Clinically Relevant Optimization Based on Simulated Annealing Algorithm in IMRT for Prostate Cancer

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DOI:_{10.12928/TELKOMNIKA.v14i2A.4356} ₃₇₉

Clinically Relevant Optimization Based on Simulated

Annealing Algorithm in IMRT for Prostate Cancer

Caiping Guo*1, Linhua Zhang2

Electronic Engineering Department, Taiyuan Institute of Technology, Taiyuan, 030008 China

Computer Engineering Department, Taiyuan Institute of Technology, Taiyuan, 030008 China

*Corresponding author, e-mail: [email protected], [email protected]

Abstract

Given the advantage and trend of the clinically relevant optimization in radiotherapy, a novel optimization method using objective function based on tumor control probability (TCP) and normal tissue complication probability (NTCP) was proposed, which can improve global optimization capacity of simulated annealing optimization algorithm (SA) by increasing poor solutions acceptance rate. The effectiveness and superiority of the new method was verified in prostate cancer cases. Experimental results show that the proposed radiotherapy optimization method, not only can improve DVH curve of organs at risk, reduce the NTCP, and improve the therapeutic gain ration, but also can decrease hot spots and gain better dose distribution uniformity of the target.

Keywords: Simulated annealing; Tumor control probability; Nature tissue control probability

1. Introduction

Intensity-modulated radiation therapy (IMRT) is an advanced mode of high-precision radiotherapy that uses computer-controlled linear accelerators to deliver precise radiation doses to a malignant tumor or specific areas within the tumor. IMRT allows for the radiation dose to conform more precisely to the three-dimensional (3-D) shape of the tumor by modulating or controlling the intensity of the radiation beam in multiple small volumes. IMRT also allows higher radiation doses to be focused to regions within the tumor while minimizing the dose to surrounding normal critical structures.

Objective function is an important index for the optimization and evaluation of treatment planning. It is not only a tool to evaluate the treatment plan, but also the connection between the input parameters and the output dose distribution. Now Objective functions used in radiotherapy optimization are based either on physical factors or on biological formulations [1]. The former employs a physical quadratic dose-based objective function, which, via a combination of weighted terms, penalizes differentially violations of the various dose and/or dose-volume constraints specified with respect to the organs and the targets considered in the optimization process [2-4]. However, this type of physics-based approach is unable, to sufficiently take into account the nonlinear response of tumors or normal structures to irradiation, especially with arbitrary inhomogeneous dose distribution [5]. In order to be meaningful the optimization process must be “clinically relevant’’ instead of being simply dose distribution oriented [6]. It is desirable to employ the biological objective function into the inverse planning, as well as balancing the various normal tissue complication probabilities (NTCP) with respect to each other and with respect to the tumor control probability (TCP).

Mainly three kinds of biological criteria can be applied: (generalized) equivalent uniform dose ((g)EUD), tumor control probability (TCP), and normal tissue complication probability (NTCP)[7].

The merits of including the gEUD concept into an objective function have been widely investigated [5, 8-13]. Based on a volume parameter a, (g)EUD corresponds, for a given non-uniform dose distribution, to the non-uniform dose that induces the same biological effect. And the NTCP biological criteria has been used in inverse treatment planning optimization [3, 6, 14-16]and incorporated into some commercial treatment planning software [14, 17]. The advantage of the TCP criteria, also, was investigated [6, 18-19]. Nevertheless the gEUD model cannot directly quantify the NTCP and TCP. In contrast, both NTCP and TCP models are more

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clinically relevant. If all constraints of the inverse planning objective function are satisfied, the NTCP-based and TCP-based treatment plan should be directly implemented, and there is no need to further optimize.

Based on Mohan et al’s research, we did some researches to improve optimized quality. Mohan et al (1992) used a single score, mixing TCP with NTCP criteria to reduce normal tissue complication rates and increase tumor control. Because both NTCP and TCP criterias are sigmoidal functions of dose distributions [20-24], and they are thus inherently nonlinear and non-convex in terms of fluence elements in fluence map optimization (FMO).The fast simulated annealing approach was applied to solve this non-convex optimization problem. In their work, calculated TCP was defined as the TCP sub-score, without paying additional penalty, while piecewise linear penalty was imposed on the computed values of each NTCP, minor importance was paid to small NTCP and only a small penalty to the score is assessed. We proposed a new objective function based on the NTCP criteria and TCP criteria during the high temperature stage, to improve the global optimization ability of simulated annealing by increasing poor solutions acceptance rate, while the same objective function proposed by Mohan et al [6] was used for low temperature stage. The effectiveness of the proposed method was assessed in 10 prostate cancer cases, and compared with optimization method proposed by Mohan et al [6].

2. Methods and Materials

2.1. Simulated Annealing Optimization Algorithm

The simulated annealing method (SA) is derived from thermodynamics and has a unique potential for finding the global minimum. It is well suited for optimization problems involving a large number of variables. The essence of the algorithm is as follows: in each step of an iterative procedure the parameter values (i.e., beam weights) are randomly changed, where the random changes are sampled from a distribution. Then the resulting change ΔF in the objective function is calculated. If the function decrease, the change is always accepted; otherwise it is randomly accepted with a probability given by the Boltzmann distribution

F)/kT] ( exp[

p  , where k is the Boltzmann constant (Mohan et al 1992), and that is the Metropolis criteria. The Metropolis criteria can be defined as follows:

    

   

   



 

rand and p F p

rand and p F

ΔF/kT) ( p

ΔF p

0 0

0 exp

0 1

(1)

Where the parameter T regulates the rate at which the optimization occurs and is analogous to the temperature in the annealing process, and the function of rand randomly generates the number located between [0, 1).

The details of the SA algorithm were described in Mohan et al [6]. 2.2. Dose Calculations

The standard pencil beam model proposed by Ahnesjö [21] was applied to calculate dose distributions. The total dose Di to a voxel i can be calculated as:

) , 2 , 1 (

T N

j j ij

i w x i N

D b









(2)

Where wij, the so-called dose depositon coefficient, represent the dose received by the

voxel i from the beamlet j of unit intensity. The number of beamlets is denoted by Nb and the

intensity of beamlets is xj,j=1,2…,Nb. NT is the total number of voxels in optimized tissues. The

dose distributions could be expressed in a matrix-vector form as Wx

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Where D(x) is the dose distribution vector (dose for each patient voxel), Wis the dose calculation matrix, and x is the fluence elements (beamlet weights).

2.3. Clinical and Biological Models

(1) Tumor Control Probability (TCP): TCP is the probability of the elimination of tumor cells, not only related to the changes in the dose, but related to the tumor size, cell oxygen status and other factors. Some domestic and foreign scholars have made some TCP models based on the same assumption that the tumor is composed of some independent clonal cells, and these clones are independent of the reaction.

In our work, we implemented the logistic function proposed by Schultheiss [18].The TCP in the form of a logistic function is expressed as

k D D D TCP ) / ( 1 1 ) ( 50 

 ₍₄₎

Where D50 is the dose required to achieve a 50% probability of control and k is a

measure of the slope of the dose response curve.

(2) Normal Tissue Complication Probability (NTCP): NTCP is the probability that the organ or tissue damage after radiation therapy, it is also associated with the volume of the normal tissue to receive irradiation. The NTCP model we applied was the LKB [22, 23] model, defined by          ) 1 ( ) 1 ( ) ( ) ( 50 50 mD D D gEUD D NTCP (5)

where 

                 





 2 1 ( ₂)

1 2 exp 2 1 ) ( 2 x erf dt t x x

 is the standard normal cumulative

distribution function; D50 represents the tolerance dose to a whole organ, causing a 50% complication probability; m is the slope of the sigmoidal function Φ; and gEUD(D(x)) is the generalized equivalent uniform dose of the dose distribution D(x), given by

a N i a i N gEUD / 1 1 1 ) ( _      



 D D (6)

N is the number of voxels in the anatomic structure of interest, Di is the dose in the ith

voxel, and a is the tissue-dependence parameter that represents the dose-volume effect. The gEUD model can be applied to both tumor area and normal tissue.

2.4. The Objective Function

We adopted the model proposed by Mohan et al for combing the TCP and NTCPs into a single score represented as:





i i

s

P

F

(7)

Where si is component scores for each normal structure,Ptc is the TCP calculated by

formula (4).

To improve the global optimization ability of simulated annealing by increasing poor solutions acceptance rate, we proposed a new objective function based on the NTCP criteria and TCP criteria during the high temperature stage, while the same objective function named piecewise linear penalty function proposed by Mohan et al [6] was used for low temperature stage. Figure 1 provides a broad overview of the method.

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Figure 1. Broad overview of the method

The operations in dotted-lined rectangle in figure 1 are the differences between our method and the method proposed by Mohan et al’s [6]. Here we first judge the current temperature T is high or low temperature stage, if it is high temperature stage, calculating the function value according to equation (9) described below, or else calculating defined in equation (8) below. In our experiments, we defined T2₃T0 as high temperature, which was determined

by trial and error method.

(1) Piecewise linear penalty function: At low temperature stage, the si in equation (7) are given by

                c ntc c ntc a a c ntc c a a ntc a ntc a P P s P P P P P P P s s P P P P s s 0 ) /( ) ( / ) 1 ( 1 (8)

Where Pa is an acceptable level for each organ, sa is the score corresponding to the

acceptable NTCP, and Pc is a defined critical level above which the treatment plan score becomes zero. In our work, we setP_a 0.05,P_c0.5.

(2) New objective function: At high temperature stage, the si in equation (7) are given by

          c ntc c ntc P P P P k e kx e s 0 )) ) 5 . 0 ( log( / 1 ) ( log( /

1 2 2

(9)

The relative relationship between the piecewise linear penalty and the new objective function described by formulas (8) and (9) was shown in figure 2.

For piecewise linear penalty function ,we can see that as long as the NTCP remains below Pa, minor importance was assigned to it and only a small penalty to the score is

set initial temparature T0 and initial solution x0

Generating new solution x

0 T 3 2

T 

Calculat ing F(x) accordi ng to new objective functi on

Calculat ing F(x) accordi ng to piecewi se linear objective function

) F(x -F(x)

F 0



F



yes no

Accept new solut ion

Accept new solut ion accordi ng to Metropolis

crit eria

Does t he termination condition meet?

end yes

Slow cooling no

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assessed; when PaPntc Pc, more importance was assigned to it and more penalty to the

score is assessed; when PcPntc the plan score becomes zero. But in the new objective

function, the biggerPntc the less importance was assigned to it and fewer penalties to the score

is assessed. According to Metropolis criteria, we can know that for the new state, corresponding to ΔF, the acceptance rate of the state located in (0, Pa) is greater than the state located in (Pa,

Pc) in piecewise linear penalty function. However, the poorer the new state is, the higher the

acceptance rate is in our proposed new objective function, since the slope of the curve becomes smaller along with the increasing Pntc. To improve the global optimization ability of simulated

annealing, at high temperature stage we applied the new objective function, and at low temperature stage piecewise linear penalty function was applied.

Figure 2. NTCP penalty function

3. Results

3.1. Experimental Parameters Setting

The effectiveness of the proposed method was assessed in 10 prostate cancer cases, and was compared with optimization method using piecewise linear penalty function. For the sake of simplicity, we randomly chose the experimental results of one patient to show the differences. All plans used identical configurations of five coplanar 6MV photon beams, with gantry angles of 36, 100, 180, 260 and 324(IEC Convention). For optimization of treatment plans, parameters for the computation of the NTCPs for rectum and bladder were obtained from Emami et al [24] and Kutcher et al [25] shown in Table 1.

To evaluate two different treatment plans, we used dose volume histogram (DVH), dose (maximum dose, minimum dose, mean dose), and biological indices (gEUD, TCP, NTCP) to comparisons.

Table 1. Parameters used for calculating TCP and NTCPs

Target (prostate） k=50 D50=76.8Gy - rectum m=0.15 n=0.12 D50=78Gy bladder m=0.15 n=0.12 D50=79Gy

3.2. Comparisons of DVH

DVH curve was defined as the volume received dose over Dmax is not more than Vmax%,

is a kind of commonly used radiotherapy planning evaluation tool, it can be used to evaluate the uniformity of the target area, but also can be used to reflect the normal tissue and the dose to the organ. Two different sets of DVH curves obtained from two different optimization methods are shown in figure 3. The solid lines and dotted line represent the plans based on new proposed function and piecewise linear penalty function respectively. It can be seen that the new plan is much better than piecewise linear penalty function based plan in terms of PTV coverage and OARs sparing. In particular, the PTV receive more uniform dose and less high dose, i.e. the volume received dose 90Gy decrease from 32% to 14%. The rectum is better

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spared. For example, the fractional volume of the rectum that receives a dose above 50Gy is dropped from 45% to 37%.

3.3. Dose Comparisons

Dose comparisons in terms of Dmax、Dmin and Dmean were done for both PTV and OARs,

as shown in table 2. It is clearly seen that the dose distributions of irradiated tissues were improved to different degrees. For example, the maximum dose to PTV, rectum and bladder dropped 2.7%, 3.99% and 2.84% respectively.

Figure 3. The whole and high dose volume histograms for PTV (a), rectum wall (b), and bladder wall (c). The solid lines represent the DVH curves of new proposed optimization, and the dotted

line the DVH curves of the piecewise linear penalty function based optimization

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90 100

Dose (Gy)

)

bladder

piecewise linear function new proposed method

60 65 70 75 80 85 90 0

2 4 6 8 10 12 14

( c )

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90 100

Dose (Gy)

)

rectum

piecewise linear function new proposed method

60 65 70 75 80 85 90 0

5 10 15 20 25 30 35

( b )

0 20 40 60 80 100 120

0 10 20 30 40 50 60 70 80 90 100

Dose (Gy)

)

PTV piecewise linear function new proposed method

80 85 90 95 100 105 110 0

10 20 30 40 50 60 70 80 90

( a )

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Table 2. Dose for PTV, rectum and bladder

organ New function (Gy) Piecewise linear penalty function (Gy) Dmax Dmin Dmean Dmax Dmin Dmean

PTV 104.87 69.525 82.71 107.78 70.43 86.63 rectum 81.97 - 40.20 84.93 - 42.25 bladder 82.52 - 14.39 85.38 - 14.61

3.4. Comparisons of Biological Indices

The biological indices results of our new method showed the lower gEUD and NTCP values for both the rectum and bladder walls, as shown in table 3. The parameter a=8, 8, -10 were respectively applied in the gEUD calculations of rectum, bladder and PTV. We used the same TCP, NTCP models and parameters as those used by Mohan et al [6] to calculate the TCP and NTCP values. The rate of change was defined as

( ) -( )

( )

new piecewise piecewise

value value rate of change

value

 (10)

It is clearly shown that the biological indices of new plan based on new method optimization are better than that gained based on piecewise linear penalty function optimization. The NTCP values of rectum and bladder dropped 11.71% and 28.75% respectively, the TCP value is similar, and therapeutic gain ratio is increased. For different optimization tissues, gEUD has been improved to different degrees shown in table 3.

Table 3. TCP, NTCP values

functions rectum bladder PTV NTCP gEUD NTCP gEUD TCP gEUD New function 2.94% 61.16 7.51% 83.48 95.65% 57.60 Piecewise linear penalty function 3.33% 63.36 10.54% 83.60 95.60% 57.26 Rate of change -11.7% -3.47% -28.8% -0.14% 0.05% 0.59%

4. Discussion

The ultimate goal of radiotherapy treatment planning is to find a treatment that will yield a high tumor control probability (TCP) with an acceptable normal tissue complication probability (NTCP). Yet most treatment planning today is not based upon optimization of TCPs and NTCPs because of uncertainties associated with the models, but rather upon meeting physical dose and volume constraints defined by the planner. It has been suggested that treatment planning evaluation and optimization would be more effective if they were biologically and not dose/volume based [26].

In our work, the essence of the new inverse planning approach presented here is the combination of clinically relevant optimization and simulated annealing optimization algorithm in a large-scale unconstrained optimization model. To improve the global optimization ability of simulated annealing by increasing poor solutions acceptance rate, we adopted different optimization model for different optimization stage as described above. To verify the effectiveness of the proposed method, experiments were conducted involving 10 prostate cancer cases, and compared with the work done by Mohan et al. The results shown in figure 3, table 2 and table 3 indicated that the new plan generated by our proposed optimization method was better than the plan based on Mohan et al’s optimization method in terms of PTV coverage and OARs sparing. The improvements attributed to the increase of poor solutions acceptance rate at high temperature stage of SA by the slope change of the optimization function depicted in figure 2.

This work is meaningful for the study of biological optimization in radiotherapy research. There are still some challenges for improvement. First, the boundary between the high temperature stage and the low temperature stage is not easily defined. In our experiments, we determined it by trial and error method, that is, we defined the temperature, which was higher

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than two-thirds of initial temperature T0 as high temperature. The boundary directly affects the

optimization results. Second, as can be seen from the DVHs in figure 3, the plans based in clinically relevant optimization provides less control of the high-dose distribution in the PTV compared with dose or dose-volume based plans [9], since a biologically based cost function would not be sensitive hot spots inside the target as these hot spots could increase tumor-cell killing[10]. More studies are needed for improve the quality of treatment plan based on biological relevant optimization.

5. Conclusions

In the work, we proposed biological relevant indices based a new optimization method, consisting of controlling the dose delivered to normal tissues and target by NTCP-based sub-scores and TCP-based sub-score respectively. The large-scale optimization problem was then solved by means of SA algorithm, and the strategy of increasing the poor solutions acceptance rate was added to improve the global optimization ability of SA. The proposed method was applied in 10 prostate cancers. Our method was proven that our method can generate better radiotherapy plans than plans attained by applying the commonly used SA algorithm.

Acknowledgements

This work was supported by department of electronic engineering, Taiyuan Institute of Technology, Education Department of Shanxi province.

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[8] B Choi, JO Deasy. The generalized equivalent uniform dose function as a basis for intensity-modulated treatment planning Phys. Med. Biol., 2002; 3579-3589.

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Figure 1. Broad overview of the method

by trial and error method.

(1) Piecewise linear penalty function: At low temperature stage, the si in equation (7) are given by

                c ntc c ntc a a c ntc c a a ntc a ntc a P P s P P P P P P P s s P P P P s s 0 ) /( ) ( / ) 1 ( 1 (8)

Where Pa is an acceptable level for each organ, sa is the score corresponding to the

acceptable NTCP, and Pc is a defined critical level above which the treatment plan score becomes zero. In our work, we setP_a 0.05,P_c0.5.

(2) New objective function: At high temperature stage, the si in equation (7) are given by

          c ntc c ntc P P P P k e kx e s 0 )) ) 5 . 0 ( log( / 1 ) ( log( /

1 2 2

(9)

The relative relationship between the piecewise linear penalty and the new objective function described by formulas (8) and (9) was shown in figure 2.

For piecewise linear penalty function ,we can see that as long as the NTCP remains below Pa, minor importance was assigned to it and only a small penalty to the score is

set initial temparature T0 and initial solution x0

Generating new solution x

0 T 3 2 T 

Calculat ing F(x) accordi ng to new objective functi on

Calculat ing F(x) accordi ng to piecewi se linear objective function

) F(x -F(x)

F 0



0 F



yes no

Accept new solut ion

Accept new solut ion accordi ng to Metropolis

crit eria

Does t he termination condition meet?

end yes

Slow cooling no

(2)

assessed; when PaPntc Pc, more importance was assigned to it and more penalty to the

score is assessed; when PcPntc the plan score becomes zero. But in the new objective

function, the biggerPntc the less importance was assigned to it and fewer penalties to the score

is assessed. According to Metropolis criteria, we can know that for the new state, corresponding to ΔF, the acceptance rate of the state located in (0, Pa) is greater than the state located in (Pa,

Pc) in piecewise linear penalty function. However, the poorer the new state is, the higher the

acceptance rate is in our proposed new objective function, since the slope of the curve becomes smaller along with the increasing Pntc. To improve the global optimization ability of simulated

annealing, at high temperature stage we applied the new objective function, and at low temperature stage piecewise linear penalty function was applied.

Figure 2. NTCP penalty function

3. Results

3.1. Experimental Parameters Setting

To evaluate two different treatment plans, we used dose volume histogram (DVH), dose (maximum dose, minimum dose, mean dose), and biological indices (gEUD, TCP, NTCP) to comparisons.

Table 1. Parameters used for calculating TCP and NTCPs

Target (prostate） k=50 D50=76.8Gy -

rectum m=0.15 n=0.12 D50=78Gy

bladder m=0.15 n=0.12 D50=79Gy

3.2. Comparisons of DVH

DVH curve was defined as the volume received dose over Dmax is not more than Vmax%,

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spared. For example, the fractional volume of the rectum that receives a dose above 50Gy is dropped from 45% to 37%.

3.3. Dose Comparisons

Dose comparisons in terms of Dmax、Dmin and Dmean were done for both PTV and OARs,

Figure 3. The whole and high dose volume histograms for PTV (a), rectum wall (b), and bladder wall (c). The solid lines represent the DVH curves of new proposed optimization, and the dotted

line the DVH curves of the piecewise linear penalty function based optimization

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90 100 Dose (Gy) Vo lu m e (% ) bladder

piecewise linear function new proposed method

60 65 70 75 80 85 90 0 2 4 6 8 10 12 14 ( c )

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90 100 Dose (Gy) Vo lu m e (% ) rectum

piecewise linear function new proposed method

60 65 70 75 80 85 90 0 5 10 15 20 25 30 35 ( b )

0 20 40 60 80 100 120

0 10 20 30 40 50 60 70 80 90 100 Dose (Gy) Vo lu m e (% ) PTV piecewise linear function new proposed method

80 85 90 95 100 105 110 0 10 20 30 40 50 60 70 80 90 ( a )

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Table 2. Dose for PTV, rectum and bladder

organ New function (Gy) Piecewise linear penalty function (Gy)

Dmax Dmin Dmean Dmax Dmin Dmean

PTV 104.87 69.525 82.71 107.78 70.43 86.63

rectum 81.97 - 40.20 84.93 - 42.25

bladder 82.52 - 14.39 85.38 - 14.61

3.4. Comparisons of Biological Indices

( ) -( )

( )

new piecewise

piecewise

value value rate of change

value

 (10)

Table 3. TCP, NTCP values

functions rectum bladder PTV

NTCP gEUD NTCP gEUD TCP gEUD

New function 2.94% 61.16 7.51% 83.48 95.65% 57.60

Piecewise linear penalty function 3.33% 63.36 10.54% 83.60 95.60% 57.26

Rate of change -11.7% -3.47% -28.8% -0.14% 0.05% 0.59%

4. Discussion

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than two-thirds of initial temperature T0 as high temperature. The boundary directly affects the

5. Conclusions

Acknowledgements

This work was supported by department of electronic engineering, Taiyuan Institute of Technology, Education Department of Shanxi province.

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