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A Patient-Specific Anisotropic Diffusion Modelfor Brain Tumor Spread

Amanda Swan · Thomas Hillen · JohnC. Bowman · Albert D. Murtha

Received: date / Accepted: date

Abstract Gliomas are primary brain tumours arising from the glial cells ofthe nervous system. The diffuse nature of spread, coupled with proximity tocritical brain structures, makes treatment a challenge. Pathological analysisconfirms that the extent of glioma spread exceeds the extent of the grosslyvisible mass visible on conventional magnetic resonance imaging (MRI) scans.Gliomas show faster spread along white matter tracts than in grey matter,leading to irregular patterns of spread. We propose a mathematical modelbased on Diffusion Tensor Imaging (DTI), a new MRI imaging technique thatoffers a methodology to delineate the major white matter tracts in the brain.We apply the anisotropic diffusion model of Painter and Hillen (2013) to datafrom 10 patients with gliomas. Moreover, we compare the anisotropic model

A. SwanDepartment of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, ABT6G 2G1, CanadaE-mail: [email protected]

T. HillenCentre for Mathematical BiologyDepartment of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, ABT6G 2G1, Canada

J.C. BowmanDepartment of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, ABT6G 2G1, Canada

A.D. MurthaCross Cancer Institute,Edmonton, AB,T6G 1Z2, Canada

2 Amanda Swan et al.

to the state-of-the-art Proliferation–Infiltration (PI) model of Swanson et al.(2000). We find that the anisotropic model offers an improvement over thestandard PI model. For tumours with low anisotropy, the predictions of thetwo models are virtually identical, but for patients whose tumours show higheranisotropy, the results differ. We also suggest using the data from the con-tralateral hemisphere to further improve the model fit. Finally, we discuss thepotential use of this model in clinical treatment planning.

Keywords Mathematical Medicine · Gliomas · Partial Differential Equations

1 Introduction

Gliomas are a type of primary brain tumour arising from the glial cells ofthe nervous system. The most aggressive subtype of glioma, grade IV astro-cytoma, or Glioblastoma multiforme (GBM), carries a poor prognosis, with amedian survival of just 14 months despite treatment with surgery, radiation,and chemotherapy [3]. Glioma cells have a tendency to spread into the sur-rounding brain tissue, making the boundaries of gliomas very diffuse, with alarge amount of spread beyond what can be seen on scans. This phenomenon,coupled with the delicate nature of brain tissue, makes treatment of gliomas achallenge. As such, clinicians typically target a treatment region that includesthe grossly visible mass plus an additional uniform margin of approximately2 cm for radiation therapy. We propose that a mathematical model can helpdefine a more accurate extension based on an individual patient’s brain struc-ture.

The heterogeneity of brain tissue has an effect on glioma spread, with dif-ferent movement mechanisms occurring in the grey matter than in the bundlesof nerve fibres that make up the white matter tracts. It has been shown thatthe cancer cells have a preferential movement direction aligned with the whitematter tracts [43,13,14], using them as “highways” for their spread. Whencancer cells move along these fibres, their pattern of spread leads to tumourshapes that often show projections in certain directions. This directed move-ment can be modelled through the use of anisotropic diffusion, where the rateof spread is allowed to vary with direction [39], with an increased rate of spreadalong the white matter tracts. The advent of Diffusion Tensor Imaging (DTI)has allowed clinicians to measure the rates of diffusion in each direction, ateach location within the brain, thereby creating a map of the white mattertracts within an individual patient’s brain. DTI technology therefore allowsus to apply an anisotropic diffusion model to simulate for each individual pa-tient the spread of glioma cells along the white matter tracts. Such a modelwas derived by Painter and Hillen [39], and in this paper we will validate thisanisotropic diffusion model on data from 10 recent glioma patients.

We find that the anisotropic diffusion model of Painter and Hillen [39]performs better than the state-of-the-art Proliferation–Infiltration (PI) modelof Swanson et al. [47], reproducing most observed tumours with a good degreeof success. However, we also identify situations where neither of these models

A Patient-Specific Anisotropic Diffusion Model for Brain Tumor Spread 3

gives a good fit and offer an explanation as to why this is the case. We thensuggest future model modifications that could further improve upon eitherglioma model. We argue that the inclusion of anisotropy, as measured throughDTI imaging, provides a significant advance in glioma modelling, although itdoes not provide a final answer, and further research is necessary.

1.1 Gliomas

Gliomas are so named because they arise from the glial cells of the central ner-vous system. Glial cells are those cells that surround the nerve cells, providingthe support necessary for proper functioning. There are three types of glialcells of particular importance: astrocytes, oligodendrocytes, and ependymalcells [27]. Each of these cell types can undergo malignant transformations toform astrocytomas, oligodentrogliomas and ependymomas, respectively. Themost aggressive subtype, grade IV astrocytomas, also called Glioblastoma mul-tiforme (GBM), are the focus of our modelling. GBMs may develop de novo, asprimary GBMs, or result from the transformation of lower grade astrocytomas,as secondary GBMs.

The main challenge in treating gliomas is the fact that a significant por-tion of the tumour is “invisible” from a clinician’s perspective. Since this isknown, treatment regions typically include not only the visible mass, but alsosome of the surrounding tissue. In Figure 1 we show several regions that aretypically used for radiation treatment. The GTV, or Gross Tumour Volumecorresponds to the visible tumour mass as it appears on a scan. The CTV,or Clinical Target Volume, is usually defined as a uniform 1.5 cm extensionthat is chosen to account for clinically occult extension. The PTV or PlanningTarget Volume includes an additional 0.5 cm margin to account for any uncer-tainties in the delivery of the prescribed dose. While using a uniform extensionof the GTV is a good starting point, the anisotropic nature of glioma spreadimplies that cancer cells will have invaded further in some directions than inothers, meaning that it may be more beneficial in terms of both survival andquality of life to treat further in some directions, and not as far in others.This is where mathematical models offer the potential to improve treatment.Through simulating the growth of a tumour, we can predict cell density levelsin regions that can not be seen on a scan.

1.2 Anisotropic Diffusion

The brain is made up of two main types of tissues: white matter and grey mat-ter. White matter has a fibrous structure, consisting of the myelinated axonsalong which nerve cells send signals, and contains relatively few cell bodies.Conversely, grey matter consists of the glial cells and nerve cell bodies, with arelatively low concentration of both myelinated and unmyelinated axons [42,28]. There is a growing body of evidence indicating that cancer cells will use the


Fig. 1: The different treatment regions for a glioma patient. The Gross TumourVolume (GTV) is the visible tumour mass delineated on a scan, the ClinicalTarget Volme (CTV) is a 1.5 cm extension of this region to account for un-detectable cancer cell invasion, and the Planning Target Volume or PTV is a0.5 cm extension to account for uncertainties in the setup or delivery of theprescribed dose.

fibrous white matter tracts to migrate, leading to spread that is decidedly notuniform [43,13,14]. The tendency of cancer cells to follow white matter tractsresults in an apparent increased rate of spread along the direction aligned withthe tract. Mathematically, we model this by assigning a higher rate of spreadin this direction than in the perpendicular directions.

Since the rate of spread is direction-dependent (anisotropic), a scalar dif-fusion coefficient is insufficient to encode the diffusion parameters; hence, weuse a symmetric, positive-definite, second-order tensor D ∈ R3×3 with entriescorresponding to the relative rates of spread in the associated directions. Theeasiest way to interpret a tensor is through its principal directions and corre-sponding principal values. The use of geometrical representations allows us tovisualize the information contained within the diffusion tensors.

The diffusion ellipsoid is defined as a level set of the inverse diffusion tensor:

Ec := v ∈ R3 : vTD−1v = c. (1)

It defines the set of equal probability of finding a random walker that startsat the origin and undergoes anisotropic diffusion according to the tensor D.

The same tensor D can be visualized via the diffusion peanut obtained byplotting the apparent diffusion coefficient, given by the map [39,24]

θ 7→ θTDθ : θ ∈ S2, (2)

where S2 is the 2-sphere consisting of all points of unit distance fromthe origin. Peanuts will have their axes aligned in the direction of highestdiffusivity, and will be pinched in directions having lower diffusivity [39]. As


(a) (b)

(c)(d)

Fig. 2: The oblate diffusion ellipsoid (a) and diffusion peanut (b) corresponding to D1 andthe prolate diffusion ellipsoid (c) and diffusion peanut (d) corresponding to D2.

an example, consider two diagonal diffusion tensors D1 and D2, given by

D1 =

5 0 00 3 00 0 1

, D2 =

8 0 00 1 00 0 0.2

. (3)

The resulting ellipsoids and peanuts, as determined via Equations 1 and 2,respectively, are shown in Figure 2. Note that D1 produces an oblate ellipsoidand a peanut that is pinched in one direction, while D2 produces a prolateellipsoid and a peanut that is pinched in two directions. For the special case ofan isotropic tensor, which has three equal principal values, both the diffusionellipsoid and the diffusion peanut are spherical.

1.3 Diffusion Tensor Imaging

Diffusion Tensor Imaging (DTI) uses magnetic resonance imaging (MRI) tech-nology to measure the rate of diffusion of water molecules in different directionswithin the brain [25]. Diffusion is a naturally occurring phenomenon, and isunaffected by the magnetic field employed in MRI [5]. In the regions aroundwhite matter tracts in the brain, the movement of water molecules along thefibres is relatively unimpeded, while movement perpendicular to these fibresis much more difficult. As such, DTI can be used to map the fibre network [1,25]. The diffusion rate is only measured in the direction of an applied gradient;thus, in order to determine the full three-dimensional diffusion tensor, gradi-ents must be applied in at least six directions, corresponding to the six degreesof freedom of the symmetric tensor [16,25]. However, applying the gradient inadditional directions results in a more accurate measurement [16].

We can quantify the degree of anisotropy of a given tensor D ∈ R3 in asingle index called a diffusion anisotropy index. While there are different op-tions available for such an index, here we use the Fractional Anisotropy (FA)


Fig. 3: A sample 2D slice from Patient 10 showing the fractional anisotropy (FA).Note that regions having high FA (white) correspond to the white matter tracts,while regions having low FA (black) correspond to the surrounding grey matter.

[26]. This is the most common choice. The FA ranges between 0, correspond-ing to fully isotropic diffusion, and 1 for fully anisotropic diffusion [26]. Thefractional anisotropy FA2 for a two-dimensional tensor is

FA2 =

√2[(λ1 −Dav)2 + (λ2 −Dav)2]

λ21 + λ2

2

(4)

and the three-dimensional fractional anisotropy FA3 is given by

FA3 =

√3[(λ1 −Dav)2 + (λ2 −Dav)2 + (λ3 −Dav)2]

2(λ21 + λ2

2 + λ23)

, (5)

where λ1, λ2, and λ3 are the principal values of a given tensor D in descendingorder, andDav = trD/2 for the 2D case, andDav = trD/3 for the 3D case [26].Note that for the isotropic case, Dav = λ1 = λ2 = λ3 and FA = 0. For the fullyanisotropic case, where FA = 1, the diffusion tensor is singular and only oneeigenvalue is non-zero. Figure 3 shows sample FA variation for an axialbrain slice taken from Patient 10. For images comparing the infor-mation available from DTI scans to that available using traditionalMagnetic Resonance Imaging (MRI), see Figure 4 of Alexander etal. [1].

1.4 Established Work

Inspired by Mosayebi et al. [35], we provide a table summarizing the state ofthe art in modelling of spatial glioma spread in Table 1. In describing previouswork, we will make several distinctions between existing glioma models. Thefirst distinction will be whether or not the model uses diffusion tensors D ∈ Rn


(anisotropic), or a diffusion coefficient d ∈ R (isotropic). Models that usediffusion tensors can be of “Fickian” form,

ut =∇ · (D∇u) + f(u), (6)

or of “Fokker–Planck” form,

ut =∇∇ : (Du) + f(u), (7)

where u(x, t) denotes the cancer cell density at time t ≥ 0 and location x ∈ Ω,and Ω is a smooth n-dimensional domain. The reaction term f(u) denotescell growth, and will be specified later. The index t denotes the partial timederivative and ∇ contains the spatial derivatives. The colon : denotes thecontraction of two tensors, specifically

∇∇ : (Du) =n∑

i=1

n∑j=1

∂

∂xi

∂

∂xj(Diju). (8)

Of course, equation (7) can be expanded into a Fickian term plus an ad-vection term:

ut =∇∇ : (Du) + f(u) =∇ · (D∇u) +∇ · ((∇TD)u) + f(u). (9)

The Fokker–Planck version of anisotropic diffusion (7) is less well known, how-ever, it has been derived in many biologically related contexts [37–39,18] andit is the basis for our anisotropic glioma model.

The cell growth term f(u) is often one of the following three types [47,39,33]: exponential growth f(u) = ru with a constant growth rate r > 0, logisticgrowth f(u) = ru(1 − u/K) for carrying capacity K > 0, or Gompertziangrowth f(u) = −ru ln(u/K).

The first application of a diffusion model to glioma spread was in 2000 bySwanson et al. [47] in the form of the PI model. While this model includesboth cell growth and diffusion, it uses, instead of the full diffusion tensor,a Fickian (6) scalar diffusion coefficient that varies between white and greymatter. The growth is modelled using an exponential function. The full PImodel is therefore given by

ut =∇ · (d(x)∇u) + ru, (10)

where d(x) is the diffusion coefficient, r is the growth rate and u(x, t) is thecancer cell density. Swanson’s group went on to lead the way in terms of apply-ing mathematical models to glioma treatment planning. They have made greatstrides in terms of applying their models in a personalized medical framework,including diagnosis and treatment planning [47,46,48,15,9,22,50,44,36].

In 2009, the group looked at quantitative measures for proliferation andinvasion rates for individual patients, and how to connect these to a prognosis[50]. In 2010, they used these results coupled with a model for cell killing by ra-diation to predict response to external beam radiotherapy on a patient-specificbasis, with a high degree of accuracy [44]. Jbabdi et al. [23] incorporated


anisotropy into the glioma growth model, replacing the diffusion coefficientd(x) with a tensor D(x). This anisotropic glioma growth model was used tosimulate brain tumour spread using a DTI scan of a healthy individual, andthe resulting shapes were compared to real gliomas.

Glioma modelling was moved along further in 2005 by Clatz et al. [8] withthe first appearance of a mass effect. The mass effect refers to the result ofphysical pressure build up and includes models for tissue strains and stresses.Growth and infiltration are modelled using the PI model with the additionof diffusion tensors [23], plus a mass effect equation derived via a momentumbalance equation. The same research group carried this work further in 2008,introducing parameters that allow for the mechanical aspects of growth, aswell as diffusion, to be tuned to a specific patient [6]. While the results of theirsimulations were eventually compared to the growth of an actual patient’stumour, the simulations used atlas data, which provides a general map of ahuman brain that is not specific to a particular patient.

In 2008, Hogea et al. [21] proposed a modified model that includes bothgrowth and infiltration, as well as a mass effect modelled with an advectionterm. Here, the cell density is modelled using a reaction-diffusion-advectionequation, given by

ut =∇ · (D∇u)−∇ · (uv) + r(u), (11)

where once again, u(x, t) represents the space- and time-dependent cancer celldensity. The advective velocity v is determined via a continuum mechanicsderivation. A major contribution of this work was the method for fitting theparameters, which involved deriving and solving a PDE constrained optimiza-tion problem.

While previous models had focused on modelling cell density, Konukogluet al. [29] developed in 2006 the first model that focussed instead on invasionmargins. This is done by matching travelling wave solutions of the classicFisher–KPP equation to the tumour mass data. Konukoglu et al. [30] extendedthis work in 2010, incorporating real patient DTI data, and validating theirmodel for two real patients. This idea was taken a step further by Mosayebiet al. in 2012 [35], who proposed a model for the tumour boundary using ageodesic distance measure on the Riemannian space induced by brain fibres.This group was able to test their model using data from 11 patients, somethingthat few of the other groups had access to.

Although some of the above models use anisotropic diffusion, the connec-tions of the tumour diffusion tensors to the DTI measurements is not welljustified. Painter and Hillen [39] developed a cell-based approach to clearlyconnect the measured water diffusion tensor (DTI) to an effective tumor dif-fusion tensor D. It is the only glioma model based on individual movement ofcells along fibrous structures and it uses a Fokker–Planck diffusion operatoras discussed in Equation 7. The derivation of the anisotropic model from thecell-level transport framework also provides a method for scaling the measuredwater diffusion tensors to cell movement anisotropies [39]. The scaling intro-duces a new anisotropy parameter that can be estimated specifically for each


Table 1: Table modified from [35] summarizing previous models and contributions. We haveadded several other relevant models, most of which have been developed since their paperwas published. For model categorization, D=Diffusion, DM=Diffusion and Mass Effect, IT=Isotropic Tensor, DT=Diffusion Tensor.

Paper Model Tensor Source of tensor Comparison

Swanson et al. (2000-16) D isotropic N/A many PatientsJbabdi et al. (2005) D DT Healthy case VisualClatz et al. (2005) DM DT Atlas 1 PatientKonukoglu et al. (2006) D DT Atlas SyntheticHogea et al. (2007) DM DT Atlas 1 PatientBondiau (2008) DM DT Atlas 1 PatientKonukoglu et al. (2010) D DT Real tensors 2 PatientsMosayebi (2012) D DT Real tensors 11 PatientsPainter and Hillen (2013) D DT Real tensors 1 PatientEngwer et al. (2014) D DT Atlas 1 PatientThis Paper D DT Real tensors 10 Patients

patient. The anisotropic glioma spread model is given by the PDE

ut =∇∇ : (Du) + ru(1− u), (12)

where r is the growth rate and D is the anisotropic diffusion tensor.The most recent model for glioma spread was developed by Engwer et al.

[11,12] in 2014 and 2016. It is an extension of the model of Painter and Hillen[39] that explicitly includes the adhesion mechanisms connecting glioma cellsto white matter tracts. The model derivation results in additional advectiveterms that are due to cell adhesions with the brain tissue. Since several de-tails that were assumed in Engwer’s model [11] have not yet been confirmedexperimentally, we prefer to use the simpler model (12).

1.5 Patient Data

We chose to work with DTI data from 10 cancer patients observed between2006–2011 at a cancer hospital. By the time we received the patient data,it had already undergone several preprocessing steps for skull stripping andtumour segmentation, making application of the anisotropic glioma growthmodel much easier. This processing was done by Dr. R. Greiner and his team[10,45].

Skull stripping serves to remove the skull from the image data. An auto-mated algorithm was used that removes the skull, creating a suitable domainfor model simulations [10,45], so that the boundary of the domain is reallythe boundary of the brain. Tumour segmentation refers to the delineation of atumour boundary using an MRI image.This process is often done manually bya clinician, however this can introduce bias and human error into the process.Additionally, these segmentations are not reproducible. Instead, an automated


tumour segmentation tool based on MRI histograms was applied to the patientdata [10,45].

All of the patient data was segmented and includes the tumour boundaries.Figure 4 displays two-dimensional cross sections through the centre of each pa-tient’s tumour, showing the slice containing the largest portion of the tumour.These images indicate the shape and size of each tumour as a black outline.The colourful background shows the fractional anisotropy (5) as computedfrom the corresponding DTI data. Light colours represent high anisotropy anddark colours represent low anisotropy.

Patient 1 Patient 2 Patient 3 Patient 4 Patient 5

Patient 6 Patient 7 Patient 8 Patient 9 Patient 10

Fig. 4: Segmentations through the largest two-dimensional slice of each patient’s glioma areshown. The segmentations are done using an automatic procedure. The underlyingcolours correspond to the fractional anisotropy with yellow=high FA, blue=low FA,and the tumour boundary is shown in black. A large variation in tumour shape andsize is observed.

Looking at these images we see already that some tumours (in Patients 1,4, 7, 8, and 10) are more spherical than others. We also observe areas of highanisotropy inside or near the tumour, where anisotropic spread is seen to besignificant.

1.6 Outline

In the next section, we will describe the cell-based modelling approach ofPainter and Hillen [39] and briefly derive the anisotropic model (12). We en-counter the von-Mises and Fisher distributions as tools to describe orientedmovement of cells in aligned tissues and discuss how to transform the mea-sured DTI data into cancer cell diffusion tensors. We then briefly outline thenumerical methods used to simulate the anisotropic model in Section 2.1.

In Section 3, we discuss how to apply the anisotropic diffusion model topatient data, and describe the details of how the model is fit. We introduce aquantitative measure to objectively evaluate model performance, by comparingthe model-predicted tumour to the real segmented tumour. Also discussed arethe initial condition and parameter selection.


The Swanson PI model is discussed in Section 3.1. We describe how the greyand white matter are delineated on actual patient data using a thresholdingvalue of the FA of Equations (4) and (5).

The main results are showcased in Section 4, where we apply the anisotropicmodel of Equation (12) to data from ten patients and compare the modelfitting with that of the established PI model of Swanson (10). The resultsof these simulations show that the anisotropic model offers an improvementover the original PI model. We also introduce the idea of estimating missingDTI data by taking advantage of the symmetry of the brain. We performsimulations on this new data for a subset of the original patient set, findingthat this technique can offer further improvement to the anisotropic model.After establishing the validity of the anisotropic model, we discuss how it maybe applied in a clinical setting.

2 The Anisotropic Spread Model

In this section, we review the derivation of the anisotropic model of Equation(12) from [39]. For this model, we are considering how cancer cells migratein a heterogenous environment by contact guidance along white matter tracts[20]. The fully anisotropic diffusion model of Equation (12) was developedfrom scaling arguments, and was built up from a cell-level transport modelincorporating the underlying biological fundamentals [17,18,20,39]. The dif-fusion tensor D(x) arises as the variance–covariance matrix of an underlyingfibre distribution q(x, t,v). Here x denotes space, t is time, and v ∈ Sn−1 de-notes a unit vector. The density q(x, t,v) measures the directional distributionof white matter fibres at location x and time t, as determined through DTIimaging. Given q(x, t,v), we can compute the diffusion tensor

D(x, t) =1µ

∫Sn−1

vvT q(x, t,v) dv, (13)

with µ denoting the average turning rate of moving cells [20].We mentioned above that DTI measures the diffusion of water molecules;

hence, while it may be used to determine q(x, t,v), it is not immediately clearhow this should be done. DTI performs an averaging of directions in each giventissue volume and doesn’t distinguish between individual fibres; the resultingmeasured tensor represents an idealized fibre distribution for each voxel. Thereare several ways to transform the measured DTI data into a fibre distribution,with both the Q-ball methodology and the diffusion peanut discussed in arecent paper by Engwer et al. [12]. Here, we choose two distributions thathave a prominent role in directional statistics: the von-Mises (2D) and theFisher (3D) distributions. The von-Mises and Fisher distributions are normaldistributions wrapped around a circle (von-Mises) or a sphere (Fisher), eachhaving a peak in a given direction. The width of this peak is determined bya concentration parameter. We assume that the dominant direction is givenby the leading eigenvector of the diffusion tensor and that the concentration


parameter is proportional to the fractional anisotropy of the DTI tensor. Theconstant of proportionality introduces a new parameter κ, the anisotropy pa-rameter, which can be estimated on a patient specific basis.

Given a preferred direction γ ∈ Sn−1 for n ∈ 2, 3, the bimodal 2D von-Mises distribution is

q(v) =1

4πI0(k)

(ekv·γ + e−kv·γ

)and the 3D Fisher distribution is

q(v) =k

8π sinh(k)

(ekv·γ + e−kv·γ

),

where I0 denotes the zero-order modified Bessel function of the first kind. Thetwo-dimensional bimodal von-Mises distribution for γ = (1, 0)T and k = 5 isshown in Figure 5. Computing the variance–covariance matrices for the von-Mises and Fisher distributions according to Equation (13) is not trivial, andhas been done in [49]. These are given as

(Var q)2D = µD(x, t) =12

(1− I2(k)

I0(k)

)I2 +

I2(k)I0(k)

γγT , (14)

(Var q)3D = µD(x, t) =(

coth kk− 1k2

)I3 +

(1− 3 coth k

k+

3k2

)γγT ,(15)

where Ij(k) denotes the modified Bessel function of order j of the first kind,and I2 and I3 denote the two- and three-dimensional identity matrices, re-spectively. The concentration parameter k is spatially dependent and is givenas

k(x) = κFA(DTI(x)), (16)where FA denotes the fractional anisotropy of Equations (4) and (5) given bythe DTI measurements DTI(x), and κ is a patient specific anisotropy param-eter. The anisotropy parameter can roughly be interpreted as the sensitivityof the cancer cells to the underlying brain structure. For some patients, thecancer cells largely ignore the fibres and the tumour grows spherically, cor-responding to a low κ value and consequently isotropic spread. Conversely,for some patients, the cancer cells closely follow the structure of the brain,resulting in highly irregular tumour shapes, corresponding to a high κ valueand anisotropic spread. If the model is applied to patient data, then γ isalso spatially dependent and denotes the leading (normalized) eigenvector ofDTI(x).

2.1 Numerics

The numerical implementation of the fully anisotropic model (12) is straight-forward. We expand the anisotropic part into individual diffusive secondderivative terms, and we use operator splitting for the resulting diffusion andreaction terms. We use mass conserving schemes for the diffusion termsand zero flux boundary conditions.


Fig. 5: An example of a bimodal von-Mises distribution. This distribution can be inter-preted as two normal distributions, each wrapped around the unit circle, andhaving peaks at γ = ±(1, 0)T .

3 Model Implementation on a Real Brain Domain

In this section we describe additional details about the Jaccard index for modelfitting and our method to find the initial condition and model parameters. Theuse of the Jaccard index is inspired by Mosayebi et al. [35]. The Jaccard indexis a measure of similarity between finite sample sets. Taking A and B to betwo sets of finite measure, the Jaccard index J(A,B) is [35]

J(A,B) =|A ∩B||A ∪B|

. (17)

When the union of the two sets approaches their intersection, the ratioapproaches 1. When the union is large relative to the intersection, the ratioapproaches 0. As such, in order to obtain the best model fit, we aim to max-imize the Jaccard index. Figure 7 shows an example of two sets with a lowJaccard index (a) and two sets with a high Jaccard index (b).

(a) (b)

Fig. 6: Two examples showing sets having (a) Jaccard index = 0.100 and (b) Jaccardindex=0.634. This demonstrates that the more two sets overlap, the higher theJaccard index will be.


(a)

Fig. 7: A sample initial condition u0(x) for Patient 10 as defined by Equation (18) withx0 = (6.2, 9.8). The cancer cell density is shown, with yellow=high density andblue=low density. This initial condition models a single cell, or small group ofcells, that initialize the tumour growth.

The Jaccard index offers a couple of advantages over other possible metrics.For one, it penalizes both undergrowth and overgrowth, so that in order toget a truly good model fit, the sets must match up everywhere. The secondadvantage is its simplicity and ease of implementation.

In order to fit the cell density models to the patient data, we must specifywhich two sets will be compared via the Jaccard index. The first set contains allof the points located within the actual segmented tumour. For the cell densityfunction u(x, t), we use a level set of u = 0.16 to represent the simulatedtumour boundary. This threshold value is chosen to be roughly the cell densitythat appears on a T2 MRI scan [48] used to segment the data. As such, thesecond set will contain all points x for which u(x, t) ≥ 0.16.

For our simulations we assume that the tumour started in a small areainside the detected tumour. We choose the initial condition

u0(x) = e−

“(x−x0)·(x−x0)

ε

”, (18)

where ε = 0.0001 cm2 controls the standard deviation of the distribution, withx0 = (x0, y0) in two dimensions and x0 = (x0, y0, z0) in three dimensions.The initial distributions are then scaled so that the maximum value of thecell density is 1, representing a cell or small group of cells where the tumourstarted to grow. A sample initial condition for Patient 1 can be seen in Figure7 (c). In this particular case, x0 = (13.52, 11.95).

For the growth and diffusion rates, it is important to use realistic values.For the growth rate r, this is straightforward, as we use r = 0.012/ day intwo dimensions [47]. In three dimensions, we use a slightly higher value ofr = 0.029/day to maintain a comparable growth/diffusion balance. The samepaper gives a biologically realistic value for the diffusion coefficient within thebrain of dref = 0.0013 cm2/day for isotropic spread. Because the anisotropicmodel (12) uses not only a spatially varying diffusion coefficient, but diffusion


tensors having different rates of diffusion in different directions, the tensorsare scaled so that the average rate of diffusion is equal to dref.

As an example, some of the tensor coefficients for Patient 1 are plotted inFigures 8 and 9. For the two-dimensional simulations, the axial slice of con-sideration is chosen to be that with the largest section of the tumour present.Figure 8 shows the two diagonal components of the two-dimensional tensor,with Images 8(a) and 8(b) showing D11(x) and D22(x), respectively, forκ = 2 and Images 8(c) and 8(d) showing D11(x) and D22(x), respectively,for κ = 20. The value for D11(x) is higher when the dominant direction ofdiffusion is horizontal, while D22(x) is higher when the dominant direction isvertical. We see that the fibre structure is more strongly defined for a largervalue of κ.

Figure 9 shows similar plots for Patient 1 in three dimensions. Slices areshown through the same axial slice of consideration, but now the three diagonalcomponents D11(x), D22(x), and D33(x) of the three-dimensional diffusiontensor are shown. These quantities correspond roughly to fibres in the x, y,and z directions. Images 9(a), 9(b) and 9(c) show the results for κ = 2,while Images 9(d), 9(e) and 9(f) show the results for κ = 20. Images9(a) and 9(d) show D11(x), Images 9(b) and 9(e) show D22(x) andImages 9(c) and 9(f) show D33(x). Again, the fibres are much more distinctfor the higher κ value. This emphasizes that the concentration parameterk(x) = κFA(x) depends on the structure of the brain, incorporated throughboth the fractional anisotropy (16) and through the patient-specific anisotropyparameter κ.

3.1 Swanson’s PI Model Implementation

Let us outline in more detail the implementation of the PI model in Equation(10). The spatially dependent diffusion coefficient d(x) is defined as

d(x) =

dg if x ∈ grey matter,dw if x ∈ white matter.

Because of the increased diffusivity in white matter, Swanson takes dw = 5dg

[47]. Note also that this model uses exponential growth, while our anisotropicmodel (12) uses logistic growth. It will be seen in the cell density plots thatthis growth will make a difference in the tumour composition.

There remains the question of how to separate the brain into white matterand grey matter. In [47], an online database that distinguishes between greyand white matter is used; however, since we apply this model to patient data,we need to make this distinction for each patient individually. This will bedone using a threshold FA value, with the assumption that white matter willhave an FA value greater than 0.25, while grey matter will have an FA valueless than 0.25. This value is consistent with that used in fibre tracking; see forexample [34] and [31]. The results of such a division are shown in Figure 10for a sample three-dimensional axial slice for Patient 1.


(a) (b)

(c) (d)

Fig. 8: The spatial distribution of the diagonal elements of the two-dimensional diffusiontensors for Patient 1. The axial slice of consideration is chosen as the slice containing thelargest portion of the tumour. Images (a) and (c) show D11(x) and Images (b) and(d) show D22(x). Images (a) and (b) correspond to κ = 2, while Images (c) and (d)correspond to κ = 20. Note that the fibres are far more pronounced for higher values of κ.

3.2 Parameter Estimation

For the anisotropic model of (12), there are either three (in two dimensions)or four (in three dimensions) parameters that can be used to fit the model.These are the initial condition (x0, y0) and (x0, y0, z0), respectively, plus theanisotropy parameter κ. This parameter controls the width of the peaks in thefibre distributions, and we tune it to each specific patient. We assume that κ isspatially consistent for each patient and is a characteristic of the cancer cells.We will also fit the PI model (10), which only depends on the initial tumourlocation (x0, y0) and (x0, y0, z0), respectively.

To fit the models to data, we sweep the parameter values over a plausibledomain and maximize the Jaccard index. In Figure 11, we give an example ofthis procedure for a fit of the PI model to a two-dimensional slice of the datafor Patient 1. The Jaccard index over a range of possible initial conditionsshows a clear maximum at (x0, y0)=(13.6,11.9), which is selected as the initialcondition.

Unfortunately, it is not known how old each tumour is and how long ittook to grow to its current size. As such, in our model, we use the Jaccardindex to determine an appropriate stopping criteria. As the simulated tumours


(a) (b) (c)

(d) (e) (f)

Fig. 9: The spatial distribution of the diagonal elements D11(x), D22(x), and D33(x) of thethree-dimensional diffusion tensors for Patient 1. These plots are shown for the same axialslice as considered for the two-dimensional case. The images are as follows: (a), (d):D11(x), (b), (e): D22(x), and (c), (f): D33(x). Images (a), (b) and (c) show resultsfor κ = 2, while Images (d), (e) and (f) show results for κ = 20. Note that the fibresare more pronounced for the higher value of κ.

Fig. 10: (a) The FA on a scale from 0 (dark blue) to 1 (yellow) for Patient 1. Note thatanything with a value above 0.25 will be classified as white matter. (b) The correspondingdistinction between grey and white matter. The white matter appears white, while the greymatter appears grey.


Fig. 11: The Jaccard index plotted over the parameter space for the PI model. Patient 1is shown as an example. A simulation is initialized at each point and run to com-pletion, and the Jaccard index corresponding to that simulation is plotted. Theoptimum value occurs at (x0, y0)=(13.6,8.1), hence this is selected as the initial condition.This is marked on the plot as a black point.

begin to grow, they will first be smaller than the actual tumour; as they growlarger, the Jaccard index will increase. At some point, this index is maximized,and after this point, the simulated tumour will outgrow the actual tumour,causing the Jaccard index to decrease. It is at this point of optimal Jaccardindex that the simulation is stopped. See Figure 12 to see how this works forPatient 1. Notice that there is some transient behaviour for small times, beforethe diffusion and growth find an appropriate balance.

4 Results

4.1 Two-Dimensional Results

Here we show the simulation results of the anisotropic model (12) and the PImodel (10) on a two-dimensional slice through the middle of each patient’stumour. Each figure in Figures 13–16 corresponds to the results for a patient,with the anisotropic simulation on the left and the PI model simulation onthe right. The actual segmented tumour boundary is plotted in black, whilethe model-predicted boundary is shown in white. The top row of each figureshows these boundaries overlaid on the FA plot to give an idea about thebrain architecture of that particular patient, and the bottom row shows thecell density function u(x, t).

In Table 2 the Jaccard index for each model and for each patient is shown,as well as the value of the anisotropy parameter, κ, for the anisotropic model.The same results are represented visually in Figure 17.


Fig. 12: (i) Figure showing how the Jaccard index of Equation (17) changes with time. As anexample, the anisotropic model is applied in two dimensions to Patient 1. As more iterationsare run, the Jaccard index increases; however, after some maximum is reached, it will beginto decrease again as the simulated tumour outgrows the actual tumour. The Jaccard scoreas a function of time is shown in the first image. The other three images show thecomputed tumour boundary (white outline) compared to the registered tumour(black outline) for t = 200 days, t = 258 days and t = 400 days. The backgroundcolours of the last three images show the fractional anisotropy.

Table 2: Jaccard indices for 10 pa-tients for the anisotropic model(12) and the PI model (10) in twodimensions. The corresponding κvalue that maximized the Jaccardindex for the anisotropic model isalso shown.

Pnt. Aniso. κ PI1 0.8811 0 0.88032 0.7809 3 0.77863 0.6467 1 0.56394 0.8633 1 0.83775 0.6424 0 0.65826 0.6263 5 0.60727 0.9241 0.5 0.90578 0.9075 0.5 0.88199 0.7994 0.5 0.768910 0.8406 2.5 0.7581

Fig. 17: Visual comparison ofJaccard indices for both theanisotropic model (blue) andthe PI model (green) in twodimensions.

We make the following observations:


Anisotropic model PI model

Patient 1

Patient 2

Patient 3

Fig. 13: Two-dimensional simulation results for Patients 1–3. The results on the left are theresults for the anisotropic model, while the results on the right are from the PI model. Ineach case, the black contour shows the segmented tumour, and the white contourthe model-predicted boundary. The first row shows the boundaries overlaid onthe FA for the patient, and the second row shows the model-predicted celldensity, with yellow=high and blue=low.



Patient 4

Patient 5

Patient 6




Patient 7

Patient 8

Patient 9




Patient 10

Fig. 16: Two-dimensional simulation results for Patient 10. The results on the left are theresults for the anisotropic model, while the results on the right are from the PI model. Ttheblack contour shows the segmented tumour, and the white contour the model-predicted boundary. The first row shows the boundaries overlaid on the FA forthe patient, and the second row shows the model-predicted cell density, withyellow=high and blue=low.

– We notice from the two-dimensional results that the anisotropic modelshows an improved fit over the PI model in nine out of ten cases.

– We also notice that the optimal value of the anisotropy parameter κ is rel-atively low in general, indicating a weak dependency of the tumour growthon the underlying structure.

– Looking at the population contours in the second row of each patient sim-ulation, we notice a difference between the anisotropic model and the PImodel, related to the different growth functions used in each model. Forthe PI model, which uses an exponential growth function, the tumourshave very dense centres that taper off very quickly. For the logistic growthfunction used in the anisotropic model, the transition in cell densities ismore gradual.

– Considering certain patients specifically, we can see that the anisotropicmodel is able to replicate certain irregularities in tumour shape. For exam-ple for Patient 2, as seen in Figure 13, the contour for the anisotropic modelclosely follows the segmented tumour boundary in the top right corner.

– Patient 3 was a particularly difficult patient to fit, due to the irregularshape of the tumour, as seen in Figure 13. While neither model obtained agreat fit, the anisotropic model offered a significant advantage over the PImodel, displaying the importance of the anisotropy. We will see later, inSection 4.3 that it is possible to achieve an even better fit for this patientusing the anisotropy of the other brain hemisphere.


– Patients 5 and 6 represent a failure of both models, as seen in Figure 14.Since the fitting was done in a completely objective, quantitative manner,the best fit was obtained by growing a tumour out from the boundary,which does not match the segmented tumour shape. A large part of thereason for this is due to the absence of mechanical effects in both theanisotropic model and the PI model. If a real tumour were growing nearthe skull, there would be a build-up of pressure, causing it to progress moreslowly in that direction. Several ideas for inclusion of the mass effect havebeen discussed in the literature (see Section 1.4), and the model needs tobe extended if the tumour grows close to the skull.

4.2 Three-Dimensional Results

In this section, we apply both the anisotropic model of Equation (12) and thePI model of Equation (10) in three dimensions to the same ten patients. Thisis a more realistic situation; however, we expect the fits to be poorer, since itis more difficult to fit full three-dimensional domains.

For brevity, we choose three patients (Patients 1,2,6) to illustrate the three-dimensional model, but the remaining patient simulations can be found in thesupplemental material. In the following Figures 18–20, the first row shows the0.16 isosurface of the cancer cell density u(x, t), intended to represent thetumour boundary as it would appear in a scan. For each patient, the firstcolumn shows the best fit of the anisotropic model, while the second columnshows the best fit of the PI model. The last column shows the actual tumouras it was segmented from the scans. The second row shows a two-dimensionalslice through the largest part of the tumour, to show specifically how closethe fit is. As in the two-dimensional results, the white line corresponds to thesimulated tumour boundary (0.16 level set of u(x, t)), while the black lineshows the actual tumour segmentation. The first column shows this image forthe anisotropic model, while the second column shows the same for the PImodel. The last image shows the fractional anisotropy in the slice in whichthe simulated anisotropic tumour was initiated. The black dot indicates theinitial condition (x0, y0) within this slice.

The results of the three-dimensional simulations are summarized in Table 3.The Jaccard index for the best fit for each patient for both the anisotropic andthe PI model are shown. Additionally, the value of the anisotropy parameterκ that achieved the best fit for the anisotropic model is shown. Figure 21 alsoshows a visual representation of these results.


Patient 1

Fig. 18: Simulation results for Patient 1 in three dimensions. The first column shows theresults for the anisotropic model, the second column for the PI model. The third columnshows the tumour segmentation in the first row and a two-dimensional slice of the FAin the second row. The first row shows the tumour boundary as an isosurface ofthe model-predicted cell density in green, along with the brain for reference inpink. The second row shows a two-dimensional slice through the largest portionof the tumour. The model-predicted cell density is plotted with yellow=highand blue=low, along with the tumour segmentation in black, and the model-predicted tumour boundary in white.

Patient 2

Fig. 19: Simulation results for Patient 2 in three dimensions. The first column shows theresults for the anisotropic model and the second column displays those for the PI model.The third column shows the tumour segmentation in the first row and a two-dimensionalslice of the FA in the second row. The first row shows the tumour boundary asan isosurface of the model-predicted cell density in green, along with the brainfor reference in pink. The second row shows a two-dimensional slice throughthe largest portion of the tumour. The model-predicted cell density is plottedwith yellow=high and blue=low, along with the tumour segmentation in black,and the model-predicted tumour boundary in white..


Patient 6

Fig. 20: Simulation results for Patient 6 in three dimensions. The first column shows theresults for the anisotropic model and the second column displays the results for the PI model.The third column shows the tumour segmentation in the first row and a two-dimensionalslice of the FA in the second row. The first row shows the tumour boundary asan isosurface of the model-predicted cell density in green, along with the brainfor reference in pink. The second row shows a two-dimensional slice throughthe largest portion of the tumour. The model-predicted cell density is plottedwith yellow=high and blue=low, along with the tumour segmentation in black,and the model-predicted tumour boundary in white.

Table 3: Jaccard indices for each of10 patients for both the anisotropicmodel and the PI model in threedimensions. The corresponding κvalue that maximized the Jaccardindex for the anisotropic model isalso shown.

Pnt. Aniso. κ PI1 0.6506 0.5 0.70022 0.6063 5.5 0.56693 0.4888 8 0.43574 0.6887 1 0.63325 0.6663 4.5 0.64536 0.4912 1 0.47947 0.6664 4 0.61498 0.6368 0 0.59309 0.6275 0.5 0.571910 0.7399 2.5 0.7156

Fig. 21: Visual comparison ofJaccard indices for both theanisotropic model (blue) andthe PI model (green) in threedimensions.

We make the following observations:


– We see directly that the anisotropic model reliably performs better than thePI model. This is to be expected, since more biological detail was included.

– We further observe that frequently a κ value of 0 optimizes the fit, meaningthat the anisotropic model does best when it is tuned to isotropic spread.Fortunately, there is a good explanation for this phenomenon. Lookingback to figure 4, we observe that the fibre networks within the tumours aredisplaced or destroyed, leaving behind tissue that is in many cases almostisotropic. Running the model using this DTI data then does not give muchadvantage over an isotropic method, which is not surprising. This issue willbe addressed and remedied in the section following the three-dimensionalresults.

– Looking deeper at these results, we can see that generally the anisotropyparameter was higher than in two dimensions, giving tumour shapes thatare more dependent on the brain architecture.

– Overall, however, we notice that the fits are quantitatively poorer than theywere in two dimensions. This is due to the difficulty in fitting the full three-dimensional model, as this amounts to simultaneously fitting many axialslices. With that said, many of the Jaccard indices achieved an impressivelevel of accuracy.

– Looking specifically at Patient 2, i.e. Figure 19, we can see that both modelfits are significantly smoother than the segmented tumour. This is due tothe fact that segmentations are done on individual two-dimensional slicesand then recombined to form the three-dimensional tumour. This high-lights the potential problems with using a two-dimensional seg-mentation technique, since any error in a two-dimensional seg-mentation creates a large discontinuity between that slice andthose adjacent to it. This suggests that a full three-dimensionalmodel may more adequately capture the real tumour shape.

– Also of note for all of the patients is the difference in cell density ranges.Due to the exponential growth of the PI model, the resultant tumourspossess very dense cores that rapidly drop off in density. The exponen-tial growth also generates very high density values, in contrast to theanisotropic model, which does not predict growth at densities greater than1.

– We encounter a failure of the model with Patient 6, as we did in twodimensions. We can see in the third column of the second row of Figure 20that the initial condition (black dot) is not actually contained within thebrain domain, which is obviously not realistic. This occurred due to theobjective nature of the model fit, as this was the configuration that gavethe highest Jaccard index. The main problem is that, as we saw in twodimensions, both the anisotropic model and the PI model do not includea mass effect.


4.3 Reflected DTI

We saw above that there is an issue with the fitting of some of the patient data.Looking back at Figure 4, we can see that within the tumour segmentations(black outline), there is not a lot of variation in the fractional anisotropy. Thisis because, as the tumours grow, they change the brain structures, pushingnerve fibres out of the way, or destroying them altogether. If we then usethis DTI data to simulate tumours, we see mostly isotropic growth, whichdoes not reflect the actual growth history of the tumour. As a remedy forthis, we use the symmetry of the brain to estimate the anisotropy of thetumour region at the time when the tumour started growing. There weresome simulations above that achieved very good fits even with the originalDTI data. We select the 5 patients (2, 3, 5, 6, and 9) with the lowest Jaccardindices from Table 2 and try to improve them by implementing the reflectedDTI technique. We don’t expect to see a large change in the resultsfor the remaining 5 patients, as their κ values are generally low. Assuch, their dependence on the underlying structure is weak. A similartechnique was employed in [40], where brain symmetry was used for tumoursegmentation using MRI scans.

We present the results for the anisotropic model, using the reflected DTIdata in two and three dimensions in Table 4. These results are also summarizedin Figure 23, where a visual representation is provided. We only show two ofthese results visually for Patients 3 and 9 in Figure 22.

Anisotropic model Original DTI Reflected DTI

Patient 3

Patient 9

Fig. 22: Two dimensional simulation results using original and reflected DTI data for theanisotropic model. Results are shown for Patient 3 (top) and Patient 9 (bottom), with boththe original FA and the reflected FA shown. The model fits can be improved for somepatients by reflecting the DTI data from the other brain hemisphere to replacethe structures that have been displaced by the tumour growth.


Table 4: Jaccard indices for Patients 2, 3, 5, 6, and 9 for the anisotropic model with bothoriginal and reflected DTI data in two and three dimensions. Using reflected DTI datafurther improved the fits in three out of the five cases.

Patient Original DTI Reflected DTI Original DTI Reflected DTI2D 2D 3D 3D

2 0.7809 0.7886 0.6063 0.61463 0.6467 0.7194 0.4888 0.49005 0.6426 0.6156 0.6643 0.65676 0.6269 0.5571 0.4912 0.45769 0.7990 0.8209 0.6275 0.6371

Fig. 23: Visual comparison of Jaccard indices for both the 2D anisotropic results (blue), the2D anisotropic results with reflection (green), the 3D anisotropic results (yellow), and the3D anisotropic results with reflection (red).

Application of the reflected DTI technique improved the results for threeout of the five patients for which it was attempted. It should be noted thatthe two for which it did not lead to an improvement were the same two pa-tients for which the models failed originally, i.e. Patients 5 and 6. Patient 3is perhaps the most successful application of this technique, and was also oneof the most challenging patients to fit initially. It is interesting to look at theFA plot in the top row of Figure 22 and to see how the tumour segmenta-tion matches up to the fibre directions. For Patient 9, a slight improvement inJaccard index was seen when the reflected DTI technique was implemented.However, the qualitative improvement in this case was significant, as can beseen in Figure 22. Overall, these results seem to indicate that for more chal-lenging, more anisotropic tumour shapes, filling in the DTI data by reflectingit over the brain’s centreline seems to offer the potential for improved modelperformance. It should be noted, however, that the use of the reflectedDTI technique should be evaluated on a patient-by-patient basis todetermine its utility.


5 Towards Anisotropic Treatment Regions

The goal of the anisotropic model is to simulate glioma invasion in a mannerthat will be useful to clinicians during treatment planning. In this section weoutline how the anisotropic model may be used to delineate clinical targetvolumes for radiation therapy. However, further model validation is neededbefore the model can be applied in a clinical setting.

As discussed in the introduction, typical treatment for glioma involvestreating the visible tumour as well as an extension to account for microscopicdisease spread. Clinically, these are referred to as the Gross Tumour Volume(GTV) and the Clinical Target Volume (CTV), respectively [7] (see Figure1). The GTV is defined as the mass that “can be seen, palpated or imaged”[7]. Defining the CTV for a glioma patient is not a trivial task, and is usuallydone in a naive, non-patient-specific manner by extending the GTV uniformlyto a distance of 2 cm. While this is a valid method, our anisotropic diffusionmodel can be used as a tool by clinicians to predict where the most invasionhas occurred, and thus advise where the CTV should extend the furthest.We compare this modified, model-suggested region to the traditional uniformextension for two of our patients, shown in Figure 24. The GTV is outlined inblack, the traditional CTV in white, and the modified CTV in red for Patients2 and 8. The modified CTV was drawn using a level set of the anisotropicsimulation results, while the traditional CTV is simply a 2 cm extension ofthe segmented tumour. The appropriate level set was chosen so that the regionscontained the same area, so that the same amount of tissue would be treatedin each case. Note that the reflected-DTI results were used to determine thetreatment region for Patient 2. For tumours with a low anisotropy coefficientκ (i.e. Patient 8), the two CTV’s are very similar, however, for patients witha more anisotropic tumour (larger κ), such as Patient 2, there may be largediscrepancies.

6 Conclusions

In this study we propose a fully anisotropic cell density model for gliomagrowth and spread, with the model given in Equation (12). While the detailedmodelling and mathematical analysis was done elsewhere [18,39,19], here wefocus on a comparison of the anisotropic model simulations to those of thestate-of-the-art PI model of Swanson et al.. We show simulation results forten patients for both the PI model of Equation (10) and the anisotropic modelof Equation (12) and compare these model simulations to the patient databy computing the optimal Jaccard index for both models in two and threedimensions. The anisotropic model improves on the PI model in nine out often cases in both two and three dimensions. We therefore conclude that theinclusion of anisotropy allows for better resolution of glioma growth.

A critical parameter in the model fit is the patient specific anisotropycoefficient κ, which indicates the significance of anisotropy in the spread of a


Fig. 24: The GTV in black, the traditional CTV in white, and the modified model-suggestedCTV in red are shown for Patients 8 (left) and 2 (right) in two dimensions. We see that forpatients whose tumours display more isotropic spread, there is little difference between thetraditional and modified CTVs; however, some patients show a large discrepancy betweenthe two, suggesting a potential benefit in using mathematical models to helpdetermine the CTV and PTV.

given brain tumour. Values below 0.5 indicate isotropic spread, while valuesabove 2 indicate anisotropic spread. We initially found that many of the fitsfeature anisotropy parameter values that are low, indicating that there is onlya small advantage over the isotropic model. The reason for this is that the DTIdata is altered within the tumour segmentations, with many of the fibre tractsmissing, as they have been dislodged by the growing tumour or even destroyedcompletely. To remedy this, we take advantage of the symmetry present in ahealthy brain, reflecting the DTI data from the other brain hemisphere toapproximate the missing data.

We considered the results of simulating both the PI model and the anisotropicmodel on the reflected data for a subset of five patients, in both two and threedimensions. To choose this subset, we took the lowest five Jaccard indices, tosee if the reflected-DTI technique could improve upon the fits. In the two-dimensional case, this process improved the Jaccard index for three out of thefive patients. In some cases, the advantage was substantial, particularly wherethe tumours had an elongated shape. In three dimensions, the advantage wasnot so substantial: although the reflected-DTI technique improved the Jaccardindex in three out of the five patients, the improvement was less pronouncedthan in the two-dimensional case. We postulate that this is due to the factthat in two dimensions, when the fibres are dislodged by the tumour, they areno longer present in the plane of consideration. In three dimensions, while thefibre network may be distorted, the displaced fibres remain within the domain.

A critical modelling component that is still missing from our model isthe mass effect. The mechanics of brain tissue is far from being trivial; theliterature offers at least three models for the mass effect in the brain (see [2,8,32,41]). It is the aim of future research to compare and contrast the differentmechanical models and to apply them to the same patient data as studied


here. In particular, we expect improvements upon accounting for the masseffect in Patients 5 and 6.

Acknowledgements AS would like to acknowledge funding from an NSERC CGS D3scholarship and an Alberta Innovates Technology Futures Graduate Student Scholarship.Both TH and JB are supported through NSERC discovery grants. The DTI images were ac-quired through an Alberta Cancer Foundation sponsored grant aimed at using DTI imagingto help predict the pattern of glioma growth.

This work is a collaboration between mathematicians at the University of Albertaand clinicians at a nearby cancer hospital. Brain tumour growth and treatments werediscussed in a weekly working group, which included members from Computer Science,Mathematics, and clinicians from the cancer hospital. The weekly meetings allowed usto learn and understand the approaches and the languages of the other disciplines. Theclinicians were able to provide specific patient data, and the initial emphasis of thegroup was on the development of machine learning methods. In parallel, we developedthe anisotropic diffusion model presented here and applied it to the same data.

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