A Nonlinear Variational Method for Improved ... - UniversitÃ¤t MÃ¼nster

Westfälische Wilhelms-Universität M¨ unster Institut f¨ ur numerische und angewandte Mathematik

A Nonlinear Variational Method for Improved Quantification of Myocardial Blood Flow Using Dynamic H215O PET

Diplomarbeit

eingereicht von

Martin Benning

betreut von

Prof. Dr. Martin Burger

M¨ unster 23.06.2008

Abstract We provide a new algorithm to quantify myocardial blood flow (MBF) from H2 15O PET data. The new algorithm differs from existing algorithms by taking into account the temporal correlation of the data sets. H2 15O as a PET-tracer offers the opportunity to examine perfusion of blood into tissue non-invasively. It features a short half-time (≈ 2 min.) and therefore adds a smaller radiation burden to the patient in comparison to other tracers. The disadvantages arising from the short half-time are noisy, low-resolution reconstructions. Previous algorithms first reconstruct images from each H2 15O dataset independently, via the standard EM-algorithm or FBP. Hence, temporal correlation is neglected. The MBF and other important parameters, like tissue fraction, arterial and venous spillover effects, are computed subsequently from these reconstructed images. Our method interprets the computation of parameters as a nonlinear inverse problem. This implies the need for inversion of a nonlinear Operator G(p) (with p denoting the parameters to compute), but allows to skip the process of generating noisy images. Therefore, our method takes into account the temporal correlation between the datasets, and not the correlation between noisy, low resolution images. The problem is transferred to a nonlinear parameter identification problem. Furthermore, regularization can be added to each parameter independently, assuring meaningful results.

Contents 1 Introduction 2 Mathematical Background 2.1 Variational Methods . . . . . . . 2.2 Convexity . . . . . . . . . . . . . 2.3 Equality Constrained Problems . 2.4 Ill-posedness and Regularization

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3 Medical Background 3.1 PET and Visualization of Physiological Effects . . . 3.1.1 PET Basics . . . . . . . . . . . . . . . . . . . 3.1.2 Examination of Physiological Effects . . . . . 3.2 Basic Heart Anatomy and Cardiovascular System . . 3.2.1 Basic Heart Anatomy . . . . . . . . . . . . . 3.2.2 Circulatory System . . . . . . . . . . . . . . . 3.2.3 Organ Oxygen Supply . . . . . . . . . . . . . 3.3 Myocardial Blood Flow and Perfusion . . . . . . . . 3.4 An ODE Model for Blood Flow Computation . . . . 3.4.1 Derivation of Ordinary Differential Equations 3.4.2 Partition Coefficient . . . . . . . . . . . . . . 3.4.3 Tissue Fraction and Spillover Effects . . . . . 3.4.4 The ODE in Banach Spaces . . . . . . . . . . 3.5 Blood Flow Quantification: State of the Art . . . . .

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4 Quantification of Myocardial Blood Flow as an Inverse Problem 4.1 Basics: The EM-Algorithm . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Inverse Problem of Myocardial Perfusion Quantification . . . . . 4.3 Variational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Image Sequence Generation . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Image Sequence Generation: A Simple Approach . . . . . . . 4.4.2 Fraction and Spillover: The More Complex Approach . . . . 4.5 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Regularization Incorporating A-priori Knowledge . . . . . . . 4.7.2 Gradient Regularization . . . . . . . . . . . . . . . . . . . . . 4.7.3 Fraction and Spillover Regularization . . . . . . . . . . . . . 4.8 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Segmentation via Factor Images . . . . . . . . . . . . . . . . 1

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4.8.2 4.8.3

A Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . Segmentation Using CT-Scan Information . . . . . . . . . . . . . . .

5 Implementation 5.1 Two-Step Algorithm . . . . . . . . . . . . . . . . . . . 5.2 EM Discretization . . . . . . . . . . . . . . . . . . . . 5.3 Discretization of the Parameter Identification Problem 5.3.1 Scaling . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Discrete Approximations . . . . . . . . . . . . . 5.4 Gradient Method . . . . . . . . . . . . . . . . . . . . . 5.5 Newton Method . . . . . . . . . . . . . . . . . . . . . . 6 Results 6.1 A Simple Synthetic Data Example . . . . . . . . 6.1.1 Parameter Identification on Exact Data . 6.1.2 The Computation of Myocardial Perfusion 6.2 The Modified Data Example . . . . . . . . . . . . 6.2.1 Parameter Identification on Exact Data . 6.2.2 The Computation of Myocardial Perfusion 6.3 Real Data . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Segmentation . . . . . . . . . . . . . . . . 6.3.2 Reconstructed Parameters . . . . . . . . . 7 Conclusions

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. . . . . . . . . . . . . . . . . . . . . . . . Synthetic PET Data . . . . . . . . . . . . . . . . . . . . . . . . Synthetic PET Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

List of Figures 1.1 1.2

Major Causes of Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2D H2 15O Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1 2.2

Example of Numerical Differentiation without Regularization . . . . . . . . Example of Numerical Differentiation with Regularization . . . . . . . . . .

16 17

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

ECAT Exact HR+ PET-Scanner . . . . . . . . . . . . . . . . Schematic PET-Scan Process . . . . . . . . . . . . . . . . . . Maximum Intensity Projection of 18F-FDG . . . . . . . . . . The Hearts Interior . . . . . . . . . . . . . . . . . . . . . . . . The Human Circulatory System . . . . . . . . . . . . . . . . . A Net of Capillaries . . . . . . . . . . . . . . . . . . . . . . . The One-Tissue-Compartmental Model . . . . . . . . . . . . The One-Tissue-Compartmental Model Extended to Terms of Fraction and Spillover Effects . . . . . . . . . . . . . . . . . . Factor Images . . . . . . . . . . . . . . . . . . . . . . . . . . . Exemplary Input Functions . . . . . . . . . . . . . . . . . . . Input Functions And Tissue Curve . . . . . . . . . . . . . . . Exemplary Computational Results with Previous Method . .

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18 19 20 21 22 23 24 25 26 31 32 32 33

4.1 4.2 4.3 4.4

Previous Computation of Myocardial Perfusion . . . . . . . . . . . . . . . . The Computation of Myocardial Perfusion as an Inverse Problem . . . . . . A Schematic Slice of a Cardiovascular Region . . . . . . . . . . . . . . . . . Two Similar Tissue Input Curves with Different Perfusion and Tissue Fraction Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of a Standard EM Reconstruction with a EM-TV Reconstruction of H2 15O-PET Data Averaged to One Frame . . . . . . . . . . . . . . . . . . Comparison of a Standard EM Reconstruction with a EM-TV Reconstruction of H2 15O-PET Data for Single Frames . . . . . . . . . . . . . . . . . . . . .

39 40 43

5.1 5.2 5.3

The Two-Step Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . EM - PI Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The PI Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.1 6.2 6.3 6.4 6.5

Example 1: Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Input Curves Describing the First Data Example . . . . . . . . . . . . The 9-th Frame of Exact Data for the First Synthetic Data Example . . . . Computed Values from the Exact Data of the First Synthetic Data Example The 9-th Frame of a Standard EM-Reconstruction of the Synthetic PET Data Generated from the First Data Example . . . . . . . . . . . . . . . . . . . .

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4.5 4.6

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6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26

Computed Values From the Synthetic PET Data of the First Data Example Comparison of Exact and Computed Data . . . . . . . . . . . . . . . . . . . Areas with Varying Perfusion . . . . . . . . . . . . . . . . . . . . . . . . . . Complete Tissue Input Curves with High and Small Perfusion Values . . . . The 9-th Frame of Exact Data for the Second Synthetic Data Example . . . Computed Values From the Exact Data of the Second Synthetic Data Example The 9-th Frame of a Standard EM-Reconstruction of the Synthetic PET Data Generated from the Second Data Example . . . . . . . . . . . . . . . . . . . Computed Values from Synthetic PET Data of the Second Data Example . Comparison of Exact and Computed Data for the Second Dataset . . . . . Real Data: Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-priori Input Curves for the Real Data Set . . . . . . . . . . . . . . . . . . Computed Values From the Real Data Set . . . . . . . . . . . . . . . . . . . The First Three Frames of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . Frames 4 - 6 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 7 - 9 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 10 - 12 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 13 - 15 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 16 - 18 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 19 - 21 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 22 - 24 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames 25 and 26 of the Standard EM Reconstruction in Comparison to the First Three Frames of G(p) . . . . . . . . . . . . . . . . . . . . . . . . . . .

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75 76 76 77 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93

Acknowledgments I want to thank Martin Burger for offering me a challenging diploma thesis topic and for giving me a lot of support, by answering my questions and introducing me to new fields of applied mathematics. Klaus Sch¨ afers for giving me the medical background I needed; for supporting me with previous programs for myocardial blood flow quantification and for taking his time for intense discussions. Frank W¨ ubbeling for supplying me with a lot of his knowledge and programs to transform synthetic data to PET data. Alex Sawatzky for helpful discussions and his EM-TV reconstructions. Thomas K¨ osters for providing me with real H2 15O-data and programs to perform standard EM reconstructions. Christoph Brune for helpful discussions and for answering a lot of questions. my family, Marie-Theres, Frank and Maximilian Benning, Sabrina and Torsten Majert and Nero; you are the best. Jahn M¨ uller, Stephanie Sillekens, Linda Kronenberg and Martin Drohmann for proofreading my thesis and for making helpful suggestions. the staff of the Institute for Computational and Applied Mathematics for all their support during the last six months. all my friends and everybody who deserves to be mentioned here.

5

Chapter 1

Introduction Cardiovascular diseases are the most common cause of death in many countries worldwide. Major causes of death in the Federal Republic of Germany are listed in Figure 1.11 . It is estimated by the World Health Organization (WHO) that cardiovascular diseases will become the most common cause of death worldwide in 2015. Up to now, about 12 million people worldwide die because of cardiovascular diseases every year. More than 50 % of these cases of death could be prevented by early diagnosis. Cardiovascular diseases Heart attack Cancer Diseases of the respiratory system Diseases of the alimentary system Injuries, accidents, toxication Suicide Miscellaneous

44,2 8 25,5 7 5,2 5,7 1,2 3,2

Major causes causes of of death death Major 3% 1% 3% 1% 6% 6% 5% 5%

7% 7% 44% 44%

Cardiovascular didiCardiovascular seases seases Heart attack attack Heart Cancer Cancer Diseases of of the the respirarespiraDiseases tory system system tory Diseases of of the the alimenalimenDiseases tary system system tary Injuries, accidents, accidents, Injuries, toxication toxication Suicide Suicide Miscellaneous Miscellaneous

26% 26%

8% 8%

Figure 1.1: The major causes of death in the Federal Republic of Germany in 2005. Many cardiovascular diseases originate in the so-called atherosclerosis, which is a calcification of an arteria. If plaque is heavily deposed on the interior wall of an artery the transport of blood is limited due to the fact that the plaque acts like a blocker. Moreover, if the plaque is unstable and detaches from the interior wall, it can cause a sudden heart attack or a stroke. All these facts point out the importance of early medical diagnosis. Investigation of blood flow in arterias, especially in the hearts arterias at an early stage could prevent 1

according to the German Federal Office for Statistics

6

latter heart attacks or other serious diseases and therefore prevent death in many cases. Examining the hearts blood flow and its amount of perfusable tissue under rest and stress conditions allows to get further insight into future mischief. If a patient suffers an aching chest or has other indications for a serious heart disease, the common procedure is to catheterize the patient. A catheter is inserted via arterial or venous vessels in the groin, the inner bend of an elbow or the neck area to investigate coronary arterias and veins. A contrast medium is injected while the heart is x-rayed simultaneously, to make blocked arterias or veins visible. If, e.g. an arteria is blocked due to plaque, a stent could be inserted to improve pathology and to reduce the risk for a stroke or heart attack. A big disadvantage of the procedure of catheterization is that it is invasive and therefore cumbersome for patients. There is also a risk of e.g. thrombosis, embolism, infections or cardiac arrythmia. Furthermore, a constriction of an arteria must not be a result of plaque, but can have different reasons. Hence, further investigation with different methods is required.

Figure 1.2: A two dimensional H2 15O reconstruction of a cardiovascular slice with 10 iterations of the standard Expectation Maximization (EM) algorithm with added Gaußian smoothing after the 4th and 8th iteration. The image represents the amount of kBq per ml in the seventh frame of a H2 15O-scan, operated in two dimensions. The small number of measured annihilations result in a low SNR; thus, these low quality reconstructions demand high experience of the medical operator. A method for non-invasive investigation of physiological processes within the body, e.g. blood flow, is via Positron Emission Tomography (PET)-imaging. Using PET, a radioactive tracer is injected into the body. Positrons of the tracer annihilate with electrons within the body. These annihilations can be measured to produce images of the physiological processes. PET has proved to be a powerful technique that allows non-invasive investigation of many different types of physiological processes. It is especially applicable to draw conclusions on blood flow in the hearts arterias and veins by examining myocardial perfusion, which is also known as myocardial blood flow. If the myocardial tissue is not perfused correctly, this indicates serious problems with arterial and venous blood flow. A tracer that could be used to visualize myocardial perfusion is radioactive water H2 15O with a half-time of about two minutes. H2 15O has several advantages as a tracer for myocardial perfusion quantification. On the one hand there is few radiation exposure. On the 7

other hand, radioactive water is highly diffusible and therefore tissue is quickly saturated with the tracer. Unfortunately, H2 15O has a huge disadvantage. Due to its short half-time the quality of reconstructed images is very poor, since fewer annihilations are measured. Reconstructed images offer limited resolution and a low signal-to-noise ratio (SNR), which makes the investigation of myocardial perfusion pretty tough. An exemplary H2 15O-reconstruction can be seen in Figure 1.2. Despite the poor quality of the reconstructions, in modern medical facilities physical models are applied to these noisy images in order to draw conclusions on myocardial blood flow and the amount of perfusable tissue. With this thesis, we want to provide a new algorithm that computes myocardial blood flow and other important parameters from the measured data instead from a noisy image sequence. In the step of image processing with the standard Expectation Maximization (EM)-algorithm all temporal correlation is neglected. The new algorithm will overcome this problem by treating the process of perfusion quantification as an inverse problem and by inverting a nonlinear operator to skip the image processing step. Since variational methods build the mathematical background of the new algorithm, we are going to describe basic variational calculus in terms of optimization in Chapter 2. This will offer a mathematical toolbox that gives further insight into the properties of the new algorithm. In Chapter 3, we will discuss the medical background needed, including the PET-principle, basic heart anatomy, the functionality of the cardiovascular system, the relation between blood flow and myocardial perfusion and simple physical models based on ordinary differential equations. Chapter 4 will provide the mathematical background of the new algorithm, introducing the standard EM-algorithm first. In the following, the variational model and the inversion of the nonlinear operator in terms of a parameter identification problem will be presented. Subsequently, regularization and segmentation have to be discussed in detail. On the theoretical background of the new algorithm its numerical realization will follow in Chapter 5, before computational results will be presented in Chapter 6. Finally, we are going to give future prospects and draw conclusions in Chapter 7.

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Chapter 2

Mathematical Background In this chapter we want to provide a short introduction to mathematical tools and their theoretical background, which will be used to derive a new approach for myocardial blood flow quantification. First of all, we are going to talk about variational calculus; in the following, we will examine the special case of convex variational calculus. Subsequently, we will give a brief introduction to constrained optimization problems.

2.1

Variational Methods

From a mathematical point of view, the most important ingredients for our approach to computing myocardial perfusion will be variational methods. The aim of this chapter is to provide a background about techniques of variational calculus for the latter use. Variational calculus mainly deals with the questions of existence, uniqueness, and computation of stationary points of functionals in Banach spaces (in particular function spaces). Finding (unique) global minima will be of highest interest and importance for us. The calculus of variations can be seen as a generalization of the computation of extreme values of functions. For that reason, we first want to derive a concept of derivatives and gradients for functionals, similar to the one for functions in Rn . Definition 2.1. Let J : U → V be a functional between Banach spaces U and V. The directional derivative (also called first variation) at position u ∈ U in direction v ∈ U is defined as dv J(u) := lim t↓0

J(u + tv) − J(u) , t

(2.1)

if that limit exists. If we define the function ϕv (τ ) := J(u + τ v), the directional derivative of J is equivalent 0 to ϕv (τ )|τ =0 . Note that by definition the direction v has to lie within the same space as the argument u. Example 2.1. Let f ∈ L2 (Ω) with Ω ⊂ Rn being compact and bounded, and J : L2 (Ω) → R+ ∪ {+∞} with J(u) := =

1 ku − f k2L2 (Ω) 2Z 1 (u − f )2 dx . 2 Ω

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(2.2)

We define ϕv (τ ) := J(u + τ v) for v ∈ L2 (Ω) and obtain Z

2 (u + τ v − f ) dx

1 d 2 dτ Ω τ =0 Z v (u − f ) dx = hv, u − f iL2 (Ω) . =

0

ϕv (τ )|τ =0 =

Ω

The permutation of integration and differentiation is justified, due to [For81, Theorem 2, p. 99]. Thus, we have obtained dv J(u) = hv, u − f iL2 (Ω) as the first variation of J at position u in direction v. Example 2.2. As a second simple example consider again f ∈ L2 (Ω) and Ω ⊂ Rn being compact and bounded. Furthermore we introduce a compact linear operator K : L2 (Ω) → L2 (Ω) and examine J : L2 (Ω) → R+ ∪ {+∞} with 1 kKu − f k2L2 (Ω) 2Z 1 (Ku − f )2 dx . 2

J(u) := =

(2.3)

Ω

In an analogous way to Example 2.1 we obtain dv J(u) = hKv, u − f iL2 (Ω) , which can be rewritten as dv J(u) = hv, K ∗ (u − f )iL2 (Ω) ,

(2.4)

with K ∗ being the adjoint operator of K. Similar to the definition of the first variation (2.1) we are able to define higher variations. Only the first and second variation will be of further interest to us. Definition 2.2. Let J : U → V be a functional between Banach spaces U and V and dv J(u) does exist. The second directional derivative (also called second variation) at position u in direction w is defined as 2 dv,w J(u) := lim t↓0

dv J(u + tw) − dv J(u) , t

(2.5)

if that limit exists. Furthermore we want to summarize two important and interesting cases of differentiability. Definition 2.3. Let J : U → V be a functional between Banach spaces U and V. The set dJ(u) = {dv J(u) < ∞ | v ∈ U} is called Gˆ ateaux-derivative. J is called Gˆ ateaux-differentiable, if (2.6) is not empty. 10

(2.6)

Henceforth we want to explore, for which cases the Gâteaux-derivative consists of only one element. This will lead to the term of Fréchet-differentiability. Definition 2.4. Let J : X → Y be a functional, X and Y Banach spaces, and suppose 0 dv J(u) exists for all v ∈ X . If there exists a continuous linear functional J (u) ∈ X ∗ , such that 0

J (u) v = dv J(u)

∀v ∈ X ,

(2.7)

and

0

J(u + v) − J(u) − J (u) v

Y

kvkX

−→ 0 , for kvkX → 0

(2.8)

0

hold, then J is called Fréchet-differentiable in u and J is called Fréchet-derivative. 0

Notice that J (u) does not need to be linear in u. In our definition u is fixed and the linearity - since we want to derive an analogous expression to classical differentiability - is a restriction to v only. The functional J is called twice Fréchet-differentiable, if the properties of Definition 2.4 2 J(u) as well. In that case, we are also going to write J 00 (u)(v, v) instead of hold for dv,v 2 J(u). dv,v Example 2.3. Consider again Example 2.1 and Example 2.2. Both functionals are Fréchetdifferentiable with the Fréchet-derivatives 0

J (u) = u − f

(2.9)

in Example 2.1 and 0

J (u) = K ∗ (u − f )

(2.10)

in Example 2.2, respectively. Besides computing potential extremal functions the question of existence and uniqueness arises. The question of existence can be answered via the fundamental theorem of optimization. First, we have to define the term of lower semi-continuity in the special case of a Banach space. Definition 2.5. Let U be a Banach space with topology τ . The functional J : (U, τ ) → R is called lower semi-continuous at u ∈ U if J(u) ≤ lim inf J(uk ) , k→∞

(2.11)

for all uk → u in the topology τ . Together with compactness this leads to the fundamental theorem (see [Zei84]). Theorem 2.1 (Fundamental theorem of optimization). Let U be a Banach space with topology τ and let J : (U, τ ) → R be lower semi-continuous. Furthermore let the level set {u ∈ U | J(u) ≤ M } be non-empty and compact in the topology τ for some M ∈ R. Then there exists a global minimum of J(u) → min . u∈U

11

Proof. Let J˜ = inf u∈U J(u). Then a subsequence (uk )k∈N exists with J(uk ) → J˜ for k → ∞. For k sufficiently large, J(uk ) ≤ M holds and hence, (uk )k∈N is contained in a compact set. As a consequence, a subsequence (ukl )l∈N exists with ukl → u ˜, for l → ∞, for some u ˜ ∈ U. From the lower semicontinuity of J we obtain J˜ ≤ J(˜ u) ≤ lim inf J(uk ) ≤ J˜ , k→∞

consequently u ˜ is a global minimizer. In finite dimensional optimization, compactness is usually caused by boundedness, which is not the case in infinite-dimensional optimization. To still conclude compactness from boundedness, we need a weaker topology. Since we are dealing with Banach spaces and their dual spaces, which contain the Fréchet-derivatives, we can use the so-called weak and weak-* topology, which is defined as follows. Definition 2.6. Let X be a Banach space, with X ∗ denoting its dual space. Then the weak topology is defined as uk * u :⇔ hv, uk iX → hv, uiX ,

(2.12)

for all v ∈ X ∗ , and the weak-* topologies are defined as vk *∗ v :⇔ hvk , uiX ∗ → hv, uiX ∗ ,

(2.13)

for all u ∈ X . For a reflexive Banach space (X = X ∗∗ ), weak and weak-* topology are the same. According to the theorem of Banach-Alaoglu, the set {v ∈ X ∗ | kvkX ∗ ≤ C}, for C ∈ R+ , is compact in the weak-* topology. Hence, we could conclude existence of a global minimum for a given infinite dimensional optimization problem, if we were able to prove lower semicontinuity in the weak-* topology. In that case, we could simply compute the Fréchetderivative to obtain the desired minimum. Unfortunately, in most cases proving lower semi-continuity in the weak-* topology is not trivial. In the following chapter we want to present a special type of functionals that actually is easy to handle: so called convex functionals. Convex functionals will offer the property of having unique global minima.

2.2

Convexity

Strictly convex functionals defined on convex subsets of Banach spaces provide interesting properties since they guarantee that all minima are global ones. We are going to make some definitions first to finally prove this statement. Definition 2.7. Let U be a Banach space. A set C ⊂ U is called convex, if αu + (1 − α)v ∈ C ,

(2.14)

for all α ∈ [0, 1] and all u, v ∈ C. Definition 2.8. Let C be a convex set. A functional J : C → R ∪ {∞} is called strictly convex, if J(αu + (1 − α)v) ≤ αJ(u) + (1 − α)J(v)

∀u, v ∈ C, ∀α ∈ [0, 1],

where equality only holds in the cases u = v or α ∈ {0, 1}. 12

(2.15)

The following theorem states that in the case of existence strict convexity of a functional leads to uniqueness of a minimum. Theorem 2.2. Let J : C → R ∪ {∞} be strictly convex. Then there exists at most one local minimum, which is a global one. Proof. Let u, v be two global minima of J. For u 6= v this implies J(αu + (1 − α)v) < αJ(u) + (1 − α)J(v) = inf J, for α ∈]0, 1[, which is a contradiction to the assumption. At first sight, Theorem 2.2 does not seem to be a helpful tool for determining global minima, since we would have to prove that (2.15) holds for all u, v ∈ C, u 6= v, and α ∈]0, 1[ for a given functional J. In order to obtain a powerful tool, we need an easily usable criterion to decide if a functional is convex. This criterion is given by the next theorem: Theorem 2.3. Let C ⊂ U be open and convex and let the functional J : C → R ∪ {∞} be 00 twice continuously Fréchet-differentiable. Then, J (u)(v, v) > 0 for all u ∈ C and v ∈ U\{0} implies strict convexity of J. Proof. We define g : [0, 1] → R with g(α) := J(αu1 + (1 − α)u2 ) , for arbitrary u1 , u2 ∈ C. With u1 , u2 ∈ C, (αu1 +(1−α)u2 ) ∈ C lies within the convex subset 00 00 of U, too. Due to the assumption, g (α) = J (αu1 + (1 − α)u2 )(v, v) > 0, with v = u1 − u2 , holds for all α ∈ [0, 1] with u1 6= u2 . Via Taylor expansion around 0 and 1 we obtain Zα g(α) = g(0) +

0

g (t)dt 0

Zα Zt

0

= g(0) + αg (0) + 0

Z1 g(α) = g(1) −

00

g (s)dsdt

and

0

0

g (t)dt α 0

Z1 Z1

= g(1) − (1 − α)g (1) +

00

g (s)dsdt . α

t

By examining a convex combination of these expansions we receive g(α) = (1 − α)g(α) + αg(α) 0

0

= (1 − α)g(0) + αg(1) + α(1 − α)(g (0) − g (1)) Zα Zt Z1 Z1 00 00 +(1 − α) g (s)dsdt + α g (s)dsdt 0

α

0

13

t

part. int.

=

Z1 (1 − α)g(0) + αg(1) − α(1 − α)

00

g (s)ds 0

Zα +(1 − α)

Z1

00

(α − s)g (s)ds + α α Zα

0

=

(1 − α)g(0) + αg(1) − (1 − α)

00

Z1

sg (s)ds − α 0

≤

00

(1 − α − s)g (s)ds

00

sg (s)ds α

(1 − α)g(0) + αg(1) ,

with equality only for α ∈ {0, 1} or v = 0. Hence, J is strictly convex. Theorem 2.3 offers an easy-to-use method to verify, if a functional is convex and therefore, in case of an existing minimum, to decide if a minimum is global. Example 2.4. Consider again Example 2.2. Computing the second directional derivative in both directions v, we obtain Z 2 (2.16) dv,v J(u) = v 2 dx ≥ 0 , Ω 00

and therefore for the second Fréchet derivative J ≥ 0 holds, with equality for v = 0 only. Hence, in case of existence, u = f is a global minimum of (2.2).

2.3

Equality Constrained Problems

To conclude this section, we want to explore optimality conditions for a special type of variational problems: so called constrained optimization problems. Definition 2.9. Let J : U → R and E : U → V, with U, V being Banach spaces. Then the optimization problem J(u) → min , u∈U

subject to E(u) = 0 , is called constrained optimization problem with equality constraint E. The most common approach to find a solution for Definition 2.9 is via Lagrange multipliers. A new variable p ∈ V ∗ , called Lagrange multiplier, is introduced and the functional J can be rewritten as L(u; p) = J(u) + hp, E(u)i ,

(2.17)

for an appropriate duality-product with respect to V and V ∗ . Now we consider what is called the dual problem. We want to minimize L with respect to u and maximize it with respect to p.

14

Existence and uniqueness of minima for equality constrained optimization problems, represented by (2.17), are not trivial to prove and heavily depend on further information, as for example convexity. But it can easily be seen that - similar to unconstrained problems - the gradient of L has to be examined, due to the so-called Karush-Kuhn-Tucker system, a necessity condition for the equality constrained optimization problem. Theorem 2.4 (Karush-Kuhn-Tucker conditions). Let J : U → R and E : U → V, with U, V 0 being Banach spaces. Furthermore, E ∗ : V ∗ → U ∗ is a surjective, bounded linear operator. If u is a solution of J(u) → min , u∈U

subject to

E(u) = 0 ,

then there exists p ∈ V ∗ with ∂L (u, p) = 0 and ∂u ∂L (u, p) = 0 , ∂p for L as defined in (2.17). Proof. See [Zei84].

2.4

Ill-posedness and Regularization

Variational calculus allows us to transfer many real-world applications to variational problems by rewriting the original problem to an operator equation of the type D(f ) = g, with D denoting a compact operator. After transferring the problem, we are able to apply variational calculus as introduced in the Sections 2.1 - 2.3, e.g. to solve an optimization problem. Unfortunately, many of these operator equations are ill-posed: Definition 2.10. Let H and K be normed spaces and D : H → K a compact operator. The problem in finding a solution f of D(f ) = g,

(2.18)

with g ∈ K is called well-posed, if 1. there exists a solution for all g ∈ K, 2. the solution is unique, 3. the solution f depends continuously on g. The problem is called ill-posed, if it is not well-posed. Most difficulties we have to face with ill-posed problems originate in the violation of condition 3. This is the infinite dimensional analogue to ill-conditioning of a matrix. But we will also discover that the violation of the other conditions can lead to serious problems, too. If we still want to deal with ill-posed operator equations, we need to compute a regularized approximation of its solution. We will just give a brief introduction to the terms of 15

regularization; for a more detailed description and definitions we suggest [Bur08]. A regularization method simply tries to approximate an ill-posed problem with a new problem close to the original one, but being well-posed. The ’closeness’ to the problem is controlled via a regularization parameter. If this parameter is chosen to be zero, we obtain the original problem. The larger it is chosen, the more the solution is regularized. We give a brief example. Example 2.5. As an easy example we want to explore the numerical differentiation of noisy data. Assuming, that our variable of differentiation lies in [0, 1] and noise can be controlled in L2 ([0, 1]), this problem can be rewritten as the inversion of a compact linear Operator A : L2 ([0, 1]) → L2 ([0, 1]) defined via Zx (Af )(x) :=

f (y)dy.

(2.19)

0

With given g(x) = (Af )(x), we are interested in computing f (x). Now assume, that not g is measured but g δ (x) = g(x) + nδ (x), with nδ (x) being the data noise. Even, if the noise is arbitrarily small we will discover that differentiating the data will increase noise dramatically (see Figure 2.1)). Hence, continuous dependency of a solution f on g is no 0.9

0.3 gδ

0.8

0.2

0.7

0.1

Distance (in m)

Velocity (in m/s)

0.6 0.5

0

-0.1

0.4

-0.2

0.3 0.2

-0.3

0.1

-0.4

Derivative fδ of gδ 0

5

10

15

20

25 30 Time (in s)

35

40

45

50

(a) The input data curve g δ

0

5

10

15

20

25 30 Time (in s)

35

40

45

50

(b) The differentiation f δ of g δ

Figure 2.1: A typical example of numerical differentiation without regularization. A measured input data curve g δ , describing the distance in z-direction over time of a jugglingexperiment, is numerically differentiated (f δ ). Little noise in g δ increases dramatically after only one differentiation. longer guaranteed. An easy way to overcome this problem is to regularize g δ by keeping its derivative bounded, i.e. we examine the following minimization problem 1 2

Z1

α (g (x) − g (x)) dx + 2 α

δ

0

2

Z1

dg α (x) dx

2 → min . α g

(2.20)

0

With the tools previously provided in chapter 2, it is easy to show that the minimization of (2.20) is characterized by the boundary-value problem α

d2 g α (x) = g δ (x) − g α (x) dx2 dg α dg α (0) = (1) = 0. dx dx 16

(2.21)

Instead of differentiating g δ , we will differentiate g α to obtain a solution f α continuously dependent on g α , reasonably close to f . Results can be seen in Figure 2.2. 0.9

0.3 gδ

0.8

gα

0.2

0.7

0.1

Distance (in m)

Velocity (in m/s)

0.6 0.5

0

-0.1

0.4

-0.2

0.3

Derivative fδ of gδ

-0.3

0.2

Derivative fα of gα 0.1

0

5

10

15

20

25 30 Time (in s)

35

40

45

-0.4

50

(a) The input data curve g δ and it’s regularized curve g α

0

5

10

15

20

25 30 Time (in s)

35

40

45

50

(b) Comparison of the derivatives f δ and f α

Figure 2.2: A typical example of numerical differentiation with gradient regularization as described in (2.20) and (2.21). In addition to g δ and f δ , the regularized solution g α and its derivative f α , with α = 0.5444 × 10−4 , are plotted for comparison. The regularized solutions appear to be flattened, but also much smoother, with no noise remaining. Now we have provided a mathematical toolbox to develop a new mathematical theory for the quantification of myocardial blood flow. Before we move on to develop this theory in chapter 4, we will first discuss the medical background needed.

17

Chapter 3

Medical Background With this chapter we are going to introduce the theoretical background of myocardial blood flow quantification from a medical point of view. This medical background is not only needed to explain existing solutions for this problem; it actually builds the basis needed to develop the new theory. First, we are going to describe the basics of Positron Emission Tomography (PET), which is the underlying imaging technique that makes examination of blood flow possible. Next, we will explain some basic anatomic and physiological background needed. Following, we will describe the connection between blood flow and perfusion in tissue, which will lead to a simple physical model that can be explained with simple mathematical formulas. Concludingly, we will discuss state-of-the-art methods for myocardial blood flow computation used in clinical applications.

Figure 3.1: A typical PET-facility equipped with an ECAT Exact HR+ PET-Scanner. c

Wikipedia

3.1

PET and Visualization of Physiological Effects

In this section we are going to briefly explain the imaging technique PET and its application to the examination of physiological processes in human and animal bodies. 18

3.1.1

PET Basics

Positron Emission Tomography (PET) is an imaging technique applied in nuclear medicine that produces 2D- or 3D-images of physiological processes in human or animal bodies. Figure 3.11 shows an ECAT Exact HR+ PET-Scanner as an example for a typical PETscanner that produces data to process images. The imaging procedure can be described as follows. First of all, a radioactive tracer isotope is injected into the body; most commonly into blood circulation. The tracer circulates throughout the body and becomes concentrated in tissue after some amount of time. The amount of time it takes and the tissue region the tracer concentrates in depends on the type of tracer chosen. Different types of applications require different types of tracer isotopes. While the radioactive tracer isotope decays, it emits positrons, which are the antimatter counterparts of electrons. While traveling a few milimetres, the positrons annihilate with electrons within the body and each produces a pair of gamma photons that travel into opposite directions. During a PET-scan, these photons are detected by photon multiplier tubes. A pair of detectors defines a line, along which the intensity of the positron/electron annihilation is measured. The imaging task is to reconstruct information about the object, in which the tracer has been concentrated in, with the use of these intensities along the lines, only. The whole process is schematically described in Figure 3.22 .

Figure 3.2: Schematic representation of a PET-scan process: positron-electron annihilations are detected and processed to a certain data format used for image reconstruction. c

Wikipedia

3.1.2

Examination of Physiological Effects

What makes PET interesting as an imaging technique in comparison to other imaging techniques with higher spatial resolution, e.g. CT (Computer Tomography) or MRI (Magnetic Resonance Imaging), is the fact that examination and visualization of physiological effects inside the body is made possible. The radioactive tracer is not only a kind of contrast 1 2

http://upload.wikimedia.org/wikipedia/commons/b/b8/ECAT-Exact-HR--PET-Scanner.jpg http://upload.wikimedia.org/wikipedia/commons/c/c1/PET-schema.png

19

medium. It interacts within the body and therefore allows further insight into functional processes that occur. The use of a particular radiotracer molecule allows to monitor its interaction with the molecules within the body. Radioactive glucose for example, can be used to monitor metabolism effects (see figure 3.33 ); blood flow can be measured by the use of radioactive water, e.g. H2 15O. The big advantage in visualizing physiological processes is the fact, that it allows noninvasive examination. Imagine a patient suffering a heart disease caused by a contradicted arteria or vein. The standard medical process would include a cardiac catheterization to determine whether an arteria or vein is blocked or simply contradicted. Catheterization is a cumbersome procedure for patients; if blood flow within the hearts arterias and veins could be estimated with PET-imaging, this procedure could be prevented.

Figure 3.3: Maximum Intensity Projection of a 18F-FDG wholebody PET acquisition. 18FFluorodeoxyglucose can be used to monitor glucose-metabolism. A typical application is the detection of tumors. The red parts describe a high, the blue parts a low absorption of c 18F-FDG into tissue. Wikipedia

3.2

Basic Heart Anatomy and Cardiovascular System

Before we continue describing the process of making flow visible, or at least quantifiable, we are going to explain relevant basics of the hearts’ anatomy, the functionality of the cardiovascular system and oxygen supply of organs.

3.2.1

Basic Heart Anatomy

The heart’s anatomy is shown in Figure 3.44 . The regions that will be of major interest for our investigation are the left and right ventricle, the septum and the myocardium. 3

http://upload.wikimedia.org/wikipedia/commons/3/3d/PET-MIPS-anim.gif By courtesy of the National Heart, Lung and Blood Institute, http://www.nhlbi.nih.gov/health/dci/ images/heart_interior.gif 4

20

c Figure 3.4: A schematic representation of the heart’s interior. National Heart, Lung and Blood Institute, as part of the National Institutes of Health and the U.S. Department of Health and Human Services

Basically, the heart performs a suction of blood from lung vein and cava into left and right atrium. The blood coming from the lung veins enter the left atrium and gets transported into the left ventricle passing the mitral valve, finally leaving throughout the aorta. The blood entering the right atrium from the superior and inferior vena cava is transported via the tricuspid valve into the right ventricle and leaves the heart via the pulmonary artery, truncus pulmonalis. The left and the right ventricle are completely separated from each other through the septum. The ventricles are surrounded by the myocardium. Thereby the myocardium surrounding the left ventricle is much thicker than the one surrounding the right ventricle, since the left part of the heart supplies the body’s blood circulation while the right part supplies the pulmonary circulation.

3.2.2

Circulatory System

The heart is the body’s engine that ensures the supply with oxygenated blood for all organs. This supply is guaranteed via the cardiovascular system. There are basically two circulations in the human circulatory system, which are shown in Figure 3.55 , the pulmonary circulation and the cardiovascular system. Starting with the cardiovascular system, blood enters the left atrium through the lung veins, moving forward from the left ventricle to leave via the aorta. The aorta transports the oxygenated blood through arterias to the body’s organs. The depleted blood is returned via the veins back to the cava, entering the right atrium and passes the right ventricle. The cardiovascular circulation is completed and the pulmonary circulation starts. Via the truncus pulmonalis arteria the blood is transported into the lungs to re-oxygenate. It returns through veins opening to the lung veins back to left atrium and the cardiovascular circulation starts again. We want to notice that if a radioactive tracer is injected into a vein (e.g. an arm’s vein), the tracer passes the heart’s right chamber first and, with a delay, enters the left chamber 5

http://upload.wikimedia.org/wikipedia/commons/0/0c/Blutkreislauf_Gleichwarme.svg

21

To Head Direction of Blood Flow Lung

Lung

Body

Figure 3.5: The human circulatory system. On the one hand, there is the pulmonary circulation, assuring the blood’s supply with oxygen. On the other hand, there is the cardiovascular c circulation, guaranteeing the organ’s supply with oxygenated blood. Wikipedia then after having passed the lungs. Next, we are going to carefully examine the functionality of the organs’ supply with oxygenated blood.

3.2.3

Organ Oxygen Supply

Let us examine the cardiovascular circulation only. Blood is transported via arterias to the body’s organs. The exchange of oxygen happens via the capillary vessels as described in Figure 3.66 . Blood coming from an arteria enters the net of capillary vessels. There, the blood is extracted into tissue and returned to the capillary again, assuring the organs’ supply with oxygen. The underpressure guarantees that the depleted blood flows back into veins and back to the heart then. Every organ is supplied with blood in that way, especially the heart itself, which leads to the terms of myocardial blood flow and perfusion.

3.3

Myocardial Blood Flow and Perfusion

As described in Subsection 3.2.3, every organ is supplied with oxygenated blood via extraction from capillaries into tissue, especially the heart itself. Now imagine a coronary disease, e.g. a blocked arteria we want to examine via PET. Therefore we would need a radioactive tracer that is highly diffusible, in order to make flow visible. An ideal candidate would be radioactive water, H2 15O. Unfortunately, as described in Chapter 1, image reconstructions will be of poor quality and low resolution. Indeed the resolution will be even too low to make capillaries visible. As a consequence, direct examination of capillary blood flow with H2 15O is impossible. Fortunately, we are able to visualize left ventricular myocardial tissue; 6

Image in [SW94]

22

Figure 3.6: A net of capillaries. Blood coming from an arteria enters via arteriolar the net of capillary vessels. Between these vessels and adjacent tissue exchange of blood happens. The depleted blood is returned via the venule to a vein. By courtesy of Prof. Dr. Erwin-Josef c Speckmann, Urban & Schwarzenberg tissue in which capillary blood extraction and contraction takes place. Hence, we are at least able to display perfusable tissue. Blood flow is usually measured as the volume of blood that passes through a blood vessel per unit time (e.g. ml/min). In comparison, perfusion is expressed as blood flow per unit mass (e.g. ml/min/mg). Thus, examining perfusion allows us to draw conclusions on blood flow; and as a consequence, on how well blood is flowing through hearts capillaries, arterias and veins. The only problem that remains is to draw conclusions from perfusable tissue on perfusion. This problem can be solved via a simple mathematical model described in the following section.

3.4

An ODE Model for Blood Flow Computation

The following parts of this chapter are mainly based on descriptions summarized in [WA04]. Before deriving a simple mathematical model to compute perfusion from a specific radioactivity concentration of H2 15O extracted into tissue, we want to introduce the terms compartment and compartment modeling. In PET, images are compositions of superimposed signals. Imagine only one signal is of interest, then we have to isolate the desired one in terms of a mathematical model, which includes all possible states of that signal given by a sequence of PET-reconstructions. In kinetic modeling, these states are known as compartments. Each compartment is characterized by its underlying radioactive concentration as a function of time, C(t). Compartments are fixed spatial areas in a PET-image-sequence. The easiest and probably most desirable example of a compartment would be a certain pixel or voxel in a generated image-sequence. Normally, larger compartments have to be chosen due to noisy, low resolution reconstructions. The reason for introducing compartments is the fact, that the underlying radioactivity concentrations of the described signals are often related to each other via physical effects. Capillary blood flow is e.g. related to perfusable tissue, as described in Section 3.3. The signal blood flow is not directly visible in an H2 15O-scan, but the signal perfusable tissue is. Hence, a compartment lying spatially in an area of perfusable tissue - equipped with a physical model relating the signals perfusable tissue and myocardial blood flow - could be used to nevertheless compute flow. The next step is to find an appropriate model relating perfusion to blood flow. This 23

so-called one-compartment model will be described in the following subsection.

3.4.1

Derivation of Ordinary Differential Equations

Let us start with a simple basic compartment-model as shown in Figure 3.7. At the bottom, Tissue

Blood

Figure 3.7: The one-tissue-compartmental model. Homogeneously distributed radioactive tracer concentrations exchange between blood (bottom) and tissue (top). there is blood in an arteria, transporting a radioactive tracer, described via a function CA (t). On top, we have blood concentrated in tissue, enriched with radioactivity, described via CT (t). In both areas the tracer is assumed to be (spatially) homogeneously distributed. Now, if we would examine the compartment over time, there would be exchange between blood and tissue, described via constants α and β. If, for each timestep t, αCA (t) is the amount of flux from arteria to tissue, while βCT (t) is the amount of flux from tissue to arteria, this process could be described via CT (t + 1) − CT (t) ≈ αCA (t) − βCT (t), or via CT (t + h) − CT (t) ≈ αCA (t) − βCT (t) h for an arbitrary timestep h, respectively. If we consider the limit h → 0 instead, we obtain the ordinary differential equation (ODE) dCT (t) = αCA (t) − βCT (t). dt

(3.1)

With initial value7 CT (0) = 0, we receive Zt CT (t) = α

CA (τ )e−β(t−τ ) dτ ,

(3.2)

0

which is easy to verify via Laplace-transformation or variation of constants. In the following, we want to extend our model in terms of blood flow, respectively perfusion. Consequently we extend the compartmental model to the one shown in Figure 3.8. The extension shows capillary and tissue. Blood flows through the capillary with a flow-rate F , transporting the radioactive tracer. It enters the capillary from the arterial side with concentration CA 7 CT (0) = 0 as an initial value is a natural choice due to the fact that an H2 15O-scan usually is started before the tracer is injected.

24

Tissue

,

Blood

,

Figure 3.8: The one-compartment model extended to terms of flow. In addition to Figure 3.7, exchange occurs between tissue (top) and a capillary vessel (bottom), transporting blood with a flow rate F . and leaves via the venous side with concentration CV . A relation between these quantities can be derived via Fick’s principle, which states that the flux of blood (or material in general) entering the compartment equals the flux that leaves the compartment (if the compartment is at steady state, which is assumed here). The arterial tracer flux simply equals the product of blood flow and arterial tracer concentration, JA (t) = F CA (t). In an analogous way we obtain JV (t) = F CV (t) for the venous flux. Hence, Fick’s principle implies JA (t) = JT (t) + JV (t). Similar to (3.1) we derive the ODE JT (t) =

3.4.2

dCT (t) = JA (t) − JV (t) = F (CA (t) − CV (t)) . dt

(3.3)

Partition Coefficient

In our applications, we will deal with H2 15O as a tracer exclusively. Since the radioactive water is highly diffusible, the tissue concentration CT (t) quickly equilibrates with the venous outflow CV (t) for all t, such that these concentrations are related to each other by the soT called partition coefficient λ = C CV . The partition coefficient is usually set to a value between 0.91 ([IKT+ 88]) and 0.92 ([LLA+ 01]). Together with (3.3) this implies dCT (t) CT (t) = F CA (t) − , (3.4) dt λ respectively its associated integral relation Zt CT (t) = F

F

CA (τ )e− λ (t−τ ) dτ .

(3.5)

0

3.4.3

Tissue Fraction and Spillover Effects

To conclude the mathematical modeling of blood flow and perfusable tissue, we have to derive so-called tissue fraction and spillover effects. By introducing the ODE (3.4), we separated the tracer concentration into arterial concentration (CA ) coming from an artery, and into venous concentration (CV ) coming from a vein. In our model illustrated in Figure 3.8, the arterial concentration is the one originating in the heart’s left chamber; the venous concentration originates in the heart’s right chamber. Therefore, from now on CA (t) denotes 25

the tracer concentration in the left, CV (t) the concentration in the right ventricle. Every tracer concentration neither belonging to the left- or right-ventricle, nor to tissue is denoted with CB (t). RV

Septum

Compartment

LV Myocardium (a) Cardiovascular slice

(b) Compartmental region

(c) Fraction and spillover due to heart motion

Figure 3.9: Fraction and spillover effects. A schematic representation of a cardiovascular slice, containing left and right ventricle (LV and RV), myocardium and septum, can be seen in 3.9(a). In 3.9(b), an exemplary compartmental region is drawn to make the problem of low resolution visible. The compartment is too big and covers LV and Septum. In 3.9(c), the problem of heart motion makes modeling fraction and spillover effects inevitable. The compartmental region is static in contrast to moving myocardium resulting from contraction and extraction of the heart. Due to the low resolution of H2 15O-scans and time-dependent effects, such as heart motion, we will be unable to determine the exact region of myocardial tissue at a certain time. A simple way to overcome this problem is to estimate a larger region, surely containing the whole myocardial region. Now we have to face spillover effects, e.g. from left ventricular blood. Moreover, we need to estimate the fraction of purely perfused tissue. These preliminary ideas lead to the following extension of our model. We replace (3.5) with a linear combination CT

real (t)

= rCT (t) + s1 CA (t) + s2 CV (t) + s3 CB (t),

(3.6)

where r denotes the fraction of CT (t) in tissue (tissue fraction) and s1 , s2 and s3 denote spillover effects from tracer concentrations in arterial, venous, and other blood. Arterial spillover effects occur for areas of tissue close to the left ventricle, venous spillovers for areas close to right ventricle (e.g. Septum) and background spillovers near the hearts exterior. 26

Obviously, (3.6) is a linear combination of radioactive tracer concentrations. Thus, r + s1 + s2 + s3 = 1

(3.7)

holds. Unfortunately, we are unable to draw any conclusions on the background concentration CB (t), since we do not have mathematical models for its description. Consequently, we cannot draw conclusions on the associated background spillover, s3 . Hence, we have to limit (3.6) to CT

real (t)

= rCT (t) + s1 CA (t) + s2 CV (t).

(3.8)

One might wonder that only myocardial tissue around the left ventricle is examined and computed via CT , and not myocardial tissue around the right ventricle. Again, the simple reason is the limited resolution of H2 15O-scans, which makes further examination impossible (up to now). Quantification of perfusion in the right ventricle is an important topic for future research.

3.4.4

The ODE in Banach Spaces

Let Ω ⊂ Rn (for n = 2 in two spatial dimensions and n = 3 in three spatial dimensions, respectively) be a bounded, compact space that denotes the compartmental space, i.e. an element x ∈ Ω denotes a compartment. Furthermore t ∈ [0, T ] ⊂ R lies within a bounded and compact set. If we extend the myocardial perfusion F (x) to a function in space, depending on a compartment x ∈ Ω, we can rewrite (3.5) to the operator CT : Dp (CT ) × Lp ([0, T ]) → Lp (Ω × [0, T ]) with Zt CT (F, CA ) (x, t) = F (x)

CA (τ )e−

F (x) (t−τ ) λ

dτ

(3.9)

0

and Dp (CT ) := {F ∈ Lp (Ω), CA ∈ Lp ([0, T ]) | F ≥ 0, CA ≥ 0 a.e.} .

(3.10)

The ouput function of CT will be denoted with CT (x, t). Since F and CA represent physiological values, they are non-negative. Hence, CT is non-negative: Theorem 3.1. Let (F, CA ) ∈ Dp (CT ). Then, CT as defined in (3.9) is non-negative. Proof. This trivial consequence follows from Zt CT (F, CA ) (x, t) = F (x) | {z } ≥0

0

F (x)

CA (τ ) e|− λ{z(t−τ}) dτ ≥ 0 . | {z } ≥0

>0

Actually, the myocardial perfusion can vary a lot between different compartments. Hence, we want to allow F to be discontinuous and therefore have chosen F to lie within Lp (Ω), for p ≥ 1. Furthermore we can assume CA to lie within Lp ([0, T ]). Under these conditions and with λ ∈ R+ being positive we can verify that CT is continuous in Lp (Ω×[0, T ]): 27

Theorem 3.2 (Lp -continuity). Let (F, CA ) ∈ Dp (CT ) and let λ ∈ R+ be positive. Then, CT (F, CA ) for CT being defined as in (3.9) is well-defined and Lp -continuous on Dp (CT ) for p ≥ 1. Proof. The well-definedness is easy to prove. We obtain kCT (F, CA )kpLp (Ω×[0,T ])

Z ZT =

|CT |p dt dx

Ω 0

p   ≥0 }| { z ≥0    F (x) z }| {   (t − τ )  −   λ   Z ZT Zt = (x) CA (τ ) e| dτ dt dx F {z } | {z } | {z } ≤1 Ω 0 ≥0 0 ≥0 p Z ZT Zt F (x) CA (τ )dτ dt dx ≤ 0 0 Ω p Zt Z ZT p = |F (x)| CA (τ )dτ dt dx 0 Ω 0 p ZT Zt Z Fubini p = |F (x)| dx CA (τ )dτ dt . 0 0 Ω {z } | =kF kpL

p (Ω)

The fact CA ∈ Lp ([0, T ]) implies that A ∈ W 1,p ([0, T ]), with A(t) :=

Rt

CA (τ )dτ . Hence, in

0

particular A ∈ Lp ([0, T ]) holds as well and this yields kCT (F, CA )kpLp (Ω×[0,T ])

p ZT Zt p ≤ kF kLp (Ω) CA (τ )dτ dt = ≤

0 0 p kF kLp (Ω) kAkpLp ([0,T ]) kF kpLp (Ω) T kCA kpLp ([0,T ])

< ∞.

Thus, CT is well-defined. Furthermore, CT is linear in CA and the derived inequality therefore implies Lp -continuity of CT in CA . The Lp -continuity in F can be shown by a Lipschitz-argument. The inequality

1

1 2 2

CT (F 1 , CA ) − CT (F 2 , CA ) Lp (Ω×[0,T ]) ≤ F 1 − F 2 Lp (Ω) + CA − CA Lp ([0,T ]) holds, due to the Lipschitz-continuity of xe−x , for positive x. Furthermore, CT : Dp (CT ) ∩ (L2p (Ω) × L2p ([0, T ])) → Lp (Ω × [0, T ]) is Fréchet-differentiable: 28

Theorem 3.3. Under the conditions of Theorem 3.2 the operator CT (F, CA ) : Dp (CT ) ∩ (L2p (Ω) × L2p ([0, T ])) → Lp (Ω × [0, T ]), for CT being defined as in (3.9), is Fréchetdifferentiable. Proof. We have to show that the partial directional derivatives ∂F CT ϕF and ∂CA CT ϕA are linear in ϕF and ϕA , respectively, and that they satisfy (2.8). We want to start with ∂CA CT ϕA . The directional derivative is given via Zt F (x) ∂CT d − λ (t−τ ) ϕA = dτ F (x) (CA (τ ) + µϕA (τ )) e ∂CA dµ 0

Zt = F (x)

µ=0

ϕA (τ )e−

F (x) (t−τ ) λ

dτ ,

0

and is therefore linear in ϕA . Furthermore CT (F, CA + ϕA ) − CT (F, CA ) −

∂ CT (F, CA )ϕA = 0 , ∂CA

and hence, (2.8) is satisfied for all ϕA ∈ Lp ([0, T ]). The directional derivative ∂F CT ϕF is given by  t Z F (x) ∂  CT (F, CA )ϕF = ϕF (x) CA (τ )e− λ (t−τ ) dτ ∂F 0

−

F (x) λ

Zt

 (t − τ )CA (τ )e−

F (x) (t−τ ) λ

dτ 

0

ZT = ϕF (x)

CA (τ )e

−

F (x) (t−τ ) λ

t−τ 1 − F (x) λ

dτ ,

0

and hence, is linear in ϕF . To prove (2.8) we are going to define the function Hxt ∈ C ∞ ([a, b] × [c, d]) via σ1

Hxt (σ1 , σ2 ) := σ1 σ2 e− λ (t−τ ) , for compact and bounded sets [a, b] ⊂ R+ and [c, d] ⊂ R+ . Then the first and second derivative in σ1 are given by σ1 σ1 ∂Hxt (σ1 , σ2 ) = σ2 e− λ (t−τ ) 1 − (t − τ ) ∂σ1 λ and σ t−τ ∂ 2 Hxt − λ1 (t−τ ) σ1 (σ , σ ) = σ e (t − τ ) − 2 . 1 2 2 λ λ ∂σ12 For arbitrary but fixed (x, t) ∈ Ω × [0, T ] the functions F , ϕF and CA can be interpreted as variables, which we will also denote with F , ϕF and CA , respectively. Then we obtain Hxt (F + ϕF , CA ) − Hxt (F, CA ) − =

∂Hxt (F, CA )ϕF ∂σ1

∂Hxt ∂Hxt (ζ, CA )ϕF − (F, CA )ϕF , ∂σ1 ∂σ1 29

for ζ ∈ [F, F + ϕF ], due to the mean-value theorem. Applying the mean-value theorem a second time yields ∂Hxt (F, CA )ϕF Hxt (F + ϕF , CA ) − Hxt (F, CA ) − ∂σ 1 ∂Hxt ∂Hxt (ζ, CA ) − (F, CA ) = ϕF ∂σ1 ∂σ1 ∂ 2 Hxt = ϕ2F (ξ, CA ) , ∂σ12 for ξ ∈ [F, ζ] = [F, F + ϕF ]. We can estimate Hxt (F + ϕF , CA ) − Hxt (F, CA ) − ∂Hxt (F, CA )ϕF ∂σ1 ∂ 2 Hxt = ϕ2F (ξ, C ) A ∂σ12 t − τ ξ ξ = CA e− λ (t−τ ) (t − τ ) − 2 ϕ2F | {z λ } λ | {z } ≤T ≤e−3 0. Suppose (4.9) has a unique minimum u ˆ. Then u ˆ is the limit of the sequence generated by (4.14). Proof. See [NW01]. In case of noisy data, convergence is not only harder to prove; even divergence might occur. In order to obtain meaningful and optimal results, stopping rules can be added. It can be proved, that these stopping rules guarantee optimal solutions, using the standard EM algorithm without modifications (see [REI07]).

4.2

The Inverse Problem of Myocardial Perfusion Quantification

In previous myocardial perfusion quantification as described in Section 3.5, the computation of parameters is based on reconstructed frames, as illustrated in Figure 4.1. The injected

f (x, t)

EM −→ algorithm

u(x, t)

nonlinear computation −→ of parameters

F,r,s1 ,s2 CA ,CV

Figure 4.1: A schematic representation of previous myocardial perfusion quantification. The inverse problem of reconstructing frames from a dynamic H2 15O-scan is solved first. Subsequently, parameters like perfusion F are computed directly from these frames. H2 15O is measured with a PET-scan. The positron/electron annihilations detected, namely f , build the basis for an image reconstruction process, for example by using the EMalgorithm described in Section 4.1. We will obtain noisy images u(x, t), due to the short

39

half-time of H2 15O, which are then used as input for the computation of the myocardial blood flow F and the remaining parameters as described in Section 3.5. We have already mentioned that the disadvantage of this concept is the huge loss of information by computing parameters from reconstructed frames instead of computing them from the underlying data. The temporal correlation between the measured data is completely neglected in this process. In Section 4.1 we have seen, that the standard EM-algorithm - in the case of reconstructing PET-data - aims at solving the inverse problem of the Radon-transform (see Definition 4.2), i.e. the computation u from Ru = f . Hence, instead of solving one inverse problem and computing the parameters directly from u, the whole computation of parameters could be treated as an inverse problem, with skipping the generation of noisy images. The inverse problem could be described as follows: a nonlinear operator G with various input functions such as F generates images over time by the use of the physicially derived ODE developed in Chapter 3. This image sequence can be computed via a linear Operator A applied to raw PET-data, named f . The computation of F can be seen as an inverse problem of the whole process, shown in Figure 4.2. By inverting the nonlinear operator G, we could skip

Figure 4.2: The Computation of myocardial perfusion as an inverse problem. By inverting the nonlinear Operator G, we could skip the step of image processing and compute the the parameters from the data. the image sequence generation and compute the parameters from the PET data. It should be mentioned here, that in our later algorithm we will still process a sequence of frames u(x, t), but as a byproduct of the nonlinear inversion process. The actual disadvantage of dealing with inverse problems is that most inverse problems are ill-posed (see Section 2.4). Thus, we have to think about appropriate regularization of the inverse problem. In the following sections we will on the one hand derive the analytical model describing the inverse problem and on the other hand derive appropriate regularizations to force well-posedness.

4.3

Variational Model

Treating the computation of myocardial perfusion as an inverse problem as described in Section 4.2 yields the following optimization problem: IM (u) + R(p) → min p

subject to u(x, t) = G(p),

(4.15)

with some functional IM representing the image reconstruction process and R denoting the regularization functional. At first, all parameters are combined to a vector p, which 40

is done only for the sake of simplicity. We will specify p and G in more detail in Section 4.4. An image reconstruction method particularly developed for PET is the EM-algorithm introduced in Section 4.1. If we choose the functional from (4.10) for IM extended to time-dependent data, we obtain ZT Z (Au − f log (Au)) dxdt + R(p) → min p

0 Ω

subject to u = G(p).

(4.16)

The easiest way to solve a constrained optimization problem consists in computing the partial Fréchet-derivatives of the associated Langrange functional ZT Z L(u, p; q) =

(Au − f log (Au)) dxdt + R(p) 0 Ω

+ hG(p) − u, qiL2 ([0,T ]×Ω) ZT Z (Au − f log (Au)) dxdt + R(p)

= 0 Ω ZT

Z (G(p) − u) q dx dt

+

(4.17)

0 Ω

and setting them to zero. Thus, we obtain a system of operator equations f ∂ L(u, p; q) = A∗ 1 − A∗ −q 0 = ∂u Au ∂ 0 0 L(u, p; q) = R (p) + G (p)∗ q 0 = ∂p ∂ 0 = L(u, p; q) = G(p) − u ∂q with G(p) being positive, which can be rewritten as (multiplying the first equation with u) f ∗ 0 = us − uA − u q, (4.18) Au 0

0

G (p)∗ q = −R (p)

(4.19)

and u = G(p),

(4.20)

with s being defined as in (4.12). Equation (4.18) contains the derivative of the EMfunctional (4.11). If we replace the minimizer by its discrete analogue (4.14), and denote its solution with uk+ 1 , (4.18) can be rewritten as 2

uk − uk+ 1 2

uk

− q = 0.

41

(4.21)

Equation (4.21) can be treated as a solution of the minimization problem 1 2

ZT Z

u − uk+ 1

2

2

dxdt − hu, qiL2 ([0,T ]×Ω) → min .

uk

(4.22)

u

0 Ω

If we use our constraint (4.20) to replace u with G(p) we obtain 1 2

ZT Z

G(p) − uk+ 1

2

2

dxdt − hG(p), qiL2 ([0,T ]×Ω) → min .

uk

(4.23)

p

0 Ω 0

It is easy to see that the Fréchet-derivative of hG(p), qi in p is simply G (p)∗ q. Therefore, using (4.19), we can replace −hG(p), qi by R(p) to obtain the reduced problem 1 2

ZT Z

G(p) − uk+ 1

2

2

dxdt + R(p) → min .

uk

p

(4.24)

0 Ω

The first-order optimality condition is given by 0

0 = G (p)∗

G(p) − uk+ 1 2

uk

! 0

+ R (p).

(4.25)

Equation (4.25) cannot be solved directly, since the parameter p contains numerous functions that have to be computed each on their own. This problem can be handled by treating the computation of p as a parameter identification problem, which will be discussed in Section 4.6. Before 4.3 is put, we will discuss different approaches for the image sequence generation operator G.

4.4

Image Sequence Generation

As a result of the poor image quality of H2 15O-reconstructions, a slice of cardiovascular region basically contains the areas described in Figure 4.3. Based on this schematic representation we make some definitions of open sets. Definition 4.4. Assume n ∈ N, typically n = 2 or n = 3. Then • Ω ⊂ Rn denotes the whole image space • A stands for the left ventricular region • V describes the right ventricular region • M represents the myocardial region, excluding the septum • S denotes the septum • T with T := M ∪ S describes the whole region of tissue • H with H := A + V + T represents the whole cardiovascular area • B := Ω\H denotes the so-called background area 42

Myocardium RV (not scannable)

Left ventricular

Right ventricular

Septum

Myocardium LV

Figure 4.3: Top-view slice of a cardiovascular region showing what remains visible from a H2 15O-PET-scan. Due to a PET-scan-resolution of currently about 5 mm, the myocardium around the RV, which is about 4 mm thick, cannot be scanned yet. LV, RV and LV’s myocardium are visible and observable. Furthermore, the sets A, V, M and S are disjoint, i.e. A ∩ V ∩ M ∩ S = ∅. If we return to the terms of compartment modelling introduced in Section 3.4, we can treat a compartment x as an element in image space, e.g. x ∈ Ω. To decide which cardiovascular region a compartment belongs to, we simply use indicator functions XΨ : Ω → {0, 1}, defined as 1 if x belongs to Ψ, (4.26) XΨ (x) := 0 else, with Ψ ⊆ Ω. These definitions and the theory derived in Chapter 3 build the basis for the following definitions of G.

4.4.1

Image Sequence Generation: A Simple Approach

Assume that we are able to perform a perfect segmentation, i.e., the indicator functions perfectly indicate to which space a compartment belongs to, for each single compartment of Ω. If we want to generate a sequence of T frames, the generation of an image sequence ˆ : L4 (A) × L4 (V) × L4 (T ) × L4 (B) × L8 ([0, T ]) × L4 ([0, T ]) × can simply be computed via G L4 (B) × L4 ([0, T ]) × L8 (T ) → L2 (Ω × [0, T ]) with ˆ A , XV , XT , XB , CA , CV , CB ; F ) = XA (x)CA (t) + XV (x)CV (t) G(X +XT (x)CT (F, CA )(x, t) +XB (x)CB (x, t),

(4.27)

with CA (t) describing the left ventricular tracer concentration, CV (t) describing the tracer concentration in the right ventricle and CT (F, CA ) : L8 (T ) × L8 ([0, T ]) → L4 (T × [0, T ])) as introduced in Section 3.4.4, representing the tracer concentration in tissue, computed via (3.4) for a given perfusion F (x), left ventricular tracer concentration CA (t), and fixed 43

partition coefficient λ (typically 0.915). The background concentration CB (x, t) simply completes the image sequence. All functions are non-negative since they represent physiological parameters. In the following, CA and CV are named arterial tracer concentration ˆ implicitly depends on F (x), and venous tracer concentration, respectively. Notice that G ˆ meets the demands of (4.24) for due to the connection between CT and F . Hence, G p = (F (x), CA (t), CV (t)). The basic problem already discussed in Section 3.4.3 is the non-existence of a perfect segmentation due to cardiovascular motion and limited resolution. Hence, no matter how small we are going to choose our compartments, we will never be able to completely indicate to which space a compartment x belongs to.

4.4.2

Fraction and Spillover: The More Complex Approach

To overcome the segmentation issue and other modelling aspects, tissue fraction and spillover terms as described in Section 3.4.3 have been introduced. This modifies (4.27) to G : L4 (A) × L4 (V) × L4 (M) × L4 (S) × L4 (B) × L8 ([0, T ]) × L4 ([0, T ]) × L4 (B) × L4 ([0, T ]) × L4 (T ) × L4 (T ) × L4 (S) × L8 (T ) → L2 (Ω × [0, T ])) with G(XA , XV , XM , XS , XB , CA , CV , CB , r, s1 , s2 ; F ) = XA (x)CA (t) + XV (x)CV (t) + XB (x)CB (x, t) + XM (x) (r(x)CT (F, CA )(x, t) + s1 (x)CA (t)) + XS (x) (r(x)CT (F, CA )(x, t) + s1 (x)CA (t) + s2 (x)CV (t)) .

(4.28)

ˆ G meets the demands of (4.24) for p = (F (x), CA (t), CV (t), r(x), s1 (x), s2 (x)). Just like G, Since the indicator functions will be fixed in the upcoming optimization process, CT implicitly depends on F , CB is not of further interest and to keep notations simple, we are going to write G(CA , CV , CT , r, s1 , s2 ) instead of G(XA , XV , XM , XS , XB , CA , CV , CB , r, s1 , s2 ; F ) in the following sections. Segmentation itself - in order to obtain accurate indicator functions X - will be discussed in more detail in Section 4.8.

4.5

Existence of a Solution

Without further regularization, the optimization problem we want to solve can simply be written as EM (u) → min subject to u = G(p) , p

(4.29)

with p denoting the parameters and EM (u) representing the time-dependent EM-functional, i.e. ZT Z Au − f log Audx dt .

EM (u) =

(4.30)

0 Ω

We can replace u with G(p) to obtain EM (G(p)) → min p

(4.31)

as the equivalent problem. With this section we want to prove that a solution for (4.31) exists indeed. In order to do so we consider the complex operator G as introduced in (4.28). 44

We want to notice that all conclusions that will be drawn throughout this section can easily ˆ satisfying (4.27). be transferred to the simple case with G First of all we want to derive some properties that are satisfied by G to subsequently prove existence with the use of Theorem 2.1. Theorem 4.4. Let F, r, s1 ∈ L8 (T ), s2 ∈ L8 (S) and CA , CV ∈ L8 ([0, T ]) be non-negative functions and let λ ∈ R+ . Then the operator G satisfying (4.28) is 1. L2 -continuous, 2. non-negative, i.e. G ≥ 0, 3. Fréchet-differentiable. Proof. All conclusions simply follow from Theorem 3.1, Theorem 3.3 and Theorem 3.4. The operator G is just a bilinear combination of non-negative L8 -functions and the operator CT (F, CA ) : L8 (Ω)×L8 ([0, T ]) → L4 (Ω×[0, T ]), which has been proved to be L4 -continuous and Fréchet-differentiable. Hence, G satisfies these conditions. In the following we want to restrict Ω ⊂ R2 to be two dimensional and we are going to write Ω × [0, T ] instead of [0, T ] × [0, T ] × T × [0, T ] × T × S for the sake of simplicity. Furthermore we will use the compact embedding H1 ,→ L8 (notice that compactness of this embedding can only be concluded in the case of one or two dimensions, respectively). We can restrict G : L8 (Ω)×L8 ([0, T ]) → L2 (Ω×[0, T ]) to G : H1 (Ω)×H1 ([0, T ]) → L2 (Ω×[0, T ]) and obtain a not only L2 -continuous, but also compact operator G. This yields existence of the optimization problem (4.31): Theorem 4.5 (Existence). Let p(x, t) ∈ H1 (Ω × [0, T ]) denote a parameter containing nonnegative H1 -input functions for G, with G being defined as in (4.28), such that p satisfies the conditions of Theorem 4.4. Furthermore let EM denote the EM-functional extended to an integral in time as defined in (4.30). Then there exists a solution of the optimization problem (4.31). Proof. In order to prove this statement we want to extend the EM-functional to the Kullback-Leibler functional. In order to do so we investigate each EM-functional for a fixed time-step t. If we add f log(f ) − f to (4.9) we obtain the Kullback-Leibler form of the EM-functional, Z Z f Au − f log(Au) + f log(f ) − f dx = Au − f + f log( )dx = d(f, Au) , Au Ω

Ω

for d(v, u) := Ω v log( uv )−v+u dx being the Kullback-Leibler divergence. Adding f log(f )− f to the EM-functional does not affect the optimal solution of (4.31) (if it exists), since the term is constant in u. Hence, we can consider d(f, Au) instead of EM (u). According to [RA07, Lemma 3.4 (iii)], d(f, Au) is lower semi-continuous with respect to the weak topology of L1 (Ω). Since t was fixed but arbitrary, we can extend all considerations to the time dependent EM operator (4.30), which is then lower semi-continuous with respect to the weak topology of L1 (Ω × [0, T ]). Furthermore the L2 -continuity of G implies L1 -continuity, since Ω × [0, T ] is compact. Due to compactness of the operator G, the composition IM (G) is lower semi-continuous as well. For a sequence pk → p in H1 (Ω × [0, T ]) compactness of G implies that uk = G(pk ) → R

45

G(p) = u holds in L1 (Ω × [0, T ]). Together with the lower semi-continuity in L1 (Ω × [0, T ]) of EM the composition EM (G(p)) is lower semi-continuous in L1 (Ω × [0, T ]). Additionally, [RA07, Lemma 3.4 (iv)] states that for C > 0 and non-negative Au ∈ L1 (Ω) the sub-level set {f ∈ L1 (Ω) | d(f, Au) ≤ C} is weakly compact in L1 (Ω). Again we can extend the theorem to the case of time-dependent data and since u = G(p) is non-negative and a L1 -function, all conditions to apply Theorem 2.1 are satisfied and there exists a solution of (4.31). Furthermore both EM and G are Fréchet-differentiable and therefore an optimal solution of (4.31) could be computed by setting the Fréchet-derivative of EM (G(p)) to zero. In practice this is not applicable and we have to move over to the parameter identification problem. But at least we can be sure that there exists a solution for our optimization problem and that Fréchet-differentiability of G and EM should allow us to compute an optimal solution within the process of parameter identification.

4.6

Parameter Identification

The remaining issue is to compute p. We cannot solve (4.25) directly for p, since p consists 0 of multiple functions and G (p)∗ is a vector of functions itself (and difficult to compute). For the inversion of G with respect to the different parameters, we have to solve a so-called parameter identification problem. The idea is to return to the constrained optimization problem with added regularization, i.e. 2 ZT Z u − u 1 ZT Z k+ 1 2 dxdt + R(p) + (G(p) − u) q dxdt → min , (4.32) p 2 uk 0 Ω

0 Ω

with the constraint G(p) = G(p(x, t)) = u(x, t), for all (x, t) ∈ Ω × [0, T ]. Considering the associated Lagrange functional, this yields 2 ZT Z u − u 1 ZT Z k+ 2 1 dxdt + R(p) + (G(p) − u) q dxdt . (4.33) L(u, p; q) = 2 uk 0 Ω

0 Ω

Additionally, we introduce a second constraint to CT , guaranteeing the connection between CT and (3.4). According to the preliminary ideas of Sections 4.2 and 4.3 and by rewriting p to its underlying functions, this finally leads to the Lagrange functional L (u(x, t), F (x), CA (t), CV (t), CT (x, t); q(x, t), µ(x, t)) 2 ZT Z u(x, t) − u 1 (x, t) k+ 2 1 = dxdt + R1 (F (x)) + R2 (CA (t)) + R3 (CV (t)) 2 uk (x, t) 0 Ω ZT Z

0 Ω ZT Z

+

+

ˆ A (t), CV (t), CT (x, t)) − u(x, t) q(x, t)dxdt G(C ∂CT CT (x, t) (x, t) − F (x) CA (t) − µ(x, t)dxdt, ∂t λ

0 T

46

(4.34)

ˆ as described in (4.27). For G as introduced in (4.28), we obtain with G L (u(x, t), F (x), CA (t), CV (t), CT (x, t), r(x), s1 (x), s2 (x); q(x, t), µ(x, t)) 2 ZT Z u(x, t) − u 1 (x, t) k+ 2 1 dxdt + R1 (F (x)) + R2 (CA (t)) + R3 (CV (t)) = 2 uk (x, t) 0 Ω

+R4 (r(x)) + R5 (s1 (x)) + R6 (s2 (x)) ZT Z (G(CA (t), CV (t), CT (x, t), r(x), s1 (x), s2 (x)) − u(x, t)) q(x, t)dxdt

+ 0 Ω ZT Z

+

∂CT CT (x, t) µ(x, t)dxdt. (x, t) − F (x) CA (t) − ∂t λ

(4.35)

0 T

In order to compute the desired optimal functions satisfying (4.32), we have to inspect the optimality conditions of the Lagrange functional. These conditions can be computed by setting the partial Fréchet-derivatives to zero. We will start with the simple case of (4.34), where we obtain u(x, t) − uk+ 1 (x, t) ∂L 2 = 0 = − q(x, t) , ∂u uk (x, t) ZT ∂L CT (x, t) 0 = 0 = R1 (F (x)) − CA (t) − µ(x, t)dt , ∂F λ 0 Z Z ∂L 0 = 0 = R2 (CA (t)) + XA (x)q(x, t)dx − F (x)µ(x, t)dx , ∂CA Ω T Z Z 0 = R2 (CA (t)) + q(x, t)dx − F (x)µ(x, t)dx , and ∂L 0 = 0 = R3 (CV (t)) + ∂CV (t)

A Z

(4.36)

(4.37)

(4.38)

T

q(x, t)dx .

(4.39)

V

The optimality condition ∂CT L = 0 is slightly more complicated to compute. Therefore, we examine the directional derivative in direction defined in (2.1), ϕT (x, t) appears ϕT (x, t). As 1 to lie within the same space as CT (x, t), i.e. f (x, t) ∈ C (T × [0, T ]) | f (x, 0) = 0 ∀x ∈ T . Hence, ϕT (x, 0) = 0, for all x ∈ T . This yields ∂L ϕT (x, t) = ∂CT

ZT Z

d ˆ G(CA (t), CV (t), CT (x, t) + τ ϕT (x, t)) dτ

0 Ω

−u(x, t)) q(x, t) dxdt ZT Z d ∂ + (CT (x, t) + τ ϕT (x, t)) − dτ ∂t 0 T

47

Fubini

=

CT (x, t) + τ ϕT (x, t) F (x) CA (t) − µ(x, t)dxdt λ τ =0 T T Z Z Z Z ˆ ∂ϕT ∂G q(x, t)ϕT (x, t)dxdt + (x, t)µ(x, t)dtdx ∂CT ∂t T

0 Ω

0

Z ZT

F (x) µ(x, t)ϕT (x, t) dtdx λ T 0  ZT Z Z  q(x, t)ϕT (x, t) dxdt +  ϕT (x, t)µ(x, t)|T0 − {z } |

+

part. int.

=

0 T

ZT

T

 ∂µ ϕT (x, t) (x, t) dtdx + ∂t

=µ(x,T )

Z ZT ϕT (x, t) T

0

F (x) µ(x, t) dtdx . λ

(4.40)

0

Setting (4.40) to zero implies, that µ has to satisfy an ODE with terminal value zero, more precisely ∂µ(x, t) F (x) = q(x, t) + µ(x, t) ∂t λ subject to µ(x, T ) = 0

(4.41) for all x ∈ T .

The remaining optimality conditions ∂L ˆ A (t), CV (t), CT (x, t)) − u(x, t) = 0 = G(C ∂q = XA (x)CA (t) + XV (x)CV (t) + XT (x)CT (x, t) − u(x, t)

(4.42)

and CT (x, t) ∂CT (x, t) − F (x) CA (t) − , 0= ∂t λ subject to CT (x, 0) = 0 for all x ∈ T ,

(4.43)

simply guarantee the fulfillment of the constraints. For the more complex case of G satisfying (4.28), the equations (4.36), (4.37) and (4.43) remain unchanged. The other equations slightly modify to Z Z Z ∂L 0 = 0 = R2 (CA (t)) + q(x, t)dx + s1 (x)q(x, t)dx − F (x)µ(x, t)dx, (4.44) ∂CA A T T Z Z ∂L 0 = 0 = R3 (CV (t)) + q(x, t)dx + s2 (x)q(x, t)dx , (4.45) ∂CV V

S

∂L = 0 = G(CA (t), CV (t), CT (x, t), r(x), s1 (x), s2 (x)) − u(x, t) ∂q = XA (x)CA (t) + XV (x)CV (t) + XM (x) (r(x)CT (t) + s1 (x)CA (t)) + XS (x) (r(x)CT (x, t) + s1 (x)CA (t) + s2 (x)CV (t)) − u(x, t)

48

(4.46)

and ∂µ F (x) (x, t) = r(x)q(x, t) + µ(x, t) ∂t λ subject to µ(x, T ) = 0 for all x ∈ T .

(4.47)

Furthermore, we obtain additional optimality conditions for r(x), s1 (x) and s2 (x), ∂L 0 = 0 = R4 (r(x)) + XT (x) ∂r

ZT CT (x, t)q(x, t)dt ,

(4.48)

0

∂L 0 = 0 = R5 (s1 (x)) + XT (x) ∂s1

ZT CA (t)q(x, t)dt , and

(4.49)

CV (t)q(x, t)dt .

(4.50)

0

∂L 0 = 0 = R6 (s1 (x)) + XS (x) ∂s1

ZT 0

The equations (4.36), (4.43), (4.46) and (4.47) can be solved analytically via u(x, t) − uk+ 1 (x, t)

2 , uk (x, t) Zt F (x) CT (x, t) = F (x) CA (τ )e− λ (t−τ ) dτ ,

q(x, t) =

(4.51)

(4.52)

0

u(x, t) = G(CA (t), CV (t), CT (x, t), r(x), s1 (x), s2 (x)) , and Zt F (x) F (x) − λ (T −t) T + e λ (t−τ ) r(x)q(x, τ )dτ . µ(x, t) = e

(4.53) (4.54)

0

The remaining equations need to be solved numerically; we will discuss different approaches in Chapter 5. First of all, we are going to talk about regularization to enforce well-posedness for our problem.

4.7

Regularization

Before we will be able to talk about (numerical) solution methods we have to specify the regularization terms Ri , i ∈ {1, . . . , 6}, to avoid ill-posedness of our problem. Remember, that for a well-posed problem, we need existence, uniqueness and continuous dependence on the data. Existence of a solution can usually be verified using Theorem 2.1 (see Theorem 4.5). Uniqueness and continuous dependency may still be problematic. As we have seen in Example 2.5, continuous dependence is very likely to become a major issue. If we do not regularize our solution by keeping it bounded in some meaningful sense, similar problems will occur as illustrated in Figure 2.1.

4.7.1

Regularization Incorporating A-priori Knowledge

Luckily, we are able to use a-priori knowledge in the regularization functional. Myocardial perfusion has a typical value of F ∗ . Hence, we are able to regularize F via Z α R1 (F (x)) = (F (x) − F ∗ )2 dx , (4.55) 2 Ω

49

with a regularization parameter α ∈ R+ . This regularization allows to favour solutions close to F ∗ and to penalize solutions being too far off. In a natural way, we can generalize this kind of a-priori-regularization to all parameters collected in p, with Z α ∗ Rα,Ψ (g(σ), g ) = (g(σ) − g ∗ )2 dσ (4.56) 2 Ψ

for a set Ψ ⊂ Ω or Ψ = [0, T ] and α ∈ R+ . The a-priori regularization demonstrates the advantages of dealing with quantification of myocardial blood flow as an inverse problem. We are not only able to derive our parameters from the data instead of an image-sequence. We can also regularize every parameter independently. In the following, we denote our a∗ , C ∗ , r ∗ , s∗ and s∗ and are, e.g., going to write R priori knowledge with F ∗ , CA α,[0,T ] (CA (t)) 1 2 V ∗ instead of Rα,[0,T ] (CA(t) , CA ), for the sake of simplicity.

4.7.2

Gradient Regularization

Furthermore, a reasonable regularization we want to add is a bound to the spatial gradients, 0 0 ∇F , ∇r, ∇s1 and ∇s2 and on the time derivatives CA and CV . Due to noise, the gradients and derivatives might become very large, comparable to the numerical differentiation in Example 2.5. Therefore we will add regularization terms of the type Z η |∇g(x)|2 dx (4.57) Rη,Φ (g) = 2 Φ

and η Rη,[a,b] (g) = 2

Zb

2 0 g (t) dt

(4.58)

a

with Φ ⊂ Ω and [a, b] ⊂ R. This type of regularization will guarantee a smoothing in space and time. The optimality conditions of (4.57) and (4.58) are computed in an analogous way to (2.21). We have Z η d 0 2 Rη,Φ ϕg (x) = 0 = |∇ (g(x) + τ ϕg (x))| dx 2 dτ Φ τ =0 2 Z X n ∂ η d = (g(x) + τ ϕg (x)) dx 2 dτ ∂xi Φ i=1

τ =0

Z Z X n ∂ ∂ =η ϕg (x) g(x)dx = η ∇ϕg (x) · ∇g(x)dx ∂xi ∂xi i=1 Φ Φ   Z Z ∂g(x) Green  = η ϕg (x)dσ − ∆g(x)ϕg (x)dx . ∂n

(4.59)

Φ

∂Φ

Hence, the derivative of Rη,Φ (g) can be identified with 0

Rη,Φ (g) = −η∆g(x) 50

(4.60)

for functions satisfying ∂g(x) =0 ∂n

on ∂Φ .

(4.61)

The boundary condition ∂g(x) ∂n = 0 for x ∈ ∂Φ guarantees that transitions of compartments between the spaces M, S, A and V are smooth. In an analogous way, we obtain 0

00

Rη,[a,b] (g) = −ηg (t)

(4.62)

for functions satisfying 0

0

g (a) = g (b) .

(4.63)

Replacing the general regularization terms in (4.35) by the a-priori and gradient regularizations, we obtain L (u(x, t), F (x), CA (t), CV (t), CT (t), r(x), s1 (x), s2 (x); q(x, t), µ(x, t)) 2 ZT Z u(x, t) − u 1 (x, t) k+ 2 1 = dxdt 2 uk (x, t) 0 Ω

+Rα,T (F (x)) + Rα,[0,T ] (CA (t)) + Rα,[0,T ] (CV (t)) + Rα,T (r(x)) +Rα,T (s1 (x)) + Rα,S (s2 (x)) + Rη,T (F (x)) + Rη,[0,T ] (CA (t)) +Rη,[0,T ] (CV (t)) + Rη,T (r(x)) + Rη,T (s1 (x)) + Rη,S (s2 (x)) ZT Z (G(CA (t), CV (t), CT (x, t), r(x), s1 (x), s2 (x)) − u(x, t)) q(x, t)dxdt

+ 0 Ω

ZT Z +

CT (x, t) ∂CT (x, t) − F (x) CA (t) − µ(x, t)dx dt . ∂t λ

(4.64)

0 T

Note that the gradient regularization guarantees that the reconstructed parameters F , CA , CV , r, s1 and s2 become parameters in the Hilbert space H1 . This is important from a theoretical point of view, since we needed H1 input parameters for G in Theorem 4.5 to prove existence of an optimal solution.

4.7.3

Fraction and Spillover Regularization

Despite a-priori and gradient regularization we will discover that we still have to face some serious problems in reconstructing the parameters. If we take a look at the two tissue input curves in Figure 4.4 we discover that both tissue input curves are similar although they were generated with completely different parameters. Especially the myocardial perfusion is much higher for the second curve, which is compensated with a lower fraction value. If we assume our data to be noisy it is likely to see that we could obtain an input curve similar to input curve 2 instead of input curve 1, even with gradient and a-priori regularization. Hence, we obtain completely different values.

51

1 Tissue Curve 1 0.9

Tissue Curve 2

0.8

kBq/ml (normalised)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5 3 Time (in min)

3.5

4

4.5

5

Figure 4.4: Two similar tissue input curves. Tissue input curve 1 is generated with the parameter values F = 5 ml/min/mg, r = 0.65, s1 = 0.21 and s2 = 0.15. Tissue input curve 2 in comparison is generated with parameter values F = 1.8613 ml/min/mg, r = 0.4107, s1 = 0.3755 and s2 = 0.2526. Both input curves are computed with the underlying arterial input curve CA (t) = sin( πt T ), for T = 320 s, and λ = 0.96. Furthermore we can witness a connection between the optimality conditions for F and r. If we start computing the optimality condition for F , for an α > 0, we receive ∂L (p) ∂F

=

0

ZT

R1 (F (x)) −

CT (x, t) CA (t) − λ

µ(x, t)dt

0 ODE

=

1 R1 (F (x)) − F (x) 0

ZT

∂CT (x, t)µ(x, t)dt ∂t

0

 part. Int.

=

1  CT (x, t)µ(x, t)|T0 − R1 (F (x)) − F (x) | {z } 0

 adj. ODE

=

0

R1 (F (x)) +

1  F (x) F (x) λ

=0 T Z

ZT

 ∂µ CT (x, t) (x, t)dt ∂t

0

CT (x, t)µ(x, t)dt 0

ZT +XT (x)r(x)

 CT (x, t)q(x, t)dt

0

=

1 R1 (F (x)) + λ 0

ZT CT (x, t)µ(x, t)dt 0 ZT

+

r(x) XT (x) CT (x, t)q(x, t)dt , F (x) 0 | {z } 0 (p)−R (r) =( ∂L ) 4 ∂r 52

(4.65)

where p = (F, CA , CV , r, s1 , s2 ) denotes the parameters, a significant connection to the optimality condition of r. The division by F is justified since perfusion examined in a living object has to be larger than zero. Assume that we are going to linearize ∂F L and ∂r L between two vectors p1 and p2 to obtain optimal values for F and r. Then, due to the possible linear dependence of linearizations, i.e. between F and r, the solution might not be unique or the sensitivity very low. This possible non-uniqueness for the linearization of the Lagrange functional (4.64) and its optimality conditions might result in wrong values of the computed parameters, comparable to the different parameter problem illustrated in Figure 4.4. As a consequence, we will introduce two additional regularization terms. We have seen that the major cause of non-uniqueness after linearization is the direct connection between F and r and their gradients. The easiest way to overcome this problem is to add some dependence to either F or r on other variables. As we have seen in Section 3.4.3, the relationship r + s1 + s2 + s3 = 1 holds for the tissue fraction r, the arterial and venous spillover s1 and s2 , respectively and the background spillover s3 . Unfortunately, we have already discovered that we could neither make a statement on the background nor on the related spillover. At least we could assume that r + s1 should be very close to one for any myocardial-compartment, excluding the septum, depending on the quality of the segmentation and the amount of background spillover. The same idea holds for any septum-compartment, but with additonal venous spillover, i.e. r + s1 + s2 shall be close to one. Concluding these preliminary ideas, we can add a penalty term  Z β Rβ (r(x), s1 (x), s2 (x)) := (r(x) + s1 (x) − 1)2 dx 2 M  Z + (r(x) + s1 (x) + s2 (x) − 1)2 dx  , (4.66) S

for β ∈ R+ . By regularizing our optimization problem this way, we not only introduce a dependence between tissue fraction and the spillover terms but furthermore regularize our solution to satisfy r + s1 ≈ 1 and r + s1 + s2 ≈ 1, respectively. Another reasonable way of regularization originates in the following considerations. From a segmentational point of view it is easy to see, that the conditions XT (x) + XA (x) + XV (x) = 1 and XT (x) XA (x) XV (x) = 0 hold for any x ∈ H, due to H = A ∪ V ∪ T and A ∩ V ∩ T = ∅. Now, tissue fraction and spillover terms simply aim in balancing the imprecise segmentation. Therefore, we could assume the product r(x) s1 (x) s2 (x) to be close to zero, which can be put into a regularization term of the form  Z γ Rγ (r(x), s1 (x), s2 (x)) := (r(x) s1 (x))2 dx 2 M  Z + (r(x) s1 (x) s2 (x))2 dx  , (4.67) S

for γ ∈ R+ . In the case of the penalty regularizations (4.66) and (4.67), β and γ are generally chosen to be much larger than one, while α and η for the a-priori and gradient regularization are chosen to be much smaller than one (β, γ >> 1 and α, η 0 is a good regularization method to solve the denoising problem. Equation (4.83) is known as the ROF (Rubin-Osher-Fatemi)-model and became one of the most popular models in image processing (cf. [ROF92]). We could expand Example 4.2 to our case of denoising a sequence of H2 15O-scan EMreconstructions. We simply replace the data term of (4.83) with our EM-functional and obtain EM (u) + α|u|T V Z = Au(x, t) − f log (Au(x, t)) dx + α|u|T V ,

(4.84)

Ω

for fixed t ∈ [0, T ]. Solving the appropriate minimization problem EM (u) + α|u|T V → minu∈BV (Ω) could on the one hand give us a better reconstruction itself (cf. [JHC98]). On the other hand we could use this TV-regularized reconstruction to segment the cardiovascular area, since the TV-regularization term, with α sufficiently high, forces a solution to be (almost) piecewise constant (cf. [Mey01]). Mκ (u) = {x ∈ Ω|u(x) < κ} ,

(4.85)

If we use a simple levelset method segmentation into different sets is no major difficulty. Some exemplary TV-reconstructions of H2 15O PET data made by Alex Sawatzky1 in comparison to standard EM reconstructions are shown in Figure 4.5 and Figure 4.6. There is also a high loss of information, but edges are preserved or even sharpened, which would be perfect for segmentation. 1

Researcher at the WWU, Institute for Computational and Applied Mathematics

58

(a) Average of 26 standard EM reconstructions of H2 15O-PET data

(b) Plotted profile curve of 4.5(a)

(c) Average of 26 EM-TV reconstructions of the same data set

(d) Plotted profile curve of 4.5(c)

Figure 4.5: Comparison of a standard EM reconstruction with a EM-TV reconstruction of H2 15O-PET data averaged to one frame. Another approach could be derived from the Mumford-Shah variational model for segmentation (cf. [MS89]), which can be described as the minimization of Z Z ν 1 δ 2 (u(x) − y (x)) dx + |∇u(x)|dx → min , (4.86) u,Γ 2 2 Ω

Ω\Γ

for Ω ⊂ Rn and a finite Hausdorff-measure Hn−1 (Γ). In the case of a smooth function, the integral over Ω\Γ equals the integral over Ω. But we allow discontinuous functions and therefore minimize (4.86) in terms of u ∈ W 1,2 (Ω\Γ). There is a significant connection between the second integral term and (4.80), except that the set of edges is excluded in the Mumford-Shah model. If (4.86) is additionally minimized not only due to u but to Γ as well, the optimal set of edges can be estimated. Again, we can expand (4.86) to ν 2

Z

1 Au(x, t) − f log (Au(x, t)) dxdt + 2

ZT Z |∇u(x, t)|dx → min , u,Γ

0 Ω\Γ

Ω

replacing the data term with the EM-functional for fixed t ∈ [0, T ]. 59

(4.87)

(a) Standard EM reconstruction of the 7-th frame (b) EM-TV reconstruction of the 7-th frame of of H2 15O-PET data H2 15O-PET data

(c) Standard EM reconstruction of the 10-th frame (d) EM-TV reconstruction of the 10-th frame of of H2 15O-PET data H2 15O-PET data

Figure 4.6: Comparison of a standard EM reconstruction with a EM-TV reconstruction of H2 15O-PET data for single frames. We discover that the EM-TV reconstructions provide objects with sharp edges and offer the option to perform a segmentation on the data. Another new approach for segmentation can be based on the new variational model derived in Section 4.3. Instead of computing the input functions CA and CV in the process of parameter identification, we could measure them in left and right atrium as in the previous method (3.5). Then, CA and CV are fixed during the remaining computation. Moreover we assume that the whole image space Ω consists of tissue, i.e. we examine the model L (u(x, t), F (x), CA (t), CV (t), CT (x, t), r(x), s1 (x), s2 (x); q(x, t), µ(x, t)) 2 ZT Z u(x, t) − u 1 (x, t) k+ 2 1 dxdt = 2 uk (x, t) 0 Ω

+ R(F (x)) + R(r(x)) + R(s1 (x)) + R(s2 (x)) ZT Z + (r(x)CT (x, t) + s1 (x)CA (t) + s2 (x)CV (t) − u(x, t)) q(x, t) dxdt 0 Ω

60

ZT Z +

∂CT CT (x, t) µ(x, t)dxdt. (x, t) − F (x) CA (t) − ∂t λ

(4.88)

0 Ω

Applying this model to our data should compute the desired parameters F , r, s1 and s2 without previous segmentation. Areas containing the left ventricular would imply a high value for s1 , but small values for r and s2 . Areas containing the right ventricular imply small values for r and s1 , but a high value for s2 . Real tissue-areas would imply higher values for r. Hence, r, s1 and s2 could be used for precise segmentation. A remaining problem are the background areas. In these areas, the values for F and r are not unique, due to the modelling. In a background area there is no blood flow (if no other organ, e.g. the liver, lies within that area), i.e. F → 0. Hence, the value of r can be close to 1, since rCT → 0, although r should be close to zero. In that case, we would indicate backround area as perfusable tissue, which is not the case. Nevertheless, using the variational problem (4.88) might offer some interesting properties and advantages in comparison to other segmentation methods.

4.8.3

Segmentation Using CT-Scan Information

Finally, we want to discuss an approach that is not based on new mathematical methods and models, but on technological progress. New PET/CT-facilities allow gated and registrated PET- and CT-scans. In this context ”gated” means that all motion between the frames can be adjusted to receive a sequence of frames with static objects. In terms of segmentation this implies that we do not need a time dependent indicator function X (x, t), since all motion has been adjusted. The PET and CT reconstructions are registrated, which means that we can assign spatial areas in one reconstruction to the associated spatial area in the other reconstruction. A CT-reconstruction offers much better opportunities to segment cardiovascular areas due to its finer resolution. This segmentation, X (x), can be applied to the PET reconstructions as well, due to the registration of frames. Hence, we are able to provide our myocardial perfusion algortihms with a highly accurate segmentation and its related indicator functions. The disadvantages of this concept are, that, first of all, a PET/CT-facillity is needed, which is not state of the art. Furthermore, the patients radiation exposure is much higher, due to the additional CT-scan.

61

Chapter 5

Implementation Finally, after deriving an appropriate model for the quantification of myocardial perfusion, we have to discuss its numerical solution. First of all, we will describe the combination of the standard EM-algorithm and the parameter identification problem that will lead to a two-step algorithm. In the following, we are going to discuss the discretization of the EMalgorithm as well as the numerical solution of the parameter identification problem. For the latter algorithm, we have to discuss the need for scaling and how to scale the variational problem appropriately. Subsequently, we are going to describe methods to rewrite the analytical formulas of chapter 4 to numerical ones. Concludingly, we want to discuss the use of numerical solvers and the basic idea of solving our parameter identification problem numerically.

5.1

Two-Step Algorithm

The basic concept of the two-step algorithm is shown in Figure 5.11 ,

EM

c

Wikipedia.

The

PI

Figure 5.1: The standard EM algorithm (EM) and the parameter identification problem (PI) are computed according to a spur gear principle, with both EM and PI representing a single gear. An iterative step of the EM algorithm is computed. Its solution is intertwined into the parameter identification process. This solution is intertwined back and builds the basis for the following EM-step. minimization problem (4.24) actually implies that the k + 12 -th step of the EM-algorithm, uk+ 1 , is computed first, building the basis for the parameter identification process. After 2 (4.33) is being solved, parameters pk+ 1 are computed. The computation of G(pk+ 1 ) = uk+1 , 2 2 with G as introduced in Section 4.4, returns the input image sequence for the next EMiteration. Instead of computing all EM-iterates first and computing the parameters then, we intertwine the parameter identification process into the EM-iterations. The whole procedure can be seen in figure 5.2. 1

http://upload.wikimedia.org/wikipedia/commons/1/14/Gears_animation.gif

62

uk

u k1

EM

u

k

1 2

PI

G  p k1/2 =u k 1 p

k

1 2

Figure 5.2: The EM - PI scheme. Starting with an iterate uk , the following EM-step, called uk+ 1 , is computed. Within the parameter identification process, parameters pk+ 1 2 2 are computed from uk+ 1 . Selecting the image sequence generation functional G, we obtain 2 uk+1 from these parameters. The image sequence uk+1 is the next iterate for the EMalgorithm, instead of uk+ 1 . 2

5.2

EM Discretization

The standard EM-algorithm is discretized with the simple iterative method as described in equation (4.14). We perform a standard EM iteration on every single frame of the data sequence. The computed images were composed to become the reconstructed image sequence u. As an initial iterate, we simply use u0 (x, t) = 1, for all (x, t) ∈ Ω × [0, T ].

5.3

Discretization of the Parameter Identification Problem

Basically we have to discuss ways to discretize our optimality conditions. Furthermore we have to develop ways to compute the optimal parameters from these discretized conditions. The basic idea is to solve this system of functional equations via linearization. For that reason, we are going to discuss iterative schemes in Section 5.4 and Section 5.5. The basic principle is illustrated in Figure 5.3. We will start with some initial values p0 and update them step by step until we will find iterates close enough to the optimal solution p˜. Iterative schemes have several advantages that make their use desirable. One of them is the easy way of implementation. Another one is that stopping criterias can be applied manually. As a reminder, the use of an iterative scheme and the involved linearization of the optimality conditions implies the need for the regularization terms described in (4.66) and (4.67). Before we move on to numerical solvers that compute pj+1 from ∂L using pj , we have to discuss the need for scaling and adequate discretization. Different variables with different variable-domains make scaling and rescaling inevitable.

63

Figure 5.3: An iterative scheme to solve the system of functional equations in the PI process. The j-th iterate of p is used as an input for a yet unspecified method, to compute the j +1-th iterate from pj and ∂L. Iterates are computed as long as they vary more than a specified factor > 0. Processing iterates is stopped after m iterations until |pm − pm−1 | < is satisfied. This iterate is estimated to be close to the optimal solution p˜, fulfilling ∂L(˜ p) = 0, and will be denoted with pk+ 1 . These parameters build the basis for the next EM-step. 2

5.3.1

Scaling

The question that arises simultaneously with suitable modeling techniques is the question of how to scale the underlying data. In an optimization problem a lot of variables and parameters with different domains may occur. If these domains vary a lot, it is likely to see that related optimality conditions vary a lot. Thus, depending on the numerical solver and initial values, it might take longer to solve the whole optimization problem. Even non-convergence of the numerical solver may be the consequence of too heavily varying parameter domains. Hence, we have to scale all our variables and parameters to new ones, all lying within the same range. After running a numerical solver and obtaining some optimal parameters, we are simply going to rescale these parameters to each fit in their original domain. The following ideas of scaling mostly base on [GMW81], chapter 7, section 7.5 and [Bur07]. We are simply going to transform our variables/parameters and their domains in an affinely linear way. Imagine a one-dimensional variable v ∈ [a, b] ⊂ R we want to scale to a variable v˜ ∈ [c, d] ⊂ R, with [a, b] 6= ∅, [c, d] 6= ∅ and [a, b] 6= [c, d]. Via affinely linear transformations we obtain v˜(v) =

c−d ad − bc v+ = k1 v + k2 , a−b a−b

(5.1)

with k1 := (c − d)/(a − b) and k2 := (ad − bc)/(a − b). In our application, we deal with physiological parameters exclusively, i.e. all parameters are nonnegative. A reasonable

64

domain to project to is simply [0, 1]. This modifies (5.1) to v˜(v) = −

1 ad v+ . a−b a−b

(5.2)

Given our set of functions F (x), CA (t), CV (t), r(x), s1 (x) and s2 (x), we have to scale two variables and six parameters. Imagine t ∈ [0, T ] and x ∈ [0, ξ]. Furthermore assume that we are able to find physiological intervals of the type [0, ζ] with ζ being the supremum value a certain parameter can map to (e.g. F ζ = kF k∞ ). We denote these suprema for F , CA , CV , ζ r, s1 and s2 with F ζ , CA , CVζ , rζ , sζ1 and sζ2 . Then we obtain linearly scaled parameters and variables 1 1 x ˜(x) = x , t˜(t) = t ξ T and F˜ (F (˜ x)) = F˜ ( 1ξ x) = r˜(r(˜ x)) =

CÃ (CA (t˜)) =

1 F (x), Fζ

1 r(x) , rζ

s˜1 (s1 (˜ x)) =

1 ζ CA (t), CA 1 s1 (x) , sζ1

C˜V (CV (t˜)) = s˜2 (s2 (˜ x)) =

1 ζ CV (t), CV 1 s2 (x) . sζ2

(5.3)

The optimization problem (4.68) almost remains the same, except that the partial derivatives of CT (x, t) and µ(x, t) to t modify to ∂CT dt ∂CT ∂CT (x, t˜) = (x, t) = T (x, t) ˜ ˜ ∂t ∂t ∂t dt

(5.4)

and ∂µ ∂µ (x, t), (x, t˜) = T ˜ ∂t ∂t

(5.5)

due to the chain rule. Furthermore, the integrals change due to the rule of substitution. If we consider the case of a time-dependent integral, ZT f (t)dt , 0

we obtain for t˜(t) =

t T

and f˜(f (t)) = ZT

f (t) fζ

f (t) dt = T fζ

0

Z1

f˜(t˜)dt˜.

(5.6)

0

Analogously, all spatial integrals can be scaled this way and every integral needed for solving the optimization problem can be scaled due to the contained parameters (5.3). In the following Section we want to describe the numerical computation of integral- and differential-equations.

65

5.3.2

Discrete Approximations

In this section, we want to describe how integrals and ODEs are numerically approximated. In general, integrals are simply approximated using the trapezoidal rule, i.e. Zb

d−1

X h f (x)dx ≈ (f (a) + f (b)) + h f (ih) , 2

(5.7)

i=1

a

where f is a one-dimensional signal, sampled at d + 1 discrete and equidistant sampling points with stepsize h = b−a d . In the case of non-equidistant sampling points a, x1 , . . . , xd−1 , b, the stepsize is given as the vector h = (xj − xj−1 )j=1...d−1 ,

(5.8)

with h(0) = x1 − a and h(d) = b − xd−1 . Equation (5.7) is modified to become Zb

d−1

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Φ

for a dataset sampled at d sampling points x1 , . . . , xd . We want to notice that scaling, as introduced in Section 5.3.1, does not affect the discretization, i.e. ZT

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˜= and h with h = The ODEs for CT (x, t) and µ(x, t) were discretized with simple Euler forward- and backward-differences, i.e. ∂CT CT (x, t) (x, t) − F (x) CA (t) − ∂t λ CT (x, t) CT (x, t + 1) − CT (x, t) − F (x) CA (t) − ≈ h λ T d−1

1 d−1 .

and ∂µ F (x) (x, t) − r(x)q(x, t) + µ(x, t) ∂t λ µ(x, t) − µ(x, t − 1) F (x) ≈ − r(x)q(x, t) + µ(x, t) , h λ 66

for the time-spacing stepsize h. Despite its simple nature, the Euler-method guarantees that the discrete adjoint actually equals the adjoint discretization of the ODE. Again, scaling does not affect the stepsize h, due to the chain rule (see (5.4) and (5.5)). An extension to non-equidistant stepsizes is possible, analogously to the case of the discrete integral approximations. For large values of the contained functions, such as F , or in order to obtain more exact results, the data has to be interpolated in time, to get a new, finer ˆ stepsize h. The last step missing to compute myocardial perfusion is the numerical solution of the optimality conditions. We are going to discuss two popular approaches; the gradient method and the Newton method.

5.4

Gradient Method

The task of solving the minimization problem (4.35) is equivalent to the task of solving the system of equations (4.51) - (4.54) and (4.69) - (4.74). Equations (4.51) - (4.54) can be solved analytically. The remaining equations have to be solved via a numerical solver. One of the easiest numerical solvers is an iterative gradient method. Let J : L2 (Ψ) → R be a differentiable functional depending on a single function a ∈ L2 (Ψ) and having a unique global minimum. To find this minimum, we could simply introduce a time parameter t and compute a gradient flow, i.e. ∂a 0 = −J (a) . (5.11) ∂t By discretizing (5.11) via finite forward-differences we obtain a simple iterative scheme to compute, 0

ak+1 = ak − τ J (ak ) ,

(5.12)

for τ > 0 reasonably small. We can apply the same strategy to the part of our system of equations that cannot be computed directly, i.e. ∂L k (p ) ∂F ∂L k k CA (t) − τ (p ) ∂CA ∂L k CVk (t) − τ (p ) ∂CV ∂L k rk (x) − τ (p ) ∂r ∂L k sk1 (x) − τ (p ) ∂s1 ∂L k sk2 (x) − τ (p ) . ∂s2

F k+1 (x) = F k (x) − τ

(5.13)

k+1 (t) = CA

(5.14)

CVk+1 (t) = rk+1 (x) = sk+1 1 (x) = sk+1 2 (x) =

(5.15) (5.16) (5.17) (5.18)

Together with the computed solutions CT (x, t), u(x, t), q(x, t) and µ(x, t) we can apply the 0 , C 0 , r0 , previously introduced iterative scheme (Figure 5.3) for some initial values F 0 , CA V s01 and s02 . If we iterate long enough, we will be able to get parameters close enough to the optimal ones; we want to denote this number of iterations with m. Then uk+1 (x, t) can be computed via m m m uk+1 (x, t) = G(CA (t), CVm (t), CTm (x, t), rm (x), sm 1 (x), s2 (x); F (x)) ,

67

(5.19)

m )(x, t) depending on F m (x) and C m (t). This solution u with CTm (F m , CA k+1 (x, t) is used as A an input for the next EM-iteration step instead of uk+ 1 , according to the scheme of Figure 2 5.2. This simple iterative scheme can be modified to find iterates in general Hilbert spaces U, not only L2 (Ω) and L2 ([0, T ]), respectively. We can generalize equation (5.11) to ∂ 0 a, v = −J (a)v ∀ v ∈ U , (5.20) ∂t U

If, for example, we want to compute differentiable parameters from the gradient flow, we could use H 1 as an appropriate Hilbert space. Furthermore, our parameters are physiological values, e.g. F ≥ 0. Hence, we want to limit the numerically computed parameters to be physiological values, too. In order to obtain parameters lying within a physiological range [0, ζ], we are going to use the projective gradient method, i.e.  0 0 if ak − τ J (ak ) ≤ 0  0 ak+1 = (5.21) ζ if ak − τ J (ak ) ≥ ζ ,  0 −J (ak ) else for τ reasonably small.

5.5

Newton Method

Another typical numerical solver is a simple linearization via the Newton method, 0 L (pk ) pk+1 − pk = −L(pk ) ,

(5.22)

with L denoting the parameter identification integral, and p denoting the parameters. If 0 we assume L not to be a-priori regularized, we could simply add the regularization to the linearized Newton method, to obtain the so-called Levenberg-Marquardt method 0 0 (5.23) L (pk )∗ L (pk ) + αId pk+1 − pk = −L(pk )L(pk )∗ , with α ∈ R+ being the a-priori regularization parameter, and Id : L2 (Ω × [0, T ]) → L2 (Ω × [0, T ]) denoting the identity operator. In our case, with CT , µ, u and q analytically computed, we obtain   F k+1 − F k  C k+1 − C k  A  A   ∂L ∂L ∂L ∂L ∂L ∂L   CVk+1 − CVk  k , CVk , rk , sk1 , sk2 , (5.24) , , , , ,  = −L F k , CA  k+1 k −r  ∂F ∂CA ∂CV ∂r ∂s1 ∂s2  r   k+1  s1 − sk1  sk+1 − sk2 2 as a system to solve in the case of (5.22). Another possible option is to apply the Newton or Levenberg-Marquardt method to the 0 gradient of L, L , i.e. 0 0 DL (pk ) pk+1 − pk = −L (pk ) , (5.25)

68

with the Jacobian       0 k DL (p ) =      

∂2L ∂F 2 ∂2L ∂CA ∂F ∂2L ∂CV ∂F ∂2L ∂r∂F ∂2L ∂s1 ∂F ∂2L ∂s2 ∂F

∂2L ∂F ∂CA ∂2L 2 ∂CA ∂2L ∂CV ∂CA ∂2L ∂r∂CA ∂2L ∂s1 ∂CA ∂2L ∂s2 ∂CA

∂2L ∂F ∂CV ∂2L ∂CA ∂CV ∂2L 2 ∂∂CV ∂2L ∂r∂CV ∂2L ∂s1 ∂CV ∂2L ∂s2 ∂CV

∂2L ∂F ∂r ∂2L ∂CA ∂r ∂2L ∂CV ∂r ∂2L ∂r2 ∂2L ∂s1 ∂r ∂2L ∂s2 ∂r

∂2L ∂F ∂s1 ∂2L ∂CA ∂s1 ∂2L ∂CV ∂s1 ∂2L ∂r∂s1 ∂2L ∂s21 ∂2L ∂s2 ∂s1

∂2L ∂F ∂s2 ∂2L ∂CA ∂s2 ∂2L ∂CV ∂s2 ∂2L ∂r∂s2 ∂2L ∂s1 ∂s2 ∂2L ∂s22

           

as in the case of the Newton method (Levenberg-Marquardt can be derived analogously). This method is called SQP method (sequential quadratic programming). In our case, without further regularization, we obtain the Jacobian   RT 0 − µ(x, t)dt 0 0 0 0    R  0 R   0 0 0 q(x, t)dx 0  − µ(x, t)dx   T  T R     0 0 0 0 0 q(x, t)dx   0 S  . DL =   0 0 0 0 0 0     T R   0 XT (x) q(x, t)dt 0 0 0 0     0   RT   0 0 XS (x) q(x, t)dt 0 0 0 0

(5.26) Regularization terms can simply be added by modifiying the specific matrix-entries of (5.26). After describing the numerical implementation of myocardial perfusion computation, we want to present computational results for the new method. This will include synthetic as well as real data.

69

Chapter 6

Results in this chapter we want to conclude this work by presenting some computed test results on synthetic as well as real data. We will develop three data phantoms for testing and will analyse one slice of real H2 15O-PET-scan data. All computations were made in two spatial dimensions exclusively. c Most computations are done with MATLAB (The MathWorksTM , Inc., Natick, MA), c except the transformation of images to artificial PET data, which is computed in C .

(a) The left ventricular area

(b) The right ventricular area

(c) The area denoting myocardium without septum

(d) The area denoting the septum

Figure 6.1: The artificial segmentation for the first synthetic data example. The generated images have a resolution of 256 × 256 pixel.

70

6.1

A Simple Synthetic Data Example

We want to start with a very simple synthetic data example. First of all, we simply define a segmentation and its related indicator functions, as described in Figure 6.1 1

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Figure 6.2: The input curves describing the first data example. Arterial and venous input curves CA and CV (Figure 6.2(a) and Figure 6.2(b)) are computed as described in the text. The tissue input curve CT in Figure 6.2(c) is computed via (3.4) with F = 1.248 ml/min/mg and λ = 0.96. Together with tissue fraction r = 0.65, arterial spillover s1 = 0.21 and venous spillover s2 = 0.15 we obtain the final tissue input curves via (3.8) for the septum area and the myocardial area without septum (Figure 6.2(d)). Now we are going to define some sample parameters. The venous input curve is simply defined as the normalised Rayleigh probability density function

f (x|b) =

x e b2

−x2 2b2

kf (x|b)k∞

,

with x ∈ [0, 25] and b = 6. The function is sampled at 26 equidistant discrete points, each representing a point of the interval [0, T ], with T = 320 s. The arterial input function is generated in a similar way, but shifted about one discrete point to simulate the timedelay between venous and arterial input curves. Blood flow, tissue fraction, arterial and 71

venous spillover are assumed to be equal all over the myocardial area with the values F = 1.248 ml/min/mg, r = 0.65, s1 = 0.21 and s2 = 0.15. With a fixed partition coefficient λ = 0.96 we obtain the input curves as described in Figure 6.2 (notice that venous spillover only appears to occur in the myocardial area denoted as septum). With the indicator functions defined in Figure 6.1 and the generated parameters and input curves, we are able to generate a 256 × 256 × 26 image sequence uexact (x, t) via the image generation functional (4.28). The 9-th frame of uexact is exemplarily shown in Figure 6.3

Figure 6.3: The 9-th frame of exact data for the first synthetic data example.

6.1.1

Parameter Identification on Exact Data

The first step to test our algorithm is to simply test the parameter identification process with the exact data uexact as an input. As a numerical solver we use the gradient method introduced in Section 5.4 with τ = 0.5. The regularization parameters as defined in Section 4.7 are set to α = 0.4×10−4 , β = 0.75, γ = 0.53 and η = 0.5444×10−4 . Furthermore we use the exact indicator functions for segmentation and the exact input functions CA and CV as apriori knowledge for regularization. In order not to be suggestive of cheating, we have chosen α to be small. The remaining a-priori values are set to F ∗ = 0.85 ml/min/mg, r∗ = 0.6, s∗1 = 0.3 and s∗2 = 0.1. Subsequently we have set the starting values to F 0 = 1 ml/min/mg, 0 (t) = C 0 (t) = sin( πt ), r 0 = 0.9, s0 = 0.3 and s0 = 0.1. CA 1 2 V T Since the gradient is not invariant to scaling, we have to scale all variables and parameters to lie within [0, 1]n , for n = 1 and n = 2, respectively. If we assume Ω to be a normalized area, the values of x are already scaled. The chosen input curves are also scaled, just like the tissue fraction and spillover effects. Hence, only t ∈ [0, T ] and F ∈ [0, F ζ ] (and F ∗ ∈ [0, F ζ ]) have yet to be scaled. We will choose F ζ = 6 ml/min/mg, since a higher perfusion is physically unrealistic. Notice that this is also a kind of regularization, since we force our solution not to exceed a certain value. Furthermore, we will regularize the parameters’ solutions by projecting them onto [0, 1]n , if they leave their domain during the computational process. 72

With these values being set we obtain the computed parameters as presented in Figure 6.4, after 1000 iterations of the parameter identification process. 1

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(c) Reconstructed perfusion Frec

(d) Reconstructed tissue fraction rrec

(e) Reconstructed arterial spillover s1 rec

(f) Reconstructed venous spillover s2 rec

Figure 6.4: Computed values from the exact data of the first synthetic data example. As we can see, we obtain parameters close to the original ones. The error values are small although we are dealing with an ill-posed inverse problem. If we investigate the scaled error, i.e.

f − f˜ ∞ , kf k∞ 73

with f denoting an exact parameter and f˜ denoting its reconstruction (and k k∞ representing the supremum-norm), we obtain the following error values: kF − Frec kL∞ (M) kF kL∞ (M) kCA − CA,rec kL∞ ([0,T ]) kCA kL∞ ([0,T ]) kCV − CV,rec kL∞ ([0,T ]) kCV kL∞ ([0,T ]) kr − rrec kL∞ (M) krkL∞ (M) ks1 − s1,rec kL∞ (M) ks1 kL∞ (M) ks2 − s2,rec kL∞ (S) ks2 kL∞ (S)

≈ 0.0861 ,

kF − Frec kL∞ (S) kF kL∞ (S)

≈ 0.0537 ,

≈ 0.0023 , ≈ 0.0014 , ≈ 0.0033 , ≈ 0.1606 ,

kr − rrec kL∞ (S) krkL∞ (S)

≈ 0.0060 ,

ks1 − s1,rec kL∞ (S) ks1 kL∞ (S)

≈ 0.1424 ,

≈ 0.0939 ,

with Frec , rrec , etc. denoting the computed reconstructions. We can see that the error values are very small, less than 20 %, even in the stronger ∞-norm, which is a very good result for an ill-posed inverse problem.

6.1.2

The Computation of Myocardial Perfusion on Synthetic PET Data

Since the parameter identification process has passed the test with exact data, we are going to test the complete algorithm with synthetic, noisy PET-reconstructions. In order to do

Figure 6.5: The 9-th frame of a standard EM-reconstruction of the synthetic PET (with a maximum number of counts of 61415) data generated from the first data example. Fifteen EM reconstructions have been made with additional Gaußian-smoothing after every fourth iteration. c so we transform the exact data uexact with a C -program, written by Frank W¨ ubbeling1 , 1


74

to a sequence of synthetic PET reconstructions, f (x, t). We limit the maximum number of counts to 61415 and obtain a noisy, low SNR reconstruction by using the standard EMalgorithm. After 15 EM iterations, we obtain a reconstruction uEM (x, t) whose 9-th frame is presented in Figure 6.5. 1

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and 5000 PI iterations. The regularization parameters are chosen to be α = 5 × 10−4 ,

(a) The 9-th frame of the exact data

(b) The 9-th frame of the standard EM reconstruction

(c) The 9-th frame of G(p) with the computed optimal parameters

Figure 6.7: The comparison of the 9-th frame of the exact data, the standard EM reconstruction of its synthetic PET data and of G(p), with p being the computed values shown in Figure 6.6.

(a) Area with lower perfusion

(b) Area with higher perfusion

Figure 6.8: Areas with varying perfusion. β = 0.75, γ = 0.53 and η = 5 × 10−4 . All other parameters such as a-priori knowledge etc. remain the same as in Section 6.1.1. The computed values are illustrated in Figure 6.6. 76

Despite the very few counts for reconstruction and the resulting noise the reconstructed values are very close to the original values. Even more noise could have been removed with a higher value for η, which we have refused to do since a higher value for η would not only remove more noise but would also smooth varying values too much. Before we 1

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Figure 6.9: The tissue input curves with high and small perfusion values, tissue fraction r = 0.65, arterial spillover s1 = 0.21 and venous spillover s2 = 0.15. move on to a second synthetic data example, the 9-th frame of the exact data, its standard EM reconstruction from synthetic PET data and the byproduct of the new algorithm are illustrated for comparison in Figure 6.7.

6.2

The Modified Data Example

Figure 6.10: The 9-th frame of exact data for the second synthetic data example. After testing our new method with a very simple example we want to modify this 77

example to feature varying perfusion. In order to do so we select two areas in which the perfusion is set to have a different value. These areas can be seen in Figure 6.8. We modify 1


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Figure 6.11: Computed values from the exact data of the second synthetic data example. the first data example to have a small perfusion value Fsmall = 0.2 ml/min/mg in the area illustrated in Figure 6.8(a) and a high perfusion value Fhigh = 5 ml/min/mg in the area illustrated in Figure 6.8(b). All other parameter values remain the same as in the first data example. Hence, we obtain two new tissue input curves that can be seen in Figure 6.9 With these parameters we obtain a new image sequence whose 9-th frame is shown in

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Figure 6.10. As with the first data example, we are going to compute optimal parameters from the exact data first and move over to compute parameters from the synthetic PET data afterwards.

6.2.1

Parameter Identification on Exact Data

To test the parameter identification process on exact data we chose to use the gradient method as a numerical solver again. In addition, the a-priori knowledge, initial values etc. remain the same. With the regularization parameters set to α = 4 × 10−5 , β = 0.75, γ = 0.53 and η = 5.444 × 10−5 we obtain the computed results as illustrated in Figure 6.11, after 1000 iterations for τ = 0.5. It is interesting to see that almost all

Figure 6.12: The 9-th frame of a standard EM-reconstruction of the synthetic PET (with a maximum number of counts of 60823) data generated from the second data example. Fifteen EM reconstructions have been made with additional Gaußian-smoothing after every fourth iteration. parameter values are reconstructed accurately, except for the area with high perfusion. The reconstructed perfusion in that area, Frec ≈ 2 ml/min/mg, is not even close to the real value F = 5 ml/min/mg. Furthermore the tissue fraction and spillover values differ a lot from the original values in that area, due to compensating the wrong perfusion value. This phenomenon has already been discussed in Section 4.7.3 and is besides all regularization still hard to handle. Even higher fraction and spillover regularization values (e.g. β, γ ≈ 10) and different initial values do not lead to more satisfying results. Luckily the parameters within the area of small myocardial perfusion have been computed very well and indicate that there is less perfusion. In practice, it does not matter if we can differ between a perfusion value of 2 or 5 but we have to be able to detect areas with small perfusion since we want to detect blocked vessels and not properly perfused tissue, respectively.

6.2.2

The Computation of Myocardial Perfusion on Synthetic PET Data

Just like in Section 6.1.2 we are going to transform the exact data image sequence to a noisy sequence of synthetic PET data with a maximum number of 60823 counts. The 9-th frame

79

of a standard EM-reconstruction with 15 iterations and additional Gaußian smoothing can be seen in Figure 6.12. We proceed analogously to the first data example and apply the new two step algorithm to the data. Again, initial values, a-priori knowledge etc. is kept the same as before. We choose our regularization parameters to be α = 5 × 10−4 , β = 0.75, γ = 0.53 and η = 5 × 10−4 and obtain the computed parameters as shown in Figure 6.13, after 15 EM iterations and each 5000 parameter identification iterations with τ = 0.05. We can witness 1


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the same effects as for the reconstruction from exact data. The values within the area of high perfusion are not computed properly but all other values - especially those within the area of small perfusion - are satisfactory, despite the high amount of noise. Hence, we are able to detect small myocardial perfusion from very poor PET data with the new algorithm. Figure 6.14 shows the 9-th frame of the exact data in comparison to the standard EM reconstruction and G(p) for the computed parameters p as being illustrated in Figure 6.13. In the following section we want to explore the new algorithms’ capability by applying

(a) The 9-th frame of the exact data

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Figure 6.14: The comparison of the 9-th frame of the exact data, the standard EM reconstruction of its synthetic PET data and of G(p), with p being the computed values shown in Figure 6.13. it to real H2 15O PET data.

6.3

Real Data

Concludingly we want to present results computed from real H2 15O PET data and compare it with standard EM reconstructions that neglect the temporal correlation of the data. The dataset we are going to reconstruct describes a two-dimensional transaxial slice containing the cardiovascular area. It has been scanned with a spatial resolution of 175×175 pixel at 26 sampling points in time. First of all, there has been made a background scan 81

lasting for 20 seconds. After injecting the tracer, 14 frames have been scanned every five seconds. To improve the SNR, the next three frames were scanned every 10, the following four frames every 20 and the last four frames every 30 seconds. If we allocate every frame with the mean value between start- and endtime of a scan, we obtain the following stepsize h (in seconds): h = (10, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5, 52.5, 57.5, 62.5, 67.5, 72.5, 77.5, 82.5, 87.5, 95, 105, 115, 130, 150, 170, 190, 215, 245, 275, 305)

(6.1)

This stepsize is needed to compute the ODEs properly. c The EM reconstructions of the dataset were computed with a Matlab -program written 2 by Thomas K¨ osters . In the following we want to discuss the way we did a segmentation and subsequently will present computational results.

6.3.1

Segmentation

A question that arises first is the question for appropriate segmentation. We could either generate factor images from the dataset as in previous algorithms, or decide for a new approach. We choose the latter option and manually generate a segmentation from EM-TV

(a) The left ventricular area

(b) The right ventricular area

(c) The area denoting myocardium without septum

(d) The area denoting the septum

Figure 6.15: The segmentation for the real data example. The generated images have a resolution of 175 × 175 pixel and were manually generated from the TV-reconstructions shown in Figure 4.5 and Figure 4.6. 2


82

reconstructions as described in Section 4.8.2. The EM-TV reconstructions illustrated in Figure 4.5 and Figure 4.6 are computed from the same dataset we want to use to test our algorithm. Hence, we can use these reconstructions for segmentation. We are going to segment the whole cardiovascular area with the help of the reconstruction over the averaged data shown in Figure 4.5 to distinguish between background and heart. Furthermore we use the reconstructions shown in Figure 4.6 to segment left and right ventricle. The myocardium is segmented by subtracting the left and right ventricular segments from the whole cardiovascular segment. The generated segmentation can be seen in Figure 6.15. 1 A-priori CA

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1 A-priori CV

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(b) A-priori venous input curve CV∗

Figure 6.16: Arterial and venous input curves used as a-priori knowledge. Both curves were generated as average curves from different test reconstructions with varying parameters.

6.3.2

Reconstructed Parameters

Since the question of segmentation is answered, we can move on to the numerical computation of optimal parameters. Again we want to use the gradient method as the numerical 83

solver for the parameter identification problem, with a fixed stepsize τ = 0.05. Initial values and a-priori knowledge remain the same as for the synthetic data examples, except for the a-priori input curves. We ran a couple of test reconstructions with different regularization parameters and different initial values to generate average input curves which have in addition been smoothed. The a-priori input curves can be seen in Figure 6.16. With the 3

× 10

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Computed CA

Computed CV

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(a) Reconstructed arterial input function CA rec (b) Reconstructed venous input function CV rec





Figure 6.17: Computed values from the real data set. regularization parameters set to α = 1 × 10−5 , β = 5, γ = 5 and η = 1.444 × 10−4 we obtain the computed parameters illustrated in Figure 6.17 after 20 EM standard iterations with each 1000 parameter identification iterations. 84

To conclude this chapter we want to compare the standard EM reconstructions with 10 iterations and additional Gaußian smoothing of each frame to the byproduct sequence generated with the new algorithm. The comparison between all frames can be seen in Figure 6.18 - Figure 6.26.

(a) Standard EM reconstruction, frame 1

(b) Modified EM reconstruction, frame 1

(c) Standard EM reconstruction, frame 2

(d) Modified EM reconstruction, frame 2

(e) Standard EM reconstruction, frame 3

(f) Modified EM reconstruction, frame 3

Figure 6.18: The first three frames of the standard EM reconstruction are shown on the left side. On the right side the first three generated frames of G(p) with the optimal parameters as described in Figure 6.17 can be seen for comparison.

85







Figure 6.19: Frames 4 - 6 of the standard EM reconstruction are shown on the left side. On the right side the first three generated frames of G(p) with the optimal parameters as described in Figure 6.17 can be seen for comparison.

86








87








88








89








90








91








92





Figure 6.26: Frames 25 and 26 of the standard EM reconstruction are shown on the left side. On the right side the first three generated frames of G(p) with the optimal parameters as described in Figure 6.17 can be seen for comparison. We discover that by taking temporal correlation into account the EM reconstruction quality is much higher, as predicted.

93

Chapter 7

Conclusions With this thesis we provided a new method for the quantification of myocardial perfusion. In comparison to existing algorithms, the new method takes into account the temporal correlation of the data. We have introduced a mathematical theory and generated computational results to demonstrate the capability of the new approach. Further medical studies and validations with different datasets remain to be conducted to compare the new algorithm to existing methods and to improve the choice of regularization parameters. The next step to improve perfusion quantification should be an extension four dimensional EM-reconstructions to take more data into account. Another related research topic to pay attention to could be the physical modeling. New and more complex models might be derived to be more accurate and to avoid different possible solutions after linearization. The segmentation process still is a major research topic. A lot of different approaches suggested with this thesis have to be tested and to be compared to existing methods. Finding appropriate regularization terms to apply to the variational problems is a challenging topic and yet at the very beginning. Only basic regularization methods have been used to improve results and avoid ill-posedness. A lot of scope is left for improved and unusual types of regularization. The practical numerical implementation can also be modified by using different numerical solvers. All computations in this thesis were done with the gradient method. Algorithms like Newton or BFGS (Broyden-Fletcher-Goldfarb-Shanno) have to be tested and might offer faster convergence. Subsequently we want to notice that the variational problem derived with this thesis can also be extended to other fields than myocardial perfusion quantification. As an example, PET in combination with radioactive glucose is used to monitor receptor-binding within the brain (cf. [WA04]). The physical model is based on compartment modeling, just as perfusion modeling is. Hence, we could replace the physical models to derive a variational method to monitor receptor bindings.

94

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[IKT+ 88] H. Iida, I. Kanno, A. Takahashi, S. Miura, M. Murakami, K. Takahashi, Y. Ono, F. Shishido, A. Inugami, and N. Tomura. Measurement of absolute myocardial blood flow with h215o and dynamic positron-emission tomography. strategy for quantification in relation to the partial-volume effect [published erratum appears in circulation 1988 oct;78(4):1078]. Circulation, 78(1):104–115, 1988. [JHC98]

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[LLA+ 01] J. S. Lee, D. S. Lee, J. Y. Ahn, G. J. Cheon, S. Kim, J. S. Yeo, K. Seo, K. S. Park, J. Chung, and M. C. Lee. Blind separation of cardiac components and extraction of input function from h215o dynamic myocardial pet using independent component analysis. J Nucl Med, 42(6):938–943, 2001. [Mey01]

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F. Natterer and F. W¨ ubbeling. Mathematical Methods in Image Reconstruction. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001.

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[SSC+ 02] K. P. Sch¨ afers, T. J. Spinks, P. G. Camici, P. M. Bloomfield, C. G. Rhodes, M. P. Law, C. S. R. Baker, and O. Rimoldi. Absolute quantification of myocardial blood flow with h215o and 3-dimensional pet: An experimental validation. J Nucl Med, 43(8):1031–1040, 2002. [SW94]

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Eidesstattliche Erkl¨ arung

Hiermit versichere ich, dass ich die vorliegende Arbeit selbstständig verfasst und neben c c den Programmen MATLAB und C keine weiteren Hilfsmittel als die im Literaturverzeichnis angegebenen verwendet habe. Alle auf der CD beigef¨ ugten Programme sind von mir selbst programmiert worden.

M¨ unster, 23.06.2008

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