Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China E-mail: [email protected]

Abstract In pharmacokinetics, the problem of solving parameters of pharmacokinetics requires algorithm to be high stability, accuracy and generality. This paper a novel swarm intelligence algorithm-glowworm swarm optimization algorithm is proposed for solving parameters of pharmacokinetics. The most important advantage of the algorithm is that it can dynamically divide into many subgroups to parallels search multiple local optima, so it is able to avoid being trapped into local optima effectively and enable the algorithm to acquire stable global optimum quickly. Compared with traditional methods, such as, artificial immune network, particle swarm optimization, the experiments result show that the run time and accuracy of algorithm is good, so the glowworm swarm optimization algorithm is a novel effective and feasible swarm intelligence algorithm for solving parameters of pharmacokinetics.

Keywords: Pharmacokinetics; Glowworm Swarm Optimization; Compartmental Model; Artificial Immune Network; Particle Swarm Optimization.

1. Introduction Pharmacokinetics is a discipline which quantificational studies the law of the medicaments, including exotic chemical matters, absorbed, distributed, excreted and metabolized in the body [1]. After medicaments enter into body, two kinds of effects appear, one is the biological effect of medicaments to the body, including therapeutically effect and toxic side effects which are called pharmacodynamics and toxicology, and the other is the effect of body to the medicaments, including absorption, distribution, metabolism and excretion of the medicaments, which is called ADME for short. Therefore, pharmacokinetics is a discipline to synthetically study the law of effect of body to the medicaments. In the process of creation of new medicaments, as to many kinds of drugs, the relation between curative effect and poisonousness and blood drug concentration is tighter than the relation between curative effect and poisonousness and its dose. The curative effect of drugs can be controlled or the poisonousness of drugs can be reduced by adjusting the blood drug concentration. One of the primary purposes of pharmacokinetics is to create particular mathematical model according to variation laws of blood drug concentration over time. Using this model, we can compute the parameters of pharmacokinetics so as to forecast the variation laws of blood drug concentration. These laws can guide the creation of dosage regimen and explain some pharmacological phenomenon so as to improve the safety and rationality of some medicaments, and meanwhile, to guide the creation, study and estimate of some new medicaments. So far, the domestic common solutions to compute parameters of pharmacokinetics are weighted residuals method (WRM), simplex method, BFGS, artificial immune algorithm, PSO and so on [2-5]. This paper, we use artificial glowworm swarm optimization algorithm [6] [7] to compute the parameters of pharmacokinetics. Artificial glowworm swarm optimization algorithm is a new swarm intelligence optimal algorithm proposed by K.N. Krishnanad and D. Ghose in 2005. This algorithm has increasingly gained more and more attention and became a new focus. This algorithm has successfully applied to the optimization of multimodal functions [8] and Multi-signal source search [9][13-16] , because it has high efficiency to capture local optima, excellent robustness, excellent versatility.

2. Compartmental Model Metabolism process of medicaments in body includes absorption, distribution, excretion, and so on. In order to master the law of those dynamic transformation processes, particular theories, solutions and

International Journal of Advancements in Computing Technology(IJACT) Volume5,Number8,April 2013 doi:10.4156/ijact.vol5.issue8.22

197

Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

mathematical models are used to demonstrate the law of the dynamic process of drug in body over time. There are many kinds of those mathematical models like compartmental model, kinetics model, nonlinear model, and so on. Traditional pharmacokinetics use compartment al model. Compartmental model regards the entire body as a whole system and divided the body into several compartments by the characteristic of pharmacokinetics. The compartmental model is shown in Figure 1.

K 21

K 12

K 13

1

K 31

Ke

2

3

4

Figure 1. Compartmental model Usually, there are three kinds of compartmental models like one-compartment model, twocompartment model and multi-compartment model. The partition of compartment is lying of the transport rate of drugs in tissues or organs of body. Different organs or tissues can be regarded in the same compartment when the transport rate in them is equal or similar, but the compartment model we describe here is only the abstract concept of mathematical model. It means that it can’t represent any tissue or organ in anatomy, so the division of compartment has abstraction and subjectivity. The onecompartment model means that drug in the whole body reaches the dynamic steady state, or the transport rate in every tissue or organ of body is equal or similar. In this moment, we regard the whole body as a compartment and call it one-compartment model which is shown in Figure 2. Twocompartment model divided the body into two compartments, the central compartment and the peripheral compartment which are shown in Figure 3. The central compartment is made up of some kinds of tissues, such as heart, liver, kidney and lung, whose membrane permeability is good and in which blood flow is rich. Medicaments first enter into this kind of tissue and achieve the dynamic balance with the medicaments in blood. Conversely, the peripheral compartment is made up of some other kinds of tissues, such as fat and muscle in static state, in which blood is not so rich, the transport rate is slow and the blood mixed with drugs hard to pour into. It needs some time for the drugs in these tissues to achieve dynamic balance with the drugs in the blood.

K12

X0

X

X

X

1

X0

K21

K Figure 2. One-compartment model

2

K 10 Figure 3. Two-compartment model

3. Basic Glowworm Swarm Optimization(GSO) In the basic Glowworm Swarm Optimization Algorithm (GSO), a swarm of glowworms are initially randomly distributed in the solution space. They carry their own luciferin respectively which has equal initial value. The glowworms emit a light whose intensity is proportional to the associated luciferin. The luciferin quantity is tightly associated with the position the glowworms locate in their movement. The glowworm whose luciferin quantity is higher has stronger attraction to the other glowworms in its neighborhood. The neighborhood whose size is decided by radius ( rd ) is called local-decision range in GSO. The size of rd dynamically changes between 0 and rs . rs is called radial sensor range in GSO.

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Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

In the movement, each glowworm moves to another glowworm which is in its neighborhood at a certain probability. Glowworm j who wants to become the neighbor of glowworm i must be located in the neighborhood of i and has higher luciferin quantity than i . Through the movement of glowworms, most glowworms will converge to the glowworms that have higher luciferin quantity. The every repeated iteration of GSO is constituted by two stages. The first one is update stage, the other one is movement stage. Luciferin update stage: In this stage, every glowworm updates their luciferin by formula (1).

li (t )  (1   )li (t  1)  J ( xi (t ))

(1)

where, li (t ) is the luciferin quantity of i at iteration t ,   (0,1) is the modulus to control luciferin quantity,



is the modulus to evaluate objective value of function, J  xi  t   and is the objective value

of function. Movement of glowworm: In this stage, glowworm i selects another glowworm j which is located in its neighborhood and move to it. The probability formula is given by (2), the next position of i is decided by (3), the update of rd , which is in the end of movement stage, is given by (4).Probability formula used to select a neighbor is as follow:

pij (t ) 



l j (t )  li (t )

(2)

l (t )  li (t )

k N i ( t ) k

Update formula of position:

 x j (t )  xi (t ) xi (t  1)  xi (t )  s *   x (t )  x (t ) i  j where, xi (t )  R

m

   

is the position of i , at time t, in the m-dimensional real space,

(3)

||  || represents the

Euclidean norm operator, and s (>0) represents moving step of glowworm.

Update formula of local-decision range:

rdi (t  1)  min{rs , max{0, rdi (t )   (nt  N i (t ) )}} where,  is the proportion modulus,

(4)

nt is the modulus used to control the number of neighbors,

| Ni (t)| is the number of neighbors of i . 4. GSO for Solving Parameters of Pharmacokinetics GSO has been used in the global optimization of multimodal functions, 0-1 knapsack problems, clustering problem, sensor noise test, simulation robot and many other application problems. This paper uses it to solve the parameters of pharmacokinetics. The key of this problem is to acquire more precision in the condition of faster convergence speed and stability. Each glowworm is constructed by

xit  [ xit1 , xit2 ,..., xidt ] , which represents the i -th glowworm in the t generation, among that vector d means that the mathematical model of pharmacokinetics has d parameters in all and xij represent the j -th parameter. In order to improve the velocity of GSO and a d -dimensional vectors

enable glowworm groups to move to the search range where the global optimum is located quickly, this

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Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

paper first use the WRM to estimate each parameter to ensure the algorithm to get more accurate search range and then use GSO to get more precise solution further so that the algorithm can get more precise solution.The detail steps of GSO to solve the parameters of pharmacokinetics are as follows: Step 1. Initialize  ,  ,  , s, l0 , rs , n and so on, and initialize the glowworm groups and maximum iteration numbers

Tmax , and then estimate each glowworm which represents a solution by WRM;

Step 2. Update luciferin of all the glowworm according to formula (1); Step 3. Calculate the neighbors of each glowworm; Step 4. Select j ( j N i (t )) as the movement direction of i by roulette, and update the position of

i by formula (3); Step 5. Update rd of each glowworm by formula (4). Step 6. Let t  t  1 ， complete an iteration, and judge whether end condition is satisfied,. If satisfied, record the result and exit iteration, or return to step 2 for the next iteration.

5. Simulations 5.1Experimental Environment and Parameter Setting In order to test and verify the feasibility and availability, simulation tested three typical optimization examples for solving parameters of pharmacokinetics. Algorithm is coded in MATLAB2008a and implemented on Intel® Core™2 Duo CPU E4500 2.20GHz PC with 2G RAM under windows XP operation system. The results of simulation are compared with References [4] and [5], which indicates that it is feasible and efficient. The parameters of GSO are set in table 1, where the size of groups n is set to 100 and the maximum iterations Tmax is set to 200.

Table 1. The parameters value of GSO







nt

l0

n

Tmax

s

0.4

0.6

0.08

5

5

100

200

0.03

rs

15

5.2 Experimental Results and Analysis The standard methods to evaluate the results of parameters of pharmacokinetics got by algorithms generally are residual sum of squares, degree of fitting and AIC (Akaike’s InformationCriterion). The essences of above standard methods are all to compute the sum of squares of the difference of the measure value and the fitted values; the value is lesser the better it is. This paper use formula (5) to evaluate fitting degree at t time point and use formula (6) to evaluate the parameter of pharmacokinetics got by different algorithms.

E (t )  min sum(C (t )  l (t )) 2 where,

(5)

C (t ) represents the blood concentration at t time point, l (t ) represents actual measure value. n

E  '

 E (t ) i

i 1

n

(6) Where, E represents the average absolute deviation, t i represents the i -th time point, there are total

n time points.

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Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

Example 1. After single-dose intravenous injections with ligustrazine hydrochloride, computing the optimized results of two-compartment model by different algorithms. The data of blood concentration over time after single-dose intravenous injections of ligustrazine hydrochloride to rat is displayed in Table 2. Table 2. The measured data of blood concentration at different time point

time / h

0.02

0.25

0.50

1.00

1.50

2.0

3.0

4.0

5.0

6.0

7.0

concentration

67.8

39.4

29.9

19.5

15.9

13.8

11.7

5.60

4.0

0.8

0.6

mg  L1

The compartmental model of this example after mathematical transformation is as follow:

C (t )  Ae  t  Be   t

(7)

where, A, B,  ,  represent the parameters to be optimized, the parameters of pharmacokinetics got by GSO and other algorithms in reference [4] are displayed in Table 3.

Table 3. The parameters of pharmacokinetics got by every algorithm Algorithm

A



B



time / s

SIA

41.093415

4.809727

30.64385

0.409432

13.98

Simplex

41.271650

0.560382

111.580405

70.967050

0.50

PKAIN

41.115602

4.802630

30.605047

0.409056

5.35

PKAIN_in

41.108291

4.808103

30.619588

0.408776

18.64

PKAIN_spx

41.086746

4.803905

30.632724

0.409310

0.53

GSO

41.084998

4.808321

30.648061

0.409457

10.64

According to the parameters got by GSO, we can identify the two-compartment model which is

C (t )  41.0850e 4.8083t  30.6480e 0.4095t . The fitting effect of above model can be seen in Figure 4 and Figure 5, where Figure 4 is the comparison chart of fitting data and measured data and Figure 6 is the error of fitting data and measured data.

Figure 4. Fitting efficient chart of GSO

Figre 5. The fitting error chart

The errors at different time points between fitting value and measured value got by GSO and other algorithms in reference [4] are displayed in Table 4.

201

Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

Table 4. The errors between fitting values and measured values 0.02

0.25

0.50

1.00

1.50

2.0

3.0

4.0

5.0

6.0

7.0

SIA

-0.0814

0.6094

-1.2192

1.1833

0.7117

-0.2854

-2.7277

0.3578

-0.0438

1.8270

1.1444

Simplex

0

-3.5236

1.2865

4.0657

1.9071

-0.3441

-4.0168

-1.2130

-1.4951

0.6303

0.2167

PKAIN

-0.0942

0.6056

-1.2309

1.1677

0.7004

-0.2923

-2.7290

0.3592

-0.0414

1.8296

1.1468

PKAIN_in

-0.0904

0.6016

-1.2263

1.1812

0.7149

-0.2784

-2.7172

0.3687

-0.0340

1.8353

1.1510

PKAIN_spx

-0.0903

0.6016

-1.2263

1.1811

0.7149

-0.2784

-2.7172

0.3687

-0.0340

1.8353

1.1510

GSO

-0.0839

0.6149

-1.2142

1.1860

0.7134

-0.2842

-2.7272

0.3580

-0.0438

1.8269

1.1443

Ttime

In order to synthetically evaluate the fitting effect of every different algorithm, we further display the absolute average errors respectively got by above algorithms displayed in Table 5.

Table 5. The absolute average error of fitting value and measured value got by every algorithm Algorithm

SIA

Simplex

PKAIN

PKAIN_in

PKAIN_spx

GSO

Absolution average error

0.926469

1.699915

0.927012

0.927181

0.927181

0.926982

In Figure 5, we can see that the fitting effect of GSO is good. From table 4 and table 5, we can see the accuracy got by GSO is better than Simplex，PKAIN，PKAIN_in and PKAIN_spx. But from table 3, it is can be seen that the run time of GSO is slower than Simplex, PKAIN and PKAIN_spx and is faster than SIA and PKAIN_in.

Example 2. The optimized results of two-compartment model of some medicaments This medicament fits the two-compartment model. The blood concentration over time after drug taking is displayed in Table 6. Table 6. The measured data of blood concentration at different time point

time / h concentration

0.1

0.3

1.0

2.5

7.5

10

15

20

25

30

50

0.417

1.190

3.342

5.861

6.205

5.318

3.785

2.783

2.139

1.704

0.810

mg  L1

The compartmental model of this example after mathematical transformation is as follow:

C (t )  A1e  kt  A2 e  t  A3e   t

（8）

In this example, A1 , A2 , A3 ,  ,  , K are the parameters to be optimized. The parameter s got by GSO and other algorithms in Reference [5] are displayed in Table 7.

Table 7. The parameters of pharmacokinetics got by every algorithm Algorithm

A1

A2

A3





K

time / s

FM

11.2928

4.7033

-15.9960

0.0352

0.4042

0.1545

—

NM-SM

10.6093

4.1696

-14.7790

0.0330

0.4000

0.1410

—

PSO

-2.3835

9.6144

-7.3239

0.0586

0.5444

0.5444

—

AHPSO

10.6146

4.1715

-14.7868

0.0330

0.4000

0.1410

—

GSO

10.6147

4.1715

-14.7868

0.0330

0.3999

0.1410

17.948

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Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

According to the parameters got by GSO, we can identify the two-compartment model, which is C(t) 10.6147e0.1410t  4.1715e0.0330t 14.7868e0.3999t . The fitting effect of above model can be seen in Figure 6 and Figure 7.

Figure 6. Fitting efficient chart of GSO

Figure 7. The fitting error chart

The errors at different time point between fitting value and measured value got by GSO and other algorithms in Reference [5] are displayed in Table 8.

Table 8. The errors between fitting values and measured values Ttime FM NM-SM PSO AHPSO GSO

1

2.5

7.5

10

15

20

25

30

40

50

-0.0269

0.1

-0.0751

0.3

-0.1973

-0.2980

-0.1799

-0.1181

-0.0641

-0.0525

-0.0482

-0.0417

-0.0224

-0.0042

0.0001

0.0002

0.0003

0.0005

0.0005

0.0005

0.0003

0.0002

0.0002

0.0002

0.0001

0

-2.3741

1.2932

0.5508

-2.1173

-2.9340

-2.2604

-1.1134

-0.4487

-0.0994

0.0781

0.02086

0.2287

0.0002

0.0003

0

0

0.0002

0.0003

0.0001

0.0002

0.0003

0

0.0002

0.0001

-0.0002

0.0003

0.0000

0.0000

0.0002

0.0003

0.0001

-0.0002

0.0003

-0.0000

-0.0002

0.0001

The absolute average error got by GSO and other algorithms in reference [5] is displayed in Table 9.

Table 9. The absolute average error of fitting value and measured value got by every algorithm Algorithm Absolute error

average

FM

NM-SM

PSO

AHPSO

GSO

0.094033

0.000258

1.126580

0.000158

0.000158

According to above experimental results, we can see that the fitting effect of GSO is good from Figure 7, nearly all the measured values are in the fitting curve. From table 8 and 9, we also can know that the fitting accuracy of GSO is nearly equal to that of AHPSO and is better than other algorithms.

Example 3. The optimized results of one-compartment model of extravascular dministration pharmacokinetics This example fit the one-compartment model. Taking drinking alcohol for instance, the data of alcohol blood concentration over time is displayed in Table 10. Table 10. The measured data of blood concentration at different time point

time / h

0.25

0.5

0.75

1

1.5

2

2.5

3

3.5

4

4.5

5

concentration

30

68

75

82

82

77

68

68

58

51

50

41

time / h

6

7

8

9

10

11

12

13

14

15

16

concentration

38

35

28

25

18

15

12

10

7

7

4

mg  L1

mg  L 1

203

Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

The compartmental model is C (t ) 

1 (e   t  e   t ) , where 1 ,  2 ,  3 2

3

are the parameters to

be optimized. The optimized parameters got by GSO and other algorithms in Reference [5] are displayed in Table 11.

Table 11. The parameters of pharmacokinetics got by every algorithms Algorithm

1

2

3

time / s

SIA Simplex PKAIN PKAIN_in PKAIN_spx GSO

114.368066 0 114.532767 114.471106 114.432551 114.432506

0.185327 0 0.185685 0.185600 0.185502 0.185502

2.007706 0 2.006240 2.007833 2.007934 2.007939

35.00 0.50 14.51 27.62 1.28 10.825149

After optimization by GSO, the compartment model is C(t )  114.4325(e fitting effect of this model can be seen in the Figure 8 and Figure 9.

Figure 8. Fitting efficient chart of GSO

0.1855t

 e2.0.079t ) , the

Figure 9. The fitting error chart

The error between the fitting value and the measured value of GSO and other algorithms in Reference [4] is displayed in Table 12.

Table 12. The errors between fitting values and measured values Time

0.25

0. 5

0.75

1

1.5

2

2.5

3

3.5

4

4.5

5

SIA Simplex PKAIN PKAIN_ in PKAIN_ spx GSO

9.9558

-5.6654

-0.8451

-2.3385

-1.0172

-0.1164

3.2037

-2.6859

1.6848

3.4581

-0.3412

4.2714

-30

-68

-75

-82

-82

-77

-68

-68

-58

-51

-50

-41

9.9781

-5.6250

-0.7931

-2.2804

-0.9596

-0.0683

3.2390

-2.6635

1.6953

3.4582

-0.3498

4.2555

9.9866

-5.6208

-0.7962

-2.2907

-0.9787

-0.0897

3.2189

-2.6806

1.6815

3.4476

-0.3575

4.2504

9.9775

-5.6346

-0.8121

-2.3067

-0.9924

-0.0997

3.2128

-2.6833

1.6819

3.4507

-0.3523

4.2574

9.9776

-5.6345

-0.8120

-2.3067

-0.9924

-0.0997

3.2127

-2.6833

1.6819

3.4507

-0.3523

4.2574

Time

6

7

8

9

10

11

12

13

14

15

16

SIA Simplex PKAIN PKAIN_ in PKAIN_ spx GSO

-0.3835

-3.7465

-2.0335

-3.4262

-0.0758

-0.1080

0.3728

0.2797

1.5407

0.0959

1.8955

-38

-35

-28

-25

-18

-15

-12

-10

-7

-7

-4

-0.4101

-3.7780

-2.0705

-3.4646

-0.1141

-0.1452

0.3375

0.2467

1.5102

0.0680

1.8702

-0.4112

-3.7781

-2.0669

-3.4598

-0.1086

-0.1393

0.3434

0.2525

1.1516

0.0733

1.8751

-0.4018

-3.7672

-2.0553

-3.4480

-0.0970

-0.1282

0.3538

0.2621

1.5246

0.0813

1.8823

-0.4018

-3.7672

-2.0552

-3.4480

-0.0970

-0.1282

0.3538

0.2621

1.5246

0.0813

1.8823

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Using Glowworm Swarm Optimization Algorithm for Solving Parameters of Pharmacokinetics Qifang Luo, Yongquan Zhou, Kai Huang

According to the data got by GSO and other algorithms in reference [4], the absolute average error is displayed in Table 13.

Table 13. The absolute average error of fitting value and measured value got by every algorithm Algorithm

SIA

Simplex

PKAIN

PKAIN_in

PKAIN_spx

GSO

Absolute average error

2.153986

41.260870

2.147094

2.148838

2.150564

2.150558

According to above experiment results of one-compartment model, We know that the fitting effect of GSO is good from figure 9. Compared with the results of reference [4], it can be seen that the fitting effect of GSO is better than SIA, Simplex and PKAIN_spx from Table 12 and Table 13. We also can know that the run time of GSO is better than SIA, PKAIN and PKAIN_in but worse than Simplex and PKAIN_spx.

6. Conclusions The key of the problem of solving parameters of pharmacokinetics is the accuracy of results. This paper first use GSO to solve the problem of parameters of pharmacokinetics. The quick parallel search capability of GSO ensures it to obtain accurate parameters of pharmacokinetics and avoid the whole groups being trapped into individual local optima. Through simulation, it is demonstrated that GSO not only can obtain stable and accurate solution quickly but also has good generality. As to the onecompartment model and two-compartment model, the fitting effect of GSO is both good, so it is feasible and effective to solve the problem of parameters of pharmacokinetics by GSO.

7. Acknowledgements This work is supported by National Science Foundation of China under Grant No. 61165015. Key Project of Guangxi Science Foundation under Grant No. 2012GXNSFDA053028, Key Project of Guangxi High School Science Foundation under Grant No. 20121ZD008 and the Funded by Open Research Fund Program of Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China under Grant No. IPIU01201100.

8. References [1] Guangji Wang. Pharmacokinetics. Beijing: Chemical Industrial Press, 2005. [2] Zhang Jiwen, Pei Xiaobing. "Feathering method for processing absorption and elimination rate constants of mono-compartm ent drugs administered with repeated doses of nonparenteral administration". Acta niversitatis Medicinalis, vol.39, no.1, pp.34-36,2004. [3] Nelder A, Mead R. “A simplex method for function optimization”. Computation J.,vol.7,no.12, pp. 308-313,1965. [4] Dan-xia Li, Liu Li, Choi-Hong Lai. "Improved parameter optimization method and its applications in pharmacokinetics". Computer Engineering and Applications, vol.46, no.26, pp.206-208,2010. [5] Aijia, OuYang Libin Liu, Guangxue Yue, et al. "Adaptive hybrid particle swarm optimization for solving parameters of pharmacokinetics". Computer Engineering and Applications (in Chinese),vol.47, no.16, pp.221-225,2011. [6] K. N. Krishnanand, D. Ghose. "Glowworm swarm optimization: a new method for optimizing multi-modalfunctions". Int. J. Computational Intelligence Studies, vol.1,no.1,pp.93-119,2009. [7] K. N. Krishnanand. "Glowworm swarm optimization: a multimodal function optimization paradigm with applications to multiple signal source localization tasks". Indian: Department of Aerospace Engineering, Indian Institute of Science, 2007. [8] K.N. Krishnanand and D. Ghose, "Glowworm swarm based optimization algorithm for multimodal functions with collective robotics applications", Multiagent and Grid Systems. Special Issue on Recent Progress in Distributed Intelligence, vol.2, no.3,pp.209-222, 2007. [9] Qiaoqiao Gong, Yongquan Zhou, Yan Yang. "Artificial glowworm swarm optimization algorithm for solving 0-1 knapsack problem". Advanced Materials Research.vol.144,pp.166-171,2011.

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[10] Zhengxin Huang, Yongquan Zhou. "Using glowworm swarm optimization algorithm for clustering analysis". Journal of Convergence Information Technology, vol.6, no.2,pp.78-85,2011. [11] K. N. Krishnanand, D.Ghose. "A glowworm swarm optimization based multi-robot system for signal source localization". Design and Control of Intelligent Robotic Systems, vol.1, pp.5374,2009. [12] K. N. Krishnanand, and D. Ghose. "Chasing multiple mobile signal sources: a glowworm swarm optimization approach". In Third Indian International Conference on Artificial Intelligence (IICAI 07), Indian ,2007. [13] Yang. X. S. "Firefly algorithms for multimodal optimization, In: Stochastic Algorithms: Foundations and Appplications". Lecture Notes in Computer Science, vol.5792, pp.169-178,2009. [14] Yang X. S., Engineering Optimization: An Introduction with Metaheuristic Applications, Wiley, 2010. [15] Guangwei Zhao , Yongquan Zhou,Yingju Wang. "Using Complex Method Guidance GSO Swarm Algorithm for Solving High Dimensional Function Optimization Problem", Journal of Convergence Information Technology, vol. 6, no. 11, pp. 352 - 360, 2011. [16] Peigang Guo, Yongquan Zhou, Huan Cheng, Jinzhao Wu. "A Hybrid Optimization Algorithm Based on Cultural Algorithm and Glowworm Swarm Optimization", Journal of Convergence Information Technology, vol. 7, no. 10, pp. 239 -247, 2012.

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