Matlab Robotics Toolbox

Information Theory and Automation of the ASCR, Prague, Czech Republic

Abstract

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch- Torrey partial differential equation. Under the assumption of negligible water exchange between compartments, the time-dependent apparent diffusion coefficient can be directly computed from the solution of a diffusion equation subject to a time-dependent Neumann boundary condition.

This paper describes a publicly available MATLAB toolbox called SpinDoctor that can be used 1) to solve the Bloch-Torrey partial differential equation in order to simulate the diffusion magnetic resonance imaging signal; 2) to solve a diffusion partial differential equation to obtain directly the apparent diffusion coefficient; 3) to compare the simulated apparent diffusion coefficient with a short-time approximation formula.

The partial differential equations are solved by P1 finite elements combined with built-in MATLAB routines for solving ordinary differential equations. The finite element mesh generation is performed using an external package called Tetgen.

SpinDoctor provides built-in options of including 1) spherical cells with a nucleus; 2) cylindrical cells with a myelin layer; 3) an extra-cellular space enclosed either a) in a box or b) in a tight wrapping around the cells; 4) deformation of canonical cells by bending and twisting; 5) permeable membranes; Built-in diffusion-encoding pulse sequences include the Pulsed Gradient Spin Echo and the Oscillating Gradient Spin Echo.

We describe in detail how to use the SpinDoctor toolbox. We validate SpinDoctor simulations using reference signals computed by the Matrix Formalism method. We compare the accuracy and computational time of SpinDoctor simulations with Monte-Carlo simulations and show significant speed-up of SpinDoctor over Monte-Carlo simulations in complex geometries. We also illustrate several extensions of SpinDoctor functionalities, including the incorporation of T2 relaxation, the simulation of non-standard diffusion-encoding sequences, as well as the use of externally generated

Arxiv:1902.01025V2 [Math.Na] 16 Sep 2019

geometrical meshes. Bloch-Torrey equation, diffusion magnetic resonance imaging, finite elements, simulation, apparent diffusion coefficient.

Ntroduction

Diffusion magnetic resonance imaging is an imaging modality that can be used to probe the tissue micro-structure by encoding the incohorent motion of water molecules with magnetic field gradient pulses. This motion during the diffusion-encoding time causes a signal attenuation from which the apparent diffusion coefficient , (and possibly higher order diffusion terms, can be calculated .

For unrestricted diffusion, the root of the mean squared displacement of molecules is given by ¯x = √2 dim σ0t, where dim is the spatial dimension, σ0 is the intrinsic diffusion coefficient, and t is the diffusion time. In biological tissue, the diffusion is usually hindered or restricted (for example, by cell membranes) and the mean square displacement is smaller than in the case of unrestricted diffusion. This deviation from unrestricted diffusion can be used to infer information about the tissue micro-structure. The experimental parameters that can be varied include 1. the diffusion time (one can choose the parameters of the diffusion-encoding sequence, such as Pulsed Gradient Spin Echo and Oscillating Gradient ).

2. the magnitude of the diffusion-encoding gradient (when the magnetic resonance imaging sig- nal is acquired at low gradient magnitudes, the signal contains only information about the apparent diffusion coefficient, at higher values, Kurtosis imaging becomes possible); 3. the direction of the diffusion-encoding gradient (many directions may be probed, as in high angular resolution diffusion imaging ).

Using diffusion magnetic resonance imaging to get tissue structural information in the mamalian brain has been the focus of much experimental and modeling work in recent years . The pre- dominant approach up to now has been adding the diffusion magnetic resonance imaging signal from simple geometrical components and extracting model parameters of interest. Numerous biophysical models subdivide the tissue into compartments described by spheres, ellipsoids, cylinders, and the extra-cellular space [7–9, 11, 12, 15–19]. Some model parameters of interest include axon diameter and orientation, neurite density, dendrite structure, the volume fraction and size distribution of cylinder and sphere components and the effective diffusion coefficient or tensor of the extra-cellular space.

Numerical simulations can help deepen the understanding of the relationship between the cellular structure and the diffusion magnetic resonance imaging signal and lead to the formulation of appro- priate models. They can be also used to investigate the effect of different pulse sequences and tissue features on the measured signal which can be used for the development, testing, and optimization of novel diffusion magnetic resonance imaging pulse sequences .

Two main groups of approaches to the numerical simulation of diffusion magnetic resonance imaging are 1) using random walkers to mimic the diffusion process in a geometrical configuration; 2) solving the Bloch-Torrey PDE , which describes the evolution of the complex transverse water proton magnetization under the influence of diffusion-encoding magnetic field gradients pulses.

The first group is referred to as Monte-Carlo simulations in the literature and previous works include [13, 24–27]. A GPU-based acceleration of Monte-Carlo simulation was proposed in [28, 29]. Some

Software Packages Using This Approach Include

1. Camino Diffusion MRI Toolkit developed at UCL (http://cmic.cs.ucl.ac.uk/camino/); 2. DIFSIM developed at UC San Diego (http://csci.ucsd.edu/projects/simulation.html); 3. Diffusion Microscopist Simulator developed at Neurospin, CEA.

The second group relies on solving the Bloch-Torrey PDE in a geometrical configuration. In [30– 32] a simplifying assumption called the narrow pulse approximation was used, where the pulse duration was assumed to be much smaller than the delay between pulses. This assumption allows the solution of the diffusion equation instead of the more complicated Bloch-Torrey PDE. More generally, numerical methods to solve the Bloch-Torrey PDE. with arbitrary temporal profiles have been proposed in . The computational domain is discretized either by a Cartesian grid [33, 34, 37] or finite elements [30–32, 35, 36]. The unstructured mesh of a finite element discretization appeared to be better than a Cartesian grid in both geometry description and signal approximation . For time discretization, both explicit and implicit methods have been used. In a second order implicit time-stepping method called the generalized α−method was used to allow for high frequency energy dissipation. An adaptive explicit Runge-Kutta Chebyshev method of second order was used in [34, 35]. It has been theoretically proven that the Runge-Kutta Chebyshev method allows for a much larger time-step compared to the standard explicit Euler method . There is an example showing that the Runge-Kutta Chebyshev method is faster than the implicit Euler method in . The Crank-Nicolson method was used in to also allow for second order convergence in time. The efficiency of diffusion magnetic resonance imaging simulations is also improved by either a high-performance FEM computing framework [39, 40] for large-scale simulations on supercomputers or a discretization on manifolds for thin-layer and thin-tube media .

In this paper, we present a MATLAB Toolbox called SpinDoctor that is a simulation pipeline going from the definition of a geometrical configuration through the numerical solution of the Bloch- Torrey PDE to the fitting of the apparent diffusion coefficient from the simulated signal. It also includes two other modules for calculating the apparent diffusion coefficient. The first module is a homogenized apparent diffusion coefficient mathematical model, which was obtained recently using homogenization techniques on the Bloch-Torrey PDE. In the homogenized model, the apparent dif- fusion coefficient of a geometrical configuration can be computed after solving a diffusion equation subject to a time-dependent Neumann boundary condition, under the assumption of negligible wa- ter exchange between compartments. The second module computes the short time approximation formula for the apparent diffusion coefficient. The short time approximation implemented in Spin- Doctor includes a recent generalization of this formula to account for finite pulse duration in the pulsed gradient spin echo. Both of these two apparent diffusion coefficient calculations are sensitive to the diffusion-encoding gradient direction, unlike many previous works where the anisotropy is neglected in analytical model development.

N Summary, Spindoctor

1. solves the Bloch-Torrey PDE in three dimensions to obtain the diffusion magnetic resonance

Imaging Signal;

2. robustly fits the diffusion magnetic resonance imaging signal to obtain the apparent diffusion

Coefficient;

3. solves the homogenized apparent diffusion coefficient model in three dimensions to obtain the

Apparent Diffusion Coefficient;

4. computes the short-time approximation of the apparent diffusion coefficient; 5. computes useful geometrical quantities such as the compartment volumes and surface areas; 6. allows permeable membranes for the Bloch-Torrey PDE (the homogenized apparent diffusion coefficient assumes negligible permeabilty).

7. displays the gradient-direction dependent apparent diffusion coefficient; in three dimensions

Using Spherical Harmonics Interpolation;

SpinDoctor provides the following built-in functionalities: 1. placement of non-overlapping spherical cells (with an optional nucleus) of different radii close

To Each Other;

2. placement of non-overlapping cylindrical cells (with an optional myelin layer) of different radii close to each other in a canonical configuration where they are parallel to the z-axis; 3. inclusion of an extra-cellular space that is enclosed either

(B) In A Rectangular Box;

4. deformation of the canonical configuration by bending and twisting; Built-in diffusion-encoding pulse sequences include

The Pulsed Gradient Spin Echo ;

2. the Oscillating Gradient Spin Echo (cos- and sin- type gradients).

Spindoctor Uses The Following Methods:

1. it generates a good quality surface triangulation of the user specified geometrical configuration by calling built-in MATLAB computational geometry functions; 2. it creates a good quality tetrehedra finite elements mesh from the above surface triangulation by calling Tetgen , an external package (executable files are included in the Toolbox

Package);

3. it constructs finite element matrices for linear finite elements on tetrahedra (P1) using routines

From ;

4. it adds additional degrees of freedom on the compartment interfaces to allow permeability conditions for the Bloch-Torrey PDE using the formalism in ; 5. it solves the semi-discretized FEM equations by calling built-in MATLAB routines for solving ordinary differential equations .

The SpinDoctor toolbox has been developed in the MATLAB R2017b and requires no additional MATLAB toolboxes. The toolbox is publicly available at:

Theory

Suppose the user would like to simulate a geometrical configuration of cells with an optional myelin layer or a nucleus. If spins will be leaving the cells or if the user wants to simulate the extra-cellular space (ECS), then the ECS will enclose the geometrical shapes. Let Ωe be the ECS, Ωin

I

the cytoplasm (or the myelin layer) of the ith cell. We denote the interface

Bloch-Torrey Pde

In diffusion MRI, a time-varying magnetic field gradient is applied to the tissue to encode water diffusion. Denoting the effective time profile of the diffusion-encoding magnetic field gradient by f(t), and letting the vector g contain the amplitude and direction information of the magnetic field gradient, the complex transverse water proton magnetization in the rotating frame satisfies the

(3)

where γ = 2.67513×108 rad s−1T−1 is the gyromagnetic ratio of the water proton, I is the imaginary unit, σl is the intrinsic diffusion coefficient in the compartment Ωl

I. The Magnetization Is A Function

of position x and time t, and depends on the diffusion gradient vector g and the time profile f(t). We denote the restriction of the magnetization in Ωin

I

and M e. Some commonly used time profiles (diffusion-encoding sequences) are: 1. The pulsed-gradient spin echo (PGSE) sequence, with two rectangular pulses of duration δ, separated by a time interval ∆−δ, for which the profile f(t) is

(4)

where t1 is the starting time of the first gradient pulse with t1 + ∆> TE/2, TE is the echo time at which the signal is measured. 2. The oscillating gradient spin echo (OGSE) sequence [4, 45] was introduced to reach short diffusion times. An OGSE sequence usually consists of two oscillating pulses of duration T, each containing n periods, hence the frequency is ω = n 2π T , separated by a time interval τ −T.

(5)

where τ = TE/2. The BTPDE needs to be supplemented by interface conditions. We recall the interface between

And Ωe Is Σi, And The Outside Boundary Of The Ecs

is Ψ. The two interface conditions on Γi are the flux continuity and a condition that incorporates

X ∈Γi,

where n is the unit outward pointing normal vector. Similarly, between Ωout

(X, 0) = Ρout,

M e(x, 0) = ρe. where ρ is the initial spin density. The dMRI signal is measured at echo time t = TE > ∆+δ for PGSE and TE > 2σ for OGSE. This

(6)

In a dMRI experiment, the pulse sequence (time profile f(t)) is usually fixed, while g is varied in amplitude (and possibly also in direction). S is usually plotted against a quantity called the b-value. The b-value depends on g and f(t) and is defined as

For Pgse, The B-Value Is :

b(g, δ, ∆) = γ2∥g∥2δ2 (∆−δ/3) .

(7)

For the cosine OGSE with integer number of periods n in each of the two durations σ, the corre-

(8)

The reason for these definitions is that in a homogeneous medium, the signal attenuation is e−σb, where σ is the intrinsic diffusion coefficient.

Fitting The Adc From The Dmri Signal

An important quantity that can be derived from the dMRI signal is the “Apparent Diffusion Co- efficient” (ADC), which gives an indication of the root mean squared distance travelled by water molecules in the gradient direction g/∥g∥, averaged over all starting positions:

Log S(B) = C0 + C1B + · · · + Cnbn,

increasing n from 1 onwards until we get the value of c1 to be stable within a numerical tolerance. 2.3.

Hadc Model

In a previous work , a PDE model for the time-dependent ADC was obtained starting from the Bloch-Torrey equation, using homogenization techniques. In the case of negligible water exchange between compartments (low permeability), there is no coupling between the compartments, at least to the quadratic order in g, which is the ADC term. The ADC in compartment Ωis given by

(11)

is a quantity related to the directional gradient of a function ω that is the solution of the homoge- neous diffusion equation with Neumann boundary condition and zero initial condition:

(12)

n being the outward normal and t ∈[0, TE], ug is the unit gradient direction. The above set of equations, (10)-(12), comprise the homogenized model that we call the HADC model.

Short Diffusion Time Approximation Of The Adc

A well-known formula for the ADC in the short diffusion time regime is the following short time

Where A

V is the surface to volume ratio and σ is the intrinsic diffusivity coefficient. In the above formula the pulse duration δ is assumed to be very small compared to ∆. A recent correction to the above formula , taking into account the finite pulse duration δ and the gradient direction

Ethod

Below is a chart describing the work flow of SpinDoctor.

Bend And Twist The Fe Mesh Nodes

by analytical transformation.

Plot Hadc

Figure 1: Flow chart describing the work flow of SpinDoctor The physical units of the quantities in the input files for SpinDoctor are shown in Table 1, in particular, the length is in µm and the time is in µs. Below we discuss the various components of SpinDoctor in more detail.

Read Cells Parameters

The user provides an input file for the cell parameters, in the format described in Table 2.

(Μsµm)−1

Table 1: Physical units of the quantities in the input files for SpinDoctor.

Height Of Cylinders

Table 2: Input file containing cells parameters.

Reate Cells (Canonical Configuration)

SpinDoctor supports the placement of a group of non-overlapping cells in close vicinity to each other. There are two proposed configurations, one composed of spheres, the other composed of cylinders. The algorithm is described in Algorithm 1.

Algorithm 1: Placing ncell non-overlapping cells. Generate a large number of possible cell centers. Compute the minimum distance, dist, between the current center and previously accepted cells.

Find the intersection of [dist −dmax × Rmean, dist −dmin × Rmean] and [Rmin, Rmax],

Where Rmean = Rmin+Rmax

. If the intersection is not empty, then take the middle of the intersection as the new radius and accept the new center. Otherwise, reject the center. Loop through the possible centers until get ncell accepted cells.

Plot Cells

SpinDoctor provides a routine to plot the cells to see if the configuration is acceptable (see Fig. 2). Figure 2: SpinDoctor plots cells in the canonical configuration.

Read Simulation Domain Parameters

The user provides an input file for the simulation domain parameters, in the format described in Table 3.

Reate Surface Triangulation

Finite element mesh generation software requires a good surface triangulation. This means the surface triangulation needs to be water-tight and does not self-intersect. How closely these require- ments are met in floating point arithmetic has a direct impact on the quality of the finite element mesh generated.

It is often difficult to produce a good surface triangulation for arbitrary geometries.

Thus, We

restrict the allowed shapes to cylinders and spheres. Below in Algorithms 2 and 3 we describe how to obtain a surface triangulation for spherical cells with nucleus, cylindrical cells with myelin layer, and the ECS (box or tightly wrapped). We describe a canonical configuration where the cylinders are placed parallel to the z-axis. More general shapes are obtained from the canonical configuration by coordinate transformation in a later step.

Path To Tetgen Cmd

Table 3: Input file of simulation domain parameters.

Plot Surface Triangulation

SpinDoctor provides a routine to plot the surface triangulation (see Fig. 3).

Finite Element Mesh Generation

SpinDoctor calls Tetgen , an external package (executable files are included in the toolbox package), to create a tetrehedra finite elements mesh from the surface triangulation generated by Algorithm 2: Surface triangulation of spherical cells and ECS.

Suppose we have ncell spherical cells with nucleus. Denote a sphere with center c and radius R by S(c, R), we use the built-in functions (convex hull, delaunnay triangulation) in MATLAB to get its surface triangulation, T(c, R). Call the radii of the nucleus r1, · · · , rncell and the radii of the cells R1, · · · , Rncell. Then the boundaries between the

{Σi = T(Ci, Ri)}, I = 1, · · · , Ncell;

For the box ECS, we find the coordinate limits of the set

S(Ci, Ri) ∈[X0, Xf] × [Y0, Yf] × [Z0, Zf]

and add a gap k = ECS gap × max{xf −x0, yf −y0, zf −z0} to make a box B = [x0 −k, xf + k] × [y0 −k, yf + k] × [z0 −k, zf + k]. We put 2 triangles on each face of B to make a surface triangulation Ψ with 12 triangles.

For the tight-wrap ECS, we increase the cell radius by a gap size and take the union

Where Rmean = Rmin+Rmax

. We use the alphaShape function in MATLAB to find a surface triangulation Ψ that contains W. Algorithms 2 and 3. The FE mesh is generated on the canonical configuration. The numbering of the compartments and boundaries used by SpinDoctor are given in Tables 4 and 5. The labels are related to the values of the intrinsic diffusion coefficient, the initial spin density, and the perme- ability requested by the user. Then the FE mesh nodes are deformed analytically by a coordinate transformation, described in Algorithm 4.

Plot Fe Mesh

SpinDoctor provides a routine to plot the FE mesh (see Fig. 4 for cylinders and ECS that have been bent and twisted).

Read Experimental Parameters

The user provides an input file for the simulation experimental parameters, in the format described in Table 6. Algorithm 3: Surface triangulation of cylindrical cells and ECS.

Suppose we have ncell cylindrical cells with a myelin layer, all with height H. Denote a disk with center c and radius R by D(c, R), and the circle with the same center and radius by C(c, R). Let the radii of the axons be r1, · · · , rncell and the radii of the cells be R1, · · · , Rncell, meaning the thickness of the myelin layer is Ri −ri.

The boundary between the axon and the myelin layer is:

(Ci, Ri) × [−H/2, H/2]

We discretize C(ci, ri) as a polygon P(ci, ri) and place one at z = −H/2 and one at z = H/2. Then we connect the corresponding vertices of P(ci, ri) × {−H/2} and P(ci, ri) × {H/2} and add a diagonal on each panel to get a surface triangulation Γi.

Between the myelin layer and the ECS we discretize C(ci, Ri) as a polygon and place one at z = −H/2 and one at z = H/2 to get a surface triangulation Σi. For the box ECS, we find the coordinate limits of the union of D(ci, Ri) and add a gap to make a rectangle in two dimensions. Then we place the rectangle at z = −H/2 and at z = H/2 to get a box. Finally, the box is given a surface triangulation with 12 triangles.

For tight-wrap ECS, we increase the cell radius by a gap size and take the union

I

D(ci, Ri + kRmean). We use the alphaShape function in MATLAB to find a two dimensional polygon Q that contains W. We place Q at z = −H/2 and at z = H/2 and connect correponding vertices, adding a diagonal on each panel. Suppose Q is a polygon with n vertices, then the surface triangulation of the side of the ECS will have 2n triangles.

The above procedure produces a surface triangulation for the boundaries that are parallel to z-axis. We now must close the top and bottom. The top and bottom boundaries is just the interior of Q. However, the surface triangulation cannot be done on Q directly. We must cut out D(ci, ri), the disk which touches the axon, and Ai = D(ci, Ri) −D(ci, ri), the annulus which touches the myelin. Then we triangulate Q −S

I Ai Using The

MATLAB built-in function that triangulates a polygon with holes to get the boundary that touches the ECS. The surface triangulation for Ai and D(ci, ri) are straightforward.

Btpde

The spatial discretization of the BTPDE is based on a finite element method where interface (ghost) elements are used to impose the permeable interface conditions. The time stepping is done using the MATLAB built-in ODE routine ode23t. See Algorithm 5.

Hadc Model

Similarly, the DE of the HADC model is discretized by finite elements. See Algorithm 6. Figure 3: SpinDoctor plots the surface triangulation of the canonical configuration. Left: spherical cells with ECS; Right: cylindrical cells with ECS.

Ncell + 1

Table 4: The labels and numbers of compartments.

Some Important Output Quantities

In Table 7 we list some useful quantities that are the outputs of SpinDoctor. The braces in the ”Size” column denote MATLAB cell data structure and the brackets denote MATLAB matrix data structure.

Ncell + 1

Table 5: The labels and numbers of boundaries.

Spindoctor Examples

In this section we show some prototypical examples using the available functionalities of SpinDoctor. 4.1. Comparison of BTPDE and HADC with Short Time Approximation In Fig. 5 we show that both BTPDE and HADC solutions match the STA values at short diffusion times for cylindrical cells (compartments 1 to 5). We also show that for the ECS (compartment 6), the STA is too low, because it does not account for the fact that spins in the ECS can diffuse around several cylinders. This also shows that when the interfaces are impermeable, the BTPDE ADC and that from the HADC model are identical. The diffusion-encoding sequence here is cosine OGSE with 6 periods.

Permeable Membranes

In Fig. 6 we show the effect of permeability: the BTPDE model includes permeable membranes (κ = 1×10−3 m/s) whereas the HADC has impermeable membranes. We see in the permeable case, the ADC in the spheres are higher than in the impermeable case, whereas the ECS show reduced Algorithm 4: Bending and twisting of the FE mesh of the canonical configuration.

The external package Tetgen generates the finite element mesh that keeps track of the different compartments and the interfaces between them. The mesh is saved in several text files.

The connectivity matrices of the finite elements and facets are not modified by the coordinates transformation described below. The nodes are transformed by bending and twisting as described next.

The set of FE mesh nodes {xi, yi, zi} are transformed in the following ways: Twisting around the z-axis with a user-chosen twisting parameter αtwist is defined by



. Bending on the x −z plane with a user-chosen bending parameter αbend is defined by



. Given [αbend, αtwist], bending is performed after twisting. ADC because the faster diffusing spins in the ECS are allowed to moved into the slowly diffusing spherical cells. We note that in the permeable case, the ADC in each compartment is obtained by using the fitting formula involving the logarithm of the dMRI signal, and we defined the ”signal” in a compartment as the total magnetization in that compartment at TE, which is just the integral of the solution of the BTPDE in that compartment.

Yelin Layer

In Fig. 7 we show the diffusion in cylindrical cells, the myelin layer, and the ECS. The ADC is higher in the myelin layer than in the cells, because for spins in the myelin layer diffusion occurs in the tangential direction (around the circle). At longer diffusion times, the ADC of both the myelin layer and the cells becomes very low. The ADC is the highest in the ECS, because the diffusion distance can be longer than the diameter of a cell, since the diffusing spins can move around multiple cells.

Twisting And Bending

In Fig. 8 we show the effect of bending and twising in cylindrical cells in multiple gradient directions. The HADC is obtained in 20 directions uniformly distributed in the sphere. We used spherical harmonics interpolation to interpolate the HADC in the entire sphere.

Then We Deformed The

radius of the unit sphere to be proportional to the interpolated HADC and plotted the 3D shape. The color axis also indicates the value of the interpolated HADC. Figure 4: FE mesh of cylinders and ECS after bending and twisting. Compartment number is 1 to 8 for the cylinders and 9 for the ECS.

Timing

In Table 8 we give the average computational times for solving the BTPDE and the HADC. All simulations were performed on a laptop computer with the processor Intel(R) Core(TM) i5-4210U 2 axons and a tight wrap ECS, the simulated sequence is PGSE (δ = 2.5ms, ∆= 5ms).

N

the impermeable case, the compartments are uncoupled, and the computational times are given separately for each compartment. In the permeable membrane case, the compartments are coupled, and the computational time is for the coupled system (relevant to the BTPDE only).

Numerical Validation Of Spindoctor

In this section, we validate SpinDoctor by comparing SpinDoctor with the Matrix Formalism method [49, 50] in a simple geometry. The Matrix Formalism method is a closed form representation of the dMRI signal based on the eigenfunctions of the Laplace operator subject to homogeneous Neumann boundary conditions. These eigenfunctions are available in explicit form for elementary

Tributed Uniformly On A Sphere;

if ngdir = 1, take the gradient direction from the

Depending On Line 13;

Table 6: Input file for simulation experiment parameters. geometries such as the line segment, the disk, and the sphere . The dMRI signal obtained using the Matrix Formalism method will be considered the reference solution in this section.

The accuracy of the SpinDoctor simulations can be tuned using three simulation parameters:

Htetgen Controls The Finite Element Mesh Size;

(a) Htetgen = −1 means the FE mesh size is determined automatically by the internal algorithm of Tetgen to ensure a good quality mesh (subject to the constraint that the radius to edge ratio of tetrahedra is no larger than 2.0).

(b) Htetgen = h requests a desired FE mesh tetrahedra height of h µm (in later versions of Tetgen, this parameter has been changed to the desired volume of the tetrahedra). Algorithm 5: BTPDE.

FE matrices are generated for each compartment by the finite element method with continuous piecewise linear basis functions (known as P1). The basis functions are denoted as ϕk for k = 1, . . , Nv, where Nv denotes the number of mesh nodes (vertices). All matrices are sparse matrices. M and S are known in the FEM literature as mass and

Ω

σi ∇ϕi · ∇ϕj dx. J has a similar form as the mass matrix but it is scaled with the coefficient g · x, we

W Φiϕj Ds

where a scalar function w is used as an interface marker. The matrices are assembled from local element matrices and the assembly process is based on vectorized routines of , which replace expensive loops over elements by operations with 3-dimensional arrays. All local elements matrices in the assembly of S, M, J are evaluated at once and stored in a full matrix of size 4 × 4 × Ne, where Ne denotes the number of tetrahedral elements. The assembly of Q is even simpler; all local matrices are stored in a full matrix of size 3 × 3 × nbe, where nbe denotes the number of boundary triangles.

Double nodes are placed at the interfaces between compartments connected by permeable membrane. Q is used to impose the interface conditions and it is associated with the interface (ghost) elements. Specifically, assume that the double nodes are defined in a pair

−Q¯I¯J

if vertex i and j belong to two different interfaces The fully coupled linear system has the following form

(14)

where ξ is the approximation of the magnetization M. SpinDoctor calls MATLAB built-in ODE routine ode23t to solve the semi-discretized system of equations. 2. rtol controls the accuracy of the ODE solve. It is the relative residual tolerance at all points of the FE mesh at each time step of the ODE solve; Algorithm 6: HADC model.

Eq. (12) can be discretized similarly as described for the BTPDE and has the matrix form

(15)

where ζ is the approximation of w and ¯ζi = σi F(t) ug · n(xi). We note that the matrices here are assembled and solved separately for each compartment. SpinDoctor calls MATLAB built-in ODE routine ode23t to solve the semi-discretized equation.

Integral Of Magnetization At Te

summed over all compartments.

Adc Accounting For All Compart-

ments in each direction. Table 7: Some important SpinDoctor output quantities.

N/A

Table 8: Computational times for solving the BTPDE and the HADC. All simulations were performed on Intel(R) 2 axons and a tight wrap ECS, the simulated sequence is PGSE (δ = 2.5ms, ∆= 5ms). 3. atol controls the accuracy of the ODE solve. It is the absolute residual tolerance at all points of the FE mesh at each time step of the ODE solve; We varied the finite element mesh size and the ODE solve accuracy of SpinDoctor and ran 6 simulations with the following simulation parameters: SpinD Simul 5-1: rtol = 10−3, atol = 10−6, Htetgen = −1;

Sta Experi 1

Figure 5: Geometry: 5 cylinders, tight wrap ECS, ECS gap = 0.2, ug = [1, 1, 1], σout = σecs = 2 × 10−3 mm2/s, κ = 0 m/s, OGSE cosine (δ = 14ms, ∆= 14ms, number of periods = 6). The vertical bars indicate the ADC in each compartment. The ADC in the rightmost position is the ADC that takes into account the diffusion in all the compartments.

SpinD Simul 5-2: rtol = 10−6, atol = 10−9, Htetgen = −1; SpinD Simul 5-3: rtol = 10−9, atol = 10−12, Htetgen = −1; SpinD Simul 5-4: rtol = 10−3, atol = 10−6, Htetgen = 1; SpinD Simul 5-5: rtol = 10−6, atol = 10−9, Htetgen = 1; SpinD Simul 5-6: rtol = 10−9, atol = 10−12, Htetgen = 1;

Hadc Experi 1

Figure 6: Geometry: 3 spheres, tight wrap ECS, ECS gap = 0.3, ug = [1, 1, 0], σin = σecs = 2 × 10−3 mm2/s, κ = 1 × 10−3 m/s (left), κ = 0 m/s (right). PGSE (δ = 5ms, ∆= 5ms). The vertical bars indicate the ADC in each compartment. The ADC in the rightmost position is the ADC that takes into account the diffusion in all the compartments.

• 3LayerCylinder is a 3-layer cylindrical geometry of height 1µm and the layer radii, R1 = 2.5µm, R2 = 5µm and R3 = 10µm. The middle layer is subject to permeable interface conditions on both the interior and the exterior interfaces, with permeability coefficient κ.

The exterior boundary R = R3 is subject to impermeable boundary conditions. The top and bottom boundaries are also subject to impermeable boundary conditions. • For this geometry, Htetgen = −1 gives finite elements mesh size (nnodes = 440, nelem = 1397).

Htetgen = 1 gives finite elements mesh size (nnodes = 718, nelem = 2088). The dMRI experimental parameters are the following: • the diffusion coefficient in all compartments is 2 × 10−3 mm2/s; • the diffusion-encoding sequence is PGSE (δ = 10ms, ∆= 13ms); • 8 b-values: b = {0, 100, 500, 1000, 2000, 3000, 6000, 10000} s/mm2;

• 1 Gradient Direction: [1, 1, 0];

In Figure 9 we show the signal differences (in percent) of the reference Matrix Formalism method and the SpinDoctor simulations, normalized by the reference signal at b = 0:

(16)

We see that the signal difference is less than 0.35% for κ = 10−5 m/s and it is less than 0.25% for κ = 10−4 m/s for all 6 SpinDoctor simulations. The signal difference becomes smaller when the ODE solve tolerances are changed from (rtol = 10−3, atol = 10−6) to (rtol = 10−6, atol = 10−9), but there is no change when the tolerances are further reduced to (rtol = 10−9, atol = 10−12). If we refine the FE mesh, but keep the ODE solve tolerances the same, the signal difference is in fact larger using the refined mesh than using the coarse mesh at the smaller b-values, though this effect disappears at higher b-values and larger permeability. This is probably due to parasitic oscillatory modes on the finer mesh that need smaller time steps to be sufficiently damped.

6. Computational time and comparison with Monte-Carlo simulation In this section, we compare SpinDoctor with Monte-Carlo simulation using the publicly available software package Camino Diffusion MRI Toolkit , downloaded from http://cmic.cs.ucl.ac.

MATLAB R2019a on the same computer. We give SpinDoctor computational times for three relatively complicated geometries. We also give Camino computational times for the first two geometries. We did not use Camino for the third The number of the degrees of freedom in the SpinDoctor simulations is the finite element mesh size (the number of nodes and the number of elements). For Camino it is the number of spins. The time stepping choice of the SpinDoctor simulations is given by the ODE solve tolerances. For Camino it is given by the number of time steps. Camino has an initialization step where it places the spins and we give the time of this initialization step separately from the Camino random walk simulation time.

Given the interest of the dMRI community in the extra-cellular space and neuron simulations,

We Chose The Following Three Geometries:

1. ECS400axons. See Figure 10. This models the extra-cellular space outside of 400 axons. We generated 400 cylinders with height 1µm and radii ranging from 2 −5µm, randomly placed according to Algorithm 1. The small height of the cylinders means that this geometry should

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.

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

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).

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.

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).

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.

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.

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 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 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.

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).

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