Sliding Mode Control Robot

Bdul. Carol I 11, Ia¸Si, Romania

(2) Dipartimento di Matematica “F. Casorati”, Universit`a di Pavia

Abstract

In the present contribution the sliding mode control (SMC) problem for a phase- field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state- feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.

Key words: phase field system, nonlinear boundary value problems, phase transi- tion, sliding mode control, state-feedback control law. AMS (MOS) Subject Classification: 34B15, 82B26, 34H05, 93B52.

Ntroduction

Sliding mode control (SMC) has for many years been recognized as one of the fundamental approaches for the systematic design of robust controllers for nonlinear complex dynamic systems that operate under uncertainty. Moreover, SMC is nowadays considered a classical tool for the regulation of continuous - or discrete - time systems in finite-dimensional settings (cf., e.g., the monographs ).

The main advantage of sliding mode control is that it allows the separation of the motion of the overall system in independent partial components of lower dimensions, and consequently it reduces the complexity of the control problem. The design of feedback control systems with sliding modes implies the design of suitable control functions en- forcing motions along ad-hoc manifolds. Hence, the main idea behind this scheme is first to identify a manifold of lower dimension (called the sliding manifold) where the control goal is fulfilled and such that the original system restricted to this sliding manifold has a desired behavior, and then to act on the system through the control in order to constrain the evolution on it, that is, to design a SMC-law that forces the trajectories of the system to reach the sliding surface and maintains them on it.

Sliding mode controls, while being relatively easy to design, feature properties of both robustness with respect to unmodelled dynamics and insensitivity to external disturbances that are quite attractive in many applications. Hence, in the last years there has been a growing interest in the extension of the well developed methods for finite-dimensional systems described by ODEs (cf., e.g., ) to the control of infinite-dimensional dy- considered, the theoretical development in a general Hilbert space setting or for PDE sys- tems has gained attention only in the last ten years. In this respect, we can quote the papers , , and dealing with sliding modes control for semilinear PDEs. In par- ticular, in the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation, powered with a Dirich- let type actuator from one of the boundaries was considered. A delay-independent SMC strategy was proposed in to control a class of quasi-linear parabolic PDE systems with time-varying delay, while in the authors study a sliding mode control law for a class of parabolic systems where the control acts through a Neumann boundary condition and the control space is finite-dimensional.

In the present contribution we would like to employ – to the best of our knowledge for the first time in the literature – a SMC technique for a nonlinear PDE system of phase- field type. In particular, we consider the following rather simple version of the phase-field

(1.2)

where Ωis the three-dimensional domain in which the evolution takes place, T is some final time, ϑ denotes the relative temperature around some critical value that is taken to be 0 without loss of generality, and ϕ is the order parameter. Moreover, ℓ, κ, ν and γ are positive constants, f is a source term and F ′ represents the derivative of a double-well

(1.5)

where c0 > 1 in (1.4) in order to produce a double well, while c0 is an arbitrary positive number in (1.5), and the function I in (1.5) is the indicator function of [−1, 1], i.e., it takes the values 0 or +∞according to whether or not r belongs to [−1, 1]. The potential (1.3) and (1.4) are the usual classical regular potential and the so-called logarithmic potential, respectively. More generally, the potential F could be just the sum

F = Bβ + Bπ,

where bβ is a convex function that is allowed to take the value +∞, and bπ is a smooth perturbation (not necessarily concave). In such a case, bβ is supposed to be proper and lower semicontinuous so that its subdifferential is well-defined and can replace the deriva- tive which might not exist. This happens in the case (1.5) and equation (1.2) becomes a differential inclusion.

The above system is complemented by initial conditions like ϑ(0) = ϑ0 and ϕ(0) = ϕ0 and suitable boundary conditions.

Oncerning The Latter, As Very Usual We Take The

homogeneous Neumann condition for both ϑ and ϕ, that is,

On Σ := (0, T) × Γ

where Γ is the boundary of Ωand ∂n is the (say, outward) normal derivative. Equations (1.1)–(1.2) yield a system of phase field type.

Such Systems Have Been

introduced (cf. ) in order to include phase dissipation effects in the dynamics of moving interfaces arising in thermally induced phase transitions. In our case, we move from the

(1.6)

where c0 and γ stand for specific heat and latent heat coefficients, respectively, with a terminology motivated by earlier studies (see ) on the Stefan problem; we refer to the monography which deals with phase change models as well. In this connection, let

Δϑ

(−the variational derivative of F with respect to ϑ) that is e = c0ϑ + γϕ. Then, the governing balance and phase equations are given by

(1.8)

where q denotes the thermal flux vector, ˜f represents some heat source and the variational derivative of F with respect to ϕ appears in (1.8). Hence, (1.8) reduces exactly to (1.2)

Sliding Modes For A Phase-Field System

along with the homogeneous Neumann boundary condition for ϕ. Moreover, if we assume the classical Fourier law q = −˜κ ∇ϑ, then (1.7) is nothing but the usual energy balance equation of the Caginalp model . By setting ℓ:= γ/c0, κ := ˜κ/c0, f := ˜f/c0, we easily see that (1.1) follows from (1.7) and the Neumann boundary condition for ϑ is a consequence of the no-flux condition q · n = 0 on the boundary. We also point out that the above phase field system has received a good deal of attention in the last decades and it can be deduced as a special gradient-flow problem (cf., e.g., and references therein).

As already noticed, the well-posedness, the long-time behavior of solutions, and also the related optimal control problems have been widely studied in the literature. We refer, without any sake of completeness, e.g., to and references therein for the well-posedness and long time behavior results and to for the related optimal The present paper is also related to the control problems, but it goes in the direction of designing sliding mode controls for the above phase-field system.

Ndeed Our Main

objective is to find out some state-feedback control laws (ϑ, ϕ) 7→u(ϑ, ϕ) that can be inserted in one of the equations in order that the dynamics of the system modified in this way forces the value (ϑ(t), ϕ(t)) of the solution to reach some manifold of the phase space in a finite time and then lie there with a sliding mode.

The first analytical difficulty consists in deriving the equations governing the sliding modes and the conditions for this motion to exist. The problem needs the development of special methods, since the conventional theorems regarding existence and uniqueness of solutions are not directly applicable. Moreover, we need to manipulate the system through the control in order to constrain the evolution on the desired sliding manifold.

In particular, we study three cases. In the first one, a feedback control is added to the internal energy balance equation (1.1) in order to force a linear relationship between ϑ and ϕ; in the second case, a pre- scribed distribution ϕ∗of the order parameter is forced by means of a feedback control added to the phase dynamics (1.2). Notice that both these choices can be considered physically meaningful in the framework of phase transition processes, since in both cases the quantities we are forcing to reach time-independent values may have a physical mean- ing. In the first problem, we can take the internal energy as a particular case, while the target ϕ∗we force for the phase parameter in the second problem could represent one of the so called pure phases (e.g., pure water or pure ice in a water-ice phase change process). Moreover, in both cases we have reduced the problem to a simplified dynamics involving only the evolution of ϕ in the first case and only of ϑ in the second one (cf. also Remark 2.8).

In each of the above problems, the control law we introduce is non-local in space, i.e., the value at (t, x) of the control depends on the whole state (ϑ(t, · ), ϕ(t, · )) at the time t and not only on the value (ϑ(t, x), ϕ(t, x)). The objective of the third problem is to design a control law that reaches the same target as in the second one and is local at the same time. However, such a problem looks much more difficult and we can ensure the existence of the desired sliding mode only under a suitable compatibility condition on Ω.

The paper is organized as follows. In the next section, we list our assumptions, state the problem in a precise form and present our results. The last two sections are devoted to the corresponding proofs. Section 3 deals with well-posedness and regularity, while the existence of the sliding modes is proved in Section 4.

Statement Of The Problem And Results

In this section, we describe the problem under study and present our results. As in the Introduction, Ωis the body where the evolution takes place. We assume

Ω⊂R3 To Be Open, Bounded, Connected, And Smooth

and write |Ω| for its Lebesgue measure. Moreover, Γ and ∂n still stand for the boundary of Ωand the outward normal derivative, respectively. Given a finite final time T > 0, we set for convenience Q := (0, T) × Ω. Now, we specify the assumptions on the structure of

Bβ : R →[0, +∞]

is convex, proper and l.s.c.

(2.4)

and denote by D(β) and D(bβ) the effective domains of β and bβ, respectively. Next, in

(2.5)

and endow the spaces V and H with their standard norms ∥· ∥V and ∥· ∥H.

On The

contrary, we write ∥· ∥W for the norm in W defined by

(2.6)

and we term CΩthe best constant realizing the inequality

(2.7)

The reason of this choice will be explained later on (see the forthcoming Remark 2.11). Now, we just notice that ∥· ∥W is equivalent to the norm induced on W by the standard one in H2(Ω) (thanks to the regularity theory of elliptic equations) and that the constant CΩactually exists due to the continuous embedding H2(Ω) ⊂C0(Ω) (since Ω⊂R3 is bounded and smooth) and only depends on Ω(see, e.g., ). Finally, for the norms both in L∞(Ω) and in L∞(Q) we use the same symbol ∥· ∥∞whenever no confusion can arise.

(2.9)

Sign 0 is the closed unit ball of H.

(2.10)

Thus, β and Sign are maximal monotone operators on R and H, respectively (see, e.g., [2, Thm. 2.8, p. 47]). In the sequel, we use the same symbol β to denote the maximal monotone operator induced on L2-spaces.

Sliding Modes For A Phase-Field System

Yosida regularizations of β and Sign.

Et Us Introduce The Yosida Regularization

βε : R →R and Signε : H →H at level ε > 0 (see, e.g., [2, formulas (2.26), p. 37]) as well as the Moreau regularization of ∥· ∥H (see, e.g., [2, formula (2.38), p. 48])

(2.11)

For the reader’s convenience, we sketch the justification of the last equality of (2.11). We write ∥· ∥instead of ∥· ∥H for simplicity. For w ∈H and y ≥0 we set

≤∥W −V∥Yields G(W) ≥G(∥W∥) For

every w ∈H. Now, from one side, one easily checks that

Y≥0 G(Y) = ∥V∥−Ε

if ∥v∥> ε. This means that miny≥0 g(y) coincides with the right-hand side of (2.11). On the other

= ∥V∥−Ε

if ∥v∥> ε. Thus, minw∈H G(w) = miny≥0 g(y) and (2.11) is proved. Next, we recall that βε and Signε are monotone and that (see, e.g., [2, Prop. 2.2 (ii), p. 38] and [2, Thm. 2.9, p. 48] for some

Of These Properties)

Signε v is the gradient at v of the C1 functional ∥· ∥H, ε

Where

β◦(r) is the element of β(r) having minimum modulus.

(2.15)

We point out that the Young inequality has been used to derive (2.14). At this point, we describe the state system modified by the state-feedback control law and we study two cases. In the first one, a feedback control is added to the first equation (1.1) in order to force a linear relationship between ϑ and ϕ; in the second case, a prescribed distribution of the order parameter is forced by means of a feedback control that is added to equation (1.2). In principle, for the data, we require that

(2.16)

Given ρ > 0 and some target that depends on the case we want to consider, we look for a quadruplet (ϑ, ϕ, ξ, σ) satisfying at least the regularity requirements

Barbu — Colli — Gilardi — Marinoschi — Rocca

and solving the related system we introduce at once. We notice that the homogeneous Neumann boundary conditions for both ϑ and ϕ are contained in (2.17) (see the definition (2.5) of W). The problems corresponding to the cases sketched above are the following.

(2.23)

In the sequel, we also term such problem Problem (A). The second problem, which we call Problem (B), depends on a given ϕ∗∈W and

(2.28)

The last case, termed Problem (C), is the same as the previous one with the following difference: the non-local operator Sign is replaced by the local sign : R →2R defined by

(2.29)

Notice that sign is the subdifferential of the real function r 7→|r| and thus is maximal monotone. For the sake of clarity, we write Problem (C), explicitly. Given ϕ∗∈W, we

(2.35)

Then, for every ρ > 0, Problem (A) has at least a solution (ϑ, ϕ, ξ, σ) satisfying (2.17)–

Sliding Modes For A Phase-Field System

where C1 and C2 depend only on the quantities involved in assumptions (2.1)–(2.3), (2.16) and (2.35). Moreover, the solution is unique if α = ℓ. Furthermore, if in addition

(2.40)

where C3 depends on the norms related to (2.38) as well. In particular, ϕ is bounded. Finally, the component ϑ of any solution satisfying all the above regularity requirements is bounded whenever ϑ0 ∈V ∩L∞(Ω) and f ∈L∞(0, T; H).

Theorem 2.2. Assume (2.1)–(2.3), (2.16), as well as

(2.41)

Then, for every ρ > 0, Problem (B) has at least a solution (ϑ, ϕ, ξ, σ) satisfying (2.17)–

(2.18) And The Estimates

∥ϑ∥L∞(0,T;H)∩L2(0,T;V ) + ∥ϕ∥L∞(0,T;H)∩L2(0,T;V ) + ∥σ∥L∞(0,T;H) ≤C4

(2.43)

where C4 and C5 depend only on the quantities involved in assumptions (2.1)–(2.3), (2.16) and (2.41). Furthermore, the components ϑ and ϕ of the solution are uniquely determined, and ξ and σ are uniquely determined as well if β is single-valued.

A similar result holds for Problem (C). We present the corresponding statement in a more accurate form for a reason that will be clear later on. Theorem 2.3. Assume (2.1)–(2.3), (2.16) and (2.41). Then, for every ρ > 0, Prob- lem (C) has at least a solution (ϑ, ϕ, ξ, σ) satisfying (2.17)–(2.18).

Furthermore, The

components ϑ and ϕ of the solution are uniquely determined, and ξ and σ are uniquely determined as well if β is single-valued. Finally, if the conditions

(2.44)

are assumed in addition, then (2.39) holds as well as ϑ ∈W 1,∞(0, T; H) ∩H1(0, T; V ) ∩L∞(0, T; W).

(2.45)

In particular, both ϕ and ϑ are bounded. Moreover, the estimates

(2.47)

hold true with a structural constant Cstr depending only on the physical parameters ℓ, κ, ν and γ, the constant CΩgiven by (2.7) and some constants C6 and C7 depending on the structure of the systems, Ω, T and on the norms of the data involved.

Barbu — Colli — Gilardi — Marinoschi — Rocca

Remark 2.4. The above results are quite general. In particular, both potentials (1.3) and (1.4) are certainly allowed and the multi-valued potential (1.5) has to be excluded just in the parts of Theorems 2.2 and 2.3 regarding uniqueness for the pair (ξ, σ), which might be not uniquely determined, in general. Concerning the constant Cstr of Theorem 2.3, we

(2.48)

However, no sharpness is guaranteed at all. For each of the first two problems, the existence of the desired sliding mode is ensured for ρ large enough. For every T > 0 we have indeed Theorem 2.5. Assume (2.1)–(2.3), (2.16), (2.35), (2.38) and f ∈L∞(0, T; H). Then, for some ρ∗> 0 and for every ρ > ρ∗, there exist a solution (ϑ, ϕ, ξ, σ) to problem (2.19)–

(2.49)

Theorem 2.6. Assume (2.1)–(2.3), (2.16) and (2.41).

Then, For Some Ρ∗> 0 And

for every ρ > ρ∗, there exist a solution (ϑ, ϕ, ξ, σ) to problem (2.24)–(2.28) and a time

(2.50)

Remark 2.7. In the proof we give in Section 4, we compute possible values of ρ∗and T ∗ that fit the conclusions of our results. For Problems (A) and (B), we can take respectively

Ρ −2Cb

where the constants CA and CB are constructed in the proofs of Theorems 2.1 and 2.2 in

∥γϑ + ν∆ϕ∗−β◦(ϕ∗) −π(ϕ)∥L∞(0,T;H) ≤CB . More precisely, we refer to (4.6)–(4.8) and (4.11)–(4.13) and we notice that our starting point in those proofs is the validity of the analogous estimates for the solutions to the approximating problems obtained by replacing the monotone operators by their Yosida regularizations. It follows that the above values of ρ∗and T ∗depend continuously on the T ∗is roughly proportional to 1/ρ in both cases, whence it tends to zero as ρ tends to infinity, i.e., the sliding mode can be forced to start as soon as one desires by prescribing a sufficiently big factor ρ in front of the feedback control.

Remark 2.8. The minimal value of T ∗of the first statement (if it is positive) also satisfies the following property: the function t 7→∥ϑ(t) + αϕ(t) −η∗∥H is strictly decreasing on [0, T ∗].

A similar remark holds for the function t 7→∥ϕ(t) −ϕ∗∥H in the second

Sliding Modes For A Phase-Field System

statement (and in the next one, at least under some reinforcement of the assumptions, as shown in Remark 4.2). In each case, the dynamics of the system is simpler after the time T ∗, since one of the unknowns can be eliminated by using the sliding mode condition.

For instance, in the second situation, the evolution of ϑ after T ∗is ruled just by the heat equation. The situation for Problem (C) is different, since we can ensure the existence of the desired sliding mode for ρ large enough only if further conditions are fulfilled. Namely, we need a restriction involving the structure of the system and the domain Ω(that is why we have written the statement of Theorem 2.3 in that form). Our result only involves the component ϕ of the solution, and we recall that ϕ is uniquely determined.

Theorem 2.9. Assume (2.1)–(2.3), (2.16), (2.41), (2.44) and

(2.51)

Let Cstr and CΩbe the constants appearing in (2.47) and in (2.7), respectively, and assume

(2.52)

Then, for some ρ∗> 0 and for every ρ > ρ∗, the following is true: if (ϑ, ϕ, ξ, σ) is a solution to problem (2.30)–(2.34), there exists a time T ∗∈[0, T) such that

(2.53)

Remark 2.10. Assume that the constants Cstr, CΩand C7 realize the inequalities (2.47) and (2.52) (i.e., in contrast with the situation of Remark 2.7, just such inequalities are required as a starting point). Then, as shown in the proof we perform in the last section, possible values of ρ∗and T ∗that fit the conclusion of the above theorem are given by

+ Ν∥∆Φ∗∥∞+ ∥Ξ∗∥∞+ M∗

π . In particular, the last two sentences of Remark 2.7 also apply to the present case. Remark 2.11. In order to understand the meaning of (2.52), let us assume that the structure of the system is chosen, so that the physical constants are fixed, and let us think of a class of open sets having the same shape. Precisely, we fix an open set Ω0 of measure 1 and assume that Ω= x0 + λR Ω0 for some x0 ∈R3, λ > 0 and some rotation R ∈SO(3). Then λ = |Ω|1/3 and one easily checks that our definition (2.6) of ∥· ∥W yields CΩ= CΩ0 |Ω|−1/2, since the H-norms of v and of ∆v are properly balanced in the norm ∥v∥W under a rescaling of a function v. Then, the smallness condition (2.52) means that |Ω| is small enough. Indeed, the left-hand side of (2.52) becomes γCstrCΩ0|Ω|2/3 in the chosen class of domains.

In performing our a priori estimates in the remainder of the paper, we often account for the H¨older inequality and the elementary inequalities (for arbitrary a, b ≥0)

Barbu — Colli — Gilardi — Marinoschi — Rocca

where δ > 0 in the latter (Young’s inequality). Moreover, we repeatedly use the notation Qt := (0, t) × Ω.

(2.55)

For simplicity, we usually omit dx, ds, etc. in integrals. More precisely, we explicitly write, e.g., ds only if the variable s actually appears in the function under the integral sign. Finally, while a particular care is taken in computing some constants, we follow a general rule to denote less important ones, in order to avoid boring calculations. The small-case symbol c stands for different constants independent of ρ but depending on Ω, the final time T, the shape of the nonlinearities and on the constants and the norms of the functions involved in the assumptions of our statements. The dependence on ρ will be always written explicitly, indeed. Hence, the meaning of c might change from line to line and even in the same chain of equalities or inequalities. On the contrary, we mark precise constants which we can refer to by using different symbols, e.g., capital letters, mainly with indices, like in (2.7).

Proof Of The Well-Posedness Results

This section is devoted to the proof of Theorems 2.1–2.3. However, as far as existence is concerned, we confine ourselves to derive the formal a priori estimates that lead to the desired regularity and just sketch how a completely rigorous proof could be performed.

Proof Of Theorem 2.1

We start with problem (2.19)–(2.23) and transform it into an equivalent system in new unknown functions. In order to argue in terms of the variable which the operator Sign

(3.1)

then, η has to satisfy the analog of (2.17) and the new problem is the following

(3.6)

First a priori estimate. We multiply (3.2) and (3.3) by η and ∂tϕ, respectively, sum up and integrate over Qt with an arbitrary t ∈(0, T].

(Ν/2)

Ω(|ϕ(t)|2 −|ϕ0|2) to both sides. With the help of (2.16) and (2.8), we infer that

Qt

ϕ∂tϕ. Now, it is straightforward to use the linear growth of π that follows from Lipschitz con- tinuity, the Young and H¨older inequalities, (2.16), (2.35), and the Gronwall lemma to

Deduce That

∥η∥L∞(0,T;H)∩L2(0,T;V ) + ∥ϕ∥H1(0,T;H)∩L∞(0,T;V ) + ∥bβ(ϕ)∥L∞(0,T;L1(Ω)) ≤c .

For A.A. T ∈(0, T)

with an obvious meaning of g1 and treat t as a parameter. We formally multiply by ∆ϕ(t) (the correct proof deals with the regularized problem) and find ∥∆ϕ(t)∥H ≤∥g1(t)∥H for a.a. t ∈(0, T). Then, we use (3.7), (2.3), (2.35) (which imply ∥g1∥L2(0,T;H) ≤c), elliptic regularity and a comparison in the above equation, in order to conclude that ∥ϕ∥L2(0,T;W ) + ∥ξ∥L2(0,T;H) ≤c .

(3.9)

where we used (3.7)–(3.8), (2.16) and (2.35) once more. Then, we multiply by ∂tη and integrate over Qt. Thanks to the chain rule property (stated, e.g., in [3, Lemme 3.3,

−Κ∆Η(T) + Ρσ(T) = G3(T) := G2(T) −∂Tη(T)

for a.a. t ∈(0, T). Then, we formally multiply by −∆η(t) and notice that ∇σ(t) · ∇η(t) ≥0 a.e. in Ω(at least formally; the inequality we need if Sign were replaced by Signε would immediately

Barbu — Colli — Gilardi — Marinoschi — Rocca

follow from (2.13)). Hence, we get κ1/2∥∆η(t)∥H ≤∥g3(t)∥H for a.a. t ∈(0, T). By owing to (3.9), (3.10) and elliptic regularity, we deduce that

(3.11)

Consequence. Estimates (3.7)–(3.11) and assumption (2.35) imply for ϑ = η −αϕ+η∗

(3.12)

Existence for Problem (A). The above a priori estimates are rigorous for the solution to the approximating problem obtained by replacing β and Sign by the corresponding

(3.13)

in place of (3.4)–(3.5). The approximating problem is more regular and has a solution (ηε, ϕε, ξε, σε). To see that, one can rewrite the approximating problem by eliminating the time derivative ∂tϕ in (3.2) on accout of (3.3). One obtains the Cauchy problem for

∂T(Η, Φ) + A(Η, Φ) + Bε(Η, Φ) = F

where A is an unbounded operator in H := H ×H, Bε : H →H is a Lipschitz continuous perturbation and F is a source term. Namely, A acts as follows

Where

λ := κα + (ℓ−α)ν. Now, let us introduce the following inner product in H

Ω

|∇ϕ|2. This shows that A is monotone in H with respect to that inner product. Then, maximal monotonicy follows since the range of A + IdH is the whole of H due to the Lax-Milgram theorem and elliptic regularity.

Therefore, the approximating problem has a solution (see, e.g., [31, Cor. 4.1 p. 181]). So, by starting from the analogs of the above formal a priori estimates (that is, from the rigorous ones, for which properties (2.12)–(2.14) have to be used) and owing to standard weak, weakstar and strong compactness results (see, e.g., [32, Sect. 8, Cor. 4]), we have for a subsequence at least

Ηε →Η

weakly star in H1(0, T; H) ∩L∞(0, T; V ) ∩L2(0, T; W)

Σε →Σ

weakly star in L∞(0, T; H).

Sliding Modes For A Phase-Field System

We stress that ξε := βε(ϕε) and σε := Signε(ηε), i.e., the same as in (3.13), where the subscripts ε were omitted for convenience. Here, ξ and σ have the meaning given by (3.16)–(3.17). Clearly, the limits ϕ, ξ and σ and the function ϑ computed from (3.1) satisfy the regularity requirements and the estimates of the statement (see also (3.12)).

Moreover, it follows that π(ϕε) converges to π(ϕ) strongly in L2(Q) and that ξ and σ satisfy (3.4)–(3.5) (because β and Sign induce maximal monotone operators on L2(Q) and L2(0, T; H), respectively, and then they are weakly-strongly closed; see, e.g., [2, Cor. 2.4, p. 41]). Hence, (η, ϕ, ξ, σ) solves the original problem (3.2)–(3.6).

Uniqueness for Problem (A). We assume α = ℓand show that the solution is unique. Let (ηi, ϕi, ξi, σi), i = 1, 2, be two solutions. We write equations (3.2)–(3.3) for both of them and take the differences. If we set η := η1 −η2 and analogously define ϕ, ξ and σ,

(3.18)

∂tϕ −ν∆ϕ + ξ = γ(η −ℓϕ) + π(ϕ2) −π(ϕ1).

(3.19)

Now, we multiply these equations by η and (κℓ2/ν)ϕ, respectively, sum up and integrate

|Η|2 + |Φ|2

. The last two terms on the left-hand side are nonnegative by monotonicity and the integral involving the gradients is estimated from below this way

(3.20)

At this point, we combine and apply the Gronwall lemma. We conclude that η = 0 and ϕ = 0, i.e., η1 = η2 and ϕ1 = ϕ2. By comparison in (3.2) and (3.3) written for both solutions, we deduce that σ1 = σ2 and ξ1 = ξ2, respectively.

Further regularity. We assume (2.38) and prove (2.39). To this end, it suffices to perform the estimate corresponding to (2.39) on the component ϕε of the solution to the approximating problem sketched above, uniformly with respect to ε. This can be done by a heavy calculation involving difference quotients. Therefore, we confine ourselves to derive a formal a priori estimate. We write equations (3.3)–(3.4) by replacing β by its Yosida regularization βε in the latter, and formally differentiate with respect to time. By writing ϕ instead of ϕε for simplicity, we have (see (3.7), (3.10), and (2.3))

(3.21)

Now, we multiply by ∂tϕ and integrate over Qt. We obtain

As Β′

ε is nonnegative by monotonicity, the only term that needs some treatment is the last one on the right-hand side. We formally have from (3.3), the modified (3.4) and the

Initial Conditions

∂tϕ(0) = ν∆ϕ0 −βε(ϕ0) −π(ϕ0) + γϑ0 .

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).

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.