development of a test bench for gamma knife optimization699782/fulltext01.pdf · 2.1.1 leksell...

Development of a test bench for

Gamma Knife Optimization

V I C T O R I A S V E D B E R G

Master of Science Thesis Stockholm, Sweden 2014

Development of a test bench for

Gamma Knife Optimization

V I C T O R I A S V E D B E R G

Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2014 Supervisor at KTH was Johan Karlsson Examiner was Johan Karlsson TRITA-MAT-E 2014:02 ISRN-KTH/MAT/E--14/02--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

The dose distribution in the Gamma Knife is optimized over the weights (or Beam-on time) usingdifferent models other than the radiosurgical one used in Leksell Gamma PlanR©. These are basedon DVH, EUD, TCP and NTCP. Also adding hypoxic regions are tested in the Gamma Knife tosee whether or not the dose can be guided to these areas. This is done in two ways. For the DVHand EUD model the hypoxic area is regarded as a organ by itself and higher constraints is definedon it. In the TCP case blood vessels are outlined and the α and β parameters are perturbed todescribe a hypoxic area. The models are tested in two cases. The first one is one tumour close to thebrainstem and the second case is two tumours located far away from each other. Finally the resultsare compared to the dose distribution computed by the Gamma Knife.

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CONTENTS CONTENTS

Contents

1 Introduction 1

2 Background 22.1 Radiosurgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Leksell Gamma Knife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 The Gamma Knife Treatment Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Leksell Gamma Plan R© . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Dose-Volume Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Radiobiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 What happens in an irradiated cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 LQ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 TCP and NTCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Equivalent Uniform Dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.5 Hypoxia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Comparison between DVH, EUD, TCP and NTCP . . . . . . . . . . . . . . . . . . . . . . 14

3 Optimization of the Beam-on time 153.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The Radiotherapeutical model-based on DVH . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Maximum and minimum dose-constraints . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Dose-Volume constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.3 The resulting DVH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 The Radiobiological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 gEUD-based objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 TCP/NTCP-based objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Results and Discussion 204.1 Case 1: One tumour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1.1 Results DVH without hypoxic region . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 Results DVH with hypoxic region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Results EUD without hypoxic region . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.4 Results EUD with hypoxic region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.5 Results TCP without hypoxic region . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.6 Results TCP with hypoxic region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Case 2: Two tumours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.1 Results DVH without hypoxic area . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Conclusion 435.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A Optimization algorithms 44A.1 Active-set algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.2 Sequential Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.3 Interior-point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.4 Trust-region-reflective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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1 INTRODUCTION

1 Introduction

The Leksell Gamma Knife (developed and produced by Elekta) is an instrument for treating braintu-mors and various malfunctions in the brain. To optimize the given dose in a treatment using the LeksellGamma Knife, a radiosurgical model is used. In radiosurgery, high-energy radiation is aimed at thetarget (most often a tumour) in order to destroy the target cells. The radiosurgical model is based oncoverage, selectivity and gradient index to achieve the aim, which is to cover the target (and only thetarget) with the planned dose and make the dose drop rapidly outside the target for it to quickly drop,thus harming the healthy tissue outside the target as little as possible. In this model no account is takenfor the riskstructures in the brain and if the target is big, the planning-time tends to become large dueto the large number of degrees of freedom.

The goal when using radiotherapy when treating a tumour is to destroy the DNA, which In radiotherapy,the models which are based on the Dose Volume Histograms (DVH), are the most common for optimizinga treatment plan. DVH describes the percentage of a target or a risk-structure which achieves a certaindose. Because of the non-convexity of the DVH and in order to use fast optimization techniques, sim-plifications have to be done. These simplifications are called DVH-surrogates and are widely used in thefield of radiotherapy, e.g. at Princess Margaret Hospital (PMH) in Toronto, researchers have replacedthe radiosurgical treatment plans in the Gamma Knife with DVH-surrogates.

Radiobiology aims to explain the biological response in the tumor tissue and in the organs at risk whenirradiatated with high-intensity ionizing radiation [12]. Radiobiological models are more and more com-mon in cancertreatment devices. These models are usually based on Tumour Control Probability (TCP),Normal Tissue Complication Probability (NTCP) and Equivalent Uniform Dose (EUD). One problemusing radiobiological models is the large uncertainties in the parameters. Along with the number ofparameters used these models may become very uncertain in predicting the outcome of the treatment.

In some tumours there may arise areas where there are less molecular oxygen than in the rest of thetissue. These areas are called hypoxic areas and are more resistant to radiation and higher dose must begiven in these areas to achieve tumour control. The model parameters for TCP change for it to expressthe stronger radio resistance.

Since combination treatments using the Gamma Knife and radiotherapy is likely to grow, having bothmodels available in the Gamma Knife and also testing and comparing the radiotherapeutical model inthe Gamma Knife against the radiosurgical one, will give an important aspect of the dose planning inthe future. Also, since radiobiological models become more common, using these models in the GammaKnife is also an important future aspect to consider.

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2 BACKGROUND

2 Background

In this thesis, models based on Radiobiology and Radiotherapy are tested in the Radiosurgical instrument,the Gamma Knife. Therefore, a brief introduction into Radiosurgery, Radiotherapy and Radiobiologywill be given and along witg a technical overview of the Gamma Knife.

2.1 Radiosurgery

Borje Larsson (professor in Radiobiology at Uppsala University) and Lars Leksell (neurosurgeon atKarolinska University Hospital) developed the concept of radiosurgery in 1951 [19]. This concept wasto direct a single high dose of radiation to the intracranial region of interest stereotacticly. The wordstereotactic is commonly used as a prefix to radiosurgery and refers to the three-dimensional coordinatesystem which makes it possible for the treatment planner to identify and use the coordinate systems ofthe diagnostic images and the actual position of the patient, thus simplifying the treatment and raisingthe accuracy of hitting the planned shot position (also called isocenter). The treatment plan is done inthe Leksell Gamma Plan.The fundamental principle of radiosurgery is that of ionization of target tissue (for instance a tumour),by means of high-energy radiation, usually gamma radiation. By ionization the tissue, the number ofions and free radicals (an atom, molecule or ion with unpaired valence electrons) which are harmful tothe cells increase (more about this in Section 2.3.1). These ions and radicals can produce irreparabledamage to DNA, proteins, and lipids, resulting in the cell’s death. Thus, biological inactivation is carriedout in a volume of the treated tissue. The radiation dose is usually measured in Grays, where one Gray(Gy) is the absorption of one joule per kilogram of mass.Radiosurgery is performed by a team of neurosurgeons, radiation oncologists, and medical physicists tooperate and maintain very sophisticated, accurate and complex instruments. The precise irradiation oftargets within the brain (and upper part of the spine) is planned using information from diagnistic imageswhich are obtained via computed tomography, magnetic resonance imaging, positron emisson tomographyand angiography. An advantage of stereotactic treatments is the delivery of the right amount of radiationto the tumour in one or a few fractions compared to traditional treatments, which is often delivered in 30fractions over a timeperiod up to 10 weeks. Also, treatments are given with extreme high accuracy, whichlimits the effect of the radiation on the surrounding healthy tissues due to the sharp dose gradients. Oneproblem with stereotactic treatments is that they are only suitable for small and medium sized tumors,since the treatment time may increase considerably for large tumours.

2.1.1 Leksell Gamma Knife

The Leksell Gamma Knife (or shortly the Gamma Knife) is a medical device for treatment of braintumours, developed by the company Elekta AB. The Gamma Knife was first developed by Borje Larsson,Ladislau Steiner and Lars Leksell in 1967 and is based on the concept of radiosurgery. The most recentmodel is Leksell Gamma Knife PERFEXION R© presented in Figure 1. It was introduced in 2006. in thissection a small technical overview of the Gamma Knife is given.

Figure 1: The Gamma Knife PERFEXION R©

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2.1 Radiosurgery 2 BACKGROUND

The most recent model is Leksell Gamma Knife PERFEXION R© presented in Figure 1. It was introducedin 2006. The Gamma Knife treats benign and malignant tumours and malfunctions in the brain byaiming gamma radiation on the tumour cells. The radiation source in the Gamma Knife PERFEXION R©

is Cobalt-60 and each of the sources is distributed in five circular rings in the device which is dividedinto 8 sectors. By radiating from each sector the gamma radiation is aimed at a target in the brain,hence focusing the rays to a so called isocenter as in Figure 2. The isocenters may have different shapes.The Gamma Knife is heavily shielded by lead, iron and tungsten to avoid leakage of gamma radiation.To avoid errors from patient movement during the treatment an aluminum frame is surgically fixed tothe skull. This frame is fixated to the Gamma Knife in order for the head to remain stationary duringthe treatment. The frame becomes a reference for the coordinate system when calculating dose (hencea Stereotactic frame).

Figure 2: Conceptual picture of the Gamma Knife.

Focusing the radiation to an isocenter is the most important objective of the Gamma Knife (and inradiosurgery overall). By focusing the radiation, the intensity outside the focus point can be low enoughto spare the tissue from any severe damage, but in the focus point become high enough to kill thetumour cells. Hence, the tumour cells die while the healthy tissue are relatively spared. This can resultin shrinkage or a complete disappearance of the tumour. Gamma Knife radiosurgery has proven effectivefor patients with benign or malignant brain tumors up to 4 centimeters in size, vascular malformationssuch as an arteriovenous malformation (AVM), pain or other functional problems [11], [17], [14], [25]. Therisks of gamma knife treatment are very low, and complications are most often related to the conditionbeing treated.

2.1.2 The Gamma Knife Treatment Plan

It is physically impossible to find an ideal treatment plan in any cancer treatment systems, giving no doseto the organs at risk and healthy tissue and 100% to the tumour. Hence one has to create a plan, whereconsiderations of the importance between the different organs at risk and healthy tissue are taken. Thetumour and organs at risk are outlined by a clinician, who also sets the constraints in the optimizationproblem and the importance of the organs to give a desired plan. A desired plan is a plan giving a highdose in the target and a low dose to the risk-structures. In order to automatically plan the optimal dosedelivery in the Gamma Knife, a possible solution strategy is solving the following two problems:

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2.1 Radiosurgery 2 BACKGROUND

• Isocenter-packing

• Full optimization

Solving the isocenter-packing problem1 gives the positions of the isocenters (the coordinates where thegamma rays are converging as well as their shape) and hence the initial number of isocenters are given.The algorithm is constructed as follows: first the target is covered by as large shot as possible and theisocenters are placed so that it touches the target volume periphery, without overlapping other shots toomuch. When no more shot positions exists, even for the smallest volume, the volume covered so far istreated as non-target and the algorithm is repeated with the reduced target volume starting with thelargest shot. Thus the target is filled from the surface and inwards [9]. A short graphical description ofthis algorithm i given in Figure 3. Only a subset of all isocenter shapes (isocenter states) are used and ashape of an isocenter is defined by x|x ∈ R3, D(p, x) ≥ 0.5Dmax, where p is the isocenter position, Dis the dose and depends on the size of the shot and Dmax is the maximal dose.

Figure 3: Illustration of the isocenter packing. A subset of all isocenter shots are used. The aim is tofill out the boundary as much as possible, then remove the isocenters, and fill the new volume. In thefirst (left) isocenter-packing, all isocenter shapes are allowed on the first boundary. In the right one thelargest shot is not allowed on the first boundary. Image from [9]

Now the optimization problem may be solved. It is based on coverage, selecticity, gradient index andBeam-on time. These are stated in Equations 1, 2, 3 and 4. Here PIV is Planning Isodose Volume, thetissue covered by the planned isodose, and TV is Target Volume, i.e. the volume of the target (for examplea tumour). ISO is the level of the isodose in percentage. Now, isocenter positions, isocenter-states andweights are varied freely.

Figure 4: Wenn-diagram of PTV and TV. Image from [9]

Coverage: C =V (PIV ∩ TV )

V (TV )(1)

1More info http://www.elekta.com/dms/elekta/elekta-assets/Elekta-Neuroscience/Gamma-Knife-Surgery/pdfs/LGP-Inverse-Planning-white-paper–Art-No–018880-02-/White%20Paper%3A%20Inverse%20Planning%20in%20Leksell%20GammaPlan%C2%AE%2010.pdf

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2.2 Radiotherapy 2 BACKGROUND

Selectivity: S =V (PIV ∩ TV )

V (PIV )(2)

Gradient index: GI =V (PIVISO/2)

V (PIVISO)(3)

Beam-on Time: Tbeam−on =

Niso∑i=1

Tbeam−on,i (4)

CostFunc =Cmin(2α,1) · Smin(2(1−α),1) + βGrad+ γT ime

1 + β + γ

where α, β, γ ∈ [0, 1] are the weights befined by the user.

(5)

Here V (A) is the volume of the set A, Niso is the number of isocenters and Tbeam−on,i is the beam-ontime from isocenter i. The objective is demonstrated in Equation 5 where Grad is ae function of thegradient index and Time is a function of the beam on times. Organ at risk are not being consideredin the objective function. Instead poor selectivity is penalized so all tissue outside the target volumeis treated as risk structures. A poor gradient index can also be penalized to create a steeper fall off oflower isodoses.For the dose calculations two different calculations are performed. Those are

• Convolution algorithm

• TMR-10 model

TMR-10 algorithm2 models all tissue in head as water [10]. This is proven to be good for targets incenter of brain, but less accurate for heterogenities in the brain and peripheral regions. The convolutionalgorithm3 takes tissue heterogenities into account in the dose calculation [8] and is only used to check thedose plan. However both give a mutually consistent result in a water phantom. During the optimizationin the Gamma Knife, only the TMR-10 algorithm is used.

2.1.3 Leksell Gamma Plan R©

The Leksell Gamma Plan R© is the software for the treatment planning system for the Gamma Knife. Itprovides a user friendly work system to the planner where all diagnostic images of the patient are collectedand the dose delivery is planned. Using the CT-scan, the MR-image is fitted into the coordinates ofthe Gamma Knife and slices of the head are displayed. In these it’s possible to outline the tumour andorgans at risk and perform an isocenter-placement (organs at risk are only consider in the isocenter-placement). Then an optimization of the dose-distribution is performed, during which one may placeout extra isocenters or remove an isocenter until a good plan is created. To investigate a plan isodosesand DVH:s are displayed. This plans are usually done by a neurosurgeon, or a medical physicist, but theplan must be approved by a neurosurgeon or an oncologist. When a plan is approved, the plan is sentto the Gamma Knife and executed.

2.2 Radiotherapy

Radiotherapy, similar to radiosurgery, also utilizes ionizing radiation to kill cells by focusing one or morebeams to a planning target volume. In Conventional Radiationtherapy (CRT) the beams have a uniformintensity field which makes it harder to shape the field to avoid organs at risk (OARs). A progress wasmade in the 3D treatment planning by the introduction of Multileaf Collimators (MLC:s) to radiotherapy(first proposed by Brahme [3]). This was the beginning of Intensity-Modulated Radiotherapy (IMRT)where the radiation intensity across the beams can be modulated. This allows a better possibility ofshaping the dose distribution of the planning target volume and sparing the OARs and normal tissue.The difference in dose shaping between CRT and IMRT is demonstrated in Figure 5.

2More info http://www.elekta.com/dms/elekta/elekta-assets/Elekta-Neuroscience/Gamma-Knife-Surgery/pdfs/LGP-TMR-dose-algorithm-white-paper/White%20Paper%3A%20A%20new%20TMR%20dose%20algorithm%20in%20Leksell%20GammaPlan%C2%AE.pdf

3More info http://www.elekta.com/dms/elekta/elekta-assets/Elekta-Neuroscience/Gamma-Knife-Surgery/pdfs/LGP-Convolution-white-paper–Art-No–018881-02/White%20Paper%3A%20Convolution%20Algorithm%20in%20Leksell%20GammaPlan%C2%AE%2010.pdf

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Figure 5: Difference in dose shaping between CRT and IMRT. Image from IAEA.org

Figure 6: Illustration of MLC:s in operation.

Before radiation from a beam reaches the patient it has to pass through a MLC. The MLC consist ofa number of pairs of metal leaves, usually made of tungsten, which can move and position itself intodifferent shapes in order to block the radiation passing through the metal leaves and hence shape it. InFigure 6, the idea of dose shaping using multileaf colimators is presented. In order to model the beamintensities mathematically one discretize the beam into small ”bixels” or ”beamlets”.There is a close resemblance between radiosurgery and radiotherapy but is in a fundamental plan notbased on the same concepts. The aim of stereotactic radiosurgery is to kill target tissue with high dosewhile preserving the surrounding normal tissue utilizinf the sharp gradients of the Gamma Knife, wherefractionated radiotherapy with less sharp gradients relies on a different sensitivity of radiation of thetarget and the surrounding normal tissue to the total accumulated radiation dose.

2.2.1 Dose-Volume Histogram

Dose Volume Histograms, abbreviated a DVH is a commonly used in clinical practice in Radiotherapyand Radiosurgery due to its economical and easy way to represent the entire dose distribution in astructure. The lost information is the spatial resolution of the dose distribution. It was first suggestedby Bortfeld [1] and it is based on the physical dose. The planning structure considered (target or organat risk) is divided into a number of voxels. The set of all voxels v in the planning structure S whichattains dose d′ can be calculated using Equation 6.

vol : d′ 7→ p(d′) :=V (v : dv = d′)

V (S)(6)

In other words the DVH provides the volume percentage of the planning structure in which at least thedose d is attained, i.e. the DVH(d) can calculated using Equation 7.

DVH(d) =V (v : dv ≥ d′)

V (S)=∑d′≥d

p(d′) (7)

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The DVH is highly non-convex and non-convave. A simple example is that if x,y is the value of the dosein two voxels repsresenting the planning structure S. If x ≤ d and y ≤ d then 1

2 (x+ y) ≤ d. If x ≥ d andy ≥ d then 1

2 (x + y) ≥ d. But what is the case of 12 (x + y) ≥ d when x ≤ d and y ≥ d or the opposite

x ≥ d and y ≤ d. Convex optimization is not applicable in this case. Because of this, transferring theDVH into more computationally desirable properties is an important field of ongoing research. Thereare some ways to control the DVH. Firstly, one simple possibility is to add a maximum or minimum doseconstraint in the target and/or in the organs at risk. In Figure 7 a maximum and minimum constrainton the dose is demonstrated in a DVH-curve.

Figure 7: A maximum and minimum constraint on the DVH-curve. Image from [1]

For some organs maximum dose constraints is not meaningful at all. Organs like lungs and kidneysdon’t have a large volume effect. If a part of these organs fail to function, the other part can stillfunction without any major problem. In other words the tolerance dose in lungs and kidneys are notvery dependent on the irradiated volume percentage. In the organs with a large volume effect, for instancethe eye nerve, if a small piece of volume is irradiated, the tolerance dose is considerably lower than forlarger irradiated volumes. In the organs with a large volume effect one can utilize the properties of DoseVolume Constraints, or DVC is also a common way to put a constraint on a DVH. The constraint is thatno more than Vmax of the volume should receive more than a dose Dmax. In Figure 8 this is visualizedas a barrier with a corner at the point (Dmax, Vmax).

Figure 8: Visualization of a DV-constraint. Image from [1]

Unfortunately DVC-constraints lead to non-convex objective function [7][29]. Although it is shown thatthe resulting local minima in the overall objective function in simulations are of minor practical relevance[20].When the constraints is set, one can decide the importance of the constraints. The more important itis that a constraints is fulfilled the higher penalty factor is put on it. A graphical illustration of this ispresented in Figure 9

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2.3 Radiobiology 2 BACKGROUND

Figure 9: By using a small penalty factor on the maximum constraints some excess dose is allowedbeyond the constraint. By a larger penalty factor prevents any larger dose than Dmax. Image from [1]

2.3 Radiobiology

The aim of radiobiology is to describe what happens when a cell is irradiated. Reasearch is made onthis area today since it can contribute to the development of radiotherapy and radiosurgery in threeimportant ways.

• It may provide an extended conceptual basis for radiotherapy and radiosurgery, by identifyingmechanisms and processes that underlie the response for tumours and normal tissue to radiationand help to explain observed phenomena. Examples are knowledge about hypoxia, reoxygeniationand repair of DNA-damage (all these are explained later in Section 2.3.1).

• Development of treatment plans for radiotherapy and radiosurgery, for instance hypoxic cell sensi-tizers, to decrease the effect of an hypoxic environment (more in Section 2.3.5).

• Advice on the choice of schedules for radiotherapy and radiosurgery. For instance conversionformulas for changes in fractionation or dose rate. It may also provide a method for predicting thebest treatment for the individual patient.

The role of oxygen is one positive example that has led to benefits in treatments today. The currentmethod is empirical and further knowledge about cellular and molecular nature of radiation effects willlead to further development of the radiotherapy and radiosurgery [12].

2.3.1 What happens in an irradiated cell

When a cell is irradiated a number of processes commence. In Figure 10 the different phases and thetime-period when they occur is graphically illustrated.

Figure 10: The three different phases occuring after irradiation. Image from [12]

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At first there is a physical phase where interactions occur between charged particles and the atoms ofwhich the tissue is composed. A charged particle passes some living tissue and as it does, it interacts withthe orbital electrons. Some is ejected from the atom (ionization) and some is raised to a higher energylevel (excitation). If these are sufficiently energetic, these secondary electrons may excite or ionize otheratoms near which they pass, giving rise to cascade of ionizing events. The damage to the cell is eitherdirect or indirect ionization of the atoms which make up the DNA chain If there is molecular oxygennearby the damage to the DNA-strand may be fixed. This phase is illustrated in Figure 11.

Figure 11: Direct and indirect action of charged particles. Image from [12]

Secondly there is a chemical phase. This is the period when these damage atoms and molecules interactwith other cellular components in rapid chemical reactions. Ionization and excitation lead to the breakageof chemical bonds and formation of broken molecules known as free radicals. These are very reactiveand start a series of reaction which damage the DNA.Last there is a biological phase, which includes all subsequent processes. These begin with enzymaticreactions that acts on the residual chemical damage. The vast majority of lesions in for instance DNAare successfully repaired. Non-cancerous cells are able to reproduce even with slightly damaged DNA.The excited state of reproduction that cancerous cells are in means that a small amount of DNA-damagerenders them incapable of reproducing. Some rare lesions fail to repair and these eventually lead to celldeath (or mutation), a phenomenon used in radiotherapy. A cell takes time to die. After being irradiatedthey may undergo a number of mitotic diversions before dying. The radiation may also give rise to cellproliferation (increase in number of cells) and a very late effect is the appearance of a second tumour(called radiation carconogenesis). In the brain the injury in the central nervous system develop a fewmonths to several years after therapy[12].By illustrating dose response curves for both probability of tumour control and normal tissue complicationin the same graph one can see therapeutic window as in Figure 12. In this area, there is a high probabilitythat the dose is high enough to kill the tumour cells but not high enough to cause any big damages tothe normal tissue. Hence the dose for the treatment plan should occur in this area.

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Figure 12: The area between the two curves are called the therapeutic window.

Another factor to take into considerations is the volume effect of the organ, or the seriality i.e. howthe functional subunits is arranged. This effect is already mentioned in section 2.2.1. In the context ofseriality, an organ is usually described as a series and parallel circuits. For instance the optic nerve is anorgan with a high seriality, which would fit into the description of a series circuit. If tissue is destroyedso that the nerve is unconnected, the patient is likely to lose its sight. The lung is an organ with lowseriality. If a part of it is destroyed there is still a probability that the function is maintained.

Figure 13: Figure illustrating the seriality. If a link is broken the first chain is no longer connected, butthe second one is. Image from [23]

2.3.2 LQ-model

The LQ-model, or Linear Quadratic model is the most widespread model used to estimate the cell survivalafter radiation. This model is based on the assumption that there is a linear and a quadratic componentof the cell survival vs. the dose as demonstrated in Figure 14.

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Figure 14: Explanatory LQ-plots. Image from [12]

According to the LQ-model, the surviving fraction of target cells after a dose per fraction d, (SFd), isgiven as.

SFd = exp(−αd− βd2) (8)

where α and β is parameters fitted into the graph. An example of one such graph is in the left picturein Figure ??. One may compare the α

β -ratio between tissues. A high αβ -ratio implies that the tissue is

early responding to radiation, while a low αβ -ratio implies that the radiation effects comes after some

time. In the right picture in Figure ?? a comparison between the LQ-curves of a structure with earlyresponding tissue and late responding tissue is made. One limitation of the LQ-model is that it workswell for doses employed by conventional fractionated radiotherapy. For higher doses per fraction, as instereotactic radiosurgery, the LQ-model is less accurate.

2.3.3 TCP and NTCP

Two common radiobiological models are TCP (Tumor Control Probability) and NTCP (Normal-TissueComplication Probability). TCP and NTCP describe probability that the cells of the tumour or normaltissue dies under the radiation treatment dose. Brahme [2] came up with the Poisson-based TCP-function.

TCP (d) =

mT∏i=1

Pi where Pi = exp(−Noexp(−(αdi + βd2i )

)(9)

Here N0 is the number of chlonogenic cells (a simplification is made assuming that the number ofchlonogens in each voxel is the same), α and β is derived from the LQ-model in Section 2.3.2, mT is thenumber of voxels in the tumour, di is the dose in the i:th voxel and d is the vector of all the di’s. TheTCP function is strictly concave and thus convenient to use in optimization [5]. The NTCP function isin Kallman [18] described by

NTCP (d) =(

1−mR∏i=1

(1− P si )1/mR

)1/swhere Pi = exp

(−Noexp(−(αdi + βd2i )

)(10)

where di is the dose in voxel i, s ∈ (0, 1] is the relative seriality of the tissue , mR is the number ofvoxels in the organ at risk and N0, α and β have the same function as in the TCP case. One issue withthe TCP and NTCP models is that there are many parameters and the value of these parameters areuncertain [26].

11


2.3.4 Equivalent Uniform Dose

Niemerko [21] made an assumption that two dose distributions are equivalent if they cause the sameprobability for tumor control, as demonstrated in Figure 15. Using this assumption and the TCP modelas a basis he developed the Equivalent Uniform Dose-model, abbreviated as the EUD and defined inEquation 11. It reduces the complex three-dimensional dose distribution into a singe number. The EUDis essentially a norm [5] which takes the radiationsensitivity of the tissue into account.

EUD(d) = −1

aln

1

|V |

|V |∑i=1

e−adi

(11)

Here, V is the set of all voxels in the structure of interest, a is a parameter specified for type of structuredepending on the radiation response of the tissue (a ≤ 0 for risk structures and a < 0 for targetstructures) and di is the dose in the i:th voxel. Originally the EUD dealt with the effect of radiation intumour structure only. This was later simplified and extended to deal with organs at risk as well, bydeveloping the generalized Equivalent Uniform Dose, the gEUD [22].

gEUD(d) =

1

|V |

|V |∑i=1

dai

1a

(12)

The gEUD is a convex function for a ≥ 1 (concave for a ≤ 1) [5] and is thus more efficient to workwith in optimization problems. While TCP and NTCP have many uncertain parameters, the EUD onlyhas one parameter (although still uncertain). The EUD is based on both physical (dose) and biologicalcriterias.

Figure 15: The EUD simplifies the dose distribution into one value. All distributions giving the sameEUD are equivalent. Image from [23]

2.3.5 Hypoxia

The response of cells to ionizing radiation depends strongly upon the supply of oxygen [12]. Tumousmay outgrow their blood supply. If a blood vessel is further away than the diffusion distance from a cell,this cell becomes more resistant to lack of molecular oxygen. A state known as Hypoxia.

12


Figure 16: Acute hypoxic, chronic hypoxic and oxic regions. Image from [12]

Oxygen makes the tumours more sensitive to radiation, i.e. it’s a strong radiosensitizer. It increasesthe effectiveness of a given dose of radiation by forming DNA-damaging free radicals. Tumor cells ina hypoxic environment may be as much as 2 to 3 times more resistant to radiation damage than thosein a normal oxygen environment [12]. In Figure 17 one can see the increases resistance of the cells in aHypoxic environment using the LQ-model.

Figure 17: Survival curve for cultured mammalian cells exposed to x-rays under oxic and hypoxic con-ditions. Image from [12]

To incorporate the resistance in the hypoxic cells one can modify the α and β parameter in the LQ-modeland take into account the distance to molecular oxygen and the tissue sensitivity [27]. By calculatingthe OER as in Equation 13 one may assess the effect molecular oxygen has on the irradiated tissue, i.e.the enhancement of radiation damage.

Oxygen Enhancement Ratio =Radiation dose in hypoxia

Radiation dose in air(13)

Based on the OER, the dose modification factor may be calculated.

f(r) =OERmax(k + pO2(r))

k +OERmax × pO2(r)(14)

Here OERmax is the maximum effect achieved in the absence of oxygen, k is the reaction constant andpO2 is the local oxygen tension, which depends on the distance r to oxygen. Oxygen effect only occurs ifoxygen is present either during irradiation of within a few milliseconds after irradiation. By performinga Positron Emission Tomography (PET) scan, the local oxygen tension in the brain can be found. Nowthe modified α and β can be calculated as in Equation 15.

13

2.4 Comparison between DVH, EUD, TCP and NTCP 2 BACKGROUND

α(r) =αoxicf(r)

, β(r) =βoxic

(f(r))2(15)

The modified TCP then becomes as in Equation 16.

TCP (d) =

mT∏i=1

Pi where Pi = exp(−N0exp(−(α(r)di + β(r)d2i )

)(16)

The only difference to the TCP stated in Equation 9 is the dependence of r in the α and β parameters.

2.4 Comparison between DVH, EUD, TCP and NTCP

How is then DVH, EUD, TCP and NTCP connected? Will a good result when optimizing a DVH-basedmodel automatically yield a good result of the EUD, TCP and NTCP and vice versa? Recall that theDVH is based solely on the physical property, the dose, and EUD, TCP and NTCP are based on bothdose and parameters derived to explain the biological response in the tissue. In these models, aspects ofhow fast the tissue responds to radiation and how many chlonogens there are in each voxel. One mightsay that a planning isodose of minimum 15 Gy are good inside the target when using DVH-based model,however in this case no regard of the density of chlongens in each voxel is taken. This number can vary alot in magnitude. It may even vary in powers of 10. A large number of chlonogens will also yield a higherdose to tumour control and a small number of chlonogens will yield a lower dose to tumour control.TCP and NTCP also take the seriality and the speed of tissue response to radiation into account, hencemaking them into very thorough models. EUD is somewhat of a simplification to both NTCP and TCP.Here a concludes all the properties described by the parameters of TCP and NTCP, hence making it abit less accurate. Overall, TCP and NTCP with the correct parameters, gives a good description on themagnitude of dose to control the tumour, keeping the spatial information, while the DVH is based onexperience.

14

3 OPTIMIZATION OF THE BEAM-ON TIME

3 Optimization of the Beam-on time

3.1 Problem set-up

The problem of finding the optimal treatment plan was divided into three parts.

1. Find the isocenter positions in the tumour.

2. Perform a dose calculation in order to find the dose rates in each voxel for a specific sector in aspecific state.

3. Optimize the Beam-on time.

To find the isocenter position the grassfire-algorithm is applied and, when the isocenters are found, asimplified version of the TMR10 algorithm is used for the dose calculation. The grass-fire algorithmstarts at the surface of the volume and goes inward to some sort of ”center”, defined by being mostfar away from the surface. When this center is found, the coordinates becomes the first isocenter. Thevolume covered by the isocenter is subtracted and the procedure is iterated. This is done until almostthe entire volume is covered. Since it may be described as putting fire to the verge of a lawn, it’s namedthe grassfire-algoritm. Some other simplifications were that the number of isocenters and the isocenterpositions were fixed throughout the optimization process. Also only the three of the different shapes ofthe isocenters were used in the procedure. All of them were spheres with a radius of either 4 mm, 8 mmor 16 mm. Fixating the positions in the isocenters renders a convex problem assuming the cost functionis convex. All this was done to make the problem convex and decrease the number of degrees of freedom.The organs at risk and tumours were also divided into voxels to speed up the computations.To calculate the dose in voxel i, di the sum below is used. This is a discretized version of the Fredholmintegral [6] and is similar the one used in Section 2.2.

di(ω) =

J∑j=1

K∑k=1

L∑l=1

ωjkldi,jkl (17)

Here ω is the vector of all the beam-on times, j is an index over the isocenterpositions, k is a index overthe sectors in the Gamma Knife (in total 8 sectors) and l is a index over the three collimator sizes. di,jklis the doserate in voxel i from sector k using collimatorsize l when the isocenter is in position j and ωjklis the beam-on time of sector k using collimatorsize l when the isocenter is in position j. Using thisestimation of the dose the treatment plan will be found by optimizing over ωjkl. The problem will haveJKL degrees of freedom where J = 3, K = 8 and L is the total number of isocenters. Throughout thisthesis, the sum in Equation 17 wil be written as di(ω) =

∑µ ωµdi,µ, where µ = 1, 2, 3, . . . JKL.

3.2 The Radiotherapeutical model-based on DVH

A DVH-based objective function is hard to solve due to the non-convexity of the DVH. Instead a sim-plification of the description of the DVH is made by using a nonlinear penalty function penalizing whenthe dose does not fulfill the constraints. Three such constraints are implemented.

3.2.1 Maximum and minimum dose-constraints

To penalize when the dose exceeds the maximum dose, dmax, the function∑mi=1(di(ω) − dmax)2+ is

implemented in the objective, where (x)+ = max(x, 0), m is the number of voxels in the structureand di is calculated as in Equation 17. The objective is to push the dose down to dmax using a squaredpenalization factor over all the voxels in the structure. The minimum dose constraint is treated similarly.

m∑i=1

(dmin − di(ω))2+

which penalizes doses below dmin.

15

3.3 The Radiobiological model 3 OPTIMIZATION OF THE BEAM-ON TIME

3.2.2 Dose-Volume constraints

The Dose-Volume constraints, or DV-constraints, are specified as V (v : dv ≥ dr) < Vr, or the volumereceiving the dose dr should be less than Vr. To implement this one needs to find dp such that V (dp) = Vr.Hence we have first the required dose dr and the present dose dp at Vr. This is illustrated in Figure 18.By penalizing all doses between dr and dp as in Equation 18 one may pull the curve closer to dr. Thismethod was first suggested by Bortfeld [1].

H(di(ω)− dr)H(dp − di(ω))(di(ω)− dr)2 (18)

where H(·) is the Heaviside function. This is done similarly for each DV-constraint. This type ofconstraints are at the moment the most clinically used one.

Figure 18: Illustration of how dp is found.

3.2.3 The resulting DVH model

The resulting and somewhat complicated model is then:

(NLP )

min∑t∈T

λt

mt

∑mt

i=1

((dmin − di(ω))2+ +

∑p∈DV C H(di(ω)− dp)H(dr − di(ω))(di(ω)− dp)2+

(di(ω)− dmax)2+

)+∑r∈R

λr

mr

∑mr

i=1

((di(ω)− dmax)2++∑

p∈DV C H(di(ω)− dp)H(dr − di(ω))(di(ω)− dp)2)

+ λ∑µ ωµ

s.t. di(ω) =∑µ ωµdi,µ, i = 1, 2, 3, . . .

ωµ ≥ 0

(19)where the λ’s is the weight on the treatment time, organ or tumour, H(·) is the Heaviside function T arethe tumours, R are the riskorgans and (x)+ = max(x, 0). In short, a minimum dose and a DV-constraintis implemented for the tumours and a maximum dose and DV-constraint is implemented for each organat risk. Also, the total treatment time is penalized by the last term in the objective function. There isalso a possibility to specify the importance of each organ by using the λ’s to weight the organs betweenone another. If a part of the tumour is hypoxic, this are is treated as a separate tumour with otherconstraints. The square term is chosen mostly because the full objective function remains convex[28].One may also penalize the constraints harder by changing the quadratic penalization function into anexponential function, (·)4 or another convex function.

3.3 The Radiobiological model

3.3.1 gEUD-based objective

Due to the convexity of the gEUD it’s easy to construct constraints and objective function based on it.A similar approach as in Section 3.2.3 is made.

16

3.4 GUI 3 OPTIMIZATION OF THE BEAM-ON TIME

(NLP )

min

∑t∈T λt(gEUD0 − gEUD(ω))2+ +

∑r∈R λr(gEUD(ω)− gEUD0)2+ + λ

∑µ ωµ

s.t. gEUD(ω) = (∑|V |i=1 di(ω)a)

1a

di =∑µ ωµdi,µ, i = 1, 2, 3, . . .

ωµ ≥ 0

(20)

Since the gEUD isn’t very intuitive, gEUD0 is derived from the desired DVH-curve. The value of theparameter a is uncertain, hence the model is solved and tested for the a parameter in an interval of 20%around the mean to study the robustness of the optimized plan.

3.3.2 TCP/NTCP-based objective

The TCP has a great advantage of being concave, which is not the case for the NTCP function. Althoughby applying the transformation h(z) = −ln(1− z) yields a convex NTCP [15], [24].

(NLP )

min

∑t∈T

λt

TCP0(TCP0 − TCP (ω))2+ +

∑r∈R

λr

ln(1−NTCPo)(−ln(1−NTCP (ω)) + ln(1−NTCP0))2+

+ λT

∑µ ωµ

s.t. di(ω) =∑µ ωµdi,µ, i = 1, 2, 3, . . .

ωµ ≥ 0

(21)Here T is a normalizing constant, which is the time it takes to irradiate the tumour with just the 8mmcollimator, serving the purpose of making the cost function dimensionless. Also here, the parameters areuncertain so a sensitivity analysis is performed for a 20% interval around the mean. The model is verysensitive to the initial solution. In order to give an initial estimate, an absolute value of a plot is chosenusing a simple solution to the DVH-model as an initial estimate.

3.4 GUI

The goal of this thesis is to construct a GUI and connect it to a optimization engine. Here one shouldbe able to open a DICOM-folder, containing structure sets of outlined target and riskorgans as well asa complete set of images of the head of the patient, outline the hypoxic areas in the tumours and chosea subset of organs at risk, solve the isocenterplacement problem and the optimization problem with thechosen algorithm and optimization model and compare the resulting dose distribution to the alreadyplanned treatment made in the Gamma Knife. Also plots of the isodose, tumours, OARs and DVH-plotsshould be presented. In the Figure below is a picture of the interface.

Figure 19: The interface of the GUI.

17


To be able to optimize in the gui some initial steps have to be taken. The first one is made in the ”Con-sidered organs” panel. By clicking the button ”New Patient” a window pops up containing the namesof all the outlined structures in the patient and an individual number to each structure. By writing avector in the editbox containing the numbers to the target structures and clicking ”Ready”, the programstarts to take the info from the DICOM-structures and create all the info needed to continue with theinitial steps and the optimization and saving the info as s.mat in the patient folder. Hence, the programcan only upload the file s.mat the next time one wishes to investigate the patient, to gain speed. Whenthis is done, one needs to define organs at risk and which target to consider. This is done by clickingthe ”New System” button. A plot is displayed with first all the targets (to chose targets), then all theoutlined organs (to chose organs at risk). Then if there is a large problem with many tumours one cansplit them into smaller subsytems by clicking the target and organs at risk and pressing ”Next Sytem”.This division will later affect the dose-matrix by making it sparse to speed up calculations during theoptimization. If one wishes to add a hypoxic area in the tumour, one can outline it by pressing ”Addhypoxic area”.

In the isocenter-panel the isocenters are defined. First the Cut-off distance is specified. The Cut-offdistance is the smallest distance to another isocenter and in this case a sort of tolerance. When pressing”Pack spheres” the grassfire-algorithm starts to calculate the isocenters. To gain degrees of freedom onlythe 4mm spheres are used. The number of isocenters will be displayed in the black box and they will beplotted in the tumour in the graph.

Options for optimization, such as maximum number of iterations and function evaluations along with thetolerance, can then be specified. To speed up the calculations decreasing tolerances may be specified in avector. MATLAB then solves the optimization for each tolerance using the results from the previous asan inittial solution to the next. This is in many cases preferred to make computations faster. By pressingthe ”Set parameters” button a window will pop-up with a button for each target and organ at risk inthe problem. There is also a box for specifying the weight on the time and a button for showing someparameters for different organs in the brain. By pressing the button for each organ parameters will showup. One may also specify model and algorithm in the pop-up boxes. The model ”TCP hypoxia” is a bitdifferent from the others. Here one can instead outline blood vessels and the parameters α and β is thenperturbed according to Equation 15. The p0(r) is, for simplicity assumed to be a linear function, wherep0(0) is the oxygen tension in blood (p0 =100mmHg), and r such that p0(r) = 0 is the diffusion distanceof blood in a muscle (rd = 39µm). The equation approximating the oxygen tension is displayed below.Other parameters used when perturbing α and β is OERmax and k, which is set to 3 and 2.5mmHgrespectively.

p0(r) =

p0(1− |r|rd ) if |r| ≤ rd0 if |r| > rd

(22)

Then every initial step is made and one can proceed to optimization by clicking ”Start Optimization”.Then dose-calculation is initiated. When this is done a simplified system is constructed containing onlytargets and organs at risk and the simplified dose-matrix. The assumption is made that a irradiat-ing a target in a subsystem doesn’t affect the dose in other subsystems considerably. The dose-matrix(diµ)i=1,..,m,µ=1,...,JKL used when calculating the dose in the voxels (in Equation 17) is made sparseby taking the mean over a specified amount of rows in the columns describing the doserate from theisocenters in the other systems. This makes calculations faster during the optimization. Every step inthe optimization is displayed in the MATLAB command window.

When the optimization has terminated a DVH plot will show along with the beam-on time. A histogramof the weights will also pop-up. By specifying a number in the editbox under the isodosesurface a plotof the dose above this value will be plotted along with targets and organs at risk. If more such areasare of interest to see one may enter more numbers in a vector. Also other Pareto-optimal solutionscan be investigated and the DVH and isodosesurface for all organs. Figure 20 shows the interface forinvestigating the isodoses in 2D. Here the countours of the target, organs at risk and dose is plotted, andine can scroll through the head in z-direction. In ”DVH using LGK” a plot will be displayed showingthe resulting DVH using the Gamma Knife.

18


Figure 20: The interface of the GUI for isodoseareas.

The N0-parameter may be estimated by first enter the values for α and β in the target and then pressing”DVH using LGK”. N0 is then calculated assuming the Gamma Knife treatment killed 90% of thetumour. Also the EUD0 may be estimated two approaches. First options is to set EUD0 to zero beforeoptimizing the EUD-model an estimate is calculated using the constraints set in maximum and minimumdose along with the DV-constraints. The second option is to first solve the DVH-model. The EUD iscalculated using the result and set as EUD0.

19

4 RESULTS AND DISCUSSION

4 Results and Discussion

Some test-cases were provided and two of them are presented here. These cases are patients alreadytreated in the Gamma Knife. The plans were derived using the interior-point algorithm and all threemodels. The DVH and EUD-based models were also solved in the case of adding a hypoxic region in thetumour, and the TCP-model, which takes hypoxia into account by perturbing the α and β-parameterswhere solved.

4.1 Case 1: One tumour

The first test-case is a tumour located close to the brainstem and the cochlea. The target and the organsat risk and the resulting DVH from the Gamma Knife is below. EUD0 is estimated using the resultsfrom solving the DVH-based model.

Figure 21: Mesh of the head, where tumour and organs at risk are outlined and the DVH from theresulting Gamma Knife dose distribution.

The constraints and the parameters used is displayed in the table below.

20

4.1 Case 1: One tumour 4 RESULTS AND DISCUSSION

Parameter Tumour Brainstem Cochleaλ 1 0.5 0.1

dmin 14 - -dmax 28 15 10DVC (18, 0.7), (23, 0.3) (8, 0.2) -a -8 7 5

EUD0 18.01 8.772 8.351α 0.22 0.0491 0.0878β 0.0272 0.0234 0.0293s - 1 0.64N0 10 5 5

TCP0 100% - -NTCP0 - 1% 1%

The Cut-off distance 1.5 mm generated 14 isocenters. Below is a plot of the chosen hypoxic area. Thishypoxic area has the same constraints as the tumour only the value of dmin is set to 25Gy. The tumoursize is 4.9cm3 and the hypoxic region is placed at the top of the tumour. The hypoxic region is usuallyin the center of the target, but is placed in the top here to display more clearly if there will be anydifference in the dose in the lower and upper part of the tumour.

Figure 22: The hypoxic region of the tumour

4.1.1 Results DVH without hypoxic region

In this case λtime is set to 10−4. Solving the optimization problem yields a value of first order optimalityof 1.296 × 10−2. 1004 iterations were made and the total iteration-time was 152 seconds. The solverstopped because a local minimum was found and the constraints were satisfied. The total beam-on timeis 54 minutes. The beam-on time is calculated by summing the beam-on times from each collimator andthen take the maximum beam-on time from each sector and finally summing the time contribution fromevery sector in every isoposition.

21


Figure 23: DVH of the target and organs at risk where each constraints is plotted.

Above is a plot of the DVH. The thicker lines are the results from the DVH model and the thinner linesis the results from Leksell Gamma Plan R©. The horizontal lines and dots are the constraints set. Theresult follow the defined soft constraints as one can see in the DVH above and there is a coverage of100%. Also the gradient of the dose in the normal tissue is steep and organs at risk are steep. The dosein the normal tissue is much higher than en the Leksell Gamma Plan R© plan due the the fact that it isnot penalized.

Figure 24: Histogram of the Beam-on times.

The histogram is over the resulting vector of all the Beam-on times. many of the smaller entries will notcontribute to the beam-on time.

22


Figure 25: Isodose of the resulting values.

In the isodoseplot the contours of the dose above 14 and 20 Gy is plotted along with contours of theorgans and for the target. One can clearly see that most of the dose is distributed so that the brainstemand the cochlea are minimally affected. One can also see this behavior in the isosurfaceplot in Figure 26.

Figure 26: Isodosesurface of the resulting weights.

Testing robustness in λ

Also the robustness was tested by varying λ below. To stress the problem further the maximum doseconstraint is set to 10Gy in the Brainstem.

23


Figure 27: The DVH for the different values of the parameter λTarget. Here the DVH from the previouslysolved plan is the thin line.

One can see that the model is fairly robust when varying λTarget.

Testing the algorithms

To test the other algorithms the DVH-problem was solved for the three other algorithms as well.

Figure 28: Results using the active-set algorithm. The resulting function value was 1.48× 10−2

24


Figure 29: Results using the SQP algorithm. The resulting function value was 1.08× 10−2

Figure 30: Results using the trust-region reflective algorithm. The resulting function value was 6.12×10−3

All algorithms give similar results. By comparing the function value (SQP gave a function value of1.08 × 10−2) the trust-region reflective provided the best result, although this algorithm is very slow(total iteration-time was 3366 seconds compared to the other algorithms which solved the problem inaround a couple of minutes). The SQP and the interior-point algorithm also have the advantage thatit can recover from infinite and not defined values of the objective function, which is needed in theTCP-model.

4.1.2 Results DVH with hypoxic region

λHypoxicregion was set to 0.2 and the minimum constraint in the hypoxic region, dmin was set to 25 Gy.The number of iterations made was 1930 and the value of first order optimality was 9.119× 10−6. The

25


calculation took 287 seconds and the beam-on time was 114 minutes.

Figure 31: DVH-curve of the target and organs at risk where each constraints is plotted.

The coverage was 100% of the target and the fall-off dose of the gradients are steep. One can also clearlysee the small bump in the DVH for the tumour due to the influence from the hypoxic region. Thecoverage of the hypoxic region is 87%. By penalizing this higher one may achieve a better coverage, butthere will be a trade-off between the minimum dose in the hypoxic region and of the maximum dose inthe tumour and organs at risk as well as the normal tissue. The integral dose in the target is 205mJ,compared to the non-hypoxic case when it was 186mJ.

Figure 32: Isodose of the result. Comparing the the non-hypoxic case the dose is higher in the upperpart of the tumour.

By looking at the isodoses in the plot above one can see a higher dose is concentrated to the top of thetumour (in the first subplot). Also by looking at the isodose surface one can see that the dose is a bithigher around the hypoxic region.

26


Figure 33: Isodosesurfaces for the dose above 14 and 10 Gy.

One remarkable observation here is the increased beam-on time. Setting the weights for all the organsat risk to 0 and redo the calculations yields a beam-on time of 66 minutes and the DVH below.

Figure 34: DVH of the resulting Beam-on times in the case when no weight is put on the organs at risk.

Due to the constraints on the organs at risk, less contribution will be given from the sectors, whosebeams pass through the organs at risk. Due to the increased beam-on time when giving higher dose toa part of the tumour which is just in the vicinity of these organs at risk a lot less contribution will begiven, hence the beam-on time increases drastically.

4.1.3 Results EUD without hypoxic region

The iterations stopped after 1167 iterations because a minimum was found and the constraints weresatisfied. The first order optimality was 2.727 × 10−3. The total iterationtime was 154 seconds andthe resulting EUD is shown in the table below. Total beam-on time was 54 minutes. Below is a tabledisplaying the resulting EUD for each organ.

27


Organ EUDTumour 18.01Cochlea 8.76

Brainstem 8.35

Figure 35: DVH of the resulting weights.

Here the resulting dose is somewhat lower then in the previous case due to he fact that either themaximum nor minimum dose is controlled. Even though minimum and maximum doses aren’t takeninto consideration in this case, the plan fulfills the DVH-constraints reasonably well.

Figure 36: Isodoses of the target and organs at risk

There is a bit less opportunity to control the dose but still the fall-off dose is steep and comparing thethe DVH-constraints the difference to he DVH-case is not large.

28


Figure 37: Isodosesurfaces of the target and organs at risk.

Testing robustness in a

The parameter a was varied in an interval of 20% around the nominal value and solved in the optimizationfor the target and brainstem to test the robustness. The basic case was when all parameters were set tothe nominal value of the parameter, i.e. the case previously solved.

Figure 38: The DVH for the different values of the parameter a. The thin line is the original result.

There are small deviations from the original solution when changing the a-values.

29


4.1.4 Results EUD with hypoxic region

Due to the hypoxic region being a part of the tumour and they will be regarded as two seperate organsthe EUD0 will be modified. The EUD0 for the tumour and hypoxic area was set to 19.19 and 26.03respectively. The number of iterations was 1145 and the first order optimality of 1.47 × 10−2. Thecalculation time was 142 seconds and the beam-on time was 50 minutes. The resulting EUD is displayedin the table below.

Organ EUDTumour 19.18

Hypoxic area 26.02Cochlea 8.77

Brainstem 8.35

One can see here that no resulting EUD deviates much from the target EUD.


The deviations to the DVH-constraints are large. Here, the highest dose is given to the target and notto the hypoxic region. This occurs due to the lack of isocenters in the hypoxic region and the fact thatthe hypoxic region lies at the surface. Because of this, isocenters in the center of the tumour and closeto the hypoxic area will be more irradiated.

30


Figure 40: Isodoses of the target and organs at risk.

Here one may notice a higher dose in the upper part of the tumour when comparing subplot one andtwo with the non-hypoxic case, which is the hypoxic region.

Figure 41: Isodosesurfaces of the target and organs at risk.

4.1.5 Results TCP without hypoxic region

Here λTumour was set to 100. The number of iteration needed to solve the problem was 2631 and it took317 seconds. The value of first order optimality was 1.786 × 10−1 and the total beam-on time was 492minutes. The resulting value of TCP and NTCP is shown in the table below.

Organ TCPTumour 99%Organ NTCP

Cochlea 3.6%Brainstem 7.8%

31



The coverage was at 95%. Even thought the dose in the organs at risk is less then in the previous casesthe dose in the tumour is very high (even too high). The same phenomenon is mentioned in [16]. Sincethe TCP don’t have any constraints on the maximum dose at all, the dose can grow very high. It willchange if the healthy tissue is taken into account in the optimization.

Figure 43: Isodosearea of the result were doses above 14 and 10 Gy are marked.

The high dose is also seen in the isodose surface and in the isodoseplot. Due to the fact that the healthytissue considered in the calculations is only the tissue in the vicinity to the tumour (made to speed upthe dose calculations) and the very high dose these plot may be deceptive.

32


Figure 44: Isodosesurfaces of the result were doses above 14 and 10 Gy are marked.

Including normal tissue in optimization

The theory about including the healthy tissue to avoid a too high dose in the target is tested. Theparameters for the healthy tissue is set to α = 0.0499, β = 0.0238, s = 0.64 and λ = 0.1. An increasewas made in λtime to 0.05. Below is the DVH of the resulting plan.

Figure 45: DVH when including the normal tissue.

918 iterations were performed and it took 132 seconds to solve the problem. The resulting optimality was1.98× 10−3 and total treatment time was 82 minutes. Now the high doses in the tumour are diminishedas expected.

33


Figure 46: Isodoses of the results.

The table below displays the resulting TCP and NTCP. The NTCP in the healthy tissue and in theorgans at risk is very high due to some parameters having erroneous values.


Cochlea 32%Brainstem 28%

Normal tissue 80%

Testing robustness in α, β and N0

Also here a robustness check were performed giving the following results. α and β were varied using threevalues, one for 0.8 and one for 1.2 of the value and one using the expected value. For N0, N0,Tumour

varied between 100 and 1000 number of chlonogenes keeping N0 for organs at risk at 5, and N0,Cochlea

and N0,Brainstem varied between 10 and 100 number of chlonogens keeping N0,Tumour at 1000 numberof chlonogenes.

34


Figure 47: The DVH for the different values of the parameter α. The thin line is the original result.

Figure 48: The DVH for the different values of the parameter β. The thin line is the original result.

35


Figure 49: The DVH for the different values of the parameter N0. The thin line is the original result.

4.1.6 Results TCP with hypoxic region

In this case no hypoxic region is outlined. Instead blood vessels are painted in the tumour. If one wouldhave a PET image or an angiogram one could use this to find the oxygen pressure in different parts ofthe tumour. Such information was not used here, hence a simplification was made. Due to the capillariesthe diffusion distance in a cell is 39µm. In this case, when an angiogram on the tumour is not available,the absence of capillaries will cause the entire tumour being hypoxic. Thus, just to show that this modelworks, the diffusion distance is set to 5mm and only one major blood vessel is marked. This bloodvesselis placed at the lower part of the tumour, causing to upper part to become more hypoxic.

36



To solve the problem 1815 iteration were performed and took 498 seconds. The first order optimalitywas 0.52× 10−6 and the total beam-on time was 259 minutes. The λ’s was same as in the TCP-problemwithout hypoxic region.


Cochlea 2.8%Brainstem 5.3%


The coverage here were also at 99% and the doses in the organs at risk is equal to that in the non-hypoxiccase.

37


Figure 52: Isodose of the result were doses above 14 and 20 Gy are marked.

The isodose plots also show no bigger difference between this case an the non-hypoxic case. Also herethey may be deceptive.

Figure 53: Isodosesurfaces of the result were doses above 14 and 10 Gy are marked.

4.1.7 Comments

The models are solved quickly and a minimum is found in each problem. Also the DVH and EUD modelsworks well when adding a hypoxic region and the largely increased beam-on time is when treating theseareas is expected. By penalizing organs at risk harder or setting larger constraints the models can becontrolled. The TCP-model is a bit harder to control. When testing the robustness one can see that theDVH-model is fairly robust in λ. The EUD model varied a little when changing the a-parameters. TheTCP model varied a lot between the different values, making it the least robust model (if robust at all).One other problem with the TCP-model is the therapeutic window, or the non-existance of it. If thegraph of TCP and NTCP looks like the figure below the problem will iterate down to ω = 0. This isbecause an initial estimate where one of the NTCP is very close to one will yield an infinite initial guess(due to the convexifiation). Also giving a too small initial estimate will result in the TCP ending up ina region where it’s constantly close to 0 and the gradient to be almost 0, hence the solver will regard theTCP-part of the model as a constant and iterate down to 0.

38

4.2 Case 2: Two tumours 4 RESULTS AND DISCUSSION

Figure 54: A TCP and NTCP graph.

However, when an initial estimate can return a value of NTCP and TCP both much larger than zero (asin the figure below), the optimization will work as it should. This is also just a numerical problem dueto the theoretical fact that these curves are not likely to reflect the true clinical situation. The tumouris in general more sensitive to irradiation than the healthy tissue and it’s always the tumour which isirradiated with the highest dose.

4.2 Case 2: Two tumours

Here a number of organs have been specified and not all are necessarily at risk. The chosen organs atrisk are the right Thamalic F and the nerve which controls the foot.

39


Figure 55: Mesh of the head, where tumour and organs at risk are outlined and the DVH from theresulting Gamma Knife dose distribution.

Parameter Right Frontal Me Left Parietal Po Right Thamalic F Footλ 1 1 0.5 1

dmin 15 15 - -dmax 30 30 15 15DV C (20 0.3),(17 0.7) (20 0.3),(17 0.7) (10, 0.5) (10, 0.5)

The estimate if N0 yields a value below 1. Instead in these calculations N0 is set to 10 for the tumoursand 5 for the organs at risk. Here EUD0 is calculated as in Case 1, and λtime is set to 10−4. 33 isocenteris distributed using the grassfire-algorithm with cut-off distance of 1.7mm.

4.2.1 Results DVH without hypoxic area

λTime = 5× 10−5. The optimization took 1051 seconds and 1709 iterations. The value of the first orderoptimality was 1.98×10−2 and the iterations stopped because a minimum was found. The total beam-ontime was 114 minutes.

40


Figure 56: DVH of the resulting values. The thin line is the result from the clinical plan.

The DVH show a steep fall-off dose and a coverage of 100%. The brainstem get a slightly higher dosethan the maximum constraint, although by penalizing it more will yield a better fit.

Figure 57: Isodoses of the dose.

In the contourlines for the isodose as 14 and 20Gy one can se the coverage and selectivity of the givendose.

41


Figure 58: Isodosesurface of the resulting dose.

4.2.2 Comments

Also here did the DVH-model work well. Hence, the conclusion is made that is this model for two tumourswork in the DVH-model it will also work on the EUD- and TCP-model and that it works equally wellwith more than two tumours.

42

5 CONCLUSION

5 Conclusion

The DVH-based model worked well according to plan and the solution is good when comparing to theclinical plan and regarding the defined soft constraints. The EUD-based model did also work well aswell and followed the soft constraints reasonably. Both models were easy to work with by giving harderconstraints or weighting the organs. There were no problems in implementing these models and they allperformed well using any MATLAB-solver algorithm. The main concern was instead when the beam-ontime was minimized. This was implemented as a L1-problem, so that it could keep the beam-on timeslow and reduce the incidence of two or more collimators irradiating from one sector. However, by penal-izing the beam-on time too much may force many sectors to be turned off, which give plans with poorselectivity. Hence one must find a balance in the time-penalty.

The TCP/NTCP-based model however was a bit more troublesome implementing. Since the gradientof TCP and NTCP are larger than zero only in a small region, choosing a initial point in this region isvital for the numeric optimization. If the initial point is too low, the TCP will have a low gradient andwill be iterated down to zero. If on the opposite the initial point is too high the initial estimate willyield an infinite objective. Also one must weight the tumours much higher than the organs at risk andthe time due to the fact that the TCP belongs to a compact interval (TCP∈ [0, 1]). Even though therelative difference to the reference value is taken, the convex transform of the NTCP is by a logarithmicfunction, causing small differences in ω to become large when calculating −ln(1 − NTCP ). Hence theproblem may once again iterate down to ω = 0 if the weight on the TCP is too low.

The method of guiding a higher dose to hypoxic regions was proven to be successful in the case of DVHand EUD. Also the method in the TCP-case were proven to be successful.

5.1 Future work

An interesting future work is to construct a complete model to include every blood vessel and capillaryin the tumour for the TCP case with a hypoxic region based on the oxygen pressure. Another interestingsubject is the minimization of beam-on time. The model used now only minimizes the sum of eachweight. The problem is that the beam-on time is not a sum of all the weight. Instead one may look atother possibilities, such as minimizing the maximum beam-on time from each sector and, if required, seta hard constraint on each collimator.

43

A OPTIMIZATION ALGORITHMS

A Optimization algorithms

Matlabs fmincon-function was used to solve the models. The algorithms used in fmincon can eitherbe the Active-set algorithm, Sequential Quadratic Programming (SQP), Interior-point method or theTrust-region relective method. A short description of these algorithms will be given in this section andthe derivations are acquired from [13]. Suppose the problem to be solved is:

(NLP )

min f(x)

s.t gi(x) ≤ 0,∀i ∈ Ix ∈ Rn

(23)

The Lagrangian function is defines as L(x, λ) = f(x) − λT g(x). The first order optimality conditionis ∇L(x, λ) = 0 and the second order optimality condition is that ∇2

xxL(x) is positive definite. For asolution to be optimal, both have to be fulfilled.

A.1 Active-set algorithm

The active-set-algorithmis is a fast algorithm since it can take large steps. The algorithm is effective onsome problems with nonsmooth constraints. It is not a large-scale algorithm and it generates feasiblepoints in each step.For x to fulfill the first order optimality condition (∇L(x, λ) = 0), Equation 24 have to be satisfied.(

∇xL∇λL

)=

(∇f(x)−A(x)Tλ

g(x)

)=

(00

)(24)

This is also called the Karusch-Kuhn-Tucker conditions. Here A(x) = (∇g1(x),∇g2(x), . . . ) and it isassumed thar A(x) has full row-rank. This is solved using the Newton iteration, setting x+ = x + p,λ+ = λ+ ν in Equation 24 and make a second order Taylor expansion. The resulting equations is.(

∇2xxL(x, λ) −A(x)T

A(x) 0

)(pν

)=

(−∇f(x) +A(x)Tλ

−g(x)

)(25)

This can be simplified into Equation 26.(∇2xxL(x, λ) A(x)T

A(x) 0

)(p−λ+

)=

(−∇f(x)−g(x)

)(26)

Compare this to the first-order optimality condition for a Quadratic Problem solved by a Newton stepin Equation 27.

(QP )

min 1

2xTHx+ cTx

s.t Ax = b⇒(H AT

A 0

)(p∗−λ∗

)= −

(Hx+ cAx− b

)(27)

By using these similarities one obtain a new optimization problem for the step p below

(QP )

min 1

2pT∇2

xxL(x, λ)p+∇f(x)T p

s.t A(x)p = −g(x)

p ∈ Rn(28)

Now, the optimal step need to be found using the Newton step setting p = p + p′. Assume that thefeasible point p is known. Guess that the constraints active at p are also active at the optimal p∗. LetA = I : aTI (x)p = −gI(x), i.e. the constraints active at p. Let W ⊆ A be such that AW(x) have fullrow rank. Keep temporarily the constraints in W active, i.e. solve

(EQPW)

min 1

2 (p− p′)T∇2xxL(x, λ)(p− p′) +∇f(x)T (p− p′)

s.t A(x)p′ = 0

p ∈ Rn(29)

The system of equations to be solved is

44

A.2 Sequential Quadratic Programming A OPTIMIZATION ALGORITHMS

(∇2xxL(x, λ) AW(x)T

AW(x) 0

)(p∗−λ∗

)= −

(∇2xxL(x, λ)p+∇f(x)

0

)(30)

Since the (QP)-problem solved is equality constrained, no further description of how to include in-equalities is made. To summarize the Active-set algorithm one iteration is described. Given a fea-sible x to Equation 23, a feasible p to Equation 29 and W such that AW(x) has full row rank andAW(x)p = −gW(x).

1. Solve

(∇2xxL(x, λ) AW(x)T

AW(x) 0

)(p∗−λ∗

)= −

(∇2xxL(x, λ)p+∇f(x)

0

)2. If λW ≥ 0 then p is optimal and the solution for the (QP)-problem is found. Let p = p + p∗.

Otherwise new (QP)-iteration.

3. Let x = x+ p and solve Equation 26.

4. If λ+ ≥ 0, x is optimal. Otherwise, make a new iteration.

A.2 Sequential Quadratic Programming

SQP satisfies bounds at all iterations and can recover from NaN or Inf results, although it is not alarge-scale algorithm. Assume the second order optimality condition is fulfilled. The proceedings are thesame as in the Active-set method until Equation 29. Then instead of following the active constraintsAssume A(x) has full row rank. One iteration with the SQP-solver for a nonlinear problem is then:

1. Complute optimal solution p and multiplier vector λ+ to

(QP )

min 1

2pT∇2

xxL(x, λ)p+∇f(x)T p

s.t A(x)p = −g(x)

p ∈ Rn

2. Change x to x+ p, and λ to λ+

The difference between SQP and active-set method is that in the subproblem for the active-set methodyou only consider the working sets, i.e. the set of constraints which is currently active, and then solvefor the other constraints. While in the SQP subproblem you solve for all constraints.

A.3 Interior-point method

The Interior method has the advantage that it can be used on very large problems. However, thesolutions can be slightly less accurate than those from other algorithms because of the barrier function,which keeps the solution away from constraint borders. Although for most practical purposes, thisinaccuracy is usually quite small. The interior-point method satisfies bounds at all iterations, and canrecover from NaN or Inf results. Also it can use special techniques for large-scale problems.First define the Barrier function as in Equation 31.

Bµ(x) = f(x)− µm∑i=1

ln(gi(x)) where µ > 0 (31)

Bµ(x) keep the solution away from the constraint border. A necessary conditions for a minimizer ofBµ(x) is fulfilling the first-order optimality condition, i.e. ∇Bµ(x) = 0 where

∇Bµ(x) = ∇f(x)− µm∑i=1

1

gi(x)∇gi(x) = ∇f(x)− µA(x)TG(x)−1e (32)

where A(x) = (∇g1(x),∇g2(x), . . . ), G(x) = diag(g(x) and e = (1, 1, . . . , 1)T . Let the Lagrangianmultiplier λ(µ) = µG(x(µ))−1e, i.e. λ(µ) = µ

g(x(µ)) . This gives∇Bµ(x(µ)) = ∇f(x(µ))−A(x(µ))Tλ(µ) =

0. To sum it all up, λ(µ) and x(µ) must solve the system of equations in Equation 33.

45

A.4 Trust-region-reflective A OPTIMIZATION ALGORITHMS

∇f(x(µ))−A(x(µ))Tλ(µ) = 0λ(µ)− µ

g(x(µ)) = 0, i = 1, . . . ,m⇐⇒ ∇f(x(µ))−A(x(µ))Tλ(µ) = 0

gi(x)λ(µ)− µ = 0, i = 1, . . . ,m(33)

Taking a Newton step and making a second order Taylor expansion yields Equation 34.(∇2xxL(x, λ) A(x)T

ΛA(x) −G(x)

)(∆x−∆λ

)=

(∇f(x)−A(x)TλG(x)λ− µe

)(34)

where Ω equals to At last, given µ > 0, x such that g(x) > 0 and λ > 0, one iteration with theiteror-method is:

1. Compute ∆x, ∆λ from(∇2xxL(x, λ) A(x)T

ΛA(x) −G(x)

)(∆x−∆λ

)= −

(∇f(x)−A(x)TλG(x)λ− µe

)2. Choose suitable steplength α such that g(x+ α∆x) > 0 and λ+ αλ > 0.

3. Change x to x+ α∆x and λ to λ+ α∆λ.

4. If (x, λ) sufficiently close to (x(µ), λ(µ)), reduce µ.

A.4 Trust-region-reflective

The main idea with the Trust-Region-reflective method is find a Trust-region at the current point andimprove the function value [4]. If you have a point x in a space and a function f(x) you want tominimize, then in a small neighborhood N of this point x you can estimate f with a simpler function q(x)instead. This neighborhood is called a Trust-Region. By knowing the Trust-Region and the approximatefunction q(x) to f(x) one may instead find a trial step s by computing the optimal to the problemminsq(s), s ∈ N. The current point is then updated to x + s if f(x + s) < f(x), otherwise x isunchanged, N is shrunk and the trial-step calculations is repeated. The question is now how to find q(x)and N . The Trust-Region subproblem is usually stated as (QP)-problem similar to Equation 29.

(QP )

min 1

2sTHs+ sT g

s.t. ‖Ds‖ ≤ ∆(35)

where g is the gradient of f and H is the Hessian of f at the current point x, D is a diagonal scalingmatrix, depending on the constraints, ∆ ∈ R+ and ‖ · ‖ is the 2-norm.An iteration with the Trust-Region-reflective algorithm is

1. State the Trust-Region subproblem

2. Solve (QP )

min 1

2sTHs+ sT g

s.t. ‖Ds‖ ≤ ∆

3. If f(x+ s) < f(x), then x = x+ s.

4. Adjust ∆.

One problem with this method is that it can only treat either bound-constraint or linear constraint.

46

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