Radiotherapy planning for glioblastoma based on a tumor growth model: improving target volume delineation

The Fisher–Kolmogorov model has been found to reproduce a wide range of glioma growth characteristics at a macroscopic scale. For a review regarding the development of the model see for example (Harpold et al 2007). The mathematical properties of the Fisher–Kolmogorov equations are discussed in detail in Murray (2002) and Murray (2003). A clinical application involves the personalization of the model to the patient specific anatomy (see e.g. Angelini et al (2007) for a review). In addition, approaches to estimate of model parameters from imaging data have been proposed (Harpold et al 2007, Konukoglu et al 2010a, Menze et al 2011), which are based on estimating the velocity of tumor growth and the steepness of the cell density fall-off. The velocity of tumor growth can be estimated from a sequence of MRI images (Konukoglu et al 2010a); estimating the steepness of the tumor density fall-off is based on comparing the size of the contrast enhancing tumor core to the size of the peritumoral edema. These observable quantities relate to the generic model parameters proliferation rate and diffusion coefficient. The model has, in particular, been used in life time prediction (Swanson et al 2008, Murray 2012). In addition, several extensions of the model have been suggested. Those extensions integrate DTI (Jbabdi et al 2005, Clatz et al 2005), tumor hypoxia (Gu et al 2012), and the effect of radiotherapy (Rockne et al 2010) and chemotherapy on the evolution of the tumor cell density (Tracqui et al 1995, Swanson et al 2002). The idea of utilizing the model for target definition in radiotherapy has been suggested by several authors (Cobzas et al 2009, Konukoglu et al 2010b, Bondiau et al 2011), who propose ways of describing tumor infiltration beyond the visible margins of the tumor. Cobzas (Cobzas et al 2009) defines a distance metric in the brain based on DTI to describe the anisotropy in tumor growth. Konukoglu (Konukoglu et al 2010b) and Bondiau (Bondiau et al 2011) uses a traveling wave approximation of the full reaction-diffusion model to estimate the spatial distribution of tumor cells in normal appearing brain tissue.

Even though the Fisher–Kolmogorov model has previously been studied in the mathematical biology and modeling community, an application for target delineation in clinical practice requires further research in multiple directions. Our work uses the previously published Fisher–Kolmogorov model and aims at characterizing the potential of the model for a true clinical application in radiotherapy treatment planning. We aim at developing a semi-automatic contouring tool for gliomas that is practically useful. The contributions of this paper are as follows.

• Systematic review of model assumptions. In section 2 we provide a review of the tumor growth model in order to systematically compile the assumptions made to apply the model to target delineation. This includes (1) highlighting the qualitative features of the model that are relevant for radiotherapy planning, (2) identify the model parameters determining those features, and (3) illustrate how the imaging data is used to personalize the model for an application to the patient at hand.
• Use case characterization. We provide a clear illustration of use cases of the tumor growth model for target delineation. In section 3 we perform a detailed case study for a tumor located in proximity to falx and corpus callosum, a situation where it is difficult to consistently account for anatomical boundaries in a manual delineation. In the appendix, we show results for four additional cases with varying tumor location. The presented results illustrate the main findings from a study involving ten glioblastoma patients previously treated at our institution.
• Sensitivity analysis. We discuss the sensitivity of the simulated tumor cell density with respect to the model inputs (section 3.5). In particular, we highlight the need for accurate brain segmentation into white matter, gray matter and anatomical barriers.
• IMRT planning. We compare radiotherapy plans optimized for intensity-modulated radiotherapy (IMRT) for model based versus manual target delineation (section 4). We illustrate to what degree differences in target delineation are translated into differences in the delivered dose distribution.

Furthermore, for one patient we compare the model predicted tumor cell density to the spatial growth patterns of the recurrent tumor (section 5). Finally, section 6 discusses the scope of the work and the potential of a practical application of the tumor growth model for target delineation.

The tumor growth calculations are based on MR images routinely acquired in clinical practice. The images incorporated into the simulation process are T1, T2, T2-FLAIR, and T1 post gadolinium. The T1 post-gadolinium image shows the vascularized GTV, and the T2-FLAIR image shows the surrounding edematous region. For illustrative purpose, these two images are shown in figure 1 for a case discussed in detail in this paper.

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In the first data processing step, all MR sequences are registered to the image with highest spatial resolution. This is done using rigid registration with six degrees of freedom, utilizing the function FLIRT (Jenkinson and Smith 2001) as part of the toolbox FSL (Smith et al 2004, Woolrich et al 2009). In the second step, a segmentation of the brain was obtained using the multimodal brain tumor segmentation algorithm published in Menze et al (2010). The algorithm is an expectation-maximization (EM) based segmentation method, which uses a probabilistic normal tissue atlas as spatial tissue prior. For every voxel, it estimates the posterior probability for three normal tissue classes (white matter, gray matter, and CSF⁵), as well as the lesion outlines on T1 post gadolinium and T2-FLAIR. The result of the EM segmentation is augmented as described in Unkelbach et al (2012) in order to facilitate a reliable identification of falx cerebri and tentorium cerebelli as anatomical barriers. The segmentation result for the patient in figure 1 is shown in figure 2. In the last step, the reference MR image is registered to the radiotherapy planning CT using rigid registration. The transformation matrix is saved and later applied to register the simulated tumor cell density to the planning CT.

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It is assumed that tumor growth is described by two processes: local proliferation of tumor cells and migration of cells into neighboring brain tissue. Mathematically, this is formalized via the Fisher–Kolmogorov equation, a partial differential equation of reaction-diffusion type for the tumor cell density $c(\boldsymbol {r},t)$ as a function of location $\boldsymbol {r}$ and time t:

$$\begin{equation} \frac{\partial }{\partial t}c(\boldsymbol {r},t)=\nabla \cdot ( D(\boldsymbol {r}) \nabla c(\boldsymbol {r},t) ) + \rho c(\boldsymbol {r},t) \left( 1-\frac{c(\boldsymbol {r},t)}{c^{\rm max}} \right) , \end{equation} \tag{ 1 }$$

where ρ is the proliferation rate which is assumed to be spatially constant, and $D(\boldsymbol {r})$ is the 3 × 3 diffusion tensor which depends on location $\boldsymbol {r}$. The first term on the right-hand side of equation (1) is the diffusion term that models tumor cell migration into neighboring tissue. The second term is a logistic growth term that describes tumor cell proliferation. The tumor cell density $c(t,\boldsymbol {r})$ takes values between zero and the carrying capacity c^max. In this paper, the diffusion tensor is constructed as

$$\begin{equation} D(\boldsymbol {r}) = \left\lbrace \begin{array}{@{}cc@{}}D_w \cdot I & \boldsymbol {r} \in \mbox{white matter} \\ D_g \cdot I & \boldsymbol {r} \in \mbox{gray matter} \end{array} \right. \end{equation} \tag{ 2 }$$

where I is the 3 × 3 identity matrix, and D_g and D_w are scaling coefficients for gray and white matter, respectively. This construction of the diffusion tensor allows for the modeling of reduced gray matter infiltration via a different diffusion coefficient D_g < D_w. Anatomical constraints are accounted for by boundary conditions. We assume that tumor cells only spread in brain tissue and do not diffuse into any tissue that is not segmented as gray or white matter⁶.

The asymptotic solution of the Fisher–Kolmogorov equation is given by a traveling wave solution. At the time of diagnosis, the tumor cell density takes high values close to the carrying capacity c^max within the enhancing tumor mass seen on the post-gadolinium T1 image. This is motivated by the observation that resected tissue from this area appears as only tumor tissue in a pathological examination (Coons 1999). At large distance from the enhancing core, the tumor cell density approaches zero. It can be shown that, within the traveling wave approximation, the following two properties hold.

• (1)  
  The fall-off of the cell density with distance from the enhancing core is described by a characteristic profile that remains approximately fixed while the tumor grows (Murray 2002, Konukoglu et al 2010b). At some distance, the tumor cell density drops approximately exponentially with distance from the visible tumor mass. For a region comprising white matter, the cell density behaves according to

  $$\begin{equation} c(\vert \boldsymbol {r} \vert ) \propto \exp \left( -\frac{\vert \boldsymbol {r} \vert }{\lambda _w} \right) , \end{equation} \tag{ 3 }$$

  where $\vert \boldsymbol {r} \vert$ denotes distance from the visible tumor. The parameter λ_w is called the infiltration length and denotes the distance at which the cell density drops by a factor of e. It can be shown that λ_w relates to the model parameters via

  $$\begin{equation} \lambda _w=\sqrt{\frac{D_w}{\rho }} . \end{equation} \tag{ 4 }$$

  Within gray matter the cell density drops with an infiltration length $\lambda _g=\sqrt{\frac{D_g}{\rho }}$ 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 constant velocity. The velocity relates to the model parameters via

  $$\begin{equation} v_w=2\sqrt{D_w \rho } . \end{equation} \tag{ 5 }$$

In a realistic brain geometry, the propagation of the wave is spatially modulated by anatomical boundaries and by different velocities in gray and white matter.

In order to apply the model to an individual patient, we have to specify how the tumor cell density resulting from the model relates to the MR images. The contrast enhancing region visible in the T1 post-gadolinium MR image is related to a breakdown of the blood-brain-barrier. Even though this only represents a surrogate for the tumor, it is commonly assumed that contrast enhancing volume corresponds to the vascularized, highly cellular part of the gross tumor. For this paper, we therefore assume that the boundary of the contrast enhancing region corresponds to an isoline of the tumor cell density close to the carrying capacity. It is used as the basis for tumor density calculations is described in subsection 2.6.

In current clinical practice, radiotherapy planning for GBM is mostly based on structural MRI imaging (T1 post gadolinium and T2-FLAIR). However, modern imaging modalities including MR spectroscopy (MRS) and positron emission tomography (PET) have been used for glioblastoma imaging and bear potential to improve the characterization of gross and infiltrative disease. In particular, tracers visualizing amino acid metabolism have been studied (Grosu et al 2005, Miwa et al 2004, Niyazi et al 2011, Götz and Grosu 2013), including FET (fluoro-ethyl-tyrosine) and MET (methionine). These PET scans are not routinely acquired for glioblastoma patients at our institution. However, if these modalities become established, the tumor growth model could be initialized using a PET based delineation of the tumor instead of using the contrast enhancing region.

For treatment planning, we want to calculate the tumor cell density in the brain at the time the MR images are taken. Using the Fisher–Kolmogorov equation directly is problematic in that context because the initial condition, which leads to a tumor consistent with the images, is unknown. One approach to circumvent this problem consists of 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 to an isoline of the tumor cell density. For this paper, we assume a value of 70% of the carrying capacity. Starting at the outline, the tumor cell density is extrapolated into the 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 Konukoglu et al (2010b).

The tumor growth model depends on four parameters: the diffusion coefficients in white 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 of parameters can be reduced to two crucial parameters.

The carrying capacity c^max can be considered as a scaling factor for the relative normalized tumor cell density that takes values between 0 and 1. It is straightforward to see that all results presented in this paper are independent of the value of c^max and only depend on the relative tumor cell density. We therefore do not have to consider the value of the carrying capacity c^max.

We are left with the model parameters D_w, D_g and ρ. Within the traveling wave approximation, D_w and ρ can be expressed through (1) the velocity of the tumor front in white matter obtained from the product of D_w and ρ according to equation (5), and (2) the infiltration length in white matter obtained from the ratio of D_w and ρ according to equation (4). In this paper, we are only interested in obtaining the distribution of tumor cells at the time when treatment starts. Thus, we do not require the velocity of the tumor front but only the infiltration length $\lambda _w = \sqrt{D_w/ \rho }$. Since the tumor primarily grows in white matter, we focus on the infiltration length in white matter instead of gray matter. If we express the diffusion coefficient in gray matter via the ratio of white and gray matter coefficient, we are left with two parameters of the tumor growth model:

• $\lambda _w = \sqrt{\frac{D_w}{\rho }}$ : infiltration length, describing how fast the tumor cell density drops with distance from the visible tumor volume.
• D_w/D_g : ratio of diffusion coefficients in white matter and gray matter, parameterizing in parts the anisotropy of tumor growth. The ratio of the infiltration lengths is consequently given by $\lambda _w /\lambda _g = \sqrt{D_w / D_g}$.

In this paper, we are concerned with the spatial definition of the target volume as an isoline of the tumor cell density. In that context, the exact value of the infiltration length λ_w is irrelevant because it does not affect the shape of the isolines. Only the absolute values of the tumor cell density associated with the isolines are determined by λ_w⁷. Thus, the only relevant model parameter for this paper is the ratio D_w/D_g, which determines the shape of the isolines of the cell density together with the brain segmentation. The literature consistently suggests that tumor cells infiltrate gray matter much less than white matter⁸. This suggests a large value for D_w/D_g ≫ 1. Most illustrations in this paper were obtained for D_w/D_g = 100. The most appropriate value is however uncertain and we discuss the impact of uncertainties in D_w/D_g in section 3.5.

Figure 4 shows the CTV (dark green contour) and the boost volume (green contour) drawn manually by the physician, i.e. these contours were the basis for the treatment plan that was actually delivered to the patient. The prescribed dose to the boost volume was 60 Gy, the prescribed dose to the CTV was 46 Gy. The boost volume 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 volumes were subsequently trimmed manually to account for anatomical boundaries (dura, ventricles, falx, and tentorium cerebelli). The dose distribution of the 3D conformal treatment plan delivered to the patient is shown in figure 6(d).

Using the tumor growth model, the target volume can be defined as an isoline of the tumor cell density. This is illustrated here for the tumor cell density shown in figure 3(a) obtained for the parameter value D_w/D_g = 100. The red and the orange contours in 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 total enclosed volume is equal to the manually delineated target. In the following two subsections, we discuss the differences between manual and model-derived target volumes in detail.

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In the manual delineation of the CTV used in the clinical plan, it is incorporated that the falx represents an anatomical barrier for the migration of tumor cells. Hence, the isotropic target expansion was trimmed manually. In the tumor growth model, the falx is modeled via a layer of CSF and is automatically accounted for through the assumption that tumor cells only migrate within white and gray matter. However, the corpus callosum connects the two hemispheres of the brain. The tumor growth model describes the migration of tumor cells through the corpus callosum (see figure 3 and 5(a)). As a consequence, the target volume based on the growth model is extended into the contralateral hemisphere. Figure 5(a) shows the tumor cell density overlaid on the coronal T1 gadolinium image. This illustrates the three-dimensional modeling of tumor spread via the model, including areas superior to the corpus callosum. This is not consistently accounted for in the manual CTV. In the manually drawn target volumes, the target is slightly extended into the contralateral hemisphere on the slices that show the corpus callosum, but not on the slices located superiorly and inferiorly (figure 5(b)). In the model-derived target volumes, the target is extended further into the contralateral hemisphere, and the spread of tumor cells in superior–inferior direction beyond the corpus callosum is modeled.

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It is typically assumed that tumor cells infiltrate gray matter much less than white matter, which is modeled by D_w/D_g ≫ 1. In the simulated tumor cell density in figure 3, 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 because the thickness of the cortical layer is only several millimeters. This can be seen in figure 3 in the posterior portion of the tumor. Here, the dura is essentially the boundary of the target volume. Technically, some gray matter surrounding the fissures is outside of the target defining isoline. However, these regions are small and are not relevant for radiotherapy planning as further discussed in section 4. For this patient, the most pronounced 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 the lateral sulcus (left-anterior part of the tumor). In this region, the large amount of gray matter surrounding 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 tissue that can be taken out of the target volume.

In order to quantitatively compare the difference of manual and model-derived target volumes, we calculate the Dice coefficient, which is given by the volume where both structures overlap, divided by the average volume of the two structures. For the boost and 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 a substantial portion of approximately one quarter is different.

The model input parameters that determine the spatial distribution of tumor cells are the segmentation of the brain (into white matter, gray matter and CSF) as well as the parameter D_w/D_g.

Brain segmentation

Gray matter infiltration parameter

Interdependence

We retrospectively analyzed ten patients with varying tumor location. All patients were treated with IMRT to a dose of 60 Gy in 30 fractions. For the clinically delivered treatment 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 where there were clear anatomical barriers to spread such as the falx or dura. A PTV expansion of 3 mm was added. For all patients, the tumor cell density is calculated for a parameter value of D_w/D_g = 100 based on the method described in section 2. The model-based target volume is defined as the isoline that encompasses the same total volume as the manually delineated CTV used in the clinically applied plan. The main findings are summarized as follows.

• The modeling of anatomical barriers leads to differences in the target volume for tumors located close to corpus callosum and falx cerebri. The use of the tumor growth model suggests a further expansion of the target into the contralateral hemisphere.
• The modeling of reduced gray matter infiltration leads to differences in the target volume for tumors located in proximity to major sulci. This effect was most pronounced in the region of the lateral sulci where the use of the tumor growth model suggests regions that can be excluded from the target volume.
• Dice coefficients between manual and model-derived target volumes were calculated for all ten patients (table 1). The overlap between manual and model-based target volumes was 79% on average, ranging from 74% to 84%. This indicates that approximately 20% of the manual target volume is not contained in the model-based volume (and vice versa).
• For all patients IMRT treatment plans were optimized using a homogeneous 60 Gy prescription dose to the CTV and the planning formulation in section 4. Dice coefficients for the 95% isodose lines (57 Gy) were calculated in order to compare the differences in the high dose region between manual and model-derived radiation plans (table 1). The Dice coefficient was 0.84 on average, ranging from 0.80 to 0.87, indicating that a substantial part of the differences in target volume also translates into differences in the high dose region.

In the appendix, we provide details for four representative cases. These four cases span a range of different tumor locations, i.e. GBMs in the parietal lobe, temporal lobe, frontal lobe, and bilateral corpus callosum.

In figure 4 we have defined the model-derived target contours such that the overall volumes of the targets equal the volumes of the targets in the clinical plan. This has been done for the sake of comparison of the shapes of the target volumes. In a practical scenario, the model is used for target definition without the existence of a manually drawn reference volume. In this case, the choice of the target defining isoline can be performed by either specifying the target expansion in white matter, or by specifying the average expansion.

• (1)  
  In the first scenario, the treatment planner is provided with the simulated tumor cell density together with the delineation of enhancing core and edema as well as the brain segmentation. The treatment planner can then manually pick a target defining isoline by measuring the distance to the enhancing core along major sheets of white matter.
• (2)  
  In the second case, a temporary reference volume can be created automatically as an isotropic expansion around the enhancing core, intersected with the brain tissue mask. The model-derived target volume can then be defined as the isoline that encompasses the same total volume as the temporary reference volume. In this case, the treatment planner has to specify the average margin added to the enhancing core.

Ideally, we would like to have a way to validate the predictions of the growth model regarding the tumor cell density. Histopathological analysis of resected brain tissue would represent one way to validate the extent of tumor cell infiltration predicted by the model. However, it is currently not realistic to obtain such data for untreated human patients at high spatial resolution. Tissue samples taken from biopsies or resected tumors may provide information at selected locations. Usually this is, however, limited to the GTV; taking biopsy samples from healthy appearing brain tissue, which would be relevant for target delineation, is not performed.

An alternative approach towards collecting evidence for the plausibility of the growth model consists in evaluating the spatial progression patterns of recurrent tumors in follow-up MRI images. The advantage of this approach is that this data is widely available since high grade glioma patients typically receive follow-up MR imaging every 2–3 months. On the downside, model validation based on follow-up imaging has limitations. This includes the difficulty in distinguishing recurrent tumor from other radiation induced changes (e.g. radiation necrosis), and the large variation in response to radiation between patients. In addition, deformations due to surgery, tumor mass effects, and radiation related brain atrophy may occur. Addressing these issues in their complexity is outside the scope of this paper. Nevertheless, we hypothesize that the analysis of progression patterns of recurrent tumors can provide some degree of evidence for the plausibility of the tumor growth model. Due to the variability in treatment and treatment 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 radiotherapy without significant tumor resection. Also, the patient did not receive additional cycles of chemotherapy or antiangiogenic drugs after the initial course of chemoradiation with concurrent temozolomide. Figure 7 shows two follow-up T2-FLAIR images for this patient. The follow-up images were registered to the reference image using rigid registration. The image in figure 7(a) was taken 3.5 months after the diagnostic image (figure 1) and 2 months after completion of radiotherapy. The image in figure 7(b) was taken 7.5 months after the diagnostic image, which was the last image taken before the patient expired. Even though there is no histopathological confirmation, the radiological appearance and the patient's death about 9 months post therapy suggest progressive disease rather than other radiation induced changes. Thus, for the remainder of this section, we make the assumption that the follow-up imaging shows the progression of the tumor. Furthermore, the tumor recurs/progresses centrally, which is the case for the vast majority of cases given standard of care treatments. Figure 6(d) shows the dose distribution of the clinically delivered 3D conformal treatment plan.

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By considering the delivered dose distribution in figure 6(d), we see that the progressive tumor is located almost entirely within the high dose region receiving approximately 60 Gy. Under the assumption that a radiation dose of 60 Gy kills a fraction of the tumors cells, the shape of the isolines after therapy is not altered within the high dose region. In this case, the Fisher–Kolmogorov model predicts that the progression of the tumor approximately follows the isolines of the tumor cell density simulated pre-therapy. This motivates to compare the simulated cell density isolines directly to the follow-up images without explicitly modeling the effect of chemoradiation on the cell density quantitatively.

In figure 7 we overlaid the tumor cell density simulated on the diagnostic image on the follow-up T2-FLAIR images. It is seen that the progression of the tumor post therapy follows 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 callosum into the contralateral hemisphere as predicted by the model. The latter is supported by the T1 post contrast images 3.5 and 7.5 months post therapy, which show contrast enhancement in the corpus callosum (T1 images not shown).

In this work, we aim at improving and automating target volume delineation for glioblastoma. We investigate the use of a phenomenological tumor growth model based on a reaction-diffusion equation for the tumor cell density. The model has been studied in the literature over the past years. Here, we aim at bringing this model closer to an application in radiotherapy treatment planning. The problem addressed here originates from the infiltrative growth characteristics of glioma and the fact that tumor cells infiltrate brain tissue beyond the abnormality visible on current imaging modalities. In clinical practice, an approximately 2 cm wide margin is added around the T2-FLAIR abnormality. It is however known that glioma growth is not isotropic. Tumor growth patterns are influenced by anatomical boundaries as well as reduced infiltration of gray matter. Hence, adding an isotropic margin will unnecessarily expose normal tissue or miss areas of concern. Anatomical constraints can partly be accounted for in the manual 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 to achieve manually.

We performed a retrospective study involving ten GBM patients treated previously at our institution. For these patients, we compare manually drawn target volumes used in the clinically delivered treatment plan to model-derived target volumes defined through an isoline of the tumor cell density. One goal of the study is to characterize the best use cases for model based target delineation, i.e. we identify anatomical situations in which the growth model leads to more consistent target volumes compared to manual delineation. As demonstrated in the paper, this may be the case for tumors located in proximity to falx and corpus callosum. In this situation, the falx represents a boundary for migrating cells, but the corpus callosum provides a route for the tumor to spread to the contralateral hemisphere. The tumor growth model represents a tool to automate and objectify target definition for such cases, which is difficult to achieve manually. In addition, it is demonstrated that modeling of reduced infiltration of gray matter may influence target delineation near major sulci, where large accumulations of gray matter surrounding 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 optimizing IMRT treatment plans based on model-derived and manually drawn target volumes, it is demonstrated that the differences in target delineation may translate into differences in the delivered dose distribution.

Technical versus clinical goal

The role of model validation

Figure A1 shows a multifocal GBM case involving the left temporal lobe as well as the frontal lobe. Figure A1(a) shows the coronal T1 post contrast image, revealing the contrast enhancing tumor mass in the temporal lobe. The axial T1 post contrast image in figure A1(b) shows the lesion in the frontal lobe. The simulated tumor cell density in A2 (a) illustrates several features discussed in sections 3 and 4: the tumor growth model describes the steep fall-off of the tumor cell density in gray matter, leading to differences in the target volume around the lateral sulcus (figure A1(b)). In addition, modeling tumor cell infiltration through the corpus callosum leads to differences in the target volume in the contralateral frontal lobe. Figure A2 shows the IMRT plan comparison for a homogeneous 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 into differences in the dose distribution. In particular, the dose difference plot in figure A2(d) shows a lower dose in the lateral sulcus region for the model-based plan, and a higher dose in the contralateral hemisphere close to the corpus callosum.

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Figure A3(a) 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 and closer to the falx. The simulated tumor cell density for D_w/D_g = 100 is shown in figure A3(b). Similar to the case discussed in section 3, the tumor growth model simulates the infiltration of the contralateral side via the corpus callosum. The comparison of model-derived target volume (red) to the manually drawn CTV (green) in figure A3(a) reveals that the tumor growth model suggests further spread into the contralateral hemisphere.

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Figure A4 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, whereas the corpus callosum allows for contralateral spread. However, in this case, the tumor is located relatively far from the corpus callosum. Therefore, the model based target contour does not differ significantly from the manually drawn CTV in the region around the midsagittal plane (figure A4(b)). The largest differences between manual and model based CTV occur in the regions near the lateral sulcus, where the tumor growth model describes the sulcus with its surrounding gray matter as a boundary (figures A4(a), (b)). Note that the left lateral sulcus is shifted anteriorly due to the tumor with mass effect.

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Figure A5 shows a GBM case involving the bilateral corpus callosum. The simulated tumor cell density in A5(c) models the fall-off of the tumor cell density in the cortical gray matter surrounding the sulci. This is most prominent in the proximity of major sulci including the lateral sulcus and occipital lobe. The effect of reduced gray matter infiltration leads to differences of up to approximately 1 cm between manual and model-derived CTV. In regions where the CTV extends all the way to the dura (see e.g. right posterior region in figure A5(a)), the differences are insignificant because of the limited dose gradients that can be achieved with therapeutic photon beams. However, in several regions, differences in the CTV contours translate into dose differences in an IMRT plan. This is visible in figure A5(b), where the red contour is further expanded into the major fiber tracts of the corona radiata, but the ventricles and regions in the occipital lobe are spared.

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