Process Induced Distortion Ansys

Quantum rate distortion, reverse Shannon theorems,

And Source-Channel Separation

Nilanjana Datta, Min-Hsiu Hsieh, and Mark M. Wilde Abstract—We derive quantum counterparts of two key the- orems of classical information theory, namely, the rate distor- tion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In the present paper, we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate distortion function. This implies that the unassisted quantum rate distortion function is non-negative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum’s conjecture that the coherent information would be relevant here). Moreover, we prove several quantum source- entanglement-assisted setting, in which we establish a necessary and sufficient codition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.

Index Terms—quantum rate distortion, reverse Shannon the- orem, quantum Shannon theory, quantum data compression,

Ntroduction

Two pillars of classical information theory are Shannon’s data compression theorem and his channel capacity theorem

, . The Former Gives A Fundamental Limit To The

compressibility of classical information, while the latter deter- mines the ultimate limit on classical communication rates over a noisy classical channel. Modern communication systems exploit these ideas in order to make the best possible use of communication resources.

Data compression is possible due to statistical redundancy in the information emitted by sources, with some signals being emitted more frequently than others. Exploiting this redun- dancy suitably allows one to compress data without losing essential information. If the data which is recovered after the compression-decompression process is an exact replica Kingdom. The contribution of M.-H. H. was mainly done when he was Sydney (UTS), PO Box 123, Broadway NSW 2007, Australia. Mark M. Wilde Canada H3A 2A7.

of the original data, then the compression is said to be lossless. The simplest example of an information source is a memoryless one. Such a source can be characterized by a random variable U with probability distribution {pU(u)} and each use of the source results in a letter u being emitted with probability pU(u). Shannon’s noiseless coding theorem states

U Pu (U) Log2 Pu (U) Of Such

an information source is the minimum rate at which we can compress signals emitted by it , . The requirement of a data compression scheme being loss- less is often too stringent a condition, in particular for the case of multimedia data, i.e., audio, video and still images or in scenarios where insufficient storage space is available.

Typically a substantial amount of data can be discarded before the information is sufficiently degraded to be noticeable. A data compression scheme is said to be lossy when the decompressed data is not required to be identical to the original one, but instead recovering a reasonably good approximation of the original data is considered to be good enough.

The theory of lossy data compression, which is also referred to as rate distortion theory, was developed by Shannon , , . This theory deals with the tradeoff between the rate of data compression and the allowed distortion. Shannon proved that, for a given memoryless information source and a distortion measure, there is a function R(D), called the rate-distortion function, such that, if the maximum allowed distortion is D then the best possible compression rate is given by R(D). He established that this rate-distortion function is equal to the minimum of the mutual information I(U; ˆU) := H (U) + H( ˆU) −H(U, ˆU) over all possible stochastic maps p ˆU|U (ˆu|u) that meet the distortion requirement on average:

(1)

In the above d(U, ˆU) denotes a suitably chosen distortion measure between the random variable U characterizing the source and the random variable ˆU characterizing the output of the stochastic map.

Whenever the distortion D = 0, the above rate-distortion function is equal to the entropy of the source. If D > 0, then the rate-distortion function is less than the entropy, implying that fewer bits are needed to transmit the source if we allow for some distortion in its reconstruction.

Alongside these developments, Shannon also contributed the theory of reliable communication of classical data over clas- sical channels , . His noisy channel coding theorem gives an explicit expression for the capacity of a memoryless

Arxiv:1108.4940V3 [Quant-Ph] 19 Aug 2012

classical channel, i.e., the maximum rate of reliable communi- cation through it. A memoryless channel N is one for which there is no correlation in the noise acting on successive inputs, and it can be modelled by a stochastic map N ≡pY |X (y|x).

Shannon proved that the capacity of such a channel is given

(N) = Max

pX(x) I (X; Y ) . Any scheme for error correction typically requires the use of redundancy in the transmitted data, so that the receiver can perfectly distinguish the received signals from one another in the limit of many uses of the channel.

Given all of the above results, we might wonder whether it is possible to transmit an information source U reliably over a noisy channel N, such that the output of the infor- mation source is recoverable with an error probability that is asymptotically small in the limit of a large number of outputs of the information source and uses of the noisy channel. An immediate corollary of Shannon’s noiseless and noisy channel coding theorems is that reliable transmission of the source is possible if the entropy of the source is smaller than the

(2)

The scheme to demonstrate sufficiency of (2) is for the sender to take the length n output of the information source, compress it down to nH (U) bits, and encode these nH (U) bits into a length n sequence for transmission over the channel. As long as H (U) ≤C (N), Shannon’s noisy channel coding theorem guarantees that it is possible to transmit the nH (U) bits over the channel reliably such that the receiver can decode them, and Shannon’s noiseless coding theorem guarantees that the decoded nH (U) bits can be decompressed reliably as well in order to recover the original length n output of the information source (all of this is in the limit as n →∞). Given that the condition in (2) is sufficient for reliable communication of the information source, is it also necessary? Shannon’s source-channel separation theorem answers this question in the affirmative , .

The most important implication of the source-channel sepa- ration theorem is that we can consider the design of compres- sion codes and channel codes separately—a two-stage encod- ing method is just as good as any other method, whenever the source and channel are memoryless. Thus we should consider data compression and error correction as independent prob- lems, and try to design the best compression scheme and the best error correction scheme. The source-channel separation theorem guarantees that this two-stage encoding and decoding with the best data compression and error correction codes will be optimal.

Now what if the entropy of the source is greater than the capacity of the channel? Our best hope in this scenario is to allow for some distortion in the output of the source such that the rate of compression is smaller than the entropy of the source. Recall that whenever D > 0, the rate-distortion function R (D) is less than the entropy H (U) of the source. In this case, we have a variation of the source-channel separation theorem which states that the condition R (D) ≤C (N) is both necessary and sufficient for the reliable transmission of an information source over a noisy channel, up to some amount of distortion D . Thus, we can consider the problems of lossy data compression and channel coding separately, and the two-stage concatenation of the best lossy compression code with the best channel code is optimal.

Considering the importance of all of the above theorems for classical information theory, it is clear that theorems in this spirit would be just as important for quantum information theory. Note, however, that in the quantum domain, there are many different information processing tasks, depending on which type of information we are trying to transmit and which resources are available to assist the transmission. For example, we could transmit classical or quantum data over a quantum channel, and such a transmission might be assisted by entanglement shared between sender and receiver before communication begins.

There have been many important advances in the above directions (some of which are summarized in the recent text ). Schumacher proved the noiseless quantum coding theorem, demonstrating that the von Neumann entropy of a quantum information source is the ultimate limit to the compressibility of information emitted by it . Hayashi et al. have also considered many ways to compress quantum information, a summary of which is available in Ref. .

Quantum rate distortion theory, that is the theory of lossy quantum data compression, was introduced by Barnum in 1998. He considered a symbol-wise entanglement fidelity as a distortion measure and, with respect to it, defined the quantum rate distortion function as the minimum rate of data compression, for any given distortion. He derived a lower bound on the quantum rate distortion function, in terms of well-known entropic quantity, namely the coherent informa- tion. The latter can be viewed as one quantum analogue of mutual information, since it is known to characterize the

Quantum Capacity Of A Channel , , , Just As The

mutual information characterizes the capacity of a classical channel. It is this analogy, and the fact that the classical rate distortion function is given in terms of the mutual information, that led Barnum to consider the coherent information as a candidate for the rate distortion function in the quantum realm.

He also conjectured that this lower bound would be achievable. Since Barnum’s paper, there have been a few papers in which the problem of quantum rate distortion has either been

, . However, Not Much Progress Has Been Made In

proving or disproving his conjecture. In fact, in the absence of a matching upper bound, it is even unclear how good Barnum’s bound is, given that the coherent information can be negative, as was pointed out in , .

There are also a plethora of results on information trans- mission over quantum channels. Holevo , Schumacher, and Westmoreland provided a characterization of the classical capacity of a quantum channel. Lloyd , Shor , and Devetak proved that the coherent information of a quantum channel is an achievable rate for quantum communication over that channel, building on prior work of

Nielsen And Coworkers , , , Who Showed That

its regularization is an upper bound on the quantum capacity (note that the coherent information of a quantum channel is always non-negative because it involves a maximization over all inputs to the channel). Bennett et al. proved that the mutual information of a quantum channel is equal to its entanglement-assisted classical capacity (the capacity whenever the sender and receiver are given a large amount of shared entanglement before communication begins).

In Ref. , the authors also introduced the idea of a reverse Shannon theorem, in which a sender and receiver simulate a noisy channel with as few noiseless resources as possible (later papers rigorously proved several quantum reverse Shannon theorems , , ). Although such a task might initially seem unmotivated, they used a particular reverse Shannon theorem to establish a strong converse for the entanglement- assisted classical capacity.1 Interestingly, the reverse Shannon theorems can also find application in rate distortion theory

, , , , And As Such, They Are Relevant For Our

purposes here. In this paper, we prove several important quantum rate distortion theorems and quantum source-channel separation complete characterization of the rate distortion function in an entanglement-assisted setting.2 This result really only makes sense in the communication paradigm (and not in a storage setting), where we give the sender and receiver shared en- tanglement before communication begins, in addition to the uses of the noiseless qubit channel. The idea here is for a sender to exploit the shared entanglement and a minimal amount of classical or quantum communication in order for the receiver to recover the output of the quantum information source up to some distortion. Our main result is a single-letter formula for the entanglement-assisted rate distortion function, expressed in terms of a minimization of the input-output mutual information over all quantum operations that meet the distortion constraint. This result implies that the computation of the entanglement-assisted rate distortion function for any quantum information source is a tractable convex optimization program. It is often the case in quantum Shannon theory that the entanglement-assisted formulas end up being formally analogous to Shannon’s classical formulas , , and our result here is no exception to this trend.

We next consider perhaps the most natural setting for quan- tum rate distortion in which a compressor tries to compress a quantum information source so that a decompressor can recover it up to some distortion D (this setting is the same as Barnum’s in Ref. ). This setting is most natural whenever sufficient quantum storage is not available, but we can equiva- lently phrase it in a communication paradigm, where a sender has access to many uses of a noiseless qubit channel and would like to minimize the use of this resource while transmitting a quantum information source up to some distortion. We 1A strong converse demonstrates that the error probability asymptotically approaches one if the rate of communication is larger than capacity. This is in contrast to a weak converse, which only demonstrates that the error probability is bounded away from zero under the same conditions.

2One might consider these entanglement-assisted rate distortion results to be part of the “quantum reverse Shannon theorem folklore,” but Ref. does not specifically discuss this topic.

prove that the quantum rate distortion function is given in terms of a regularized entanglement of purification in this case. In spite of our characterization being an intractable, regularized formula, our result at the very least shows that the quantum rate distortion function is always non-negative, demonstrating that Barnum’s conjecture from Ref. does not hold since his proposed rate-distortion function can become negative. Furthermore, we prove that the entanglement-assisted quantum rate distortion function is a single-letter lower bound on the unassisted quantum rate distortion function (one might suspect that this should hold because additional resources such as shared entanglement should only be able to improve compression rates). This bound implies that the coherent information between the source and distorted output is not relevant for unassisted quantum rate distortion, in spite of Barnum’s conjecture that it would be.

We finally prove three source-channel separation theorems that apply to the transmission of a classical source over a quantum channel, the transmission of a quantum source over a quantum channel, and the transmission of a quantum source over an entanglement-assisted quantum channel, respectively.

The first two source-channel separation theorems are single- letter, in the sense that they do not involve any regularised quantities, whenever the Holevo capacity or the coherent information of the channel are additive, respectively. The third theorem is single-letter in all cases because the entanglement- assisted quantum capacity is given by a single-letter expression for all quantum channels , . We also prove a related set of source-channel separation theorems that allow for some distortion in the reconstruction of the output of the information source. From these theorems we infer that it is best to search for the best quantum data compression protocols , ,

Assisted Quantum Error-Correcting Codes , , ,

independently of each other whenever the source and channel are memoryless. The theorems then guarantee that combining these protocols in a two-stage encoding and decoding is optimal.

We structure this paper as follows. We first overview rel- evant notation and definitions in the next section. Section III introduces the information processing task relevant for quan- tum rate distortion and then presents all of our quantum rate distortion results in detail. Section IV presents our various quantum source-channel separation theorems for memoryless sources and channels. Finally, we conclude in Section V and discuss important open questions.

Notation And Definitions

Let H denote a finite-dimensional Hilbert space and let D(H) denote the set of density matrices or states (i.e., positive operators of unit trace) acting on H. Let ρA ∈D(HA) denote the state characterizing a memoryless quantum information source, the subscript A being used to denote the underlying quantum system. We refer to it as the source state. Let

Ra|

is a pure state density matrix of a larger composite system RA, such that its restriction on the system A is given by ρA,

Ra, With Trr Denoting The Partial Trace Over

the Hilbert space HR of a purifying reference system R. The

Ra⟩Is Entangled If Ρ Is A Mixed State. The Von

Neumann entropy of ρA, and hence of the source, is defined

(3)

The quantum mutual information of a bipartite state ωAB is

Defined As

I (A; B)ω ≡H (A)ω + H (B)ω −H (AB)ω . The coherent information I(A⟩B)σ of a bipartite state σAB is

(4)

In quantum information theory, the most general mathemat- ical description of any allowed physical operation is given by a completely positive trace-preserving (CPTP) map, which is a map between states. We let idA denote the trivial (or identity) CPTP map which keeps the state of a quantum system A unchanged, and we let N ≡N A→B denote the CPTP map

N A→B : D(Ha) 7→D(Hb).

The entanglement of purification of a bipartite state ωAB is a measure of correlations , having an operational interpre- tation as the entanglement cost of creating ωAB asymptotically from ebits, while consuming a negligible amount of classical communication. It is equivalent to the following expression:

Abe Is Some Purification

of ωAB, and the minimization is over all CPTP maps NE acting on the system E. (The original definition in Ref. is different from the above, but one can check that the definition

Given Here Is Equivalent To The One Given There.)

In this paper we make use of resource inequalities (see e.g., ), to express information-processing tasks as inter- conversions between resources. Let [c →c] denote one for- ward use of a noiseless classical bit channel, [q →q] one forward use of a noiseless qubit channel, and [qq] one ebit of shared entanglement (a Bell state). A simple example of a

[Q →Q] ≥[Qq] ,

meaning that Alice can consume one noiseless qubit channel in order to generate one ebit between her and Bob. Teleportation is a more interesting way in which all three resources interact 2 [c →c] + [qq] ≥[q →q] .

The above resource inequalities are finite and exact, but we can also express quantum Shannon theoretic protocols as resource inequalities. For example, the resource inequality for the protocol achieving the entanglement-assisted classical

Capacity Of A Quantum Channel Is As Follows:

⟨N⟩+ H (A) [qq] ≥I (A; B) [c →c] . The meaning of the above resource inequality is that there exists a protocol exploiting n uses of a memoryless quantum channel N and nH (A) ebits in order to transmit nI (A; B) classical bits from sender to receiver. The resource inequality becomes exact in the asymptotic limit n →∞because it is possible to show that the error probability of decoding these classical bits correctly approaches zero as n →∞.

A. The Information Processing Task

The objective of any quantum rate distortion protocol is to compress a quantum information source such that the decom- pressor can reconstruct the original state up to some distortion.

Like Barnum , we consider the following distortion measure d(ρ, N) for a state ρA ∈D(HA) with purification |ψρ

(5)

where Fe is the entanglement fidelity of the map N:

(6)

The entanglement fidelity is not only a natural distortion measure, but it also possesses several analytical properties which prove useful in our analysis.

A ) Characterizes N Succes-

sive outputs of a memoryless quantum information source. A source coding (or compression-decompression) scheme of rate R is defined by a block code, which consists of two quantum operations—the encoding and decoding maps. The encoding En is a map from n copies of the source space to a subspace

Hqn),

and the decoding Dn is a map from the compressed subspace

A ).

The average distortion resulting from this compression-

N

is the “marginal operation” on the i-th copy of the source space induced by the overall operation Fn ≡Dn ◦En,

F(I)

n (ρ) ≡TrA1,A2,··· ,Ai−1,Ai+1,··· ,An[Fn(ρ⊗n)].

(7)

The quantum operations Dn and En define an (n, R) quantum rate distortion code.

For Any R, D ≥0, The Pair (R, D) Is Said To Be An

achievable rate distortion pair if there exists a sequence of (n, R) quantum rate distortion codes (En, Dn) such that

(8)

The quantum rate distortion function is then defined as Rq(D) = inf{R : (R, D) is achievable}.

(B)

Fig. 1. The most general protocols for (a) unassisted and (b) assisted quantum rate distortion coding. In (a), Alice acts on the tensor power output of the quantum information source with a compression encoding E. She sends the compressed qubits over noiseless quantum channels (labeled by “id”) to Bob, who then performs a decompression map D to recover the quantum data that Alice sent. In (b), the task is similar, though this time we assume that Alice and Bob share entanglement before communication begins.

In the communication model, if the sender and receiver have unlimited prior shared entanglement at their disposal, then the corresponding quantum rate distortion function is denoted

Eaq(D), Depending On Whether The Noiseless

channel between the sender and the receiver is classical or quantum. Figure 1 depicts the most general protocols for unassisted and assisted quantum rate distortion coding.

B. Reverse Shannon Theorems and Quantum Rate-Distortion

Oding

Before we begin with our main results, we first prove Lemma 1 below. This lemma is similar in spirit to Lemma 26

Of Ref. And Theorem 19 Of Ref. , And Like Them,

it shows that to generate a rate-distortion code, it suffices to simulate the action of a noisy channel on a source state such that the resulting output state meets the desired distortion criterion. Unlike them, however, it is specifically tailored to the entanglement fidelity distortion measure.

Lemma 1: Fix ε > 0 and 0 ≤D < 1. Consider a state ρA

Ra.

Furthermore, let {Fn}n denote a sequence of quantum oper-

Ra)⊗N

. Then for n large enough, the average distortion under the

(10)

By monotonicity of the trace distance under partial trace, we

(11)

Hence, the average distortion under the quantum operation Fn

(13)

where 0 ≤P ≤I is any positive operator and (A −B)− denotes the negative spectral part of the operator (A −B).

(14)

where the inequality follows from (13) and the definition of

(15)

which concludes the proof of the lemma. The above lemma illustrates a fundamental connection between quantum reverse Shannon theorems and quantum rate-distortion protocols. In particular, if a reverse Shannon theorem is available in a given context, then it immediately leads to a rate-distortion protocol. This is done simply by choosing the simulated channel to be the one which, when acting on the source state, yields an output state which meets the distortion criterion for the desired rate-distortion task. This is our approach in all of the quantum rate-distortion theorems that follow, and it was also the approach in Refs. , , .

There is, however, one caveat with the above approach. The reverse Shannon theorems often require extra correlated resources such as shared randomness or shared entanglement

, , , , And The Demands Of A Reverse Shannon

theorem are much more stringent than those of a rate-distortion protocol. A reverse Shannon theorem requires the simulation of a channel to be asymptotically exact, whereas a rate- distortion protocol only demands that a source be recon- structed up to some average distortion constraint. The differ- ences in these goals can impact resulting rates if sufficient correlated resources are not available .

In the entanglement-assisted setting considered in the next subsection, the assumption is that an unlimited supply of entanglement is available, and thus the entanglement-assisted quantum reverse Shannon theorem suffices for producing a good entanglement-assisted rate-distortion protocol. In the unassisted setting, no correlation is available, and exploiting the unassisted reverse Shannon theorem leads to rates that are possibly larger than necessary for the task of quantum rate distortion. Nevertheless, we still employ this approach and dis- cuss the ramifications further in the forthcoming subsections.

Entanglement-Assisted Rate-Distortion Coding

1) Rate-Distortion with noiseless classical communica-

Eac(D), For

entanglement-assisted lossy source coding with noiseless clas- sical communication, is given by the following theorem. Theorem 2: For a memoryless quantum information source defined by the density matrix ρA′, with a purification |ψρ

Aa′⟩,

and any given distortion 0 ≤D < 1, the quantum rate dis- tortion function for entanglement-assisted lossy source coding with noiseless classical communication, is given by

Aa′),

and I (A; B)ω denotes the mutual information. Proof: We first prove the converse (optimality). Consider the most general protocol for entanglement-assisted lossy source coding that acts on many copies (ρ⊗n) of the state ρ ∈D(HA) (depicted in Figure 1(b)). We take a purification

Of Ρ As |Ψρ

RA⟩. Let ΦTATB denote an entangled state, with the system TA being with Alice and the system TB being with Bob. Alice then acts on the state ρ⊗n and her share TA of the entangled state with a compression map En ≡EAnTA→W , where W is a classical system of size ≈2nr, with r being the rate of compression (in Figure 1(b), W corresponds to the outputs of the noiseless quantum channels). Then Bob acts on both the classical system W that he receives and his share TB of the entangled state with the decoding map Dn ≡DW TB→Bn. The final state should be such that it is distorted by at most D according to the average distortion criterion in the limit n →∞(8). With these steps in mind,

= I (Wtb; Rn)

≥I (Bn; Rn) . The first inequality follows because the entropy nr of the uniform distribution is the largest that the entropy H (W) can be. The second inequality follows because conditioning cannot increase entropy. The third inequality follows because H (W|RnTB) ≥0 from the assumption that W is classical.

The first equality follows from the definition of mutual infor- mation, and the second equality follows from the fact that Rn and TB are in a product state. The third equality is the chain rule for quantum mutual information. The final inequality is

Is The Marginal Operation On The I-Th Copy

of the source space induced by the overall operation Fn ≡ Dn ◦En, and is given by (7). The first inequality follows from superadditivity of quantum mutual information (see Lemma 15 in the appendix). The second inequality follows from the fact that the map Di ◦Ei has distortion d (ρ, Di ◦Ei) and the information rate-distortion function is the minimum of the mutual information over all maps with this distortion. The last two inequalities follow from convexity of the quantum rate-

Eac (D), (See Lemma 14 In The Appendix),

from the assumption that the average distortion of the protocol

Eac (D), Is Non-Increasing As A Function

of D (see Lemma 14 in the appendix). The direct part of Theorem 2 follows from the quantum reverse Shannon theorem, which states that it is possible to simulate (asymptotically perfectly) the action of a quantum channel N on an arbitrary state ρ, by exploiting noiseless clas- sical communication and prior shared entanglement between a sender and receiver , , , . The resource inequality

(18)

where the entropies are with respect to a state of the following

Is An Isometric Ex-

tension of the channel N A′→B. Our protocol simply exploits this theorem. More specifically, for a given distortion D, we take N to be the CPTP map which achieves the minimum

Eac(D). Then We Exploit Classical

communication at the rate given in the resource inequality (18) to simulate the action of the channel N on the source state ρ. For any arbitrarily small ε > 0 and n large enough, the protocol for the quantum reverse Shannon theorem simulates the action of the channel up to the constant ε (in the sense of (9)). This allows us to invoke Lemma 1 to show that the resulting average distortion is no larger than D + ε.

The main reason that we can use the quantum reverse Shannon theorem as a “black box” for the purpose of quantum rate distortion is from our assumption of unlimited shared entanglement. It is likely that this protocol uses much more entanglement than necessary for the purpose of entanglement- assisted quantum rate distortion coding with classical channels, and it should be worthwhile to study the trade-off between classical communication and entanglement consumption in more detail, as previous authors have done in the context of

Channel Coding , , , . Such A Study Might Lead

to a better protocol for entanglement-assisted rate distortion coding and might further illuminate better protocols for other quantum rate distortion tasks.

We think that our protocol exploits more entanglement than necessary from considering what is known in the classical case regarding reverse Shannon theorems and rate-distortion coding , , . First, as reviewed in (1), the classical mutual information minimized over all stochastic maps that meet the distortion criterion is equal to Shannon’s classical rate-distortion function . Bennett et al. have shown that the classical mutual information is also equal to the minimum rate needed to simulate a classical channel whenever free common randomness is available . Thus, a simple strategy for achieving the task of rate distortion is for the parties to choose the stochastic map that minimizes the rate distortion function and simulate it with the classical reverse Shannon theorem.

But this strategy uses far more classical bits than necessary whenever sufficient common randomness is not available . Meanwhile, we already know that the mutual information is achievable without any common randomness if the goal is rate distortion .

2) Rate-Distortion with noiseless quantum communica-

Eaq(D), For

entanglement-assisted lossy source coding with noiseless quantum communication, is given by the following theorem. Theorem 3: For a memoryless quantum information source defined by the density matrix ρA′, with a purification |ψρ

Aa′⟩,

and any given distortion 0 ≤D < 1, the quantum rate dis- tortion function for entanglement-assisted lossy source coding

(20)

and I (A; B)ω denotes its mutual information. Proof: We first prove the converse (optimality). The setup is similar to that in the converse proof of Theorem 2, with the exception that W is now a quantum system and we let E denote the environment of the compressor. Consider the

(21)

The first inequality is because the entropy nr of the uniform distribution is the largest that the entropy H (W) can be. The first equality follows from the fact that the state on systems WRnTBE is pure. The second inequality follows by subtracting the positive quantity H (WRnTBE). The second equality is from the definition of quantum mutual information.

The third inequality is from quantum data processing (tracing over system E). The third equality is a useful identity for quantum mutual information. The fourth equality follows from I (Rn; TB) = 0 since Rn and TB are in a product state. The second-to-last inequality is from I (W; TB) ≥0, and the final inequality is from the quantum data processing inequality. The rest of the proof proceeds as in (17).

The direct part follows from a variant of the quantum reverse Shannon theorem known as the fully quantum reverse Shannon theorem (FQRS) , . This theorem states that it is possible to simulate (asymptotically perfectly) the action of a channel N on an arbitrary state ρ, by exploiting noise- less quantum communication and prior shared entanglement between a sender and receiver. It has the following resource

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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