Radiotherapy planning for glioblastoma based on a tumor growth model: Improving target volume delineation
Jan Unkelbach, Bjoern H. Menze, Ender Konukoglu, Florian Dittmann, Matthieu Le, Nicholas Ayache, Helen A. Shih
RRadiotherapy planning for glioblastoma based on atumor growth model:Improving target volume delineation
Jan Unkelbach , Bjoern H. Menze , , Ender Konukoglu , FlorianDittmann , Matthieu Le , , Nicholas Ayache , Helen A. Shih Department of Radiation Oncology, Massachusetts General Hospitaland Harvard Medical School, Boston, MA, USA Martino’s Center for Biomedical Imaging, Massachusetts GeneralHospital and Harvard Medical School, Boston, MA, USA Asclepios Project, INRIA Sophia Antipolis, France Computer Vision Laboratory, ETH Z¨urich, SwitzerlandDecember 20, 2013
Abstract
Glioblastoma differ from many other tumors in the sense that they grow infil-tratively into the brain tissue instead of forming a solid tumor mass with a definedboundary. Only the part of the tumor with high tumor cell density can be local-ized through imaging directly. In contrast, brain tissue infiltrated by tumor cellsat low density appears normal on current imaging modalities. In current clinicalpractice, a uniform margin, typically two centimeters, is applied to account formicroscopic spread of disease that is not directly assessable through imaging.The current treatment planning procedure can potentially be improved by ac-counting for the anisotropy of tumor growth, which arises from different factors:Anatomical barriers such as the falx cerebri represent boundaries for migratingtumor cells. In addition, tumor cells primarily spread in white matter and infil-trate gray matter at lower rate. We investigate the use of a phenomenologicaltumor growth model for treatment planning. The model is based on the Fisher-Kolmogorov equation, which formalizes these growth characteristics and estimates a r X i v : . [ phy s i c s . m e d - ph ] D ec he spatial distribution of tumor cells in normal appearing regions of the brain.The target volume for radiotherapy planning can be defined as an isoline of thesimulated tumor cell density.This paper analyzes the model with respect to implications for target volumedefinition and identifies its most critical components. A retrospective study in-volving 10 glioblastoma patients treated at our institution has been performed.To illustrate the main findings of the study, a detailed case study is presented fora glioblastoma located close to the falx. In this situation, the falx represents aboundary for migrating tumor cells, whereas the corpus callosum provides a routefor the tumor to spread to the contralateral hemisphere. We further discuss thesensitivity of the model with respect to the input parameters. Correct segmenta-tion of the brain appears to be the most crucial model input. We conclude thatthe tumor growth model provides a method to account for anisotropic growthpatterns of glioma, and may therefore provide a tool to make target delineationmore objective and automated. Gliomas are the most common primary brain tumors. For glioblastoma, the most ag-gressive form, median survival is a little more than one year after diagnosis, despitetreatment schedules combining surgery, external beam radiotherapy, and chemotherapy.The clinical experience suggests that improvements in radiotherapy alone will not leadto patient cure. On the other hand, radiotherapy is the single most effective therapy.Hence, the benefit of radiotherapy to prolong survival is undoubted [1, 2, 3]. Conse-quently, we hypothesize that better targeting of the radiation can improve the efficacyof radiotherapy and lead to prolonged survival or reduced side effects.Glioblastoma differ from many solid tumors in the sense that they grow infiltratively.Instead of forming a solid tumor mass with a defined boundary, glioblastoma are char-acterized by a smooth gradient of the tumor cell density. It is well known that tumorcells infiltrate the adjacent brain tissue and can be found several centimeters beyond theenhancing tumor mass that is visible on MRI [4, 5, 6]. Functional imaging modalitiesincluding amino acid Positron Emission Tomography (PET), including FET (Fluoro-Ethyl-Tyrosine) and MET (Methionine) [7, 8, 9, 10], have potential to improve thedefinition of the gross tumor volume. However, these modalities also fail to identifyareas of low tumor cell infiltration.In current practice, radiotherapy planning is typically based on the gross tumor vol-ume (GTV) visible on MRI. Many practitioners account for the infiltrative growth byexpanding the GTV with a 1-3 centimeter margin to form the clinical target volume(CTV), which is irradiated to a homogeneous dose of 60 Gy. The current treatmentplanning procedure can potentially be improved by accounting for two growth charac-teristics of gliomas that are currently not or not consistently incorporated in treatmentplanning: 2
An anisotropic growth of gliomas. MRI data as well as histological analysis af-ter autopsy or resection shows that glioma growth is anisotropic [11, 12]. Thisaccounts for the complex shapes of the visible tumor and is due to mainly threeaspects:- Anatomical boundaries: The dura, including its extensions falx cerebri andtentorium cerebelli, represents a boundary for migrating tumor cells. Also,except for rare cases of CSF seeding, gliomas do not infiltrate the ventricles.- Tumor cells infiltrate gray matter much less than white matter.- Tumor cells seem to migrate primarily along white matter fiber tracts. • A spatially varying tumor cell density. Most gliomas lack a defined boundary andtumor cells can be found several centimeters away from the T1 contrast enhancingtumor and outside of the peritumoral edema visible on T2. The tumor is rathercharacterized by a continuous fall-off of the tumor cell density [4, 5].These macroscopic growth characteristics are partly known from histopathological anal-ysis after autopsy or resection. A comprehensive review can be found in [12]. Math-ematically, these macroscopic patterns of tumor evolution can be modeled by partialdifferential equations of reaction-diffusion type. In one of the most popular models, theFisher-Kolmogorov model, tumor growth is described via two processes: local prolifer-ation of tumor cells and migration into neighboring tissue [13]. The anisotropic growthdue to anatomical boundaries and reduced gray matter infiltration is considered via asegmentation of the brain into white matter, gray matter and cerebrospinal fluid (CSF).Preferential spread of tumor cells along white matter fibers can be modeled based on acell diffusion tensor derived from Diffusion Tensor Imaging (DTI) [14, 15]. This leadsto a phenomenological tumor growth model that estimates the spatial distribution oftumor cells in the brain tissue in regions that appear normal using current imagingmodalities. Incorporating the tumor growth model into radiotherapy planning can beapproached in three steps: • The shape of the target volume is modified as to match an isoline of the simulatedtumor cell density while the total volume is kept identical to the conventionaltarget based on an isotropic margin. • The prescription dose within the target is redistributed based on the varying tumorcell density, in order to deliver less dose to regions of low tumor cell density, and(possibly) boost the regions with high tumor cell density. • Unavoidable dose outside the target is pushed into regions that are more likely tobe infiltrated by tumor cells.This paper primarily addresses the first aspect, i.e. the use of the tumor growth modelfor automatic delineation of a radiotherapy target volume, taking into account spatial3rowth characteristics. An accompanying paper [16] discusses the implications of themodel for redistribution of dose, i.e. the second aspect.
The Fisher-Kolmogorov model has been found to reproduce a wide range of gliomagrowth characteristics at a macroscopic scale. For a review regarding the development ofthe model see for example [13]. The mathematical properties of the Fisher-Kolmogorovequations are discussed in detail in [17] and [18]. A clinical application involves thepersonalization of the model to the patient specific anatomy (see e.g. [19] for a review).In addition, approaches to estimate of model parameters from imaging data have beenproposed [13, 20, 21], which are based on estimating the velocity of tumor growth andthe steepness of the cell density fall-off. The velocity of tumor growth can be estimatedfrom a sequence of MRI images [20]; estimating the steepness of the tumor densityfall-off is based on comparing the size of the contrast enhancing tumor core to thesize of the peritumoral edema. These observable quantities relate to the generic modelparameters proliferation rate and diffusion coefficient. The model has, in particular,been used in life time prediction [22, 23]. In addition, several extensions of the modelhave been suggested. Those extensions integrate diffusion tensor imaging [14, 15], tumorhypoxia [24], and the effect of radiotherapy [25] and chemotherapy on the evolution ofthe tumor cell density [26, 27]. The idea of utilizing the model for target definitionin radiotherapy has been suggested by several authors [28, 29, 30], who propose waysof describing tumor infiltration beyond the visible margins of the tumor. Cobzas [28]defines a distance metric in the brain based on DTI to describe the anisotropy in tumorgrowth. Konukoglu [29] and Bondiau [30] uses a travelling wave approximation of thefull reaction diffusion model to estimate the spatial distribution of tumor cells in normalappearing brain tissue.
Even though the Fisher-Kolmogorov model has previously been studied in the mathe-matical biology and modeling community, an application for target delineation in clinicalpractice requires further research in multiple directions. Our work uses the previouslypublished Fisher-Kolmogorov model and aims at characterizing the potential of themodel for a true clinical application in radiotherapy treatment planning. We aim atdeveloping a semi-automatic contouring tool for gliomas that is practically useful. Thecontributions of this paper are as follows: • Systematic review of model assumptions:
In section 2 we provide a reviewof the tumor growth model in order to systematically compile the assumptionsmade to apply the model to target delineation. This includes 1) highlighting the4ualitative features of the model that are relevant for radiotherapy planning, 2)identify the model parameters determining those features, and 3) illustrate howthe imaging data is used to personalize the model for an application to the patientat hand. • Use case characterization:
We provide a clear illustration of use cases of thetumor growth model for target delineation. In section 3 we perform a detailed casestudy for a tumor located in proximity to falx and corpus callosum, a situationwhere it is difficult to consistently account for anatomical boundaries in a manualdelineation. In A, we show results for 4 additional cases with varying tumorlocation. The presented results illustrate the main findings from a study involving10 glioblastoma patients previously treated at our institution. • Sensitivity analysis:
We discuss the sensitivity of the simulated tumor celldensity with respect to the model inputs (section 3.5). In particular, we highlightthe need for accurate brain segmentation into white matter, gray matter andanatomical barriers. • IMRT planning:
We compare radiotherapy plans optimized for intensity-modulatedradiotherapy (IMRT) for model based versus manual target delineation (section4). We illustrate to what degree differences in target delineation are translatedinto differences in the delivered dose distribution.Furthermore, for one patient we compare the model predicted tumor cell density to thespatial growth patterns of the recurrent tumor (section 5). Finally, section 6 discussesthe scope of the work and the potential of a practical application of the tumor growthmodel for target delineation.
The purpose of the tumor growth model amounts to estimating the density of tumorcells in the regions of the brain that appear normal on MRI. It is assumed that thegrowth of the tumor is described phenomenologically by local proliferation of tumorcells and migration into neighboring tissue.
The tumor growth calculations are based on MR images routinely acquired in clinicalpractice. The images incorporated into the simulation process are T1, T2, T2-FLAIR,and T1 post gadolinium. The T1 post gadolinium image shows the vascularized grosstumor volume, and the T2-FLAIR image shows the surrounding edematous region. Forillustrative purpose, these two images are shown in figure 1 for a case discussed in detailin this paper. 5 a) (b)
Figure 1: (a) post-gadolinium T1 weighted image of a glioblastoma located in the leftparietal lobe. (b) T2-FLAIR image of the same tumor, showing the surrounding peri-tumoral edema. (Note that the right side of the image corresponds to the left side ofthe brain to follow brain imaging conventions.)
In the first data processing step, all MR sequences are registered to the image withhighest spatial resolution. This is done using rigid registration with 6 degrees of freedom,utilizing the function FLIRT [31] as part of the toolbox FSL [32, 33]. In the second step, asegmentation of the brain was obtained using the multimodal brain tumor segmentationalgorithm published in [34]. The algorithm is an Expectation-Maximization (EM) basedsegmentation method, which uses a probabilistic normal tissue atlas as spatial tissueprior. For every voxel, it estimates the posterior probability for three normal tissueclasses (white matter, gray matter, and CSF ), as well as the lesion outlines on T1post gadolinium and T2-FLAIR. The result of the EM segmentation is augmented asdescribed in [35] in order to facilitate a reliable identification of falx cerebri and tentoriumcerebelli as anatomical barriers. The segmentation result for the patient in figure 1is shown in figure 2. In the last step, the reference MR image is registered to theradiotherapy planning CT using rigid registration. The transformation matrix is savedand later applied to register the simulated tumor cell density to the planning CT. In this work, we refer to all brain tissue that is neither white matter, gray matter, nor tumor asCSF.
It is assumed that tumor growth is described by two processes: local proliferationof tumor cells and migration of cells into neighboring brain tissue. Mathematically,this is formalized via the Fisher-Kolmogorov equation, a partial differential equation ofreaction-diffusion type for the tumor cell density c ( r , t ) as a function of location r andtime t : ∂∂t c ( r , t ) = ∇ · ( D ( r ) ∇ c ( r , t )) + ρc ( r , t ) (cid:18) − c ( r , t ) c max (cid:19) (1)where ρ is the proliferation rate which is assumed to be spatially constant, and D ( r )is the 3 × r . The first term on the righthand side of equation (1) is the diffusion term that models tumor cell migration intoneighboring tissue. The second term is a logistic growth term that describes tumor cellproliferation. The tumor cell density c ( t, r ) takes values between zero and the carryingcapacity c max . In this paper, the diffusion tensor is constructed as D ( r ) = (cid:26) D w · I r ∈ white matter D g · I r ∈ gray matter (2)where I is the 3 × D g and D w are scaling coefficients for grayand white matter, respectively. This construction of the diffusion tensor allows for themodeling of reduced gray matter infiltration via a different diffusion coefficient D g < D w .Anatomical constraints are accounted for by boundary conditions: We assume thattumor cells only spread in brain tissue and do not diffuse into any tissue that is notsegmented as gray or white matter. The model can be extended towards anisotropic tumor cell migration within white matter. Thisis achieved by replacing the identity matrix I in equation (2) by a tensor that is constructed based on .4 Qualitative features of the solution The asymptotic solution of the Fisher-Kolmogorov equation is given by a traveling wavesolution. At the time of diagnosis, the tumor cell density takes high values close to thecarrying capacity c max within the enhancing tumor mass seen on the post-gadolinium T1image. This is motivated by the observation that resected tissue from this area appearsas only tumor tissue in a pathological examination [12]. At large distance from theenhancing core, the tumor cell density approaches zero. It can be shown that, withinthe traveling wave approximation, the following two properties hold:1. The fall-off of the cell density with distance from the enhancing core is described bya characteristic profile that remains approximately fixed while the tumor grows [17,29]. At some distance, the tumor cell density drops approximately exponentiallywith distance from the visible tumor mass. For a region comprising white matter,the cell density behaves according to c ( | r | ) ∝ exp (cid:18) − | r | λ w (cid:19) (3)where | r | denotes distance from the visible tumor. The parameter λ w is called theinfiltration length and denotes the distance at which the cell density drops by afactor of e . It can be shown that λ w relates to the model parameters via λ w = (cid:115) D w ρ (4)Within gray matter the cell density drops with an infiltration length λ g = (cid:113) D g ρ which is believed to be smaller than the infiltration length in white matter.2. Over time, the tumor front moves outwards into the brain tissue at a constantvelocity. The velocity relates to the model parameters via v w = 2 (cid:112) D w ρ (5)In a realistic brain geometry, the propagation of the wave is spatially modulated byanatomical boundaries and by different velocities in gray and white matter. the water diffusion tensor obtained through DTI imaging. Although this idea has been conceptuallyintroduced [15, 14], it is related to several difficulties that are outside the scope of this paper. Problemsrelated to DTI in this context include: (1) even though histopathology provides anecdotal evidence forpreferential spread along white matter fibers, there is no data that quantitatively evaluates this effect;(2) the DTI signal is corrupted within the edematous region where the increased water content leads toreduced fractional anisotropy (FA) and increased apparent diffusion coefficients (ADC). In this paper,we only consider the effects of anatomical boundaries and reduced infiltration of gray matter, becausethese effects are widely agreed upon, whereas the use of DTI is more speculative at this stage. .5 Relating the tumor model to imaging data In order to apply the model to an individual patient, we have to specify how the tumorcell density resulting from the model relates to the MR images. The contrast enhancingregion visible in the T1 post gadolinium MR image is related to a breakdown of theblood-brain-barrier (BBB). Even though this only represents a surrogate for the tumor,it is commonly assumed that contrast enhancing volume corresponds to the vascular-ized, highly cellular part of the gross tumor. For this paper, we therefore assume thatthe boundary of the contrast enhancing region corresponds to an isoline of the tumorcell density close to the carrying capacity. It is used as the basis for tumor densitycalculations is described in subsection 2.6.In current clinical practice, radiotherapy planning for GBM is mostly based on structuralMRI imaging (T1 post gadolinium and T2-FLAIR). However, modern imaging modali-ties including MR spectroscopy (MRS) and positron emission tomography (PET) havebeen used for glioblastoma imaging and bear potential to improve the characterizationof gross and infiltrative disease. In particular, tracers visualizing amino acid metabolismhave been studied [7, 8, 9, 10], including FET (Fluoro-Ethyl-Tyrosine) and MET (Me-thionine). These PET scans are not routinely acquired for glioblastoma patients at ourinstitution. However, if these modalities become established, the tumor growth modelcould be initialized using a PET based delineation of the tumor instead of using thecontrast enhancing region.
For treatment planning, we want to calculate the tumor cell density in the brain atthe time the MR images are taken. Using the Fisher-Kolmogorov equation directly isproblematic in that context because the initial condition, which leads to a tumor con-sistent with the images, is unknown. One approach to circumvent this problem consistsof using the traveling wave approximation. It is assumed that, at the time of imaging,the tumor cell density is characterized by its asymptotic solution. In this approach,it is assumed that the tumor cell density at the T1 gadolinium outline corresponds toan isoline of the tumor cell density. For this paper, we assume a value of 70% of thecarrying capacity. Starting at the outline, the tumor cell density is extrapolated intothe brain tissue, assuming the fall-off profile from the traveling wave approximation,and taking into account the anatomical constraints from the brain tissue segmentation.Mathematically, this is achieved by approximating the partial differential equation in(1) by a static Hamilton-Jacobi equation that is solved using a Fast-Marching method.The details of this approach are described in [29].9 −10−9−8−7−6−5−4−3−2−10 (a) −10−9−8−7−6−5−4−3−2−10 (b)
Figure 3: Simulated tumor cell density based on the segmentation in figure 2 for pa-rameters λ w = 4 . D w /D g = 100 (a) and D w /D g = 10 (b) . The color scalerefers to the log cell density normalized to the carrying capacity. The tumor growth model depends on four parameters: The diffusion coefficients inwhite matter D w and gray matter D g , the proliferation rate ρ , and the carrying capacity c max . We now argue that, for our application in radiotherapy planning, the number ofparameters can be reduced to two crucial parameters.The carrying capacity c max can be considered as a scaling factor for the relative nor-malized tumor cell density that takes values between zero and one. It is straightforwardto see that all results presented in this paper are independent of the value of c max andonly depend on the relative tumor cell density. We therefore do not have to considerthe value of the carrying capacity c max .We are left with the model parameters D w , D g , and ρ . Within the traveling waveapproximation, D w and ρ can be expressed through 1) the velocity of the tumor frontin white matter obtained from the product of D w and ρ according to equation (5), and2) the infiltration length in white matter obtained from the ratio of D w and ρ accordingto equation (4). In this paper, we are only interested in obtaining the distribution oftumor cells at the time when treatment starts. Thus, we do not require the velocity ofthe tumor front but only the infiltration length λ w = (cid:112) D w /ρ . Since the tumor primarilygrows in white matter, we focus on the infiltration length in white matter instead ofgray matter. If we express the diffusion coefficient in gray matter via the ratio of whiteand gray matter coefficient, we are left with two parameters of the tumor growth model: λ w = (cid:113) D w ρ : Infiltration length, describing how fast the tumor cell density dropswith distance from the visible tumor volume.10 w /D g : Ratio of diffusion coefficients in white matter and gray matter, param-eterizing in parts the anisotropy of tumor growth. The ratio of the infiltrationlengths is consequently given by λ w /λ g = (cid:112) D w /D g .In this paper, we are concerned with the spatial definition of the target volume as anisoline of the tumor cell density. In that context, the exact value of the infiltrationlength λ w is irrelevant because it does not affect the shape of the isolines. Only theabolute values of the tumor cell density associated with the isolines are determined by λ w . Thus, the only relevant model parameter for this paper is the ratio D w /D g , whichdetermines the shape of the isolines of the cell density together with the brain segmen-tation. The literature consistently suggests that tumor cells infiltrate gray matter muchless than white matter This suggests a large value for D w /D g (cid:29)
1. Most illustrationsin this paper were obtained for D w /D g = 100. The most appropriate value is howeveruncertain and we discuss the impact of uncertainties in D w /D g in section 3.5. In this section we demonstrate the use of the tumor growth model for target definition.In sections 3.1-3.5, we provide a detailed discussion of the patient shown in figure 1 inorder to illustrate the main findings of the study. We first compare the model-derivedtarget volumes to the manual delineation used in the clinical treatment plan. We thendiscuss the impact of anatomical boundaries and reduced gray matter infiltration in moredetail, and finally discuss sensitivity to model inputs. We analyzed 10 GBM patientswith varying tumor locations previously treated at our institution. In section 3.6, wesummarize the main findings for these patients; further details are provided in A.
Figure 4 shows the clinical target volume (dark green contour) and the boost volume(green contour) drawn manually by the physician, i.e. these contours were the basis forthe treatment plan that was actually delivered to the patient. The prescribed dose tothe boost volume was 60 Gy, the prescribed dose to the CTV was 46 Gy. The boostvolume is defined based on a 2 cm isotropic extension of the contrast enhancing lesion.The CTV is defined via a 1.5 cm expansion of the T2-FLAIR abnormality. Both vol-umes were subsequently trimmed manually to account for anatomical boundaries (dura,ventricles, falx, and tentorium cerebelli). The dose distribution of the 3D conformaltreatment plan delivered to the patient is shown in figure 6d. The value of λ w will be crucial for dose prescription considered in the accompanying paper [16]. This holds for the most common case of astrocytomas, not necessarily for oligodentrogliomas (see[12] for a review of glioma growth patterns). D w /D g = 100. The red and the orange contoursin figure 4 show the CTV and the boost volume derived from the tumor growth model,respectively. In this example, the target defining isolines are chosen such that the totalenclosed volume is equal to the manually delineated target. In the following two sub-sections, we discuss the differences between manual and model-derived target volumesin detail.Figure 4: Comparison of manually defined targets (light/dark green) and model de-rived targets (orange/red). In the clinical treatment plan, the light green volume wasprescribed to 60 Gy, the dark green volume to 46 Gy. The yellow contour shows theabnormality on T2-FLAIR. In the manual delineation of the CTV used in the clinical plan, it is incorporated thatthe falx represents an anatomical barrier for the migration of tumor cells. Hence, theisotropic target expansion was trimmed manually. In the tumor growth model, thefalx is modeled via a layer of CSF and is automatically accounted for through theassumption that tumor cells only migrate within white and gray matter. However, thecorpus callosum connects the two hemispheres of the brain. The tumor growth modeldescribes the migration of tumor cells through the corpus callosum (see figure 3 and 5a).As a consequence, the target volume based on the growth model is extended into thecontralateral hemisphere. Figure 5a shows the tumor cell density overlaid on the coronalT1 gadolinium image. This illustrates the three-dimensional modeling of tumor spreadvia the model, including areas superior to the corpus callosum. This is not consistentlyaccounted for in the manual CTV. In the manually drawn target volumes, the target is12lightly extended into the contralateral hemisphere on the slices that show the corpuscallosum, but not on the slices located superiorly and inferiorly (figure 5b). In the modelderived target volumes, the target is extended further into the contralateral hemisphere,and the spread of tumor cells in superior-inferior direction beyond the corpus callosumis modeled. −10−9−8−7−6−5−4−3−2−10 (a) (b)
Figure 5: (a) Log tumor cell density overlayed on the coronal T1 image showing spreadof the tumor in the contralateral hemisphere ( λ w = 4 . D w /D g = 100). (b)Comparison of manual CTV (green) and model based CTV (red) (same contours as infigure 4. It is typically assumed that tumor cells infiltrate gray matter much less than whitematter, which is modeled by D w /D g (cid:29)
1. In the simulated tumor cell density in figure3, this leads to a sharp drop of the cell density towards the cortex. In most regions,this effect does not alter the global shape of the target volume substantially becausethe thickness of the cortical layer is only several millimeters. This can be seen in figure3 in the posterior portion of the tumor. Here, the dura is essentially the boundary ofthe target volume. Technically, some gray matter surrounding the fissures is outsideof the target defining isoline. However, these regions are small and are not relevantfor radiotherapy planning as further discussed in section 4. For this patient, the mostpronounced differences between the model derived target and the manually drawn target,which can be attributed to reduced gray matter infiltration, arise in the region of thelateral sulcus (left-anterior part of the tumor). In this region, the large amount of graymatter surounding the lateral sulcus represents a soft barrier for migrating tumor cells.Thus, the tumor growth model can be used to identify regions of functioning brain tissue13hat can be taken out of the target volume.
In order to quantitatively compare the difference of manual and model-derived targetvolumes, we calculate the Dice coefficient, which is given by the volume where bothstructures overlap, divided by the average volume of the two structures. For the boostand CTV volumes in figure 4 the Dice coefficients are 0.78 and 0.77, respectively. Thus,three quarters of the manual target is also contained in the model-derived target, but asubstantial portion of approximately one quarter is different.
The model input parameters that determine the spatial distribution of tumor cells arethe segmentation of the brain (into white matter, gray matter and CSF) as well as theparameter D w /D g . Brain segmentation:
Anatomical boundaries like the ventricles and the falx are mod-eled via boundary conditions. As a consequence, correct segmentation of the brain isa crucial input to the model. This can be illustrated for the falx: Correct modeling ofthe anatomical boundary in our approach relies on identifying a layer of CSF along thefalx. If due to limitations in the automatic segmentation, the two hemispheres are con-nected through white or gray matter bridges (outside of the corpus callosum), the falxis not correctly established as a barrier. In this case, the growth model will incorrectlypredict interhemispherical cell migration outside of the corpus callosum. This problemcan in particular arise if, due to a tumor with mass effect, the brain tissue is compressedand the falx is pushed towards the contralateral hemisphere. A reliable application ofthe model for target definition therefore requires reliable segmentation of the ventriclesand the extensions of the dura, i.e. falx cerebri and tentorium cerebelli. In addition,correct modeling of contralateral spread requires a reliable segmentation of the corpuscallosum as white matter. This can, for example, be problematic in the superior regionof the corpus callosum when the slice distance in the MRI scan is large. Approaches tocustomize brain segmentation methods for this application can be found in [35].
Gray matter infiltration parameter:
In addition to the segmentation, the ratio ofwhite and gray matter diffusion coefficient D w /D g influences the shape of the isolinesof the tumor cell density. For D g = 0, gray matter represents a hard boundary andtumor cells only spread in white matter. For D w /D g = 1, tumor cells spread equallyin white and gray matter and the shape of the target is solely influenced by anatomicalconstraints. In figure 3 the simulated tumor cell density is compared for D w /D g = 10and D w /D g = 100. For smaller D w /D g , the cell density is more washed out (figure 3b)14ompared to a larger D w /D g where the tumor cell density follows more closely the whitematter structure (figure 3a). It has been discussed above that the cortical gray matterhas a thickness of only a few millimeters. As a consequence, varying D w /D g has littleimpact on the global shape of the target volume. The most significant changes for thispatient are around the lateral sulcus. Interdependence:
There is some interdependence regarding the sensitivity against D w /D g versus segmentation errors. For small values of D w /D g , the result may be sen-sitive to errors in the segmentation of CSF, whereas errors in the segmentation betweenwhite and gray matter become irrelevant. For large values of D w /D g , the segmentationof CSF becomes less crucial because falx and tentorium cerebelli are surrounded by alayer of gray matter which models the boundary. However, errors in the segmentationof white matter versus gray matter become more crucial. We retrospectively analysed 10 patients with varying tumor location. All patients weretreated with IMRT to a dose of 60 Gy in 30 fractions. For the clinically deliveredtreatment plan, the CTV was defined using isotropic expansions of 1-2 cm to the T2-FLAIR hyperintensity. Manual editing was performed to trim the CTV expansion wherethere were clear anatomical barriers to spread such as the falx or dura. A PTV expansionof 3 mm was added. For all patients, the tumor cell density is calculated for a parametervalue of D w /D g = 100 based on the method described in section 2. The model-basedtarget volume is defined as the isoline that encompasses the same total volume as themanually delineated CTV used in the clinically applied plan. The main findings aresummarized as follows: • The modeling of anatomical barriers leads to differences in the target volume fortumors located close to corpus callosum and falx cerebri. The use of the tumorgrowth model suggests a further expansion of the target into the contralateralhemisphere. • The modeling of reduced gray matter infiltration leads to differences in the targetvolume for tumors located in proximity to major sulci. This effect was mostpronounced in the region of the lateral sulci where the use of the tumor growthmodel suggests regions that can be excluded from the target volume. • Dice coefficients between manual and model derived target volumes were calculatedfor all 10 patients (table 1). The overlap between manual and model-based targetvolumes was 79% on average, ranging from 74% to 84%. This indicates thatapproximately 20% of the manual target volume is not contained in the model-based volume (and vice versa). 15
For all patients IMRT treatment plans were optimized using a homogeneous 60Gy prescription dose to the CTV and the planning formulation in section 4. Dicecoefficients for the 95% isodose lines (57 Gy) were calculated in order to comparethe differences in the high dose region between manual and model derived radiationplans (table 1). The Dice coefficient was 0.84 on average, ranging from 0.80 to0.87, indicating that a substantial part of the differences in target volume alsotranslates into differences in the high dose region.In the appendix A, we provide details for four representative cases. These four casesspan a range of different tumor locations, i.e. GBMs in the parietal lobe, temporal lobe,frontal lobe, and bilateral corpus callosum.Location Dice CTV contour Dice 57 Gy isoline1 Parietal (Fig. 4) 0.77 0.822 Temporal/Frontal (Fig. 8) 0.74 0.813 Parietal (Fig. 10) 0.81 0.824 Temporal (Fig. 11) 0.84 0.875 Corpus Callosum (Fig. 12) 0.76 0.826 Temporal (not shown) 0.81 0.857 Temporal (not shown) 0.83 0.868 Parietal (not shown) 0.80 0.849 Parietal (not shown) 0.84 0.8510 Temporal (not shown) 0.74 0.80Table 1: Dice coefficients for the CTV contours and the 57 Gy isoline of an IMRT planfor different tumor locations.
In figure 4 we have defined the model derived target contours such that the overallvolumes of the targets equal the volumes of the targets in the clinical plan. This hasbeen done for the sake of comparison of the shapes of the target volumes. In a practicalscenario, the model is used for target definition without the existence of a manuallydrawn reference volume. In this case, the choice of the target defining isoline can beperformed by either specifying the target expansion in white matter, or by specifyingthe average expansion:1. In the first scenario, the treatment planner is provided with the simulated tumorcell density together with the delineation of enhancing core and edema as well asthe brain segmentation. The treatment planner can then manually pick a targetdefining isoline by measuring the distance to the enhancing core along major sheetsof white matter. 16. In the second case, a temporary reference volume can be created automaticallyas an isotropic expansion around the enhancing core, intersected with the braintissue mask. The model-derived target volume can then be defined as the isolinethat encompasses the same total volume as the temporary reference volume. Inthis case, the treatment planner has to specify the average margin added to theenhancing core.
In this section we illustrate the impact of the model derived target volumes on the dosedistribution. A treatment plan for intensity modulated radiotherapy was optimized,using 9 equally spaced coplanar 6MV photon beams. Dose calculation was performedusing the Quadrant Infinite Beam (QIB) algorithm in CERR [36], and the optimizationof IMRT treatment plans was performed using our own implementation of the L-BFGSquasi-newton method. More specifically, we minimize the following piece-wise quadraticobjective function f ( d ) = (cid:88) η w oη N η (cid:88) i ∈ V η (cid:0) d i − d maxη (cid:1) (6)+ w uT N T (cid:88) i ∈ T ( d presi − d i ) + w oT N T (cid:88) i ∈ T ( d i − d maxT ) (7)+ w oH N H (cid:88) i ∈ H ( d i − d maxi ) (8)The first term (6) denotes overdose objectives for the organs at risk (OAR). For themaximum doses d maxη and weighting factors w oη , we use the values summarized in table2. The second term (7) represents under and over dose objectives for all target voxels.The prescribed dose d presi is set to 60 Gy for all voxels i in the boost volume and 46 Gy inthe CTV. The maximum dose d maxT in all target voxels is set to 60 Gy. For all unclassifiednormal tissues surrounding the target volume (including skull, brain tissue, ventricles),we use a conformity objective in which d maxi decreases linearly with the distance of voxel i from the target volumes. Here, we define a generalization of the conformity objectiveto the case of an inhomogeneous prescription dose. More specifically, for every voxel i we define d maxi = max (cid:20) d low , max j (cid:0) d presj − z ij d grad (cid:1)(cid:21) (9)where z ij denotes the euclidean distance of the voxel i from another voxel j that hasa non-zero prescription dose d presj . The parameter d grad is specified in Gy per cm anddescribes the desired dose falloff in healthy tissue; d low is a lower dose threshold belowwhich dose is not penalized. The optimization parameters used are summarized in table2. 17 (a) (b) −30−20−100102030 (c) (d) Figure 6: Comparison of dose distributions (in units of Gy): (a) for a plan based onmanual target volumes; (b) for model-derived target volumes. The difference of the dosedistribution is shown in (c). Figure (d) shows the planned dose distribution for the 3Dconformal plan the patient was treated with.18 o w u d max d pres d grad d low CTV 10 20 60 46 - -Boost 10 20 60 60 - -Brainstem 10 - 45 - - -Optic nerves 10 - 30 - - -Chiasm 10 - 30 - - -Eye lenses 10 - 10 - - -Eye balls 10 - 10 - - -Unclassified 10 - - - 40 20Table 2: Objective function parameters used for IMRT optimizationFigure 6 shows the dose distributions obtained for two treatment plans. The plan infigure 6a is based on the manually drawn target volumes, whereas the plan in figure6b is based on the target volumes derived from the growth model. For the sake ofcomparison, the plan based on model-derived targets was optimized by imposing anadditional constraint on the integral dose delivered to the patient, i.e. we minimizeobjective function (6-8) subject to the constraint (cid:88) i ∈ V d i ≤ d int (10)where V denotes the set of all voxels in the patient, and d int denotes the integral doseobtained for the treatment plan based on the manual target volumes. Figure 6c showsthe difference of the plans in figures 6a and 6b. It can be seen that in plan 6b more doseis delivered to the contralateral hemisphere. Under the constraint that integral dose isnot increased, this is compensated for by delivering less dose to other regions, e.g. nearthe lateral sulcus.To quantify the difference in dose distributions in figure 6, we calculate the Dice coeffi-cients for the volumes enclosed by selective isodose lines. The Dice coefficients for the57 Gy and the 46 Gy isodose line evaluate to 0.84 and 0.83, respectively. In section 3.4we calculated the Dice coefficient between manual and model-derived boost and CTVvolumes, which are 0.78 and 0.77, respectively. It is apparent that differences in targetvolumes only partly translate into differences in dose distributions. The rugged shapeof the model-derived target volume that arises from reduced infiltration of gray matteris mostly not of relevance for radiotherapy planning. Due to the physical characteristicsof photon beams it is not possible to spare thin layers of gray matter surrounding smallfissures. 19 Comparison to follow-up imaging
Ideally, we would like to have a way to validate the predictions of the growth modelregarding the tumor cell density. Histopathological analysis of resected brain tissuewould represent one way to validate the extent of tumor cell infiltration predicted bythe model. However, it is currently not realistic to obtain such data for untreated humanpatients at high spatial resolution. Tissue samples taken from biopsies or resected tumorsmay provide information at selected locations. Usually this is, however, limited to thegross tumor volume; taking biopsy samples from healthy appearing brain tissue, whichwould be relevant for target delineation, is not performed.An alternative approach towards collecting evidence for the plausibility of the growthmodel consists in evaluating the spatial progression patterns of recurrent tumors infollow-up MRI images. The advantage of this approach is that this data is widely avail-able since high grade glioma patients typically receive follow-up MR imaging every 2-3months. On the downside, model validation based on follow-up imaging has limitations.This includes the difficulty in distinguishing recurrent tumor from other radiation in-duced changes (e.g. radiation necrosis), and the large variation in response to radiationbetween patients. In addition, deformations due to surgery, tumor mass effects, andradiation related brain atrophy may occur. Addressing these issues in their complex-ity is outside the scope of this paper. Nevertheless, we hypothesize that the analysisof progression patterns of recurrent tumors can provide some degree of evidence forthe plausibility of the tumor growth model. Due to the variability in treatment andtreatment response, this is not the case for all patients - but for individual cases.
The patient discussed in sections 3 and 4 only had a biopsy taken prior to radiother-apy without significant tumor resection. Also, the patient did not receive additionalcycles of chemotherapy or antiangiogenic drugs after the initial course of chemoradia-tion with concurrent temozolomide. Figure 7 shows two follow-up T2-FLAIR imagesfor this patient. The follow-up images were registered to the reference image using rigidregistration. The image in figure 7a was taken 3.5 months after the diagnostic image(figure 1) and 2 months after completion of radiotherapy. The image in figure 7b wastaken 7.5 months after the diagnostic image, which was the last image taken before thepatient expired. Even though there is no histopathological confirmation, the radiologicalappearance and the patient’s death about 9 months post therapy suggest progressivedisease rather than other radiation induced changes. Thus, for the remainder of thissection, we make the assumption that the follow-up imaging shows the progression ofthe tumor. Furthermore, the tumor recurs/progresses centrally, which is the case forthe vast majority of cases given standard of care treatments. Figure 6d shows the dose20istribution of the clinically delivered 3D conformal treatment plan. (a) 3.5 month (b) 7.5 months
Figure 7: T2-FLAIR images acquired after therapy, overlaid with isolines of the tumorcell density simulated on the diagnostic images prior to radiotherapy (figure 3a): (a) 3.5month after diagnosis with isolines for tumor cell densities of 10 − , 10 − , and 10 − , and(b) 7.5 months after diagnosis with isolines for tumor cell densities of 10 − , 10 − , and10 − . By considering the delivered dose distribution in figure 6d, we see that the progressivetumor is located almost entirely within the high dose region receiving approximately60 Gy. Under the assumption that a radiation dose of 60 Gy kills a fraction of thetumors cells, the shape of the isolines after therapy is not altered within the high doseregion. In this case, the Fisher-Kolmogorov model predicts that the progression ofthe tumor approximately follows the isolines of the tumor cell density simulated pretherapy. This motivates to compare the simulated cell density isolines directly to thefollow-up images without explicitly modeling the effect of chemoradiation on the celldensity quantitatively.In figure 7 we overlaid the tumor cell density simulated on the diagnostic image on thefollow-up T2-FLAIR images. It is seen that the progression of the tumor post therapyfollows the isolines of the simulated tumor cell density relatively closely. In particular,the follow-up images show the progression of the tumor through the corpus callosuminto the contralateral hemisphere as predicted by the model. The latter is supportedby the T1 post contrast images 3.5 and 7.5 months post therapy, which show contrastenhancement in the corpus callosum (T1 images not shown).21
Discussion
In this work, we aim at improving and automating target volume delineation for glioblas-toma. We investigate the use of a phenomenological tumor growth model based on areaction-diffusion equation for the tumor cell density. The model has been studied inthe literature over the past years. Here, we aim at bringing this model closer to an appli-cation in radiotherapy treatment planning. The problem addressed here originates fromthe infiltrative growth characteristics of glioma and the fact that tumor cells infiltratebrain tissue beyond the abnormality visible on current imaging modalities. In clinicalpractice, an approximately 2 centimeter wide margin is added around the T2-FLAIRabnormality. It is however known that glioma growth is not isotropic. Tumor growthpatterns are influenced by anatomical boundaries as well as reduced infiltration of graymatter. Hence, adding an isotropic margin will unnecessarily expose normal tissue ormiss areas of concern. Anatomical constraints can partly be accounted for in the man-ual definition of the CTV, e.g. by excluding ventricles and regions outside of the dura.However, a consistent modeling of more complex 3D anatomical features is difficult toachieve manually.
We performed a retrospective study involving 10 GBM patients treated previously atour institution. For these patients, we compare manually drawn target volumes used inthe clinically delivered treatment plan to model-derived target volumes defined throughan isoline of the tumor cell density. One goal of the study is to characterize the bestuse cases for model based target delineation, i.e. we identify anatomical situations inwhich the growth model leads to more consistent target volumes compared to manualdelineation. As demonstrated in the paper, this may be the case for tumors located inproximity to falx and corpus callosum. In this situation, the falx represents a boundaryfor migrating cells, but the corpus callosum provides a route for the tumor to spread tothe contralateral hemisphere. The tumor growth model represents a tool to automateand objectify target definition for such cases, which is difficult to achieve manually. Inaddition, it is demonstrated that modeling of reduced infiltration of gray matter mayinfluence target delineation near major sulci, where large accumulations of gray mattersurrounding the sulci represent a soft boundary for migrating tumor cells. In particular,this effect is observed for tumors located in proximity to the lateral sulcus. By optimizingIMRT treatment plans based on model-derived and manually drawn target volumes, itis demonstrated that the differences in target delineation may translate into differencesin the delivered dose distribution. 22 .3 Scope of the work
Technical versus clinical goal:
The work presented has both a technical goal anda clinical goal. The technical goal of this project is to develop automated delineationof the CTV while accounting for anisotropic growth patters. Currently, anatomicalbarriers are to some degree taken into account manually. In complex situations, this isa time consuming process and often leads to inconsistent CTVs. Automated GTV toCTV expansion will greatly accelerate treatment planning and make treatment volumesmore objective. In addition to the technical goal, more accurate identification of thehighest risk regions of microscopic disease may translate into a clinical advantage. Thelarge isotropic CTV expansions currently used include neighboring low risk normal braintissue, which may receive unnecessary and potentially harmful irradiation. However, byusing smaller CTV margins, clinicians risk missing areas of tumor extension, which maylead to earlier locoregional failure. Due to the heterogeneity among GBM patients,the clinical advantage will be difficult to demonstrate, however, the technical goal ofautomating CTV delineation is itself worthwhile to pursue.
The role of model validation:
Common criticisms to the use of tumor growth mod-els for radiotherapy planning are that models are oversimplified, that a validation of themodel based on clinical data is possible only to a limited extent, or that model pa-rameters are unknown or uncertain. For this model, the analysis of growth patternsof recurrent tumors may provide some indication for the plausibility of the model forindividual patients. However, further research on model validation based on follow-upimaging, including MRI, MR Spectroscopy, and PET is needed. Nevertheless, we arguethat an application of the model may be helpful in radiotherapy target volume defi-nition despite these limitations. The model parameterizes features of glioma growththat are commonly agreed upon. The model can be viewed as a component in an au-tomatic GTV to CTV expansion tool, which defines a distance measure in the brainthat accounts for the patient specific anatomy. It allows treatment planners to incorpo-rate their belief on glioma growth characteristics in multiple stages: In the first stage,anatomical boundaries can be accounted for, which is achieved by segmentation of theventricles, falx cerebri and tentorium cerebelli. This is a promising application because,clearly, the falx represents a barrier for migrating tumor cells while the corpus callosumallows for inter-hemispheric spread of disease. In the second stage, reduced gray matterinfiltration can be incorporated by choosing D w /D g >
1. If integrated into contouringsoftware used in clinical practice, this tool can suggest an initial target volume to thephysician, which optionally can be modified manually.23
Conclusions
The Fisher-Kolmogorov equation represents a model for the spatial distribution of infil-trating tumor cells in normal appearing brain tissue. It can be applied in radiotherapytarget delineation by defining target volumes as isolines of the tumor cell density. Themodel can incorporate spatial growth characteristics of infiltrating gliomas: the effectof anatomical barriers and a reduced tumor infiltration in gray matter. Therefore, thetumor growth model can provide the basis for an automatic contouring tool that pro-vides the option to account for widely agreed growth patterns of glioma. The approachappears particularly useful for tumors located close to the falx and the corpus callosum.The most crucial input to the model is the segmentation of the brain tissue, in particularthe reliable segmentation of the anatomical barriers falx cerebri and tentorium cerebelli.
A Target volume comparison for different tumor lo-cations
In section 3.6, we summarized the results for model based target delineation for 10 GBMpatients that were analyzed. In this appendix, we provide details on four representativecases, which span a range of different tumor locations, i.e. GBMs in the parietal lobe,temporal lobe, frontal lobe, and bilateral corpus callosum. The Dice coefficients betweenthe manual and model-derived target volumes are summarized in table 1, together withthe corresponding Dice coefficient between the 95% isodose lines of the IMRT plans. (a) (b)
Figure 8: Multifocal GBM located in the left temporal and frontal lobe: (a) coronal T1post contrast image; (b) axial T1 post contrast image. The manually drawn CTV isshown in green, the model derived CTV in red.24 −14−12−10−8−6−4−20 (a) cell density (b) model-derived target (c) manual target −30−20−100102030 (d) dose difference
Figure 9: (a) Simulated tumor cell density for the patient shown in figure 8; (b-c)Comparison of IMRT dose distributions (in units of Gy) of the plan based on manualtarget volume (c) and the model-derived target volume (b). The difference of the dosedistributions is shown in (d). 25 .1 Multifocal temporal/frontal lobe case
Figure 8 shows a multifocal GBM case involving the left temporal lobe as well as thefrontal lobe. Figure 8a shows the coronal T1 post contrast image, revealing the contrastenhancing tumor mass in the temporal lobe. The axial T1 post contrast image in figure8b shows the lesion in the frontal lobe. The simulated tumor cell density in figure9a illustrates several features discussed in sections 3 and 4: The tumor growth modeldescribes the steep fall-off of the tumor cell density in gray matter, leading to differencesin the target volume around the lateral sulcus (figure 8b). In addition, modeling tumorcell infiltration through the corpus callosum leads to differences in the target volumein the contralateral frontal lobe. Figure 9 shows the IMRT plan comparison for ahomogeneous 60 Gy prescription to the manual CTV (c) and the model-derived CTV(b). The figure shows that the differences in the target volume partly translate intodifferences in the dose distribution. In particular, the dose difference plot in figure 9dshows a lower dose in the lateral sulcus region for the model-based plan, and a higherdose in the contralateral hemisphere close to the corpus callosum.
A.2 Parietal lobe case
Figure 10a shows the T2-FLAIR image of a GBM located in the left parietal lobe.Compared to the case discussed in section 3, the tumor is located more posteriorly andcloser to the falx. The simulated tumor cell density for D w /D g = 100 is shown in figure10b. Similar to the case discussed in section 3, the tumor growth model simulates theinfiltration of the contralateral side via the corpus callosum. The comparison of modelderived target volume (red) to the manually drawn CTV (green) in figure 10a revealsthat the tumor growth model suggests further spread into the contralateral hemisphere. (a) (b) Figure 10: GBM located in the left parietal lobe adjacent to the falx: (a) T2-FLAIRimage with manual CTV (green) and model derived CTV (red); (b) simulated tumorcell density. 26 .3 Temporal lobe case
Figure 11 shows a GBM located in the left temporal lobe. As for the other patients,the ventricles and the falx represent a boundary for infiltrating tumor cells, whereasthe corpus callosum allows for contralateral spread. However, in this case, the tumoris located relatively far from the corpus callosum. Therefore, the model based targetcontour does not differ significantly from the manually drawn CTV in the region aroundthe midsagittal plane (figure 11b). The largest differences between manual and modelbased CTV occur in the regions near the lateral sulcus, where the tumor growth modeldescribes the sulcus with its surrounding gray matter as a boundary (figures 11a/b).Note that the left lateral sulcus is shifted anteriorly due to the tumor with mass effect. (a) (b) (c)
Figure 11: GBM located in the left temporal lobe: (a) T1 post contrast image; (b)T2-FLAIR image; (c) simulated tumor cell density. Note that (b) and (c) represent thesame slice whereas (a) is located 7.5 mm inferiorly. The manually drawn CTV is shownin green, the model derived CTV in red.
A.4 Corpus callosum case
Figure 12 shows a GBM case involving the bilateral corpus callosum. The simulatedtumor cell density in 12c models the fall-off of the tumor cell density in the corticalgray matter surrounding the sulci. This is most prominent in the proximity of majorsulci including the lateral sulcus and occipital lobe. The effect of reduced gray matterinfiltration leads to differences of up to approximately 1 cm between manual and modelderived CTV. In regions where the CTV extends all the way to the dura (see e.g. rightposterior region in figure 12a), the differences are insignificant because of the limiteddose gradients that can be achieved with therapeutic photon beams. However, in severalregions, differences in the CTV contours translate into dose differences in an IMRT plan.This is visible in figure 12b, where the red contour is further expanded into the major27ber tracts of the corona radiata, but the ventricles and regions in the occipital lobe arespared. (a) (b) (c)
Figure 12: GBM located in the corpus callosum: (a) T1 post contrast image; (b) T2-FLAIR image; (c) simulated tumor cell density. Note that (b) and (c) represent thesame slice whereas (a) is located 12.5 mm superiorly. The manually drawn CTV isshown in green, the model derived CTV in red.
Acknowledgment
The project was supported by the Federal Share of program income earned by Mas-sachusetts General Hospital on C06 CA059267, Proton Therapy Research and TreatmentCenter. Additional support was provided by the ERC Advanced Grant MedYMA.
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