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This paper uses a molecular dynamics (MD) method for intensity-modulated proton therapy treatment planning optimization to overcome the problem of gradient-based optimization methods such as quasi-Newton, which is sensitive to starting conditions and is easily trapped in local minima. We implemented a molecular dynamics (MD) method and a quasi-Newton method for plan optimization. Three types of cancer cases, prostate, head-and-neck and lung cancer, were tested with three starting different initial conditions. Overall, the MD method consistently resulted in solutions with lower objective function values (OFVs) compared to those from the quasi-Newton algorithm. Furthermore, the MD method converged on the same OFV regardless of its initial starting points used for the prostate cancer case.

Radiation therapy uses ionizing radiation to treat malignant tumors by damaging the DNA of cancer cells to block their ability to proliferate. The purpose is to deliver a prescribed dosage of radiation to the targeted tumor while minimizing the dose deposition in healthy tissues [

feat not typically feasible for photon beams commonly used in intensity modulated radiation therapy (IMRT). The most advanced proton therapy delivery technique is the intensity-modulated proton therapy (IMPT) which uses scanning beamlets (pencil beams) whose weights (intensities) can be modulated independently for treatment. Using data from imaging technology such as CT (computed tomography) or MRI (magnetic resonance imaging), the beamlets weight profile can be optimized according to the prescribed treatment dose. The optimization problem to determine the optimal beamlets weight is called a fluence map optimization (FMO). As a result, IMPT can deliver a highly conformal radiation dose to the desired target tumor while sparing the surrounding organs at risk (OAR), especially for structures with complex shapes [

The FMO problem for intensity-modulated radiation therapy (IMRT) planning has been extensively studied and has been addressed by various solution strategies. These strategies can be grossly classified into two groups: global optimization (GO) and local optimization (LO). GO approaches include linear programming [

Despite the fact that many solution approaches have been proposed to solve the FMO problem, a method of avoiding local minima while converging to a solution within a practical time limit has rarely been reported. Hou et al. [

In this study, a MD method was implemented for the FMO problem in IMPT and tested on three patient cases with different starting points. The primary objective of this study is to demonstrate the MD method can be a viable alternative for solving the IMPT FMO problem and evaluate whether this method can alleviate the impact of the starting points, a major issue of gradient-based methods.

The remainder of the paper is organized as followed: Section 2 describes the optimization model, the solution algorithms are in Section 3 which includes an introduction to the MD method and its formulation of the IMPT FMO problem and an existing quasi-Newton method formulation is shown for the purpose of algorithmic performance comparison. The data used in the experiment and the initial configuration setups are listed in Section 4. Last but not least, Section 5 provides the results regarding the convergence properties of the MD method and quasi-Newton algorithm.

The main purpose of IMPT is to deliver the prescribed conformal radiation dose to the targeted tumor while sparing normal tissues. To achieve this goal, we define a quadratic objective function to quantify the difference between the prescribed dose ( D n * , where n is the index of each organ of interest) and the actual dose ( D i , where i is the voxel index) delivered to the patient. Although different types of objective functions are reported in the literature [

A dose-based objective function F is composed of two parts: F n t u m o r for the n^{th} target, n = 1 , 2 , ⋯ , and F m o a r for the m^{th} OAR, m = 1 , 2 , ⋯ . Because the primary goal of treatment planning is to obtain an actual radiation dose profile that is identical or nearly identical to the prescribed dose level on the target, F n t u m o r can be defined as the deviation of the resulting actual dose D i on voxel i from the target prescription dose D n T :

F n t u m o r = 1 N n T ∑ i = 1 N n T ( D i − D n T ) 2 , (1)

where N n T is the total numbers of voxels in the n^{th} target. Similarly, F m o a r can be defined as

F m o a r = 1 N m o a r ∑ i = 1 N m o a r ( D i − D m o a r ) + 2 , (2)

where N m o a r is the total number of voxels in the m^{th} OAR, D m o a r is the specified tolerance dose for the organ, and ( δ ) + is defined as ( δ ) + = max ( δ , 0 ) . Note, we introduced the step function F m o a r because healthy organs are often allowed to receive radiation doses up to a certain amount. However, once the amount is over a tolerance value, a penalty will be imposed on the voxel according to the degree of deviation from the tolerance.

The dose D i in voxel i can be calculated as:

D i = ∑ j N k i j ω j , (3)

where ω j is the weight or intensity of beamlet j, which is the decision variable of our IMPT optimization model. Notation N denotes the total number of beamlets and k i j is the unit dose contribution of the j^{th} beamlet to the i^{th} voxel; k i j is also known as the dose deposition coefficient. Here the values of k i j are calculated using an in-house dose calculation engine for proton beamlets [

Using the notation described above, the dose-based objective function for our optimization model is:

F = ∑ n p n t u m o r F n t u m o r + ∑ m p m o a r F m o a r , (4)

where p n t u m o r and p m o a r denote the penalty weights of the n^{th }tumor site and m^{th} OAR, respectively. These weights are often obtained by trial and error by planners (dosimetrists, physicists, etc.), to find a balance between tumor dose coverage and OAR dose sparing to satisfy the clinical criteria. In this study, the model follows the common practice in the medical community of containing a physical constraint: the beamlet weight cannot be negative, i.e., ω j ≥ 0 , j = 1 , 2 , ⋯ , N . Hence, our optimization model for the IMPT FMO problem is:

min F s .t . w j ≥ 0 , j = 1 , 2 , ⋯ , N (5)

In the following sections, we briefly describe the general concept of MD followed by the use of MD for radiation therapy. For the purpose of algorithm performance comparison, we describe a quasi-Newton method for FMO, introduced by Lomax [

MD is a computational technique for many-body system simulation that has been widely applied in the material sciences community. In a classical MD model, the physical movements of particles in the system follow Newton’s Laws of Motion [

f j = m j ⋅ a j = m j d v j d t = m j d 2 x j d t 2 f j = − ∇ j E (6)

where t is the time of the system and E is the potential energy of the system. The force f j is related to the acceleration and can be expressed as the gradient of the potential energy of the particle.

Based on Newton’s Laws of Motion, the position, velocity and acceleration of the particle can be described as functions of time t;

v j = d x ( t ) j d t , a j = d v ( t ) j d t = d 2 x ( t ) j d t 2 (7)

Therefore, the continuous motion configuration of the system can be calculated by integrating Newton’s Laws of Motion. When the system is under the influence of continuous potential energy, the positions and velocities can be approximated using a Taylor series expansion for a small time step Δ t , Δ t > 0 :

x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 1 2 a ( t ) Δ t 2 + 1 6 b ( t ) Δ t 3 + ⋯ (8)

v ( t + Δ t ) = v ( t ) + a ( t ) Δ t + 1 2 b ( t ) Δ t 2 + 1 6 c ( t ) Δ t 3 + ⋯ (9)

where a, b and c are the second, third and fourth time derivatives of the coordinates.

This Taylor expansion serves as the basis for the most common integrators used in MD calculations. Many classical methods for integrating equations require information from both current and previous steps to update the system. This means the information from these steps must be stored in memory and the system cannot self-start at the beginning [

x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 1 2 a ( t ) Δ t 2 , v ( t + Δ t ) = v ( t ) + 1 2 Δ t [ a ( t ) + a ( t + Δ t ) ] . (10)

Therefore, this approach is selected in our algorithm to update the MD system to solve the IMPT FMO problem.

In IMPT, the optimization problem can be formulated as a dynamic system with N virtual atoms [

v j = d ω j d t , a j = d v j d t = d 2 ω j d t 2 , f j = m j d v j d t = ∂ F d ω j . (11)

In this study, we followed the approach of Hou et al. [

ω j ( t + Δ t ) = ω j ( t ) + v j ( t ) Δ t + 1 2 m j Δ t 2 f j ( t ) v j ( t + Δ t ) = v j ( t ) + 1 2 m j Δ t [ f j ( t ) + f j ( t + Δ t ) ] (12)

Hence, we calculate the updating beamlet weights by Equation (12) and we can combine Equation (12) with Equation (4) to calculate the trajectory of the OFVs.

In physics, temperature is used to specify the thermodynamic state of a system. In the MD system, temperature (T) is related to the kinetic energy of the system and can be calculated as:

T = 1 3 N k ∑ j = 1 N m j v j 2 (13)

where k is the Boltzman constant and N is the total number of particles in the system.

The MD system will converge to an equilibrium state with the lowest free energy. Note that free energy consists of kinetic energy and potential energy. Therefore, the potential energy equals the free energy only when kinetic energy is zero, i.e., the temperature of such system is zero. Thus, the objective function (potential energy) of our FMO model is minimized when the system reaches an equilibrium state with zero system temperature. However, in physics, the kinetic energy and potential energy of a dynamic system follow the law of energy conservation. Although kinetic and potential energy will interchange continuously, the total energy will remain unchanged. This can create an issue of convergence to a specific point because the atoms can still carry significant speed when they reach that point. As a result, atoms may move away from an optimal point with the lowest potential energy. In order to solve this problem, a “friction” to the system (i.e., a damping factor to the MD system) is added to slow the movements of atoms as they get closer to an optimal point. The following damping function is applied in our algorithm:

v j ( t ) = { λ v j ( t ) , if v j ( t ) f j ( t ) < 0 v j ( t ) , otherwise . where 0 < λ < 1 (14)

As we mentioned above, their speed may cause the atoms to pass the optimal point and create an issue of convergence. On the other hand, when the virtual atoms become trapped in local minima, a proper speed may help them continue to move and get out of those local minimum points. Using this feature, we employ temperature scaling to adjust the velocities to help the atoms move out from the local minimum points. From the updating function, Equation (12), we define the scaling function as:

v j ( t + Δ t ) = T d T 0 v j ( t ) + 1 2 m j Δ t [ f j ( t ) + f j ( t + Δ t ) ] , (15)

where T 0 is the initial temperature and T d is the desired temperature.

An important physical constraint of IMPT planning is that the beamlet weight ω j cannot be negative. Hou and Wang [

The proposed MD method to optimize the IMPT FMO problem is outlined below. The algorithm stops when either of these following conditions are met: 1) a certain number of iterations is reached or 2) there is no change (smaller than ε , i.e. 10^{−4}) in the objective value for a certain number of consecutive iterations.

In our implementation, if a new solution increases the OFV to a value larger than , it replaces the old one. Otherwise, if the new solution increases the OFV but the value remains smaller than ε , it replaces the old solution and the system is heated by rescaling the temperature by T d = β T 0 , where β is the heating rate and typically lies between 1.1 and 2. If there is no OFV improvement, we keep the old solution and also heat the system. When the stopping criteria are satisfied, we stop the algorithm and return the final solution.

The quasi-Newton algorithm is often used for solving the FMO problem in radiation treatment planning; it has been implemented in Eclipse, a leading radiation treatment planning system [

For a standard Newton method, the updating function of a beamlet weight at each iteration is defined as:

ω ( k + 1 ) = ω ( k ) − α k ( ∇ 2 F ( ω ( k ) ) ) − 1 ∇ F ( ω ( k ) ) . (16)

Because calculating the Hessian matrix ∇ 2 F ( ω ( k ) ) and its inverse can be computationally expensive due to the large size of the beamlet vector, the matrix is often approximated by its diagonal matrix [

a i j ( k ) = ω j ( k ) k i j D i ( k ) . (17)

Therefore, the iterative updating function is:

ω j ( k + 1 ) = ω j ( k ) − α k ( d F d ω j d 2 F d ω j 2 ) = ω j ( k ) ∑ n p n t u m o r N t n ∑ i N t n k i j 2 D 0 T D i + ∑ m p m o a r N o a r m ∑ i N o a r m k i j 2 D 0 o a r D i ∑ n p n t u m o r N t n ∑ i N t n k i j 2 + ∑ m p m o a r N o a r m ∑ i N o a r m k i j 2 (18)

Similar to the MD method, the quasi-Newton algorithm stops when either one of the following two conditions is met: 1) a preset number of iterations is reached or 2) there is no improvement in the OFV after a certain number of consecutive iterations.

All algorithms were implemented in C++ and all experiments were performed on a 64-bit Linux workstation with 128 GB of memory and quad Intel Xeon E5649 2.53 GHz processor.

Both MD method and quasi-Newton method were used to solve the FMO problems for plan optimization. Different types of cancer patient cases were tested to represent varying complexity and tumor size for the study. There were three different point generation methods used to create different initial conditions for each case. The final objective function value and the objective function values during the optimization process were recorded for result analysis.

The three clinical cancer cases from The University of Texas MD Anderson Cancer Center selected for this study was a prostate cancer, head-and-neck cancer and a lung cancer case (see

The beam angles, number of beamlets in each beam, volumes of interest and number of voxels of each volume for each case are listed in

The planned doses and penalty weights for the corresponding VOIs in a dose-based objective function for the three IMPT cases are listed in

The parameter values of each algorithm were assigned the same values for all

Cancer type | Beam angle | Number of beamlets | VOI | Number of voxels |
---|---|---|---|---|

Prostate | 90˚ | 599 | STV | 4916 |

270˚ | 605 | Bladder | 15,189 | |

Femoral heads | 23,908 | |||

Rectum | 8570 | |||

Head-and-neck | 75˚ | 374 | CTV | 2603 |

240˚ | 356 | Brain | 96,536 | |

300˚ | 365 | Brainstem | 2506 | |

Optic chiasm | 110 | |||

Lung | 205˚ | 1539 | PTV | 11,161 |

275˚ | 1218 | Esophagus | 1435 | |

345˚ | 1042 | Spinal cord | 2030 | |

Total lung | 159,188 | |||

Heart | 18,148 |

Abbreviations: VOI: volume of interest, STV: scanning target volume, CTV: clinical target volume and PTV: planning target volume.

Cancer type | VOI | Dose (Gy) | Weight |
---|---|---|---|

Prostate | STV | 78 | 200 |

Bladder | 0 | 1 | |

Femoral heads | 0 | 1 | |

Rectum | 0 | 1 | |

Head-and-neck | CTV | 74 | 200 |

Brain | 0 | 1 | |

Brainstem | 0 | 1 | |

Optic chiasm | 0 | 1 | |

Lung | PTV | 74 | 200 |

Esophagus | 0 | 1 | |

Spinal cord | 0 | 1 | |

Total lung | 0 | 1 | |

Heart | 0 | 1 |

cancer cases. The stopping criteria were either: 1) the number of iterations reached 10,000 or 2) there was no change in the OFV for 20 consecutive iterations, where ε = 10 e − 4 .

In this study, the IMPT plans of each of three cases were obtained from three different initial conditions (

1) Forward wedge (FW). All beamlet weights are set the same creating a wedge-shaped dose that has a high dose at the proximal edge and a low dose at the distal edge (

2) Inverse wedge (IW). The beamlet weights are set to distal tracking creating an inverse wedge-shaped dose that has a very low dose to the proximal edge and a high dose to the distal edge (

3) Spread-out Bragg peak (SOBP). The beamlet weights are arranged to deliver a flat dose on the targeted area (

Choosing the prostate cancer case as a representative, the plots of OFV as a function of the number of iterations using the two algorithms are shown in

along the optimization process of MD. This indicates the OFVs were not monotonically decreased over the iteration which was not the case for the quasi-Newton method where the OFV decreased smoothly and monotonically over time. Similar results were also observed in the other two cases.

Observation 1: In all cases, the MD method outperformed the quasi-Newton method in terms of OFV.

Observation 2: For each problem instance, the MD method consistently converged on the same objective value regardless of the starting conditions while

OFVs obtained by the quasi-Newton method varied substantially when different starting conditions were utilized.

Observation 3: The IW condition for the quasi-Newton method yielded the worst OFVs.

Overall, the MD method required longer computation times before it reached one of the stopping criteria. For the prostate and head-and-neck cases, the quasi-Newton method had a greater time variation among the initial conditions while the computation times for the MD method differed slightly depending on the initial conditions. For the lung case, the quasi-Newton method reached convergence points about 25% faster than the MD method in three initial conditions.

In

The dose-volume histograms (DVHs) of the scanning target volume (STV) and femoral heads for the prostate case, the DVHs of the clinical target volume (CTV), brain, brainstem, and optic chiasm for the head-and-neck case and the DVHs of the planning target volume (PTV), spinal cord and esophagus for the lung case are shown in

Cancer type | OFV | MD Method No. Iterations | Quasi-Newton Method No. Iterations | ||||
---|---|---|---|---|---|---|---|

FW | IW | SOBP | FW | IW | SOBP | ||

Prostate | 5000 | 16 | 16 | 15 | 10 | 130 | 8 |

3000 | 62 | 64 | 61 | 249 | 725 | 247 | |

2500 | 213 | 231 | 204 | 1880 | NA | 1860 | |

Final | 3750 | 3334 | 3210 | 3921 | 5210 | 1952 | |

Head-and-neck | 3000 | 14 | 185 | 15 | 8 | 182 | 6 |

2000 | 24 | 199 | 24 | 36 | 319 | 25 | |

1000 | 55 | 326 | 76 | 243 | 1541 | 157 | |

Final | 3432 | 2879 | 3378 | 5534 | 2659 | 3076 | |

Lung | 5000 | 20 | 15 | 9 | 32 | 27 | 8 |

3000 | 51 | 44 | 23 | 153 | 142 | 16 | |

2500 | 113 | 104 | 52 | 401 | 385 | 47 | |

Final | 3690 | 3304 | 3582 | 3654 | 3256 | 2879 |

Abbreviations: NA means the stopping criteria were met before the OFV was reached.

method starting from different initial conditions were almost identical. In contrast, the results using the quasi-Newton method varied from case to case. The higher OFVs may reflect worse target coverage and OARs sparing in DVHs.

The dose distributions in the prostate cancer case using the MD and quasi-Newton methods are shown in

Our MD method was developed to overcome the local entrapment issue observed in many gradient-based algorithms that are extensively used in radiotherapy planning systems in clinics. A previously reported quasi-Newton method for fluence map optimization was implemented as a benchmark for our proposed MD method. The performance of the MD method was compared to the quasi-Newton method using three representative clinical cancer cases. For each method, three popular initial conditions were used for each case. We found that the MD method consistently produced solutions that were the same or within a negligible margin of error regardless of the initial conditions used. The quasi-Newton method, however, was sensitive to its initial conditions by converging on noticeably different solutions. Compared to the quasi-Newton method, the MD method requires longer total computation time to reach the stopping points

especially in large tumor sized case. However, to reach the same OFVs, fewer number of iterations were required for the MD method. This indicates that the MD method can not only solve the FMO problem effectively, but also has the potential to obtain a clinically feasible solution in a short time. Future work can include the implementation of the MD method using parallel computation in a multi-core desktop environment.

The authors declare no conflicts of interest regarding the publication of this paper.

Liao, L., Lim, G.J. and Zhang, X.D. (2019) A Molecular Dynamics Approach for Optimizing Beam Intensities in IMPT Treatment Planning. Journal of Applied Mathematics and Physics, 7, 2130-2147. https://doi.org/10.4236/jamp.2019.79146