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Personalized Radiotherapy Planning Based on a Computational Tumor Growth Model

TL;DR: Two methods to derive the radiotherapy prescription dose distribution are introduced, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density and it is shown how the method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration.
Abstract: In this article, we propose a proof of concept for the automatic planning of personalized radiotherapy for brain tumors. A computational model of glioblastoma growth is combined with an exponential cell survival model to describe the effect of radiotherapy. The model is personalized to the magnetic resonance images (MRIs) of a given patient. It takes into account the uncertainty in the model parameters, together with the uncertainty in the MRI segmentations. The computed probability distribution over tumor cell densities, together with the cell survival model, is used to define the prescription dose distribution, which is the basis for subsequent Intensity Modulated Radiation Therapy (IMRT) planning. Depending on the clinical data available, we compare three different scenarios to personalize the model. First, we consider a single MRI acquisition before therapy, as it would usually be the case in clinical routine. Second, we use two MRI acquisitions at two distinct time points in order to personalize the model and plan radiotherapy. Third, we include the uncertainty in the segmentation process. We present the application of our approach on two patients diagnosed with high grade glioma. We introduce two methods to derive the radiotherapy prescription dose distribution, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density. We show how our method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration. We further present extensions of the method in order to spare adjacent organs at risk by re-distributing the dose. The presented approach and its proof of concept may help in the future to better target the tumor and spare organs at risk.

Summary (3 min read)

Introduction

  • Previous works on computational growth models for gliomas have focused on reaction-diffusion equations to model cell proliferation and infiltration into surrounding brain tissue [1].
  • In order to account for the infiltrative nature of the tumor, several studies recently proposed to personalize radiotherapy planning based on a computational growth model.
  • They showed that personalizing the delivered dose could improve therapy in terms of days gained by the patients.
  • In the third scenario, the authors include the uncertainty in the segmentation of the abnormality visible on the different MRIs to the personalization strategy.

II. SEGMENTATION SAMPLES

  • The T1Gd abnormality, which is the active part of the tumor, and the larger T2-FLAIR abnormality, which is usually called the edema, were segmented by a clinician.
  • In order to take into account the uncertainty in the segmentation, the authors propose to randomly modify the original clinician segmentations.
  • The samples are efficiently produced on the regular grid using the separability and stationary properties of the squared exponential covariance function (see [18] for details).
  • Segmentation samples for the T1Gd and T2-FLAIR abnormalities at the first and second time points are generated.
  • One could compare the output of the tumor growth model with probabilistic segmentation approaches which have been proposed for glioblastoma [20].

III. TUMOR GROWTH MODEL

  • Below, the authors identify the scalar parameter dw with D. The solution of the reaction-diffusion equation (1) is a tumor cell density u computed over the whole brain domain.
  • The method is based on the assumption that the solution of equation (1) at the first time point has converged to its asymptotic, traveling wave type solution.
  • Thereby, the tumor cell density is propagated outward (and inward), starting from the T1Gd segmentation, and drops approximately exponentially with distance.
  • The reaction-diffusion equation is solved using the Lattice Boltzmann Method [21], [23], [24] which allows for easy parallelization and fast computations.
  • The common approach taken in these works is to add a death term to the reaction-diffusion equation, which allows to model the shrinkage of the tumor due to the therapy.

IV. PERSONALIZATION

  • The personalization of the tumor growth model is combined with a dose response model in order to define the radiotherapy planning.
  • First the authors only use a single time point (the second acquisition) to personalize the model such that the radiotherapy plan will be defined using a single acquisition, similarly to what is being done in clinic.
  • Second the authors use two time points in order to personalize the model.
  • The radiotherapy plan will then be defined on the latest acquisition.
  • Third, the authors use two time points and include the uncertainty in the segmentation.

A. Scenario 1: One time point only

  • The authors further model the prior as log-uniform and independent between the parameters, P (θ) = P (D)P (ρ) (4) The authors sample from the posterior distribution using a Metropolis-Hasting algorithm.
  • Note that this section only uses the initialization algorithm (see Section III) which only depends on the invisibility index λ = √ D/ρ.
  • Note that this section can be related to the method described in [16], where a single time point is used to propose a dose planning.
  • Moreover, the Bayesian methodology allows to take into account the uncertainty in the personalization.

V. RADIOTHERAPY PLANNING

  • The authors detail how they use the personalization of the tumor growth model in order to define the best radiotherapy plan at the time of the second acquisition.
  • The authors start by coupling the growth model with a cell survival model (Section V-A).
  • The authors then detail how to compute the prescription doses in Section V-B, and how to compute the delivered dose in Section V-C.

A. Cell survival

  • Cell survival after irradiation is often modeled using the linear-quadratic model.
  • The authors follow the derivations of [16], and consider the linear approximation of the linear-quadratic model.

B. Prescription Dose Optimization

  • A prescription dose can be defined as the dose minimizing the surviving fraction of tumor cells.
  • The personalization of the tumor growth model provides samples {θl} from the posterior distribution P (θ|S).
  • The probabilistic dose is defined as the dose minimizing the expectation of the surviving fraction of tumor cells.

VI. RESULTS

  • The authors first present the results for one high grade glioma patient.
  • For scenarios 2 and 3, 2000 samples are drawn from the posterior distribution, leading to an acceptance rate of 60%.
  • Including the second time point, and the uncertainty in the segmentation, increases the uncertainty in the invisibility index.
  • Those two figures highlights the two sources of uncertainty for scenario 3: the segmentation, and the tumor infiltration.
  • This can be more clearly observed by looking at the dose volume histograms on Figure 15.

VII. DISCUSSION

  • Extensions of the model could be considered in order to include the noise level σ, the thresholds τ1, and τ2, as free parameters of the model.
  • This level of noise allows to explore the parameter space, and to focus on a region of interest which is in accordance with the lowest distances found.
  • It was set manually after a few experiments.
  • The authors did not include it in this study for several reasons.
  • Second, this adds complexity and increases the computational cost of the method.

VIII. CONCLUSION

  • The authors presented the proof of concept for a method combining a computational model of tumor growth with a dose response model in order to optimize radiotherapy planning, which takes into account the uncertainty in the model parameters and the clinical segmentations.
  • In the third one, the authors include uncertainty in the segmentation process.
  • The MAP dose minimizes surviving tumor cells after irradiation of the most probable situation, while the probabilistic dose allows one to take into account the uncertainty by minimizing the expected surviving tumor cells.
  • To that end, the model should be extended in order to take into account the complex therapy the patient is undergoing.
  • Finally, it should be investigated if more conformal dose delivery techniques such as proton therapy lead to IMRT planning more conformal to the prescribed dose.

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Personalized Radiotherapy Planning Based on a
Computational Tumor Growth Model
Matthieu Lê, Hervé Delingette, Jayashree Kalpathy-Cramer, Elizabeth R
Gerstner, Tracy Batchelor, Jan Unkelbach, Nicholas Ayache
To cite this version:
Matthieu Lê, Hervé Delingette, Jayashree Kalpathy-Cramer, Elizabeth R Gerstner, Tracy Batche-
lor, et al.. Personalized Radiotherapy Planning Based on a Computational Tumor Growth Model.
IEEE Transactions on Medical Imaging, Institute of Electrical and Electronics Engineers, 2016, pp.11.
�10.1109/TMI.2016.2626443�. �hal-01403847�

JOURNAL SUBMISSION 1
Personalized Radiotherapy Planning
Based on a Computational Tumor Growth Model
Matthieu L
ˆ
e
1
, Herv
´
e Delingette
1
, Jayashree Kalpathy-Cramer
2
, Elizabeth R. Gerstner
3
,
Tracy Batchelor
3
, Jan Unkelbach
4
, Nicholas Ayache
1
Abstract—In this article, we propose a proof of concept
for the automatic planning of personalized radiotherapy for
brain tumors. A computational model of glioblastoma growth
is combined with an exponential cell survival model to describe
the effect of radiotherapy. The model is personalized to the
magnetic resonance images (MRIs) of a given patient. It takes
into account the uncertainty in the model parameters, together
with the uncertainty in the MRI segmentations. The computed
probability distribution over tumor cell densities, together with
the cell survival model, is used to define the prescription
dose distribution, which is the basis for subsequent Intensity
Modulated Radiation Therapy (IMRT) planning. Depending on
the clinical data available, we compare three different scenarios to
personalize the model. First, we consider a single MRI acquisition
before therapy, as it would usually be the case in clinical routine.
Second, we use two MRI acquisitions at two distinct time points in
order to personalize the model and plan radiotherapy. Third, we
include the uncertainty in the segmentation process. We present
the application of our approach on two patients diagnosed with
high grade glioma. We introduce two methods to derive the
radiotherapy prescription dose distribution, which are based on
minimizing integral tumor cell survival using the maximum a
posteriori or the expected tumor cell density. We show how our
method allows the user to compute a patient specific radiotherapy
planning conformal to the tumor infiltration. We further present
extensions of the method in order to spare adjacent organs at
risk by re-distributing the dose. The presented approach and its
proof of concept may help in the future to better target the tumor
and spare organs at risk.
Index Terms—Radiotherapy planning, computational tu-
mor growth model, personalization, uncertainty, segmentation,
glioblastoma
I. INTRODUCTION
H
IGH grade glioma is one of the most common and
aggressive types of primary brain tumors. The treatment
of high grade glioma usually involves resection when possible,
followed by concurrent chemotherapy and radiotherapy.
Previous works on computational growth models for
gliomas have focused on reaction-diffusion equations to model
cell proliferation and infiltration into surrounding brain tis-
sue [1]. The model has been extended to model response
to chemotherapy, surgical resection, and radiotherapy. For
instance, a sink term can be added to the reaction-diffusion
1
Asclepios Project, Inria Sophia Antipolis, France.
2
Martinos Center for Biomedical Imaging, Harvard-MIT Division of Health
Sciences and Technology, Charlestown, MA, USA.
3
Department of Neurology, Massachusetts General Hospital, Boston, MA,
USA.
4
Department of Radiation Oncology, Massachusetts General Hospital and
Harvard Medical School, Boston, MA, USA.
Figure 1. The clinical segmentation of the T1Gd abnormality (Top, orange
line) is used to define the clinical target volume (CTV, white dashed line) as a
2 cm expansion of the segmentation. In clinical settings, 60 Gy is prescribed
to the CTV. We propose to personalize the prescription dose (Bottom) to
account for tumor infiltration and segmentation uncertainty.
equation in order to model the impact of chemo or radio-
therapy [2], [3]. The resection of a brain tumor can also be
modeled by deleting the tumor cells in the resected region
[4], [5]. More advanced therapy schedules using for instance
anti-angiogenic drugs can also be studied with more complex
models [6], [7], [8].
In this article, we provide proof of concept of a method
for the automatic planning of personalized radiotherapy for
glioblastoma (Figure 1). The beneficial impact of radiotherapy
for glioblastoma patients has been clearly demonstrated [9],
[10]. However, its planning is made difficult by the infiltrative
nature of the disease, and the uncertainty in delineating the
abnormality in Magnetic Resonance Images (MRI). To account
for the tumor infiltration, a margin of 1 to 3 cm is added to the
abnormality visible on MRI to define the clinical target volume
(CTV) [11] (Figure 1). The exact extent of this margin is left
at the discretion of the clinician.

JOURNAL SUBMISSION 2
Figure 2. Summary of the method: the segmentation of the tumor on the different MRIs is used to personalize the tumor growth model. This is combined
with a dose response model to define the prescription dose. Finally, the delivered dose is optimized using 9 equally spaced coplanar photon beams. The color
code indicates which data is used for the different scenarios: one or two MRI acquisition at two different time points, the clinical segmentations or plausible
samples to take into account the segmentation uncertainty.
Figure 3. First time point on the left, second time point on the right. (Top)
The proliferative rim is outlined in orange on the T1Gd MRI. (Middle Top)
The edema is outlined in red on the T2-FLAIR MRI. The edema encloses
the proliferative rim. (Middle Bottom) Tumor cell density computed with the
reaction-diffusion model. The black (resp. white) line is the threshold values
τ
1
(resp. τ
2
) corresponding to the T1Gd (resp. T2-FLAIR) abnormality.
(Bottom) Comparison between the clinician segmentation and the contours
from the model.
In order to account for the infiltrative nature of the tumor,
several studies recently proposed to personalize radiotherapy
planning based on a computational growth model. Corwin et
al. [12], [13] personalized spherically symmetric doses based
on a 1D reaction-diffusion tumor growth model using the
T1Gd and T2-FLAIR abnormalities radius as observations
[14], [15]. In this framework, they showed that personalizing
the delivered dose could improve therapy in terms of days
gained by the patients. However, this spherically symmetric
assumption prevents taking into account boundaries of the
tumor progression such as the ventricles. Unkelbach et al.
[16], [17] studied the optimization of the radiotherapy planning
based on a tumor growth model in order to automatically
define realistic 3D prescription dose distributions, taking into
account the natural boundaries and privileged pathways of the
tumor progression. The proposed planning was personalized
to the patients geometry, but without personalizing the tumor
growth model parameters.
In this article, we extend previous works by personalizing
a 3D tumor growth model in order to define radiotherapy
prescription doses. This allows one to automatically compute
realistic 3D prescription doses conformal to the tumor infiltra-
tion (see Figure 1). Moreover, we study the impact of taking
into account the uncertainty in the different inputs of the model
(segmentations and model parameters). We use a tumor growth
model based on a reaction diffusion equation, which models
the infiltrative spread of tumor cells in the surrounding white
and gray matter. A Bayesian approach is taken to estimate
the posterior distribution over the model parameters based
on the MRIs of the patient. A recently proposed method to
sample plausible image segmentations is used to incorporate
uncertainty in the segmentation of the tumor in the MR
images [18]. The tumor growth model is then combined
with an exponential cell survival model to describe the effect
of radiotherapy. The probability distribution over tumor cell
densities, together with the cell survival model, is used to
define the prescription dose distribution, which is the basis
for subsequent Intensity Modulated Radiation Therapy (IMRT)
planning. The scope of this paper is the personalization of
radiotherapy planning. As such, we focus on patients which
were not treated with surgical resection. The proposed model
could however be extended in order to included the impact of
such therapy following the developments done in [4], [5].

JOURNAL SUBMISSION 3
In this article, we consider three different scenarios. In
the first one, we only consider a single MRI acquisition of
the T1Gd and T2-FLAIR MRI before therapy planning. This
scenario is the closest to the clinical setting where radiotherapy
planning is usually based on a single MRI acquisition. In the
second, we consider two MRI acquisition at two time points
for a total of four MRIs: the T1Gd and T2-FLAIR at the first
and second time point (see Figure 3). In the third scenario, we
include the uncertainty in the segmentation of the abnormality
visible on the different MRIs to the personalization strategy.
The second and third scenarios are proofs of concept of a
method to include additional information to the personalized
therapy pipeline. We acknowledge that patients are usually
subject to therapy between the two time points, and as such,
the growth model personalization is biased by the impact of
therapy. Note however that if the therapy does not result in a
decrease of the tumor volume, its impact is implicitly taken
into account in the personalization of the growth parameter.
Based on those different scenarios, we propose three prin-
cipled approaches to compute the prescription dose. First, we
minimize the surviving fraction of tumor cells after irradiation
for the most probable tumor cell density. Second, we minimize
the expected survival fraction tumor cells after irradiation.
Third, we present an approach to correct the prescription dose
to take into account the presence of adjacent organs at risk.
The generation of different plausible segmentations based
on the clinical ones is presented in Section II. The forward
model of tumor growth is presented in Section III. The
personalization method for the three different scenarios is
presented in Section IV. The three principled approach for
the personalization of the dose response model to define the
prescription dose and the IMRT is detailed in Section V. A
summary of the method is illustrated in Figure 2. To our
knowledge, this is the first work that uses a personalized model
of brain tumor growth taking into account the uncertainty in
tumor growth parameters and the clinician’s segmentations in
order to optimize radiotherapy planning.
II. SEGMENTATION SAMPLES
The T1Gd abnormality, which is the active part of the tumor,
and the larger T2-FLAIR abnormality, which is usually called
the edema, were segmented by a clinician. In order to take
into account the uncertainty in the segmentation, we propose
to randomly modify the original clinician segmentations. The
method is based on [18], where samples of such segmentations
are generated from a high dimensional Gaussian process,
as the zero crossing of a level function. The samples are
efficiently produced on the regular grid using the separability
and stationary properties of the squared exponential covariance
function (see [18] for details). The samples take into account
the image intensity information using the signed geodesic
distance as the mean of the Gaussian process.
Segmentation samples for the T1Gd and T2-FLAIR abnor-
malities at the first and second time points are generated. Let
S
0
i
denote the clinical segmentations for the T1Gd and T2-
FLAIR abnormalities at the first and second time points, where
the index i = 1, ..., 4 refers to the 4 available images (see
Figure 3). Let S
i
=
S
k
i
k=1,...,K
denote sets of K plausible
segmentations per modality and time point, where each S
k
i
is
a plausible sample from S
0
i
, the i-th clinician segmentation.
Figure 4 shows examples of such samples for K = 5. The
samples automatically respect the boundaries of the tumor
progression such as the ventricles, because of the presence
of large intensity gradients. The five presented samples per
abnormality correspond to an average DICE of 87%, which is
comparable to the inter-expert DICE measured in the BraTS
Challenge for brain tumors delineation [19]. Comparing the
output of the forward tumor growth model with these plausible
noisy segmentations allows to include the uncertainty of the
original clinician segmentations.
Note that other approaches could allow the handling of
segmentation uncertainty. For instance, one could compare the
output of the tumor growth model with probabilistic segmen-
tation approaches which have been proposed for glioblastoma
[20].
III. TUMOR GROWTH MODEL
The tumor growth model is based on the reaction-diffusion
equation,
u
t
= (D.u)
| {z }
Diffusion
+ ρu(1 u)
| {z }
Logistic Proliferation
(1)
Du.
n
= 0 (2)
Equation (1) describes the spatio-temporal evolution of the
tumor cell density u, which infiltrates neighboring tissues with
a diffusion tensor D, and proliferates with a net proliferation
rate ρ. Equation (2) enforces Neumann boundary conditions
on the brain domain . Following [21], we define the diffusion
tensor as D = d
w
I in the white matter, and D = d
w
/10 I in
the gray matter, where I is the 3x3 identity matrix. Below, we
identify the scalar parameter d
w
with D.
The solution of the reaction-diffusion equation (1) is a
tumor cell density u computed over the whole brain domain.
However, parts of the brain that glioblastomas usually do not
invade were excluded from the tumor simulation such as the
CSF or the cerebellum. In order to relate the tumor cell density
u to the MRIs, the frontier of the visible abnormalities is
assumed to correspond to a threshold value of the tumor cell
density u. We note τ
1
the value of the tumor cell density u
corresponding to the frontier of the T1Gd abnormality, and
τ
2
the value corresponding to the frontier of the T2-FLAIR
abnormality (see Figure 3).
The initialization of the tumor cell density u(t = t
1
, x) at
the time of the first acquisition is of particular importance, as it
impacts the rest of the simulation. In this work, the tumor tail
extrapolation algorithm described in [22] is used. The method
is based on the assumption that the solution of equation
(1) at the first time point has converged to its asymptotic,
traveling wave type solution. Thereby, the tumor cell density
is propagated outward (and inward), starting from the T1Gd
segmentation, and drops approximately exponentially with
distance. The steepness of the falloff, i.e. the distance at which
the cell density drops by a factor 1/e is given by the invisibility

JOURNAL SUBMISSION 4
index λ =
p
D . By construction of the initialization, the
T1Gd abnormality falls exactly on the threshold τ
1
of the
tumor cell density at the first time point.
The reaction-diffusion equation is solved using the Lattice
Boltzmann Method [21], [23], [24] which allows for easy
parallelization and fast computations. On a 1mm×1mm×1mm
resampled MRI, simulating 30 days of growth takes approxi-
mately 50 seconds on a 2.3Ghz 50 core machine.
Note that this model is an approximation of the complex
growth of the disease. For instance, it could be extended
in order to include mass effect [25], or a more detailed
description of the disease [6]. In other works, this model
has been extended to model different types of therapy such
as resection [26], [5], chemotherapy [2], or anti-angiogenic
therapy [7]. The common approach taken in these works is
to add a death term to the reaction-diffusion equation, which
allows to model the shrinkage of the tumor due to the therapy.
It was also shown in [27] that the personalized parameters
of a reaction-diffusion model were good predictors of certain
mutations status of the patient.
IV. PERSONALIZATION
The personalization of the tumor growth model is combined
with a dose response model in order to define the radiotherapy
planning. We compare three different scenarios. First we only
use a single time point (the second acquisition) to personalize
the model such that the radiotherapy plan will be defined using
a single acquisition, similarly to what is being done in clinic.
Second we use two time points in order to personalize the
model. The radiotherapy plan will then be defined on the latest
acquisition. Third, we use two time points and include the
uncertainty in the segmentation.
A. Scenario 1: One time point only
In this section, we are interested in the posterior probability
of the model parameter θ = (D, ρ), knowing the clinical
segmentations S
0
3
on the T1Gd and S
0
4
on the T2-FLAIR at
the second time point. To cast the problem in a probabilis-
tic framework, we follow the Bayes rule: P (θ|S
0
3
, S
0
4
)
P (S
0
3
, S
0
4
|θ) P (θ). The likelihood is modeled as
P (S
0
3
, S
0
4
|θ) exp
H(D, ρ, S
0
3
, S
0
4
)
2
σ
2
(3)
where H(D, ρ, S
0
3
, S
0
4
) is the 95th percentile of the symmet-
ric Hausdorff distance between the border of the segmentation
S
0
4
, and the isoline at τ
2
of the simulated tumor cell density
u using (D, ρ), and initialized with the segmentation S
0
3
.
We further model the prior as log-uniform and independent
between the parameters,
P (θ) = P (D)P (ρ) (4)
We sample from the posterior distribution using a
Metropolis-Hasting algorithm. Note that this section only
uses the initialization algorithm (see Section III) which only
depends on the invisibility index λ =
p
D. Note that this
section can be related to the method described in [16], where a
single time point is used to propose a dose planning. However,
Unkelbach et al. [16] use a nominal value of the invisibility
index whereas it is personalized in this scenario. Moreover,
the Bayesian methodology allows to take into account the
uncertainty in the personalization.
B. Scenario 2: Two time points
In this section, we are interested in the posterior probability
of the model parameter θ = (D, ρ), knowing the clinical
segmentations S
0
i
for i = 1, 2, 3, 4 on the T1Gd and T2-FLAIR
at the first and second time point respectively. In this case, the
likelihood is model as
P ({S
0
i
}
i=1,2,3,4
|θ) exp
1
σ
2
P
4
i=2
H
i
(D, ρ, S
0
1
, S
0
i
)
3
!
2
(5)
where H
i
(D, ρ, S
0
1
, S
0
i
) is the 95th percentile of the sym-
metric Hausdorff distance between the border of the segmen-
tation S
0
i
for i = 2, 3, 4, and the isoline of the simulated
tumor cell density u using (D, ρ), and initialized with the
segmentation S
0
1
. We model the prior as described in Section
IV-A.
We sample from the posterior distribution using the Gaus-
sian Process Hamiltonian Monte Carlo (GPHMC) algorithm
first described by [28], and used for tumor growth personal-
ization in [21].
C. Scenario 3: Two time points and segmentation uncertainty
In this section, we want to include the uncertainty in the
segmentation to the personalization process. We denote the set
of plausible segmentations by S = {S
i
}
i=1,2,3,4
(see Section
II). We introduce the random variables Z
i
= (Z
i1
, ..., Z
iK
)
for i = 1, 2, 3, 4, which are one-hot binary vectors where
P (Z
ij
= 1|S) P (S
j
i
), and Z
il
= 0 for l 6= j when Z
ij
= 1.
The random variable Z
i
is a measure of the plausibility
of the samples: P (Z
i
) =
Q
K
i=1
P (Z
ij
= 1)
Z
ij
. We are
interested in the posterior probability of the model parameter
θ = (D, ρ, Z
1
, Z
2
, Z
3
, Z
4
), knowing the observations S. We
model the likelihood as
P (S|θ) exp
1
σ
2
P
4
i=2
H
i
(D, ρ, Z
1
, Z
i
)
3
!
2
(6)
where H
i
(D, ρ, Z
1
, Z
i
) is the 95th percentile of the sym-
metric Hausdorff distance between the border of the segmen-
tation indexed by Z
i
, and the isolines of the simulated tumor
cell density u using (D, ρ), and initialized with the contour
selected with Z
1
. We model the prior independent between
the parameters, log-uniform for D and ρ, and uniform for Z
i
(i.e. P (Z
ij
= 1) = 1/K),
P (θ) = P (D)P (ρ)
4
Y
i=1
P (Z
i
) (7)

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Journal ArticleDOI
TL;DR: The experimental results demonstrate that the proposed edge-aware generative adversarial networks (Ea-GANs) outperform multiple state-of-the-art methods for cross-modality MR image synthesis in both qualitative and quantitative measures.
Abstract: Magnetic resonance (MR) imaging is a widely used medical imaging protocol that can be configured to provide different contrasts between the tissues in human body. By setting different scanning parameters, each MR imaging modality reflects the unique visual characteristic of scanned body part, benefiting the subsequent analysis from multiple perspectives. To utilize the complementary information from multiple imaging modalities, cross-modality MR image synthesis has aroused increasing research interest recently. However, most existing methods only focus on minimizing pixel/voxel-wise intensity difference but ignore the textural details of image content structure, which affects the quality of synthesized images. In this paper, we propose edge-aware generative adversarial networks (Ea-GANs) for cross-modality MR image synthesis. Specifically, we integrate edge information, which reflects the textural structure of image content and depicts the boundaries of different objects in images, to reduce this gap. Corresponding to different learning strategies, two frameworks are proposed, i.e., a generator-induced Ea-GAN (gEa-GAN) and a discriminator-induced Ea-GAN (dEa-GAN). The gEa-GAN incorporates the edge information via its generator, while the dEa-GAN further does this from both the generator and the discriminator so that the edge similarity is also adversarially learned. In addition, the proposed Ea-GANs are 3D-based and utilize hierarchical features to capture contextual information. The experimental results demonstrate that the proposed Ea-GANs, especially the dEa-GAN, outperform multiple state-of-the-art methods for cross-modality MR image synthesis in both qualitative and quantitative measures. Moreover, the dEa-GAN also shows excellent generality to generic image synthesis tasks on benchmark datasets about facades, maps, and cityscapes.

177 citations


Cites background from "Personalized Radiotherapy Planning ..."

  • ...For example, multi-modality MR images have been collectively utilized to study the neuroanatomy of human brains for disease diagnosis [3] or therapy planning [4]....

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Journal ArticleDOI
01 Oct 2019
TL;DR: The opportunities of a model‐based approach as discussed in this review can be of great benefit for future optimizing and personalizing anticancer treatment.
Abstract: Increasing knowledge of intertumor heterogeneity, intratumor heterogeneity, and cancer evolution has improved the understanding of anticancer treatment resistance. A better characterization of cancer evolution and subsequent use of this knowledge for personalized treatment would increase the chance to overcome cancer treatment resistance. Model-based approaches may help achieve this goal. In this review, we comprehensively summarized mathematical models of tumor dynamics for solid tumors and of drug resistance evolution. Models displayed by ordinary differential equations, algebraic equations, and partial differential equations for characterizing tumor burden dynamics are introduced and discussed. As for tumor resistance evolution, stochastic and deterministic models are introduced and discussed. The results may facilitate a novel model-based analysis on anticancer treatment response and the occurrence of resistance, which incorporates both tumor dynamics and resistance evolution. The opportunities of a model-based approach as discussed in this review can be of great benefit for future optimizing and personalizing anticancer treatment.

84 citations


Cites methods from "Personalized Radiotherapy Planning ..."

  • ...This model structure was also recently used to investigate the personalization of radiotherapy strategy for brain cancer patients.(66) Setting f (c (x,t))= ⋅c (x,t), a similar model structure was also used to simulate the growth of glioblastoma based on previous reported parameters estimated from patients and estimated the survival times of patients under different parameter settings....

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Journal ArticleDOI
TL;DR: In this paper, the authors present a summary of biophysical growth modeling and simulation, inverse problems for model calibration, integration with imaging workflows, and their application to clinically relevant studies.
Abstract: Central nervous system (CNS) tumors come with vastly heterogeneous histologic, molecular, and radiographic landscapes, rendering their precise characterization challenging. The rapidly growing fields of biophysical modeling and radiomics have shown promise in better characterizing the molecular, spatial, and temporal heterogeneity of tumors. Integrative analysis of CNS tumors, including clinically acquired multi-parametric magnetic resonance imaging (mpMRI) and the inverse problem of calibrating biophysical models to mpMRI data, assists in identifying macroscopic quantifiable tumor patterns of invasion and proliferation, potentially leading to improved (a) detection/segmentation of tumor subregions and (b) computer-aided diagnostic/prognostic/predictive modeling. This article presents a summary of (a) biophysical growth modeling and simulation,(b) inverse problems for model calibration, (c) these models' integration with imaging workflows, and (d) their application to clinically relevant studies. We anticipate that such quantitative integrative analysis may even be beneficial in a future revision of the World Health Organization (WHO) classification for CNS tumors, ultimately improving patient survival prospects.

41 citations

Journal ArticleDOI
TL;DR: In this paper, a contrast-adaptive generative model for whole-brain segmentation with a new spatial regularization model of tumor shape using convolutional restricted Boltzmann machines is presented.

35 citations

Journal ArticleDOI
TL;DR: The ultimate goal of the work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making, which requires reliable, high-fidelity algorithms with a short time-to-solution.
Abstract: PDE-constrained optimization problems find many applications in medical image analysis, for example, neuroimaging, cardiovascular imaging, and oncologic imaging. We review the related literature and give examples of the formulation, discretization, and numerical solution of PDE-constrained optimization problems for medical imaging. We discuss three examples. The first is image registration, the second is data assimilation for brain tumor patients, and the third is data assimilation in cardiovascular imaging. The image registration problem is a classical task in medical image analysis and seeks to find pointwise correspondences between two or more images. Data assimilation problems use a PDE-constrained formulation to link a biophysical model to patient-specific data obtained from medical images. The associated optimality systems turn out to be sets of nonlinear, multicomponent PDEs that are challenging to solve in an efficient way. The ultimate goal of our work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making. This requires reliable, high-fidelity algorithms with a short time-to-solution. This task is complicated by model and data uncertainties, and by the fact that PDE-constrained optimization problems are ill-posed in nature, and in general yield high-dimensional, severely ill-conditioned systems after discretization. These features make regularization, effective preconditioners, and iterative solvers that, in many cases, have to be implemented on distributed-memory architectures to be practical, a prerequisite. We showcase state-of-the-art techniques in scientific computing to tackle these challenges.

34 citations


Cites background or methods from "Personalized Radiotherapy Planning ..."

  • ...They have been used to (i) study tumor growth patterns in individual patients [45, 127, 135, 167, 189], (ii) extrapolate the physiological boundary of tumors [113, 148], (iii) or study the effects of clinical intervention [116, 117, 139, 164, 169, 191, 203]....

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  • ...finite-difference approximations to the gradient [101], or tackle the parameter estimation problem within a Bayesian framework [48, 91, 116, 117, 122, 123, 138, 154] (see Rem....

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  • ...A common assumption in many models is that the rate of migration depends on the tissue composition; typically, it is modeled to be faster in white than in gray matter [116, 117, 189]....

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  • ...such example is to relate (manual) segmentations of imaging abnormalities to a detection thresholds for the computed population density of cancerous cells [70, 89, 116, 117, 127, 133, 191]....

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References
More filters
Journal ArticleDOI
TL;DR: The tumor growth model provides a method to account for anisotropic growth patterns of glioma, and may therefore provide a tool to make target delineation more objective and automated.
Abstract: Glioblastoma di ffer from many other tumors in the sense that they grow in filtratively into the brain tissue instead of forming a solid tumor mass with a de fined boundary. Only the part of the tumor with high tumor cell density can be localized through imaging directly. In contrast, brain tissue in filtrated by tumor cells at low density appears normal on current imaging modalities. In current clinical practice, a uniform margin, typically two centimeters, is applied to account for microscopic spread of disease that is not directly assessable through imaging. The current treatment planning procedure can potentially be improved by accounting for the anisotropy of tumor growth, which arises from di erent factors: Anatomical barriers such as the falx cerebri represent boundaries for migrating tumor cells. In addition, tumor cells primarily spread in white matter and in ltrate gray matter at lower rate. We investigate the use of a phenomenological tumor growth model for treatment planning. The model is based on the Fisher-Kolmogorov equation, which formalizes these growth characteristics and estimates the spatial distribution of tumor cells in normal appearing regions of the brain. The target volume for radiotherapy planning can be de fined as an isoline of the simulated tumor cell density. This paper analyzes the model with respect to implications for target volume de finition and identi fies its most critical components. A retrospective study involving 10 glioblastoma patients treated at our institution has been performed. To illustrate the main findings of the study, a detailed case study is presented for a glioblastoma located close to the falx. In this situation, the falx represents a boundary for migrating tumor cells, whereas the corpus callosum provides a route for the tumor to spread to the contralateral hemisphere. We further discuss the sensitivity of the model with respect to the input parameters. Correct segmentation of the brain appears to be the most crucial model input. We conclude that the tumor growth model provides a method to account for anisotropic growth patterns of glioma, and may therefore provide a tool to make target delineation more objective and automated.

71 citations


"Personalized Radiotherapy Planning ..." refers methods in this paper

  • ...[16], [17] studied the optimization of the radiotherapy planning based on a tumor growth model in order to automatically define realistic 3D prescription dose...

    [...]

  • ...We optimize an Intensity Modulated Radiation Therapy (IMRT) plan using 9 equally spaced coplanar 6 MV photon beams and a piece-wise quadratic objective function, as detailed in [16], [17]....

    [...]

Journal ArticleDOI
TL;DR: A mathematical model is developed to describe the growth and invasion of glioma cells throughout an anatomically accurate virtual human brain as well as the effects of operation on these lesions.

63 citations


"Personalized Radiotherapy Planning ..." refers background or methods in this paper

  • ...The resection of a brain tumor can also be modeled by deleting the tumor cells in the resected region [4], [5]....

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  • ...The proposed model could however be extended in order to included the impact of such therapy following the developments done in [4], [5]....

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Journal ArticleDOI
TL;DR: A new mathematical model is derived that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis.

57 citations


"Personalized Radiotherapy Planning ..." refers background in this paper

  • ...More advanced therapy schedules using for instance anti-angiogenic drugs can also be studied with more complex models [6]–[8]....

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  • ...For instance, it could be extended in order to include mass effect [25], or a more detailed description of the disease [6]....

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Journal ArticleDOI
15 Dec 2014-PLOS ONE
TL;DR: A clinical-scale model of GBM is constructed whose predictions uncover a new pattern of recurrence in 11/70 bevacizumab-treated patients and support an exception to the Folkman hypothesis: GBM grows in the absence of angiogenesis by a cycle of proliferation and brain invasion that expands necrosis.
Abstract: Glioblastoma multiforme (GBM) causes significant neurological morbidity and short survival times. Brain invasion by GBM is associated with poor prognosis. Recent clinical trials of bevacizumab in newly-diagnosed GBM found no beneficial effects on overall survival times; however, the baseline health-related quality of life and performance status were maintained longer in the bevacizumab group and the glucocorticoid requirement was lower. Here, we construct a clinical-scale model of GBM whose predictions uncover a new pattern of recurrence in 11/70 bevacizumab-treated patients. The findings support an exception to the Folkman hypothesis: GBM grows in the absence of angiogenesis by a cycle of proliferation and brain invasion that expands necrosis. Furthermore, necrosis is positively correlated with brain invasion in 26 newly-diagnosed GBM. The unintuitive results explain the unusual clinical effects of bevacizumab and suggest new hypotheses on the dynamic clinical effects of migration by active transport, a mechanism of hypoxia-driven brain invasion.

56 citations


"Personalized Radiotherapy Planning ..." refers methods in this paper

  • ...In other works, this model has been extended to model different types of therapy such as resection [5], [26], chemotherapy [2], or anti-angiogenic therapy [7]....

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Journal ArticleDOI
TL;DR: The dose fall-off rate concept reflects the idea that infiltrating gliomas lack a defined boundary and are characterized by a continuous fall-offs of the density of infiltrating tumor cells, and can potentially be used to individualize the prescribed dose distribution.
Abstract: Gliomas differ from many other tumors as they grow infiltratively into the brain parenchyma rather than forming a solid tumor mass with a well-defined boundary. Tumor cells can be found several centimeters away from the central tumor mass that is visible using current imaging techniques. The infiltrative growth characteristics of gliomas question the concept of a radiotherapy target volume that is irradiated to a homogeneous dose—the standard in current clinical practice. We discuss the use of the Fisher–Kolmogorov glioma growth model in radiotherapy treatment planning. The phenomenological tumor growth model assumes that tumor cells proliferate locally and migrate into neighboring brain tissue, which is mathematically described via a partial differential equation for the spatio-temporal evolution of the tumor cell density. In this model, the tumor cell density drops approximately exponentially with distance from the visible gross tumor volume, which is quantified by the infiltration length, a parameter describing the distance at which the tumor cell density drops by a factor of e. This paper discusses the implications for the prescribed dose distribution in the periphery of the tumor. In the context of the exponential cell kill model, an exponential fall-off of the cell density suggests a linear fall-off of the prescription dose with distance. We introduce the dose fall-off rate, which quantifies the steepness of the prescription dose fall-off in units of Gy mm−1. It is shown that the dose fall-off rate is given by the inverse of the product of radiosensitivity and infiltration length. For an infiltration length of 3 mm and a surviving fraction of 50% at 2 Gy, this suggests a dose fall-off of approximately 1 Gy mm−1. The concept is illustrated for two glioblastoma patients by optimizing intensity-modulated radiotherapy plans. The dose fall-off rate concept reflects the idea that infiltrating gliomas lack a defined boundary and are characterized by a continuous fall-off of the density of infiltrating tumor cells. The approach can potentially be used to individualize the prescribed dose distribution if better methods to estimate radiosensitivity and infiltration length on a patient by patient basis become available.

38 citations


"Personalized Radiotherapy Planning ..." refers background or methods in this paper

  • ...This is formally defined as the dose solving the following optimization problem [16],...

    [...]

  • ...We optimize an Intensity Modulated Radiation Therapy (IMRT) plan using 9 equally spaced coplanar 6 MV photon beams and a piece-wise quadratic objective function, as detailed in [16], [17]....

    [...]

  • ...[16] use a nominal value of the invisibility index whereas it is personalized in this scenario....

    [...]

  • ...Note that this section can be related to the method described in [16], where a single time point is used to propose a dose planning....

    [...]

  • ...[16], [17] studied the optimization of the radiotherapy planning based on a tumor growth model in order to automatically define realistic 3D prescription dose...

    [...]

Related Papers (5)
Frequently Asked Questions (2)
Q1. What have the authors contributed in "Personalized radiotherapy planning based on a computational tumor growth model" ?

In this article, the authors propose a proof of concept for the automatic planning of personalized radiotherapy for brain tumors. First, the authors consider a single MRI acquisition before therapy, as it would usually be the case in clinical routine. The authors present the application of their approach on two patients diagnosed with high grade glioma. The authors introduce two methods to derive the radiotherapy prescription dose distribution, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density. The authors show how their method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration. The authors further present extensions of the method in order to spare adjacent organs at risk by re-distributing the dose. 

The authors presented the proof of concept for a method combining a computational model of tumor growth with a dose response model in order to optimize radiotherapy planning, which takes into account the uncertainty in the model parameters and the clinical segmentations. In the second one, the authors use two time points in order to personalize the model and plan radiotherapy. In the future, the inclusion of the fractionation scheme of the delivered dose could be optimized. To that end, the model should be extended in order to take into account the complex therapy the patient is undergoing.