Biophysical Modeling of the fMRI signals
Jonathan R. Polimeni1,2,3
1Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA, United States, 2Department of Radiology, Harvard Medical School, Boston, MA, United States, 3Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, MA, United States

Synopsis

Keywords: Contrast mechanisms: fMRI, Neuro: Brain function

This educational lecture will review classic and modern biophysical models of the fMRI signals, with a focus on models of the BOLD response. We will consider the “balloon” model framework, which seeks to model the BOLD response within a single voxel and its nonlinearities, as well as extensions such as hierarchical models based on mass balance inspired by newly-available high-resolution fMRI that account for the coupling of hemodynamics between adjacent voxels sampling across cortical depths. We will conclude with newer approaches based on realistic Vascular Anatomical Network or VAN models informed by optical imaging measurements of microvascular anatomy and dynamics.

Syllabus

All fMRI methods in use today infer brain activity from observing the associated localized changes in blood flow, volume, and oxygenation. Because fMRI provides this indirect measure of neuronal activity through observing these hemodynamic changes, proper interpretation of fMRI data requires a model for how the underlying neural activity manifests as fMRI signal changes. Most fMRI practitioners utilize the simplest models for neural activity based on the stimulus or task paradigm timing, and models of the hemodynamics based on a convolutional kernel—the hemodynamic response function or HRF—to predict fMRI time-courses of voxels responding to the stimulus or task in order to detect activation.

While these basic models are useful for detecting neural responses in measured fMRI data in the presence of noise (Bandettini et al., 1993; Friston et al., 2000), biophysical models attempt to directly account for the physiological changes with neural activity and the physics of the fMRI signal generation. This includes modeling the neuronal signaling and metabolism, neurovascular coupling, the coordinated vascular and hemodynamic response, and their impact on relaxation rates and MRI contrast. Biophysical modeling can be used for forward modeling, used to test hypotheses about how the unobservable neural activity is captured by the observable fMRI response and determine how closely theoretical predictions match experimental observations, or for model inversion (sometimes called “deconvolution”) used to interpret the measured fMRI data or use these data to infer the amplitude, timing, or spatial distribution of neural activity. Such biophysical models have been proposed since the advent of fMRI (Bandettini and Wong, 1995; Boxerman et al., 1995; Fisel et al., 1991; Ogawa et al., 1993), however over the past decade new classes of models—motivated by the availability of high-resolution fMRI data and informed by newly-available invasive microscopy data from experimental animals—have been proposed. There is thus a spectrum of realism and practical usability in these models, with different modeling approaches suitable to different research questions.

The goal of this lecture is to review the most common approaches for biophysical modeling of the fMRI signals, with an emphasis on methods that are suited to modern high-resolution fMRI data, including BOLD and non-BOLD contrasts. The goal of this syllabus is to enumerate several key concepts that will appear in this lecture, categorize the biophysical models that have been proposed, and provide citations to recommended literature for further reading. (See the recent review article by Polimeni & Lewis (2021) for a more comprehensive bibliography; some text below is excerpted from this review article.)

Single-compartment lumped models

The original models of the fMRI signal that incorporated physiologically plausible biophysical principles were the “Balloon model” (Buxton et al., 2004, 1998; Mildner et al., 2001; Obata et al., 2004) and the related post-arteriole “Windkessel model” (Mandeville et al., 1999). These single-compartment lumped models encapsulated the full vascular network from arteries to capillaries to veins presumed to be contained within each fMRI voxel, thus modeled the hemodynamic signals at the voxel scale. These initial models mainly accounted for the steady-state and transient relationships between CBF and CBV, derived based on empirical relationships observed from data available at the time. Both the Balloon and Windkessel models postulated that observations of a delayed component of the CBV response could be explained by a compliant venous “balloon”, which could also explain the post-stimulus undershoot often observed in the BOLD response. It is important to note that the fully-developed Balloon Model framework (Buxton et al., 2004) was later expanded to include multiple individual sub-models accounting for the full cascade—from neuronal activity to the BOLD response: a model for neuronal activity, independent models for CBF and CMRO2 responses representing neuronal activity, the venous compliance or “balloon” model proper, and a model that relates changes in deoxyhemoglobin concentration in the vasculature and blood volume to the observed changes in R2* needed to generate a BOLD signal. While the venous ballooning is at the heart of the model, this ballooning is not needed to generate a realistic BOLD response per se when considering conventional slow block-design paradigms. Still, the Balloon Model framework provided many key insights into the interrelationship between the various hemodynamic components and was the first model to use biomechanical principles to explain the fMRI signal.

The Balloon Model framework can also account for the temporal nonlinearities in the transformation between the stimulus/task timing and the observed BOLD response (Buxton et al., 2004; Miller et al., 2001), and can incorporate transient decoupling between the various hemodynamic components to generate features such as the initial dip and the post-stimulus undershoot (Simon and Buxton, 2015). The Balloon Model can fit a remarkably wide variety of data with a small number of parameters, although the exact physiological interpretation of the model parameters is not straightforward (Buxton, 2012; Gagnon et al., 2016a; Griffeth and Buxton, 2011).

Multi-compartment lumped models

With the availability of invasive microscopy recordings of blood flow and volume at the individual microvessel scale (Devor et al., 2007; Hillman et al., 2007; Kleinfeld et al., 1998), the vascular compartments comprising the original lumped models could be resolved, providing the first opportunity to directly measure dynamics that had been indirectly inferred based on sound physical reasoning and coarse resolution data. With these newly available data, one assumption of the original lumped models—that there should be negligible arterial response to neuronal activity—appeared to not match the experimental data, and arteries exhibited larger-than-expected dilations in vivo (Buxton, 2009). In fact, in many cases venous CBV changes (i.e., venous ballooning) was instead found to be negligible: intracortical veins did not appear to dilate in response to short-duration stimuli (Berwick et al., 2005; Fernández-Klett et al., 2010; Hillman et al., 2007), but may only dilate in response to longer, sustained stimulation (Barrett et al., 2012; Drew et al., 2011) that presumably reflected a gradual build-up of pressure on the venous side of the vascular network. These findings inspired multi-compartment lumped models such as the “bagpipe” model that allowed for independent arterial and venous dilation (Drew et al., 2011), and other models that included distinct arterial, capillary, and venous compartments (Griffeth and Buxton, 2011; Kim et al., 2013; Kim and Ress, 2016; Zheng et al., 2005) that could better account for the new empirical findings.

Hierarchical lumped models

Most conventional lumped models consider only large fMRI voxels that each contain the entire vascular tree from arteries to veins. Motivated by the increasing availability of fMRI data with high spatial resolution, recent extensions of the multi-compartment lumped hemodynamic models explicitly account for the inter-dependence between the hemodynamics of adjacent voxels and even the hierarchy of vascular compartments within the cerebral cortical gray matter. With conventional voxel sizes, each voxel contains a similar mix of vascular elements spanning the large supplying pial surface arteries, the intracortical diving arterioles, the small pre-capillary arterioles, the capillary bed, the small post-capillary venules, the intracortical ascending venules, and the large draining pial surface veins. However, with voxel sizes far smaller than the cortical thickness, there is less heterogeneity within any given voxel, and there is more heterogeneity across voxels, as neighboring voxels will sample from different levels of the vascular hierarchy. To address this, newer hierarchical lumped models attempt to account for the differences seen in the hemodynamic responses across different cortical depths, which correspond roughly to different levels of the vascular hierarchy, and also account for the coupling of these hemodynamic responses across neighboring voxels (Havlíček and Uludağ, 2020; Heinzle et al., 2015; Markuerkiaga et al., 2016). (In a way, this shift from classic lumped models, in which each voxel is modeled in isolation, to hierarchical lumped models, in which groups of voxels are modeled jointly, is somewhat akin to the shift from univariate to multivariate analysis of fMRI data.) The increased degrees of freedom in these extended, spatially-coupled models provide additional flexibility to account for the vascular anatomy, albeit with more parameter values to determine. These models are typically built on principles of mass balance, such that blood flow and oxygen is conserved between the individual vascular inputs and outputs (Havlíček and Uludağ, 2020).

Vascular anatomical models based on random cylinders

While lumped models are often derived directly from fMRI data, there is another class of models, vascular anatomical models, that attempt to generate biophysical simulations of the fMRI signals from first principles using explicit representations of the unseen vascular anatomy within each fMRI voxel. The original such models represented blood vessels as randomly spaced and oriented infinite cylinders (Bandettini and Wong, 1995; Boxerman et al., 1995; Fisel et al., 1991; Ogawa et al., 1993; Uludağ et al., 2009) and have provided many fundamental insights into physics of the fMRI signal. These random cylinder models are used either as the basis for numerical simulations (Bandettini and Wong, 1995; Boxerman et al., 1995; Fisel et al., 1991; Ogawa et al., 1993; Uludağ et al., 2009) or analytic expressions (He and Yablonskiy, 2007; Kiselev et al., 2005; Kiselev and Posse, 1999; Troprès et al., 2001; Yablonskiy and Haacke, 1994) to calculate the BOLD fMRI signal arising from vessels with specified magnetic susceptibilities relative to tissue. Because in random cylinder models the individual vessel segments are not connected to one another they cannot be used to simulate blood flow through the vasculature or other aspects of the hemodynamic response.

Vascular anatomical network models

Vascular anatomical models based on random cylinders were later extended into vascular anatomical network (VAN) models that account for the coupling of hemodynamics between connected vascular segments. Although the hemodynamic response and BOLD signal components in the hierarchical lumped models described above are typically phrased in terms of either macro- or mesoscopic quantities that are measurable non-invasively with MRI such as CBF and CBV (Havlíček and Uludağ, 2020; Heinzle et al., 2015), the hemodynamics represented in VAN models are expressed in terms of microscopic quantities that can only be derived from optical microscopy that resolves individual microvessels such as blood velocity, vessel diameter, and the partial pressure of oxygen of blood and surrounding tissue.

Early, simplified VAN models (Boas et al., 2008) were used to account for coupled blood flow and volume dynamics within an extended vascular network covering a large brain region. Perhaps the most significant recent step for fMRI modeling was the extension of the simplified VAN model into the realistic VAN model in which a full reconstruction of the complete vascular anatomy within a voxel, based on optical microscopy data, served as the basis of the modeling (Gagnon et al., 2016a, 2015). These realistic VAN models combine explicit reconstructions of the microvascular network geometry and topology (Blinder et al., 2013) with in vivo measures of blood flow and vessel diameter both in the baseline and active states recorded across the vascular hierarchy (Tian et al., 2010; Uhlirova et al., 2016), then use computational fluid dynamics simulations to estimate blood flow throughout the reconstructed vascular network and oxygen transport through the blood and tissue (Gould and Linninger, 2015; Lorthois et al., 2011a, 2011b; Peyrounette et al., 2018; Reichold et al., 2009; Secomb et al., 2004) from which the BOLD fMRI signal can be computed numerically using techniques originally developed for random cylinder models (Bandettini and Wong, 1995; Boxerman et al., 1995; Fisel et al., 1991; Ogawa et al., 1993).

These realistic VAN models have been applied to simulate baseline physiology (Gould et al., 2017; Schmid et al., 2017) and MR signal characteristics to derive model parameters used in simpler lumped models (Báez-Yánez et al., 2017; Cheng et al., 2019; Gagnon et al., 2016a; Pfannmoeller et al., 2019; Pouliot et al., 2017). They have also been applied to modeling BOLD dynamics directly (Báez-Yáñez et al., 2020; Gagnon et al., 2016a, 2016b, 2015; Hartung et al., 2022; Pfannmoeller et al., 2020) and, recently, non-BOLD fMRI such as VASO (Genois et al., 2020, 2019). Remarkably, these models—based on microvascular anatomy and physiology—were able to predict previously undiscovered BOLD signal characteristics that were subsequently confirmed experimentally in conventional human fMRI data (Gagnon et al., 2015; Viessmann et al., 2019), demonstrating how the physics and physiology represented in these models are able to link unobservable microvascular dynamics with observable fMRI responses at the voxel level.

Outlook

Several themes emerge when considering biophysical models for fMRI data. While BOLD time-course predictions based on the HRF implicitly assume linearity, biophysical models attempt to capture the various nonlinearities present in the transformation from stimulus/task timing to neural activity, hemodynamics, and the fMRI signals. While nonlinearity adds complexity, these nonlinearities are often advantageous to fMRI practitioners since small changes in neural activity may cause large changes in the fMRI signal properties, so this can enhance discriminability of different brain functions. A common form of nonlinearity causes the timing of the hemodynamic response to vary with stimulus/task timing (Glover, 1999; Miller et al., 2001; Vazquez and Noll, 1998; Yeşilyurt et al., 2008), with stimulus/task intensity (Chen et al., 2020; Liu et al., 2010; Yeşilyurt et al., 2010), and other features of the stimulus/task configuration (Havlíček et al., 2017; Mullinger et al., 2017; Sadaghiani et al., 2009).

While some models seek to capture the behavior of the fMRI signals at the scale of a single voxel with adjustable parameter values that can be fit to data, other models such as the realistic VAN approach attempt to predict fMRI signals from “first principles” using explicit representations of the full microvascular network within each voxel, setting parameter values from independently measurable physical and physiological properties. These two approaches differ greatly in their degrees of freedom. While the fully realistic bottom-up approach is appealing, there are many unknowns that must be assumed—especially when trying to build such models for the human fMRI. The appropriate model complexity depends on the goal of the research, with no one model suitable for all applications. Having fewer degrees of freedom can be advantageous for model inversion, for example, and models with fewer free parameters may generalize better and be less prone to “over-fitting”. Nevertheless, one hopes that the simple models are compatible with the complex models, and sophisticated models such as the realistic VAN approach can guide the development of more accurate lumped models that may be easier to use in practice. Generally, models that are as simple as possible—but “no simpler”—should be favored. Finally, when applying these models to our fMRI data, it is important to keep in mind the adage “all models are wrong, but some are useful.” Even though these models inevitably make simplifying assumptions, when used properly they can provide deep insights into how to relate fMRI measurements to the underlying neural activity.

Acknowledgements

I thank Drs. Kâmil Uludağ, Jean Chen, Avery Berman, and Grant Hartung for providing materials for this presentation and for helpful discussions.

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