A particle filtering approach tailored to functional MRI data for tracking changes in connectivity between brain regions

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A particle filtering approach tailored to functional

MRI data for tracking changes in connectivity

between brain regions

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Abstract

The general idea behind brain organization is that the brain is divided into compartments that coordinate a given function. Studies on brain or-ganization also revealed that these functionally localized sites in the brain continuously share information with each other, and the brain is therefore a complex network of mutually interacting nodes. In the past decades, neu-roscientists have extensively studied the connections between regions of the brain, and a growing body of evidence have related disorders in these connec-tions to neurological diseases, suggesting that connectivity could be used as a clinical biomarker for these pathologies. However, most studies assumed the connections as stationary, and recently the interest switched to determining their variations through time, since these changes, in addition to clarify the dynamic organization of the brain, may also have a clinical interest. Our purpose is to detect these time-varying brain connectivity through the pro-cessing of functional Magnetic Resonance Imaging (fMRI) time series. fMRI is a widespread imaging technique that permits to track brain activity with almost any disturbance in the process itself. The fMRI signal depends on magnetic properties of the tissue under examination, and a specific depen-dency, called Blood Oxygenation Level Dependent (BOLD) effect, can be particularly suited to our objective. Deoxy-hemoglobin molecules in blood have an iron atom that is not fully shielded by their structure, so there is a difference in magnetic susceptibility between deoxy- and oxy-hemoglobin. This causes the MR signal to be dependent on the level of blood oxygenation, allowing the tracking of activated brain regions, because operating brain cells have a greater requirement of oxygenated blood. Variations in the BOLD signal reflects activation/deactivation of brain areas, so it can be used to establish which area is functionally connected to which and to what extent, allowing to define a connectivity map. Our proposal is to apply a relatively new technique, named Particle Filter (PF), to evaluate the time-dependent causal influence between BOLD fMRI time series. Sequential Monte-Carlo techniques like the PF were first developed in the 1950s, but due to their high computational cost their use was not feasible on ordinary computers until the 1990s. Since then, they have found many applications, like mod-eling of chemical processes, mobile communication channels and biomedical signals. The main feature of the PF is the possibility to track variation in

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time of hidden parameters of a certain model describing the recursive de-pendency between data. In our case, this hidden parameters represent an estimate on the degree of connection between two or more brain regions. In this study we analyze the performances of the PF technique both in simu-lated and experimental data. The first chapter depicts the fMRI technique, addressing the biophysical basis and interpretation of the BOLD effect, with a preliminary, but necessary, introduction to conventional Magnetic Reso-nance Imaging. The second chapter reviews the state-of-the-art knowledge on brain connectivity and the suggestions for its clinical relevance. The third chapter introduces the basic formulations of Sequential Monte Carlo algo-rithms, and the Particle Filter algorithm is described in detail. The fourth and final chapter exposes our results and conclusions on real and simulated data.

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Magnetic Resonance Imaging

Magnetic Resonance Imaging (MRI) is an imaging technique based on the Nuclear Magnetic Resonance (NMR) effect. This effect consists in the pre-cession of nuclear spins of the sample in presence of a magnetic field. The frequency of this precession depends on the spin of the nuclei and on the strength of the applied magnetic fields. Identical nuclei will precess at dif-ferent rates in locations with difdif-ferent magnetic field, and the spatial dis-tribution of the nuclei can be encoded in the spatial disdis-tribution of these frequencies. This first chapter starts with a brief overview of the NMR phe-nomenon, which is discussed more specifically in the appendix. Later we are going to describe how the measured signal frequencies can be encoded to track the spatial distribution of a given nuclear species within the sample and the fundamental notion of k-space will be discussed. Then the acqui-sition scheme used in our measurements, called Gradient Echo (GRE), will be introduced. Later on, the discussion will be generalized to more than one spatial dimension. Finally the Echo-Planar Imaging (EPI) technique will be reviewed. This technique improved the temporal sampling of ac-quisitions, and allowed the development of functional Magnetic Resonance Imaging which can be used to observe biological functionalities, and that is the topic of the last section.

1.1 Nuclear Magnetic Resonance

This section summarizes the basic NMR phenomenon, which is discussed more extensively in the appendix. Nuclei and particles have an intrinsic an-gular momentum ~J , called spin, associated with a magnetic dipole moment

~ µ = γ ~J

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6 CHAPTER 1. MAGNETIC RESONANCE IMAGING The proportionality constant is called gyromagnetic ratio and its value de-pends on the kind of particle under examination. In presence of a constant magnetic field ~B0, the magnetic dipole precesses around the direction of ~B0,

with precession frequency given by the Larmor equation ω0= γB0

In real life NMR experiments, single spin contribution are too small to be detectable, and in a macroscopic sample the discussion must be generalized referring to a macroscopic magnetization vector, resulting from the com-bination of all the spins in the sample. The equation of motion of this magnetization ~M under the effect of a general external magnetic field ~Bext

is given by the Bloch equations [2] dMz dt = 1 T1 (M0− Mz) d ~M⊥ dt = γ ~M⊥× ~Bext− 1 T₂∗M~⊥

where Mz and ~M⊥ are the components of the magnetization vector

respec-tively along ~Bext and perpendicular to it. M0 is the magnetization

equi-librium value, T1 is the experimental spin-lattice relaxation time, T2∗ is the

combination of the spin-spin relaxation characteristic time (T2) and the

char-acteristic time of the decay of the magnetization due to field inhomogeneities (T₂0) 1 T₂∗ = 1 T2 + 1 T₂0

These parameters are fundamental in every NMR acquisition: T1

parame-terizes the signal loss due to interactions between spins and the lattice in a macroscopic sample, while T₂∗ takes into account losses due to interac-tions between spins (T2) and due to inhomogeneities in the external field

(T₂0), so that the spins are not precessing on phase (see appendix). Then the NMR phenomenon can be summed up as follows: the Bloch equations state that in presence of an external magnetic field the magnetization is time-varying, and its changes depend on magnetic features of the material. Therefore through the measurement of the behavior of the magnetization through time1 it is possible to infer the composition of the sample through the evaluation of these characteristic times, which are the objective of every NMR examination.

1

This is possible since a time-varying magnetization is related to a time-varying mag-netic field, which induces a current in an external receiving coil (see appendix).

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1.2. MAGNETIC RESONANCE IMAGING 7

1.2 Magnetic Resonance Imaging

1.2.1 Frequency Encoding and K-Space

The aim of MRI is to determine the spin density ρ(z) of a sample through time-dependent MR signals. First of all, we need a way to connect the spin precession frequency to its position. This can be done superimposing a linear field gradient G to the static magnetic field B0, thanks to the

lin-ear dependency between the precession frequency and B0. The maximum

amplitude of this position-dependent field is in general much smaller than B0. Before the gradient field is turned on, the magnetization is precessing

with frequency ω0. Relaxation effects are neglected, meaning that we are

supposing that the data sampling takes place over times much smaller than T₂∗. In the beginning the discussion will be restricted to one dimensional imaging. From now on, we suppose that the magnetization vector has been initially tipped into the plane perpendicular to B0 (the transverse plane).

This is done to push the magnetization out of equilibrium and it can be performed adding to the constant field a smaller radiofrequency (RF) B1

perpendicular to it. The tipping angle, or flip angle, is given by ∆θ = γB1τ

where τ is the duration of the RF.

The magnetic field resulting from the superposition of the static field and the gradient field is

Bz(z, t) = B0+ zG(t)

where G is the constant amplitude of the linearly varying magnetic field, i.e. G = ∂Bz/∂z. The variation in the spin angular frequency is

ω(z, t) = ω0+ γzG(t)

This equation shows that the use of a gradient establishes a relation between position and precessional rate and this is called frequency encoding along a direction (in this case, along the field direction ˆz). The phase accumulated due to the gradient is

φG(z, t) = − Z t 0 dt0ωG(z, t0) = −γz Z t 0 dt0G(z, t0) (1.1)

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8 CHAPTER 1. MAGNETIC RESONANCE IMAGING the spatial dependency is only caused by the receiving field and the trans-verse magnetization, the signal is (see A.23)

s(t) = Λω0B⊥

Z

dzM⊥(z, 0)ei(Ωt+φ(z,t)

Λ is a constant factor that takes into account the gain factors from the electronic detection system, Ω is the demodulation frequency and

φ(z, t) = − Z t

0

dt0ω(z, t0)

is the accumulated phase. The transverse magnetization in terms of the spin density ρ0 is (see A.17)

M = ργ

2

~2B0

4kBT

and then we have

s(t) = Z

dzρ(z)eiΩt+φ(z,t)

where ρ(z) is the effective spin density

ρ(z) ≡ ω0ΛB⊥M0(z) = ω0ΛB⊥ρ0(z)

γ2~2 kBT

B0

If the demodulating frequency has no offset, i.e. Ω = ω0, the signal with

precession frequency (1.1) is s(t) =

Z

dzρ(z)eiφG(z,t) _(1.2)

This equation is called the 1D imaging equation and it is valid for every spatial dependence of the varying field. Expliciting the z-dependence for a linear field we have

s(k) = Z

dzρ(z)e−i2πkz where k is the spatial frequency

k(t) = γ Z t

0

dt0G(t0) (1.3)

and for a constant gradient the time-dependence of the spatial frequency k is

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1.2. MAGNETIC RESONANCE IMAGING 9 This important equation shows that for linear gradients the signal s(k) is the Fourier transform of the spin density of the sample, so the spin density is said to be Fourier encoded along ˆz by the gradient. Therefore, the spin density can be found taking the inverse Fourier transform of the signal

ρ(z) = Z

dks(k)e+i2πkz (1.4)

Since the signal and the spin density are a Fourier transform pair, it is useful to speak in terms of the image space (z) and the k-space (k). In practice, k-space is the habitat of the complex raw data. Each point in k-space is a sampled data point in the acquisition. The data value at the center of k-space (k=0) arises from the signal from the entire field of view (FOV), as it is acquired with no additional gradient-dephasing. The center of k-space therefore represents the overall signal intensity of the image. The minimum distance ∆k = γ G∆t between two points in k-space, where ∆t is the read sampling time interval, defines the FOV

FOV = 1

∆k

The maximum spatial frequency kmax = γ GT where T is the total time

the gradient is turned on, defines the size of the sampling step in the given direction in the image space, i.e. ∆z = 1/2kmax. Extending this to more

than one dimension, the maximum spatial frequency for every encoding direction in space defines the pixel (2D) or voxel (3D) [3].

Solving equation (1.4) requires a good coverage of k-space, i.e. a sufficient amount of data point in the k-space. In the case of constant gradient a uniform coverage of k-space requires only a constant sampling rate. Simple as it seems, this way to measure the spin density has limitation: for example, to obtain a faithful picture of the spin density we would need a large amount of k-space data, requiring a long acquisition time; but a too long acquisition time introduces signal decay due to relaxation effects that were neglected in the previous discussion. Moreover, a finite acquisition time will result in data truncation, meaning that only a discretized version of s(k) can be found, leading to an approximated measure of the spin density.

1.2.2 Gradient Echo

To reconstruct the spin density signal measurements over a sufficient range of k are needed. The principal technique for obtaining both negative and positive values of k is the Gradient Echo (GRE) method. In this method a

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10 CHAPTER 1. MAGNETIC RESONANCE IMAGING series of gradient lobes is used in order to recover the signal losses due to the gradient itself: because the gradient is a field inhomogeneity, the spins precess inhomogeneously, resulting in a loss of phase coherence and therefore a loss of signal intensity. This dephasing can be reversed by the application of a second gradient with opposite polarity with respect to the first, gener-ating a signal echo at time TE when the phase coherence is restored (see appendix).

Suppose we apply a negative constant gradient Gz = −G in the time

interval (t1, t2). The phase accumulation at a time t is

φG(z, t) = γGz(t − t1) t1 < t < t2

The second gradient is a positive constant gradient applied in the time in-terval (t3, t4) The phase at a time t is then

φG(z, t) = γGz(t2− t1) − γGz(t − t3) t3 < t < t4

The echo will appear when the phase goes back to zero, so the echo-time is

TE = t3+ t2− t1 (1.5)

The echo corresponds to that time during the second gradient where the area under the second lobe cancels the total area under the first one, so the general condition for an echo is

Z

G(t)dt = 0 (1.6)

With this criterion we can choose the time interval for the second lobe so that the echo appears in its center: (t4− t3)/2 = t2− t1. Then we can define

t0 = t − t3− (t2− t1) = t − TE

that is a time coordinate with the origin at the echo. The phase is then φG= −γGzt0 and the signal (1.2) when the second gradient is on becomes

s(t0) = Z

dzρ(z)e−iγGzt0 = Z

dzρ(z)e−i2πk(t0)z

where equation (1.3) was used for k. This experiment leads then to a range of both negative and positive spatial frequencies symmetric about the echo. The GRE is then an acquisition method that allows frequency encoding

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1.2. MAGNETIC RESONANCE IMAGING 11 through the application of gradients, whose size and duration is tuned ac-cording to (1.6) in order to refocus the dephasing introduced by the gra-dients. However, the GRE does not refocus the dephasing due to intrinsic field inhomogeneities, so a RF with a 180◦ flip-angle (called a π pulse) is usually employed to compensate this effect. There are two ways this can be made which are displayed in figure 1.1: with the π pulse between the two gradient lobes, with the same polarity, or with the π pulse before the two lobes with opposite polarity.

Figure 1.1: 1D GRE experiment with π pulse between two lobes of the same polarity (left) and with the π pulse before the two lobes with opposite polarity (right)

Both methods have to suffice (1.6) to achieve the formation of the gradi-ent echo: in the first case the lobes have the same polarity because the RF reverses the phase of the spins after the first lobe is applied, while in the second case the polarities of the two lobes are different because the pulse is applied before both of them, so the second gradient has to compensate the dephasing produced by the first one. In both cases the phase induced by both the gradients and the inhomogeneity effects must be simultaneously refocused. The echo due to the π pulse appears at

TE = 2τ (1.7)

where τ is the time of application of the RF pulse, so the refocusing effects are simultaneous if both conditions (1.5) and (1.7) are met.

In practice, the two methods coincide, because the effect of the RF in the first case is to change the polarity of the first gradient. Still, the second method is preferable because the position encoding takes place at a time closer to the sampling of the data, reducing motion artifacts.

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12 CHAPTER 1. MAGNETIC RESONANCE IMAGING

1.2.3 Imaging in more Dimensions

Previously we have found that the signal from an MR experiment with a linearly varying magnetic field is the Fourier transform of the proton density of the sample. The position of the signal source is said to be frequency encoded in the direction of the gradient, because spins in different position precess with different frequencies. Now we want to generalize this discussion to more than one dimension. Quite obviously, this can be done with the addition of magnetic field gradient in the second and third dimensions, so that the frequency will have a 3D-position dependency, but this is not the only or the most efficient way to do this.

We start generalizing the imaging equation (1.2) in three dimensions. the signal from a single RF excitation in presence of a set of three orthogonal gradients is

s(~k) = Z

d3rρ(~r)e−i2π~k·~r (1.8) that in a more explicit form is

s(kx, ky, kz) =

Z Z Z

dx dy dzρ(x, y, z)e−i2π(kxx+kyy+kzz)

= F [ρ(x, y, z)]

This equation is called the 3D imaging equation and it states that the signal is the 3D Fourier tranform of the 3D proton density in presence of linearly varying gradients in every direction in space. Each spatial frequency is related to its respective gradient component by

ki = γ

Z t

Gi(t0)dt0 i = x, y, z

The generalization to 3D imaging is then quite simple, but the real difficulty is to find a way to sufficiently sample the 3D k-space to obtain an accurate estimate of the proton density. This is done selecting a trajectory in k-space, i.e. choosing how the gradients must be applied in order to cover k-space in the most efficient way.

In the 1D case, there is just one way to explore k-space. In 2D, the k-space is a plane, so there are many ways to explore it. The simplest way is to cover this plane with a series of parallel lines as shown in figure 1.2. This is obtained alternately turning on the gradient along x (the read direction) and y (the phase encoding direction) as shown in figure 1.3.

The gradient along x must be reversed every time the gradient along y is turned on, so that the direction of data collecting is reversed. The amplitude

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1.2. MAGNETIC RESONANCE IMAGING 13

Figure 1.2: 2D coverage of k-space with parallel lines.

Figure 1.3: Sequence diagram for a 2D k-space coverage with parallel lines through a single RF excitation.

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14 CHAPTER 1. MAGNETIC RESONANCE IMAGING of the gradient along y determines the magnitude of the encoding step in the phase encoding direction. The data are collected during the application of Gx. This is just the most standard way to 2D k-space coverage, but many

trajectories are possible, like spiral trajectories. In general, the trajectory is determined by how the gradients are combined, i.e. the sampling step is a vector in k-space with components

(∆kx, ∆ky) = ( γ Gx∆t, γ Gy∆t)

where ∆t is the time interval between two consecutive measurements. In 3D, the k-space can be thought as a stack of 2D planes and each of them is covered as in the 2D case. The third dimension is covered by the introduction of a phase-encoding gradient in the third direction. There are two ways this can be achieved, depending on how the RF pulses are carried out: multi-slice 2D imaging and 3D imaging. In the first one the third dimension is filled through slice selection, i.e. a series of RF pulses which excite one slice of the sample at a time is used and 2D imaging is sequentially applied to each slice. In the second one, the third direction is frequency encoded as the other directions: a RF pulse excites a slice, thicker than the slices of the first method, and that slice is phase encoded in the slice select gradient direction. This is often referred to as partition encoding. For both methods, a series of RF pulses are needed. The second method has the advantage that the thickness, and therefore the number, of slices can be varied to fit the requests upon the acquisition time, so shorter pulses can be used, leading to a reduction in the echo-time and then an improvement in spatial resolution, Signal-to-Noise ratio and signal losses due to T₂∗ dephasing. Moreover, in the first method we need to avoid slice crosstalk: if two adjacent slices are excited one after the other, spins in the intermediate region will be excited two times, due to imperfect RF profiles, so these spins would not have time to reach equilibrium, resulting in signal alteration. This can be avoided leaving a gap between adjacent slices or exciting all the odd-numbered slices and then all the even-numbered slices. This is not necessary in the second method. The advantage of the first method is that, if we are interested in viewing an image from different orientations, it undergoes minor quality degradation than the first method because of different resolution in each direction. In conventional imaging only one line of k-space data is collected following a RF excitation, because collecting more lines requires a longer acquisition time, resulting in signal decay due to T2 or T2∗ relaxation.If TR is the time required to acquire a line

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1.2. MAGNETIC RESONANCE IMAGING 15 along the read direction, the acquisition time in 3D imaging is

Tacq = NyNzTR

where Ny, Nz are the number of steps in the phase encoding and slice

selection direction necessary to cover the volume of interest.

A brief focus on slice selection is now useful: slice selection is done using a combination of gradient field and spatially selective RF pulses, i.e. pulses with finite bandwidth which excite a small portion of the sample. This spatially dependent fields introduce dephasing in the selected slice, resulting in signal losses. As usual these losses are recoverable with the introduction of a rephasing gradient. The phase accumulated due to the slice selection gradient Gss is

φ(z, t) = −γGsszt

This dispersion can be corrected with a reversed gradient applied when the RF is turned off so that the spins have zero phase at the end of this refocusing lobe. If we suppose that the spins are instantaneously tipped into the transverse plane at the center of th RF pulse, the dephasing occurs only for a time that is half the total duration of the pulse. As in (1.6), the total area under the gradients must vanish to reach refocusing and since only half of the slice selection gradient is responsible of refocusing we have

R Grephasedt R Gssdt = 1 2

This result is valid only under the assumption of instantaneous tipping, that is reasonable only for small flip-angles. In general there will be corrections depending on the characteristics of the RF pulse.

1.2.4 Echo-Planar Imaging

At last, we introduce an imaging technique called Echo-Planar Imaging (EPI), which in the last decades has become one of the most significant techniques in clinical imaging, thanks to the important reduction in acqui-sition time. Only the basic aspects of this technique will be discussed, but a more detailed description of EPI-MRI can be found in [1] and [4]. Consid-erations about hardware requirements, image reconstruction and artifacts correction in EPI are beyond the scope of this text and information can be found in [5], [6] and [7].

EPI was first developed in 1977 by Sir P. Mansfield, but the heavy hard-ware demands of this technique impeded its use until 1987. Since then, it

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16 CHAPTER 1. MAGNETIC RESONANCE IMAGING

Figure 1.4: a) Sequence diagram for a single shot GRE-EPI, with constant phase encoding gradient. b) k-space coverage of the sequence in (a). c) Sequence diagram of single shot GRE-EPI with a blipped phase-encoding gradient. d) k-space coverage produced by (c)

has been widely used and it is considered one of the fastest and most efficient encoding techniques, permitting the acquisition of a single image in times of the order of 30 - 100 ms and a full study can be done in a few seconds. EPI’s short acquisition times have broadened the number of possible appli-cations for functional MRI, because physiological processes can be studied with a better time resolution. Unlike conventional imaging, where lines are acquired one at a time after the RF excitation, EPI encodes all the infor-mation necessary to reconstruct an image after a single RF excitation pulse. Briefly, an oscillatory gradient, called EP readout, is applied along the fre-quency encoding direction so that a train of echoes is generated, in the same way they were generated in a conventional GRE technique. Each echo can be independently phase-encoded along any other orthogonal axis and the en-tire k-space can be acquired. Several waveforms for the EP readout can be used, as long as they can generate the train of echoes. Each echo is acquired for a very short time, so T₂∗ relaxation and susceptibility artifacts cannot be appreciated. The time interval between two consecutive echoes, called Echo

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1.2. MAGNETIC RESONANCE IMAGING 17 Spacing (ES), should be minimized so that the time needed to cover the whole k-space is reduced and consequently T₂∗ losses are minimized over a full image. Two possible sequence diagrams for a Gradient Echo-EPI (GRE-EPI) are shown in figure 1.4. Basically, the EPI technique is analogous to conventional imaging, with the main difference that the whole k-space is covered after a single RF pulse, resulting in faster acquisition. Then the arguments exposed previously about excitation with RF pulses, frequency encoding, echo formation and k-space coverage still hold. Therefore, any imaging strategy can generally be adapted to EPI, with the advantage of immunity to motion distortion. The main disadvantage to EPI techniques are the required gradient power and duty cycle, because the gradients used need to be high to cover the whole k-space in a short amount of time, and they need to be turned on and off with a very high speed. Gradients mag-nitude and slew rate define the spatial resolution of the system. Another problem of standard EPI, called Single-Shot EPI, is that the readout pe-riod is several times longer (30-100 ms) than in conventional imaging (5-15 ms), so this technique is extremely sensitive to static field inhomogeneities, that cause susceptibility artifacts and geometrical distortions. For this rea-son, the EPI readout time is limited by signal decay due to T₂∗ relaxation. This problems can be overcome by accumulating the data over multiple ex-citations. This technique is called Multi-shot EPI and it can significantly reduce the required gradient performances or improve spatial resolution with the same gradient hardware and it reduces T₂∗ effects preserving the image quality with shorter acquisition times.

Gradient-Echo Echo-Planar Imaging

A π/2 pulse, i.e. a RF that produces a 90◦ flip-angle, is applied to collect the signal encoded using the EPI module, that is shown in figure 1.4. If the echo time TE is short, the image is essentially proton-density weighted, without any T1 contamination. Short TE can be difficult to achieve due to

hard-ware limitations. With longer TEs the image becomes more T₂∗ weighted or susceptibility-weighted, i.e. it becomes more sensitive to local field inhomo-geneities, and a sequence design like this can be useful in obtaining Blood-Oxygenated Level Dependent (BOLD) effect, that will be introduced in the next section. It must be taken into account that longer TEs cause signal loss, basically due to T₂∗ decay, and if the TE is too long the signal will decay to zero. This means that for T₂∗ or susceptibility weighted measurements a compromise on the echo time must be found. Echo times typically chosen range from tens to hundreds of ms, depending on the T₂∗ of the tissue

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un-18 CHAPTER 1. MAGNETIC RESONANCE IMAGING der examination and on experimental conditions [8]. In conclusion, the EPI short acquisition times result in great advantages in MR experiments that track the dynamics of biological processes, called functional MRI (fMRI), that will be discussed in the following section.

1.3 Functional Magnetic Resonance Imaging

Functional magnetic resonance imaging (fMRI) is a relatively young technol-ogy that began in the early 1990s and since then has become the major form of noninvasive functional imaging in humans in many fields, like psychology, cognitive science and neuroscience. The development of fMRI began with the discovery that deoxyhemoglobin (dhB) has different magnetic properties than oxyhemoglobin, i.e. it is paramagnetic, meaning that the MR signal is different from voxels with different oxygen concentrations. The use of these property of hemoglobin was not employed until the early 1990s [9], when dHb was first used as a endogenous contrast agent for fMRI signal in a technique that was later termed Blood Oxygenation Level Dependent (BOLD). It is fundamentally based on the principle that the brain oversup-plies blood and oxygen to regions that increase their activity relative to the state before the stimulus is presented, but it has to be reminded that it is not a direct measurement, so results need to be interpreted cautiously. The aim of this section is to describe the BOLD-fMRI technique, so an introduc-tion to signal changes due to different magnetic properties in tissue and to biophysical processes involved in neural activity is necessary. A brief review of disturbances in the signal concludes the section.

1.3.1 Magnetic Properties of Tissue: Paramagnetism,

Dia-magnetism, Ferromagnetism and Susceptibility

Inside a body, a spin is not isolated and it is subject to an internal field due to it neighbors in addition to any external field [1]. The internal field is dominated by nearby electrons and their contribution can be approximated by magnetic dipole fields corresponding to dipole moments associated with their orbital and spin degrees of freedom. Nuclear magnetic moments have a minor importance due to the inverse-mass dependency in the gyromagnetic factor

γ ≈ q

2m (1.9)

The electron magnetic moments are intrinsic or can be induced and materials are classified in paramagentic, diamagnetic or ferromagnetic according to the

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 19 different kinds of magnetic dipole moments.

Paramagnetism

The quantum stacking of electrons in an atom or molecule involves a system-atic cancellation of spin moments for each pair. An atom with an unpaired electron has a nonvanishing permanent magnetic moment with an associ-ated nonzero dipole magnetic field and is referred to as paramagnetic. In absence of external perturbations, these moments are randomly distributed and do not produce a macroscopic magnetization. When an external per-turbation is present the moments tend to align with the external magnetic field producing a bulk magnetic moment and a corresponding magnetic field augmenting the external field, similarly to the nuclear macroscopic magne-tization discussed previously. Since the atomic moment is much larger than the nuclear one, as stated by (1.9), the local field can variate substantially from the applied field value in absence of material. As for the proton spins, thermal motion reduces the tendency of the atomic moments to line up along an external field.

Diamagnetism

Whether or not a permanent dipole moment is present, all materials will have induced dipole moments in the presence of a time-varying external magnetic field. In an analogous way as for induced currents in a coil as depicted by Faraday’s law of induction, atomic currents produce a counter magnetic field tending to cancel the external field, caused by slight micro-scopic shifts in the orbital motion of electrons of a given atom or molecule. This effect is weaker than paramagnetism and it is called diamagnetism. Its weakness is such that, even if it is an omnipresent phenomenon, diamagnetic effects are not negligible only in the case that all the electrons of a given material are paired, i.e. when paramegnetism is not present. The macro-scopic effect of diamagnetism is a tendency to an anti-parallel alignment of the induced moment with the external magnetic field, which locally reduces the field strength. Differently from paramagnetism, it is not a temperature-dependent effect.

Ferromagnetism

Certain materials have permanent domains of electron spin magnetic mo-ment which concur to produce very strong macroscopic self fields existing independently of an external magnetic field. This effect is present in many

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20 CHAPTER 1. MAGNETIC RESONANCE IMAGING materials (iron, gadolinium, nickel, cobalt and so on), which are called fer-romagnetic. The main cause is a strong spin-spin coupling between nearby atoms, which is limited only by the range of spin-spin forces and its do-main can span distances of millimeters and contain a huge number of spin moments in a material. These domains are randomly distributed until an external field is turned on. When this happens, they align and a permanent macroscopic magnetic moment is produced. This alignment is called mag-netic saturation when it involves all the domains, producing the maximum magnetic moment density. As long as the temperature is not too high and depending on the material, a residual magnetization is left when the external field is turned off. This dependency of magnetization on what has happened in the past is called hysteresis. There are also antiferromagnetic and fer-rimagnetic materials, where neighboring spins completely or incompletely cancel, respectively.

Susceptibility

These characteristics of materials necessitate to correct the external field to take into account variations due to electrons dipole moments, because the precession frequency and the phase of the local spin density is modified by this local disturbances. To parameterize the field changes due to material’s magnetism the permeability µ is defined through the relation between the physical magnetic field B and the vector field H. In linear materials, i.e. isotropic materials where there are no special directions in space, B and H are related by

~ B = µ ~H

A relative permeability is defined µr = µ/µ0 where µ0 = 4π × 10−7 is

the permeability of empty space. With these definitions, paramagnetic, diamagnetic and ferromagnetic materials correspond to µr> 1, < 1 and 1

respectively. With the aforementioned magnetic dipole approximation, the macroscopic source of local field disturbances is the current corresponding to the magnetization M, given by

~

JM = ~∇ × ~M

Through the combination of Maxwell’s equation corresponding to the Am-pere’s law after separating the total current into a free current and the magnetization current written above we have the definition of the H field

~

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 21 In linear materials the fields and magnetization are proportional to each other through the magnetic susceptibility χ

~

M = χ ~H

and from the definition of H and (A.15) we find the relation between χ and µ χ = µ µ0 − 1 and ~ B = 1 + χ χ µ0 ~ M (χ)

Susceptibility is then a dimensionless quantity. It is positive for para-magnetic, negative for diamagnetic and zero for nonmagnetic materials. Tissues in human body are mostly made up of water, which has2 χ ≈ −7.2 · 10−7cm3g−1 and susceptibility variations between materials due to paramagnetic effects are usually so small that the augmented susceptibility is still negative. The main interest relies on phase changes due to susceptibil-ity effects: a spin moving from regions with different susceptibilsusceptibil-ity is subject to different field strengths and the phase shift for a given field variation ∆B is

∆φ = −γ∆BTE

at the echo time TE when the acquisition is made. It is important then to address how field and susceptibility variations are related and phase infor-mation can be used to learn about changes in magnetic properties of tissues relative to each other.

1.3.2 Biophysics of Energy Metabolism in the Brain

At this point a brief description of the basic processes that determine brain activity can be useful. A more detailed description can be found in [10].

Brain needs energy to generate electrical activity required for neuronal signaling. In the neuron there is an electrical potential difference across the cell membrane, with Vin< Vout, called resting potential. An action potential

is a transient disturbance of that potential, i.e. a rapid depolarization of the

2_{The dimensionless magnetic susceptibility explained above is often referred to as}

vol-ume magnetic susceptibility. For experimental purposes it is often more practical to refer to the mass magnetic susceptibility

χM =

χ ρ where ρ is the density of the material.

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22 CHAPTER 1. MAGNETIC RESONANCE IMAGING membrane. It propagates down the axon until it reaches a junction with another neuron, called a synapse, and it influences the firing of the post-synaptic neuron by creating a local fluctuation in the post-post-synaptic resting potential. With an excitatory post-synaptic potential (EPSP) the resting potential is raised, moving the second neuron closer to be activated. Instead, an inhibitory post-synaptic potential (IPSP) decreases the resting potential, inhibiting the firing of the second neuron. Each neuron has the capacity to integrate inputs from many other neurons through their cumulative effect on its resting potential.

The functioning of resting potential, action potential and post-synaptic potential critically depends on mantaining the intracellular and extracellular concentration of several species of ions in a state far from equilibrium: at rest there is an excess concentration of sodium (Na+_{) and calcium (Ca}2+₎

ions in the extracellular space and an excess of potassium ions (K+) inside the membrane and this concentration gradient would tend to balance the ions inside and outside the membrane. However, potassium ions’ tendency to exit the cell is reduced by the electrostatic force caused by the negative potential inside the membrane and sodium is kept outside the cell by the low permeability of the membrane to these ions. But this permeability is sensitive to the potential difference across the membrane and when the voltage decreases the permeability increases. This growth is slow until a critical threshold is reached and then the permeability increases sharply as the potential becomes less negative. The result is a rapid depolarization of the membrane caused by the growing flux of ions through the membrane. This process is then a passive way to equilibrium that requires no energy expense and the result is a net flux of sodium into the cell and potassium out the cell. The action potential drives down the axon triggering a change of sodium permeability through its path. The need to a continuous leak of ions is minimized by the myelin sheaths that surround the axon: myelin is a poor conductor, so the ion currents are small and the action potential has to be restored by the ion flux only in the myelin interruptions called nodes of Ranvier.

At the synapse the arrival of the action potential initiates an increase of the membrane permeability to calcium, allowing the leak of calcium in-side the presynaptic terminal where neurotransmitters are stored in pack-ages called vescicles. The influx of calcium causes this vescicles to merge with the membrane and release the neurotransmitter in the synaptic gap. Neurotransmitters drift through the gap and bind to receptor sites in the postsynaptic terminal of another neuron. When this happens the postsy-naptic potential is slightly depolarized, moving the neuron closer to produce

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 23 its own action potential. In inhibitory synapses, on the contrary, the binding of the neurotransmitter causes a repolarization of the presynaptic terminal preventing the formation of the action potential. This process too requires no energy and results in a net flux of calcium into the membrane and of neurotransmitters out the membrane. Then for brain activity the energy is only needed to keep the neuronal system out of equilibrium, so that there is an excess free energy in the system that can be used to trigger the action potential.

Everytime a neuron is fired there is a change in the ion concentrations inside and outside the membrane and energy is needed to restore the previous concentrations. This is done by the cell through active transport, when enzymes move ions against their concentration gradients with an energy expense. This energy is provided by the conversion of adenosine triphosphate (ATP) in adenosine diphosphate (ADP) in the Na-K pump, an enzyme that transports three sodium ions out of the cell and two potassium ions into the cell for each ATP consumed. This allows the restoring of the ion concentrations and of the resting potential, since there is a net flux of charge in this process. In the synapses energy is required to restore the calcium concentration and to repack neurotransmitters into vescicles.

The ATP molecules needed to provide this energy is obtained through the metabolism of glucose and oxygen through two main processes: glycolisis and trans-carboxilic acid (TCA) cycle. Glycolisis does not require oxygen, but produces only a small amount of ATP which is mainly produced in the TCA cycle, that requires oxygen. The overall mechanism of consumption of glucose and oxygen can be summarized as

C6H12O6+ 6O2→ 6CO2+ 6H2O

+ 38ATP

Then brain activity is linked to glucose and oxygen metabolism and a greater activity requires a greater amount of oxygen and glucose [11]. When this happens, astrocytes and neuronal cells are induced to send vasoactive signals into nearby arterioles capillaries, dilating the upstream arterial vessels [12]. This rush of blood to activated areas is the physiological basis for most of the modern techniques for functional imaging. Many techniques focus on the variations of Cerebral Blood Flow (CBF) and Cerebral Metabolism Rate for Glucose (CMRGlc), like functional Positron Emission Tomography (PET) techniques, and it has been proven that the glucose consumption rate and the blood flow to activated areas increase in a similar way, i.e. if there is a 50% growth of CBF it is accompanied by a 50% increase of CMRGlc. However, this correlation does not necessarily mean that there is

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24 CHAPTER 1. MAGNETIC RESONANCE IMAGING a link between these two quantities: for example, it has been discovered by Gjedde et al. [14] that at rest glucose is delivered in excess to brain cells, so the CMRGlc could increase without any variation in CBF. In addition, it was shown that a change in glucose delivery does not affect the CBF and that a suppression of CBF change during activation leaves the CMRGlc unchanged. These studies then show that there is no need for a match between CBF and CMRGlc during activation as it would be expected from our knowledge of the underlying mechanism of brain activity and this means that there still is a lot to understand in the correlation of brain activity and glucose consumption.

In a similar way, CBF and Cerebral Metabolic Rate of Oxygen (CMRO) were discovered to be not so strictly related as they would be expected to be: for example in [13] it is shown through a functional PET experiment that a 29% increase in CBF can result in a 5% increase in CMRO, which means that during activation the extraction rate of oxygen drops substantially. This imbalance was extensively confirmed by fMRI experiments [15], which are more suited than PET to track oxygen concentration levels in human subjects, because no tracers are needed. The cause of this imbalance is still object of studies, but its existence is widely recognized and it is the cornerstone of fMRI techniques like Blood Oxygenation Level Dependent (BOLD): if CBF and CMRO changed in the same way, there would not be any variations in oxygen concentration in activated areas and therefore there would not be any signal dependency on oxygenation. Many explanations of this disproportion have been proposed and some of them can be found in [10].

1.3.3 Blood Oxygenation Level Dependent Response

In 1990, Ogawa et al. [16] reported that functional brain mapping was possible using venous blood oxygenation level-dependent(BOLD) magnetic resonance imaging contrast. This BOLD contrast relies on changes in dHb concentration, which acts like an endogenous paramagnetic contrast agent. Therefore changes in the local dHb concentration in the brain lead to al-terations in MRI signal intensity. As already pointed out, this alteration is caused by a mismatch in CBF and CMRO during activation, causing an increase in capillary and venous oxygenation levels during activation. It has been reported that BOLD fMRI response correlates with underlying local field potentials (LFP), which are believed to represent synaptic ac-tivity including neural input, and with spiking acac-tivity, which represents the suprathreshold neural output, but since they occur together it is very

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 25 difficult to separate their contribution to BOLD response [12]. The exact relation between BOLD signals and the underlying neural activity is still a matter of debate [10, 17, 18].

Now we examine the contribution of physiological parameters to mag-netic resonance imaging signals. In a typical fMRI experiment pixel sizes are a few millimeters, so they can include blood, vascular tissue and cere-brospinal fluid (CSF) components. Arterial and venous blood have different T2 values, so they are considered separately, with the assumption that blood

in capillaries is partly arterial and partly venous. The MR signal is then a sum of signals originating from different compartments with different spin density and relaxation parameters [12]:

S =X

i

ρi× Vi× Mss,i× e−T E/T

∗ 2,i

The subscript i indicates the compartment, whose spin density is ρi and

volume fraction is Vi. TE is the echo-time and T2∗is the apparent transverse

relaxation time including BOLD effects. Mssis the steady-state

magnetiza-tion of each compartment, which is Mss= (1 − e−T R/T

∗

1_{) sin θ/(1 − cos θ × e}−T R/T1∗₎

where θ is the flip angle, TR is the repetition time and T₁∗ is the apparent longitudinal relaxation time in the presence of inflow. Then the signal is de-pendent both on imaging parameters and biophysical responses that change any parameter and not only on BOLD effects.

Contributions of physiological parameters to baseline MRI signal The inflow and CBF can decrease the longitudinal relaxation time T₁∗ of blood and extravascular tissue components. The flow of blood moves the spin outside the imaging plane into the slice pixels and the signal from unsaturated blood is enhanced relative to the surrounding stationary spins. The magnitude of this inflow contribution depends on imaging parameters, like the flip angle θ and the repetition time TR. In the extravascular tissue pool the relaxation time with this contribution is [12]

1/T₁∗= 1/T1+ f /λ

where f is the CBF in mL of blood/g of tissue per second and λ is the blood-to-tissue partition coefficient in mL of blood/g of tissue. This value is the baseline value for T₁∗. Also, stimulus-induced increases in blood velocity or

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26 CHAPTER 1. MAGNETIC RESONANCE IMAGING inflow in pixel containing large vessels act to reduce T₁∗, increasing the fMRI response and decreasing spatial specificity. However, in fMRI studies with typical spatial (≈ mm3) and temporal resolutions (≈ 2-3s) this effects do not contribute substantially to fMRI signal changes.

The other fundamental parameter of MRI is the transverse relaxation time T₂∗, which is the superposition of the intrinsic T2and of the

susceptibility-induced T₂0:

1/T₂∗ = 1/T2+ 1/T20

This values are directly correlated with the dHb content of blood. Water rapidly exchanges between deoxyhemoglobin-containing red blood cells and plasma and also diffuses in presence of field gradients generated by dHb inside vessels, so there is a dynamic loss of phase coherence which is not recoverable. In addition a frequency change is also observed, depending on magnetic field, oxygen saturation level and the angle between vessels and B0: since within any given pixel there are multiple vessels with different

orientations, their frequency shift causes a phase dispersion that reduces blood T₂∗ [19].

The field gradient generated by dHb decreases by (r/a)2, where r is the distance and a the vessel radius, so its susceptibility effects extend to extravascular tissue and CSF. It also has a dependence on vessel orientation and there are no effects on vessels perpendicular to the main field. This spatial dependence of susceptibility effects causes the presence of multiple frequency shifts within a pixel which result in phase dispersion and signal loss. This dephasing effect can be approximated as R0₂ = 1/T₂0 and it is given by [12]

R0₂ = A · CBVv∆χ0ω0(1 − Y )β

CBVv is the venous cerebral blood volume, Y is the oxygenation level, ∆χ0

is the susceptibility difference between fully oxygenated and fully deoxy-genated blood, ω0 is the resonance frequency and A and β are constants:

A depends on vessel size and orientation, while β has values between 1, for large vessels, and 2 for capillaries. From this equation it is clear that an increase in susceptibility effects will arise if there is an increase in venous CBV, resulting in an increase of dHb content in vessels, if there is a decrease in venous oxygen saturation levels or if there is an increase in the magnetic field, through ω0= γB0.

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 27 BOLD contribution from intravascular and extravascular compo-nents

Now we examine the contributions to BOLD signal from intravascular and extravascular compartments. A more detailed review can be found in [20] and [21].

Intravascular BOLD signal increases almost linearly when TE<T2 of

blood for Spin-Echo (SE) studies and when TE<T∗₂in Gradient-Echo (GRE) studies. It reaches a maximum when TE=T2 for SE and when TE=T∗2 for

GRE and when TE is increased again the signal decreases. This contribution can be estimated through the measurement of relaxation times in different experimental conditions. Blood R2 has a quadratic dependence on

deoxy-genation level 1-Y and on B0 [22]. As the magnetic field is increased, the

relaxation times of venous blood decrease much faster than corresponding tissue relaxation times, so if the field is sufficiently high the contribution to the signal of the intravascular component can be significantly reduced at ultra-high fields.

The relative contribution of intravascular and extravascular components can be examined using small diffusion gradients which induce velocity de-pendent phase shifts in presence of flow and consequently reduces signal from blood due to inhomogeneous velocities within a vessel and the pres-ence of blood vessels at different orientations within a pixel. Increasing the magnitude of diffusion weighting progressively attenuates the intravascular signal from faster flowing to slower flowing blood [12].

Spatial specificity and sensitivity of BOLD fMRI

Spatial resolution in BOLD fMRI depends both on the hemodynamic re-sponse and the vascular structure. One important issue for very high reso-lution studies is that pixels with different baselines will have different BOLD change for the same oxygenation variation, so signals across pixels need sim-ilar vascular properties to be compared.

A major problem in BOLD fMRI is the combination of blood coming from different areas, resulting in a strong dependence of the signal on the distance from the area of activation and on the spatial extent of neural re-sponse: if the activation area is small, the deoxygenated blood is quickly di-luted by blood from inactive regions, reducing the oxygenation level change from adjacent regions. On the contrary, if the activation area is large much of the blood draining into the large downstream vessels also has origin from activated regions, resulting in a spatially non specific contribution. These

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28 CHAPTER 1. MAGNETIC RESONANCE IMAGING effects can be reduced with SE BOLD fMRI, improving the spatial speci-ficity to small-sized vessels and nearby tissue. SE BOLD fMRI with small diffusion gradients improves spatial specificity, but since SE BOLD fMRI signal sensitivity is reduced by the refocusing of the dephasing effect around large vessels, and since the additional refocusing radiofrequency pulse in-creases power deposition to tissue, the GRE BOLD fMRI technique is often preferred. Specificity of hemodynamic based fMRI ultimately depends on how finely the blood flow is regulated: if intracortical arteries are indepen-dently regulated spatial resolution can be as fine as 0.33 to 2 mm for arterial blood based fMRI measurement [23]; if precapillary arterioles regulate the hemodynamic changes, even better resolution can be achieved.

Intrinsic spatial resolution for venous vessel-based BOLD response is ≈ 1 mm. If only small sized venous vessels are detected, resolution can be even better, but specificity is poorer than in capillary changes-based techniques [12].

Physiological sources of BOLD fMRI signals

The functional change in blood oxygenation level Y is dependent on the mismatch between changes in CMRO and CBF. Assuming Yart = 1.0 for

arteries and constant hematocrit, the relative change in Y for veins can be determined by [12]

∆Y 1 − Y =

∆CBF/CBF − ∆CMRO/CMRO ∆CBF/CBF + 1

where ∆ refers to the stimulus-induced change. Increasing venous oxygena-tion level decreases the blood oxygen extracoxygena-tion fracoxygena-tion OEF= 1 − Y from blood to tissue. When the relative CBF change is much larger than the CMRO change, ∆Y will be highly correlated with the CBF change.

In extravascular tissue, the BOLD signal change for constant hematocrit level in venous blood can be approximated [24]

%BOLD = M 1 − 1 + ∆CMRO/CMRO 1 + ∆CBF/CBF β 1 +∆CBV CBV ! (1.11)

where M is an overall scaling parameter related to baseline, vascular and imaging parameters and β is a constant parameter assumed to be ≈ 1.5 [24]. The parameter β describes a nonlinear dependence on venous oxygenation, reflecting the idea that dHb has a weak effect on the signal change in the smallest vessels compared to larger veins because of the effect of diffusion

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 29 [19]. The first term relates to the mismatch between CBF and CMRO and the second one relates to the relative CBV change, which is calculated from Grubb’s formula [25]

(1 + ∆CBV/CBV) = (1 + ∆CBF/CBF)α

where α = 0.38 is the flow-volume converting power term. Equation (1.11) is commonly used for calibration to determine oxygen metabolic changes. One important assumption of (1.11) is that the total CBV change obtained from the CBF change is the same as the BOLD-induced venous CBV change [26]. If this assumption does not hold, quantitative functional CMRO2 changes

obtained with BOLD fMRI contain errors.

Equation (1.11) can be linearly approximated to the first order as [26]

%BOLD = M ∆CBF/CBF − ∆CMRO/CMRO ∆CBF/CBF − 1 β · ∆CBVv CBVv ! = = M ∆Y 1 − Y − 1 β · ∆CBVv CBVv ! (1.12)

This shows that increasing Y will increase the signal, while an increase in CBVv decreases the signal. The first term is larger than the second and

when this second term is negligible, i.e. venous CBV changes are small, the BOLD signal change is simply related to blood oxygenation level change weighted by baseline venous blood volume.

Following equation (1.12), CBF, CMRO and CBV contribute quanti-tatively to BOLD fMRI dynamics. Usually, stimulus onset causes CBF to initially rise to an overshoot level, after which the increase in CBF decreases to a slightly lower plateau level during stimulus continuance. This increased CBF level will increase venous oxygen saturation, which is directly related to a positive BOLD signal. Dynamic CBF responses with high temporal resolution can be obtained with a conventional laser Doppler flowmeter or from fMRI. Since arterial blood response precedes venous response, the CBF signal is 0.5 to 1 s earlier than the BOLD response [27].

Regarding CMRO2, it is generally assumed that it increases rapidly and

peaks quickly after a stimulus onset [12]. Calibration studies have suggested that functional increases in CMRO2 are concomitant with increases in CBF

[24]. This is due to the overwhelming contribution of stimulus-induced CBF increases to blood oxygenation changes [28]. To measure dynamic changes produced only by the changes in CMRO2 a vasodilatator can be used to

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30 CHAPTER 1. MAGNETIC RESONANCE IMAGING highlighted an earlier CMRO2 response which peaks later than the BOLD

response. The delayed peak can be explained by the astrocyte-neuron-lactate-shuttle hypothesis [30], in which lactate is produced by increased glycolysis in astrocytes and then becomes the primary substrate for oxida-tive metabolism in neurons.

Dynamic CBF and CBV changes are intercorrelated, since CBF is de-pendent on CBV and velocity changes [12]. In some early studies, the BOLD contribution to CBV weighted fMRI was not corrected, leading to errors in quantification of CBV dynamic changes. More recently, it was observed that the total CBV response occurs earlier than the BOLD response [29]. Com-partment specific CBV studies revealed that venous CBV change is much less than total CBV change. It has been shown that in short stimulus-induced changes, i.e. less than 15 seconds long stimulus, the arterial CBV change is dominant over the venous CBV change, which is minimum [31], while for longer stimuli the venous CBV increases slowly, eventually reach-ing a magnitude similar to the arterial CBV change [32]. After stimulus offset, arterial CBV rapidly decreases to baseline values or exhibits a small undershoot, while venous CBV slowly returns to baseline. Recent venous CBV studies have important implications for BOLD quantification: first, since during stimulus the relative venous CBV change is less than relative total CBV change, relative CMRO2 changes are underestimated from (1.11)

with α = 0.38 [24]; second, a decrease in BOLD signals with constant CBF during the stimulus period can be explained by a slow increase in venous CBV [33]; third, the time independent BOLD linearity may not hold due to time dependent venous CBV contribution when short versus long stimulus data are compared [12].

Sources of initial dips, poststimulus undershoot and prolonged negative signals in BOLD fMRI

With the precedent discussion the main features which influence the BOLD response have been addressed. The theoretical BOLD response which arises from such influences is shown in figure 1.5. In the following discussion the typical shape of the BOLD response are explained.

A small initial BOLD dip has been commonly detected in humans, but the existence of this response is still debated. It may reflect an increase in dHb concentration due to CMRO2 increases before the CBF response or

an increase in venous CBV preceding the CBF response. Measurements of CBV fMRI during stimulus show a slow increase in venous CBV, with no significant vessel dilatation during the initial period [32], suggesting that the

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1.3. FUNCTIONAL MAGNETIC RESONANCE IMAGING 31

Figure 1.5: Theoretical BOLD response. It shows an initial dip, which re-flects the initial decreased oxygen concentration due to activation. Then the signal assumes a shape which reflects the increasing of oxygen concentration during activation and the subsequent return to its resting value. The resting value is reached after a brief undershoot, whose nature is still a matter of debate.

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32 CHAPTER 1. MAGNETIC RESONANCE IMAGING initial BOLD dip may be due to an early mismatch between CMRO2 and

CBF changes.

The poststimulus BOLD undershoot may be due to many physiologi-cal factors, including poststimulus CBF undershoot [34], slow poststimu-lus return of venous CBV to baseline [32] and slow poststimupoststimu-lus return of CMRO2-related oxygenation values to baseline [35]. Early result have to

be carefully interpreted, because total CBV responses are often measured instead of venous CBV response, because at high magnetic fields both CBF and CBV responses contaminate the BOLD signal and because CMRO2

dy-namics cannot be measured directly. In [35] it was suggested that ≈ 80% of BOLD undershoot arises from a slow return of CMRO2-related contributions

and the rest is from venous CBV contributions, with no CBF contributions. On the contrary, in [34] a CBF undershoot and a slow return of venous CBV to baseline were found responsible of the poststimulus undershoot. Another contribution may come from stimulus type and duration.

Negative BOLD signals occurring during a stimulus period may be ex-plained by a decrease in CBF due to neural inhibition [36], a decrease in CBF due to redistribution of CBF into nearby negative regions, called steal effect [37], an increase in CMRO2 without concomitant CBF increase [38]

and an increase in vasoconstrictive neurotransmitter with increased neural activity in subcortical areas [39]. In areas where vascular reactivity is slower, a negative BOLD response is probably due to an increase in CMRO2

with-out a corresponding increase in CBF. The physiological basis of negative BOLD signals is then dependent on stimulus type and brain region.

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Chapter 2

Brain Connectivity

Studies on how the brain works and how it is organized began in the 19th century. Initially, the idea was that the brain is organized into compartments which are responsible for coordinating a given function. This idea, known as functional localization [40], was abandoned in favor of the idea that the brain is also a complex integrative network consisting of spatially distributed but functionally linked regions that share information with each other [41]. This means that brain organization relies on the two principles of functional segre-gation and integration [42]: segresegre-gation refers to the existence of specialized neurons and brain areas, organized into distinct neuronal populations and grouped together to form segregated cortical areas, while integration allows the coordinated activation of distributed neuronal populations, enabling the emergence of coherent cognitive and behavioral states. In the past decades many studies have given insights about which region is responsible for a given function. In more recent years, neurosciences have changed their fo-cus from determining the function of each regions to determine how they communicate and how this communication influences human behavior. This matter has largely been studied through various neuroimaging techniques, and fMRI is one of the most employed thanks to its non-invasive nature. The present chapter reviews the main arguments concerning brain connectivity: the first section describes structural connectivity, which is how brain regions are anatomically connected, and focuses on its relationship with functional connectivity; the second and third section describe functional and effective connectivity respectively, with a focus on detection methods and on their clinical applications.

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34 CHAPTER 2. BRAIN CONNECTIVITY

2.1 Structural Connectivity

Structural connectivity is the anatomical connection through nerve cells axons between brain regions and it is often referred to as the connectome [40]. Results demonstrate that the cerebral cortex is made up of clusters of densely and reciprocally coupled cortical areas that are globally interconnected [43]. These connectivity patterns are neither completely regular nor completely random, but combine structural aspects of both of these extremes [40]. The main current interest to structural connectivity regards its connection to functional and effective connectivity: structural connectivity can be seen as a constraint or a prior belief on effective connectivity [40]; it has also been demonstrated that functional connectivity strength is positively correlated with structural connectivity strength [44]. This means that functional and effective connectivity depend on structural connection, but the anatomical linkage is not sufficient to fully describe connectivity: two areas can be functionally connected even though they are not structurally connected, for example if they are structurally connected through a third region.

2.1.1 Methods for structural connectivity detection

Structural connectivity is most commonly measured through diffusion ten-sor imaging (DTI). DTI is an MRI-based imaging technique that measures diffusion of water molecules in the brain: in an unrestricted environment water molecules diffuse freely in any direction; if their motion is constrained by brain architecture, water molecules will diffuse more along axons than across them, so measuring the direction of diffusivity can infer the orien-tation of white matter tracts in the brain [44]. This technique also allows to measure the degree of myelination and fiber density of cerebral white matter tracts, which is given by the anisotropy of the diffusion tensor [46]. The success of this technique is due to a much higher resolving power than other imaging techniques, like standard MRI, and to the feasibility to study structural connectivity in vivo in humans. Limitations to this technique are due to susceptibility induced signal loss and difficulties in resolving small tracts as they cross large tracts. Also, a voxel can contain many fiber tracts with different orientations, causing an incorrect estimation of the principal diffusion direction [44].

Recently, to overcome this issues, another MRI methodology called Dif-fusion Spectrum Imaging (DSI), or q-space imaging (QSI), has been used to study in-vivo structural connectivity. DSI is a model-free methodol-ogy which allows to resolve intravoxel diffusion heterogeneity by measuring

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2.2. FUNCTIONAL CONNECTIVITY 35 its diffusion density spectra estimator [45]. Basically, this method phase-encodes spin displacements using a strong pulse gradient on each side of the π RF pulse in a conventional spin-echo experiment. This allows a measure of the probability for a spin in a given voxel to diffuse of a certain quantity in a given experimental time, which is called the diffusion spectrum.

2.2 Functional Connectivity

Functional connectivity is defined as the temporal dependency between spa-tially remote neurophysiological events and it describes the relationship be-tween the neuronal activation patterns of anatomically separated brain re-gions, reflecting the level of functional communication between regions [41]. Recently, new neuroimaging techniques allowed the examination of func-tional connectivity in vivo on a whole-brain scale through the study of coac-tivation patterns between functional time-series of anatomically separated regions and one of the most employed techniques is BOLD fMRI. In short, functional connectivity is an observable phenomenon that can be quanti-fied through measures of statistical dependencies, such as correlation [40]. Then, by definition, functional connectivity does not depend on the method chosen to measure it, because it is a statistical theoretical measure that depends only on probability distribution over observed responses. Analysis of functional connectivity does not make any inference about coupling be-tween regions, but it only compares the statistical dependency hypothesis with the null hypothesis of no dependency. This is usually made through determination of predominant patterns of correlation or establishing that a particular correlation between two areas is significant. In recent years, functional connectivity patterns have been suggested to be possible markers for mental diseases.

2.2.1 Functional Connectivity during resting state

In the last two decades, fMRI studies of brain connectivity have focused on measuring functional connectivity as the coactivation of spontaneous fluc-tuations recorded during rest. These experiments are called resting-state fMRI, because during the examination subjects are instructed not to move or think to anything in particular, without falling asleep. This was first made by Biswal and colleagues in 1995: they demonstrated that during rest there are regions which are not silent, but show a high level of correlation between their BOLD fMRI time-series [47], but their activity decreases as cognitive tasks are performed. This suggests that even during rest information are

A particle filtering approach tailored to functional MRI data for tracking changes in connectivity between brain regions