _{4}), we assume that each drug can take one of the 11 distinct values that evenly divide [-3, 3]. In addition, we also defined the following 5-dimensional drug response functions

and

For the first function _{5a}(_{1}, _{2}, _{3}, _{4}, _{5}), _{1 }and _{2 }are allowed to take any value from the set of values obtained by evenly dividing the range [-2, 2] into 11 discrete concentrations. The remaining drug concentrations (_{3}, _{4}, and _{5}) can take any of the 11 concentrations that evenly divide [-4.5, 4.5]. For the second drug response function _{5b}(_{1}, _{2}, _{3}, _{4}, _{5}), we assume that each drug can take one of the 11 discrete values that evenly divide [-3, 3]. Finally, we also defined two 6-dimensional drug response functions

and

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

We first evaluated the overall performance of the ARU stochastic search algorithm based on four two-dimensional drug response functions (see Methods). (i) HIV inhibitor response _{2a}(_{1}, _{2}), (ii) second De Jong function (Rosenbrock's saddle) _{2b}(_{1}, _{2}), (iii) normalized lung cancer inhibition response _{2c}(_{1}, _{2}), and (iv) bacterial (_{2d}(_{1}, _{2}). These functions are shown in Figure _{1 }and _{2}. The parameter _{1}, _{2}) ≥ 0.95. The second metric is defined as the average number of unique drug combinations that have to be tested until a potent drug combination is identified, in case the search is successful. These performance assessment results are summarized in Table

Performance for optimizing the combination of two drugs.

**Gur Game (simultaneous)**

**Gur Game (sequential)**

**Previous search algorithm **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

**success rate**

**unique comb**.

**success rate**

**unique comb**.

**success rate**

**unique comb**.

**success rate**

**unique comb**.

_{2a}(**x**) : HIV

(

97%

13.2

95%

17.2

100%

13.4

**100%**

**12.1**

_{2b}(**x**) :

(

9%

3.6

10%

4.5

99%

56.2

**99%**

**46.2**

_{2c}(**x**) :

(

58%

11.3

53%

13.0

98%

13.2

**98%**

**12.4**

_{2d}(**x**):

(

96%

5.9

91%

6.8

100%

4.8

**100%**

**4.5**

**Performance of the ARU algorithm**. Further performance evaluation results of the proposed Adaptive Reference Update (ARU) algorithm.

Click here for file

Optimizing multi-drug cocktails

Next, we tested the performance of the ARU algorithm for optimizing multi-drug cocktails that consist of three to six drugs. For this purpose, we used the eight hypothetical drug response functions that were defined before (see Methods). As in our previous experiments, for each drug response function, we used the proposed ARU algorithm (with **x**) ≥ 0.95). In each search experiment, we started from an randomly selected initial concentrations, and continued the search up to 1,000 steps for 3 drugs, 2,000 steps for 4 drugs, 3,000 steps for 5 drugs, and 4,000 steps for 6 drugs. This experiment was repeated 5,000 times to obtain reliable performance assessment results. The simulation results are summarized in Table

Performance for optimizing multi-drug cocktails.

**Gur Game (simultaneous)**

**Gur Game (sequential)**

**Previous search algorithm **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

**success rate**

**unique comb**.

**success rate**

**unique comb**.

**success rate**

**unique comb**.

**success rate**

**unique comb**.

_{3a}(**x**)

(^{3})

1%

4.3

2%

5.5

100%

105.3

**100%**

**74.0**

_{3b}(**x**)

(^{3})

83%

229.4

58%

204.8

100%

88.5

**100%**

**79.4**

_{4a}(**x**)

(^{4})

20%

823.9

11%

666.6

100%

177.9

**100%**

**136.8**

_{4b}(**x**)

(^{4})

52%

706.7

24%

520.9

100%

117.9

**100%**

**91.6**

_{5a}(**x**)

(^{5})

8%

2.1

2%

4.8

100%

138.1

**100%**

**80.6**

_{5b}(**x**)

(^{5})

89%

1013.4

54%

976.2

100%

252.9

**100%**

**216.8**

_{6a}(**x**)

(^{6})

90%

1269.1

44%

1260.8

100%

191.9

**100%**

**178.1**

_{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 **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.

In order to evaluate the performance of the different search algorithms in the presence of noise, we performed similar search experiments as before at three different levels of additive noise. 2%, 5%, and 8%. More precisely, we assume that

where _{obs}(**x**) is the observed drug response, _{true}(**x**) is the true response, and **type-A **search, we evaluated the success rate and the average number of unique drug combinations that have to be tested until a potent drug combination is identified. For **type-B **search, we evaluated the success rate and the average number of iterations, instead of the number of unique drug combinations, until an effective combination is identified. This is because, in a **type-B **search, the search algorithm re-evaluates the drug response even if it revisits the same drug combination that was previously tested. The simulation results are shown in Table

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 **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

_{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**

_{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**

_{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**

_{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 **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

_{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**

_{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 **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

_{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**

_{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 **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

_{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**

_{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 **
**( α = 1)**

**ARU algorithm (proposed) ( α = 1)**

_{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**

_{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

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

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Conceived and developed the algorithm: MK, BJY. Performed the experiments: MK. Analyzed the data and wrote the paper: MK, BJY.

Acknowledgements

Based on “Efficient combinatorial drug optimization through stochastic search”, by Mansuck Kim and Byung-Jun Yoon which appeared in

This work was supported in part by the National Science Foundation, through NSF Award CCF-1149544.

This article has been published as part of