The pial arterial vasculature of the cerebrum consists of smaller distal arteries and arterioles that cover the cerebral cortical surface, and connects the branches of the three major supplying arteries of the cerebrum—the anterior, middle, and posterior cerebral arteries—with the penetrating intracortical arterioles, which deliver blood into the cortical grey matter (Cipolla, 2009; Jones, 1970). Notably, the pial arterial vasculature is the only source of blood supply to the neocortex (Mchedlishvili and Kuridze, 1984), and its redundancies are organized in three tiers: global re-routing of blood through the circle of Willis, intermediate anastomoses between branches originating from the three major arteries, and local loops formed by pial anastomoses within the same branch (Blinder et al., 2010; Duvernoy et al., 1981).

Although extensive anatomical studies have described topological properties and the relevant constituents of the pial arterial vasculature (Pfeifer, 1930; Szikla et al., 1977), quantitative data of the human pial arterial vasculature remain scarce (Cassot et al., 2006; Helthuis et al., 2019; Hirsch et al., 2012; Payne, 2017; Schmid et al., 2019). The by far still ‘most comprehensive and influential work’ (Hirsch et al., 2012) is the detailed description of the pial vasculature by Duvernoy et al., 1981, which examined 25 brains using intravascular ink injections. As indispensable as this dataset has been, 3D reconstructions of the vascular network and surrounding anatomy were not provided in this study. A second recent analysis performed by Helthuis et al., 2019, used corrosion casts from four brain specimens and provided valuable insights into the branching pattern of the arterial vasculature. However, only limited information can be obtained in this way about the morphometry of vessels, in particular their position and geometric relationship with the cortex. Further, the elaborate preparation and the limitation to ex vivo samples restrict the applicability of these methods when, for example, one wants to characterize the large variability across individuals in pial arterial morphometry (Alpers et al., 1959; Beevor and Ferrier, 1997; Cilliers and Page, 2017; Gomes et al., 1986; Papantchev et al., 2013; Stefani et al., 2000; van der Zwan et al., 1993) and function (Baumbach and Heistad, 1985; van Laar et al., 2006).

Given the central role of the pial arterial vasculature for healthy human brain function (Hirsch et al., 2012; Iliff et al., 2013; Xie et al., 2013), its impact on functional magnetic resonance imaging (fMRI) signals (Bright et al., 2020; Chen et al., 2020), and its involvement in numerous cerebrovascular diseases (Hetts et al., 2017; McConnell et al., 2016), there is a clear need to be able to image the pial arterial vasculature in individual subjects to characterize its morphometry and topology including arterial diameter and geometric relationship with the cortex (Mut et al., 2014). For example, many intracranial pathologies have selective involvement of superficial vessels (Ginat et al., 2013; Herz et al., 1975; Hetts et al., 2017; McConnell et al., 2016; Song et al., 2010; Uhl et al., 2003), and the outcome of stroke patients is heavily dependent on the status of the collateral circulation (Ginsberg, 2018; Raymond and Schaefer, 2017). Yet, existing hemodynamic modelling approaches have to synthesize the pial arterial vasculature either implicitly (Park et al., 2020) or explicitly (Ii et al., 2020) from an incomplete quantitative understanding of its morphometry and variability. In addition, modelling of hemodynamic changes in response to neural activity needs to account for the effect of pial vessels (Markuerkiaga et al., 2016; Polimeni et al., 2010; Uludağ and Blinder, 2018). Although signal changes in fMRI data using a blood-oxygenation-level-dependent (BOLD) contrast arise predominantly on the venous side of the vascular hierarchy (Ogawa et al., 1990), the strongest vascular response to neural activity is located in intracortical arteries and arterioles (Hillman et al., 2007; Vanzetta et al., 2005), and significant diameter changes in upstream arteries have been observed (Bizeau et al., 2018; Cho et al., 2012; Cho et al., 2008). With the recent interest in cerebral blood volume-based fMRI (Huber et al., 2014) and the numerous accounts of vascular contributions found in BOLD fMRI signals (Amemiya et al., 2020; Bright et al., 2020; Chen et al., 2020; Drew et al., 2020), including a detailed, subject-specific depiction of the underlying angio-architecture, would immensely augment forthcoming modelling initiatives (Havlicek and Uludağ, 2020; Tak et al., 2014).

So far, numerous approaches exist to image large, macroscopic proximal intracranial arteries such as the circle of Willis including its branches and the basal arteries with magnetic resonance imaging (MRI) (Carr and Carroll, 2012). Similarly, the density of microscopic, parenchymal vessels can be quantified through techniques such as vessel size imaging and capillary density imaging (Kiselev et al., 2005; Troprès et al., 2001; Xu et al., 2011), and newer methods for estimates of cerebral blood volume of the arteriolar side of the microvascular tree (Hua et al., 2019). However, direct non-invasive imaging of the mesoscopic vessels of the pial arterial network, that is, arteries with a diameter between 50 and 300 µm, is not in current use, either in clinical practice or in basic research.

Recent studies have shown the potential of time-of-flight (TOF)-based magnetic resonance angiography (MRA) at 7 Tesla (T) in subcortical areas (Bouvy et al., 2016; Bouvy et al., 2014; Ladd, 2007; Mattern et al., 2018; Schulz et al., 2016; Von Morze et al., 2007). In brief, TOF-MRA uses the high signal intensity caused by inflowing water protons in the blood to generate contrast, rather than an exogenous contrast agent. By adjusting the imaging parameters of a gradient-recalled echo (GRE) sequence, namely the repetition time (T_R) and flip angle, the signal from static tissue in the background can be suppressed, and high image intensities are only present in blood vessels freshly filled with non-saturated inflowing blood. As the blood flows through the vasculature within the imaging volume, its signal intensity slowly decreases. (For a comprehensive introduction to the principles of MRA, see for example Carr and Carroll, 2012.) At ultra-high field, the increased signal-to-noise ratio (SNR), the longer $T_1$ relaxation times of blood and grey matter, and the potential for higher resolution are key benefits (Von Morze et al., 2007). However, the current description of the magnetic resonance physics underlying TOF-MRA (Brown et al., 2014a; Carr and Carroll, 2012) is not tailored to imaging pial arteries, because their small diameter and complex branching pattern require particular considerations. For example, while the advantage of reduced voxel size for imaging small arteries has empirically been shown numerous times (Haacke et al., 1990; Mattern et al., 2018; Von Morze et al., 2007), the sentiment prevails (Chen et al., 2018; Masaryk et al., 1989; Mut et al., 2014; Park et al., 2020; Parker et al., 1991; Wilms et al., 2001; Wright et al., 2013) that the slower flow in small arteries should significantly diminish the TOF effect (Haacke et al., 1990; Pipe, 2001), and perhaps for that reason imaging the pial arterial vasculature in vivo has received little attention. Here, we revisit the topic of high-resolution TOF-MRA to investigate the feasibility of imaging the pial arterial vasculature in vivo at 7 T. We demonstrate, based on simulations and empirical data, that pial arteries can be detected reliably. Note that while some, particularly larger pial arterials, have undoubtedly been visualized incidentally in high-resolution images, we present here the first targeted, comprehensive account of imaging the pial arteries.

In the following, we first summarize the relevant properties of the pial arterial vasculature, in particular vessel diameter, blood velocity, and blood delivery time. With this in mind, we revisit the theory behind 3D TOF-MRA and derive optimal parameters for imaging these vessels. We then demonstrate the supra-linear relationship between vessel contrast and voxel size and explore the requirements for accurate vessel detection. This work argues that, from a physiological and theoretical perspective, it is indeed possible to image the pial arterial network using a TOF-based contrast. Subsequently, we present results from several experiments supporting our hypothesis and ultimately employ high-resolution TOF-MRA with prospective motion correction to image the pial arterial vasculature at 140 μm isotropic resolution.

Theory

Anatomical architecture of the pial arterial vasculature

The pial arterial vasculature consists of smaller arteries and arterioles on the cortical surface (Figure 1, left) (Duvernoy, 1999). Pial arteries range in diameter from 280 to 50 μm and connect to the smaller (<50 μm) intracortical arterioles. Similar values of 700–30 μm in pial arterial diameter were also reported by Nonaka et al., 2003b, who exclusively characterized the calcarine fissure. The recent work on the topology of the vascular system by Hirsch et al., 2012, might create the impression that mesoscopic pial vessels entirely cover the cortical surface (e.g. Figure 2 in Hirsch et al., 2012). However, the segmentation provided by Duvernoy, 1999, shows that pial cerebral blood vessels are rather sparse, and only approximately 17% of the human cortex is covered by arteries and 11% by veins. (To obtain a rough estimate of their density, we segmented the stylized drawing accompanying Figure 1 (Duvernoy, 1999); see also subsection Data analysis below.) Pial arteries exhibit a distinctive branching pattern on the cortical surface, in that an arterial branch arises at a nearly right angle from its parent vessel (Rowbotham and Little, 1965), usually with a significantly reduced diameter. In the absence of ground-truth data, this conspicuous feature constitutes an important prior that can be used as an image quality marker: simply, the more right-angled branches are detected, the higher the sensitivity to small arteries.

Figure 1

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From a cross-sectional point of view cutting through the cortex, pial arteries are embedded in the meningeal layers between the pia and arachnoid mater, that is, the subarachnoid space (Figure 1, right). Thus, their immediately surrounding tissue is not only grey matter, but a mixture of grey matter, cerebrospinal and interstitial fluid, and meningeal and glial tissue (Marin-Padilla, 2012). In the next section, we will propose a simplified partial-volume model to account for the small diameter of pial arteries. Nevertheless, one should keep in mind that the actual surrounding of pial arteries consists of a number of different tissue classes and fluids, with considerable differences in $T_1$ values, especially between grey matter, arterial blood, and cerebrospinal fluid (Rooney et al., 2007; Wright et al., 2008).

Along with their diameter, the blood velocity within pial arteries is an important factor determining the TOF contrast. However, blood velocity values in human pial arteries are difficult to obtain due to their small size. Using optical imaging in cats, Kobari et al., 1984, measured blood velocities in pial arteries ranging from 20 to 200 μm diameter. For example, in pial arteries with a diameter between 100 and 150 μm, the centreline velocity was 42.1 mm/s. Similar values were obtained by Nagaoka and Yoshida, 2006, for arterioles in the human retina using laser Doppler velocimetry, where the average centreline velocity for first-order arterioles with an average diameter of 107.9 μm was 41.1 mm/s. The only study in humans measuring blood velocity in small cerebral arteries was performed by Bouvy et al., 2016, using quantitative flow MRI. They estimated mean blood velocities of 39–51 mm/s in the basal ganglia, where arterial diameters range from 175 to 668 μm (Djulejić et al., 2015). Figure 2 provides a comprehensive overview of all blood velocity values and vessel diameters reported for pial arteries. Note that optical imaging methods (Kobari et al., 1984; Nagaoka and Yoshida, 2006) are expected to considerably overestimate the mean velocity within the vessel, whereas MRI-based measurements can severely underestimate the velocity values due to partial-volume effects (Bouvy et al., 2016; Hofman et al., 1995; Tang et al., 1993). This might explain the large discrepancies that are apparent in Figure 2 between optical and MRI-based methods. Due to the gradual reduction in vessel diameter along the vascular tree, the reported velocity values only apply to the very last branch, and faster velocities are expected along much of the vasculature. For example, blood velocities in the three major cerebral arteries exceed the velocity values depicted here by at least an order of magnitude (Lee et al., 1997; Molinari et al., 2006).

Figure 2

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The blood delivery time for a given voxel containing a vessel is the time it takes for the blood water spins to reach that voxel after they first enter the imaging slab, and is a crucial imaging parameter when estimating blood signal enhancement in TOF-MRA. To directly compute blood delivery time from blood velocity, one would need a detailed velocity profile of the blood and its path along the arterial tree together with the exact slab prescription for each scan. While this is not feasible, we can find an upper bound on the blood delivery time expected for pial arteries using estimates of arterial transit time used in arterial spin labelling (Detre et al., 1992). The arterial transit time, which includes the transport of blood water spins from the large carotid arteries in the neck through the pial arterial vasculature and the capillaries into the tissue, is estimated to be between 500 and 1500 ms in healthy grey matter (Alsop et al., 2015). Since the coverage or slab thickness in TOF-MRA is usually kept small to minimize blood delivery time by shortening the path length of the vessel contained within the slab (Parker et al., 1991), and because we are focused here on the pial vasculature, we have limited our considerations to a maximum blood delivery time of 1000 ms, with values of few hundreds of milliseconds being more likely.

In summary, targeting pial arteries places us into an imaging regime of mesoscopic vessels (50–300 μm diameter) in a complex branching pattern on a folded surface with blood velocities of 10–50 mm/s and blood delivery times of 200–700 ms. When revisiting the theory behind TOF-MRA for mesoscopic pial arteries, whose diameters are at the size of the imaging voxels or smaller, we thus need to consider partial-volume effects, and take into account that: (i) there is no preferential vessel orientation; (ii) while the blood delivery times are long relative to macroscopic arteries, they are still sufficiently short that the blood arrives in the pial arteries in a small fraction of the overall acquisition time; and (iii) the blood passes swiftly through each voxel (i.e. in a time which is short relative to commonly used repetition times).

Flow-related enhancement

Before discussing the effects of vessel size, we briefly revisit the fundamental theory of the flow-related enhancement (FRE) effect used in TOF-MRA. Taking into account the specific properties of pial arteries, we will then extend the classical description to this new regime. In general, TOF-MRA creates high signal intensities in arteries using inflowing blood as an endogenous contrast agent. The object magnetization—created through the interaction between the quantum mechanical spins of water protons and the magnetic field—provides the signal source (or magnetization) accessed via excitation with radiofrequency (RF) waves (called RF pulses) and the reception of ‘echo’ signals emitted by the sample around the same frequency. The ${\displaystyle T_1}$ contrast in TOF-MRA is based on the difference in the steady-state magnetization of static tissue, which is continuously saturated by RF pulses during the imaging, and the increased or enhanced longitudinal magnetization of inflowing blood water spins, which have experienced no or few RF pulses. In other words, in TOF-MRA, we see enhancement for blood that flows into the imaging volume. The steady-state longitudinal magnetization ${\displaystyle M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}$ of static tissue imaged with a (spoiled) FLASH sequence can be calculated (Brown et al., 2014a) using the Ernst equation:

(1) $M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}=\frac{M_0\left(1-{\mathrm{e}}^{-T_{\mathrm{R}}/T_1^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}\right)}{1-{\mathrm{e}}^{-T_{\mathrm{R}}/T_1^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}\cdot \cos \theta },$

with $M_0$ being the thermal equilibrium magnetization before any RF pulses, $T_{\mathrm{R}}$ the repetition time, $\theta$ the excitation flip angle, and $T_1^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}$ the longitudinal relaxation time of the tissue.

To generate a TOF contrast, a higher flip angle than the optimal Ernst angle (Ernst and Anderson, 1966) is chosen to decrease or suppress the tissue longitudinal magnetization relative to the thermal equilibrium value. The flowing blood water spins entering into the excited volume are fully relaxed, and as they flow through the imaging volume their longitudinal magnetization $M_z^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}$ just before the nth RF pulse is (Brown et al., 2014a):

(2) $M_z^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}\left(n_{\mathrm{R}\mathrm{F}}\right)=M_{z\mathrm{S}}^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}+{\left(\left(e^{-\frac{T_{\mathrm{R}}}{T_1^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}}}\right)\cdot \cos \theta \right)}^{n_{\mathrm{R}\mathrm{F}}-1}\cdot \left(M_0-M_{z\mathrm{S}}^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}\right).$

The number of RF pulses experienced by the blood, $n_{\mathrm{R}\mathrm{F}}=t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}}{/T}_{\mathrm{R}}$, depends on the blood delivery time $t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}}$, that is, the time it takes the blood spins from entering the imaging volume to reaching the target vessel, and the repetition time $T_{\mathrm{R}}$. Hence, the longitudinal magnetization of the blood water spins is an exponentially decaying function of the number of applied RF pulses, and inversely related to the blood delivery time. We define the FRE, which is the fundamental contrast mechanism in TOF-MRA, as the difference in blood longitudinal magnetization and surrounding tissue signal relative to the tissue signal (Al-Kwifi et al., 2002):

(3) $\mathrm{F}\mathrm{R}\mathrm{E}\left(n_{\mathrm{R}\mathrm{F}}\right)=\frac{M_z^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}\left(n_{\mathrm{R}\mathrm{F}}\right)-M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}{M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}.$

In classical descriptions of the FRE effect (Brown et al., 2014a; Carr and Carroll, 2012), significant emphasis is placed on the effect of multiple ‘velocity segments’ within a slice in the 2D imaging case. Using the simplified plug-flow model, where the cross-sectional profile of blood velocity within the vessel is constant and effects such as drag along the vessel wall are not considered, these segments can be described as ‘disks’ of blood that do not completely traverse through the full slice within one $T_{\mathrm{R}}$, and, thus, only a fraction of the blood in the slice is replaced. Consequently, estimation of the FRE effect would then need to accommodate contribution from multiple ‘disks’ that have experienced 1 to $k$ RF pulses. In the case of 3D imaging as employed here, multiple velocity segments within 1 voxel are generally not considered, as the voxel sizes in 3D are often smaller than the slice thickness in 2D imaging and it is assumed that the blood completely traverses through a voxel each $T_{\mathrm{R}}$. However, the question arises whether this assumption holds for pial arteries, where blood velocity is considerably lower than in intracranial arteries (Figure 2). To comprehensively answer this question, we have computed the blood dwell time (note that the blood dwell time is inversely related to the critical velocity in 2D MRA but does not need to assume a specific $T_{\mathrm{R}}$ value), that is, the average time it takes the blood to traverse a voxel, as a function of blood velocity and voxel size (Figure 2). For reference, the blood velocity estimates from the three studies mentioned above Bouvy et al., 2016; Kobari et al., 1984; Nagaoka and Yoshida, 2006 have been added in this plot as horizontal white lines. For the voxel sizes of interest here, that is, 50–300 μm, blood dwell times are, for all but the slowest flows, well below commonly used repetition times (Brown et al., 2014a; Carr and Carroll, 2012; Ladd, 2007; Von Morze et al., 2007). Thus, in a first approximation using the plug-flow model, it is not necessary to include several velocity segments for the voxel sizes of interest when considering pial arteries, as one might expect from classical treatments, and the FRE effect can be described by Equations 1–3, simplifying our characterization of FRE for these vessels. When considering the effect of more complex flow patterns, it is important to bear in mind that the arteries targeted here are only 1 voxel thick, and signals are integrated across the whole artery.

Based on these equations, optimal ${\displaystyle T_{\mathrm{R}}}$ and excitation flip angle values ($\theta$) can be calculated for the blood delivery times under consideration (Figure 3). To better illustrate the regime of small arteries, we have illustrated the effect of either flip angle or ${\displaystyle T_{\mathrm{R}}}$ while keeping the other parameter values fixed to the value that was ultimately used in the experiments; although both parameters can also be optimized simultaneously (Haacke et al., 1990). Figure 3—figure supplement 1 further delineates the interdependency between flip angle and ${\displaystyle T_{\mathrm{R}}}$ within a parameter range commonly used for TOF imaging at ultra-high field (Kang et al., 2010; Stamm et al., 2013; Von Morze et al., 2007). Note how longer $T_{\mathrm{R}}$ values still provide an FRE effect even at very long blood delivery times, whereas using shorter $T_{\mathrm{R}}$ values can suppress the FRE effect (Figure 3, left). Similarly, at lower flip angles, the FRE effect is still present for long blood delivery times, but it is not available anymore at larger flip angles, which, however, would give maximum FRE for shorter blood delivery times (Figure 3, right). Due to the non-linear relationships of both blood delivery time and flip angle with FRE, the optimal imaging parameters deviate considerably when comparing blood delivery times of 100 and 300 ms, but the differences between 300 and 1000 ms are less pronounced. In the following simulations and measurements, we have thus used a ${\displaystyle T_{\mathrm{R}}}$ value of 20 ms, that is, a value only slightly longer than the readout of the high-resolution TOF acquisitions, which allowed time-efficient data acquisition, and a nominal excitation flip angle of 18°. From a practical standpoint, these values are also favourable as the low flip angle reduces the SAR (Fiedler et al., 2018) and the long $T_{\mathrm{R}}$ value decreases the potential for peripheral nerve stimulation (Mansfield and Harvey, 1993).

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The optimizations presented so far have assumed that the voxel is fully filled with blood. However, in the case of pial arteries, vessel diameters are often comparable to or smaller than the voxel size. In the next section, we will therefore investigate the individual contributions of blood delivery times and voxel size to the overall FRE in pial arteries. We will introduce a model that consider the partial-volume effects of blood and tissue to allow us to decide whether imaging of pial arteries is limited by the physiological properties of the brain vasculature, that is, blood flow rates, or the available image resolution.

Introducing a partial-volume model

To account for the effect of voxel volume on the FRE, the total longitudinal magnetization $M_z$ needs to also consider the number of spins contained within in a voxel (Du et al., 1996; Venkatesan and Haacke, 1997). A simple approximation can be obtained by scaling the longitudinal magnetization with the voxel volume (Venkatesan and Haacke, 1997). (Note that this does not affect the relative FRE definition presented in Equation (3), as the effect of total voxel volume cancels out.) To then include partial-volume effects, the total longitudinal magnetization in a voxel $M_z^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ becomes the sum of the contributions from the stationary tissue $M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}$ and the inflowing blood $M_z^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}$, weighted by their respective volume fractions $V_{\mathrm{r}\mathrm{e}\mathrm{l}}$:

(4) $M_z^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}\cdot M_z^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}+V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}\cdot M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}.$

For simplicity, we assume a single vessel is located at the centre of the voxel and approximate it to be a cylinder with diameter $d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}}$ and length $l_{\mathrm{v}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{l}}$ of an assumed isotropic voxel along one side. The relative volume fraction of blood $V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}$ is the ratio of vessel volume within the voxel to total voxel volume (see section Estimation of vessel-volume fraction and Figure 4—figure supplement 1), and the tissue volume fraction $V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}$ is the remainder that is not filled with blood, or

(5) $V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}=1-V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}.$

We can now replace the blood magnetization in Equation (3) with the total longitudinal magnetization of the voxel to compute the FRE as a function of vessel-volume fraction:

(6) $\mathrm{F}\mathrm{R}{\mathrm{E}}_{\mathrm{P}\mathrm{V}}\left(n_{\mathrm{R}\mathrm{F}},l_{\mathrm{v}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{l}},d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}}\right)=\frac{M_z^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(n_{\mathrm{R}\mathrm{F}},l_{\mathrm{v}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{l}},d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}}\right)-M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}{M_{z\mathrm{S}}^{\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{e}}}.$

Note that this can also be written as $\mathrm{F}\mathrm{R}{\mathrm{E}}_{2\mathrm{C}}\left(n_{\mathrm{R}\mathrm{F}},l_{\mathrm{v}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{l}},d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}}\right)=V_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}(l_{\mathrm{v}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{l}},d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}})\cdot \mathrm{F}\mathrm{R}\mathrm{E}\left(n_{\mathrm{R}\mathrm{F}}\right)$.

Figure 4 illustrates the partial-volume FRE ($\mathrm{F}\mathrm{R}{\mathrm{E}}_{\mathrm{P}\mathrm{V}}$) assuming a pial artery diameter ($d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}}$) of 200 μm, a $T_{\mathrm{R}}$ value of 20 ms, an excitation flip angle of 18°, and longitudinal relaxation times of blood and tissue of 2100 and 1950 ms, respectively (Huber, 2014, Table 2.1). Therein, two prominent regimes appear along the voxel size axis: (i) the blood-delivery-time dominated regime, where as long as the voxel is completely filled with blood, that is, ${\displaystyle l_{\mathrm{v}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{l}}<\cos \frac{\pi }{4}\cdot d_{\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}}}$, the voxel size has no impact on the FRE, and the FRE is solely an exponential function of the blood delivery time (Equation (2), Figure 3); (ii) the voxel-size dominated regime, where if the vessel is smaller than the voxel, the FRE also depends quadratically on the voxel size (see Estimation of vessel-volume fraction). Note that our partial-volume model encompasses both regimes, and because the $\mathrm{F}\mathrm{R}{\mathrm{E}}_{2\mathrm{C}}$ definition reverts to the classical FRE definition for larger-than-voxel-sized vessels, we will from now on use FRE synonymously with $\mathrm{F}\mathrm{R}{\mathrm{E}}_{2\mathrm{C}}$. In theory, a reduction in blood delivery time increases the FRE in both regimes, and—if the vessel is smaller than the voxel—so would a reduction in voxel size. In practice, a reduction in slab thickness—which is the default strategy in classical TOF-MRA to reduce blood delivery time—might not provide substantial FRE increases for pial arteries. This is due to their convoluted geometry (see section Anatomical architecture of the pial arterial vasculature), where a reduction in slab thickness may not necessarily reduce the vessel segment length if the majority of the artery is still contained within the smaller slab. Thus, given the small arterial diameter, reducing the voxel size is a much more promising avenue when imaging the pial arterial vasculature.

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Inflow artefacts in sinuses and pial veins

Inflow in large pial veins and the sagittal and transverse sinuses can cause FRE in these non-arterial vessels. One common strategy to remove this unwanted signal enhancement is to apply venous suppression pulses during the data acquisition, which saturate bloods spins outside the imaging slab. Disadvantages of this technique are the technical challenges of applying these pulses at ultra-high field due to constraints of the specific absorption rate (SAR) and the necessary increase in acquisition time (Conolly et al., 1988; Heverhagen et al., 2008; Johst et al., 2012; Maderwald et al., 2008; Schmitter et al., 2012; Zhang et al., 2015). In addition, optimal positioning of the saturation slab in the case of pial arteries requires further investigation, and in particular supressing signal from the superior sagittal sinus without interfering in the imaging of the pial arteries vasculature at the top of the cortex might prove challenging. Furthermore, this venous saturation strategy is based on the assumption that arterial blood is traveling head-wards while venous blood is drained foot-wards. For the complex and convoluted trajectory of pial vessels, this directionality-based saturation might be oversimplified, particularly when considering the higher-order branches of the pial arteries and veins on the cortical surface.

Inspired by techniques to simultaneously acquire a TOF image for angiography and a susceptibility-weighted image (SWI) for venography (Bae et al., 2010; Deistung et al., 2009; Du et al., 1994; Du and Jin, 2008), we set out to explore the possibility of removing unwanted venous structures from the segmentation of the pial arterial vasculature during data post-processing. Because arteries filled with oxygenated blood have ${\displaystyle T_2{}^{\ast }}$ values similar to tissue, while veins have much shorter ${\displaystyle T_2{}^{\ast }}$ values due to the presence of deoxygenated blood (Pauling and Coryell, 1936; Peters et al., 2007; Uludağ et al., 2009; Zhao et al., 2007), we used this criterion to remove vessels with short ${\displaystyle T_2{}^{\ast }}$ values from the segmentation (see Data analysis for details). In addition, we also explored whether unwanted venous structures in the high-resolution TOF images—where a two-echo acquisition is not feasible due to the longer readout—can be removed based on detecting them in a lower-resolution image.

Velocity- and T_E-dependent vessel displacement artefacts

While isotropic 3D TOF-MRA provides detailed information on the vessel position, the continuous motion of blood during image acquisition can lead to vessel displacement artefacts in the phase-encoding directions (Brown et al., 2014b; Parker et al., 2003). The magnitude of this misregistration artefact depends on the time elapsed between the time of the phase-encoding blip ($t_{\mathrm{p}\mathrm{e}}$) and the echo time ($T_{\mathrm{E}}$) and on the blood velocity in the primary phase-encoding direction $y$, $v_y$:

(7) $\mathrm{\Delta }{y}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}}=-v_y\cdot \left(T_{\mathrm{E}}-t_{\mathrm{p}\mathrm{e}}\right).$

A similar relationship for $\mathrm{\Delta }{z}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}}$ exists for the secondary phase-encoding direction $z$. According to this relationship, and based on the reported velocity range in Figure 2, vessel displacements in the phase-encoding directions between 25 and 250 μm for a time delay $\left(T_{\mathrm{E}}-t_{\mathrm{p}\mathrm{e}}\right)$ of 5 ms can be expected. Flow compensation gradients can correct for the effect of constant velocities (Parker et al., 2003), but the required large gradient lobes for high-resolution applications would substantially increase the $T_{\mathrm{E}}$, and, consequently, reduce the SNR. For these reasons, no flow compensation in the phase-encoding-directions was used for the high-resolution images in this study. However, we do assess the extent of vessel displacement using a two-echo low-resolution acquisition.

Considerations for imaging the pial arterial vasculature

We have identified voxel size as the key parameter for imaging pial arteries. While the benefit of smaller voxels was conceptually known and empirically shown before (Haacke et al., 1990; Mattern et al., 2018; Von Morze et al., 2007), clear guidelines on how small is sufficiently small and how strongly the FRE increases with decreasing voxel size were not established. We can now conclude that the voxel size should be at least the size of the pial artery, that is, somewhere between 50 and 280 μm in our case, and ideally around 70% of that, that is, 35–200 μm. While this is challenging in practice, the strong increase in FRE especially when voxel sizes are similar to vessel size makes a reduction of even only 10 or 20 μm worthwhile for imaging pial arteries. Further, we have seen that slow blood velocities are an unlikely explanation for loss of FRE in small vessels, because (i) blood delivery times to pial arteries are sufficiently short due to the perhaps surprisingly fast blood flow in these mesoscopic vessels, (ii) the effect of blood delivery time on FRE decreases with increasing blood delivery times, and (iii) the addition of partial-volume effects to the FRE estimation predicts the often observed drastic loss of contrast in small vessels. Thus, we expect a reduction in voxel size to directly translate into an increased sensitivity for small pial arteries.

The anonymized imaging data presented in this manuscript are stored in OSF (OSF, Center for Open Science, Inc, Charlottesville, Virginia, USA) accessible via https://doi.org/10.17605/OSF.IO/NR6GC.

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Author details

1. Saskia Bollmann

   1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, United States
   2. Department of Radiology, Harvard Medical School, Boston, United States
   3. Centre for Advanced Imaging, The University of Queensland, Brisbane, Australia

   Contribution

   Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Visualization, Writing - original draft, Writing – review and editing

   For correspondence

   saskia.bollmann@cai.uq.edu.au

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-8242-8008

2. Hendrik Mattern

   Department of Biomedical Magnetic Resonance, Institute of Experimental Physics, Otto-von- Guericke-University, Magdeburg, Germany

   Contribution

   Formal analysis, Investigation, Methodology, Software, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5740-4522

3. Michaël Bernier

   1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, United States
   2. Department of Radiology, Harvard Medical School, Boston, United States

   Contribution

   Conceptualization, Software, Writing – review and editing

   Competing interests

   No competing interests declared

4. Simon D Robinson

   1. Centre for Advanced Imaging, The University of Queensland, Brisbane, Australia
   2. High Field MR Centre, Department of Biomedical Imaging and Image-guided Therapy, Medical University of Vienna, Vienna, Austria
   3. Karl Landsteiner Institute for Clinical Molecular MR in Musculoskeletal Imaging, Vienna, Austria
   4. Department of Neurology, Medical University of Graz, Graz, Austria

   Contribution

   Resources, Software, Writing – review and editing

   Competing interests

   No competing interests declared

5. Daniel Park

   Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, United States

   Contribution

   Resources, Software, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-7263-9327

6. Oliver Speck

   1. Department of Biomedical Magnetic Resonance, Institute of Experimental Physics, Otto-von- Guericke-University, Magdeburg, Germany
   2. German Center for Neurodegenerative Diseases, Magdeburg, Germany
   3. Center for Behavioral Brain Sciences, Magdeburg, Germany
   4. Leibniz Institute for Neurobiology, Magdeburg, Germany

   Contribution

   Funding acquisition, Methodology, Project administration, Resources, Writing – review and editing

   Competing interests

   No competing interests declared

7. Jonathan R Polimeni

   1. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, United States
   2. Department of Radiology, Harvard Medical School, Boston, United States
   3. Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, United States

   Contribution

   Conceptualization, Funding acquisition, Project administration, Resources, Supervision, Writing - original draft

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1348-1179

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