What tools and platforms are used for Equivalent Single Layer Ansys?

Microcontrollers, specialized sensors, circuit design software (Proteus/MATLAB), and standard simulation platforms for Equivalent Single Layer Ansys.

Do you provide hardware, simulation code, and project reports for Equivalent Single Layer Ansys?

Yes. Full simulation packages, source code/RTL, circuit schematics, university-format report, presentation slides, and viva Q&A are provided for Equivalent Single Layer Ansys.

Is this Equivalent Single Layer Ansys project suitable for academic submission?

Yes, all project documentation for Equivalent Single Layer Ansys is prepared according to VTU, Anna University, JNTU, and autonomous college evaluation guidelines.

Equivalent Single Layer Ansys Project | Equivalent Single Layer Ansys Design and Implementation | Equivalent Single Layer Ansys IEEE Project

Formation

Michael MacDonald* · Marcin J.

Received: Dd Month Year / Accepted: Dd Month Year

Abstract We investigate the effects of condensation and liquid water loading on the stably stratiﬁed surface layer, with an eye towards understanding the inﬂuence of turbulent mixing on fog formation. Direct numerical simulations (DNS) of dry and moist open channel ﬂows are conducted, where in both a constant cooling rate is applied at the ground to mimic long- wave radiative cooling. Depending on the cooling rate, it can lead to either turbulent (weakly stable) or laminar (very stable) ﬂows. Compared to the completely dry case, the condensa- tion of liquid water in the moist case enables slightly higher cooling rates to be achieved before leading to turbulence collapse. In the very stable cases, runaway cooling leads to the substantial condensation of liquid water close to the ground and fog (visibility less than 1 km) results over much of the domain. In the weakly stable cases, turbulent mixing narrowly yields visibilities of 1 km close to the ground over a similar time period. However, despite the idealized nature of the system, the present results suggest that turbulence impedes, al- though will not necessarily inhibit, fog formation. A possible mechanism for fog formation within turbulent ﬂows is identiﬁed, wherein regions of increased liquid water content form within the low-speed streaks of the near-wall cycle. These streaks are energized in the moist cases due to reduced dissipation of turbulence kinetic energy compared to the dry case, al- though in both cases the streaks are less energetic and persistent than in neutrally stratiﬁed ﬂow.

edged. Keywords Direct numerical simulation · Fog formation · Stable surface layer

Ntroduction

Fog is an important meteorological phenomenon that can have wide-reaching inﬂuences on human lives, transportation, and the economy (Gultepe et al. 2007). In the case of radiation fog, it forms within the stably stratiﬁed atmospheric boundary layer (SBL), wherein suf- ﬁcient surface cooling leads to the air reaching its saturation point. Typically, this cooling occurs during nocturnal clear-sky conditions due to longwave radiation. Fog is deﬁned when the resulting suspension of water droplets reduces visibility to below 1 km, while visibilities above this threshold but below approximately 11 km are called mist (NOAA 2017). Despite the importance and several decades of extensive research on fog, accurate forecasts remain challenging (Steeneveld et al. 2015). This is in part due to a lack of understanding of some of the fundamental physical processes involved in fog formation and the early stages of its growth (Gultepe et al.

Taylor (1917) provided a simple thermodynamic analysis of fog formation using a Clausius–Clapeyron diagram, in which cooling, mixing, and moistening are the three prin- ciple processes involved. All three processes may occur simultaneously during fog forma- tion, although mixing alone is often not signiﬁcant enough to cause fog (Teixeira 1999).

Moreover, the role of turbulence and mixing on fog formation has often received a range of interpretations (Gultepe et al. 2007). One hypothesis (e.g. Brown and Roach 1976; Roach et al. 1976) is that very low, or absent, winds are necessary for radiation fog to form, as otherwise the vertical mixing produced by turbulence draws drier, warmer air from aloft and prevents the air from reaching its saturation point. An alternative view (e.g. Rodhe 1962; Welch et al. 1986; Duynkerke 1999) is that turbulent mixing is essential, as it combines air- masses of different humidity and temperature such that saturation is achieved. Furthermore, both mechanisms might be responsible for fog formation depending on the speciﬁc condi- tions. Here, we use direct numerical simulations (DNS) of a moist, stably stratiﬁed surface layer undergoing cooling as an idealized system to study the fundamental mechanisms and relationship between turbulent mixing and fog formation.

Recently, large-eddy simulations (LES) of fog have had some success in simulating the main characteristics and qualitative behaviour of the fog life cycle in stably stratiﬁed envi- ronments (Nakanishi 2000; Porson et al. 2011; Bergot 2013). Further LES studies (Bergot 2016; Maronga and Bosveld 2017; Mazoyer et al. 2017) have attempted to quantify the ef- fect of turbulence on fog formation and its life cycle. However, a core uncertainty here is the ability of LES to simulate stably stratiﬁed ﬂows, especially when the ground cooling is sufﬁcient to lead to partial or complete turbulence collapse. Ideally, LES resolves the largest, energy-containing eddies and uses a subgrid-scale (SGS) model to account for the dissipative actions of the smallest eddies. In stably stratiﬁed ﬂows, however, the largest ed- dies become suppressed due to the effort of drawing up heavier (cooler) ﬂuid from below and pulling down lighter (warmer) ﬂuid from above. With increasing stability the largest energy-containing scales approach the grid size, potentially leading the SGS model to pre- dict near-zero turbulent ﬂuxes (Chung and Matheou 2014). Alternatively, some models ar- tiﬁcially predict excessive mixing (de Roode et al. 2017). While there are SGS models that can account for the effects of buoyancy in the weakly stable regime (e.g. Lilly 1962; Dear- dorff 1980; Moeng 1984; Bou-Zeid et al. 2010; Chung and Matheou 2014), there are still challenges when turbulence collapse is considered.

Further complications are introduced to considering fog when the non-linear mixing pro- cesses during fog formation occur at small length scales. For example, the buoyancy length scale, Lb = wrms/N, gives the level of suppression of vertical motions due to the sta- DNS of the Moist SBL: Turbulence and Fog Formation ﬂuctuations and N is the buoyancy frequency. From the ﬁeld observations of Price (2011), fog occurred with N ≈0.08 s−1 and the vertical velocity ﬂuctuations were near zero with wrms ≲0.1 m s−1. This therefore leads to Lb ∼1.2 m. Similarly, the Ozmidov length,

P

ϵ/N 3, gives the smallest scale inﬂuenced by buoyancy, where ϵ is the turbulence dissipation. For typical values of ϵ ∼10−3 m2 s−3 then LOz is of the same order as Lb. Recent high-resolution ﬁeld observations have also noted substantial temperature gradients close to the ground during fog formation with N ≈0.4 s−1 (Izett et al. 2019), suggesting LB and LOz may be even smaller. Ultimately, this implies that LES, with horizontal grid spacings of several metres and at best vertical resolutions of 1 m (e.g. Bergot 2013; Maronga and Bosveld 2017), may not properly resolve the small-scale mixing in fog formation. For this reason, we will use DNS of the moist SBL in the present study, which to our knowl- edge is the ﬁrst time such a technique has been used to study an idealized representation of fog formation. DNS solves the Navier–Stokes equations, directly resolving the dissipative Kolmogorov length scale with no turbulence parametrization.

The structure of the dry SBL was described by Monin (1970) as follows. The overall boundary layer, of height δ, is split into two regions, with the lowermost region termed the surface layer. Here, Coriolis effects can be neglected and the thickness of the surface layer, h, is on the order of tens of metres. The surface layer is further divided, wherein buoyancy forces can be neglected in the so-called dynamic sublayer, with thickness on the order of the Obukhov length, L. The buffer layer (z ≪L) exists close to the ground and accounts for viscous effects in the case of a smooth surface, or roughness effects otherwise. Above the buffer layer, viscosity or roughness becomes irrelevant and the only remaining length scale is the distance to the ground, z. This yields the familiar logarithmic mean velocity proﬁle, as in neutrally stratiﬁed wall-bounded turbulence. The ﬂow remains turbulent so long as the ground cooling is sufﬁciently weak, such that L is large, in the so-called weakly stable regime (Mahrt 1999). However, under sufﬁciently strong ground cooling, L becomes small and turbulence can collapse completely. This results in a laminar ﬂow with so-called runaway cooling (Van de Wiel et al. 2007), and is termed the very stable regime (Mahrt 1999).

The dry SBL has been simulated in a variety of conﬁgurations using DNS. The surface layer can be simulated using an open channel ﬂow of height h, driven by a constant pressure gradient. Two alternative approaches are typically used, in which either a constant cooling ﬂux is applied to the ground and the transient response is studied (e.g. Nieuwstadt 2005; Flores and Riley 2011), or a constant temperature difference is applied between the bottom and top boundaries yielding a statistically steady ﬂow (e.g. Garc´ıa-Villalba and del ´Alamo 2011). The Coriolis force is neglected and low-level jets and other large-scale features of the SBL are not observed. Simulations of the Ekman layer under stable stratiﬁcation, mean- while, attempt to represent the full SBL and include the Coriolis force (Shah and Bou-Zeid 2014; Ansorge and Mellado 2014; Gohari and Sarkar 2017). While some differences be- tween Ekman layers and open channel ﬂows were reported in the outer-layer of the ﬂow (where z ∼h), Ansorge and Mellado (2014) and Flores and Riley (2018) explicitly com- pared the near-wall region (buffer and logarithmic layers) of the two ﬂows. They showed that the logarithmic velocity proﬁle and the turbulence kinetic energy (TKE) budget were com- parable in this region, suggesting that the outer-layer does not have a signiﬁcant inﬂuence on the near-wall ﬂow.

While the Reynolds numbers of DNS are relatively low, appropriate non-dimensionalization of turbulent ﬂows often exhibits Reynolds number similarity scaling. In particular, Ansorge and Mellado (2014) demonstrated that the velocity proﬁles, TKE budget, and intermittency factor do not vary signiﬁcantly with Reynolds number in the neutrally stratiﬁed Ekman Michael MacDonald* et al.

layer. Furthermore, the stably stratiﬁed surface-layer simulations of Flores and Riley (2011) demonstrated that the time evolution of the total mass ﬂux and density gradient at the ground were similar for different Reynolds numbers with matched cooling rates. Similar validation will be performed here, which will enable extrapolation and comparison of the results of the present study at relatively low Reynolds numbers to those found in the atmosphere.

In this paper we perform DNS of both dry and moist horizontally homogeneous open- channel ﬂows, in which the ﬂow is initialized from a dry turbulent neutrally stratiﬁed ﬂow and a constant cooling ﬂux is then applied to the ground. This represents the surface layer just after sunset undergoing radiative cooling (as in Nieuwstadt 2005), wherein the resulting condensation and small-scale turbulent mixing are studied in the context of fog formation.

The numerical procedure and simulation set-up are described and validated in Sect. 2. Par- ticular attention is given in Sect. 2.2 to how this system can be treated as an idealized repre- sentation of the early stages of fog formation. Results are presented in Sect. 3, including an analysis of the fog development (Sect. 3.2) and of the turbulence collapse in dry and moist ﬂows (Sect. 3.3). Finally, conclusions are offered in Sect. 4.

Governing Equations And Simulation Set-Up

We simulate a horizontally homogeneous open-channel ﬂow driven by a constant pressure gradient and an imposed ground-cooling ﬂux. This represents the same set-up as Nieuw- stadt (2005) and Flores and Riley (2011), although in addition we include moisture effects to study the inﬂuence of condensation of liquid water. The incompressible Navier–Stokes equations with Boussinesq approximation are solved in Cartesian coordinates, along with transport equations for temperature, T, and water vapour mixing ratio, qv. A bulk approxi- mation for the liquid water mixing ratio, ql is used, yielding

(5)

where u = (u, v, w) is velocity in the streamwise (or mean ﬂow, x), spanwise (y) and ver- tical (z) directions; π is the pressure perturbation; ρ is the constant air density; t is time; i and k are the unit vectors in the streamwise and vertical directions, respectively; dP/dx is the imposed constant streamwise pressure gradient that drives the ﬂow; ν is the constant molecular kinematic viscosity; νT and νv are the thermal and water vapour molecular diffu- sivities, which are constant and related to viscosity through the molecular Prandtl number, Prm = ν/νT = ν/νv = 0.71; and Cd is the condensation rate, which maintains thermo- dynamic equilibrium with zero supersaturation (Grabowski and Smolarkiewicz 1990). B is

(6)

DNS of the Moist SBL: Turbulence and Fog Formation where g = 9.81 m s−2, T0 and qv0 are the reference temperature and water vapour mixing ratio, respectively, and ϵ+1 = Rv/Rd ≈1.61 is ratio of the gas constants for water vapour and dry air. The latent heat of condensation is Lv = 2.5 × 106 J kg−1 and speciﬁc heat at constant pressure is cp = 1005 J kg−1 K−1.

The governing equations above are solved using the ﬁnite volume, non-hydrostatic anelas- tic model EULAG, broadly documented in the literature (Grabowski and Smolarkiewicz 1990; Smolarkiewicz and Margolin 1997, 1998; Grabowski and Smolarkiewicz 2002; An- drejczuk et al. 2004; Kurowski et al. 2014), with a review in Prusa et al. (2008). The second- order accurate Eulerian (ﬂux form) mode of EULAG is used in the present work. The present bulk approximation for ql, along with a more detailed microphysical scheme that models supersaturation and the size dependence of multiple cloud droplets on sedimentation and evaporation, have previously been tested with EULAG in Andrejczuk et al. (2004, 2006, 2009). There, the authors performed DNS of cloud ﬁlaments embedded within decaying turbulence in a triply periodic domain. They only observed moderate differences between the bulk and detailed microphysics schemes, indicating that the bulk liquid water scheme used for the present DNS of the moist SBL is likely adequate. A similar bulk scheme has also been used in DNS studies of stratocumulus cloud tops (Mellado 2010; Mellado et al.

2010, 2014) as well as in the LES study of radiation fog in Nakanishi (2000). Simulations are conducted in an open channel, consisting of periodic boundary condi- tions in the lateral directions. The lower wall (or ground, subscript g) at z = 0 consists of no- slip (u = v = 0) and impermeability (w = 0) conditions with an imposed constant (cool- ing) heat ﬂux (Hg < 0, described in Sect. 2.3) and zero total water ﬂux (d(qv+ql)/dz = 0).

The upper boundary at z = h is a free-slip (du/dz = dv/dz = 0) impermeable wall with zero total water ﬂux and constant temperature (T −T0 = 0). Both dry (only Eqs. 1–3, with no water vapour and liquid water) and moist (Eqs. 1–5) ﬂows are simulated with the same imposed ground heat ﬂux, so that the effects of liquid water can be analyzed independently of the ground cooling.

A snapshot from the turbulent quasi-stationary neutrally stratiﬁed dry case is used to initialize the ﬂow, after which the ground cooling ﬂux is applied. The initial thermodynamic state is set to air at standard atmospheric pressure (1013 hPa) with density ρ = 1.265 kg m−3 and viscosity ν = 1.38×10−5 m2 s−1. The temperature is initialized as T0 = 279.15 K everywhere, although a case with higher temperature of T0 = 285.15 K is also simulated and shows little difference (see Appendix 2). The initial relative humidity is set to RH0 = 99.9% throughout the domain, corresponding to qv0 ≈6.24 g kg−1. This large value of RH0 is used as lower values are computationally inefﬁcient at reaching saturation, and is discussed further in Sect. 2.2. The air is initialized with no liquid water, ql0 = 0.

In subsequent sections, temporal averaging is denoted with an over bar, ·, while spatial averaging in the horizontal plane is denoted by angled brackets, ⟨·⟩. Velocity ﬂuctuations are deﬁned based on the difference between the instantaneous, spatially dependent velocity and its spatially averaged velocity at a given vertical location and time, u′(x, y, z, t) =

Applicability To Fog Formation

The simulation set-up detailed above enables us to study the inﬂuence of moisture on the stable surface layer. More generally however, this system can also be treated as an ideal- ized representation of fog formation within the atmosphere. Here, we are isolating just the Michael MacDonald* et al.

inﬂuence of the small-scale turbulent mixing on fog formation over the relatively short tur- bulence time scales. As such we deliberately neglect other competing processes; some of the key assumptions and limitations of this idealized system in the context of fog formation are discussed below.

The present bulk liquid water approximation assumes that the disperse liquid phase, ql, can be modelled as a continuum. As discussed in Mellado et al. (2010), this condition is gen- erally not met at the tops of stratocumulus clouds, as the cloud droplets of diameter d ≈10 µm and number density Nd = 1000 cm−3 are too sparse within a volume of the order of the Kolmogorov length scale, lη = 1 mm (i.e., that used in DNS). A similar problem exists for fog, as the Kolmogorov length scale is of the same order, however the droplets are even sparser with Nd ≈10–100 cm−3 and diameter 1–10 µm (Roach et al. 1976; Price 2011).

In addition, we assume the liquid water diffusivity is equal to that of water vapour. We also assume thermodynamic equilibrium, wherein phase changes occur instantaneously to maintain saturation through the condensation rate Cd. These assumptions may not hold for fog formation and, furthermore, predicting their impact on the present results is not trivial.

However, they form useful approximations which are commonly employed in DNS stud- ies of cloudy boundaries, notably stratocumulus cloud tops (e.g., Mellado 2010; Mellado et al. 2014; de Lozar and Mellado 2015, 2017). As this is a dynamically similar problem to fog, which involves the mixing of saturated and unsaturated air within a stably stratiﬁed environment, we employ the same assumptions in the present study.

Direct numerical simulation of the top of stratocumulus clouds have also shown radia- tion and droplet sedimentation to be an important process governing the mixing of clear and cloudy air (de Lozar and Mellado 2015, 2017). While stratocumulus cloud tops are a dy- namically similar problem, fog in its early formation stages is unlikely to be optically thick enough for radiation effects to be signiﬁcant. It is only once the fog is a few metres thick (i.e. on the scale of the present domain) that radiation is thought to become important (Oliver et al. 1978), wherein the radiative cooling enhances fog growth. In the present simulations the liquid water mixing ratio is typically less than 0.01 g kg−1 and so we neglect radiation effects. For simplicity, we also neglect droplet sedimentation effects, similar to the DNS study of stratocumulus tops by Mellado et al. (2014), which only considered shear and evap- orative cooling in isolation. Moreover, it has been suggested that droplet sedimentation and radiative cooling are strongly dependent on one another, based on simple one-dimensional fog models (Brown and Roach 1976; Bott et al. 1990) and studies of individual cloud or fog droplets (Roach 1976; Barkstrom 1978). This suggests that both processes would need to be modelled together.

The present simulations neglect any moisture ﬂux through the ground and more broadly there are no soil-vegetation effects, which are important in fog formation (Gultepe et al. 2007). Additional simulations were run with an imposed downwards moisture ﬂux, repre- senting dew deposition and hygroscopic absorption. The magnitude of this moisture ﬂux was approximately equivalent to 20 g m−2 hr−1, similar to that observed during one of the fog formation cases in the ﬁeld study of Price and Clark (2014). However, in the present system this caused a signiﬁcant drying inﬂuence and completely inhibited all condensa- tion. Similarly, reducing the initial relative humidity to a uniform value of 99.5% (with zero moisture ﬂux boundaries) to enable more realistic water vapour gradients to develop also inhibited condensation. These two results are presumably due to a lack of moisture enter- ing the system, so that the ground cooling alone is then insufﬁcient to lead to condensation within reasonable simulation times. The present simulations therefore use the simpler con- ﬁguration of zero moisture ﬂux and a high initial relative humidity, rather than attempting to specify the complex moisture ﬂuxes that occur with the ground or the atmosphere (either DNS of the Moist SBL: Turbulence and Fog Formation due to horizontal heterogeneity or from higher up in the boundary layer) at such small scales necessary for DNS.

Scaling Variables

The characteristic velocity scale within the surface layer is the friction velocity (Monin 1970). Following Flores and Riley (2011), we distinguish between the (constant) friction velocity obtained from the imposed driving pressure gradient, U 2

Τ(T) = Τw/Ρ = Νd⟨U⟩/Dz|G. The Latter Can

vary with time due to accelerations in the bulk velocity, ub(t) = (1/h)

⟨U⟩Dz, Caused

by the ground cooling. For the neutrally stratiﬁed dry cases, a statistically steady state is obtained such that the bulk velocity is approximately constant over time and uτ = U⋆. As such, U⋆can be regarded as the reference friction velocity of the neutrally stratiﬁed case, which is matched for all simulations with the same Re⋆.

The characteristic temperature, water vapour, and length scales will be related to Hg < 0, the imposed (cooling) heat ﬂux that is applied to the ground at t = 0. This heat ﬂux is matched between respective dry and moist cases. In response, sensible and, when saturation occurs, latent heat ﬂuxes will be directed toward the surface. These ﬂuxes, when horizontally

(8)

In meteorology, the conduction (gradient) terms of temperature and water vapour are often ignored as they are considered negligible relative to the turbulent ﬂuxes. However, as we are resolving the viscous sublayer with impermeability constraint in the present DNS, w(z = 0) = 0, the gradient terms become signiﬁcant at the ground (Businger 1982) whereas the turbulent ﬂuxes vanish. As such, the imposed ground heat ﬂux can be seen to prescribe the

(10)

In the SBL, the characteristic length scale is taken to be the Obukhov length, L, as dis- cussed in Sect. 1. This can be interpreted as the height at which buoyancy effects dominate over mechanical (shear) production of turbulence kinetic energy (Stull 1988). The Obukhov length is a function of the ground cooling and is deﬁned as

(11)

where κ = 0.41 is the von K´arm´an constant, included for historical reasons. In LES models and ﬁeld studies the heat ﬂux at the ground is often taken to be the vertical turbulent heat ﬂux, ⟨w′T ′⟩, at the lowest level. However, as mentioned above, these ﬂuxes vanish at the ground so that Eq. 10 is used to deﬁne Hg. The ratio between the surface layer height (or channel half height) and the (imposed) Obukhov length, h/L, is often reported in stably Michael MacDonald* et al.

Fig. 1

Cross-section of the streamwise–vertical plane showing an instantaneous snapshot of the liquid water mixing ratio for case 395M06. The saturation interface, δf, where ql > 0, varies in space and grows in time as the ground continues to be cooled through the imposed heat ﬂux, Hg. Variations in the Bowen ratio, β, are shown on the right stratiﬁed DNS studies; when h/L ≲1 the cooling is relatively weak while h/L ≳1 corresponds to stronger cooling. Note, however, that the ratio h/L is not a suitable measure for turbulence collapse as it is Reynolds number dependent (Nieuwstadt 2005; Flores and Riley 2011). In addition, there are two characteristic length scales in neutrally stratiﬁed wall turbulence, namely h, the outer-layer or large-scale length scale and ν/U⋆, the inner-layer, near-ground or viscous length scale. Both of these will be used here; superscript + indicates non-dimensionalization on ν and U⋆. For example, z+ = zU⋆/ν is the vertical position non-dimensionalized on the viscous length scale.

The horizontally averaged Bowen ratio, β(z, t) = Hs/Hl, gives the ratio between the sensible and latent heat ﬂuxes. At a given vertical height, z, there could exist regions of unsaturated air (Hl = 0) alongside regions of non-zero liquid water mixing ratios (Hl > 0).

Hence, as with the heat ﬂuxes, the Bowen ratio is a function of both vertical position and time. When a vertical level is completely saturated, β will tend towards the equilibrium Bowen ratio βe due to the use of the bulk condensation model that maintains thermodynamic equilibrium (Sect. 2.1). Based on the present thermodynamic state, this can be estimated as βe = (cp/Lv)/(∂qsat/∂T) ≈0.92 (Stull 1988). Meanwhile, in the dry cases and moist cases where the air is completely unsaturated at a given z, there is no latent heat ﬂux and

/Β = 0 (Cf. Fig. 1)

The instantaneous friction temperature and water vapour can now be deﬁned as

(13)

Crucially, these can not be known a priori as they depend on the evolution of β(z, t) (or the saturation interface δf) and instantaneous wall-shear stress through uτ(t). These quan- tities vary with height and time, which can be interpreted using the internal boundary layer framework (Elliott 1958; Panofsky and Townsend 1964). For example, after a step change in roughness, the ﬂow above the newly formed internal boundary layer (IBL) still depends on the friction velocity of the original surface upstream, while that within the IBL scales with the friction velocity of the new surface. In the present moist simulations, the ﬂow at the top of the domain when unsaturated would depend on Tτ deﬁned with 1/β = 0, as the information regarding the saturated condition has not yet reached this height. The ﬂow close to the ground, where liquid water has condensed, would meanwhile depend on Tτ deﬁned DNS of the Moist SBL: Turbulence and Fog Formation Table 1 Details of the simulations performed. Nruns is the number of runs using unique initialization snapshots from the dry, neutrally stratiﬁed case

Turbulent

with β →βe ≈0.92. Note that rather than growing spatially, as in conventional IBL de- pictions, the saturation interface grows temporally due to the periodic boundary conditions in the streamwise direction, which imposes horizontal homogeneity (Mellado 2012; Kozul et al. 2016). We can deﬁne a constant friction temperature and water vapour, analogous to the U⋆friction velocity, using the equilibrium Bowen ratio,

(15)

however this assumes fully saturated conditions across the entire domain. The value of Tτ and qτ would therefore converge to T⋆and q⋆(for all z), although only when t becomes sufﬁciently large.

Escription Of Cases

Table 1 details the different simulations conducted. We independently vary the friction Reynolds number, Re⋆= U⋆h/ν and Obukhov length, L. Cases are referred to with an ID of their friction Reynolds number, whether the case is dry or moist, and the ratio between channel half height and Obukhov length. For example, case 395M06 is a moist case per- formed at Re⋆= 395 with h/L = 0.6. A constant grid-spacing is used in the horizontal directions, with ∆x+ = ∆xU⋆/ν ≈9.7 and ∆y+ ≈4.8. A hyperbolic tangent grid stretch- ing is used in the vertical direction (Moin and Kim 1982), resulting in grid spacings at the

Z=H ≈7.0, Respectively. These Grid

spacings are in good agreement with those used in previous stably stratiﬁed DNS studies (e.g. Flores and Riley 2011; Garc´ıa-Villalba and del ´Alamo 2011; Gohari and Sarkar 2017). The DNS studies of Ansorge and Mellado (2014) and Shah and Bou-Zeid (2014), mean- while, had matched streamwise and spanwise grid spacings with ∆x+ = ∆y+ ≳4 and thus had a ﬁner streamwise grid spacing than the present case. However, if we assume that vis- cous dissipation equals the production of total turbulence kinetic energy, ϵ = −ubdP/dx (e.g. Nieuwstadt 2005), then the present grid spacings can be related to the Kolmogorov Michael MacDonald* et al.

(B)

Fig. 2 Proﬁles of (a) velocity and (b) TKE budget for the present dry neutrally stratiﬁed case (solid lines) and the DNS data of Moser et al. (1999) (magenta dashed lines) at Re⋆= hU⋆/ν = 395. In (a), the dotted black line shows the viscous sublayer velocity, U+ = z+, while the solid black line shows the logarithmic velocity proﬁle, U+ = (1/0.41) log(z+) + 5.2. The TKE budget terms in (b) are given in Eq. 16 and normalized on U4

Production, B, Is Zero

length, η = (ν3/ϵ)1/4, as ∆x = 4.0η, ∆y = 2.0η, ∆z|z=0 = 0.15η and ∆z|z=h = 2.9η. They are therefore all O(η), in agreement with the grid-spacing recommendations for con- ventional DNS (Moin and Mahesh 1998). For the cases with Re⋆= 395 and h/L = 0.6, multiple runs are performed in which unique initialization snapshots from the dry neutrally stratiﬁed case are used for both dry and moist cases (395D06 and 395M06). The compu- tational domain size is set to Lx × Ly = 2πh × πh, which is commonly employed for neutrally stratiﬁed dry simulations (Lozano-Dur´an and Jim´enez 2014; Munters et al. 2016).

The effect of the domain size is investigated in Appendix 1, showing that while the domain is relatively small for stably stratiﬁed ﬂows, it should not alter the conclusions of this paper. The present simulations use 256 × 256 × 128 grid points for the Re⋆= 395 cases and 384×384×192 grid points for the Re⋆= 590 cases to obtain the grid spacings mentioned above.

Alidation Of Eulag

The neutrally stratiﬁed dry case, 395D00, is ﬁrst validated with the DNS data of Moser et al. (1999, herein MKM99) at a friction Reynolds number Re⋆= hU⋆/ν = 395. This case obtains a statistically steady state independent of the initial conditions, identiﬁed by a linear proﬁle of the total stress proﬁle, −⟨u′w′⟩+ νd⟨u⟩/dz (Kim et al. 1987). The ﬂow is then temporally averaged over tU⋆/h ≈20 large-eddy turnover times. Figure 2 shows the mean velocity proﬁle and (resolved) TKE budget for the present data from EULAG, along with the DNS data of MKM99. Good agreement is observed between the two datasets, indicating that EULAG can be readily used for surface-layer simulations. The budget for the horizontally DNS of the Moist SBL: Turbulence and Fog Formation

H/L

z/h ≈2/3. Line colours deﬁned in (c), corresponding to cases listed in Table 1. Solid lines represent dry cases, while dashed lines represent moist cases. Additional runs for Re⋆= 395, h/L = 0.6 with unique initialization snapshots are indicated with symbols; ⋄and × are dry and moist cases, respectively, for the same snapshot; likewise □□and + for the other snapshot.

Note the time normalization is diﬀerent in (a) and (b) following Flores and Riley (2011); as such the neutrally stratiﬁed case (L = ∞) is not plotted in (a)

Iu′

i⟩/2, is shown in Fig. 2b, where the terms correspond to

(16)

where Pr is the mechanical (or shear) production, D the viscous diffusion, Tr the turbu- lent transport, Π the pressure correlation, B the buoyant production and ϵ the (pseudo-) dissipation. The left hand side, referred to as the tendency or residual, is zero for this neu- trally stratiﬁed ﬂow due to it achieving a statistically steady state. Conventional summation notation is used for repeated subscripts of i and j. The Reynolds stress terms, ⟨u′

J⟩(Not

shown) were also in good agreement with MKM99, although the vertical Reynolds stress, ⟨w′w′⟩went to zero at z = h due to the use of the slip, impermeable boundary condition. This reduction of ⟨w′w′⟩began at approximately z/h ≈0.8 in agreement with other open channel simulations (e.g. MacDonald et al. 2017).

Temporal Evolution Of Turbulent Fluctuations

The imposed ground cooling can either lead to complete turbulence collapse, resulting in laminar ﬂow (very stable regime), or maintenance of the turbulent ﬂow (weakly stable Michael MacDonald* et al.

vertical velocity ﬂuctuations at two different heights, z+ ≈15 ≈0.04h+ (i.e. close to the ground, within the buffer layer) and z/h ≈2/3. A moving average ﬁlter of size 1h/U⋆is applied to the present time series for clarity. Figure 3 matches Fig. 3 of Flores and Riley (2011) and, as discussed in Flores and Riley (2011), the temporal adjustment to the imposed cooling scales with L/U⋆in the buffer layer (Fig. 3a), while in the outer layer (Fig. 3b) it scales with h/U⋆. The neutrally stratiﬁed case is therefore not shown in Fig. 3a as L = ∞.

In general, the cooling causes a reduction in the vertical velocity ﬂuctuations within the ﬁrst 5–10 large-eddy turnover times. In the weakly stable cases the turbulence then recovers to magnitudes close to the neutrally stratiﬁed case, while for very stable cases the turbulence collapses and the ﬂuctuations tend to zero. As will be seen later, in the moist cases (dashed lines) saturation begins at the ground almost immediately. The associated latent heat release and buoyancy effects due to condensation appear to cause increased mixing in the moist case, leading to the turbulent ﬂuctuations remaining larger compared to the dry cases (solid lines) at matched h/L. This is especially evident when the ﬂow is close to laminarization.

In particular, for cases 395D06 and 395M06 three different initialization snapshots were run with cooling of h/L = 0.6, wherein the cooling is sufﬁcient to lead to complete turbulence collapse for one of the dry runs (red solid line with diamond symbols) but not the other two.

Meanwhile, all three moist runs remained weakly stable and maintained turbulence. The exact critical value of h/L that leads to turbulence collapse has been recognized as being Reynolds number dependent, with Nieuwstadt (2005) observing a critical value of h/L = 0.51 for Re⋆≈360, while Flores and Riley (2011) observed the critical value to be h/L = 0.82 for Re⋆≈560. The present values of h/L ≈0.6 for Re⋆≈395 and h/L ≳0.85 for Re⋆≈590 for the dry cases therefore agree with this trend. Due to the Reynolds number dependence in h/L, Flores and Riley (2011) suggested that Lτuτ/ν is a better measure for the critical cooling level, where Lτ is the Obukhov length deﬁned using the instantaneous wall-shear stress uτ (as opposed to the constant driving U⋆, as in Eq. 11). The two dry runs of 395D06 which sustained turbulent ﬂow were observed to have a minimum Lτuτ/ν ≈95, while the other initialization snapshot, which lead to turbulence collapse, had Lτuτ/ν ≤75. This agrees with the observation in Flores and Riley (2011) that turbulence collapses when Lτuτ/ν ≲100, due to insufﬁcient scale separation between the length scale of turbulent production in the buffer region (approximately 100ν/uτ) and the buoyancy length scale (Lτ). The moist cases with Re⋆= 395 and h/L = 0.6 had a minimum Lτuτ/ν ≈225 for all three initialization snapshots, although for h/L = 0.7 it was Lτuτ/ν ≈70 and led to turbulence collapse. Compared to the dry case with criti- cal h/L = 0.6, the production of liquid water enables a slightly larger cooling rate to be achieved before leading to turbulence collapse, with critical h/L in the range 0.6–0.7.

Note that, as discussed in Flores and Riley (2011) and Garc´ıa-Villalba and del ´Alamo (2011), the asymptotic laminar friction Richardson number, Riτ,l = (2/κ)(h/L)Re⋆Prm for all the present cases is less than the linear stability limit (Gage and Reid 1968). Therefore we would expect the cases with laminar ﬂow to eventually recover a turbulent state given enough time. Also, purely laminar ﬂow across the entire surface layer is unlikely to occur in the real atmosphere due to turbulence generated by large-scale structures that are not captured in the present idealized system, such as low-level jets and the breaking of gravity waves (Flores and Riley 2011).

The effects of the relatively small computational domain size are evident in the dry higher Reynolds number cases (particularly 590D085, green solid line), where it takes longer to return to the statistically steady turbulent state. This is due to the existence of ‘locked’ turbulent structures (Flores and Riley 2011; Garc´ıa-Villalba and del ´Alamo 2011), where adjacent turbulent and laminar patches become locked in place as a result of the periodic DNS of the Moist SBL: Turbulence and Fog Formation

Fig. 4

Time series of (a) sensible (solid) and latent (dashed) heat ﬂuxes normalized on the imposed ground heat ﬂux, Hg, and (b) inverse of the Bowen ratio, 1/β = Hl/Hs. These are shown at z+ ≈15 (black), z/h = 2/3 (blue), and z = h (red) for case 395M06. Equilibrium Bowen ratio 1/βe ≈1.09 shown by horizontal dashed line in (b) boundary conditions (see also Appendix 1). The moist case, meanwhile, does not exhibit this behaviour for the same cooling rate. This is possibly due to the latent heat released during condensation, which would enhance mixing due to buoyancy production. Moreover, a positive feedback system exists wherein any tendency toward a laminar state with reduced mixing would result in enhanced, or runaway, cooling (Van de Wiel et al. 2007). This would therefore promote condensation-induced mixing and thus avoid the spatially locked laminar patches associated with turbulence collapse. Note that this effect may be sensitive to the domain size (Garc´ıa-Villalba and del ´Alamo 2011; Ansorge and Mellado 2014) and would require further investigation.

the sensible and latent heat ﬂuxes transported by the ﬂuid change over time for case 395M06. Close to the ground, at z+ ≈15, the sensible heat ﬂux (black solid line) initially increases rapidly due to the imposed ground heat ﬂux. However, soon thereafter the temperature reaches saturation point and liquid water condenses, resulting in an increase in the latent heat ﬂux (black dashed line) and decrease in the sensible heat ﬂux. Within ﬁve large-eddy turnover times, the latent and sensible heat ﬂuxes reach a statistically steady state and the inverse of the Bowen ratio (Fig. 4b) is exactly equal to its equilibrium value (horizontal dashed line). This does not vary with time due to the use of the bulk condensation model, which maintains equilibrium conditions. A similar effect occurs for the heat ﬂuxes higher up at z/h = 2/3 (blue lines), although this is delayed as the interface between saturated and unsaturated air requires time to grow upwards (see sketch in Fig. 1). Note that the sum of Hs and Hl at this height does not equal Hg as the ﬂow has not achieved a true statistically steady state; this would only occur when the heat ﬂux at the top of the domain balances the imposed ground heat ﬂux. No signiﬁcant amount of liquid water reaches the top of the domain (z = h, red line) so the latent heat ﬂux and Bowen ratio remains essentially zero.

as T is uniformly set to T0 at t = 0. The temperature in Fig. 5 is normalized on the in- stantaneous, vertically dependent friction temperature Tτ(z, t) from Eq. 12. This choice of normalization results in good agreement between dry and moist turbulent cases with dif- ferent cooling rates, as well as for the different Reynolds number cases. Figure 5a shows tU⋆/h ≈10 for the dry cases 395D06 and 590D085 (solid red and green lines, Fig. 5a)

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

●

Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

●

Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

Figure: System Model & Simulation Flow for Ansys Mri Compatible Device

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

(1)

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

8

13C-bicarbonate doped with dimethyl silicone, various

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.

Spectre & Simulation

Gazebo, cloud twin and Webots worlds with navigation, SLAM and control stacks.

Control & Planning

Compliance, deep learning control, path planning and behavior trees.

Hardware Bring-up

Motors, sensors, ESP32/STM32 firmware and HIL validation paths.

Report & Viva

University-format documentation, PPT and viva preparation.

FAQ

Spectre, Gazebo, NVIDIA cloud twin, MATLAB/Simulink, Webots, Blynk / ThingSpeak, plus Arduino/STM32/ESP32, cameras, LiDAR and motor drivers.

Yes — simulation packages, hardware guidance, report, PPT and viva Q&A.

CFD Lab — Bangalore

Simulation, control and hardware support for final-year robotics projects.

Spectre
Stacks

Nav · SLAM

Gazebo
Worlds

Simulation

cloud twin
Digital Twin

Omniverse

MATLAB
Control

Simulink

Webots
Robots

Controllers

Blynk / ThingSpeak
Offline

Manipulators

Hardware
Bring-up

Sensors · Motors

Viva & Report

Docs