Estimating Volumetric Motion in Human Thorax with

IEEE TRANSACTIONS ON MEDICAL IMAGING

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Estimating Volumetric Motion in Human Thorax with Parametric Matching Constraints Luis Weruaga, Associate Member, IEEE, Juan Morales, Luis N´ un ˜ez, and Rafael Verd´ u

Abstract—In Radiotherapy (RT) organ motion caused by breathing prevents accurate patient positioning, radiation dose and target volume determination. Most of the motion-compensated trial techniques require collaboration of the patient and expensive equipment. Estimating the motion between two Computed Tomography (CT) three-dimensional scans at the extremes of the breathing cycle and including this information in the RT planning has been shyly considered, mainly because that is a tedious manual task. This paper proposes a method to compute in a fully automatic fashion the spatial correspondence between those sets of volumetric CT data. Given the large ambiguity present in this problem, the method aims to reduce gradually this uncertainty through two main phases: a similarity-parametrization data analysis and a projection-regularization phase. Results on a real study show a high accuracy in establishing the spatial correspondence between both sets. Embedding this method in RT planning tools is foreseen, after making some suggested improvements and proving the validity of the two-scan approach. Index Terms—Volumetric motion, parametric model fitting, regularization, radiotherapy.

I. Introduction ADIOTHERAPY (RT) is an essential technique for the treatment and cure of tumors and other lesions when these are confined in a closed region. The main aspect in the RT planning is the accurate determination of the target or the volume that confines a lesion or a tumor. Once this region is determined several highenergy radiation beams from different angular views are focused precisely in the region of interest so that the dose is high only on the tumor. Although currently the RT treatment is frequently used in the initial and advanced stages of a tumor with acceptable success, the technique is far from having reached a steady state of technology. An important drawback encountered in this treatment is the physiological movement of organs and patient, which by the one hand affects to the alignment, positioning and dose of the radiation beam, and on the other hand to the determination of the target volume [1],[2]. This problem is becoming more important due to the natural tendency in the RT techniques to get better accuracy in the delivery of the treatments. This means that margins as narrow as

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L. Weruaga, J. Morales and R. Verd´ u are with the Information and Communication Technologies Department, Cartagena University of Technology, Cartagena, Spain (e-mail: [email protected]). L. N´ un ˜ez is with the Department of Radiophysics, Clinica Puerta de Hierro, Madrid, Spain (e-mail: [email protected]).

possible along with well adapted treatment beams pursue to get the minimum effective dose distribution conformed to the shape of the target volume. This paper makes first a revision of the main studies dealing with organ motion for RT planning and treatment. Since the range of cases is quite broad [2], this study is focused on the thorax, which on the other hand is the most important cause of organ motion. Most of the proposed techniques either require patient training and collaboration, not feasible in many cases, or expensive medical equipment. Special attention lies on planning methods that use the motion information drawn from each patient Computer Tomography (CT) scans. This paper includes a proposal in this direction: an automatic method for estimating the spatial correspondence between two volumetric CT sets acquired at the extremes of the breathing cycle. This novel method may represent an important contribution for the further development of more precise RT planning tools. This paper is structured as follows: in Section II the state of the art of organ motion in RT is revised, as well as the most significant contributions to volumetric motion estimation; Section III proposes a method to estimate automatically the motion in the human thorax from separate volumetric CT sets of the exhale and inhale positions; in Section IV results over a real human thorax are shown; Section V suggest some research avenues for the improvement of the method; the conclusions close the paper. II. Motivation A. Motion in Radiotherapy planning Patient’s anatomy and position during the course of radiation therapy usually varies to some degree from those used for therapy planning purposes. This is mainly due to patient movement, inaccurate patient positioning, and organ motion. Consequently, the actual received absorbed dose distribution differs from the planned dose distribution. The two scenarios of relevance are the insufficient dose coverage of the targeted tumor volume and the overdosage of normal tissues. ICRU Reports 50 and 62 [3] define the relevant terminology. First, the gross tumor volume (GTV) is defined as the volume containing demonstrated tumor. Second, the clinical target volume (CTV) is defined to enclose the GTV plus a margin to account for suspected tumor involvement. The planning target volume (PTV) is defined by the CTV


plus a margin to allow for geometrical variations such as patient movement, positioning uncertainties, and organ motion. Information about the amount and nature of the CTV motion is essential for the determination of the internal margin size. For some anatomical sites, the internal motion of organs due to physiological processes presents a challenge [2]. The problem is very significant below the base of the skull as, in addition to organ motion, the surrounding soft tissue thicknesses change and this affects the radiation path length. The position-related organ motion is most severe in the abdominal region [4]. Significant changes in organ position were found in the thorax and abdominal region with a caudal shift of abdominal organs. Additionally, internal structures can change in thickness and shape. The use of immobilization devices can also cause organ motion. Studies on several organs influenced by breathing motion have been profusely performed: motion in liver from CT images was reported in [5], determining a movement of 17 mm; in another study [6], larger margins were found; by using fluoroscopy [7] the diaphragm movement in free-breathing conditions in five supine lung cancer patients was measured, 18.8-32.3 mm; in [8] and also by using fluoroscopy the Superior-Inferior (SI) excursion of the right diaphragm in several patients under normal breathing conditions was of 9.1 ± 2.4mm; displacements in the pancreas were also studied [9] and measurements ranged 0-35 mm. Intrathoracic tumor movement with respiratory and cardiac motion was investigated [10] using an ultrafast computerized tomography (UFCT) scanner. During the study, patients were asked to breath normally. The measurements were done manually on the set of images, and in some cases the measurements were not accurate since SI motion causes apparent anterior-posterior (AP) and lateral motion. The results of the study were very heterogeneous, depending on the location of the tumor within the lungs and on the patient. In [11] fluoroscopy was used to evaluate the motion in lung cancer patients. The maximal observed movements in the AP, lateral, and SI directions were 5, 5, and 12 mm, respectively. A number of techniques that aim to synchronize the diagnostic and treatment procedures with breathing have been investigated. These include voluntary breath holding [12], forced breath holding [13],[14], synchronized equipment gating [15]-[20], and combinations of the above. These methods of motion control require training and cooperation of the patient or the implementation of techniques that are not widely available in clinical environments. In synchronized techniques, for instance, gating in linear accelerators is the main drawback, since fast switch-off-on limits the equipment performance. A different approach to organ motion is related to the position of the organ in the initial planning CT. In [8],[21] the use of static exhale-inhale CT images in the determination of the motion margins was investigated. Measurements were done in a manual basis, and, as reported in [10], determination of SI, lateral and AP displacements is an

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ill-posed problem. Whereas there is an common understanding that organ motion is necessary to determine the internal margin, there is no agreement in the method to determine the size of the margin and the relationship between this and the motion magnitude. And the PTV margins is not the only issue: the therapeutical beam energy is computed based on the different tissues encountered in the beam path; motion of the surrounding structures, specially, ribs and bones, which have high radiation absorption levels, is not being considered in the RT planning studies. Since these studies are currently performed on one CT scan, and the CTV margins do not necessarily correspond to viewable structures on the data, trying to determine the PTV, manually, from two sets of CT data, would add more confusion to the problem. An automatic tool that could assist the analyst, without modifying the current clinical protocols, for determining the PTV margins and the required radiation dosage to perform a treatment of minimum collateral effects, would be gladly accepted. B. Volumetric motion estimation Motion is commonly described by a displacement vector field associated with the spatial transform which links the location of each point in a given volumetric frame to its location in the next frame. Works dedicated to motion analysis can be found in [22],[23]. Matching of deformable images and volumetric data has gathered an important attention in the last decade [24]. Potential applications include image segmentation and labelling [25], image registration [26],[27], estimation of three-dimensional (3D) motion and anatomical variations [28]-[31], etc. The most important approaches for matching or motion estimation can be split into correlation and differential methods. The principle of correlation methods is to compute the vector displacement field which optimizes the similarity between two images. Block-matching techniques are the most popular algorithms of this type. In differential approaches, object motion is described as an optical flow and, by analogy to fluid mechanics and the associated mass conservation equation, a differential equation linking the time and the spatial variations is used [32]. The application of such approaches to the computation of 3D velocity fields from 3D cine CT images of a human heart has been reported in [33]. In [34], a stochastic approach to compute regularized motion fields is described. A special group of differential methods is the so-called ”snakes” (and their muldimensional extensions) [35]. In general, differential methods are better suited for tracking deformable motion than correlation ones. On the contrary, differential methods are more sensitive to noise in the data. Obtaining automatically the spatial matching between two volumes of CT data acquired at different moments of the breathing cycle (usually at the exhale and inhale or mid-full lung position) is considered valuable for RT planning tasks, to determine more precisely the PTV and the treatment radiation dose levels. In the following sections it


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is intended to analyze only the motion estimation problem, and not to discuss its possible integration in RT planning tasks. III. Proposed method Let U and V be two sets of volumetric data, that correspond to the same scanned object at different times. In order to deal with the high degree of ambiguity present when estimating the spatial correspondence between U and V , it is hereby proposed a method that aims to reduce gradually this uncertainty throughout three main operators 1 : • the similarity operator, Γ1 : U, V → ρ, which determines the likeness, 0 ≤ ρ ≤ 1, between volumetric blocks in U and V : the closer ρ to 1 is, the higher the similarity between U and V , • the parametrization operator, Γ2 : ρv → Ωv , which translates a volumetric map of scattered data ρv into a parametric model Ωv that contains the relevant information in ρv , ′ • the projection operator, Γ3 : X, Ωv → X , which projects point X onto the region defined by parametric model Ωv , and outputs the projected point X ′ . Operator Γ1 establishes the Similarity Map (SM) for each voxel, operator Γ2 reduces the uncertainty in the SM by fitting a parametric model, and, finally, operator Γ3 combined with a regularization process translates the parametric models into motion field vectors, which relate spatially the two volumes U , V . A. Similarity operator Design of the similarity operator depends on the data source: if the source is the same in both volumes (that is the case of this work, two CT explorations) correlation and/or difference conform Γ1 . For each voxel v the similarity coefficient is obtained from the following tradeoff γv (x) =

rv (x) 1 + α dv (x)

(1)

where α is a positive constant that sets the importance of the block difference. Cross-correlation and square difference are computed respectively as follows PL n=−L (Uv (n) − U v )(Vv+x (n) − V v+x ) q rv (x) = (2) b b Uv Vv+x PL 2 n=−L (Uv (n) − Vv+x (n)) dv (x) = (3) bv U

bv are respectively the mean value and where U v and U the variance of block Uv ; searching index, x, is confined for each dimension in the interval [−N N ] (this makes 1 The

notation used in this section is: voxel indexes, such as x, v, n, are represented in bold small letters; V (x) represents the value of volume V at voxel x; ρv (x) represents a volumetric set, indexed by x and computed for each voxel v; ρv (x) is equivalent to ρ(v + x).

similarity map, γv (x), 2N + 1 voxels wide), and parameter L controls the block size (2L + 1 voxels wide) 2 . Cross-correlation detects accurately similar shapes and is robust to noise and artifacts, and square difference is sensitive to the difference between the intensity in the data (intensity in CT is related to X-ray absorption and thus corresponds to certain tissues). The combination of these properties (as done in (1)) is required to detect shape and tissue of each structure present in both CT volumes. A clipped (1) is actually used as similarity coefficient, ( γv (x) if γv (x) > λ (4) ρv (x) = 0 if γv (x) ≤ λ where threshold 0 < λ ≤ 1 is used to discard low similarity values. Map ρv (x) informs about the similarity between a block of volume U centered at v and the surroundings of V at the location v + x. B. Parametrization of the similarity map Similarity map ρv (x) is a volumetric mass distribution, whose values range in the interval [0 1] (0 being dominant). This mass distribution is usually spread and its shape depends generally on the contents of the searching block and its surroundings: • it is surface-shaped, if the searching block is located on a surface (such as heart-lung or thorax-lung), • it is line-shaped, if the searching block is located close to a thin structure (such as ribs, bronchial tubes), • it is a small cluster, if the searching block is located on a small and clear detail, • and it shows an amorphous shape, if the block is located on low-gradient areas or on textured regions. According to the previous facts, a procedure for extracting the relevant information in the SM is that based on fitting a simple parametric model in ρv (x). According to the previous points, the fuzzy models are respectively a surface (Model 2), a 3D curve (Model 1), a 3D point (Model 0), or undefined (Model 3). The parametric functions that represent each model are described below. Fitting surfaces and curves in scattered volumetric data is a key problem in computer vision and medical imaging. Several research lines cope this problem: active models [35], polynomial fitting [36], etc. The approach used here is a combination of 3D rotation followed by explicit polynomial fitting: azimuth and elevation of the mass distribution are computed, and over the de-rotated space a Taylor’s expansion is fitted. SMs are usually smooth and a secondorder fitting is sufficient to follow the mass distribution. The limited capabilities of this order is also advantageous for detecting anomalous cases. Model 2 consists of the combination of 3D rotation plus an explicit quadratic surface, this last defined by z = axx x2 + ayy y 2 + axy xy + ax x + ay y + a0

(5)

2 Searching window and block size may take different values for each dimension. For sake of simplicity, this fact is omitted.


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A point that lies on Model 2 fulfills that, after 3D rotation, is in agreement with (5). The procedure for obtaining from scattered volumetric data the six parameters in (5) and azimuth and elevation angles, θ and φ, is detailed in Appendix A. Another result is the fitting error, E2 , which is the mean standard deviation (in voxel units) of the scattered data respect to the surface model. Model 1 consists of the combination of 3D rotation plus a curve defined with two explicit quadratic surfaces, 2

y = a2 x + a1 x + a0

(6)

z = b2 x2 + b1 x + b0

(7)

A point that lies on this curve fulfills that, after rotation, is in agreement with (6),(7). Computation of the parameters is based on a method similar to that used in Model 2. Model 0 is obtained by computing the SM center of mass. Finally, Model 3 represents no model and thus it does not impose any further constraint. The four previous models are candidates to represent the mass distribution in the SM. The diagram in Fig. 1 shows the procedure to choose the right model. Curve Fitting

E1