Leader Follower Robot

Autonomous and non-autonomous fixed-time leader-follower

Abstract

This paper addresses the problem of consensus tracking with fixed-time convergence, for leader-follower multi-agent systems with double-integrator dynamics, where only a subset of followers has access to the state of the leader. The control scheme is divided into two steps. The first one is dedicated to the estimation of the leader state by each follower in a distributed way and in a fixed-time. Then, based on the estimate of the leader state, each follower computes its control law to track the leader in a fixed-time. In this paper, two control strategies are investigated and compared to solve the two mentioned steps. The first one is an autonomous protocol which ensures a fixed-time convergence for the observer and for the controller parts where the Upper Bound of the Settling-Time (UBST) is set a priory by the user. Then, the previous strategy is redesigned using time-varying gains to obtain a non-autonomous protocol. This enables to obtain less conservative estimates of the UBST while guaranteeing that the time-varying gains remain bounded. Some numerical examples show the effectiveness of the proposed consensus protocols.

Ntroduction

In the last years, the problems of coordination and control of Multi-Agent System (MAS) have been widely studied (see for instance [1, 2, 3, 4, 5]), due mainly to the ability of a MAS to face complex tasks that a single agent is not able to handle. Distributed control approaches applied to a MAS require a communication network allowing to share information with a subset of agents (neighbors). In this context, several interesting problems and applications have been investigated in the literature, for instance, synchronization of complex networks , distributed resource allocation , consensus and formation control of multiple agents . Among all the mentioned problems, an interesting one is the leader-follower consensus problem where a set of agents, through local interaction, converge to the state of a leader, even though the leader may not be accessible for all agents.

The consensus problem consists in reaching a common agreement state by exchanging only local information [8, 10]. Linear average consensus protocols with asymptotic convergence were proposed in [8, 10]. It has been demonstrated that the second smallest eigenvalue of the Laplacian graph (i.e. the algebraic connectivity) determines the convergence rate of the MAS.

aThis is the accepted version of the manuscript: Trujillo, M.A., Aldana-L´opez, R., G´omez-Guti´errez, D. et al. Autonomous and non-autonomous fixed-time leader–follower consensus for second-order multi-agent sys- bCINVESTAV Unidad Guadalajara, Av. del Bosque 1145, Zapopan, 45019, Jalisco, Mexico dIntel Tecnolog´ıa de M´exico, Intel Labs, Multi agent autonomous systems lab, Jalisco, M´exico. Tecnol´ogico fCentro de Investigaci´on en Matem´aticas (CIMAT), Jalisco S N, Col. Valenciana, 36023, Guanajuato, Mexico

Arxiv:2602.16260V1 [Eess.Sy] 18 Feb 2026

Furthermore, the problem of tracking a reference by a MAS (i.e. leader-follower consensus prob- lem) has been investigated where the common agreement to reach is the state of a reference imposed by a leader which evolves independently of the MAS [11, 12, 13, 14]. In , the con- sensus problem has been addressed where the agents reach a time-varying reference. However, the control protocol has been derived for first-order MAS. The problem for second-order MAS has been studied in and extended to high-order MAS in [11, 13]. Furthermore, has considered the consensus tracking control problem of uncertain nonlinear MAS with predefined accuracy. Nevertheless, in these works, the convergence is only asymptotic.

To improve the convergence rate of a MAS, finite-time consensus protocols have been inves- tigated in . Finite-time stability has been studied in [17, 18, 19]. However, the settling time is an unbounded function of the initial conditions of the system. Therefore, the concept of fixed- time stability has been introduced and applied to systems with time constraints [20, 21, 22].

In this case, the settling time is bounded by a constant which is independent of the initial conditions of the system. In the literature, there are several contributions on algorithms with fixed-time convergence property, such as stabilizing controllers [21, 23], state observers , multi-agent coordination [25, 26], online differentiation algorithms [27, 28], etc. Nevertheless, one can mention that the fixed-time stabilization problem of second-order systems is not an easy task since usually the settling time is not provided or is overestimated. Indeed, there are several works for second-order systems stabilization based on block-control techniques ([29, 21, 30, 31]) or on the homogeneity in the bi-limit (). However, the homogeneity-based algorithms do not provide an estimate of the settling time and many block-control-based algorithms neglect some transient when the system trajectories stay on a region around a manifold. Moreover, the works [33, 34, 35, 36] deal with the problem of leader-follower consensus. Nevertheless, these algorithms require that each follower know the inputs of its neighbors simultaneously, which problem of a MAS, where each agent of the MAS estimates and tracks the trajectory of the leader using local available information even when just a subset of MAS has access to the leader state, and we provide the necessary conditions to achieve the convergence in a fixed-time.

A Lyapunov differential inequality for an autonomous system to exhibit fixed-time stability was presented in . Based on this methodology using autonomous systems, the consensus problem with fixed-time convergence property has been derived for first-order MAS in [25, 37, 38]. Nevertheless, in , the UBST has been estimated from design parameters, algebraic connectivity and group order. Thus, it cannot be easily tuned. In , the UBST was a design parameter which was established a priory by the user. However, the settling time becomes over- estimated and the slack between the settling time and the UBST is conservative. Furthermore, the works [39, 40, 41] have addressed the consensus tracking problem, i.e., the MAS follows a trajectory imposed by the leader. The scheme presented in has introduced a fixed-time algorithm considering inherent dynamics for the agents. However, disturbances were not taken into account. The leader-follower consensus problem for agents with second-order and high order integrator dynamics has been addressed in [42, 41], respectively. The approach was based on a fixed-time observer to estimate the leader state and a fixed-time controller to drive the state of the agent to the estimated leader state. Unfortunately, although the observer can be designed to converge at a desired UBST (with a conservative estimate of the UBST), the controller is based on the homogeneity theory and no methodology has been provided to estimate an UBST. Thus, although the algorithm is fixed-time convergent, the desired convergence time cannot be set a priory by the user. To address this issue, autonomous algorithms were proposed in [44, 45, 46] with an estimation of the UBST. Unfortunately, such estimate of the UBST results very conservative leading to over-engineered consensus protocols. Therefore, the design of fixed- time leader-follower consensus algorithms where the UBST is set explicitly as a parameter of the system, as well as the reduction of the conservativeness of the estimate of the UBST is of a great interest.

An approach to derive predefined-time consensus algorithms has been addressed via a linear function of the sum of the errors between neighboring nodes together with a time-varying gain, using time base generators , see e.g., [48, 49, 50, 51, 52, 53, 54, 55]. This approach ensures that the convergence is obtained exactly at a predefined time. However, such time-varying gain becomes singular at the predefined time, either because the gain goes to infinite as the time tends to the predefined time [53, 54, 55] or because it produces Zeno behavior (infinite number of switching in a finite-time interval) as the time tends to the predefined time .

In this paper, we present a methodology to achieve leader-follower consensus with fixed-time convergence. It consists in two steps. The first one estimates the leader state (position and velocity) using a fixed-time observer that only requires information of the neighbors. Then, the second step computes the control law to drive the followers to the observer states in a protocol, called autonomous protocol, the convergence for the observer and for the controller is in fixed-time, where the UBST is established a priory by the user. In the second protocol, called non-autonomous protocol, we redesign the previous one by adding time-varying gains to obtain a less conservative estimate of the UBST while guaranteeing that the time-varying gains remain bounded. The contribution lies in the following. A novel protocol is derived for second-order MAS with fixed-time stability where the UBST is a design parameter. Moreover, a non-autonomous protocol is presented to achieve the convergence in a predefined-time with less conservative estimates of the UBST compared to existing results in the literature. In fact, the resulting UBST can be made arbitrarily tight. At last, our algorithm yields a bounded time- varying gain, thus we avoid the drawbacks present in the existing algorithms with time-varying gains where the gain goes to infinity.

This work is structured as follows. Section 2 recalls some definitions and results from graph theory, and preliminaries on finite-time and fixed-time convergence are presented. In Section 3, the problem of consensus tracking with fixed-time convergence is formulated.

Section 4

introduces two methodologies to solve the consensus tracking problem. The first (resp. second) one is based on algorithms to obtain a fixed-time stable autonomous (resp. non autonomous) system with an UBST function independent of the initial conditions of the system. Numerical results using both methodologies are shown in Section 6. Finally, the conclusions are presented in Section 7.

Graph Theory

In this section, some notations and preliminaries about graph and consensus theory are pre- sented. One can refer to [56, 57] for a deeper insight in these fields. This paper is only focused on undirected graphs for the follower agents.

Definition 1. A graph consists of a set of vertices V(X) and a set of agents E(X) where an edge is an unordered pair of distinct vertices of X.

Ij Denotes An Edge, If Vertex I And

vertex j are adjacent or neighbors. The set of neighbors of i in graph X is expressed by Ni(X) = {j ∈X : ji ∈E(X)}.

Definition 2. A path from i to j in a graph is a sequence of distinct vertices starting with i and ending with j such that consecutive vertices are adjacent. If there is a path between any two vertices of graph X, then X is said to be connected.

Definition 3. Let X be a weighted graph such that ij ∈E has weight aij and let N = |V(X)|. Then, the adjacency matrix A(X) (or simply A when the graph is clear from the context) is an N × N matrix where A = [aij] and, the Laplacian is denoted by Q(X) (or simply Q) and is defined as Q(X) = ∆(X) −A(X) where ∆(X) = diag(d1, ..., dN) with di = P j∈Ni aij.

Through this work, it is assumed that aij = aji, i.e. only undirected and balanced graphs are considered.

ˆ

X be a weighted graph among all the agents (i.e.

The Leader And The

followers). Then, the communication matrix between all the agents is represented by M( ˆ

) =

Q (X) + B where B = diag(b1 . . bN) ∈RN×N with bi > 0 when there is an edge from the leader to the i-agent and Q (X) is the weighted graph associated to the communication topology of the followers.

Emma 1. [58, 13] Let ˆ

X be the communication graph among all the agents with the leader as

The Root. Then, Matrix M( ˆ

X) is symmetric positive definite.

(1)

where x ∈Rn is the system state, the vector ρ ∈Rb stands for more parameters of system (1) which are assumed to be constant, i.e., ˙ρ = 0.

Furthermore, There Is No Limit For The

number of parameters, so b can take any value in the natural number set N. The function f : Rn × R+ →Rn is nonlinear and the origin is assumed to be an equilibrium point of system (1), so that f(0, t; ρ) = 0.

Besides, when function f does not depend explicitly on t, the system is said to be autonomous or time-invariant. Otherwise, it is called non-autonomous or time-varying .

Definition 5. The origin of (1) is globally finite-time stable if it is globally asymptotically stable and any solution x(t; x0) of (1) reaches the equilibrium point at some finite time moment i.e. x(t; x0) = 0, ∀t ≥T(x0) where T : Rn →R+ ∪{0} is called the settling-time function.

Definition 6. The origin of (1) is fixed-time stable if it is globally finite-time stable and the settling function is bounded, i.e., ∃Tmax > 0 : T(x0) ≤Tmax, ∀x0 ∈Rn.

(2)

with x ∈R. The parameters of the system are the real numbers α, β, p, q, k > 0 which satisfy the constraints kp < 1 and kq > 1. Let ρ = [α, β, p, q, k]T ∈R5. Then, the origin x = 0 of system (2) is fixed-time stable and the settling time function satisfies T(x0) ≤Tf = γ(ρ), where

(4)

where x ∈Rn is the system state, the vector ρ ∈Rb stands for the system parameters which are assumed to be constant. The function f : Rn × R+ →Rn is such that f(0, t; ρ) = 0. Assume that there exists a continuous radially unbounded function V : Rn →R such that:

∀X ∈Rn\{0}

and the derivative of V along the trajectories of (4) satisfies

(Αv (X)P + Βv (X)Q)K, ∀X ∈Rn\{0}

where α, β, p, q, k > 0, kp < 1, kq > 1, γ is given in (3) and ˙V is the upper right-hand time- derivative of V . Then, the origin of (4) is predefined-time stable with predefined-time Tc. Definition 7. For any real number r, the function x 7→⌊x⌉r is defined as ⌊x⌉r = |x|rsign (x) for any x ∈R if r > 0, and for any x ∈R \ 0 if r ≤0. Moreover, if r > 0,⌊0⌉r = 0.

Problem Statement

Let us consider a group of N + 1 agents with one leader and N followers labeled 0 and i ∈ {1, . . , N}, respectively. The dynamics of the leader is described by

U0(T)

where X0 = [x0, v0]T ∈R2 is the state of the leader and u0 ∈R is the control input of the

(5)

where Xi = [xi, vi]T ∈R2 is the state of agent i, ui ∈R is the control input of agent i and ∆i is an unknown external disturbance which is assumed to satisfy |∆i(t)| ≤δi, ∀t ≥0 with δi a known constant. Besides, each agent estimates the leader states, represented by ˆxi (position) and ˆvi (velocity). The communication topology is represented by an undirected graph, which is assumed to contain a spanning tree with the leader agent as the root. The i−th agent shares its estimated states of the leader with its neighbors, defined by the neighbor set Ni.

The control objective is to design a distributed control ui such that the consensus is achieved in a fixed-time Tc, i.e.

I(T) = X0(T),

∀t > Tc. This goal is achieved into two stages. An “observer”, based on consensus algorithms, allows each agent to obtain an estimate of the leader state in a distributed manner in a fixed-time.

Then, after the observer converges, a controller drives the state of the agent towards the state trajectory of the leader. Two protocols are investigated hereafter. In the first one, known as an autonomous protocol, we guarantee that each agent is driven towards the leader state in a fixed- time, where the Upper Bound of the Settling-Time (UBST) is specified a priory by the user. In the second one, known as a non-autonomous protocol, we redesign the previous one by adding time-varying gains to obtain a less conservative estimate of the UBST while guaranteeing that the time-varying gains remain bounded.

Fixed-time leader-follower consensus using autonomous pro-

Istributed Fixed-Time Observer

Since the leader state is not available to all followers, for each agent, an observer is designed to estimate the state of the leader in a fixed-time. The observer has the following structure:

With E1,I = P

j∈Ni aij(ˆxj(t) −ˆxi(t)) + bi(x0(t) −ˆxi(t)) and e2,i = P

J∈Ni Aij(ˆVj(T) −ˆVi(T)) +

bi(v0(t) −ˆvi(t)), ˆxi (resp. ˆvi) is the estimate of the leader position (resp. velocity) for the i-th follower. κi,x, κi,v, α, β, k, p, q, ζx and ζv are positive constants to be defined later. For each

(7)

Therefore, the observation error dynamics can be expressed as:

J∈Ni Aij(˜Xj(T) −˜Xi(T)) −Bi˜Xi(T)) And E2,I = P

j∈Ni aij(˜vj(t) −˜vi(t)) −bi˜vi(t)). In a compact form, with ˜x = [˜x1 · · · ˜xN]T ∈RN and ˜v = [˜v1 · · · ˜vN]T ∈RN, system (8)

Where M( ˆ

X) represents the connection matrix of the graph describing the network between the followers and the leader, and for z = [z1 · · · zN]T ∈RN, the functions Φx : RN →RN and



. Theorem 3. If the observer parameters are selected as α, β, p, q, k > 0, kp < 1, kq > 1, ζx ≥0,

I∈{1...N} Κi,V

and γ(ρ) is defined in Equation (3), then under the distributed observer (6), the observer error dynamics (8) is fixed-time stable with a predefined-time To = Tc1 + Tc2. Proof. Consider the radially unbounded Lyapunov function candidate

˜Vt M( ˆ

X)˜v. Its time-derivative along the trajectories of system (9) is

Et Us Denote E2 = M( ˆ

X)˜v = [e2,1 · · · e2,N]T . Then, Equation (10) can be written as follows

(11)

Now, using the inequality (28) of Lemma 3 in Appendix, the first term of Equation (11) can

!Q!K

with κv = min{κ1,v, . . , κN,v}. Since ∥e2∥1 = PN

+ Βv Q

1 )k . Now, for the last two terms of Equation (11), one can obtain

(Κvζv −Umax

) ≤0. Therefore, the following inequality can be obtained

+ Βv Q

1 )k . From Theorem 2, the observation error in velocity ˜v converges to the origin in a fixed-time before the predefined-time Tc1 where γ(ρ) is given by Eq. (3).

Once the observation error in velocity ˜v converges to zero (i.e. after time Tc1), the observation

I

sign (e1,i) . Similarly to the previous analysis, one can easily show that

+ Βv Q

2 )k , ∀t ≥Tc2. From Theorem 2, the observation error in position ˜x converges to the origin in a fixed-time before the predefined-time Tc2.

Therefore, the proposed distributed observer guarantees the estimation of the leader states in a fixed-time before the predefined-time To = Tc1 + Tc2.

A Fixed-Time Tracking Controller

After time To, each agent has an accurate estimation of the leader state. For each agent, the

(12)

or, equivalently, after the convergence of the observation error:

=

ui + ∆i −u0. Here, the objective is to design the control input ui such that the origin (ex,i, ev,i) = (0, 0) is fixed-time stable where the Upper Bound of the Settling-Time (UBST) is set a priory by the user, in spite of the unknown but bounded perturbation term ∆i −u0. Herefater, we present the following results motivated by the work .

Theorem 4. If for each agent, the controller is selected as

(15)

where parameters are selected as α1, α2, β1, β2, T ′ o, ˆTc1, ˆTc2 > 0, p′, q′, k′ > 0, k′p′ < 1, k′q′ > 1,

K′Q′−1

q′−p′ , then the leader-follower consensus is achieved in a predefined-time ˆTc =

T ′

o + ˆTc1 + ˆTc2. Proof. First, the time derivative of σi along the trajectory of the system solution is given by

/2

. Using the control input ui given by (14), one obtains

(16)

Let us consider the candidate Lyapunov function V1(σi) = |σi| with its time derivative as

+ Δi

one can easily rewrite the Lyapunov function derivative, using (16), in the following inequality

Α2V1(Σ)P′ + Β2V1(Σ)Q′K′

. From Theorem 2, one can deduce that σi converges to zero in a fixed-time ˆTc2.

Α1 |Ex,I| + Β1 |Ex,I|31/2

sign (ex,i) . From Theorem 1, it is clear that ex,i converges to zero in a fixed-time before the settling time ˆTc1.

Moreover, from (15), since σi = 0 and ex,i = 0, then ev,i = 0.

Hence, We Can

conclude that system (5) with (14) as the control input, is fixed-time stable with predefined- time ˆTc1+ ˆTc2. Moreover, due to Theorem 3, where the leader states are estimated in a fixed-time with the predefined settling time To. Hence, if T ′

O = To, One Can Deduce That Leader-Follower

consensus is achieved in fixed-time before the predefined-time ˆTc = T ′

T ′

o = 0 and To < ˆTc1 + ˆTc2, the leader-follower consensus is achieved before the predefined-time ˆTc = ˆTc1 + ˆTc2. Fixed-time leader-follower consensus with improved estimate

For The Ubst Using Non-Autonomous Protocol

The autonomous leader-follower protocol presented in Section 4 allows a fixed-time convergence. However, the estimate of the UBST for the observer and controller are both too conservative. This is a common drawback on existing fixed-time consensus protocols, see e.g., [26, 40] for the leader -follower problem for agents with first-order integrator dynamics and [64, 45] for agents with second-order integrator dynamics. To address this issue, we present new protocols, based on the class of time-varying gains proposed in , to significantly reduce such conservatism.

Contrary to some existing protocols such as [52, 47, 54], where the time-varying gains become singular when consensus is reached, in our approach the convergence is achieved with bounded time-varying gains in a user-defined time.

Before designing the proposed fixed-time leader-follower consensus protocol, let us define

Efinition 8. Let Us Define The Following

• Φ : R+ →R+ ∪{+∞} \ {0} is a continuous function on R+ \ {0} that satisfies

– Φ(Τ) < +∞, ∀Τ ∈R+ \ {0},

– is either non-increasing or locally Lipschitz on R+ \ {0}.

Tc

Φ(τ)−1. Definition 9. (Definition in ). A regular parametrized curve, with parameter t, is a C1(I) inmersion c : I 7→R, defined on a real interval I ⊆R. This means that dc

Dt̸ = 0 Holds Everywhere

Definition 10. (Pg. 8 in ). A regular curve is an equivalence class of regular parametrized curves, where the equivalence relation is given by regular (orientarion preserving) parameter transformation ψ, where ψ : I →I′ is C1(I), bijective, and dψ

Dt > 0. Therefore, If C : I →R Is

a regular parametrized curve and ψ : I →I′ is a regular parameter transformation, then c and c ◦ψ : I′ →R are considered to be equivalent. Lemma 2. Let t0 be the initial time. The function t = ψ(τ) + t0, defines a parameter transformation with τ = ψ−1(t −t0) as its inverse mapping.

Proof. It Follows From Definition 10

To derive the fixed-time non-autonomous scheme, we define the following time-varying gain for each predefined settling time Tci as follows using the previously defined functions:

T ∈[T0, T0 + Ηi(T)Tci)

otherwise. Remark 1. Notice that if T < +∞, then ˆκi(t; t0, Tci, T) is bounded. Such bound can be user- defined by tuning T.

Now, we are ready to present our main result. Theorem 5. Let us consider the same observer parameters as in Theorem 3 (i.e. α, β, p, q, k >

, Tc2, Tβ) Be Time-

varying gains. Tα and Tβ are positive parameters and t′ 0 = t0 + η1(Tα)Tc1.

−1

1 (t)ζv, the observer error dynamics is fixed-time stable with the UBST given by To = t0 + η1(Tα)Tc1 + η2(Tβ)Tc2, with Tα, Tβ > 0. Let us consider the same control parameters as in Theorem 4 (i.e. α1, α2, β1, β2, ˆTc1, ˆTc2 > 0, p′, q′, k′ > 0, ζi, γ1, γ2) and set Tc3 = ˆTc1 + ˆTc2. Let̺

(T) = ˆΚ3(T; To, Tc3, Tγ) Be A Time-Varying

gain with Tγ a positive parameter.

−1

3 (t)ev,i) is given by (14), the leader-follower consensus is achieved in fixed time

With The Ubst Given By ˆT = T ′

o + η(Tγ)Tc3. The proof of Theorem 5 will be divided into two parts. The first part focuses on the observer stability whereas the second one focuses on the controller stability.

Proof. First, let us study the observer error dynamics using the non-autonomous observer (17). Let us consider the observer errors as in (7). Using (17), the observation error dynamics is

I

sign (e2,i) −u0. Now, considering the observer error dynamics of the velocity ˜vi in the new τ-time variable as

(19)

and according to the parameter transformation given in Lemma 2,

(20)

= ˆκi(t; t0, Tci, T)−1. Thus, the observation error dynamics of the velocity given by (19) is rewritten, using (20), as

(Τ) = ˆΚ1(T; T0, Tc1, Tα)−1U0 With |U0| < Umax

is the disturbance term and ˆκ1(t; t0, Tc1, Tα)−1 =̺

(21)

Note that by the definition of the function Φ1(τ), the time-varying gain ˆκ1(t; t0, Tc1, Tα)−1 for t ∈[t0, t0 + η1(Tα)Tc1), can be written as ρ1(τ; Tc1)−1 ∀τ, and function ρ1(τ; Tc1)−1 is non- increasing. Besides, ρ1(τ; Tc1)−1 is bounded and ρ1(τ; Tc1)−1 →0 as τ →+∞. Then, the

Disturbance̟

(τ) = ρ1(τ; Tc1)−1uo is vanishing. Furthermore, notice that ˆζv(t) =̺

(Τ)| < ˆΖv, ∀Τ Since Umax

≤κvζv. Then, similarly to (9), the compact form of (21) is

(22)

with ˜v = [˜v1 · · · ˜vN]T ∈RN. Furthermore, from Theorem 3, the observation error dynamics of velocity (22) is fixed-time stable in the time-variable τ and, converges to the origin with a settling time T ′ c1 < +∞.

Then, the observation error in velocity reaches the origin at T(˜v0) = limτ→T ′c1(ψ1(τ) + t0) ≤ t0 + η1(Tα)Tc1; ∀˜v0 ∈RN as the initial conditions.

N A Similar Way, The Observation Error

dynamics of the position ˜xi in the time-variable τ is written as follows

(23)

where υi(τ) = ˆκ2(t; t0 + η1(Tα)Tc1, Tc2, Tβ)−1˜vi. The compact form of (23) is

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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