Computational Models and Simulators of functional MRI Marwan Abdellah
[email protected] Blue Brain Project (BBP) École Polytechnique Fédérale de Lausanne (EPFL)
A literature review report submitted to
Prof. Rolf Gruetter
[email protected] Laboratory for Functional and Metabolic Imaging (LIFMET) Centre d’Imagerie Biomédical (CIBM) École Polytechnique Fédérale de Lausanne (EPFL)
Advanced Biomedical Imaging Methods and Instrumentation Doctoral School of Neuroscience (EDNE) École Polytechnique Fédérale de Lausanne (EPFL)
Frbruary 2015 35 pages
Computational Models and Simulators of fMRI
Acknowledgments I would like to acknowledge and express my deepest gratitude to Prof. Gruetter, his team, all the presenters and the researchers who have contributed to this outstanding course.
Marwan Abdellah February 2nd , 2015
1
Contents 1
Introduction 1.1 fMRI Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Report Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 7 7
2
The Biophysical Basis of BOLD fMRI
7
3
Data Generation Models for BOLD fMRI 3.1 Baseline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 BOLD Activation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Noise Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 11 11 11
4
BOLD Activation Models 4.1 Models for the Estimated HRF . . . . . . . . . . . . . . . . 4.2 Biophysically-plausible Models of the BOLD HR . . . . . . 4.2.1 Models for Neural Basis . . . . . . . . . . . . . . . . 4.2.2 Models for Energy Consumption and Metabolism . 4.2.3 Models for Blood Flow (CBF) . . . . . . . . . . . . . 4.2.4 Models for Vasculature Dynamics (CBV) . . . . . . 4.2.5 Models for Oxygen Extraction . . . . . . . . . . . . 4.2.6 BOLD Signal Models . . . . . . . . . . . . . . . . . . 4.3 Other BOLD HR Models . . . . . . . . . . . . . . . . . . . .
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11 12 13 15 15 15 16 16 17 19
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19 19 20 22 22
6
Simulation Software Applications 6.1 fMRI Simulation Toolbox (SimTB) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 neuRosim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 24 24
7
Conclusion
26
5
fMRI Simulation Studies 5.1 Single Voxel fMRI Simulations 5.2 Multi-Voxel fMRI Simulations . 5.3 Multi-resolution BOLD Models 5.4 fMRI artifacts Simulations . . .
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Computational Models and Simulators of fMRI
List of Figures 1
2
3 4 5
6 7 8
9
The structural content of a unit volume (voxel) in the visual cortex of a monkey. The stained image to the left shows a Nissl slice extracted from the cortex. The central image depicts the relative density of the blood vessels in the same area of the cortex. The vessel diameter is color coded. The three-dimensional reconstructions demonstrate the difference in the diameters of the blood vessels and the spatial distribution of a group of neurons between the blood vessels. Image courtesy of Nikos K. Logothetis (Macmillan Publishers) [Log08]. . . . . . A simplified block diagram of the underlying principle of fMRI within a single voxel starting from the stimulation activity and until the generation of the BOLD signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated models of the BOLD HRF . . . . . . . . . . . . . . . . . . . . . . . . . . The underlying physiological processes that account for the generation of the BOLD response. Image source: [Den06] . . . . . . . . . . . . . . . . . . . . . . . . The two-stage fMRI computational model of Chen and Calhoun. The neurophysiology model simulates the neurovascular coupling process and the fMRI technology model simulates the acquisition of the BOLD fMRI signal. . . . . . . A description of the multi-resolution computational BOLD fMRI model presented by Chen and Calhoun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data generation sequence of synthetic fMRI datasets in SimTB. Image courtesy of Allen et al. [AEW∗ 11b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The simulated BOLD signals based on the three convolution functions implemented in neuRosim for a 20-seconds ON/OFF block design. Image courtesy of Welvaert et al. [WDM∗ 11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A block diagram summarizing the data generation models for fMRI simulations.
3
9
10 13 14
21 23 25
26 28
Computational Models and Simulators of fMRI
List of Acronyms BOLD Blood Oxygenation-level Dependent CBF Cerebral Blood Flow CBM Cerebral Blood Magnetism CBV Cerebral Blood Volume CMRGL Cerebral Metabolic Rate of Glucose CMRO2 Cerebral Metabolic Rate of Oxygen DCM Dynamic Causal Modelling deoxyHb Deoxygenated Hemoglobin FPM Finite Perturber Method fMRI functional Magnetic Resonance Imaging GLM General Linear Model GRE Gradient-recalled Echo GUI Graphical User Interface HR Hemodynamic Response HRF Hemodynamic Response Function ICA Independent Component Analysis iEEG intra-cranial Electroencephalography LFP Local Field Potential MR Magnetic Resonance MRI Magnetic Resonance Imaging oxyHb Oxygenated Hemoglobin pixel Picture Element voxel Volume Element
4
Computational Models and Simulators of fMRI
Abstract The introduction of functional Magnetic Resonance Imaging (fMRI) has pioneered the neuroscience research to address the mapping of the functional aspects of the brain. Since its inception about twenty years ago, several models and simulation studies for blood oxygen-level dependent (BOLD) fMRI have been proposed and extensively discussed. Most of these research studies have been focused on understanding the underlying mechanisms of the technique. The complexity of fMRI data has challenged the development of common, precise and physiologically-plausible fMRI simulators, and consequently the data generation models in the majority of the current simulators are ad hoc. This report reviews the modelling work and numerical simulation studies of BOLD fMRI that have been covered in the literature to date. Also, the review lists multiple software simulation packages that have been developed recently to meet the increasing demands of the fMRI community.
Keywords fMRI, BOLD contrast, fMRI simulation, fMRI models, numerical simulations.
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Computational Models and Simulators of fMRI
1
Introduction
There is no doubt that the inception of the functional magnetic resonance imaging (fMRI) 20 years ago was a significant scientific breakthrough for brain research and neuroimaging [Ban12]. During this period, fMRI was growing exponentially until it became a standard tool for cognitive studies. The impact of this advanced technology was associated with the emergence of new scientific fields such as social neuroscience, neuroeconomics and neuromarketing [Ashss]. The main advantage of fMRI is its capability of mapping brain activity noninvasively. It measures the blood oxygen-level dependent (BOLD) perturbations that arise in response to a neuronal activity that follows a change in brain state [Jez99, Gor03, HSM09]. The principle idea of BOLD fMRI is based on the fact that a neuronal activity will incur perturbations in the cerebral blood magnetism (CBM) that can be measured by a T2∗ -weighted MRI sequence. By and large, this process can be characterized by a neurovascular coupling model. This model expresses the vascular response to a neuronal activity in terms of the cerebral blood flow (CBF), cerebral blood volume (CBV) and the cerebral metabolic rate of oxygen (CMRO2 ) [CC12]. Despite its importance and the wide acceptance the technique has gained since its introduction, the underlying mechanisms of BOLD fMRI in general and the complicated neurovascular process in particular are not fully understood [HB99, MFPK07, MB08, HGCDB12]. This limitation makes the generated datasets with fMRI have no ground truth that can be exploited to validate the statistical techniques being developed to analyse the measured signals and assess their properties quantitatively. Moreover, fMRI signals are contaminated by different sources of noise mainly due to electronic and physiological artifacts, chemical shift and the magnetic susceptibility induced by magnetic field inhomogeneities [Dro07]. Several methods have been developed to establish the missing ground truth such as intra-cranial electroencephalography(iEEG). Nevertheless, most of theses iEEG studies were not feasible to reconstruct a precise ground truth [DGS∗ 08]. Additionally, various artifact-correction techniques have been developed to suppress or at least minimize the artifacts that contaminate the fMRI signal, but these techniques were based on the analysis of empirically measured fMRI data, which fail to identify a valid ground truth.
6
Computational Models and Simulators of fMRI
1.1
fMRI Modeling and Simulation
Ultimately, the modeling and simulation of fMRI can be used to facilitate the understanding the complex dynamics and the different mechanisms involved in the generation of the BOLD effect quantitatively. Though, the realistic simulation of fMRI data is known to be an extremely challenging mission due to the complexity that characterize the data itself. This simulation requires very accurate computational models for the neurovascular responses, different sources of noise and artifacts, a complete MR acquisition system and above all geometric definition of the anatomy of the brain at different scales. Even if detailed models for each item in this list were available, combining these features into a single simulation system is still an inspiring task that require the collaborations of interdisciplinary research teams.
1.2
Report Objectives
In this report, the different modeling and simulation contributions of fMRI that have been presented and discussed in the literature are reviewed. We also refer and highlight the features of different fMRI simulation packages that have been developed recently to meet the demands of the community for flexible and standard simulation software applications.
2
The Biophysical Basis of BOLD fMRI
Obviously, the signaling mechanisms involved in this process cannot be extensively discussed in this sequel. Thus, this section is just a brief review of the basic physiological basis of the underlying mechanisms of fMRI. It is though to be very helpful to identify and highlight the different parameters that are required to model and simulate the BOLD response. For further details, this set of references will be very helpful in this regard [FM10, Gor03, Bux02, Duy05, UDB05]. Moreover, since the majority of fMRI experiments uses BOLD contrast to measure the brain activity [Ashss], our coverage for the biophysical principles of fMRI will be limited to BOLD-contrast fMRI only. BOLD fMRI does not measure the neural activity itself. Instead, it provides an indirect way to address the functional activity of certain brain region by measuring the changes in blood
7
Computational Models and Simulators of fMRI
oxygenation levels that happen in response to the local activation of certain functional network of neurons in this brain region. The BOLD signal emerges within the vasculature as a result of the concentration changes of the de-oxygenated hemoglobin (deoxyHb) that depends directly on the CBF, CBV, and CMRO2 . During the neural activity, the CBF increases much more than the CMRO2 . This results in more oxygenation of the venous blood that is accompanied with susceptibility changes of the blood and creates magnetic field gradient around the vessel (see Figure 1). This happens because the deoxyHb is more magnetic, i.e. paramagnetic, than oxygenated hemoglobin (oxyHb), which is virtually resistant to magnetism, or diamagnetic. This effect gradually changes the T2∗ relaxation time and finally causes changes in the detected MRI signal over time. This improvement can be mapped to show which regions of the brain are active at a time. Figure 2 summarizes the biophysical principles of BOLD fMRI in a simplifying block diagram.
3
Data Generation Models for BOLD fMRI
There are two approaches to generate fMRI simulation data. The first and most common one considers pure synthetic simulations and the other uses a hybrid simulation strategy. The first approach relies completely on rigours or simplified mathematical models to reflect the characteristics of the measured fMRI signals. In the hybrid approach, the generated fMRI combines a realistic dataset acquired at resting state added to synthetic activation data. Although the noise in this case is representative for real data, the parametrization or the manipulation of this noise in the resulting simulated fMRI data is not applicable. This reason obviously accounts for the widespread acceptance of the first approach in the literature [WR14]. Bellec et al. have discussed two simulation models for fMRI data and evaluated their performance relying on the bootstrap framework in a completely controlled environment [BPE09]. The first model, Equation (1), adds a mixture of multiple components, mainly a baseline signal b(t), the BOLD activation signal BAct (t) and random noise n(t) (for example [DMCPNM02, MP04]).
BSim (t) = b(t) + BAct (t) + n(t)
(1)
The second model adds a non-linear term to account for the motion artifacts either using 8
Computational Models and Simulators of fMRI
Figure 1: The structural content of a unit volume (voxel) in the visual cortex of a monkey. The stained image to the left shows a Nissl slice extracted from the cortex. The central image depicts the relative density of the blood vessels in the same area of the cortex. The vessel diameter is color coded. The three-dimensional reconstructions demonstrate the difference in the diameters of the blood vessels and the spatial distribution of a group of neurons between the blood vessels. Image courtesy of Nikos K. Logothetis (Macmillan Publishers) [Log08]. spatial interpolation [ABH01, PLMD05] or a first principle model of MR physics [DGS∗ 06a, XWR∗ 07, KYB08]. The advantage of the additive model is its flexibility that allows control over the signal-to-noise ratio and the profile of the activated region. The majority of the simulation studies have relied on the pure synthetic approach to generate the fMRI simulation data [WR14].
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Computational Models and Simulators of fMRI
Stimulation Activity
Neural
Neuronal Response (Activity)
Vasculature O2 Demande
Vasculature Response
CBF
CBV
CMRO2
Magnetic Magnetization Susceptibility Response
Magnetic Filed Changes Signal
Single Voxel BOLD Response
Figure 2: A simplified block diagram of the underlying principle of fMRI within a single voxel starting from the stimulation activity and until the generation of the BOLD signal.
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Computational Models and Simulators of fMRI
3.1
Baseline Models
The modeling of the baseline signal was quite straightforward. It was assumed to be zero or constant value in some cases, and time-varying in other cases to reflect the intensity variations of the signal. The time-varying case was used by Backfrieder et al. to model thermal shifts [BBS∗ 96].
3.2
BOLD Activation Models
The classification of the BOLD activation models will be covered in depth in Section 4.
3.3
Noise Models
Roughly, the noise models were always involved in the generation of synthetic fMRI simulation data. Gaussian noise was used in most of the models and other models have added to it some drift. A little number of simulations have used experiment-specific noise models to account for some artifacts of the BOLD signal such as physiological noise or motion-correlated noise.
4
BOLD Activation Models
Due to the fact that the BOLD response detected in most of the fMRI studies is merely an indirect measure of the neuronal activation, precise models of the BOLD response do not exist so far. A clear understanding of the exact underlying biophysical principles of the neural activation and the observable BOLD response requires a rigours mathematical model that can link the two variables. Consequently, most of the modeling studies of fMRI do not attempt to model the neuronal activation at such biophysical detail and only assume that the neuronal activation can be addressed by an abstract latent variable [Ashss, Den06]. Their hypothesis was set such that a linear time-invariant relationship governs the transformation of the neuronal activation directly into the BOLD response. This model allows exploiting powerful analysis tools such as linear regressions and statistical tests [FHW∗ 94]. Several studies have addressed the validity of this linear relationship. Some of them have confirmed that its deviation from linearity is not severe as long as the triggering events 11
Computational Models and Simulators of fMRI
are separated by few seconds and the duration of the exposure is larger than a certain threshold [Ashss, BEGH96, VN98]. The others studies which involved different measurement modalities have accounted for this non-linearity mainly due to the hemodynamic effects [MLL∗ 01, OLM∗ 04]. These conditions were enough such that the BOLD response Bi (t) can be described by a linear system in terms of the input neuronal activation Ni (t). The transfer function of this model f TF represents all the complex mathematical transformations that potentially contribute to the conversion of the neural activation Ni (t) into the BOLD response Bi (t) where Bi (t) = f TF [ Ni (t)]
(2)
Relying on the superposition principle of linear systems, the BOLD response Bi (t) can be obtained if the neural activation can be represented by an impulse function. In this case, the BOLD response is called the hemodynamic response function (HRF) h(t). This is called the general linear model (GLM), where the HRF is seen to be a hypothetical BOLD response to an impulse neuronal activation. This means that the BOLD response Bi (t) to any complex neuronal activation Ni (t) can be computed from the convolution integral described by Equation (3) if the HRF is known a priori.
B(t) =
Z t 0
N (τ )h(t − τ )dτ
(3)
It is obvious from this previous integral in Equation (3) that the HRF can be simply obtained if the input activation was an impulse. Nevertheless, such experiment is not practically feasible because we do not have a direct control over the input neuronal activation.
4.1
Models for the Estimated HRF
Specific estimated models for the HRF can be used based on experimental observations and previous knowledge characteristics of the function. There are two basic and famous models that have been used to estimate the HRF: the gamma function [BEGH96, Coh97] and the difference of two gamma functions1 [FFJ∗ 98, Glo99]. The two models are depicted in Figures 3(a) and 3(b). 1 Also
known by the canonical HRF.
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Computational Models and Simulators of fMRI
(a) The Gamma function model of the HRF
(b) The canonical model of the HRF
Figure 3: Estimated models of the BOLD HRF
4.2
Biophysically-plausible Models of the BOLD HR
The standard convolution models and the BOLD HR estimations presented in the previous section are blind to any physiological processes that result in the BOLD signal. They can only reflect a simplification of a chain of physiological effects and complex mechanisms that extend between the neuronal activation and the generated BOLD response. A more realistic approach is to investigate the causal models of how BOLD responses are generated. These causal models require deep and crystal-clear understanding of the different biophysical mechanisms that translate the neuronal activation into a detected BOLD signal, as seen in Figure 4. Following a neuronal activation, these physiological mechanisms can be briefly summarized in the following points: 1. The demands of each activated neuron for the oxygen and glucose increase. 2. The cerebral metabolic rate of consumption of the glucose (CMRGL) and the oxygen CMRO2 increase. 3. The extraction rate of glucose and the oxygen molecules from the blood stream increases. 4. The higher supply of glucose and oxygen causes the CBF to increase. 5. The increased CBF cause the venous vessels to dilate according to their visco-elastic properties. 6. The CBF effect dominates over those of the CMRO2 and CBV, and consequently, blood becomes more oxygenated.
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Computational Models and Simulators of fMRI
7. The susceptibility is changed and the MR signal increases.
Figure 4: The underlying physiological processes that account for the generation of the BOLD response. Image source: [Den06] . Each of the previous points requires a complex physiological model in order to be able to formulate an accurate and complete biophysically-plausible model of the BOLD HR. During the last decade, there have been sustained efforts to model each individual mechanism that is suspected to contribute to this chain of effects. According to the classification proposed by Buxton et al. [BUDL04] and Deneux and Faugeras [Den06], the biophysical models of the BOLD response can be divided into the following categories: 1. Models for neural basis. 2. Models for energy consumption and metabolism. 3. Models for blood flow (CBF). 4. Models for vasculature dynamics (CBV). 14
Computational Models and Simulators of fMRI
5. Models for oxygen extraction. 6. BOLD signal models. 4.2.1
Models for Neural Basis
A neural basis model would be valuable to identify the main neuronal activity that affects the BOLD response. Mathiesen et al. have discussed a sigmoid dependence of the CBF on LFP [MCAL98]. Logothetis et al. have concluded that the BOLD response is proportional to intra-cortical processing of a given brain area and not due to its spiking activity [Log02]. Kim et al. have shown a high correlation between the LFP and the amplitude of the BOLD response measured in the same region [KRO∗ 04]. Friston et al. have presented the dynamic causal modeling (DCM) that was considered the first step to present comprehensive forward models that take into account parameters at the neuronal level to model the causal chain starting from the external stimulus to the BOLD response [FHP03]. 4.2.2
Models for Energy Consumption and Metabolism
Aubert and Costalat have presented a relatively sophisticated mathematical model that accounts for the different mechanisms involved in the oxygen metabolism for a given neural activity [AC02]. This model was used to simulate the BOLD responses for several patterns of neuronal activities and different assumptions for the changes of the CMRO2 signal. Davis et al. have presented another model that only accounts for the CMRO2 response using a step function [DWK∗ 94]. Buxton et al. have proposed a more relaxed model by convolving the neuronal activity with a gamma-variate function [BUDL04]. 4.2.3
Models for Blood Flow (CBF)
Four neurovascular coupling models were discussed extensively in the literature [Blo07, Den06]. Based on an empirical approach, Miller et al. have considered a non-linear relation between the CBF and the BOLD signal [MLL∗ 01]. Friston et al. have proposed a damped oscillator model that relates the CBF to a flow-inducing signaling mechanism [Blo07]. Their model accounts for the CBF in terms of the neuronal activity, blood flow value at resting state, neural efficacy and auto-regulatory feedback time constant. Behzadi et al. have extended Friston’s model to account for the changes in the muscular compliance of the cerebral arteriole [BL05]. 15
Computational Models and Simulators of fMRI
The previous models provide a more realistic reflection of the blood flow that that presented by Buxton et al. that convolve the neuronal activity with a gamma-variate function [BWF98]. 4.2.4
Models for Vasculature Dynamics (CBV)
Vasculature models have to account for the density of blood in a fixed volume, or the perfusion (CBV). In general, the dynamics of CBV and CBF depend on each others according to the visco-elastic properties of the vasculature. Grubb has proposed a very simplified model to link the two variables relying on a power law V = F α . The Grubb’s constant α was estimated between 0.25 and 0.4 [Den06]. This model is only valid in steady state and thus, a dynamic model is necessary to precisely reflect the vasculature response. Mainly, two reference dynamic models exist. The first one is the Balloon model presented by Buxton et al. [BWF98] and the other one was proposed by Mandeville et al. and is called the Windkessel model [MMA∗ 99]. The Balloon model has expressed the output flow as the sum of a linear component and a power law to determine the volume dynamics. The Windkessel model has discussed an analogy between the blood pressions and the CBF. Kong et al. have improved the Windkessel model to account for the delayed compliance by modelling it using an additional state variable [KZJ∗ 04]. 4.2.5
Models for Oxygen Extraction
Since deoxyHb molecules cause the paramagnetism responsible for the BOLD signal, there should be some biophysical models for describing the oxygen extraction from the hemoglobin molecules in the blood stream. Buxton and Frank have proposed a model of the dynamics of oxygen delivery to the tissue within the context of the Balloon model [BF97, BWF98]. It was called the Oxygen limitation model. This model has assumed no capillary recruitment, negligible arteriolar contribution to the volume, zero oxygen concentration in the tissue and finally the presence of the all the deoxyhemoglobin in the venous vasculature. As a consequence of the simplification of this model, due to the previous assumptions, the oxygen extraction rate was found to depend only on the CBF. A more elaborated model was presented by Zheng et al. called Oxygen tissue transport model [ZMJ∗ 02]. This model has discarded several assumptions from Buxton’s model. It incorporated non-zero concentration of the oxygen into the tissue and established a direct relation between tissue oxygenation and the metabolic demand. 16
Computational Models and Simulators of fMRI
4.2.6
BOLD Signal Models
The BOLD signal is generated due to the paramagnetic properties of deoxyHb. This affects the T2∗ relaxation time of the MR signal as a consequence of the increased inhomogeneity of the induced magnetic field. Ogawa et al. have presented a first model to this effect in [OMT∗ 93]. Afterwards, several studies have been developed to introduce improved models of this effect, but the first attempt of proposing a precise model for the BOLD signal was taken by Davis et al. [DKWR98, UDB05]. Their model have introduced the relationship between the BOLD signal changes and the underlying physiological variables of the neurovascular coupling process and few other parameters that characterize the local tissue. This simplified model was based on several reasonable approximations and previous outcomes of different numerical simulations [DKWR98]. Due to the simplicity of this model, it was proven to be a very significant step towards the understanding of the BOLD effect in a quantitative manner. In that regard, Buxton et al. have presented another simplified model of the MR signal acquisition that links the solution of the Balloon model to an approximation of the measured BOLD signal [BWF98]. This model computes the changes in the BOLD signal as a function of the changes in the deoxyHb content in the blood and the blood volume. The formalism of this model is discussed below. The basic MR signal in Buxton’s model was represented with the following decaying exponential function in terms of the echo time TE ∗
S = S0 · e−TE· R2
(4)
where S0 is the effective spin density2 and R2∗ is the transverse relaxation rate constant. This constant is expressed in terms of the T2∗ relaxation time by the following formula R2∗ =
1 = R0 + R T2∗
(5)
R2∗ can be also expressed by the sum of the two terms R0 and R which represent the value of R2∗ if no deoxyHb is present and the additional relaxation produced by the deoxyHb respectively. Few years later, Buxton’s model was updated by Obata et al. to account for the difference in 2 Signal
measured at TE = 0
17
Computational Models and Simulators of fMRI
the intra-vascular and extra-vascular signal contributions to the final BOLD signal [OLM∗ 04]. In this model, the signal measured from a single voxel at resting state S0 is approximated by a volume weighted sum of the intra-vascular signal SI and the extra-vascular one SE (Equation 6). One of the critical assumptions of this model was the random and isotropic distribution of the blood vessels in the tissue. Based on this assumption, the intra-vascular signal has neglected the effect of the arterial blood and it was only assumed to reflect a venous volume fraction V0 merged with the contribution of the capillaries. The intra- and extra-vascular signals were represented by a decaying exponential function in terms of R2∗ of each component (Equation 7).
S0 = (1 − V0 ) · SE + V0 · S I
S0 = (1 − V0 ) · SE0 · e
− TE· R2∗
E
(6)
+ V0 · S I0 · e
− TE· R2∗
I
(7)
The final BOLD signal that corresponds a neuronal activation was given by
S = (1 − V ) · S E · e
− TE· R2∗
E
+ V · SI · e
− TE· R2∗
I
(8)
and the fractional signal changes is given by i h δS 1 − TE· R2∗ − TE· R2∗ I E = + es · V · e (1 − V ) · e − (1 − V0 ) − es · V0 ) S0 1 − V0 + eS V0
(9)
where es = S I /SE represents the ratio of the blood to tissue signal, and the venous volume fraction V0 was changed to V for generalization. Assuming a first-order linear approximation, this model was simplified to δS ≈ − TE · δR2∗E − eS V · TE · δR2∗I + (V0 − V )(1 − eS ) S0
(10)
where by δR2∗I and δR2∗E represent an increment in the R2∗ of both signal components. The modelling of these two terms was based on the numerical simulations of Ogawa et al. [OMT∗ 93].
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Computational Models and Simulators of fMRI
4.3
Other BOLD HR Models
According to [WR14], there are many fMRI simulations that have not used any of the previous models for representing the BOLD activation in the synthetic fMRI data. Some of them have ignored the simulation of the BOLD activation component3 BAct (t) entirely and others have selected a boxcar function Π(t) to reflect it.
5
fMRI Simulation Studies
5.1
Single Voxel fMRI Simulations
Boxerman et al. have presented an fMRI model to evaluate the intra- and extra-vascular contributions of a single voxel to the fMRI signal at 1.5 Tesla using Monte Carlo simulation for modelling the susceptibility-based contrast and physiological model for the neurovascular coupling process [BBK∗ 95]. The results of their model have shown good agreements with the experimental designs, and therefore they have confirmed that the intra-vascular components have major contributions to the fMRI signal changes. In 2008, Pathak et al. have developed a novel modelling technique called the Finite Perturber Method (FPM) to model the different susceptibility-based contrast mechanisms for arbitrary vasculature morphologies at the micro-scale [PWS08]. FPM has shown excellent agreement with the theory and the extant susceptibility modelling data. Later the same year, Marques and Bowtell have applied a three-dimensional finite element method to a realistic model of the cortical vasculature to evaluate the reliability of the infinite cylinder model used previously in the literature [MB08]. They have used the realistic model to highlight the relative importance of intra- and extra-vascular contributions to the BOLD signal. Martindale et al. have developed a generalization of the Monte Carlo models presented by Boxernamn et al. [BBK∗ 95] that can accurately model the intra- and extra-vascular signals for an arbitrary tissue with various imaging parameters [MKJ∗ 08]. This model was verified against combined measurements of fMRI and optical imaging. 3 Refer
to Equation (1)
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Computational Models and Simulators of fMRI
5.2
Multi-Voxel fMRI Simulations
The majority of fMRI simulations were performed to compute the BOLD signal in a single voxel and others have considered only a few number of voxels in their simulations. This limited approach was taken due to the computational complexity of the algorithm needed or the memory requirements to express the BOLD effect for a four-dimensional fMRI volume. Such approach cannot be useful if a complete simulation of three-dimensional spatial distribution of the BOLD signal is required. In 2012, Chen and Calhoun have proposed an overall multi-voxel BOLD fMRI simulation model. This model was decomposed into two cascaded stages. The first stages reflects the neurophysiology of the neurovascular coupling process. It characterizes the neurovascular coupling mechanism in terms of susceptibility perturbations for MR signal detection, i.e. it generates a series of snapshots of three-dimensional susceptibility distributions. The susceptibility change is expressed in terms of the susceptibility difference between the oxyHb and deoxyHb χdo , CBV, CBF, CMRO2 and the neuronal activation NA ( x, y, z, t) as seen in Equation (11). This stage is based on previous models presented in the literature [AB02, AI02, Bux10, BF97, ZLHC08]. The other stage simulates the MRI technology that is used to image the susceptibility changes based on similar ideas for single voxel BOLD simulations [CC10, CCC11]. The field map due to the susceptibility map computed in the first stage can be computed by the convolution operation in Equation (12), where h( x, y, z) is represented by a point dipole filed three-dimensional kernel. The final BOLD signal is generated based on the volume integral in Equation (13) in terms of the filed distribution δB, the gyromagnetic ratio γ, and the echo time of the sequence TE. The block diagram of this computational model is illustrated in Figure 5.
δχ( x, y, z, t) = χdo · f (CBV, CMRO2 ) · CBV ( x, y, z, t) · NA ( x, y, z, t)
(11)
δB( x, y, z, t) = conv(δχ( x, y, z, t), h( x, y, z))
(12)
SBOLD ( xi , yi , zi ) =
Z Z Z
exp(i · γ · δB · TE)dxdydz
20
(13)
Computational Models and Simulators of fMRI
Figure 5: The two-stage fMRI computational model of Chen and Calhoun. The neurophysiology model simulates the neurovascular coupling process and the fMRI technology model simulates the acquisition of the BOLD fMRI signal.
21
Computational Models and Simulators of fMRI
5.3
Multi-resolution BOLD Models
Chen and Calhoun have presented a computational multi-resolution (four levels of resolution) model for BOLD fMRI [CC11]. They were motivated to use this model in a numerical simulation in order to capture and understand several aspects of the BOLD signal formation at different scales. This model has considered multiple spatial scales that range from the atomic scale (10−6 m) and up to the millimeter scale (10−6 m) based on a spin packet model presented in their previous study [CC10]. Their model has involved bulk nuclear spin precession in a BOLD-induced inhomogeneous magnetic field with a millimeter-resolution voxel. The model has considered the partitioning of the nuclear spin pool into spin packets at a macroscopic scale and calculating the multi-resolution voxel signal by grouping the spin packets at a mesoscopic scale (Figure 6). Based on this multi-resolution model, and assuming a small-angle approximation, they have concluded that the BOLD signal intensity is related to its phase counterpart by a two-resolution-level relationship.
5.4
fMRI artifacts Simulations
According to Drobnjak, early fMRI simulators have used naïve techniques to generate the synthetic fMRI data [Dro07]. These techniques were based on using existing MRI data and artificially integrate into this data an activation time-series to reflect the behaviour of the activated brain regions in response to an external stimulus. The spatial distributions of the assigned activations were pre-defined to certain voxels in the brain image. In some studies, these voxels were determined relying on the results of previous experiments [EFS∗ 02] and in others they were manually selected based on the region of interest [FM01, GSS∗ 02, LWS02, WEM∗ 92]. The objective of these simulators was only dedicated to generate synthetic fMRI data for static objects without considering any motion artifacts. Drobnjak and his colleagues have presented an advanced computational model for the fMRI acquisition process that takes into account different sources of artifacts in the fMRI signal. This model was used later in the development of an extensive software toolbox that can simulate these artifacts including rigid-body motion, magnetic field inhomogeneities, chemical shift and eddy currents [DGS∗ 06b, DGS∗ 06a]. Their simulator was dependent on three main components: (1) a mathematical model of the BOLD activation, (2) a three-dimensional
22
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FOV
10
10
10
5
m
Sproton Spin Pool
m
S✏ Spin Packet
10
6
10
3
Sj=i Multi-resolution
m
Voxel Tree 10
4
10
3
m
Voxel S0
10
3
10
1
m
Image Array
Figure 6: A description of the multi-resolution computational BOLD fMRI model presented by Chen and Calhoun. geometric representation to model the brain and (3) a model to solve Bloch equations. Further details on the development of the different models for the simulator, its numerical implementation and its validation procedures are present in [Dro07].
6
Simulation Software Applications
The main obstacle that have blocked the generalization and sharing the simulation results of fMRI to the community was the absence of flexible software architectures that could be used to unify and standardize the simulation procedures. Major contributions and research studies have created their synthetic fMRI datasets mainly relying on their in-house fMRI simulators that would not fit for other studies [MBM14, WR14]. Fortunately, during the last four years, the introduction of several open software packages have reduced this gap and
23
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met the demands of the community. This section highlights the features of two software applications that are continuously developed and released to the community on a regular basis.
6.1
fMRI Simulation Toolbox (SimTB)
Allen et al. have developed a flexible MATLAB-based fMRI simulation environment called SimTB for generating synthetic fMRI datasets and verifying a huge variety of statistical analytic methods [AEW∗ 11b, EOW∗ 12]. The underlying structure of this toolbox is based on spatio-temporal separability assumptions of independent component analysis (ICA) such that the resulting data can be expressed by the product of time courses and spatial maps. This package is capable of systematically capturing the characteristics of realistic fMRI data. The flexibility of this software package comes due to the full parametrization and user controllability of the data generation model employed to generate the synthetic datasets. SimTB is designed to allow MATLAB beginners to easily design and execute different types of simulations. For this reason, it was provided with friendly graphical user interface (GUI) and also comes with batch scripting support to automate and customize the simulation process. Figure 7 illustrates a flowchart for the data generation model in SimTB. The software package is freely downloadable at [AEW∗ 11a] and its documentation is available online at [AEW∗ 13] .
6.2
neuRosim
Welvaert et al. have developed an R-based software package for simulating fMRI data called neuRosim [WDM∗ 11]. They aimed to provide an open tool for fMRI simulation that can spark the researchers to focus their studies on further validated simulation methods for fMRI data to generalize the results of the simulation studies. The application of neuRosim can be of importance for two kinds of researchers. On one hand, it can be used to evaluate the performance of fMRI designs for particular research question. On the other hand, it is a very useful tool for theoretical modelling scientists and statisticians to validate existing and new fMRI interpretation methods based on the analysis of the generated data. Currently, there are three different response functions (Figure 8) that can be simulated in neuRosim:
24
Computational Models and Simulators of fMRI
Figure 7: Data generation sequence of synthetic fMRI datasets in SimTB. Image courtesy of Allen et al. [AEW∗ 11b] 1. A gamma-variate HRF based on the model presented by Buxton et al. [BUDL04]. This function is implemented in the procedure gammaHRF. 2. A double-gamma HRF based on the model presented by Friston et al. [FFJ∗ 98]. This function is implemented in the procedure canonicalHRF. 3. The Balloon model presented by Buxton et al. [BUDL04] is implemented in the balloon procedure. Moreover, neuRosim supports several various functions to model different types of noise in fMRI signals. 1. Drift signals based due to system noise can be simulated with the lowfreqdrift function. 2. Physiological noise due to the cardiac and respiratory artifacts is implemented in the physnoise function. 25
Computational Models and Simulators of fMRI
Figure 8: The simulated BOLD signals based on the three convolution functions implemented in neuRosim for a 20-seconds ON/OFF block design. Image courtesy of Welvaert et al. [WDM∗ 11] 3. Task-related noise due to the spontaneous neuronal activity is implemented in the tasknoise function.
7
Conclusion
The modelling and simulation of fMRI is known to be an extremely challenging research topic due to the complexity of the data and its underlying causal mechanisms. Realistic, or biophysically-plausible, simulations of fMRI are subject to the development of sophisticated models that can capture and reflect the potential signalling mechanisms that account for the generated BOLD response. This review has summarized the different modelling work, numerical simulation studies and software simulation packages of BOLD fMRI that have been
26
Computational Models and Simulators of fMRI
reported in the literature to date. The categories of the underlying models are highlighted in Figure 9.
27
Computational Models and Simulators of fMRI
fMRI Data Generation Models for Simulations
Hybrid Data Generation Models
Pure Synthetic Data Generation Models Baseline Models Constant Time Varying Neural Basis Models BOLD Activation Models
Energy Consumption and Metabolism Models
Estimated HRF Models
Blood Flow (CBF) Models
Physiologically-Plausible Models
Vasculature Dynamics (CBV) Models Noise Models Vasculature Geometric Models
Random Experiment Specific
Oxygen Extraction Models BOLD Signal Models
Figure 9: A block diagram summarizing the data generation models for fMRI simulations.
28
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