Cell Balancing Bms Matlab

The nuts and bolts of the BMS Bootstrap.

Arjun Bagchi,A Mirah Gary,B And Zodinmawia.A

aIndian Institute of Technology Kanpur, Kalyanpur, Kanpur 208016. INDIA. Abstract: In this paper, we elaborate on aspects of the recently introduced BMS bootstrap programme. We consider two-dimensional (2d) field theories with BMS3 symmetry and ex- tensively use highest weight representations to uncover the BMS version of crossing symmetry in 4-point functions that are constrained by symmetry. The BMS bootstrap equation is for- mulated and then analytic expressions for BMS blocks are constructed by looking at the limit of large central charges. These results are also applicable to 2d Galilean Conformal Field Theories through the isomorphism between the BMS3 and 2d Galilean Conformal Algebras.

We recover our previously obtained results in the non-relativistic limit of the corresponding ones in 2d relativistic CFTs. This provides a comprehensive check of our previous analysis. We also explore the chiral limit of BMS3 where the BMS algebra reduces to a single copy of the Virasoro algebra and show that our analysis is consistent with earlier work in this direction.

3.6

Differential equations for global blocks from quadratic Casimirs

4.3

Differential equation for blocks from the limiting case

36

A Level 2 coefficients: Detailed calculations in intrinsic method

39

B Level 2 coefficients: Detailed calculations in limiting method

– 1 –

C Level 2 analysis of coefficients in the Chiral limit

Introduction

The modern way of understanding relativistic quantum field theories (QFTs) is through renormalization group flows away from conformal field theories (CFTs). In the parameter space of all QFTs, CFTs arise as fixed points with enhanced scale and conformal symmetry.

The very ambitions programme of understanding all QFTs thus is intimately related to the classification of all consistent CFTs. Conformal bootstrap [1, 2] has emerged as the leading tool in this endeavour.

Any conformal field theory is determined by what has now come to be known as “CFT data”, viz. the spectrum of primary operators in the theory, the structure constants that are the constants of the three-point functions of primary operators not fixed by conformal invariance and the central charge of the theory (in case of 2d CFTs). But any random set of data does not constitute a consistent theory. The theory has to obey associativity of the operator algebra. Conformal bootstrap uses conformal symmetry and the consistency of the operator product expansion (OPE) to constrain possible CFTs.

The use of the conformal bootstrap programme was initially limited to two dimensional conformal field theories . Here one has the additional power of infinite dimensional sym- metries of the two copies of the underlying Virasoro algebra.

(1.1C)

The bootstrap equation in 2d CFTs help us solve some CFTs explicitly. The analytical handle that the Virasoro symmetry provides helps put in powerful constraints on the mathematical consistency of theories in 2d. For values of the central charges between 0 and 1, there is a discrete number of CFTs with finite number of primary fields and these are called the minimal models. The bootstrap equations leads to a complete solution of the 2d minimal series.

Following the seminal work of Rattazzi, Rychkov, Tonni and Vichi in 2008 , who build on earlier work by Dolan and Osborn [5, 6], there has been a great flurry of activity in applying conformal bootstrap techniques to spacetime dimensions higher than two. The method of conformal bootstrap has emerged as a very effective tool in calculating things like the critical exponents of Ising model or the O(N) model in 3 dimensions. We refer the reader to the excellent reviews [7, 8] for a more detailed account of the excitement in this emerging field. See also for a very well written overview.

– 2 –

Our present goal is to generalise the ideas and methods of the conformal bootstrap programme to theories with symmetries other than conformal invariance.

In This Present

work, which is a continuation and elaboration of an earlier shorter piece of work , we will concentrate on 2d field theories which are invariant under the following symmetry algebra:

(1.2C)

This algebra arises as a contraction of two copies of the Virasoro algebra (1.1) and is called the 2d Galilean Conformal Algebra (GCA) [11, 12]. The algebra also arises as asymptotic symmetries of 3d flat spacetimes and is called the 3d Bondi-Metzner-Sachs (BMS) algebra . This isomorphism was first noticed in and goes under the name of the BMS/GCA correspondence.

We will find that we will be able to construct, in a spirit very similar to that of CFTs, a BMS version of an OPE and then by considering four point functions, we will define the notion of BMS blocks and a BMS crossing equation. This will then lead us to the BMS bootstrap equation. In the limit of large central charge, we will find closed form expressions for these BMS blocks that form the basis for the solution of the bootstrap equation. We will then go on to recover all our answers as contractions of appropriate quantities in a 2d relativistic CFT. This forms a comprehensive check of our results obtained in the intrinsic method, some of which were first reported in .

To the best of our knowledge, this constitutes the first successful attempt at the con- struction and concrete steps towards the solution of a bootstrap equation in a theory that is not a relativistic conformal field theory.

Our motivations for addressing field theories with the symmetry algebra (1.2) are man- ifold. This algebra has recently surfaced in various contexts, viz. as symmetries of putative dual field theories to 3d flat space, as conformal symmetries in non-relativistic systems and also as the residual symmetry algebra on the worldsheet of the tensionless closed bosonic string [17, 18]. Below we address the first two of these applications.

Holography For Flat Spacetimes

The notion of asymptotic symmetries is a very important concept in the study of gravitational theories, and especially in the context of holographic theories. For a fixed set of boundary conditions, the Asymptotic Symmetry Group (ASG) is the group of allowed diffeomorphisms modded out by the trivial ones (trivial diffeomorphisms are ones that lead to vanishing canon- ical charges). In a quantum theory of gravity, the states of the theory form representations of the ASG. The ASG also dictates the symmetries of the putative holographically dual field theory.

Infinite dimensional ASGs turn out to be very effective in understanding aspects of the dual field theory. The most studied example of this is the ASG of AdS3, which turns out to

– 3 –

be two copies of the infinite dimensional Virasoro algebra. This leads to the conclusion that the dual field theory is a 2d CFT. The canonical analysis by Brown and Henneaux can actually be looked upon as the birth of the AdS/CFT correspondence .

Interestingly, infinite dimensional ASGs have been known to exist in the context of Minkowski spacetimes long before the discovery of Brown and Henneaux. Bondi, van der Burg, Metzner and independently Sachs studied the asymptotic structure of Minkowski spacetime in 4 dimensions at its null boundary and found to their surprise that the symmetries were not dictated by the Poincare group, but an infinite dimensional group which included over and above the Poincare generators, translations of the null direction that depended on the angles of the sphere at infinity. These were called supertranslations and in spite of many efforts to do away with them, it was found that the algebra could not be truncated to just the Poincare algebra. The asymptotic symmetry algebra takes the form

(1.3C)

Here n, m range from −1 to +1 while the other variables can take all integral values. The generators Mr,s are the super-translation generators, the translations that depend on the angles of the sphere at infinity.

Later, inspired by possible links to holography, Barnich and Troessaert proposed an extension of the ASG of 4d Minkowski space to include what they called super-rotations. Superrotations are group of all the conformal generators of the sphere at infinity and this extension is essentially the same as the extension of the 2d conformal algebra to include all the generators of the Virasoro algebra from the globally well-defined ones L0,±1. In the above algebra, this means that n, m now take all integral values.

This Extended Asg Of

4d flatspace is now what is commonly known as the Bondi-Metzner-Sachs (BMS) group. Recently, following Strominger and collaborators , a beautiful story has emerged linking BMS symmetries to soft theorems and memory effects . We refer the reader to for a detailed discussion of these aspects.

In the present paper, we are interested in the ASG of 3d Minkowski spacetimes. At null infinity, the ASG is given by the BMS3 algebra , which, as we have mentioned above, takes the form (1.2). For Einstein gravity, the central terms are cL = 0, cM =

4G. When One

considers modifications to Einstein gravity with a gravitational Chern-Simons term, a theory that goes under the name of Topological Massive Gravity, the ASG remains the same but central terms change and cL and cM are now both non-zero. Putative duals to theories with 3d gravity with asymptotically flat boundary conditions would thus be given by (1.2) with two non-zero central terms. A review of some progress in flat holography in general and in 3d in particular can be found in [30, 31]. An incomplete list of interesting directions that have been explored in this context are – .

– 4 –

Our principle goal in this paper is the following. We would like to attempt to constrain 2d field theories with BMS3 symmetry and hence chart out a parameter space for all possible putatively dual theories to asymptotically Minkowskian spacetimes in 3d.

Non-Relativistic Conformal Symmetries

We live in a world where the everyday things are governed principally by non-relativistic physics. Galilean invariant theories thus are a very good approximation for many real life applications. Thus it is vitally important to understand Galilean field theories. In analogy with relativistic QFTs, it is thus interesting to answer whether all Galilean QFTs can be understood as renormalization group flows away from fixed points governed by the analogue of conformal symmetry. Galilean Conformal Field Theories (GCFT), i.e. field theories with GCA as their symmetry algebra, arise as contractions from relativistic CFTs . It is thus very natural to expect that these non-relativistic fixed points in the parameter space of all Galilean QFTs will be governed by the GCA.

Through the intriguing link of the BMS/GCA correspondence , our programme thus is very useful when we consider applications to non-relativistic QFTs in 2d. This analysis, carried out to its conclusion, would thus help classify all 2d GCFTs and hence lead to an understanding of all 2d Galilean QFTs.

It is interesting here to comment on possible extensions to higher dimensions. It has been claimed in that the GCA is infinite dimensional in all spacetime dimensions. This follows from the observation that the finite contracted algebra can be written in a suggestive form and given an infinite lift in any dimensions. The rather astounding claim is that the non-relativistic limit of a CFT leads to a theory which has an infinite dimensional symmetry.

The infinite-dimensional GCA in any arbitrary spacetime dimensions is given by

M] = (N −M)Mi

n+m.

(1.4B)

Interestingly, it has been shown that field theories like Maxwell’s theory and Yang-Mills theory, which are classically conformally invariant in D = 4, have non-relativistic versions that exhibit this infinite dimensional symmetry in the Galilean regime [34, 35] 1. Some recent investigations reveal that this classical symmetry enhancement is rather generic and happens in many cases where there are relativistic conformal symmetries to begin with. If there are field theories which also exhibit this infinite dimensional symmetry quantum mechanically, then these systems would be extremely interesting. They could be looked upon as closed sub-sectors in relativistic CFTs that perhaps have the promise to becoming integrable.

In the context of the bootstrap in these higher dimensional theories, it is very possible that our methods here would generalise in a rather simple way to any dimensions. The additional power of infinite symmetries would help in the restriction of the higher dimensional theories.

1The reader is referred to for a slightly different take on infinite symmetries in non-relativistic electro- dynamics. Here the authors claim to have a bigger infinity of symmetries that include the GCA and in all dimensions, not only D = 4.

Outline Of The Paper

The rest of the paper is organised as follows. In Sec. 2, we give a short summary of the conformal bootstrap programme, specifically focussing on 2d CFTs. This forms a basis for the analysis we will perform for the 2d field theories with BMS symmetry.

In Sec. 3, we look at the 2d field theories with BMS3 symmetries in an intrinsic way. This means that we formulate the analogues of the conformal bootstrap analysis by relying solely on the symmetry structure of the field theory. Some of the results in this section have been reported earlier in . Here we provide a detailed analysis of those results as well as some more new results which were promised but not presented in .

In Sec. 4, we first discuss the two different limits, viz. the non-relativistic and the ultra- relativistic, of the two copies of the Virasoro algebra to BMS3. We then concentrate on the non-relativistic limit and recover many of the results of Sec. 3 in terms of this limit of the relativistic CFT answers. This serves as a comprehensive check of our results and also stresses the importance of the existence of this limit.

In Sec. 5, we look at a specific subsector of the BMS3 algebra, where the symmetry algebra has previously been shown to reduce to the Virasoro sub-algebra . We observe that with the specific restrictions on the operator weights and central charges, the bootstrap analysis is consistent with this earlier claim.

We conclude in Sec. 6 with a summary of the paper, some discussions and a list of future directions.

The Conformal Bootstrap

In this section, we revisit some aspects of the conformal bootstrap, which we will specifically need for our analysis in the BMS bootstrap. We will confine ourselves to 2d CFTs, which are governed by two copies of the Virasoro algebra (1.1). More details can be found in the original BPZ paper or in some standard CFT text books [37, 38].

We will be work exclusively on the plane and hence the form of the generators of the 2d

(2.1)

We define a unique vacuum state in the theory |0⟩. One defines a state-operator correspon-

(2.2)

The states in a CFT are labelled by their weights under L0 and ¯L0:

(2.3)

One defines a notion of primary fields as the ones which are annihilated by all positively

(2.4)

The representations of the Virasoro algebra, called Verma modules, are built by acting on primary fields by raising operators L−n, ¯L−n. A general state in a CFT is given by:



(z, ¯z).

Operator Product Expansion

The two and three point functions of primary states are fixed up to constants by invariance under the global part of the algebra L0,±1, ¯L0,±1. The two-point function is given by:

(2.6)

The three point function of primary fields is given by:

(2.7)

where hijk = −(hi + hj −hk). The operator product expansion (OPE) of two primary

I Iki, ¯K = P

j j¯kj. Using the OPE to find the three-point function it can be

12

.

(2.9)

The coefficients B can be obtained by demanding that both sides of the OPE transform the same way under the action of Lm and ¯Ln. These coefficients for level one and level two are shown in Table (1).

¯C−10¯Hp+2¯C¯Hp+16¯H2P

Table 1. Coefficients of OPE at level 1 and level 2.

Conformal Blocks And Crossing Symmetry

Invariance under global conformal symmetry is not enough to fix the four-point functions of primary fields. Global invariance can help fix the form of these correlators up to a function of the conformally invariant cross ratios given below. The four-point function has the form

(2.10)

where FCFT (z, ¯z) is an arbitrary coefficients of the cross-ratios z and ¯z

(Z1 −Z3)(Z2 −Z4), ¯Z = (¯Z1 −¯Z2)(¯Z3 −¯Z4)

(¯z1 −¯z3)(¯z2 −¯z4).

(2.11)

We can always do a global conformal transformation such that {(zi, ¯zi)} →{(∞, ∞), (1, 1), (z, ¯z), (0, 0)}.

G21

34(z, ¯z) = ⟨h1, ¯h1|φ2(1, 1)φ3(z, ¯z)|h4, ¯h4⟩.

(2.14)

Using the OPE on φ3 and φ4 inside the correlator, the function G21

34(Z, ¯Z) Can Be Written In

terms of three-point functions of primaries and their descendants. Specifically, we have

12A12

34(p|z, ¯z).

The Blocks A21

34(p|z, ¯z) factorizes into a holomorphic and an anti-holomorphic parts

⟨H1|Φ2(1)|Hp⟩

.

(2.17)

Inside the correlator we can move the operators around which does not matter except for fermions which would introduce a sign. So apart from G21

⟨Φ1(Z1, ¯Z1)Φ4(1, 1)Φ3(Z, ¯Z)Φ2(0, 0)⟩= G41

32(z, ¯z) = ⟨h1, ¯h1|φ4(1, 1)φ3(z, ¯z)|h2, ¯h2⟩.

34(Z, ¯Z) = G41

32(1 −z, 1 −¯z).

(2.19)

If we expand both sides in term of the conformal blocks we have the bootstrap equation

32A41

32(p|1 −z, 1 −¯z).

Global Conformal Blocks

The large central charge limit of the Virasoro algebra simplifies a lot of the analysis in 2d CFTs and has been recently pursued actively starting from . There are particular simplifications for the conformal blocks. The global conformal block is the large central charge limit of the

⟨¯H1|Φ2(1)|¯Hp⟩

.

(2.21)

The closed form expression of this can be obtained by using the constraint that both sides of the OPE (2.8) transform the same way under the quadratic Casimirs [5, 6]

(2.22)

of the global subgroup generated by {L0,±1, ¯L0,±1}.

For Simplicity, Take All The External

operators to be identical and to be a scalar with dimension ∆φ. We may write the global

G∆Φ(P|Z, ¯Z) = Z−∆Φ ¯Z−∆Φkhp, ¯

hp(z, ¯z).

(2.23)

The constraint that both sides of the OPE transform the same way under the two quadratic Casimirs gives two differential equations for Khp, ¯



Khp,¯hp(z, ¯z) = ¯hp(¯hp + 1)Khp,¯hp(z, ¯z).

(2.25)

the solution are given in terms of gauss hypergeometric function Khp(z) = αz−hp 2F1 (−hp, −hp; −2hp; z) + γzhp+1 2F1 (hp + 1, hp + 1; 2hp + 2; z)

(2.26)

and similarly for the anti-holomorphic sector. We expand around z = 0 and match to the



.

.

34(P|Z, ¯Z) = Zhp−H1−H2 ¯Z

¯hp−¯h1−¯h2 2F1(hp + h12, hp + h34; 2hp; z) 2F1(¯hp + ¯h12, ¯hp + ¯h34; 2¯hp; ¯z),

(2.29)

where hij = hi −hj, ¯hij = ¯hi −¯hj.

Bootstrapping Bms Symmetries: Intrinsic Analysis

In this section, we will construct the bootstrap programme for field theories with BMS sym- metries through an intrinsic method. This just means that we will be inspired by the methods of 2d CFTs that we outlined in the previous section, but there will be many crucial differences, as the symmetry algebra (1.2) is fundamentally different from two copies of the Virasoro alge- bra (1.1). In the subsequent section we will provide a limiting analysis where we consider the contraction of (1.1) to (1.2) and we will recover some of the answers of this section through the limit. Some of the central results of this section have already appeared in . In this paper, and particularly in this section, we provide a much more detailed exposition of the basic analysis presented earlier. There are a number of new mathematical details and results that are presented here.

Highest Weight Representations

We consider 2d field theories that are invariant under the BMS3 algebra. We will call the directions of the field theory (u, v). We will be interested in representation of the algebra

(3.1)

This will be called the “plane” representation of the BMS3 algebra. The states of the BMS invariant 2d field theory are by their weights under L0. Since M0 and L0 commute, the states get an additional label under M0 as well.

(3.2)

Like in usual 2d CFTs, we will build the representation theory by first defining BMS primary operators. We do this by demanding that the spectrum (defined with respect to ∆) be bounded from below. Then the BMS primary operators |∆, ξ⟩p are the ones for which

Ln|∆, Ξ⟩P = Mn|∆, Ξ⟩P = 0

∀n > 0.

(3.3)

We will assume a state-operator correspondence in the case of BMS field theories as well. While this is not strictly necessary for our analysis, it would be good to have the freedom to talk about operators and states interchangeably. The BMS modules, very much like the Verma modules in the case of the Virasoro algebra, are built by acting creation operators on the BMS primary states.

Operator Product Expansion

The main objects of physical interest in field theory are the correlation functions.

If We

know all the correlation functions, we may say that we have completely solved the theory. In finding the form of these functions, symmetries play an important role. It is interesting to know which part of the correlation functions is fixed by symmeties alone and what other parts depend on the dynamics of the theory. In particular, for BMS-invariant theories, the

– 11 –

co-ordinate dependence of the two-point and three-point functions are completely fixed simply by invariance under the global subgroup of the BMS group i.e., co-ordinate transformation generated by L0,±1, M0,±1. The two-point function is given by [12, 42]

(3.4)

The normalisation of the 2-point function has been fixed to δ12. The three-point function is

U13 . (3.5)

Here ∆ijk = −(∆i + ∆j −∆k) and ξijk is defined similarly. C123 is an arbitrary parameter called the structure constant. It is not fixed by symmetry but depends on the dynamics (or the details) of the field theory under consideration. So, if these constants are given to us, we can completely determine the three-point function by symmetry consideration alone.

We can also consider higher correlation functions and see how much of their form are fixed by symmetry alone and what other dynamical inputs are needed to fixed the rest. Now, all information about the correlation functions are contained in the operator product algebra, which gives the operator product expansion (OPE) of two primary fields as summation over the primaries and towers of their descendants.

So, In Order To Know How The Correlation

functions are constrained by symmetries, it is enough to study constraints on the OPE. Indeed, considering these symmetries, we make the following ansatz for the OPE of two

P

(u2, v2).

(3.6)

Our notation is that for vectors −→k = (k1, k2, ...kr) and −→q = (q1, q2, ...qs), descendant fields

(U, V) Is A Descendant Field At Level K + Q. For

ease of calculation we can take the point (u2, v2) in (3.6) to be the origin, giving us

– 12 –

Here the form of the factor u−∆1−∆2+∆p e(ξ1+ξ2−ξp) v

U Is Fixed By The Requirement That The

OPE gives the correct two-point function and the factor PK+Q

Uk+Q−Αvα Is To

ensure that both sides of the OPE transform the same way under the action of L0. To verify this second requirement, let us act both sides of (3.8) on the the vacuum |0, 0⟩and then see the action of L0 on the resulting state. On the LHS we have, L0φ1(u, v)φ2(0, 0)|0, 0⟩= ([L0, φ1(u, v)] + φ1(u, v)L0)φ2(0, 0)|0, 0⟩ = (u∂u + v∂v + ∆1 + ∆2)φ1(u, v)φ2(0, 0)|0, 0⟩.

(3.9)

So, if the OPE is correct, the RHS of equation (3.8) must also transform as above

K M−

→q |∆p, ξp⟩.

K M−

→q |∆p, ξp⟩.

(3.12)

Thus, equation (3.10) is satisfied, which means that both sides of OPE transform the same way under the action of L0. Furthermore, using the OPE inside the three-point functions and comparing the coefficients with (3.5) it can be seen that

≡Cp

12 = Cp12.

12

= 1.

12

can be calculated by demanding that both sides of (3.8) trans- form the same way under the other generators Lm and Mn. Thus, the form of the OPE is

– 13 –

completely constrained by symmetries to depend only on external inputs, such as the struc- ture constants, the spectrum of primary operators appearing in the OPE, and the central charge. In other words, if these dynamical inputs are given to us, we can use symmetries to calculate all the correlation functions in a BMS-invariant field theory. These dynamical inputs can be used to classify and completely specify a given BMS-invariant field theory.

However, any random sets of these dynamical inputs need not constitute a consistent field theory; they must satisfy a constrain equation given by the BMS bootstrap equation, which arises as a condition for the associativity of the operator product algebra.

Recursion Relations

Now let us try to find recursion relations for evaluating the coefficients βp{−

. For The

sake of simplicity we will consider the case ∆1 = ∆2 = ∆, ξ1 = ξ2 = ξ. Applying both sides

(3.17)

is a descendant state at level N in the BMS module, L0|N, α⟩p = (∆p + N)|N, α⟩p.

(3.18)

We now act with the generators Ln>0 on both sides of sides of equation (3.8) and demand that they should transform in the same way. On the LHS, we have

Lnφ1(U, V)|∆, Ξ⟩= [Ln, Φ1(U, V)]|∆, Ξ⟩

= [un+1∂u + (n + 1)unv∂v + (n + 1)(∆un −nξun−1v)]φ1(u, v)|∆, ξ⟩.

(3.19)

Substituting the RHS of (3.8) in the above equation, we have

|N, α⟩p.

|N, α −1⟩p.

(3.21)

Similarly, demanding that both sides of the OPE transform the same way under M0 and

(3.22)

Mn|N + n, α⟩p = ((n −1)ξ + ξp) |N, α⟩p −(α + 1)|N, α + 1⟩p.

(3.23)

These three recursion relations can be used to find all the coefficients βp{−

. We Have

shown this calculation for level 1 and level 2 in the next section.

12

|∆p, ξp⟩= |∆p, ξp⟩.

12

M−1|∆p, ξp⟩, α = 0, 1.



|∆p, ξp⟩.

2

Table 2. Coefficients of OPE at level 1.

12

.

(3.31)

Now, using the recursion relation (3.23) with N = 0, n = 1, α = 0, we have

= −1

2.

(3.33)

With N = 0, n = 1, α = 0, (3.21) gives the recursion relation

12

= 0.

(3.35)

The various coefficients are collected above in Table (2). We can see that these match with the coefficients in (A.7) of .

Level 2

The details of the relevant calculations at level 2 are presented in Appendix A. We collect all these coefficients in Table (3).

= −36C2

M(1+∆p)+24cM(3ξ+∆p(3ξ−2ξp)+(1−3∆)ξp)+ξp(−60ξ+∆p(96ξ−4ξp)+5ξp−48∆ξp+18cL(4ξ+ξp))

12

= 36ξ−24ξcL−18cM+24∆cM−16ξ∆p+6cM∆p−3ξp+16∆ξp−6cLξp

8

Table 3. Coefficients of OPE at level 2.

Bms Blocks, Crossing Symmetry And Bootstrap

We have seen that BMS-invariant theories are completely specified by the structure constants, the spectrum of primary fields, and the central charge. However any given sets of these inputs need not always constitute a consistent theory; they have to satisfy an infinite set of equations analogous to the conformal case which we will call the BMS bootstrap equation. This equation comes from self consistency of the OPE, namely that it has to be associative when applying inside the correlator. More precisely, if we use the OPE inside the correlator, the resulting correlator should not depend on which two neighbouring primary operators we applied the OPE. We will study this requirement by considering the four-point function, which for a

(3.36)

where the BMS analogues of the cross ratio u and v given by

(3.37)

are invariant under the global coordinate transformation generated by L0,±1, M0,±1. We can conveniently do a global coordinate transformation such that

(3.39)

which in terms of the in and out states is given by

G21

34(u, v) = ⟨∆1, ξ1|φ2(1, 0)φ3(u, v)|∆4, ξ4⟩.

(3.42)

So, the four-point function can be expressed in terms of G21

Φi(Ui, Vi)⟩= P({∆I, Ξi, Uij, Vij})F(U, V)−1G21

34(u, v).

(3.44)

and it can be easily seen that these functions Gkl

34(U, V) = G41

32(1 −u, −v).

(3.45)

It is important to emphasise here that the crossing equation that we have obtained above is not the same as the usual conformal crossing equation (2.19). If we use the OPE between the fields φ3 and φ4 in G21

34(U, V) We Can See That The Function

can be expressed in terms of the three-point functions of primary fields and their descendants. More precisely, using the OPE, it can be decomposed as

Where The Four-Point Conformal Block A21

34(p|u, v) is the sum of all contributions coming from the primary field φp and its descendants and is given by2

Can Be Calculated Recursively Using Bms

symmetry. Thus, the closed form expression of the BMS blocks are completely determined by symmetry and the only dynamical inputs needed to find the four point functions are the structure constants and the spectrum of primary operators appearing in the OPE.

For The Function G41

32(u, v) we may use the OPE on φ2 and φ3 giving us the expansion

⟨∆1, Ξ1|Φ4(1, 0)|∆P, Ξp⟩

.

(3.49)

2It should be noted that we can only apply the OPE between neighbouring primary fields, so it is understood that the point (u, v) lies between the origin and a circle of radius 1.

– 18 –

Now, (3.45) must be satisfied, even if we expand both sides using OPE in terms of the BMS

41A41

32(q|1 −u, −v).

(3.50)

This is one of the main initial results of our analysis. Knowing the BMS blocks, the above equation put a constrain on the structure constants and weights of primary operators in a consistent field theory with BMS symmetry. We can try to solve the bootstrap equation to find all such possible consistent field theories. The only problem is that we do not have a closed form expression of the blocks even though they are fixed by symmetry alone. However, we can find the leading term in a

Cl,M Expansion Of The

blocks. Using this expansion on both sides of (3.50), the equation has to be satisfied order by order. The leading order give us the constraint

Where Gkl

ij (p|u, v) are the large central charge limit of the blocks Akl

Cl,M→∞Akl

ij(p|u, v).

We Will Find Gkl

ij (p|u, v) in the next section.

3.6

Differential equations for global blocks from quadratic Casimirs For even dimensional CFTs with d ≥4, the closed form expression of the four point conformal blocks was obtained for scalar operators by Dolan and Osborn in [5, 6]. For 2d CFTs, their method gives the global conformal blocks, which is the large central charge limit of the full Virasoro conformal blocks, as we have mentioned in the previous section. In this section we will employ this method to obtain the global blocks for BMS algebra, assuming that such a limit will act in a similar manner.

If we take the asymptotic limit cL, cM →∞in the OPE (3.8), (3.6), the leading terms are given by the descendant fields generated by L−1 and M−1. This can be explicitly seen by looking at the coefficients β in the limit cL, cM →∞. For levels 1 and 2, this can be verified by the results obtained in previous sections and outlined in Table (2) and Table (3). More

Where The Global Block G21

34(p|u, v), which is the large central charge limit of G21

⟨∆1, Ξ1|Φ2(1, 0)|∆P, Ξp⟩

.

(3.56)

with the four-point function given by the 1/cL,M expansion

34(P|U, V) = P({∆I, Ξi, Uij, Vij})F(U, V)−1G21

34(p|u, v).

It Is Possible To Find The Blocks G21

34(p|u, v) by demanding that both sides of the OPE transform the same way under the action of the quadratic Casimirs belonging to the global algebra generated by {L−1, L0, L1, M−1, M0, M1}. These Casimirs are given by

(3.60)

It can be seen that the states (L−1)k(M−1)qφ4(u4, v4)|0, 0⟩are eigenstates of C1 and C2 since

Λp

2 = (2∆pξp −2ξp).



+ ....

(3.63)

After taking the inner product on both sides with ⟨φ1(u1, v1)φ2(u2, v2)|, we have



+ ....

(3.64)

On the LHS of the above equation, C1,2 act as differential operators D1,2. More precisely,

0 −M−1M1)Φ3(Y3)Φ4(Y4)|0⟩

= [(−u3∂v3 + ξ3 −u4∂v4 + ξ4)(−u3∂v3 + ξ3 −u4∂v4 + ξ4)

4∂V4 + 2Ξ4U4)]Φ3(Y3)Φ4(Y4)|0⟩

= [2ξ3(u3 −u4)∂v4 −2ξ4(u3 −u4)∂v3 + (ξ3 + ξ4)2 −(u3 −u4)2∂v3∂v4](φ3(y3)φ4(y4))|0⟩

2(L−1M1 + L1M−1 + M1L−1 + M−1L1)]Φ3(Y3)Φ4(Y4)|0⟩

= [2(∆3 + ∆4 −1)(ξ3 + ξ4) + (−2u3ξ3 + 2u4ξ3)∂u4 + (2u3ξ4 −2u4ξ4)∂u3 +(−2u3∆4 + 2u4∆4 + 2v3ξ4 −2v4ξ4)∂v3 + (2u3∆3 −2u4∆3 −2v3ξ3 + 2v4ξ3)∂v4

4)∂V3∂U4

+(2u3v3 −2u4v3 −2u3v4 + 2u4v4)∂v3∂v4]φ3(y3)φ4(y4)|0⟩ ≡D2(φ3(y3)φ4(y4))|0⟩.

(3.66)

Pulling the differential operator outside the four-point function, we have



+....



+ ....

– 21 –

This equation has to be satisfied order by order. The leading order give us a differential

(3.69)

We can decouple this to get differential equations for each block ˜g21

1,2 ˜G21

34(p|u, v).

1,2 P({∆I, Ξi, Uij, Vij})F(U, V)−1G21

34(p|u, v).

(3.71)

Let us first look at the differential equation associated with C1, which is given by

K=1 Ξk−Ξi−Ξj) F(U, V)−1G21

34(p|u, v).

(3.72)

For simplicity let us consider the case where ∆i=1,2,3,4 = ∆, ξi=1,2,3,4 = ξ. Then the above

(3.73)

where we have used the notation g∆,ξ(p|u, v) for the blocks g21

34(P|U, V) In This Special Case And

D1 is also taken with ξi=1,2,3,4 = ξ. If we combine the functions of u and v into

(3.76)

where uij = ui −uj. Under the global conformal transformation (3.38), the above equation

ˆG∆,Ξ(P|U, V) = 0.(3.77)

In terms of the global blocks g∆,ξ(p|u, v), the above equation is given by

P) −4Ξu(U −1)∂V + U2(U −1)∂2

v]g∆,ξ(p|u, v) = 0.

(3.78)

The differential equation gets simpler if we introduce a function

K(P|U, V) = U2∆E−2Ξv

u g∆,ξ(p|u, v).

(3.79)

Plugging this back into the above equation, we have the simplified version:

#

h(p|u, v) = 0.

(3.80)

Now let us look at the differential equation associated with C2. For simplicity we again only consider the case where ∆i=1,2,3,4 = ∆, ξi=1,2,3,4 = ξ. We have,

2Ξ

3 ˆg∆,ξ(p|u, v).

(3.81)

Under the global conformal transformation (3.38), the above differential equation reduces to [(2ξ(−6 + 8∆−2u3∆−2u(−6 + 8∆+ vξ) + u2(−6 + 10∆+ vξ))

−9(−1 + U)2(−1 + ∆P)Ξp) −12(−2 + U)(−1 + U)2Uξ∂U

+3(−1 + u)2(u2(6 + 4∆) + 8vξ −8u(∆+ vξ))∂v + 18(−1 + u)3u2∂v∂u

+9(−1 + U)2U(−2 + 3U)V∂2

v] ˆg∆,ξ(p|u, v) = 0.

(3.82)

In terms of the global blocks g∆,ξ(p|u, v) = (1 −u)−2∆

3U ˆG∆,Ξ(P|U, V), We Have

[2(2ξ(−1 + 2∆−2u∆+ vξ) −(−1 + ∆p)ξp) + 2(u2(1 + 2∆) + 2vξ −2u(∆+ 2vξ))∂v

+U(−2 + 3U)V∂2

v −4(−1 + u)uξ∂u + 2(−1 + u)u2∂v∂u]g∆,ξ(p|u, v) = 0.

(3.83)

In terms of the function k(p|u, v), the differential equation again gets simpler



k(p|u, v) = (∆p −1)ξp k(p|u, v).

Solution Of The Bms Global Block

In this subsection, we find the explicit solution for the differential equation for the global BMS block. The general solutions of the above differential equation (3.80) are given by

Ξp

t√1−u v.

(3.85)

Substituting (3.85) in the second differential equation (3.84) we have

Du

= 0.

(3.87)

The solutions of the above differential equations are given by

(3.88)

where KA and KB are constant of integration. So the most general solution of h(p|u, v) is

U√1−U V. (3.89)

Note again that the blocks are defined only for u2 + v2 < 1. So we don’t have to consider the case u > 1, where the above equation becomes oscillatory. Now, we need boundary conditions to find the constant of integration. Looking at (3.55),

⟨∆, Ξ|Φ(1, 0)|∆P, Ξp⟩

.

(3.90)

Let us show a few of the terms in the summation. We know that βp{0,0},0

⟨∆, Ξ|Φ(1, 0)|∆P, Ξp⟩

= ξp.

2 U −Ξp

2 v + ...



.

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

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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