External beam radiotherapy with high energy photons or electrons plays ... the use of intensity modulated beams to realize a .... eye view (BEV) of such volume.
Ann. Ist. Super. Sanità, vol. 37, n. 2 (2001), pp. 225-230
Optimization of intensity modulated radiation therapy: assessing the complexity of the problem
Simona MARZI, Maurizio MATTIA, Paolo DEL GIUDICE, Barbara CACCIA and Marcello BENASSI Laboratorio di Fisica, Istituto Superiore di Sanità, Roma
Summary. - Intensity modulated radiation therapy (IMRT) is one of the most innovative techniques in oncological radiotherapy, allowing to conform the dose delivery to the tumoral target, preserving the normal tissue. The high number of parameters involved in the IMRT treatment planning requires an automated approach to the beam modulation. Such optimization process consists in the search of the global minimum of a cost function representing a quality index for the treatment. The complexity of this task, has been analyzed with a statistical approach for three clinical cases of particular interest in IMRT. Our main result is that a cost function based on dose-volume constraints entails lower complexity of the optimization process, in terms of the choice of the parameters defining the cost function and in a smaller sensitivity to the initial conditions for the optimization algorithm. Key words: IMRT, radiotherapy, optimization, inverse problem. Riassunto (Ottimizzazione della radioterapia con fasci ad intensità modulata: valutazione della complessità del problema). - La radioterapia con fasci ad intensità modulata IMRT (intensity modulated radiation therapy) è una delle tecniche più innovative nel settore della radioterapia oncologica, permettendo di conformare il rilascio di dose al bersaglio tumorale, risparmiando i tessuti sani. L’elevato numero di parametri, caratteristico della tecnica IMRT, richiede un sistema di ottimizzazione automatico che possa determinare la modulazione del fascio. Il processo di ottimizzazione si traduce quindi nella ricerca del minimo globale della funzione costo che rappresenta un indice di qualità del trattamento. Sono stati analizzati tre casi clinici utilizzando un approccio di tipo statistico. E’ stata utilizzata una funzione costo di tipo fisico basata su vincoli di sola dose e di tipo dosevolume. I risultati ottenuti in questo lavoro mostrano sia in termini di scelta dei parametri che la definiscono, sia in una minore sensibilità rispetto alle condizioni iniziali per il processo di ottimizzazione che una funzione costo basata su vincoli dose-volume offre una minore complessità del processo di ottimizzazione. Parole chiave: IMRT, radioterapia, ottimizzazione, problema inverso.
Introduction External beam radiotherapy with high energy photons or electrons plays an important role in the treatment of tumor pathologies. Every year in Italy about 140 000 new oncological patients are treated with ionizing radiation [1]. The accuracy and the precision of the execution of the treatment has a crucial role for the success of the therapy and for a good quality of life of the patient. The goal of the radiotherapy is to deliver a high and homogeneous dose to the diseased tissue, minimizing the dose to the normal tissue. The possibility of conforming the dose to the tumor shape allows to escalate the dose to the target volume, increasing the tumor control probability (TCP) and reducing the risks of recidives [2]. In recent years there has been a growing interest in the use of intensity modulated beams to realize a conformal radiotherapy (IMRT - intensity modulated radiation therapy). IMRT allows a treatment delivery in which photon fluence is varied over time and/or space
as an aid to dose conformity and normal tissue avoidance [3]. The dose distribution can be thought of as produced by dividing each beam into “beamlets” each one with a different intensity. For the prostate tumor recent studies [4] indicate that a dose escalation of about 20% reduces the risk of metastatic diseases. Traditional radiation therapy has many limitations in delivering high dose to the prostate, because of the relatively low radiation tolerance of rectum and bladder, frequently overlapped with the safety margins of target volume. Radiotherapy of esophagus and antrum maxillary cancers is often complex due to the target doses of 70 Gy or higher and the proximity of organs at risk (OAR), such as lungs and spinal cord for the esophagus case and optic pathway structures for the maxillary antrum case, with tolerance doses of 45-55 Gy or less [5, 6]. So the adoption of IMRT allows to deliver a uniform target dose with steep dose gradients at tumor-normal tissue interfaces and reduce doses over OAR. IMRT planning is characterized by a huge number of parameters, the modulations of the beamlet intensity (in
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a typical situation the number of beamlets can be about 1000), so that the optimal beam configuration, suited for producing the desired dose distribution on the target and OAR, cannot be achieved through the “trial and error” method typically applied in conventional radiotherapy [2]. To solve such an optimization problem in an effective way it is necessary to use computer aided procedures, as discussed in some detail in the next section. Objective functions and complexity of optimization The optimization procedures can be classified into two broad classes: analytic and heuristic. Analytic techniques try to solve a non-linear integral equation in order to obtain the photon fluence modulation necessary to deliver to the patient the desired dose. On the other hand the heuristic approach is based on iterative adjustment of the intensity pattern of the beams and assessing the resulting dose distribution until an acceptable configuration is obtained. To quantify how close a given intensity modulation is to the optimal choice, a cost function can be defined as a quality index of the treatment plan. Such a cost (objective) function associates an “error” to a given choice of the intensity modulation. The error represents the “distance” of the resulting dose distribution from the ideal one. The optimization problem is solved if one can find the intensity modulation associated to the global minimum, that corresponds to the absolute minimal error between the resulting dose distribution and desired one. Heuristic procedures themselves can be subdivided in those adopting “global” and “local” search strategies; however, the vast majority of the optimization studies related to radiation therapy adopt local strategies except for some works [6, 7]. In the literature two different classes of cost function are described: a “physical” one, based on dose calculation, and a “biological” function, based on the evaluation of the tumor control probability, TCP, and on the normal tissue complication probability, NTCP. The biological cost function represents a more clinically relevant approach, because it allows to forecast the radiation effects on tissues and to estimate the real efficacy of radiation therapy. The lack of accurate models and response parameters does not allow an effective application of the biological approach in clinical practice and the use of a physical cost function is today more realistic. A physical cost function requires to establish some dose constraints to control the dose delivery inside the regions of interest (ROI), maximizing the dose level and homogeneity in the target while minimizing the damage to the normal tissues and organs at risk. These constraints are explicitly incorporated in the model and can be expressed as maximum and minimum dose levels
inside a single “voxel” of a ROI or through dose-volume (DV) limits, specified as maximum volumes that cannot receive doses greater than specified tolerance values. The specific form of the cost function, the multiplicity of “valleys” in its “landscape” and the height of the hills separating them, essentially determine how hard the optimization problem is. In this respect, it must be stressed that for IMRT the number of degrees of freedom is huge; in other terms, the cost function is embedded in a very high dimensional space. If many valleys (local minima) are present, sophisticated, possibly stochastic, optimization algorithms are called for since, whatever the initial condition of the iterative optimization process, a pure downhill motion of the representative point of the system (the set of beamlets intensities) in the cost function landscape would trap it in the first reached (possibly local) minimum. In fact, there has been a number of studies in which the “simulated annealing” [8], or some variants of the many stochastic optimization algorithms available [9, 10] have been exploited in the context of treatment plans optimization. On the other hand, gaining systematic indications as to the cases in which to expect multiple minima has proved to be hard; the problem is relevant in view of the computational burden entailed in the stochastic optimization therapy, which casts serious doubts on their applicability as a routinely used tool. Also, even when one has reasons to expect that multiple minima exist in the cost function, it remains to be assessed which is a clinically relevant measure of the price paid for having possibly missed the absolute minimum and having been trapped in a local one. Of course, using a computationally heavy algorithm would not be worth the effort if the improvement of the resulting plan were not appreciable with respect to the many sources of errors and uncertainties. The simplest dose-based cost function consists of a quadratic difference between the ideal dose distribution and the actual one inside each region of interest [3, 8]. Actually, dose-volume objective function is the most significant way to approach the optimization problem in IMRT, because provides more flexibility for the optimization process and greater control over the dose distribution [11, 12]. This type of function is based on the results of the dose-volume histogram (DVH) analysis of the treatment plan. A DV histogram represents the fraction of the volume of a critical structure receiving a fixed dose value. The analysis of the DVH provides the most common tool to evaluate the quality of a treatment plan, showing the dose distribution over the volumes involved. Few indications are available in literature on the general features of the dose and dose-volume (DV) based cost function in optimization properties related to radiotherapy. There is a wide consensus on the fact that, optimizing with respect to the fields orientations (with
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or without modulating their intensity) brings about local minima in the cost function. On the other hand it has been argued in a simplified setting that no local minima should appear when minimizing the quadratic cost function for fixed angles [13]. In the present work we have undertaken to devise a systematic strategy to expose the minima structure of the cost function in a compact way, in a realistic scenario in which the cost function embodies multiple constraints for different ROI; the investigation was performed for both the quadratic, dose-based cost function, and for the one based on DVH. Since a global characterization of the cost function landscape is not possible in realistic cases, to expose the structure of the minima we try to formulate a statistical statement on the end points of the trajectories followed by the representative point. Materials and methods The process of optimization was performed with respect to the intensity modulation of beams, incident from stationary gantry angles. The size of each field was planned to enclose the target according to the beam’s eye view (BEV) of such volume. For this work an homemade algorithm for the dose calculation, CARO [14, 15], was used. All the software routines for the optimization and data analysis have been implemented using the Matlab programming environment (http://www. mathworks.com). To probe the minima distribution of the cost functions, we sampled many, purely downhill trajectories on the cost function’s hypersurface, starting from randomly chosen initial conditions. To this end, a gradient descent method was performed, starting from each initial condition, and the end points were recorded. Therefore, to infer the multiplicity of minima from the stored end points of the gradient descent, we introduced, as a statistical observable, the distribution of the values of the cost function vs the distances from a reference configuration. This scatter plot (Fig. 1) is obtained by computing at the end points the Euclidean distance between the modulation vector m (mi gives the intensity of the beamlet i) found starting from an arbitrary initial condition, and that one corresponding to the minimum of the cloud of the statistical sample. By observing how many separated clouds of points appear in such a plot, it can be deduced if only one minimum or more that one “optimal” solution exists. We also checked that the distribution of points in the scatter plot is consistent with the hypothesis of two separate valleys, by inspecting the curvature of the cost function around the lower minima of each cloud. Minima appear as clouds because of the finite resolution with which the optimization algorithm probes the cost function surface.
This plot can also provide some information about the curvature of the cost function around the minima, considering that a “vertical” cloud of points suggests an essentially flat valley, while a cloud along an oblique line would mean a steeper valley. Results and discussion After the analysis of the minima structure of the cost function, we evaluated the treatment plan corresponding to the global minimum and to the possible local minima, visualizing the dose distribution inside each ROI: the target and the organs at risk. From the isodose plot and from the DVH, we estimated the homogeneity and the dose level inside the target, and the involvement of the critical structures. Fig. 1 shows the scatter plots expressing the distribution of minima for the esophagus case, obtained using the two cost functions described above. The plot
Fig. 1. - Scatter plots obtained using a dose-based (top) and a DVH-based (bottom) cost function models for the esophagus cancer. The plot related to the dose-based cost function suggests the presence of both global and local minimum. The other plot shows a unique cloud of points, suggesting the presence of only one minimum.
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on the top, corresponding to the quadratic dose based function, has been obtained using about 500 random initial modulations, and suggests the presence of both a global and a local minimum. By analyzing the dose distribution relative to the two minima, through the
isodose representation and the DVHs, we found a more conformal release in the global minimum, due to a significant less involvement of the critical structures (lungs and spinal cord). The dose level inside the target ranges between 93% and 98% of the prescribed dose.
Fig. 2. - Isodose distribution and the intensity modulation of the fields for the case of prostate cancer. The shown data correspond to the global minimum of the dose based cost function (top) and the DVH based cost function (bottom). By comparing the isodose plots, it is evident that a more conformal delivery can be obtained using dosevolume constraints.
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The scatter plot at the bottom of the figure has been obtained using the DVH based cost function, starting from 500 random initial modulations. This plot shows an unique clouds of points, suggesting the presence of only one minimum. The sparing of the OAR corresponding to the best modulation found, is comparable with the one resulting from the global minimum of the dose based cost function. The dose distribution inside the target is characterized instead by a higher and more homogeneous dose delivery (95%100%). The above analysis suggests that the use of dosevolume constraints reduces the complexity of the cost function, giving at the same time a better solution in terms of DVH in the ROI. It should be stressed that, while the dose-volume constraints are specified by the physician, according to the dose-response curves for each tissue [6], the dose constraints are commonly established on the basis of empirical knowledge, and have to be carefully selected before running the optimization procedure [16]. In the prostate case, the same statistical analysis (results are not shown in this paper) indicates that a unique minimum exists, for both the cost functions. Fig. 2 shows the isodose distribution and the intensity modulation of the fields, corresponding to the global minimum of the dose based cost function (top) and the DVH based cost function (bottom). By comparing the isodose plots, we can conclude that a more conformal delivery can be obtained using dose-volume constraints: in fact the dose released to the normal tissues is significantly less, while the homogeneity and the dose level inside the target is comparable in the two cases. The last clinical case we analyzed is a tumor of the maxillary antrum. As we found for the esophagus, it turned out that the dose based cost function has two minima, while the DVH based cost function has only one minimum. This result confirms that a suitable choice of the cost function model can reduce the complexity of the treatment plan optimization. Conclusions The discussed statistical analyses clearly shows that the heuristic optimization approach to the inverse planning IMRT has unavoidable intrinsic uncertainty in the finding of the best intensity modulation. Such uncertainty has to be attributed to different elements. One of these is the stopping criteria of the minimization algorithms which determines the convergence time: the smaller the number of computational steps to reach the valley, the worst the estimate of the optimal solution. Furthermore, at least for deterministic minimization algorithms, like gradient descent methods, the optimal modulations crucially depend on the starting pencil beams configuration. On the other side stochastic heuristics produce themselves
many different trajectories in the configuration space due to the intrinsic noisy nature of the algorithm which allows to avoid non-optimal solutions, a feature characterized by the inability to provide an unambiguous answer to the problem of the minimization in a finite time. A more serious contribution to the uncertainty is a possible complex landscape of the space where the cost function is analyzed, showing local minima that can potentially trap the optimization algorithm and in principle can provide larger variability in the possible solutions. Our work has quantified such uncertainty as a statistical distribution (the “clouds” in Fig. 1) of the found solutions for different cases, and has shown how much the results crucially depends on the cost function chosen. We found that, with respect to a dose-based costfunction, the DV-based one, though intrinsically a highly non linear function of the modulation vector, “typically” exhibits a simpler landscape of the cost function, without local minima. The interest in this observation is further strenghtened by the recognition that setting up a meaningful, DV-based cost function is a fairly straightforward process, from the point of view of clinical assessment. Indeed, choosing a meaningful DV-based cost function essentially amounts to implement a few, more or less established dose constraints to macroscopically relevant volumes, and minimal trial-and-error is needed in order to find an acceptable relative balance of the different ROI components of the cost function (penalties). Rephrased in other words, one virtue of this approach is that one starts stating the optimization problem in terms of the same physical observable (DVH) that provides a figure of merit for the clinical evaluation of the plane; as a bonus, the simpler setup is associated with a more compact set of solutions for the optimization problem. Even if a careful choice of the cost function can help in reducing the uncertainty of the optimal solution, the relevance of such uncertainty remains to be assessed, in view of the many sources of errors associated with the whole TPS. The great variability of realistic situations, even for the same kind of treatment, due to varying anatomy among patients, and/or temporal changes for the same patient, suggests to consider with due caution the elements emerging from the three cases presented in the present work. Received on 26 February 2001. Accepted on 16 June 2001. REFERENCES 1.
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