Vericut Composite Drilling

Abstract

Motivated by engineering applications of subsea installation by deepwater construction vessels in oil drilling, and of aid delivery by unmanned aerial vehicles in disaster relief, we develop output-feedback boundary control of heterodirectional coupled hyperbolic PDEs sandwiched between two ODEs, where the measurement is the output state of one ODE and suffers a time delay. After rewriting the time-delay dynamics as a transport PDE of which the left boundary connects with the sandwiched system, a state observer is built to estimate the states of the overall system of ODE-heterodirectional coupled hyperbolic PDEs-ODE-transport PDE using the right boundary state of the last transport PDE. An observer-based output-feedback controller acting at the first ODE is designed to stabilize the overall system using backstepping transformations and frequency-domain designs. The exponential stability results of the closed-loop system, boundedness and exponential convergence of the control input are proved. The obtained theoretical result is applied to control of a deepwater oil drilling construction vessel as a simulation case, where the simulation results show the proposed control design reduces cable oscillations and places the oil drilling equipment to be installed in the target area on the sea floor. Performance deterioration under extreme and unmodeled disturbances is also illustrated.

Key words: Distributed parameter system; delay; boundary control; oil drilling; unmanned aerial vehicles.

Otivation

The first motivation of this work arises from off-shore oil drilling, where some equipment, such as a subsea manifold, a subsea pump station, a subsea distribution unit along with associated foundations, flowlines and umbilicals should be installed at designated locations [35, 36] around the drill center on the seafloor. The installation of the equipment is completed by deepwater construction vessels (DCVs) , because the installation sites are located outside a radius 45 m of the floating drilling platform (Fig.2 in ) and can- not be accessed by the huge floating drilling platform which has limited access and mobility , and some of the equip- ment, such as flowlines, umbilicals, should be installed in advance to prepare to hook up the floating drilling platform when it arrives. The DCV is shown in Fig. 3, where the top of the cable is attached to a crane on a vessel at the ocean surface and the bottom attached to equipment to be installed at the sea floor, referred to as payloads hereafter. The tra- ditional method in underwater installation by DCVs is reg- ⋆The material in this paper was not presented at any conference.

ulating the vessel dynamics position and manipulating the crane to obtain the desired heading for the payload . It is not suitable for the deeper water construction in offshore oil drilling (more than a thousand meter) because the cable is very long when the payload is near the seabed, which would increase the natural period of the cable-payload system and introduce large oscillations [15, 42]. The cable oscillations would cause large offset between the payload and the de- sired heading position of the crane, namely the designated installation location. In addition to large oscillations of the long cable, another challenge in the subsea installation is the existence of a sensor delay which is due to the fact that the sensor signal is transmitted over a large distance from the seafloor to the vessel on the ocean surface through a set of acoustics devices (Ch. 10.6.6 in ). It would result in information distortion or even make the control system lose stability. It is vital to design a delay-compensated control force at the onboard crane to reduce the cable oscillations and then place the equipment in the target area on the sea floor.

The second motivation is aid delivery to dangerous and in- accessible areas, such as flood, earthquake, fire, and indus- trial disaster victims via unmanned aerial vehicles (UAVs) , where food, first-aid kits, referred to as suspended objects or payloads hereafter, are tied to the bottom of a ca- ble, of which the other end is hanged to an UAV, i.e., a struc-

Arxiv:1910.05948V3 [Math.Oc] 16 Jul 2020

ture of UAV-cable-payload. The swing/oscillation of cable- payload would appear during the transportation motion due to the properties of the cable and external disturbances, such as wind, which may cause damage to the suspended ob- ject, the environment and the people around . At the end of the transport motion, when the UAV arrives at the location directly over to the rescue site and is ready to land the aid supplies, the suspended object naturally continues to swing which makes precisely placing these aid supplies at the target position difficult. Therefore, rapid suppression of oscillations of the cable and suspended object through a control force provided by rotor wings of the UAV is required.

The measurement can be the oscillation acceleration of the suspended object by an accelerometer placed at the bottom end of the cable. Sensor delay would exist in the process of data acquisition, transmission and integration calculation to obtain the payload oscillation displacement which is used in constructing the observer and controller. In addition to aid delivery in disaster relief, UAV delivery is also used in some commercial cases to reduce labor cost. For example, some companies use UAVs to transport cargos in storehouses or lift and position building elements in architectural construc- tion . Some logistics companies have also begun to use UAVs to deliver packages in a small area .

The vibration/oscillation dynamics of cables are distributed

Parameter Systems Modeled By Wave Pdes , And

the crane/equipment and UAV/supplies at two ends of the cable can be regarded as lumped tip payloads described by ODEs [18, 19]. For the sake of order reduction, the wave PDEs with viscous damping terms describing the cable ma- terial damping can be converted to a class of heterodirec- tional coupled hyperbolic PDE systems , - via Rie- mann transformations . Therefore, the control problems in the aforementioned two applications come down to a the- oretical problem about delay-compensated boundary control of a sandwiched coupled hyperbolic PDE system.

Ontrol Of Pde Sandwiched System

Boundary control designs of a transport PDE sandwiched

By Two Odes , , , Viscous Burgers Pde Or

heat PDE sandwiched systems were developed in the previous research. Control design of the coupled hyperbolic PDE sandwiched system mentioned in the last section is more challenging because of the in-domain instability which comes from in-domain couplings between PDEs. Recently, some results about state-feedback control of a coupled hy- perbolic sandwiched system was proposed in [31, 32, 38].

Based on observer designs, output-feedback control of the coupled hyperbolic PDE sandwiched system was designed in . However, the aforementioned research has not in- vestigated delay compensation ( [29, 32] only achieve ro- bustness to a small delay) in boundary control of sandwiched practical engineering, especially the sensor delay, which ex- ering time-delay compensation in the control design is an important step to apply theoretical results into practice.

Sensor Delay Compensation

The topic of sensor delay compensation has received much attention in the past three decades. In an advanced result presented in [25, 26], the sensor delay was captured as a transport PDE and then the original plant of ODE with sensor delay was rewritten as an ODE-transport PDE cas- caded system without delay, before the observer/controller designs were conducted via backstepping. Therein, the ob- server was built as a “full-order” type which estimated both plant states and sensor states, compared with some clas- sical results about sensor-delay-compensated observer de- signs , , which only estimated plant states, namely “reduced-order” type. Using a model-based predictor, ob- server design for ODE systems with a time-varying sen- sor delay was presented in . Time-varying sensor delay compensation was also considered in which designed a delay-compensated observer to estimate vibration states of a wave PDE modeled cable elevator. Boundary stabiliza- tion of a wave PDE whose boundary observation suffers a time delay was also proposed in . In the aforementioned work, the sensor delay was considered in the plant which is an ODE or a simple form PDE while the sensor delay in this paper exists in a more complex plant which is a sandwiched PDE system.

Ain Contribution

• Some restrictions on the proximal ODE structure in the previous results about boundary control of ODE- hyperbolic PDE-ODE sandwiched systems are relieved, such as the first-order and scalar form , a chain of integrators , det(C0B0)̸ = 0 and B0 being invert- ible .

• Compared with which first addressed the left invert- ible type proximal ODE in sandwiched PDEs, this paper further proposes observer-based output-feedback control design using a delayed measurement of which the delay length is constant and arbitrary. Compensation of sensor delay has not been investigated in control of sandwiched systems before. This is a more challenging task because the plant is extended to ODE-coupled hyperbolic PDEs- ODE-transport PDE after rewriting the delay as a trans- port PDE.

• The obtained theoretical result is applied to oscillation suppression of a DCV with compensating the sensor delay arising from large-distance transmission of the sensing signal via acoustics devices, where only one control force at the onboard crane is required while one more control force applied at the payload is required in .

For complete clarity, the comparisons with the recent results of boundary control of sandwiched systems are summarized in Tab. 1.

√

√denotes “included” and × denotes “not included”.

Organization

The concerned model and the control task is described in Section 2. Observer design is proposed in Section 3. Therein, three transformations are used to convert the observer error system to a target observer error system whose exponential stability is straightforward to obtain, where all the dynamics output injections required in constructing the observer are determined. Observer-based output-feedback control design is proposed in Section 4, where two transformations are applied to transform the observer to a so-called target system in a “stable-like” form except for the proximal ODE which is influenced by perturbations originating from PDEs and the distal ODE. After representing this target system in the frequency domain to obtain the relationships between the states of the proximal ODE and those perturbation states, the proximal ODE is reformulated as a new ODE without external perturbations in the frequency domain, and then the stabilizing control input is designed. The exponential stability of the closed-loop system and the boundedness and exponential convergence to zero of the control input are proved in Section 5. The obtained theoretical result is applied to oscillation suppression and position control of a DCV used for seabed installation as a simulation case in Section 6.

The conclusions and a discussion of future work are provided in Section 7.

(7)

∀(x,t) ∈×[0,∞). The block diagram of (1)-(7) is shown in Fig. 1. X(t) ∈Rn×1, Y(t) ∈Rm×1 are ODE states. z(x,t) ∈ R,w(x,t) ∈R are states of the 2 × 2 coupled hyperbolic PDEs with initial conditions (z(x,0),w(x,0)) ∈L2(0,1) × L2(0,1). τ is an arbitrary constant denoting the time delay in the measurement. U(t) is the control input to be designed.

c1,c2 ∈R and E0 ∈Rn×1 are arbitrary. q1 and q2 are positive transport velocities. q, p ∈R satisfy Assumption 1. A0 ∈ Rn×n, B0 ∈Rn×1, C0 ∈R1×n, A1 ∈Rm×m, B1 ∈Rm×1, C1 ∈ R1×m satisfy Assumptions 2-3.

(8)

and q̸ = 0. This assumption will be used in the output-feedback control design in Section 4.

Assumption 2 The pairs (A0,B0), (A1,B1) are stabilizable and (A0,C0), (A1,C1) are detectable. According to Assumption 2, there exist constant matrices L0, L1, F0, F1 to make the following matrices Hurwitz:

(12)

Note that A1 −eτA1L1C1e−τA1 has the same eigenvalues as

A1 −L1C1 .

Fig. 1. Block diagram of the plant (1)-(7).

(13)

for all s ∈C,ℜ(s)≥0. Assumption 3 is about matrices of the proximal ODE-X(t), namely actuator dynamics. Even though zeros in the closed right-half plane are excluded here while the zeros are al- lowed in some previous results on control of sandwiched PDE systems, such as , this assumption relieves some re- strictions on the structure of the proximal ODE in the exist- ing literature (such as A0,B0,C0 being scalar in [2, 29], B0 being invertible in , det(C0B0)̸ = 0 in , or a form of a chain of integrators in [38, 39]) . This assumption is equal to the existence of a stable left inversion system of (1) and is used in the control input design in Section 4.3.

(14)

for all s ∈C,ℜ(s)≥0. Assumption 4 is about matrices of the distal ODE-Y(t) with a sensor delay τ in the measurement output state. This as- sumption also prohibits the zeros of the ODE subsystem (C1,A1,B1) are located in the closed right-half plane. It is not particularly restrictive and (C1,A1,B1) is still quite gen- eral covering many application cases. This assumption is used in the observer design for the overall sandwiched sys- tem with the delayed measurement in Section 3. Note that if the sensor delay is zero, this assumption has the same form as Assumption 3.

Remark 1 The design in this paper also can be suitable for collocated control, namely the measurement is the output state of the proximal ODE X(t) with a time delay τ, if

= 0

for all s ∈C,ℜ(s)≥0. The control objective of this paper: exponentially stabilize the overall sandwiched system, i.e.,the ODE statesY(t),X(t) and the PDE states u(x,t),v(x,t), by constructing an output- feedback control input U(t) applied at the proximal ODE X(t), using the delayed measurement yout(t).

(18)

∀(x,t) ∈×[τ,∞), to describe the time delay in the mea- surement (7). Replacing (7) by (16)-(18), we obtain a sandwiched hyper- bolic PDE-ODE system connecting with another transport PDE, i.e., the following ODE-coupled hyperbolic PDEs-

(27)

for t ≥τ. Note that the time delay is “removed” at a cost of adding a transport PDE into the plant (1)-(7). Now, the control task is equivalent to exponentially stabilizing overall sys- tem (19)-(27), i.e., ODE(X)-PDE(z,w)-ODE(Y)-PDE(v), by constructing an output-feedback control input U(t) at the first ODE (19), using the right boundary state of the last

Observer Design

In order to build the observer-based output-feedback con- troller of the plant (1)-(7), in this section, we design a state- observer to track the overall system (1)-(7) only using the delayed measurement yout(t). Through the reformulation in Section 2.2, the estimation task is equivalent to designing a state-observer to recover the overall system (19)-(27) only using measurements at the right boundary x = 2 of the last transport PDE v.

The observer is built as a copy of the plant (19)-(27) plus

(35)

where a constant matrix Γ1 and dynamics h1,h2,h3,h4,h5 are to be determined. Initial conditions are taken as (ˆz(x,0), ˆw(x,0), ˆv(x,0)) ∈L2(0,1) × L2(0,1) × L2(1,2).

(36)

according to (19)-(27) and (28)-(35), the observer error sys-

(44)

where Γ1 ˜v(2,t) is an output injection, and h1(˜v(2,t)), h2(˜v(2,t);x), h3(˜v(2,t);x), h4(˜v(2,t)), h5(˜v(2,t);x) are dy-

(49)

where L −1 denotes inverse Laplace transform and transfer functions H1(s), H2(s;x), H3(s;x), H4(s), H5(s;x) are to be determined later. Note that x in H2,H3,H5 is only a param- eter.

Introducing (45)-(49) is helpful in constructing the dynam- ics hi(·) in (28)-(35), because the algebraic relationships between ˜v(2,s) and other states can be obtained by using Laplace transform, and the transfer functions in (45)-(49) can be solved in algebraic equations after rewriting the re- quired conditions of achieving an exponentially stable ob- server error system in the frequency domain.

Determination of H1(s), H2(s;x), H3(s;x), H4(s), H5(s;x) and Γ1 in the observer (28)-(35), will be completed through the following three transformations which convert the ob- server error system (37)-(44) to a target observer error sys- tem whose exponential stability is straightforward to obtain.

(50)

where ϕ(x) is to be determined, we intend to convert (42)-

(53)

where ¯A1 is a Hurwitz matrix defined in (10). In what fol- lows, ϕ(x), Γ1, H5(s;x) are determined in matching (42)- (44) and (51)-(53) via (50).

Inserting the transformation (50) into (42), we have ˙˜Y(t) = (A1 −Γ1ϕ(2))˜Y(t)+B1˜z(1,t)−Γ1 ˜η(2,t).

(55)

Evaluating (50) at x = 1 and applying (43), (52), we have

(57)

Taking the time and spatial derivatives of (50) and submitting

Τ ˜Vx(X,T)+H5(˜V(2,T);X)

= ˜ηt(x,t)+ϕ(x)A1 ˜Y(t)+ϕ(x)B1˜z(1,t)−ϕ(x)Γ1 ˜v(2,t)

(58)

where (42), (53) are used, and ϕ(x) should satisfy

Τ Φ′(X)]˜Y(T) Zero, And H5(S;X) Which De-

termines h5(˜v(2,t);x) via (49) should be defined to ensure

The Rest Term In (58) Is Zero, I.E.,

ϕ(x)B1˜z(1,t)−ϕ(x)Γ1 ˜v(2,t)+h5(˜v(2,t);x) = 0.

(60)

Before determining H5(s;x), we solve conditions (55), (57),

(64)

for t ≥τ. Taking Laplace transform of (64), we have

(65)

where I is an identity matrix with appropriate dimension. Note: for brevity, we consider all zero initial conditions while taking Laplace transform (arbitrary initial conditions could be incorporated into the stability statement through an expanded analysis which is routine). Recalling ¯A1 be- ing Hurwitz, det(sI −¯A1) does not have any zeros in the closed right-half plane. Then the matrix sI −¯A1 is invertible for any s ∈C, ℜ(s) ≥0. Multiplying both sides of (65) by

(Si −¯A1)−1, We Have

˜Y(s) = (sI −¯A1)−1B1˜z(1,s).

According To (50) And (63), We Have

˜v(2,t) = ϕ(2)˜Y(t), t ≥τ.

(67)

Writing (67) in the frequency domain, inserting (61), (66)

Where

r(s) = C1e−τA1(sI −¯A1)−1B1.

(69)

Notice r(s) ∈R due to C1 ∈R1×m and B1 ∈Rm×1. Lemma 1 r(s) = C1e−τA1(sI −¯A1)−1B1 ∈R is nonzero for any s ∈C, ℜ(s)≥0 under Assumptions 2 and 4.

= 0

for any s ∈C, ℜ(s) ≥0. Thereby C1e−τA1(sI −¯A1)−1B1̸ = 0. The proof of this lemma is completed.

■

According to Lemma 1, we know the existence of r(s)−1 = r(s). Let us go back to (60) to determine H5(s;x) now. Taking Laplace transform of (60) and recalling (49), inserting (61)

, (72)

where Lemma 1 is used. Thereby, (71) holds. Then (60) holds by rewriting (71) in the time domain. Together with (59), then (58) holds for t ≥τ.

h5(˜v(2,t);x) can then be defined via (72) and (49). In the above, we have completed the conversion be- tween (42)-(44) and (51)-(53) through (50) and determined Γ1,h5(˜v(2,t);x) needed in the observer.

In what follows, H4(s) is determined to make the boundary

Condition (41) As Zero, I.E.,

˜w(1,t) = q˜z(1,t)+C1 ˜Y(t)−h4(˜v(2,t)) = 0.

(73)

Taking Laplace transform of (73) and recalling (48), insert-

(75)

to make (74) hold. It follows that ˜w(1,t) = 0 in (73) by rewriting ˜w(1,s) = 0 in the time domain. h4(˜v(2,t)) is thus determined by (48), (75).

Therefore, through the first transformation (50) with determining the dynamics output injections h4(˜v(2,t)), h5(˜v(2,t);x), (37)-(44) can be converted to the first inter-

(83)

for t ≥τ, where (80)-(83) are in a “stable-like” form while couplings, i.e., sources terms, exist in the domain x ∈, i.e., (78)-(79). In the next subsection, we would introduce the second transformation to decouple the couplings in (78)- (79).

(89)

in which K1(y) will be defined in the next subsection where the well-posedness of (86)-(89) will be shown, to convert the first intermediate system (76)-(81) to the second inter-

Z Y

x ψ(x,δ) ¯M(δ,y)dδ −c1ψ(x,y).

(97)

Note that ˜η(·,t) (82)-(83) is removed for brevity because ˜η(·,t) ≡0, t ≥τ. In what follows, H2(s;x), H3(s;x) are determined in match- ing the first intermediate system (76)-(81) and the second intermediate system (90)-(95) via (84)-(85).

Inserting (84)-(85) into (79) along (92)-(93), and applying

(98)

of which the detailed calculation is shown in (B.1). We thus

(99)

where ˜α(1,t) = ˜z(1,t) according to (85) is used. Writing (99) in the frequency domain and applying (47), (68), we

Q1Ψ(X,1)˜Z(1,S)+H3(S;X)˜V(2,S)

= (q1ψ(x,1)+H3(s;x)r(s)) ˜z(1,s) = 0.

(101)

to make (100) hold. It follows that (98) holds by rewriting (100) in the time domain. h3(˜v(2,t);x) can then be obtained by (47), (101).

Inserting (84)-(85) into (78) along (92)-(93), applying (89),

(96), We Have

˜zt(x,t)+q1˜zx(x,t)+c1 ˜w(x,t)+c1˜z(x,t)+h2(˜v(2,t);x)

(102)

of which the detailed calculation is shown in (B.2). There-

(103)

where ˜α(1,t) = ˜z(1,t) according to (85) is used. Taking Laplace transform of (103), and recalling (46), (68), we have

Q1Φ(X,1)˜Z(1,S)+H2(S;X)˜V(2,S)

= (q1φ(x,1)+H2(s;x)r(s)) ˜z(1,s) = 0.

(105)

to ensure (102) holds. h2(˜v(2,t);x) can thus be defined by (105), (46). Boundary conditions (77), (80) follow directly from insert- ing x = 0, x = 1 into (84)-(85) and applying (87), (91), (94).

ODEs (76), (81) are obtained directly from (90), (95) via (84), (85) respectively. The second conversion is thus completed and two PDEs (78)-(79) are decoupled now, which can be seen in (92)-(93).

Third Transformation

In order to decouple the ODE (90) with the PDEs and re- build the ODE in a stable form, we intend to convert the second intermediate system (90)-(95) to the following target

Q2 Recalling

(93)-(94), and then ˜β(x,t) can be removed for brevity. (90)-

(113)

for t ≥t0. Note that (112)-(113) are the same as (108)-(109). We thus only need to convert (110)-(111) to (106)-(107).

(114)

is applied to complete the conversion, where K1(y) satisfies

(116)

Note that (86)-(89) and (115)-(116) is a 2 × 2 hyperbolic PDE-ODE system, which is a scalar case of the well-posed kernel equations (17)-(23) in (setting dimensions in as 1). Therefore, the conditions of the kernels ψ(x,y),φ(x,y) in (84)-(85) and K1(y) in (114), i.e., (86)-(89), (115)-(116) are well-posed.

In what follows, H1(s) is determined in matching (110)- (111) and (106)-(107) via (114). Submitting (114) into (106),

˜Α(Y,T)Dy

=h1(˜v(2,t))+q1K1(1) ˜α(1,t) = 0, t ≥t0.

(117)

H1(s) which determines h1(˜v(2,t)) by (45) can be solved

(118)

where ˜α(1,t) = ˜z(1,t) according to (85) is used. Writing (118) in the frequency domain and applying (45), (68) yield

H1(S)˜V(2,S)+Q1K1(1)˜Z(1,S)

= (H1(s)r(s)+q1K1(1)) ˜z(1,s) = 0.

(120)

It follows that (117) holds by rewriting (119) in the time domain. h1(˜v(2,t)) can then be defined via (45), (120). Inserting (114) into (111), it is straightforward to obtain (107). Therefore, (106)-(107) is converted from (110)-(111) through (114) for t ≥t0. The third transformation is com- pleted and the ODE (106) is independent and exponentially stable now.

After the above three transformations, we have converted the original observer error system (37)-(44) to the target ob- server error system (106)-(109) (for t ∈[t0,∞), ˜η(x,t) ≡0 (82)-(83) and ˜β(x,t) ≡0 (93)-(94) are removed for brevity).

Because the original observer error system (37)-(44) is bounded in the finite time t ∈[0,t0), we prove the exponen- tial stability of (37)-(44) for t ∈[t0,∞) in the next subsection.

Note that dynamics Hi determined above are some dynamic extensions, of which the states are dynamic output injections

(123)

which are proved as exponentially convergent to zero as well in the next subsection. Remark 2 Hi(s)˜v(2,s) usually generates time-derivatives of ˜v(2,t) in the time domain, which often appears in us- ing the frequency-domain design approach. In practice, one way to avoid taking the time derivatives which may lead to measurement noise amplification is measuring n-order time-

Sn To Make Hi(S) Proper. As Shown In An Appli-

cation case of control of DCV in Section 6, payload oscilla- tion acceleration is measured and the velocity is calculated by integrating with the known initial conditions. Measur- ing acceleration is a prevalent method in many mechanical systems, because the acceleration sensor is cheaper and far easier to manufacture and install .

D1 U(X,T)2Dx Denotes The L2 Norm

and ∥u(·,t)∥∞= supx∈[d1,d2]{|u(x,t)|} denotes the ∞-norm. |·| denotes the Euclidean norm. Theorem 1 For any initial data (˜z(x,0), ˜w(x,0), ˜v(x,0), ˜X(0), ˜Y(0)) ∈L2(0,1) × L2(0,1) × L2(1,2) × Rn × Rm, in- ternal exponential stability of the observer error system

(124)

with the decay rate being adjustable by L0,L1. Proof. The stability of the original observer error system can be obtained by analyzing the stability of the target observer error system (106)-(109) and using the invertibility of the transformations. (106)-(109) is a cascade of ˜Z(t) into ˜α(·,t) into ˜Y(t). From (106), ˜Z(t) is exponentially convergent to zero because ¯A0 is Hurwitz. With the method of characteris- tics as it is easy to show that ˜α(x,t) in the PDE subsystem (106)-(107) are exponentially convergent to zero. Because ¯A1 is Hurwitz, we have ˜Y(t) is exponentially convergent to zero. The decay rate λe of (106)-(109) depends on the de- cay rate of the ODEs ˜Z(t), ˜Y(t). In other words, the decay rate λe is adjustable by L0,L1 according to (9)-(10). Recall-

Q2 +Τ, We Obtain

¯Ω(t) = ∥˜α(·,t)∥∞+∥˜β(·,t)∥∞+∥˜η(·,t)∥∞+| ˜Z(t)|+|˜Y(t)| is bounded by an exponential decay with the decay rate λe for t ≥t0. Note that the transient in the finite time [0,t0) can be bounded by an arbitrarily fast decay rate considering a trade off between the decay rate and the overshoot coeffi- cient, i.e., the higher the decay rate, the higher the overshoot coefficient. Therefore, we conclude the exponential stabil- ity in the sense of ¯Ω(t) being bounded by an exponential decay rate λe with some overshoot coefficients for t ≥0.

Applying the transformation (50), (114) and (84)-(85), we

,

for some positive ϒ1a,ϒ1b,ϒ1c. According to (45)-(49), (120), (105), (101), (75), (72), we know the output injection states y1(t), y2(x,t), y3(x,t), y4(t), y5(x,t) are the output states of the following extended dy-

(129)

of which the input signal is ˜v(2,t) which is exponentially convergent to zero. Recalling Lemma 1, we know there is not pole in the closed right-half plane in the trans- fer function (125)-(129), the exponential convergence of |y1(t)|,∥y2(x,t)∥∞,∥y3(x,t)∥∞,|y4(t)|,∥y5(x,t)∥∞are thus obtained. Note that x ∈ is just a parameter in the transfer functions (126),(127),(129) and the stability result would not be affected.

Output-Feedback Control Design

In the last section, we have built the observer which can compensate the time-delay in the output measurement yout(t) of the distal ODE, which is the only one measurement used in the observer, to track the states of the overall sandwiched PDE system (1)-(7). In this section, we design an output- feedback control lawU(t) based on the observer (28)-(35) by using backstepping transformations and frequency-domain designs.

First, two transformations are introduced to transform the observer (28)-(35) to a target system (170)-(177), which is in a stable-like form except for the proximal ODE which is influenced by perturbations originating from the PDEs and distal ODE. Representing this “target system” in the fre- quency domain by using Laplace transform, the algebraic relationships (191)-(197) between the states of the proximal ODE and the states of the PDEs and distal ODE are ob- tained. Inserting these algebraic relationships to rewrite the perturbations in the proximal ODE, a new ODE (203) with- out external perturbations can be built in the frequency do- main, where the control input to exponentially stabilize this ODE can be designed.

First Transformation

The aim of the first transformation is to remove the source terms in the PDE domain x ∈, i.e., couplings in (30)- (31), and to build the state matrix of the distal ODE (33) as a Hurwitz matrix. A PDE backstepping transformation in

(131)

is introduced, where the kernels K3(x,y), J3(x,y), γ(x), K2(x,y), J2(x,y), λ(x) are to be determined later, to convert

(139)

where ˆA1 is a Hurwitz matrix by choosing the control pa- rameter F1 according to Assumption 2. ¯K1(x), ¯K2(x), ¯K3,

(145)

which are obtained by matching (132)-(133) and (28)-(29) via (130)-(131) (Please see Step 4 in the Appendix-A for the details). The following conditions of the kernels in the transformations (130)-(131) are obtained by matching (134)- (137) and (30)-(33) (the detailed process is shown in Steps.

(157)

The well-posedness of (146)-(157) is shown in the following lemma. Lemma 2 The kernel equations (146)-(151) have a unique

Have A Unique Solution K2,J2 ∈C1(D), Λ(X) ∈C1(),

with D = {(x,y)|0 ≤x ≤y ≤1}. Proof. (146)-(151) and (152)-(157) have the analogous structure with (19)-(24) in . Following the proof of Lemma 1 in , we can obtain this lemma.

■

Similarly, the inverse transformation can be obtained as

(159)

where M (x,y),N (x,y),G (x),D(x,y),T (x,y),P(x) are kernels which can be determined through a similar process in the Appendix-A. The first transformation in the control design is completed.

Second Transformation

In order to remove the last three terms in the boundary condition (133) and form a Hurwitz matrix of the proximal

(160)

where C0+ denotes the Moore-Penrose right inverse of C0. Note that because C0 is full-row rank (with rank equal to 1), a right inverse exists for C0. i.e., C0C0+ = I. A choice of

+N1Α(1,T)+N2Β(0,T)+H [H1(˜V(2,T)),H2(˜V(2,T);X),

h3(˜v(2,t);x),h4(˜v(2,t)),h5(˜v(2,t);x), ˜v(2,t)],

Where

H [h1(˜v(2,t)),h2(˜v(2,t);x),h3(˜v(2,t);x),h4(˜v(2,t)),

H5(˜V(2,T);X), ˜V(2,T)]

=h1(˜v(2,t))+C0+ ¯K3Γ1 ˜v(2,t)+q2C0+ ¯K2(1)qh4(˜v(2,t))

(164)

ˆA0 is Hurwitz by choosing the control parameter F0 consid- ering Assumption 2. N1,N2,Mα(x),Mβ(x),MY in (161) are

(169)

We thus arrive at the target system consisting of (134)- (139), (161)-(162), which includes dynamic output injec- tions in (163). Considering Theorem 1 and (121)-(123), we know h1(˜v(2,t)), h2(˜v(2,t);x), h3(˜v(2,t);x), h4(˜v(2,t)), h5(˜v(2,t);x) and Γ1 ˜v(2,t) in the target system (134)-(139), (161)-(162) can be regarded as zero, i.e., H = 0, for brevity.

Therefore, the target system (134)-(139), (161)-(162) can be

Ontrol Design In Frequency Domain

In the last two subsections, the system (28)-(35) is converted to the target system (170)-(177), through the two transfor- mations (130)-(131) and (160). In this section, the control ¯U(t) in (170) of the target system (170)-(177) will be de- signed in the frequency domain by using Laplace transform.

(185)

Note: for brevity, we consider all zero initial conditions while taking Laplace transform (arbitrary initial condition- s could be incorporated into the stability statement through an expanded analysis which is routine).

(186)

according to (179)-(185) and Sec. 3.2 in , we obtain the following algebraic relationships between C0 ˆZ(s) and other

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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