Robust Control Robotic Arm

Abstract

Control barrier functions guarantee safety but require accurate system models; parametric uncertainty in- validates these guarantees. Existing robust methods maintain safety via worst-case bounds at the cost of performance, while modular learning schemes decouple estimation from safety and risk constraint vio- lations during transients. This paper presents the composite adaptive control barrier function (CaCBF) algorithm for nonlinear control-affine systems with linear parametric uncertainty. The adaptation law is de- rived from a composite energy function integrating a logarithmic safety barrier, a control Lyapunov function, and a parameter-error term, creating a direct coupling between estimation accuracy and the safety margin.

We prove three main results: (i) the safe set is forward invariant for all bounded parameters, without re- quiring persistence of excitation; (ii) the safety guarantee is robust to bounded errors in the state-derivative measurement; and (iii) all closed-loop signals are uniformly ultimately bounded. We further prove that the CaCBF admissible control set always contains the robust counterpart as a subset. Simulations of adaptive cruise control, an omnidirectional robot, and a planar drone traversing a narrow gate confirm that CaCBF recovers the performance margin surrendered by robust methods while maintaining strict safety throughout.

K E Y W O R D S

control barrier functions, adaptive control, parametric uncertainty, safety-critical systems, control Lyapunov

Ntroduction

Control barrier functions (CBFs) provide a mathematical mechanism for enforcing safety in autonomous systems . By acting as a pointwise filter, a CBF monitors a nominal con- trol signal and modulates it only when the system approaches the boundary of a safe set. This operation renders the safe set forward invariant if the underlying mathematical model accurately represents the physical system. However, physical systems inevitably deviate from their models due to parametric uncertainties, such as variable payloads, unmodeled friction, or aerodynamic drag. This discrepancy challenges the formal safety guarantees of standard CBFs. When the model parame- ters are incorrect, the calculated safe control input may drive the system into the unsafe region. This uncertainty is typically addressed through two primary paradigms: robust control, bounding the uncertainty, or adaptive control, estimating the parameters.

Robust formulations, such as those established by Jankovic and Zhao et al. , maintain safety by enforcing con- straints against the worst-case realization of the uncertainty.

Recent developments have extended robust CBFs to accommo- date complex dynamics. For instance, Buch et al. addressed sector-bounded inputs via second-order cone programs, Xiao and Belta developed formulations for high-order relative degree systems, and Ames et al. introduced input-to-state safety conditions for bounded disturbances. While theoreti- cally rigorous, robustness imposes a geometric cost. As the bounds on parameter uncertainty expand, the subset of control inputs certified as safe contracts. In scenarios with significant uncertainty, this conservatism can render the safe control set empty, causing the solver to fail even when a safe solution physically exists. Furthermore, robust methods compel the system to operate with reduced efficiency, braking earlier, or moving more slowly, regardless of whether the environment is hostile or benign .

To mitigate this conservatism, researchers utilize online es- timation to reduce uncertainty bounds. Modular architectures, such as the robust estimator framework proposed by Das and

Arxiv:2601.17683V3 [Eess.Sy] 28 Sep 2026

Kamaldar, M. Burdick or the measurement-robust CBFs developed by Dean et al. , employ learning modules to refine parameter estimates over time. These methods separate the estimation process from the control logic. The estimator typically min- imizes a prediction error metric, such as the mean squared error between predicted and observed states . While this improves model accuracy, the separation of timescales allows for transient estimation errors independent of the safety bound- ary. A transient estimation error that occurs while the system is near the boundary can induce a safety violation before the estimator converges.

Integrated approaches attempt to couple estimation and control more tightly. Taylor and Ames introduced Adap- tive CBFs to preserve safety under parametric uncertainty, a formulation extended by Lopez and Slotine to handle unmatched uncertainties via certainty equivalence. Other varia- tions include the dynamic penalty functions employed by Xiao et al. , the high-order adaptive formulations by Cohen et al.

, and the auxiliary systems used by Cheng et al. and Zhang et al. for specific applications like electric vehicles and marine vessels. Recently, Cheng et al. combined adap- tive CBFs with disturbance observers to manage unstructured noise. Despite these advances, many state-of-the-art methods adopt a modular structure where the adaptation law (e.g., re- cursive least squares or gradient descent) remains agnostic to the barrier function, as seen in [18, 19]. The learning pro- cess reduces parameter error globally rather than prioritizing accuracy in the critical regions where safety is threatened.

Parallel to these efforts, data-driven and nonlinear control techniques seek to tighten robust bounds through experience. Zeng et al. applied composite learning for robotic ma- nipulators, Gutierrez et al. utilized real-time Gaussian Process modeling, and Chriat et al. leveraged reinforce- ment learning to optimize class-K functions. Similarly, Barrier Lyapunov Functions (BLFs), explored by Jiang et al. and others , enforce constraints on tracking errors. How- ever, BLF methods generally require the control architecture to follow a strict backstepping structure , which constrains their application to general, optimization-based safety filters for arbitrary nominal controllers.

This paper introduces the composite adaptive control bar- rier function (CaCBF) framework to resolve the conflict between robust conservatism and transient learning risks. In- stead of appending an estimator to a fixed controller, we unify parameter estimation and safety certification into a sin- gle Lyapunov-based design. We construct a composite energy function that sums a logarithmic barrier potential, representing the proximity to constraint violation, a control Lyapunov func- tion, and a quadratic parameter error term. We then derive an update law that strictly dissipates the total energy. This estab- lishes a feedback loop where the adaptation is driven not only by prediction error but also by the gradient of the safety barrier.

As the system approaches the boundary, the effective learning rate adjusts to prioritize the parameters required to maintain safety.

This work makes the following contributions. We con- struct a composite energy function that unifies a logarithmic safety barrier, a control Lyapunov function, and a quadratic parameter-errorterm, and we derive the adaptation law directly from this function (Section 3). Building on this construction, we prove that forward invariance of the safe set holds for all bounded parametric uncertainties without persistence of exci- tation, and for any κ > 0 (Theorem 3 and Remark 5). We further establish that this safety guarantee is unconditionally robust to bounded errors in the state-derivative measurement, with no restriction on the noise magnitude (Theorem 5). Be- yond safety, we prove that all closed-loop signals are uni- formly ultimately bounded for any κ satisfying the explicit computable condition (50) (Theorem 4 and Corollary 1). Fi- nally, we prove that the CaCBF admissible control set always contains the robust one as a subset, which quantifies the reduction in conservatism relative to robust CBF (Theorem 6).

We validate the framework through three numerical case studies: adaptive cruise control with unknown drag, an omni- directional robot avoiding obstacles under unknown drift, and a planar drone navigating a narrow gate under unknown cross- wind. The results demonstrate that CaCBF achieves set uti- lization comparable to exact-model methods and outperforms robust baselines, confirming its ability to recover performance in uncertain environments without compromising safety. A comparison of the CaCBF architecture against representative prior adaptive CBF methods is given in Remark 12.

The remainder of the paper is organized as follows. Section 2 formalizes the problem. Section 3 derives the CaCBF framework and control law. Section 4 proves the main theoretical results. Section 6 presents numerical examples, and Section 7 discusses conclusions and future directions.

Notation. We denote the set of real numbers by R and the n-dimensional Euclidean space by Rn. The Euclidean norm is denoted by ∥· ∥. For a symmetric matrix P ∈Rn×n, we write P ≻0 (resp. P ⪰0) if P is positive definite (resp. positive semidefinite). Given P ≻0, the weighted Euclidean norm is

√

zTPz. The n × n identity matrix is denoted by In. We denote the limit from the left by limε↑0 and from the right by limε↓0. A continuous function α: [0, a) →[0, ∞) is of class K if it is strictly increasing and α(0) = 0. If a = +∞ and limr→+∞α(r) = +∞, it is of class K∞. In addition, for b > 0, a continuous function α: (−b, a) →R is of extended class Ke if α(0) = 0 and it is strictly increasing.

Et H: Rn →R, F : Rn →Rn, And G: Rn →Rn×M Be Con-

tinuously differentiable functions. For x ∈Rn, let ∇h(x) ≜

∂X(N) ]T ∈Rn Denote The Gradient Vector Of H With

Composite Adaptive Control Barrier Functions for Safety-Critical Systems respect to x. In addition, the Lie derivative of h along f is Lf h(x) ≜∇h(x)Tf(x) ∈R. Similarly, the Lie derivative of h along G is LGh(x) ≜∇h(x)TG(x) ∈R1×m. Finally, for an op- timization problem subject to g(z) ≤0, a point z∗is strictly feasible if g(z∗) < 0.

Problem Formulation

Consider the nonlinear control-affine system subject to linear

⊆Rn Is The State, And

u(t) ∈U ⊆Rm is the control input. The functions f : X →Rn,

F: X →Rn×P, And G: X →Rn×M Are Known And Con-

tinuously differentiable, while θ∗∈Rp represents a vector of constant, unknown parameters. We impose the following

Assumption:

(A1) There exists a known constant θmax > 0 such that ∥θ∗∥≤θmax. Assumption (A1) implies the existence of a known compact

(2)

such that θ∗∈Θ. Remark 1. The requirement that θ∗be constant and bounded is standard in adaptive control and is physically mean- parameters arise naturally when the uncertain quantities are structural properties of the plant that are time-invariant over the operational horizon of interest: the aerodynamic drag and rolling-resistance coefficients in adaptive cruise control, the viscous friction or payload mass of a robotic manipulator, and the crosswind forces acting on a drone are all well- modeled this way, and we study each in Section 6. When true time-variation is present but slow, the constant-parameter assumption remains a valid approximation whose error can be bounded and treated as a residual disturbance; extending CaCBF to slowly time-varying parameters via a leakage mod- ification is a direction for future work. Known bounds are rou- tinely available in practice: physical conservation laws (e.g., drag forces are bounded by dynamic pressure), manufacturer datasheets (e.g., maximum payload mass), or offline system- identification experiments all provide conservative envelopes θmax without requiring exact values. In fact, if the parame- ters were known exactly, no adaptation would be needed; the bound θmax does not enter the control law but only initializes the projection set Θ and prevents parameter drift, thus a conser- vative overestimate is both safe and acceptable. The selection guideline for θmax is discussed further in Remark 3.

The dynamics in (1) thus capture a broad class of physical systems characterized by linear-in-parameter uncertainty. To encode the safety objective, let the safe set C ⊂X be the superlevel set of a continuously differentiable function

(3)

and let bd C ≜{x ∈X : h(x) = 0} and int C ≜{x ∈X : h(x) > 0} denote the boundary and interior of C, respectively. Let s: [0, ∞) × X →X be such that, for all t ≥0, s(t, x(0)) is the trajectory of (1) corresponding to each initial condition x(0) ∈X. The primary objective is to synthesize a control law u: X →U such that C is forward invariant with respect to the dynamics (1), that is, for all initial conditions x(0) ∈C and all t > 0, s(t, x(0)) ∈C. Mathematically, a sufficient condition for forward invariance of C with respect to (1) is provided by Nagumo’s Theorem , which requires the vector field ˙x to point into the set at the boundary, that is, for all x ∈bd C,

(4)

then, C is forward invariant with respect to (1). To ensure (4) holds and to regularize the closed-loop behavior within int C, CBFs enforce the stronger condition that, for all x ∈C,

(5)

where α ∈Ke is locally Lipschitz. Note that (5) implies that as the state x approaches the boundary of C (i.e., as h(x) ↓0), the permissible rate of decay of the safety margin approaches zero (i.e., −α(h(x)) ↑0). Based on the invariance condition (5), we define a CBF for the system (1) as follows.

Definition 1. A continuously differentiable function h: X → R is a CBF for the system (1) and the safe set C defined by (3), if there exists a locally Lipschitz function α ∈Ke such that,

(6)

is non-empty, where the time derivative ˙h is along the trajecto-

Ries Of (1), And Is Given By

˙h(x, ¯u) = Lf h(x) + LFh(x)θ∗+ LGh(x)¯u.

(7)

The result connecting Definition 1 to safety is summarized in Theorem 1 in Section 4. Note that (6) relies on ˙h, which explicitly depends on the parameter θ∗. Since θ∗is unknown, Kamaldar, M.

the set Ucbf is not directly computable. This motivates the de- sign of the composite adaptive CBF in the following section, which simultaneously estimates θ∗, drives the state toward a desired equilibrium, and enforces safety. To characterize the control objective of stabilizing the system states to the origin, we consider the following definition.

Definition 2. A continuously differentiable function V : X → [0, ∞) is a CLF for the system (1) if V is positive definite, radially unbounded, and there exists λ > 0 such that, for all

(8)

Note that condition (8) ensures the existence of control inputs that drive the system state to the origin. To ensure the stabilization objective is compatible with the safety con- straints, we consider the standard feasibility condition that the equilibrium point lies within the interior of the safe set, i.e., h(0) > 0. In the following section, we unify this stabilization objective with the safety guarantees of Definition 1 to derive the proposed composite adaptive framework.

Barrier Functions

In this section, we develop the composite adaptive control bar- rier function (CaCBF) framework. Unlike robust approaches that rely on static, worst-case bounds, we design an adaptation law that dynamically reduces conservatism while guaranteeing safety. We term this framework composite because it unifies three coupled objectives: safety, which enforces forward in- variance via a logarithmic barrier; stability, which drives the state to equilibrium; and adaptation, driven by the minimiza- tion of an error-estimation cost, to allow the system to operate closer to the safety boundary without violating it.

Let ˆθ: [0, ∞) →Rp be such that, for all t ≥0, ˆθ(t) denotes the instantaneous estimate of the unknown parameter θ∗. In ad- dition, let ˜θ: [0, ∞) →Rp, defined by ˜θ(t) ≜ˆθ(t)−θ∗, denote the instantaneous error. Using Lie derivatives and suppressing time dependence for brevity, the time derivative of h along the

Trajectories Of (1) Is Given By

˙h(x) = Lf h(x) + LFh(x)θ∗+ LGh(x)u.

(10)

where ψ: X →R1×p is the safety regressor defined by

(12)

which represents the derivative of the barrier function based on the current parameter estimate ˆθ. To simultaneously ensure safety, promote stability, and bound the parameter error, we introduce the CaCBF, denoted

(13)

where h is the CBF used to define the safe set C in (3), V : X → [0, ∞) is a candidate CLF for the system (1), κ > 0 is a weight- ing parameter, and Γ ∈Rp×p is a positive-definite adaptation gain matrix. Note that Vc contains a logarithmic barrier func- tion. This construction relies on the property that the barrier term becomes unbounded as the system approaches the safety



= −∞. Consequently, if Vc remains bounded, the state x must remain within C—since any approach to bd C would force Vc →+∞.

This implication is the engine of the safety proof in Section 4 (Theorems 3 and 5). We now derive the adaptation law ˙ˆθ by analyzing the time evolution of (13). First, the time derivative of V along the

(14)

where φ: X →R1×p is the stability regressor defined by

(16)

which represents the time derivative of the CLF based on the current parameter estimate ˆθ. Differentiating the composite

˙H(X)

h(x)(1 + h(x)) + κ ˙V(x) + ˜θTΓ−1 ˙˜θ.

(17)

Composite Adaptive Control Barrier Functions for Safety-Critical Systems Since θ∗is constant, it follows that ˙˜θ = ˙ˆθ. Thus, substituting

, (18)

where the terms in the parenthesis capture the sign-indefinite effect of parameter uncertainty on both safety and stability. We design ˙ˆθ to not only cancel these terms but also minimize the estimation error.

Define the instantaneous estimation error e: X × Rp →Rn

(19)

which represents the discrepancy between the measured or es- timated state derivative and the estimated state derivative using the current parameter estimate ˆθ. Substituting (1) into (19)

(20)

To minimize the magnitude of e, we incorporate a gradient descent term based on the instantaneous estimation-error cost

∥E(X, ˆΘ)∥2. Treat-

ing the measured state derivative ˙x as independent of ˆθ, the

(21)

To cancel the sign-indefinite bracket in (18) and simultane- ously reduce the estimation error, we design the adaptation law to explicitly cancel the safety and stability regressors while injecting the gradient update (21). Furthermore, to ensure Assumption (A1) is satisfied, we apply a projection operator.

(22)

where ˆθ(0) ∈Θ, γ ≥0 is the adaptation gain, and PΘ : Rp × Rp →Rp is a smooth projection operator that confines ˆθ to the

(23)

where τ ∈Rp. The projection operator PΘ(τ, ˆθ) modifies the update direction τ only when the estimate reaches the bound- ary of Θ (i.e., ∥ˆθ∥= θmax) and the update direction τ points outside the set Θ (i.e., ˆθTτ > 0). To satisfy the Lyapunov dis- sipation property with a general positive-definite gain matrix Γ ≻0, the projection is defined as an orthogonal projection onto the tangent plane of the boundary with respect to the Γ−1- weighted Euclidean metric used in (13). As shown in Lemma 1 in the next section, for all τ ∈Rp, PΘ satisfies the projection

(24)

Substituting (22) back into (18) and employing (20) and (24)

(25)

which highlights the core advantage of the formulation: the CaCBF evolution depends only on known or estimated quan- tities, while the estimation error provides a nonpositive contri- bution.

Remark 2. The adaptation law (22) uses the state derivative ˙x through the prediction error e, which is often not directly mea- sured. Two complementary observations address this. First, when ˙x is approximated by finite differences, ˙x ≈(xk − xk−1)/Ts, the approximation error is bounded, and Theorem 5 in the next section establishes that forward invariance is pre- served for all bounded errors ¯w with no smallness condition.

Second, the dependence on ˙x can be removed entirely by the filtered-regressor technique used in . Specifically, convolv- ing the dynamics (1) with the impulse response of a stable, strictly proper filter and integrating by parts yields a filtered prediction error computable from x and u alone, without differ- entiation. Incorporating this filtered error into the composite energy (13) while preserving the dissipation inequality (27) is a direction for future work. We further note that Proposition 1 is a continuous-time result. When ˙x is obtained by finite dif- ferencing, the estimator is implemented on a sampled and delayed signal, and the rate σ does not transfer directly from the continuous-time argument: the sampled implementation induces a delayed composition of contraction maps whose convergence rate depends on the sampling period Ts and the delay, and degrades as either grows. Quantifying this depen- dence requires a sampled-data analysis; general treatments of sampled-data models appear in , and related techniques Kamaldar, M.

for bounding convergence rates of delayed compositions of contraction maps have been developed in the consensus litera- ture . Such an analysis is beyond the scope of the present continuous-time development. The safety guarantees of Theo- rems 3 and 5 are unaffected, since they require only that the resulting error be bounded.

Leveraging (25), we synthesize a controller that bounds Vc by constraining the estimated safety and stability derivatives.

(26)

where λ, δ > 0 and α is a class K function, transforms (25)

H(X)(1 + H(X)) −Κλv(X) −Γ∥E(X, ˆΘ)∥2 + Κδ, (27)

where the upper bound ensures that Vc is bounded, thereby guaranteeing safety, as proven via contradiction in Section 4 (Theorem 3).

To strictly enforce the conditions in (26) while minimizing control effort and determining the optimal relaxation δ, we for- mulate the control synthesis as a convex optimization problem that integrates the parameter estimates directly into the con- straints. For all x ∈X and all ˆθ ∈Rp evolving according to the adaptation law (22), the control input u∗: X × Rp is the

(30)

where R: X →Rm×m is positive definite, δ ≥0 is the relax- ation variable for the stability constraint, ρ > 0 is the stability- relaxation penalty, and λ > 0 is the nominal convergence rate of the CLF.

Remark 3. The CaCBF has seven design parameters, namely, Γ, γ, κ, λ, ρ, α, and θmax. The adaptation gain Γ ≻0 governs how fast ˆθ evolves; larger eigenvalues accelerate estimation but may excite high-frequency oscillations, so a diagonal Γ with entries scaled to the expected magnitude of each θ∗- component is recommended as a starting point. The gradient gain γ ≥0 scales the prediction-error term in (22): setting γ = 0 disables the identification signal, while increasing γ improves estimation accuracy when persistency of excitation is present but does not degrade safety when it is absent. The barrier weight κ > 0 does not affect forward invariance (See Theorems 3 and 5), but must satisfy κ > cα/ε for uniform

Apply U∗And Integrate ˙ˆΘ

boundedness (See Theorem 4, condition (50)); under Lemma 3 with condition i), the explicit sufficient bound κ > α0λ/(2A2) is given in Corollary 1, and in practice values κ ∈are effective when the CLF and CBF are normalized to similar scales. The CLF decay rate λ > 0 is the desired exponential convergence rate for the nominal, known-parametersystem; λ should reflect the desired closed-loop speed and must satisfy λ > 2cFθmax if condition ii) (i.e., linear-regressor growth) of Lemma 3 is in- voked. The stability penalty ρ > 0 penalizes relaxation of the CLF constraint, with large ρ prioritizing stability and small ρ allowing larger relaxation when the CBF constraint is tight, which can be important near the safety boundary. The class-K function α determines how quickly the CBF condition is en- forced near the boundary; the linear choice α(r) = α0r with α0 > 0 is the most common, where smaller α0 yields a more conservative safety margin and larger α0 allows the system to operate closer to the boundary. Finally, the bound θmax should be any known conservative upper bound on ∥θ∗∥: an overes- timate increases initial conservatism but does not affect the safety proof, whereas an underestimate that violates θ∗∈Θ would invalidate the projection argument and must be avoided.

The closed-loop system formed by Algorithm 1 applied to (1) is analyzed in the following section, where we prove the safety, noise-robustness, uniform ultimate boundedness (UUB), and parameter-convergence properties stated in the introduction.

Composite Adaptive Control Barrier Functions for Safety-Critical Systems

Osed-Loop Analysis

This section analyzes the closed-loop system formed by the plant dynamics (1), the adaptation law (22), and the adap- tive controller (28)–(30). We establish feasibility of the op- timization (Theorem 2), forward invariance of the safe set (Theorems 3 and 5), and uniform boundedness of all signals (Theorem 4); under persistent excitation, the parameter error additionally converges to zero exponentially (Proposition 1).

To ensure the design is well-posed, we impose the following structural assumptions. (A2) The safe set C is non-empty, and h has relative degree 1 with respect to (1); that is, for all x ∈C, LGh(x)̸ = 0.

(A3) R is continuous and uniformly positive definite; that is, there exist r, r > 0 such that, for all x ∈C, rIm ⪯ R(x) ⪯rIm.

(A4) h is a CBF for the system (1) and the safe set C, and h(0) > 0. (A5) V is a CLF for the system (1).

Assumption (A2) grants the control input direct authority over the safety barrier. By requiring the relative degree to be 1, we guarantee that the actuator can instantaneously in- fluence the time derivative ˙h to steer the system away from the boundary. If this condition fails (i.e., if LGh(x) = 0), the safety constraint becomes locally independent of u, poten- tially rendering the optimization infeasible. We address this specific challenge in the planar drone scenario of Example 3, where the position-based constraints exhibit relative degree 2.

In such cases, we employ backstepping techniques [6, 34] to construct extended barrier functions that recover the required relative degree 1 property, thereby restoring direct control au- thority over the safety condition. Assumption (A3) ensures the optimization problem remains strictly convex. By bound- ing the eigenvalues of R away from zero, we guarantee a unique optimal solution and prevent the control effort from be- coming unbounded. Finally, Assumptions (A4) and (A5) align the safety and stability objectives. Specifically, the condition h(0) > 0 implies the target equilibrium lies strictly within the safe set, ensuring the CLF drives the state to the origin without conflicting with the safety barrier.

The following result, which follows from standard invari- ance arguments, links the algebraic constraint in Definition 1 to the physical safety of the system. See for a proof.

Theorem 1. Consider the system (1) and the safe set C. As- sume that (A2) is satisfied, and let h : X →R be a CBF for (1) and C. In addition, assume that, for all t ≥0, u(t) ∈Ucbf(x(t)).

Then, C is forward invariant with respect to (1). Remark 4. Theorem 1 is a classical result that requires that, for all t ≥0, u(t) ∈Ucbf(x(t)), where Ucbf is defined using the unknown θ∗. Because θ∗is not available, the adaptive con- troller cannot directly enforce membership in Ucbf. The CaCBF framework addresses this by providing an independent safety proof (i.e., Theorem 3) that does not invoke Theorem 1 at all.

Instead, forward invariance follows from the fact that the com- posite Lyapunov function Vc remains bounded, which in turn forces h(x(t)) > 0 for all t ≥0. Theorem 1 is included for completeness and to motivate the CBF framework, while the actual safety guarantee of the adaptive scheme is established via Theorem 3.

Having established the conditions for safety via Theorems 3 and 5, the next theorem confirms that the control law remains feasible and Lipschitz continuous.

Theorem 2. Consider the CLF-CBF-QP defined in (28)–(30). Assume that (A2) and (A3) are satisfied. Then, the following

Statements Hold:

i) For all x ∈int C and all ˆθ ∈Rp, the CLF-CBF-QP is strictly feasible and has a unique global minimizer. ii) The optimal control u∗and relaxation δ∗are locally Lipschitz continuous on int C × Rp.

Proof. Let z ≜[uT δ]T ∈Rm+1. We write the CLF-CBF-QP as

B1(X, ˆΘ) ≜Lf H(X) + Lfh(X)ˆΘ + Α(H(X)) ∈R,

b2(x, ˆθ) ≜−Lf V(x) −LFV(x)ˆθ −λV(x) ∈R.

(33)

To prove i), let x ∈int C and ˆθ ∈Rp. First, consider the safety constraint a1(x)z < b1(x, ˆθ). Since (A2) is satis- fied, it follows that a1(x)̸ = 0, and thus the strict inequality a1(x)z < b1(x, ˆθ) defines a non-empty, open half-space in Rm.

Therefore, there exists u∗∈Rm such that, for all δ0 ≥0,

Z0 ≜[Ut

∗δ0]T satisfies the strict safety constraint a1(x)z0 < b1(x, ˆθ). Now, let δ1 ≜LGV(x)u∗−b2(x, ˆθ) and z2 ≜[uT

∗Δ2]T,

where δ2 > max(0, δ1). It thus follows that a2(x)z2 < b2(x, ˆθ). Therefore, since z2 satisfies both constraints, the CLF-CBF-QP Kamaldar, M.

problem is strictly feasible. Moreover, since ρ > 0 and (A3) is satisfied, it follows that Q(x) ≻0, ensuring a unique global minimizer z∗.

To prove ii), we use the sensitivity theorem in [35, Theorem 1], which requires the linear independence constraint qualifi- cation. Note that Q, A, and b are composed of smooth vector fields f, F, G and continuously differentiable functions h and V. Since continuously differentiable functions are locally Lip- schitz on compact sets, it follows that Q, A, and b are locally Lipschitz on compact sets. In addition, R ≻0 and ρ > 0 im- ply strict convexity. Next, let x ∈C and ˆθ ∈Rp. We prove that the gradients of the active constraints are linearly independent.

First, consider the case where exactly one constraint is active. Since (A2) is satisfied, it follows that a1(x)̸ = 0. Since, in addi- tion, a2(x)̸ = 0, it follows that the active constraint is linearly independent. Second, consider the case where both constraints are active. Let c1, c2 ∈R, and consider the linear combination

(34)

Since (A2) is satisfied, (34) implies c1 = c2 = 0. Thus, the gra- dients of the active constraints are linearly independent. Since all conditions of [35, Theorem 1] are satisfied, z∗is locally Lipschitz continuous, which confirms ii) because u∗and δ∗are linear projections of z∗.

We now confirm that the projection operator (23) enforces parameter boundedness without corrupting the adaptation di- rection. This property ensures that the projection mechanism does not counteract the adaptation process. The following result is an extension of the standard projection-operator property from [29, 36] to the weighted inner product induced by Γ−1.

Lemma 1. Assume that (A1) is satisfied. Then, for all τ ∈Rp and all ˆθ ∈Θ, the projection operator PΘ defined in (23)

(35)

The following lemma guarantees that the projection opera- tor confines the parameter estimates to the compact set Θ. Its proof follows the standard argument in adapted to the Γ−1-weighted metric.

Lemma 2. Assume that (A1) is satisfied. Consider the param- eter update law given by (22), where ˆθ(0) ∈Θ. Then, for all t ≥0, ˆθ(t) ∈Θ.

With the parameter estimates strictly bounded, we turn to the safety of the system. The following theorem proves that the adaptive controller renders the safe set C forward invari- ant with respect to the system (1), ensuring the constraints are satisfied for all time.

Theorem 3. Consider the closed-loop system comprising the system (1), the parameter update law (22), and the adaptive control law u∗defined by (28)–(30). Assume that (A1)–(A5) are satisfied, and let x(0) ∈int C and ˆθ(0) ∈Θ. Then, the safe set C is forward invariant with respect to (1).

Proof. Consider the composite function Vc defined in (13). Taking the time derivative along the closed-loop trajectories yields (18). Note that since (A1) is satisfied and ˆθ(0) ∈Θ, Lemma 2 implies that for all t ≥0, ˆθ(t) ∈Θ. Therefore, sub- stituting the adaptation law (22) into (18), and using Lemma 1

˙HˆΘ(X, ˆΘ)

h(x)(1 + h(x)) + κ ˙Vˆθ(x, ˆθ) + γ˜θTF(x)Te(x, ˆθ)

(36)

where the inequality follows from Lemma 1, that is, (35) im- plies ˜θTΓ−1(PΘ(τ, ˆθ)−τ) ≤0, making the subtracted bracket in the second line of the derivation nonnegative; and the last step uses ˜θTFTe = ˜θTFT(−F˜θ) = −∥e∥2 from (20). Note that (A4) and (A5) are satisfied. Thus, using (12) and (16), and

Α(H(X))

h(x)(1 + h(x)) −κλV(x) −γ∥e(x, ˆθ)∥2 + κδ∗(x, ˆθ).

(37)

Next, suppose for contradiction, that C is not forward in- variant with respect to (1). Then, there exists t1 > 0 such that h(x(t1)) = 0 and, for all t ∈[0, t1), h(x(t)) > 0. It thus follows

Lim

t↑t1 Vc(x(t), ˆθ(t)) = +∞.

Α(H(X(T)))

h(x(t))(1 + h(x(t))) + κδ∗(x(t), ˆθ(t)).

(39)

Since α is Lipschitz with α(0) = 0 and h(x(t1)) = 0, there exists cα > 0 such that α(h(x(t))) ≤cαh(x(t)) for all t ∈[0, t1]. Combined with h(x(t)) ≥0 on this interval, (39) gives, for all

T ∈[0, T1],

˙Vc(x(t), ˆθ(t)) ≤cα + κδ∗(x(t), ˆθ(t)).

(40)

Composite Adaptive Control Barrier Functions for Safety-Critical Systems Next, note that since x is continuous on the closed interval [0, t1], it follows that, for all t ∈[0, t1], x(t) belongs to a com- pact set. Moreover, since (A1) is satisfied, Lemma 2 implies that ˆθ is bounded. Furthermore, since (A2) and (A3) are satis- fied, part ii) of Theorem 2 implies δ∗is Lipschitz continuous.

It thus follows that δmax ≜supτ∈[0,t1] δ∗(x(τ), ˆθ(τ)) exists. It

Thus Follows From (40) That, For All T ∈[0, T1],

˙Vc(x(t), ˆθ(t)) ≤cα + κδmax.

Ntegrating (41) Over The Interval [0, T1] Yields

Vc(x(t1), ˆθ(t1)) ≤Vc(x(0), ˆθ(0)) + t1(cα + κδmax) < +∞, (42) which contradicts (38). Thus, for all t ≥0, h(x(t)) > 0, which confirms the result.

Remark 5. Theorem 3 places no condition on κ: forward in- variance holds for any κ > 0. This is because the proof uses a finite-time contradiction argument in which κ appears only in the bound cα +κδmax, which is finite for any fixed κ since δ∗is Lipschitz continuous (Theorem 2). The role of κ is therefore purely to govern the tightness of the UUB set in Theorem 4 (via condition (50)), and to weight the relative priority of the CLF term in Vc (13). A larger κ drives the state more aggres- sively toward the origin outside S but does not add or remove the safety guarantee.

To guarantee the uniform boundedness of all closed-loop signals, we impose the following structural assumption on the system’s asymptotic behavior.

(A6) There exists a compact set S ⊂X containing the ori- gin and a constant ε > 0 such that, for all x ∈X \ S

(43)

Assumption (A6) ensures that outside the set S, the drive for stability λV outpaces the cost of safety δ∗. Physically, the relaxation term δ∗represents the control effort “wasted” to sat- isfy safety constraints or compensate for parameter errors. By requiring the stability term to dominate, we ensure the con- troller always retains enough authority to pull the system back toward the origin. This structure is typical of mechanical sys- tems where the CLF grows quadratically, while the uncertainty and safety conflicts grow only linearly or sub-quadratically.

The following result identifies sufficient conditions for this property to hold. Lemma 3. Assume that (A1)–(A3) and (A5) are satisfied, and let λ > 0 and un : X →Rm be such that, for all x ∈X, Lf V(x) + φ(x)θ∗+ LGV(x)un(x) ≤−λV(x).

(44)

In addition, assume that there exists cu > 0 such that, for all

(45)

Furthermore, assume that there exists cF > 0 such that, for all x ∈X, at least one of the following conditions holds:

I) ∥Φ(X)∥≤Cf

√V(x). ii) ∥φ(x)∥≤cFV(x) and λ > 2cFθmax. Then, (A6) is satisfied.

Proof. Note that since (A1) is satisfied, it follows that Θ de- fined by (2) exists. Let ˆθ ∈Θ. Since (A5) is satisfied, using

(46)

where δn(x, ˆθ) ≜max{0, φ(x)˜θ}. It thus follows that, for all x ∈X, the pair (un(x), δn(x, ˆθ)) satisfies the stability constraint (30). Moreover, since (A2) and (A3) are satisfied, it follows that, for all x ∈X, (u∗(x, ˆθ), δ∗(x, ˆθ)) is the optimal solution minimizing the cost (28). It thus follows that, for all x ∈X,

(47)

Note that, since ˆθ, θ∗∈Θ, using the triangle inequality ∥˜θ∥≤

∥ˆΘ∥+ ∥Θ∗∥≤2Θmax Implies That, For All X ∈X,

δn(x, ˆθ) ≤∥φ(x)∥∥˜θ∥≤2∥φ(x)∥θmax.

(48)

Taking the square root of both sides of (47), and dividing by

(49)

where the last inequality follows from (45) and (48). First, consider the case where i) is satisfied, and it follows

That

lim∥x∥→∞(λV(x) −δ∗(x, ˆθ)) = +∞. Kamaldar, M. Next, consider the case where ii) is satisfied, and it follows

(X),

which, since V is radially unbounded and λ > 2cFθmax, implies that lim∥x∥→∞(λV(x) −δ∗(x, ˆθ)) = +∞. Therefore, since in both cases i) and ii), lim∥x∥→∞(λV(x) −δ∗(x, ˆθ)) = +∞, it follows that, for all ε > 0, there exists a compact level set S ⊂X such that for all x ∈X \S, λV(x)−δ∗(x, ˆθ) ≥ε, which confirms (A6).

Building on this asymptotic dominance, the following theorem guarantees that all closed-loop signals remain uni- formly bounded.

Theorem 4. Consider the closed-loop system comprising the system (1), the parameter update law (22), and the adaptive control law u∗defined by (28)–(30). Assume that (A1)–(A6) are satisfied, and let x(0) ∈int C and ˆθ(0) ∈Θ. Let cα > 0 satisfy α(r) ≤cαr for all r ≥0, and let ε > 0 be as in (A6). If

(50)

then all closed-loop signals x, ˆθ, u, and δ are uniformly ultimately bounded. Proof. Consider the time derivative of Vc. Since (A1) is satisfied, using the derivation in (37) implies that

(51)

By hypothesis, cα > 0 satisfies α(r) ≤cαr for all r ≥0.

(52)

where the second inequality uses h(x(t)) > 0 for all t ≥0 (Theorem 3), so cα/(1 + h) ≤cα. Next, note that since (A6) is satisfied, it follows from (52)

(53)

Since (50) holds, it follows from (53) that, for all x ∈X \ S, ˙Vc(x, ˆθ) ≤cα −κε < 0.

(54)

We now establish the forward invariance of a specific sub- level set. Let s0 ≜maxx∈S,ˆθ∈Θ Vc(x, ˆθ) and define the sublevel set Ωs ≜{(x, ˆθ) ∈X × Θ | Vc(x, ˆθ) ≤s}, where s ≥ max{Vc(x(0), ˆθ(0)), s0}. Since s ≥s0, it follows that, for all (x, ˆθ) ∈X \ int Ωs, (x, ˆθ) ∈X \ S × Θ. It thus follows from (??)Vcdda that Ωs is forward invariant with respect to (1).

Thus, for all t ≥0, (x(t), ˆθ(t)) ∈Ωs, which implies Vc(t) ≤s. Finally, note that since (A5) implies that V is radially un- bounded, there exists a class K function α such that, for all t ≥0, α(∥x(t)∥) ≤V(x(t)). In addition, since ˜θTΓ−1˜θ ≥0, the definition (13) gives Vc(x, ˆθ) ≥V(x) for all x ∈C and ˆθ ∈Θ.

It thus follows that α(∥x(t)∥) ≤V(x(t)) ≤Vc(t) ≤s . There- fore, for all t ≥0, ∥x(t)∥≤α−1(s), which confirms that x is uniformly bounded. Since, in addition, Lemma 2 implies, for all t ≥0, ˆθ(t) ∈Θ, it follows that the joint state (x(t), ˆθ(t)) evolves within a compact domain. Therefore, since the image of a compact set under a continuous map is compact, part ii) of Theorem 2 implies that the control input u∗and relaxation δ∗are uniformly bounded, which confirms the result.

Remark 6. Theorem 4 establishes that all closed-loop signals are uniformly ultimately bounded within the sublevel set Ωs, where s ≥max{Vc(x(0), ˆθ(0)), s0}. The quantity s0 depends on the maximum of Vc over the compact set S (determined by ε from (A6) and the system dynamics), while the initial value Vc(x(0), ˆθ(0)) reflects the starting conditions. Consequently, a large initial parameter error ∥˜θ(0)∥increases Vc(x(0), ˆθ(0)) and therefore increases s. In practice, better initial guesses ˆθ(0) (from offline identification, for example) directly reduce the size of the invariant set and improve transient behavior. The relaxation variable δ∗is also bounded on compact sets; how- ever, the bound on δ∗grows with the degree of safety-stability conflict and the magnitude of ˜θ.

Lemma 3 establishes that for large states, the drive for stabil- ity outpaces the relaxation penalties required for safety. Under the conditions of Lemma 3, the required gain κ in (50) can be computed explicitly, as shown in the following corollary.

Corollary 1. Assume that (A1)–(A3) and (A5) are satisfied, and let λ > 0 and un : X →Rm be such that, for all x ∈X, (44) is satisfied. In addition, assume that there exists cu > 0 such that, for all x ∈X, (45) is satisfied. Furthermore, assume that there exists cF > 0 such that condition i) of Lemma 3 is

Then, For All Κ > Α0Λ

2A2 , (50) is satisfied. Proof. Using the same process as in the proof of Lemma 3 confirms (49), which together with condition i) of Lemma 3

(56)

Composite Adaptive Control Barrier Functions for Safety-Critical Systems Define g: [0, ∞) →R by g(r) ≜λr2 −Ar. Using r = √V(x),

(57)

Since g′(r) = 2λr −A, the function g is increasing on (A/(2λ), ∞). Define the compact set S ≜{x ∈X : V(x) ≤ 4(A/λ)2}, and note that, for all x /∈S, √V(x) ≥2A/λ > A/(2λ).

Since g is increasing on this interval, its minimum over {r ≥ 2A/λ} is attained at r = 2A/λ, that is, for all x /∈S,

(58)

Combining (57) and (58) implies that, for all x /∈S and all ˆθ ∈Θ, λV(x) −δ∗(x, ˆθ) ≥2A2/λ. Hence (A6) holds with ε = 2A2/λ. Since α(r) = α0r gives cα = α0, the gain condition

(2A2/Λ) = Α0Λ/(2A2), Which Is The Stated

bound. Remark 7. Corollary 1 makes the gain condition explicit and computable from design parameters: cu (i.e., nominal control bound (45)), ρ (i.e., stability-relaxation penalty), cF (i.e., re- gressor bound from condition i)), and θmax (i.e., parameter bound from (A1)). Notably, κ does not appear in Theorems 3 or 5: forward invariance holds for any κ > 0. The lower bound

Κ > Α0Λ

2A2 is needed only for uniform boundedness of signals. The adaptation law (22) contains the term −ψ(x)T/[h(x)(1 + h(x))], which is unbounded as h(x) ↓0, and the projection (23) constrains ˆθ but not ˙ˆθ. The following result shows that ˙ˆθ is nevertheless uniformly bounded. The mechanism is that the logarithmic barrier in (13) is self-regulating: invariance of the sublevel set Ωs established in Theorem 4 forces a uniform pos- itive lower bound on h along closed-loop trajectories, which in turn bounds the barrier gradient.

Corollary 2. Consider the closed-loop system of Theorem 4, and assume that (A1)–(A6) and (50) are satisfied. Let s > 0 and Ωs be as in the proof of Theorem 4. Then, the following

E−S

1 −e−s > 0. ii) There exists cθ > 0 such that, for all t ≥0, ∥˙ˆθ(t) ≤cθ. Proof. To prove i), note that Theorem 4 implies that, for all t ≥0, Vc(x(t), ˆθ(t)) ≤s. Since, for all x ∈X, κV(x) ≥0 and 2 ˜θTΓ−1˜θ ≥0, it follows from (13) that, for all t ≥0,

(59)

which implies h(x(t))/(1 + h(x(t))) ≥e−s. Since h(x(t)) > 0 by Theorem 3 and e−s ∈(0, 1), rearranging yields h(x(t))(1 − e−s) ≥e−s, which confirms i).

To prove ii), note that Theorem 4 implies that x(t) evolves in the compact set Xs ≜{x ∈X : ∥x∥≤α−1(s)}, and Lemma 2 implies ˆθ(t) ∈Θ. Since ψ, φ, and F are continuous, there exist ¯ψ, ¯φ, ¯F > 0 such that, for all x ∈Xs, ∥ψ(x)∥≤¯ψ, ∥φ(x)∥≤¯φ, and ∥F(x)∥≤¯F. Moreover, since ˆθ, θ∗∈Θ, it follows that ∥˜θ∥≤2θmax, and thus (20) implies ∥e(x, ˆθ)∥≤2¯Fθmax. Let τ denote the unprojected update direction in (22). Using i) and the triangle inequality yields that, for all t ≥0, ∥τ∥≤¯τ,

(60)

If the projection is inactive, then ˙ˆθ = τ and ∥˙ˆθ∥≤¯τ. If the projection is active, then ∥ˆθ∥= θmax, and it follows from (23)

¯τ. Remark 8. Corollary 2 shows that the apparent singularity of the barrier gradient in (22) is never encountered along closed- loop trajectories. The bound h degrades as s increases, so larger initial energy Vc(x(0), ˆθ(0)), which includes larger ini- tial parameter error, permits closer approach to the boundary and correspondingly larger ∥˙ˆθ∥. This is consistent with the intended behavior of the design: proximity to the safety bound- ary accelerates adaptation, and Corollary 2 certifies that this acceleration remains bounded.

Definition 3. The regressor F is persistently exciting (PE) along the trajectory of (1) if there exist constants T0, µ > 0

(62)

Although the main results guarantee safety without persis- tence of excitation, it is worth recording what happens when the trajectory is informative, since this directly addresses the practical concern that the relaxation δ∗may remain large. If F is persistently exciting, the gradient term γF(x)Te in (22) drives the parameter error to zero, which in turn tightens the bound (49) on δ∗. The following result formalizes this.

Its proof follows from the standard exponential-stability the- ory for persistently excited gradient estimators [29, Sec. 4.3], applied once the projection (23) becomes inactive.

Proposition 1. In addition to the assumptions of Theorem 4,

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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