Quasi Isotropic Laminate Ansys

Arxiv:1810.00080V4 [Math.Dg] 15 Nov 2020

Preprint: The final version will appear as Math. J. Okayama Univ. 63 (2021), 15–52

Uiz C. B. Da Silva

Abstract. We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions.

Basically,

the simply and pseudo isotropic geometries are the study of a three- dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then gene- ralizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Gaussian and mean curvatures, we solve the corresponding problem of prescribed cur- vature for invariant surfaces whose generating curves lie on a plane con- taining the degenerated direction.

Ntroduction

Needless to say, there is a great interest in manifolds equipped with a metric which is not necessarily positive definite.

However, Less Attention

has been paid to cases where the metric is allowed to be degenerated. For- tunately, there has been a recent and growing interest in the geometry of spaces equipped with a degenerated metric from both applied and pure mathematical viewpoints. In this work, we study simply isotropic I3 and

Pseudo-Isotropic I3

p spaces which are the affine space R3 equipped with the degenerated metric ds2 = dx2 + dy2 and ds2 = dx2 −dy2, respectively. The study of I3 has been initiated by the Austrian geometer Karl Strubecker in the 1930’s [17, 18, 19, 20, 21] (see also and references therein), while

That Of I3

p began only recently [3, 7].

Besides Its Mathematical Interest

[1, 3, 9, 23, 24], see also the recent contributions by this Author [6, 7], isotropic geometry finds applications in economics [4, 5], image processing , and shape interrogation . In addition, this theory may prove use- ful in understanding the geometry of surfaces with zero mean curvature in semi-Riemannian spaces [15, 16].

Here we are mainly interested in the geometry of invariant surfaces, which are generated by the action of 1-parameter subgroups of simply and pseudo Mathematics Subject Classification. Primary 53A35; Secondary 53A10; 53A40.

Key words and phrases. Simply isotropic space, pseudo-isotropic space, singular metric, invariant surface, prescribed Gaussian curvature, prescribed mean curvature.

Onsequently, Our Investigation Generalizes The

study of revolution and helicoidal surfaces in Euclidean and Lorentzian ge- ometries to the context of spaces with a singular metric. The enumeration of 1-parameter subgroups of rigid motions in I3 has been already done by Strubecker , see also Chap. 2 of . However, in I3 much attention has been paid only to helicoidal and revolution surfaces, and [2, 9, 24],

P Were Studied In . Here We Re-

visit the 1-parameter subgroups of simply isotropic rigid motions by exploit- ing their linear representation in the group of invertible matrices GL(4, R), which we believe offers the advantage of being simpler and easier to follow.

We also study 1-parameter subgroups of pseudo-isotropic rigid motions and the invariant surfaces generated by them.

After Listing All Invariant Sur-

faces, which are divided into 7 basic types, we compute their mean and Gaussian curvatures and we also solve the problem of prescribed curvatures for the so-called invariant surfaces of i-type (see Definition 1; for the ni-type we solve the prescribed Gaussian curvature problem for helicoidal surfaces only). These findings generalize the study of helicoidal surfaces in I3 [2, 9]

And Revolution Surfaces In I3

p with constant curvatures. This work is divided as follows. In Sect. 2 we present background ma- terial: definition of isotropic spaces in Subsect. 2.1; isotropic surfaces in Subsect. 2.2; and the notion of invariant surfaces in Subsect. 2.3. 3 we describe 1-parameter subgroups of simply (Subsect. 3.1) and of pseudo (Subsect. 3.2) isotropic isometries, while in Sect. 4 we describe their invari- ant surfaces. Finally, in Sect. 5 and 6 we study the geometry of simply and pseudo isotropic invariant surfaces, respectively, and solve the corresponding We mention that here we shall follow the Einstein convention of summing

In I3

p are omitted, the reader is then referred to the corresponding results in I3 to devise a proof.

Preliminaries

Simply and pseudo isotropic spaces are examples of Cayley-Klein geome- tries , i.e., the study of properties in projective space P3 invariant by the action of the projectivities that fix the so-called absolute figure, which for our isotropic geometries are given by a plane at infinity, identified with ω : x0 = 0, and a degenerate quadric of index zero or one, identified with

+ Δ X2

2 = 0: δ = 1 for the I3 absolute figure [14, 17] and δ = −1

For The I3

p one . (Euclidean and Lorentzian geometries stand for the choice

Fferential Geometry Of Isotropic Invariant Surfaces

2.1. Simply and pseudo isotropic three-dimensional spaces. In the Cayley-Klein framework, the simply isotropic I3 and pseudo-isotropic I3

P

geometries are the study of those properties in R3 invariant by the action of

,

respectively, where a, b, c, c1, c2, φ ∈R.

E2

1 plane geometries. The projection of a vector u = (u1, u2, u3) on the xy plane is called the top view of u and it is denoted by eu = (u1, u2, 0).

Notice that the z-direction is preserved by the action of B6 and Bp

A Line

P + t(0, 0, a) is an isotropic line and a plane containing an isotropic line is an isotropic plane. One may respectively introduce a simply isotropic and a pseudo-isotropic inner products between two vectors u = (u1, u2, u3) and v = (v1, v2, v3) as

(2.2)

⟨u, v⟩z = u1v1 + u2v2 and ⟨u, v⟩pz = u1v1 −u2v2.

⟨U, U⟩Pz = ∥Eu∥1,

respectively: ∥· ∥and ∥· ∥1 are the norms in E3 and E3

(2.4)

⟨u, v⟩= u1v1 + u2v2 + u3v3 and ⟨u, v⟩1 = u1v1 −u2v2 + u3v3. In addition, the corresponding vector products in E3 and E3

Shall Be Denoted

here by × and ×1, respectively: notice that ×1 should be computed as u ×1 v = (u2v3 −u3v2, u1v3 −u3v1, u1v2 −u2v1). The isotropic inner products are both degenerated: for any u = (0, 0, a)

P). To Exert Some Control On Isotropic Vec-

tors, we introduce the co-metric ⟨⟨u, v⟩⟩= u3v3 coming from the codistance cd(u, v) = |b3 −a3|, which is preserved by B6 and Bp

Only When Applied To

isotropic vectors. In short, we think of I3 and I3

P As R3 Equipped With An

“hierarchy” of metrics: when a metric fails to “control” tangent vectors, we descend one level and start to use the next one. Remark 1. The hierarchy of metrics (⟨·, ·⟩z, ⟨⟨·, ·⟩⟩) in I3 and I3

P Should Not

be confused with a single metric defined by parts, since a function like G(u, v) = ⟨u, v⟩z, if ˜u̸ = 0 or ˜u̸ = 0, and G(u, v) = ⟨⟨u, v⟩⟩, if ˜u = ˜v = 0, is not bilinear and can not be a metric. For example, we have G((1, 0, 1) + (0, 0, 1), (0, 0, 1)) = 0, but G((1, 0, 1), (0, 0, 1)) + G(0, 0, 1), (0, 0, 1)) = 1̸ = 0.

Uiz C. B. Da Silva

2.2. Simply and pseudo isotropic surfaces. When dealing with surfaces M2 in isotropic geometry we must distinguish between two cases. We say that M2 is an admissible surface when the metric in M2 induced by ⟨·, ·⟩z, or ⟨·, ·⟩pz, has rank 2. Otherwise, it is not admissible. If M2 is parameterized by a C2 map x(u1, u2) = (x1(u1, u2), x2(u1, u2), x3(u1, u2)), then it is admissible



. As a consequence, every admissible C2 surface M2 can be locally param- eterized as x(u1, u2) = (u1, u2, Z(u1, u2)): we say that M is in its normal form.

Simply isotropic and pseudo-isotropic spheres are connected and irre- ducible surfaces of degree 2 given by the 4-parameter family

P . Up To A Rigid

motion, we can express an isotropic sphere in one of the two normal forms

P,

where p̸ = 0 is the radius of the sphere and it is invariant under rigid motions (the translation by p/2 above, if we compare our normal form with that of

At The Origin Of The Coordinate System); And

(2) Spheres of cylindrical type: x2 + y2 = r2 in I3 and x2 −y2 = ± r2 in I3

P,

where the radius r > 0 is a constant and invariant under rigid motions. Only spheres of parabolic type are admissible and we use them to intro-

P : The Normal With Respect

to ⟨·, ·⟩z, or ⟨·, ·⟩pz, is necessarily the co-unit vector field N = (0, 0, ±1) and, needless to say, a constant Gauss map adds nothing interesting to the theory.

(2.6)

I = gijduiduj, gij = ⟨xi, xj⟩z or gij = ⟨xi, xj⟩pz. When M2 is parameterized in its normal form, I = gijduiduj takes a simpler

(2.7)

I = (du1)2 + (du2)2, in I3, and I = (du1)2 −(du2)2, in I3

Fferential Geometry Of Isotropic Invariant Surfaces

which, in particular, shows that every admissible surface is flat, i.e., its intrinsic curvature vanishes identically, see Eq. (8.50) of in I3 and Eq.

P Is Timelike, I.E.,

gij is non-degenerated with index 1 [3, 7]. Regular admissible curves may be either spacelike, ⟨α′, α′⟩pz > 0, or timelike, ⟨α′, α′⟩pz < 0, while lightlike curves, ⟨α′, α′⟩pz = 0, are non-admissible .

Below Will Play A Role In

isotropic geometries similar to that of S2 in Euclidean geometry E3 and of

Where {Ei}3

i=1 is the canonical basis of the vector space R3.

N E3 And E3

1 the Gauss maps are ξeucl = x1 × x2 ∥x1 × x2∥−1 and ξlor =

E1 + X31

e2 + e3. (The z-coordinate of ξ was then adjusted to give ξ ◦x ∈Σ.) In I3

E2 + E3 ⇒E

Nh = eξp. The isotropic shape operator Lq : TqM2 →TqM2 is defined as

Q(Wq) = −Dwqξ Or Lq(Wq) = −Dwq Ξp,

where D denotes the usual (flat connection) directional derivative in R3. The shape operator Lq maps TqM2 in Tξ(q)Σ2, but as in Euclidean space, TqM2 and Tξ(q)Σ2 are parallel and, therefore, they can be identified. The same

Reasoning Applies To I3

p . (In other words, ξ is an equiaffine transversally

Vector Field On M2 .)

The second fundamental form II in simply and pseudo-isotropic spaces is

(2.14)

II(uq, vq) = I(Lq(uq), vq) ⇒II = hijduiduj, hij = II(xi, xj).

Uiz C. B. Da Silva

The coefficients of II can be easily computed via Nh in Eqs. (2.11) and

P,

where ⟨·, ·⟩and ⟨·, ·⟩1 are the inner products from E3 and E3 1 in Eq. (2.4). Finally, the isotropic Gaussian and mean curvatures are defined as

Tr(Lq) ,

respectively. In addition, if in local coordinates Lq(xi) = −Ak

G11G22 −G2

. The reader is referred to for examples and more information about surfaces in isotropic geometries. (The geometry of curves can be found, e.g.,

In .)

2.3. One-parameter subgroups and invariant surfaces. Let (G, ◦) be a (Lie) group. A 1-parameter subgroup H of G is a subgroup of G such that there exists a surjective continuous group homomorphism ψ : (R, +) → (H, ◦): ψr+s = ψr ◦ψs, ψt = ψ(t). Despite the non-uniqueness of ψ, it is common to identify H with ψ, since ψ(R) = H.

Sometimes It Is Useful

to work with a linear representation of the group of interest, i.e., see the elements of G in a subgroup of invertible matrices of GL(n, R). In this case,

We Have The Following Useful Property

Proposition 2.1. Let H be a 1-parameter subgroup of G ⊆GL(n, R) and h ∈H, then det h > 0. Proof. If h ∈H, then there exists a map ψ such that ψ(1) = h (if ψ(r) = h, then Ψ(s) = ψ(rs) defines a new group homomorphism which sends 1 to h).

□

When G happens to be a group of isometries of a certain geometry, its 1-parameter subgroups ψt give rise to invariant surfaces: a C2 surface M2 is said to be an invariant surface when there exists a 1-parameter subgroup

(2.19)

∀t ∈R, M = ψt(M). Intuitively, we can approximate an invariant surface by successive appli- cations, to a given curve α(s), of a certain kind of rigid motion (isometry): M ∼= {. . , ψt0(α(s)), ψt0+∆t(α(s)), . , ψt0+n∆t(α(s)), . }.

Fferential Geometry Of Isotropic Invariant Surfaces

In the limit ∆t →0, we generate M by continuously moving the curve α under the action of ψt. We call such α the generating curve of M and, for the geometries modeled in R3, it may be assumed to be planar. Usually, α is obtained by intersecting M with the xz- or the xy-plane. Notice, if α(u) is the generating curve of M, then one may parameterize M by x(u, t) = ψt(α(u)).

An important feature of invariant surfaces is that the values of geometric quantities, such as the Gaussian and mean curvatures, only depend on their values assumed along the generating curve.

3. One-parameter subgroups of simply and pseudo isotropic

Isometries

Let us introduce the following group of 4 × 4 invertible real matrices

, A ∈I3 , A ∈Oi(3)},

where OI(3) denotes the set of simply isotropic orthogonal matrices:

Cdz(Au, Av) = Cdz(U, V), If ∥U∥Z = ∥V∥Z = 0

. Note that every element in OI(3) can be written as



, φ, a, b ∈R and σ, τ ∈{−1, +1}. Together with translations, OI(3) gives us the group of simply isotropic isometries ISO(I3). Now we construct a linear representation Ψ : ISO(I3) → GI(4) for the group ISO(I3). Thus, we can see isometries as linear trans- formations by identifying R3 with a hyperplane in R4 via the inclusion map (x, y, z) ∈R3 7→(x, y, z, 1) ∈R4.

Proposition 3.1. Let ISO(I3) be the group of simply isotropic isometries.

Then,

(1) T ∈ISO(I3) ⇔T(x) = Ax + a, where A ∈OI(3) and a ∈I3;



. Proof. (1) Given T ∈ISO(I3), define A(x) = T(x) −T(0). Using that T preserves ⟨·, ·⟩z, and also cdz(·, ·) for isotropic vectors, one can deduce that

Uiz C. B. Da Silva

A ∈OI(3). Indeed, expanding L = ⟨A(x + λy) −A(x) −λA(y), A(x + λy) − A(x) −λA(y)⟩z, and using that T preserves ⟨·, ·⟩z, we conclude that L = 0. Applying the same reasoning to the codistance, we deduce that A is a linear isometry, since x = 0 ⇔⟨x, x⟩z = 0 = cdz(x, 0). Finally, writing a = T(0) we get the desired result. Conversely, since by definition any translation and any A ∈OI(3) give rise to isotropic isometries, every map Ax + a defines an isotropic rigid motion.

(2) Ψ is clearly injective. Using item 1 we also see that Ψ is surjective. Finally, if Ti(x) = Aix + ai, then (T1 ◦T2)(x) = (A1A2)x + (A1a2 + a1) and

= Ψ(T1 ◦T2),

which shows that Ψ is a group isomorphism.

□

Notice that the same concepts above apply to rigid motions in I3

P , A ∈Oip(3)},

where OIp(3) denotes the set of pseudo-isotropic orthogonal matrices:

⟨Au, Av⟩Z,P = ⟨U, V⟩Z,P

cdz,p(Au, Av) = cdpz(u, v), if ∥u∥pz = ∥v∥pz = 0 . Therefore, every element in OIp(3) can be written either as

,

where φ, a, b ∈R and σ, τ ∈{−1, +1}. As in I3, we have in I3

Proposition 3.2. Let Iso(I3

p) be the group of pseudo-isotropic isometries.

Fferential Geometry Of Isotropic Invariant Surfaces

3.1. One-parameter subgroups of simply isotropic isometries. First, notice one may write an arbitrary element h ∈H ⊆GI(4) as



. Indeed, since det h > 0 (Prop. 2.1), it follows that στ = +1 in Eq. (3.3). In addition, given ψ : R →GI(4), we may generate a 1-parameter subgroup of plane Euclidean isometries by composing ψt with the map F that associates

, Where Bij = Tij,

i, j = 1, 2. Then, σ = +1 and, consequently, τ = +1. For a simply isotropic rigid motion what happens in the top view plane is independent from what happens in the isotropic z-direction. Then, we may investigate the effect of a 1-parameter subgroup on the top view plane and on the isotropic direction separately. It will follow that the 1-parameter subgroups of simply isotropic isometries are distributed along 7 types, di- vided into two main categories [14, 17]: (a) helicoidal motions, which in the isotropic direction act either as a pure translation or as the identity map; and (b) limit motions (Grenzbewegungen ), which in the top view plane act either as a pure translation or as the identity map.

By noticing that simply isotropic helicoidal motions are in one-to-one correspondence with Euclidean helicoidal motions with Oz as the screw axis, the category of simply isotropic helicoidal motions is given by

∈Gi(4),

which can be divided in two classes [14, 17]: (I) Euclidean rotations in the top view plane when c = 0; and (II) helicoidal motions around an isotropic axis.

On the other hand, for the category of limit motions we have Theorem 3.1. The group G5 of simply isotropic limit motions, i.e., motions that in the top view plane act either as a pure translation or as the identity



∈GI(4), ci, c, a, b ∈R. Proof. First, notice that the elements of G5 are characterized by setting φ = 0 in Eq. (3.9). Now, if h ∈G5 and ψ : R →H is a continuous surjective map such that ψ(1) = h, then one can prove by induction that



. In addition, since h = ψ(1) = ψ(m/m) = ψ(1/m)m, we also have



. Finally, for any rational q = n/m we conclude that



. Now, using the continuity of ψ and, in addition, redefining c to be c + (ac1 + bc2)/2, we find Eq. (3.11). (The (3, 4)-entry converges to ct + (ac1 + bc2)t(t −1)/2, which under our redefinition of c is equal to the (3, 4)-entry

□

The 1-parameter group of limit motions may be divided in five types [14, 17]: (III) parabolic rotations, when (ac1 + bc2)̸ = 0; (IV) warped trans- lations, when c = (ac1 + bc2) = 0, but (a, b)̸ = (0, 0) and (c1, c2)̸ = (0, 0);

Fferential Geometry Of Isotropic Invariant Surfaces

(V) isotropic shears, when (a, b) = (0, 0) and c = 0, but (c1, c2)̸ = (0, 0); (VI) translations along a non-isotropic direction, when c, c1, c2 = 0, but (a, b)̸ = (0, 0); and (VII) translations along an isotropic direction, when (a, b), (c1, c2) = (0, 0), but c̸ = 0. As we shall see in Sect. 4, only types (I), (II), (III), (IV), and (VI) lead to admissible surfaces.

Finally, let us mention that the division into two classes according to the action of a 1-parameter subgroup in the top view and isotropic direction allowed us to classify the simpler contributions to each motion. However, if one does not impose any restriction on the action of ψt, the most general 1-parameter subgroup of isotropic isometries is given by

,

where D1 = (ac1 +bc2), D2 = (ac2 −bc1), and we have defined the functions

[1 −Cos(Φ)]

. Notice that limφ→0 Ct(φ) = t, limφ→0 St(φ) = 0, limφ→0 ˜Ct(φ) = t(t −1)/2, and limφ→0 ˜St(φ) = 0, which then allow us to recover the known expression for a limit motion corresponding to φ = 0. On the other hand, it is easily seen that setting a, b, c1, c2 = 0 leads to helicoidal motions.

To prove Eq. (3.12), let ψ : R →H ⊆GI(4) be a 1-parameter subgroup such that ψ(1) = h, with h as in Eq. (3.9). Using that ψ(n) = ψ(1)n = hn,

K=0

sin(kφ) = c2Cn(φ) −c1Sn(φ). Applying the same reasoning, we can find an expression for φ(1/m) = h1/m (which allows us to find φ(r) for any rational r), extend it to the real numbers by continuity, and finally deduce Eq. (3.12).

To deduce closed expressions for the sums P cos(kφ), P sin(kφ), P(n − 1−k) cos(kφ), and P(n−1−k) sin(kφ) above, we may use complex numbers and then apply the known techniques of summing power series:

P Cos(Kφ) + I P Sin(Kφ) = P Zk

P(n −1 −k) cos(kφ) + i P(n −1 −j) sin(kφ) = P(n −1 −k)zk

,

where z = cos(φ) + i sin(φ). 3.2. One-parameter subgroups of pseudo-isotropic isometries. With a similar reasoning as in the previous subsection, we may write any h ∈



. The 1-parameter subgroups of pseudo-isotropic isometries are also dis- tributed along 7 types, divided into two categories: (a) causal helicoidal motions, which in the isotropic direction act either as a pure translations or as the identity map; and (b) limit motions (Grenzbewegungen), which in the top view plane act either as a pure translation or as the identity map.

∈Gip(4)

can be divided in two types: (I) Lorentzian rotations in the top view plane, when c = 0 (here the orbits can be either time- or space-like Lorentzian circles, i.e., hyperbolas); and (II) helicoidal motions around an isotropic axis (here the orbits can be either time- or space-like Lorentzian helices).

With a proof completely similar to that of Theorem 3.1, the category of pseudo-isotropic limit motions can be described by

Of Pseudo-Isotropic Limit Motions, I.E., Mo-

tions that in the top view plane act either as a pure translation or as the



∈GIp(4), ci, c, a, b ∈R. The 1-parameter subgroups of pseudo-isotropic limit motions are divided in five types: (III) pseudo-parabolic rotations, when (ac1 + bc2)̸ = 0; (IV) warped translations, when (ac1 + bc2) = 0 and c = 0, but (a, b)̸ = (0, 0) and (c1, c2)̸ = (0, 0); (V) pseudo-isotropic shears, when (a, b) = (0, 0) and c = 0, but (c1, c2)̸ = (0, 0); (VI) translations along non-isotropic direction, when c = c1 = c2 = 0, but (a, b)̸ = (0, 0); and (VII) translations along an isotropic direction, when (a, b), (c1, c2) = (0, 0), but c̸ = 0. As we shall see in Sect. 4, only types (I), (II), (III), (IV), and (VI) lead to admissible surfaces.

Finally, let us mention that without any restriction on c, c1, c2, a, b, and φ, the most general 1-parameter subgroup of pseudo-isotropic isometries is

[1 −Cosh(Φ)]

. Notice that limφ→0 Cht(φ) = t, limφ→0 Sht(φ) = 0, limφ→0 ˜

)/2, And Limφ→0 ˜

Sht(φ) = 0, which allow us to recover the expression for a limit motion corresponding to φ = 0. To deduce the expressions above, we can follow steps similar to those employed for GI(4) by working with hyperbolic trigonometric functions. To find expressions for P cosh(kφ), P sinh(kφ), P(n −1 −k) cosh(kφ), and P(n −1 −k) sinh(kφ), we may use the ring of Lorentz numbers L (also known as hyperbolic or double numbers ) and then apply the known

P Cosh(Kφ) + ℓp Sinh(Kφ) = P Wk

P(n −1 −k) cosh(kφ) + ℓP(n −1 −j) sinh(kφ) = P(n −1 −k)wk

,

where w = cosh(φ) + ℓsinh(φ) ∈L = {a + ℓb : a, b ∈R, ℓ̸ ∈R, ℓ2 = 1}.

Simply And Pseudo Isotropic Invariant Surfaces

In this section, it is described the invariant surfaces obtained from the 1-parameter subgroups of ISO(I3) (Subsect. 4.1) and of ISO(I3 p) (Subsect.

4.2). We also identify those invariant surfaces that are admissible and show, as expected, that spheres of parabolic type are invariant surfaces under all types of revolutions.

The generating curve of an invariant surface can be assumed to be a plane curve by intersecting the surface with a plane, in which case we should distinguish between isotropic and non-isotropic planes. Thus, we have the Definition 1. When the generating curve of an invariant surface, assumed to be at least C2, lies on a non-isotropic plane (here we choose the xy-plane), we say that the corresponding invariant surface is of non-isotropic type, or of ni-type, for short. On the other hand, when the generating curve lies on an isotropic plane (here we choose the xz-plane), we say that the corresponding invariant surface is of isotropic type, or of i-type, for short.

We shall see in Subsects. 5.1 and 6.1 that the mean curvature of helicoidal surfaces of ni-type depends on the value of c, while for the i-type there is no dependence on c. This shows that the distinction between invariant surfaces of ni- and i-types in simply and pseudo isotropic geometries is meaningful.

Fferential Geometry Of Isotropic Invariant Surfaces

4.1. Simply isotropic invariant surfaces. We now use the 1-parameter subgroups of simply and pseudo isotropic isometries to investigate invariant surfaces, Eq. (2.19). For each 1-parameter subgroup we have two classes of surfaces, i.e., invariant surfaces of ni- and of i-types.

We first describe helicoidal invariant surfaces in I3, i.e., when a, b, ci = 0:

(I) Euclidean Revolution Surfaces (C = 0):

• If α(u) = (x(u), y(u), 0), we have the revolution surface of ni-type (4.1) Y1(u, t) = (x(u) cos(tφ) −y(u) sin(tφ), x(u) sin(tφ) + y(u) cos(tφ), 0), that represents a portion of a non-isotropic plane; • If α(u) = (x(u), 0, z(u)), we have the revolution surface of i-type

(4.2)

Z1(u, t) = (x(u) cos(tφ), x(u) sin(tφ), z(u)).

(Ii) Helicoidal Surfaces:

• If α(u) = (x(u), y(u), 0), we have the helicoidal surface of ni-type (4.3) Y2(u, t) = (x(u) cos(tφ) −y(u) sin(tφ), x(u) sin(tφ) + y(u) cos(tφ), ct); • If α(u) = (x(u), 0, z(u)), we have the helicoidal surface of i-type

(4.4)

Z2(u, t) = (x(u) cos(tφ), x(u) sin(tφ), z(u) + ct). Remark 2. When α is α(u) = R(cos u, sin u, 0) in the ni-type or an isotropic line α(u) = (±R, 0, z(u)) in the i-type (R constant), the resulting helicoidal surface is a sphere of cylindrical type, which is not an admissible surface.

Proposition 4.1. Assume that the generating curve is neither a circle cen- tered at the origin (in the ni-type) nor an isotropic line (in the i-type). Then, all simply isotropic Euclidean revolution and helicoidal surfaces are admissible.

Proof. It is enough to investigate helicoidal surfaces, type (II) above, since revolution surfaces are obtained by imposing c = 0. In the ni-type we have

=

(x′ cos(tφ) −y′ sin(tφ), x′ sin(tφ) + y′ cos(tφ), 0)

=

(−φx sin(tφ) −φy cos(tφ), φx cos(tφ) −φy sin(tφ), c)

(4.5)

I = (x′ 2 + y′ 2)du2 + 2φ(xy′ −x′y)dudt + φ2(x2 + y2)dt2.

(4.6)

det gij = φ2(xx′ + yy′)2 = 0 ⇔x2 + y2 = R2. By hypothesis, α(u) is not a circle centered at the origin and, therefore, the corresponding helicoidal surface of ni-type is admissible.

(4.7)

I = x′ 2du2 + φ2x2dt2 ⇒det gij = φ2(xx′)2 = 0 ⇔x2 = R2. By hypothesis, α(u) is not an isotropic line and, therefore, the corresponding helicoidal surface of i-type is admissible.

□

Remark 3. It is known that when written in its normal form, the first fun- damental form of any simply isotropic surface are given by Eq. (2.7). Now notice that the first fundamental form of helicoidal surfaces of ni- and i-types can be written as I = dU 2 + dT 2 under the change of coordinates

T(U, T) = Tφ X(U) + T0 ,

respectively (U0, T0 constants). Indeed, completing squares in Eq. (4.5),

=

(x′2du2 −2φx′ydudt + φ2y2dt2) + (y′2du2 + 2φxy′dudt + φ2x2dt2)

=

(x′du −φydt)2 + (y′du + φxdt)2.

= Φ(Xx′ + Yy′)̸ = 0,

we see that (u, t) 7→(U, T) defines a smooth coordinate change, which can be easily integrated to give U = x −tφy + U0 and T = y + tφx + T0. On the other hand, for helicoidal surfaces of i-type it is seen that, under (u, t) 7→ (U = x + U0, T = tφx + T0), I in Eq. (4.7) is rewritten as I = dU 2 + dT 2.

Simply isotropic invariant surfaces resulting from limit motions are:

(Iii) Parabolic Revolution Surfaces:

• If α(u) = (x(u), y(u), 0), we have the parabolic revolution surface of

Ni-Type

(4.8) Y3(u, t) = (a t+x(u), b t+y(u), c t+ ac1 + bc2

T2 +C1T X(U)+C2T Y(U));

• If α(u) = (x(u), 0, z(u)), we have the parabolic revolution surface of

Z3(U, T) = (A T + X(U), B T, C T + Ac1 + Bc2

t2 + c1t x(u) + z(u)). (IV) Warped translation surfaces (c = (ac1 + bc2) = 0; (a, b), (c1, c2)̸ =

Fferential Geometry Of Isotropic Invariant Surfaces

• If α(u) = (x(u), y(u), 0), we have the warped translation surface of

(4.10)

Y4(u, t) = (a t + x(u), b t + y(u), c1t x(u) + c2t y(u)); • If α(u) = (x(u), 0, z(u)), we have the warped translation surface of

(4.11)

Z4(u, t) = (a t + x(u), b t, c1t x(u) + z(u)). (V) Isotropic shear surfaces (a, b, c = 0; (c1, c2)̸ = (0, 0)): • If α(u) = (x(u), y(u), 0), we have the isotropic shear surface of ni-

Y5(U, T) = (X(U), Y(U), C1T X(U) + C2T Y(U));

• If α(u) = (x(u), 0, z(u)), we have the isotropic shear surface of i-type

(4.13)

Z5(u, t) = (x(u), 0, c1t x(u) + z(u)). (VI) Translation surfaces along non-isotropic directions (c, ci = 0; (a, b)̸ =

(0, 0)):

• If α(u) = (x(u), y(u), 0), we have the translation surface of ni-type

Y6(U, T) = (A T + X(U), B T + Y(U), 0),

that represents a portion of a non-isotropic plane; • If α(u) = (x(u), 0, z(u)), we have the translation surface of i-type

Z6(U, T) = (A T + X(U), B T, Z(U)),

that represents a cylindrical surface with non-isotropic rulings. (VII) Translation surfaces along isotropic directions (a, b, ci = 0; c̸ = 0): • If α(u) = (x(u), y(u), 0), we have the translation surface of ni-type

Y7(U, T) = (X(U), Y(U), C T),

that represents a cylindrical surface with isotropic rulings; • If α(u) = (x(u), 0, z(u)), we have the translation surface of i-type

Z7(U, T) = (X(U), 0, Z(U) + C T),

that represents a portion of an isotropic plane. Remark 4. Notice that isotropic shear surfaces and translation surfaces along isotropic directions, types (V) and (VII) above, are not admissible (here a = 0 = b).

In addition, if α(u) = (au, bu, 0), then the corresponding parabolic surface of ni-type is of the form Y3 = (aU, bU, Z(U, V )), where U = t + u and V = t −u. Rewriting it as Y3 = U (a, b, 0) + Z e3 clearly shows Y3 is a portion of an isotropic plane. Similar conclusions apply for parabolic revolution surfaces of i-type when α(u) = (k, 0, z(u)) or when

Uiz C. B. Da Silva

b = 0. Indeed, in the former, Z3 = (k, 0, 0) + t(a, b, 0) + Z(t, u)e3, while in the latter Z3 = X(u, t)e1 + Z(u, t)e3. Proposition 4.2. Assume that a, b do not vanish simultaneously and also that the generating curve α is neither a part of a straight line {bX −aY = k, Z = 0} (in the ni-type) nor a part of an isotropic line or b̸ = 0 (in the i-type).

Then, all simply isotropic parabolic revolution surfaces, warped translations surfaces, and translation surfaces along non-isotropic directions are admissible.

Proof. It is enough to investigate the parabolic revolution surfaces, since the other surfaces are obtained by imposing some kind of restriction on the constants a, b, c, c1, and c2. In the ni-type we have

(4.18)

I = (x′ 2 + y′ 2)du2 + 2(ax′ + by′)dudt + (a2 + b2)dt2.

(4.19)

det gij = (bx′ −ay′)2 = 0 ⇔bx −ay = k. By hypothesis, α(u) is not a straight line with neither slope b/a nor a/b and, therefore, the corresponding parabolic revolution surface of ni-type is admissible.

(4.20)

I = x′ 2du2+2ax′dudt+(a2+b2)dt2 ⇒det gij = b2x′ 2 = 0 ⇔b = 0 or x = k. By hypothesis, α(u) is neither an isotropic line nor b̸ = 0 and, therefore, the corresponding parabolic revolution surface of i-type is admissible.

□

Remark 5. In analogy to Remark 3, the first fundamental form of parabolic surfaces of ni- and i-types can be written as I = dU 2 + dT 2 under the map

,

respectively (U0, T0 constants). Indeed, completing squares in Eq. (4.18),

=

(x′2du2 + 2ax′du dt + a2dt2) + (y′2du2 + 2by′du dt + b2dt2)

=

(x′du + a dt)2 + (y′du + b dt)2.

= (Bx′ −Ay′)̸ = 0,

we see that (u, t) 7→(U, T) defines a smooth coordinate change, which can be easily integrated to give U = x + at + U0 and T = y + bt + T0. On the other hand, proceeding similarly for parabolic surfaces of i-type, we see that, under (u, t) 7→(U = x + at + U0, T = bt + T0), I in Eq. (4.20) is rewritten as I = dU 2 + dT 2.

Concerning the most general 1-parameter subgroup of simply isotropic motion in Eq. (3.12), we have the following invariant surfaces of ni- and

+

[x sin(tφ) + y cos(tφ) + bCt + aSt]e2 + [c t + (ac1 + bc2) ˜Ct +

+

(ac2 −bc1) ˜St + (c1Ct + c2St)x + (c2Ct −c1St)y]e3,

(4.21)

if the generating curve is α(u) = (x(u), y(u), 0) and

+

[x(u) sin(tφ) + bCt + aSt]e2 + [c t + (ac1 + bc2) ˜Ct +

(4.22)

if the generating curve is α(u) = (x(u), 0, z(u)), respectively. Given a point q ∈I3, the orbits of the 1-parameter subgroups ψt of type (I), (II), and (III), i.e., the curves t 7→ψt(q), are Euclidean circles, Euclidean helices with an isotropic screw axis, and parabolas (isotropic circles), respec- tively. For the other types of 1-parameter subgroups, i.e., when φ = 0 and (ac1 + bc2) = 0, the orbits of the corresponding 1-parameter subgroups of

Limit Motions Are Straight Lines And, Therefore,

Proposition 4.3. Invariant surfaces in I3 of type (IV) −(VII) are ruled. Example 4.1 (Spheres as invariant surfaces). Let p̸ = 0 be a constant and

Α(U) = (Pu, 0, Pu2

2 ) be an isotropic circle (the induced geometry in the xz- plane is I2, whose circles of parabolic type are precisely parabolas with vertex and focus lying on an isotropic line ). Applying an elliptic revolution to

),

which parameterizes the sphere of parabolic type Σ2(p) : z = 2p(x2 + y2). On the other hand, applying a parabolic revolution to α(u) with c = 0,

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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