Composite Rotor Blade Ansys

Abstract

In this report, we present the theory on aerodynamics of quadrotors using the well established momentum and blade element theories. From a robotics perspective, the theoretical development of the models for thrust and horizontal forces and torque (therefore power) are carried out in the body fixed frame of the quadrotor. Using momentum theory, we propose and model the existence of a horizontal force along with its associated power.

Given the limitations associated with momentum theory and the inadequacy of the theory to account for the different powers represented in a proposed bond graph lead to the use of blade element theory. Using this theory, models are then developed for the different quadrotor rotor geometries and aerodynamic properties including the optimum hovering rotor used on the majority of quadrotors. Though this rotor is proven to be the most optimum rotor, we show that geometric variations are necessary for manufacturing of the blades. The geometric variations are also dictated by a desired thrust to horizontal force ratio which is based on the available motor torque (hence power) and desired flight envelope of the vehicle. The detailed aerodynamic models obtained using blade element theory for different geometric configurations and aerodynamic properties of the aerofoil sections are then converted to lumped parameter models that can be used for robotic applications. These applications include but not limited to body fixed frame velocity estimation and individual rotor thrust regulation [1, 2].

Ntroduction

In recent years, there has been an increased interest within the robotics community in understanding the aerody- written for helicopters by aerodynamicists. Some of these references include [6, 9, 17, 21] and contain theories developed for aerodynamicists and helicopter performance analysts. From a robotic point of view, many of the parameters in the theories are immeasurable and therefore not available for control purposes. Hence the arguments in the report are driven by robotic and not aerodynamic arguments. In addition, the non-linear scaling of the forces and their effects and mechanisms (e.g. Bell-Hiller system), blade geometries and disc loading are different between quadrotors and full-sized helicopters. Quadrotors are designed based on simplicity, ease of maintenance and cost. For these reasons, the majority of quadrotors have fixed pitched blades contrary to the variable pitch and flapping hinges of the rotor blades typical of helicopters. An example of a helicopter rotor mechanism is shown in Figure 1. In this report, we present the relevant aerodynamic models for quadrotors based on the well established momentum and blade element theories developed primarily for helicopters. From these theories, we extend our analysis to produce simplified or lumped aerodynamic models that can be used in robotic applications.

The quadrotor used in the analysis is that used in which weighs 1.2kg and has 10in diameter propellers. The report starts by describing the different frames of reference on quadrotors and their rotor blades. By choosing the body fixed frame, Section 2 uses momentum theory to model thrust and horizontal force and torque.

By looking at a bond graph representation of power and axial flights, the limitations of momentum theory which include its inability to model some of the vortex states and a distribution of the forces across the rotor implies that another modelling technique has to be used. In Section 3 to 6, blade element theory is applied to individual elements to produce models for the forces and torque for different blade geometries and aerodynamic properties of the aerofoil sections used on quadrotors.

In Section 3, the modelling framework for elemental forces and torque is developed. With the assumption of constant chord and pitch, infinite aspect ratio (AR) and zero-lift

Arxiv:1601.00733V1 [Physics.Flu-Dyn] 5 Jan 2016

Figure 1: A helicopter rotor mechanism with Bell-Hiller flybar and swash plates mechanism for controlling cyclic and collective pitch. angle of attack, which represent the simplest rotor blades used predominantly in the helicopter literature, lumped parameter models for thrust, horizontal force and torque (hence power) are developed in Section 4. In Section 5, the assumptions of zero-lift angle of attack aerofoil and infinite aspect ratio are removed and a more complicated model is developed. In Section 6, ideal twist and chord are considered in developing models for the so-called ideal rotors used on the majority of quadrotors. In the development of the models for the ideal rotor, we show that the geometry is a design parameter that must be optimised with respect to the flight envelope of the quadrotor and the mechanical properties of the material used in manufacturing the blades.

Omentum Theory Of Rotors

In this section, we present the aerodynamics of the rotor blades on quadrotors using momentum theory. The models for thrust, horizontal force and power are derived with the necessary assumptions contained in the theory. Starting with frames of reference, we carry out detailed momentum theory analysis of quadrotor rotor blades. To account for the fact that we are modelling in the body fixed frame of the quadrotor and not the rotor plane used in helicopter literature, we propose the existence of a horizontal force with an associated required power. The section concludes with a bond graph representation of the different powers and the limitations of momentum theory.

Frames Of Reference

Consider Figure 2 which shows a quadrotor along with a rotor and the different planes and frames of reference. We consider that there exists a fixed frame on the Earth’s surface termed inertial frame {A}. Attached to the quadrotor which is assumed rigid is the body fixed frame {B}. If we let the relative velocity of {B} to {A} be V ∈{B} where V ∈R3. If there is wind blowing at a velocity of W ∈R3 relative to {A}, then the total air relative velocity seen by the quadrotor is −W + V expressed in {B}. Throughout the report Ð→e 1,Ð→e 2,Ð→e 3 ∈R3 are used to denote unit vectors in x,y and z directions respectively.

Taking a closer look at the rotor shaft and rotor plane, the rotor shaft is always aligned with the Ð→e 3 axis of {B}. This implies that the hub reference frame and the body fixed frame {B} are equivalent. We define the rotor reference frame {C} which has its Ð→e 3 aligned with that of {B} and the Ð→e 1,Ð→e 2 directions in rotation at the speed of the rotor (ϖ) relative to {A}. In the sequel, the disposition of Ð→e 1 of {C} from that of {B} is referred to as the azimuthal angle ψ. For the rotating rotor, a plane on its spinning tips is referred to as the tip path plane denoted {D} and is otherwise known as the axis of zero flapping. In addition, it is the plane on which an observer does not see the conning or tilting of the rotor disc. It is a known phenomenon that as the spinning rotor translates, it tilts sideways and backwards and forms a cone around the rotor hub (shown later in Figure 8) thus moving this plane ({D}) above the rotor hub and putting it at an angle to the hub and therefore {B}. This phenomenon is referred to as blade flapping and in the sequel will be modelled by the angle β. If the quadrotor has variable pitch propellers such as helicopters, there is an additional plane known as the no-feathering plane {NFF} and is normal to the plane of the swash plates. To minimise the effect of blade flapping, quadrotors are designed with most often fixed pitch rotor blades that are very stiffand rigid. For consistency with current quadrotor dynamic models, the analysis and theoretical developments will be carried out in the rotor hub frame which aligns with the quadrotor body fixed frame {B}. It should be noted that derotating Ð→e 1,Ð→e 2 of {C} gives the Ð→e 1,Ð→e 2 of {B} through the Figure 2: A quadrotor and the different frames of reference. In addition to {A} and {B} defined for quadrotors, the rotor and rotor hub also have additional frames of reference which include the rotor reference frame {C}, the tip path plane {D} and the no-feathering plane NFF. These are also shown in the figure.

azimuthal angle (ψ) and the flapping angle (β) which depends on the rotor azimuthal position as such is denoted by β(ψ). This is the approach taken in the blade element theory (Sections 3 to 6) in developing models for the different forces, torque and power for the different rotor geometric and aerodynamic configurations. The frames {C} and {D} along with β(ψ) will also be explained and modelled in detail in the next sections.

Remark 1 We choose to model the thrust force T ∈R in the Ð→e 3 direction of {B}. It is the component of force along the rotor shaft/hub used in the quadrotor dynamic modelling literature [4, 5]. As the rotor translates, a force perpendicular to T lying on the plane containing Ð→e 1,Ð→e 2 in {B} is generated to oppose the motion. This force we model by the so-called in-plane horizontal force (H ∈R3). In common practice for slow moving quadrotors, H⊺Ð→e 3 = 0. As the rotor translates, there is tilting of the tip path plane ({D}) from the rotor hub/shaft as a result of blade flapping and this causes the misalignment of the tip path plane and the rotor shaft ({B}). This creates one of the components of the H-force. In addition, there is an induced drag term associated with forward motion. Blade flapping and induced drag have been lumped in the quadrotor literature and used to improve the control performance of multirotor vehicles .

Remark 2 For helicopters, thrust is changed by using the feathering mechanism to change the blade pitch either collectively or cyclically using a swash plate and the Bell-Hiller mechanism shown in Figure 1. For the majority of small-scale electrically powered fixed pitch quadrotors, thrust changes are achieved by changing the speed of the rotor using an electronic speed controller (ESC). This is because quadrotors have a low moment of inertia blades relative to helicopters that make rotor speed changes easily achievable.

Ntroduction To Momentum Theory

Momentum theory for rotary wing vehicles was developed by Glauert based on earlier work by Froude for aircraft propellers. It is one of the two most popular theories for propeller analysis. Its simplistic approach has made it the starting point for modelling aerodynamic forces on rotors. The theory considers a rotor as an actuator disc with the accelerating air forming a streamtube. The control volume shown in Figure 3 shows this streamtube. As the air is sucked and accelerated by the rotor as it goes through it, it generates a virtual induced airflow with

Y,Vi

z)⊺. Glauert made several assumptions which are stated in Assumption 2.1.

Assumption 2.1 These Assumptions Include

• The rotor disc has an infinite number of rotor blades such that there is a uniform constant distribution of aerodynamic forces over the rotor disc. • The rotor disc is an infinitely thin disc of area A which offers no resistance to air passing through it.

• The flow is irrotational and therefore no swirl is imparted to it. • The air outside the streamtube remains undisturbed by the actuator disc. It has been observed that the higher the disc loading, the more these assumptions hold. The disc loading (DL) is

= T

Noting the quadrotor considered in this report (and quadrotors in general) has a higher disc loading than the majority of helicopters, indicates that momentum theory holds better for quadrotors than for helicopters. Remark 3 As we are considering the entire rotor disc, the forces generated are in the rotor hub plane, i.e. {B}, and any effects resulting from blade flapping are not considered in the control volume but should be seen as a validation of the existence of the proposed H-force. Details on blade flapping and its effects on the generated forces and torque will be covered in detail in Section 3. It should be noted that blade flapping is responsible for the misalignment of the rotor plane and the rotor hub shown in Figure 4.

Omentum Theory For Z-Direction Or Axial Motion

To develop the thrust and power models using momentum theory, consider the simplest of control volumes shown in Figure 3 where the velocity of the vehicle is V = (0,0,Vz)⊺. It should be noted that in {B}, Vz > 0 indicates that the vehicle is moving downwards based on our right hand coordinate system shown in Figure 2. For the rotor in hover or undergoing purely axial motion, we make the following additional assumptions Assumption 2.2 The flow through the rotor is one-dimensional, quasi-steady, incompressible, inviscid, and be- haves as an ideal fluid and the radius of a plane perpendicular to the control volume at the rotor disc equals the rotor radius.

From Froude’s theory, the airflow through the propeller disc is continuous and characterised by a constant speed which at hover is the induced velocity denoted by vi. The propeller disc introduces a discontinuity in the pressure. This discontinuity is denoted by ∆P and can also be thought of as the increment in the static pressure across the disc. As shown in Figure 3, we consider also a virtual cylindrical surface of radius R0 > R containing the propeller disc of radius R and displaced along the propeller spin axis. This surface will be employed only to compute the amount of air flowing inside and outside the propeller disc. More specifically, let us denote as upstream plane the disc of radius R0 and as downstream plane the disc of radius r2 < R which are located at the beginning and at the end of the cylinder respectively. By assuming that they are infinitely far from the propeller disc implies that the streamlines are parallel to the propeller spin axis. By applying momentum theory, the thrust force T (in the axial direction i.e. Ð→e 3 of {B}) can be computed as the difference between the momentum of the flux going out and the momentum of the flux coming into the streamtube i.e.

,

Figure 3: Vertical streamtube for hover and axial analysis. In the figure, there are three sections: upstream (0), rotor disc (1) and downstream (2) to help with the analysis. The figure also shows the velocities at each of these sections. Please note for ease of analysis, V ∞is pointing downwards and therefore positive in {B}.

is the total air velocity at the rotor disc within the streamtube. The area of the streamtube is that of the rotor disc given by A = πR2. The power supplied can then be computed as the product of the thrust and the local

Velocity Across The Disc (I.E., Vi

z −Vz in the direction of T along Ð→e 3) and is given by

(2)

This power is also the rate of kinetic power imparted into the air across the streamtube and is given by

(3)

Substituting for T using (1) in the power equation (2), and comparing it to (3), we get

Thus Substituting For ˙M And Vi

z in (1), the aerodynamic thrust and power in the air for a given rotor in axial or

(5)

Figure 4: Rotor control volume for generalised motion. The figure shows the streamtube and generated forces, the velocity of the rotor V and the resultant air velocity V a seen by the rotor along with the induced velocity at the disc. Please note for ease of analysis, V ∞is pointing downwards and therefore positive in {B}.

Remark 4 It is worth noting that at hover (Vz = 0), using (4) and (5), the known static relationship between

P =

√2ρA. In designing helicopters, large rotor blades are used since for the generation of the same thrust, there is less power required thus implying that helicopters have low disc loading. This equation can also be applied to the design of heavy lifting quadrotors. However as will be shown later, larger and heavier blades are disadvantageous for fast and agile quadrotor vehicles due to reduced transient performance of the rotor speed as a result of rotor mass moment of inertia. Hence high performance quadrotors have rotor blades with higher disc loading than helicopters.

Omentum Theory For Generalised Motion

Consider now Figure 4 which shows a slightly tilted actuator disc to that of Figure 3. In this case, the rotor experiences both translational and vertical airflow. As such the induced airflow now has all the components in x,y,z. Figure 4 also shows a well known phenomenon of rotor blades, blade flapping which is responsible for the coning and backwards tilting of rotor blades. Its net effect is to create/increase any force in the plane of Ð→e 1,Ð→e 2 opposing the motion of the rotor. As a result of the coning and tilting, the Ð→e 3 direction of {B} and {D} are misaligned. Hence the induced airflow (vi)⊺Ð→e 3 ∈{D} now has components in the horizontal plane and vertical

X,Vi

y,0)⊺is ignored as it is relatively small compared to vi

Z And

its existence is as a result of Vh and exists only in {B} and not in {D}. This is because they assume a completely vertical flow and carry out their analysis in the tip path plane {D}. As shown previously in Section 2.3, for a

Purely Vertical Flight, There Is Only Vi

z. However, for a purely horizontal flight, there are vi

X,Vi

y. Remark 5 In the hub frame ({B}), the rotor experiences the resultant horizontal/planar velocity. This velocity is represented by the subscript h. For example Vh = (Vx,Vy,0)⊺∈{B} is seen as the resultant ∣Vh∣. Hence in the sequel, derivations are for a 2-D flow. To minimise notational confusion, we use Vh,vi

H∣,∣H∣

respectively. From Figure 4 and more obvious in Figure 5, it is quite clear that there is an additional force to T. This is the force (H ∈R3) in the plane of the rotor acting against the motion of the plane that is perpendicular to the rotor Figure 5: Rotor reference frame {C}.The figure shows the resolution of T into components parallel and perpen- dicular to the airflow thus clearly showing the existence of H as it is required for steady state flight.

hub. This force exists as we are modelling in the rotor hub or {B} of the vehicle and not in {D} as in traditional helicopter literature [6, 9, 21]. Before continuing with the analysis, we make the following assumption on the wind velocity W.

Assumption 2.3 The wind velocity ∣W∣= 0. It is however not tedious to incorporate the wind velocity W into the computations by setting the air velocity to (−Wx + Vx,−Wy + Vy,Wz −Vz)⊺.

We reapply the same analysis as the axial flight in Section 2.3 and carry out the analysis in the rotor hub or {B}. It should be noted that because our Ð→e 3 is pointing downwards, we introduce a new variable V s = (Vx,Vy,−Vz)⊺ to represent the velocity at the farstream seen by the rotor. Starting with the application of momentum theory in the direction of motion of air around the rotor hub or {B} by first considering the momentum of the fluid entering and leaving the control volume and letting the forces be F = (Hx,Hy,T)⊺with H⊺Ð→e 3 = 0 for slow moving vehicles,

Then

F = ˙m(V ∞+ V s) −˙m(V s).

Z −Vz),

be the total velocity of the air at the rotor hub, the power to generate this force is P = F ⊺V a.

(6)

To determine the relationship between V ∞and vi, it is worth noting that power is a scalar as such we can consider it as a result of horizontal and vertical motion separately and sum them to get the total power. Consider first the axial direction (Ð→e 3) which contains the thrust force T. Applying Newton’s second law or the conservation of momentum at the farstream and downstream and along the rotor hub ({B}),

(8)

Power is also the rate of kinetic power imparted into the air across the streamtube. This is given by

(9)

Comparing (8) and (9) and substituting for T using (7), we get

(10)

The same relationship for the induced and downstream velocity components for the horizontal motion can be

∞

h . Thus in a similar manner to T, the horizontal force H generated that is acting against the direction of motion of the plane at the rotor hub and power in generating this force are given by

H + Vh),

respectively. The original generalised momentum theory equations which do not include the H force and vi

H Though

were originally developed for vertical or axial flights as shown in Section 2.3 have not been theoretically verified. There are however experimental evidence supporting the theory [21, pg. 51-52]. In the literature, the H force is the drag force and it has been used in the estimation of the body fixed frame velocity of quadrotors [1, 2]. The modified generalised momentum theory equations are presented in (11) to (14).

(14)

In Figure 6, we show the different powers generated and consumed at the rotor hub. In the diagram Pp represents the profile power dissipated as a result of the spinning of the rotor blades. It should be noted that momentum theory does not account for this power and will be later modelled in Section 3. Intuitively, profile power can be seen as Pp = τaϖ, where τa is the air resistance torque and ϖ the speed of the rotor. From experiments performed in [3, 4], the aerodynamic power (13) can be rewritten as

H + Vh)),

Figure 6: The different powers on a spinning rotor. where FoM is the figure of merit and is defined as an efficiency factor to account for non-ideal losses and the region not modelled by momentum theory as shown in the bond diagram in Figure 7. If Pm is the mechanical power supplied to the rotor shaft, then the FoM is defined as

(15)

In helicopter literature, losses occur in two parts: rotor wake and tip vortices. Typical values are 10% for wake flow and 15% for tip vortices and other losses [9, pg. 45-46]. From experiments performed in [3, 4], the FoM was found to be between 60 and 70% for most rotors. The theoretical maximum of FoM at hover is 81% [9, pg. 47] though values as high as 77% have been recorded [15, 16]. It should be noted that the FoM is the same for both axial and planar axis because the properties of the fluid are uniform and so the non-ideal loses are the same in every direction.

Carrying out a power balance at the rotor hub and ignoring the power due to profile blade losses and if Pr is the power dissipated in accelerating the rotor to a constant rotor speed (ϖ), the mechanical power Pm defined as

The Power Supplied To The Rotor Shaft Is Given By

Pm = Pr + Pa. In reality, this is the power that is controlled and can be set to a desired value. The Pr is estimated using methods described in . A bond diagram showing the power flow from the rotor shaft to the air and the generated forces on the vehicle are shown in Figure 7. Starting from the rotor shaft which acts as a source of effort producing for the entire vehicle the torque τ and rotor speed ϖ with the power lost through the resistance Rl = Pr modelled by a 1-junction. Through algebraic equations i.e. a modulated transformer MTF, the distribution of forces f(r,ψ) and velocity U(r,ψ) are obtained. These then go through a 0-junction and an MTF to produce the force F and velocity V a with power losses in the wake represented by the conductance

Rl1 . Applying Momentum Theory To

this region through the 0-junction, the power used in moving the vehicle (FV ) and power lost in the induced flow

(Fvi) Are Seen Through The Conductance

Rl2 . The distribution of forces f(r,ψ) and velocities U(r,ψ) will be dealt with using blade element theory which is covered in the remainder of the report.

Rotor Wake And Vorticity

The induced velocity distribution is not constant across the rotor as would be expected especially for forward flights where its value at the leading edge (face directed into wind) is greater than at the trailing edge (face away from wind). To account for such a non-constant induced velocity distribution, let vi0 ∈R3 be the mean induced

∑

Momentum theory fails. Momentum theory region. Figure 7: Bond diagram for a motor/rotor system. The source of effort (Se) is the motor shaft which makes the vehicle to fly and overcome vortices and wake. The figure also shows that the forces on a rotor should be modelled by a distribution f(r,ψ) and velocity U(r,ψ) at every radial position and azimuthal angle. In addition the figure also shows losses represented by conductance and resistances (Rl) which are embedded in the FoM.

velocity at the disc centre and its components are given by (based on (11) and (12) respectively)

H =

2ρA∣V a∣. Glauert proposed the following induced velocity model

R Cosψ),

where r ∈[0,R] is the radial position, ψ is the azimuthal angular position and κe is a number greater than unity (usually 1.2 for helicopters) [21, pg. 54]. For our modelling application, we believe that this model also holds for

The Proposed Vi

h with the same κe value. Details on a better induced velocity model are presented in Section 3.4 using the Mangler and Squire’s method. A drawback of these models is the vio dependence on itself through V a and on thrust both of which are unknown for small-scale open-source quadrotors. From a robotics perspective, the induced power (power associated with induced velocities) in the power model (13) can be modified to account for the non-uniform and non-constant induced velocities. Thus the modified power equation is

Z −Vz) + H(Κevi

h + Vh). From static tests performed in [3, 4], without any exact measurements of V a, it can be said that κe is embedded in the FoM value obtained. Hence proper wind tunnel tests where accurate measurements of V,V a are necessary to experimentally determine κe and FoM. In the sequel, it will be shown that unlike helicopters, quadrotor blades are designed to produce substantial amount of lift across the different sections of the blades and not only at the outer blade sections as a result of blade geometry which suffer from significant tip vortices in the case of helicopters.

Imitations Of Momentum Theory

There are flight regimes in which momentum theory fail. These regimes are associated with axial motion corre- sponding to cases where the streamtube model and Assumptions 2.1 and 2.2 are no longer valid. These axial flight regimes which rotors of quadrotors and helicopters experience in addition to the normal state are outlined below .

Vortex Ring State (VRS): The rotor is said to be in this state when the rate of descent is half the vertical induced velocity. In this state, vortex ring encircles the disc causing the flow to become unsteady resulting in high levels of vibration. It should be noted that momentum theory cannot be used to model in this state.

Turbulent Wake State (TWS): This state occurs when the rate of descent of the rotor equals that of the vertical induced velocity as such there is no net flow of air through the rotor. From (11) and (13), it is easily seen that there is no thrust generated and the power required is zero. This contradicts the fact that power has to be supplied to maintain the spinning of the rotors. Thus indicating that momentum theory cannot be applied. However, the vibrations are less in this state than those of the VRS.

Windmill Brake State (WBS): This state occurs when the rate of descent of the rotor is more than twice the induced velocity. At this rate, the blade sections are likely to stall. In this state, the net flow of air is entirely upwards thereby creating negative thrust which causes a consumption of power from the air. Of the three axial descent states mentioned, momentum theory models for T and P are only valid in this state ,[9, pg. 60], [21, pg. 10-13].

Unlike axial flights, there are no limitations caused by the horizontal/planar motion as both ∣vi

H∣And ∣Vh∣Are

always positive in the direction of motion. Contrary to helicopters, to avoid such axial states for quadrotors, more the subsequent sections, it will be shown that most quadrotor blades are designed to have optimal chord with the entire blade span generating lift and drag forces required to generate T,H and P. Hence the tip vortices have minimal effects on the aerodynamic forces generated compared to helicopters which have the majority of the aerodynamic forces generated in the outer portion of the blade.

In the next sections, blade element theory is presented. It uses an element of a blade to model forces and torque (hence power) irrespective of the axial flight condition. In addition, it makes use of the fact that the elemental forces and velocities along the elements of a rotor are functions of the elemental radial and azimuthal position.

Ntroduction To Blade Element Momentum Theory

Blade element theory considers the individual elements of a rotor blade, the aerodynamic properties (lift and drag coefficients) of the aerofoil, blade geometry and uses elemental forces and torque. The overall model for thrust (T), in-plane horizontal force (H) and torque (τ) and power (P) are obtained by integrating along the entire blade and over a rotor revolution of these elemental forces and torques. In this section, we examine the necessary assumptions and models for induced velocity and blade flapping that are used in the development of the elemental forces and torque. These models are then incorporated into the model for the total velocity along with its horizontal and vertical components. This then led to the examination of the different elemental aerodynamic forces in the tip path plane {D} and {C} and their resultants which is the thrust and horizontal force in the body fixed frame {B}. The final elemental results will be used in the subsequent sections to model T,H and P for a variety of rotor geometries and aerodynamic properties.

The following assumptions form the basis of blade element theory. Assumption 3.1 The outward centripetal force acting on the blades ensure that they can be assumed rigid and do not stall.

It should be noted that similar to Section 2, the development of the models for T,H and P are carried out in the rotor hub frame or body fixed frame {B}. The airflow consists of the vertical component and a horizontal component which is the resultant of the airflow in the Ð→e 1 and Ð→e 2 directions. For this reason, we only model the thrust T acting in Ð→e 3 and the magnitude of the horizontal force H = ∣H∣in the plane containing Ð→e 1,Ð→e 2.

Theoretical Framework

Since blade element theory considers individual elements of the rotor, it is necessary to obtain the total of these elements along the span of the rotor which defines the physical quantity of interest. In addition, the rotor spins resulting in azimuthal changes from 0 to 2π. Hence we can model the elemental contribution of a quantity say force F as dF(r,ψ) for an element located at a distance r from the rotor hub (r ∈[0,R]) and ψ is its azimuthal Figure 8: A view of the rotor-shaft system of quadrotors. The figure shows a side view with the coning angle a0 and sideways tilt a1 relative to {B} which are part of the flapping angle β. In addition, there is the top view which shows the TPP relative to the⃗e1 of {B} and its disposition, the azimuthal angle ψ.

angular displacement. The total force is then given by the sum of the individual elements across all the azimuthal positions for the blade. For Nb number of blades, the total force is given by

(16)

Remark 6 From Section 2.5, it was pointed out that rotors shed tip vortices which are accounted for by using a tip loss model. If ct is the chord of the rotor at the tip, if B ∈ where BR is the effective radius of the rotor,

R.

Glauert used this to model tip loss such that the radial limits of the integrand (16) are [0,BR]. In the sequel B = 1 is used to obtain a simplified model. BR will be a coefficient to be determined experimentally as will be shown in the lumped parameter model presented in Section 4.

Relevant Definitions

Before presenting blade element theory, it is worth defining some non-dimensional variables in line with Remark 1 which states that the forces and torque are modelled in {B} i.e. along and perpendicular to the rotor shaft/hub. If Ib is the mass moment of inertia of the blades, Nb is the number of blades, c is the chord length of the blade, R is the blade radius and ϖ is the speed of the rotor, then Definition 1 The rotor hub/shaft advance ratio µ in the direction of Vh

Πr

= ∣µi + µh∣. Definition 2 The rotor hub/shaft vertical inflow ratio λ in the Ð→e 3 of {B}

Σ = Nbc

πR . Definition 4 The Lock number for a constant chord blade

B

. We define two further coefficient variables which will be shown in the sequel to be dependent on the aerodynamic state of the rotor.

Odel For Blade Flapping Angle (Β(Ψ))

With the assumption of steady state flight, we model the blade flapping angle β at an azimuth ψ using the following Fourier series expression with harmonic terms (a0,a1,...,an,b1,...,bn) by

(17)

Further details on the treatment of blade flapping as a drag term is presented in . In this work, we consider only the first harmonics, i.e. n = 1, since the higher harmonic terms a2,b2,a3 ..., can be ignored as they are very small or negligible for short rigid rotors. The blade flapping model is thus given by β(ψ) = a0 −a1 cosψ −b1 sinψ.

(20)

From Figure 8, a0 can be seen as the coning angle of the blade and a1 and b1 as the −Ð→e 1 ∈{B} and −Ð→e 2 ∈{B} tilt of the rotor disc or the tip path plane. The coefficient a0 is strongly linked to blade rigidity and is very small for quadrotors since they have high rigidity and stiffness rotor blades. It should be noted that even at hover, there is an a0 for long, slender and fully flexible rotor blades especially those used on helicopters. The presence of a0 does not cause any misalignment of the TPP or {D} from {B}. The other coefficients a1 and b1 are the backward and sideways tilting of the TPP from {B} and therefore results in the misalignment of {D} from {B}.

If θ0 is the blade pitch/twist, then the flapping coefficients for a no flapping hinge offset are defined by [6, pg.

Given That Μ Is Small Such That

1±µ2/2 can be approximated to 1, the flapping angle coefficients are

(26)

Multiples and higher powers of these coefficients are such that they can be neglected. This is used in the compu- tations of the H-force and torque. One of the root causes of blade flapping is the dissymetry in the lift during forward flights. This dissymetry creates a moment at the rotor hub. To minimise this moment, rotors are either connected to a hinge at the hub or rigidly attached and cyclically feathered by decreasing the pitch of the advancing and increasing the pitch of the retreating blade thus removing the lift imbalance (see Figure 1). Furthermore as was pointed out, quadrotors neither have flapping hinges nor swash plates but have short rigid and stiffblades to minimise the flapping effect.

Nduced Velocity (Vi(R,Ψ)) Distribution

In order to apply blade element theory, the induced velocity distribution along the span and different azimuth angles must be known. The estimation of the distribution of the vertical component of the induced velocity (and therefore proposed horizontal induced velocity) distribution on the rotor is a complex problem. In the theoretical development of the forces and torque, we will assume a constant vi distribution and therefore have the induced and translational velocities modelled through λ and µ. Furthermore, a good approximation for vi was proposed by Mangler and Squire . The model treats the rotor as a lifting surface with a pressure jump. The induced velocity field is modelled as a small perturbation superimposed upon an otherwise uniform velocity field. The velocity distribution is expressed by a Fourier series of harmonic terms given by (27)

(27)

where αD is the rotor disc incidence and vi0 ∈R3 is the mean induced velocity. To obtain vi0, recall from Section 2

Z −Vz),

where V is the velocity of the vehicle. Using momentum theory, we restate the mean induced velocities are given

(29)

This use of momentum theory in determining the mean induced velocity vi0 is the reason the theoretical develop- ment is also referred to as blade element momentum theory (BEMT). The harmonic terms or coefficients di are

N/2

. Figure 9: An aerofoil section of a blade at a location r from the rotor hub. The figure also shows the different elemental forces which include lift and drag and the horizontal and vertical forces on the aerofoil in {D}. The different angles and velocity components are also shown.

For N = 2K + 1,K ≥2,

dn = 0. As has been stated in Section 3.3, we are only concerned with the primary modes and therefore first harmonic flapping motion. However, d1 << d0 and αD introduces another unknown which is not measured as it is a function of β(ψ). The final simplified model for the induced velocity is given by

(30)

It will shown later in the sequel that (30) can be represented by a constant i.e. vi(r,ψ) = vi and later on changed as it does not change the computations. Hence, the modelling process does not make use of the Mangler and Squire model for vi(r,ψ). By choosing a constant vi implies that we are not required to know the thrust before hand (through (28) and (29)) as for present day quadrotor technology, T as well as the induced velocity vi are unknown. Furthermore, given the near hovering flight envelope for which quadrotors are designed and their use of ideal rotors (see Section 6) implies that this assumption is valid.

Similarly, the current helicopter literature, uses these assumptions in the development of the models for T,H and P. A consequence of this is a model for a lower power as the induced power is underestimated but can however be compensated for during the calibration process and the introduction of a scaling factor in the final induced power model.

Elocity Components At A Blade Section

Consider a blade element at a distance r from the rotor hub shown in Figure 9. For ease of analysis (mainly for 2-D flow assumptions to hold), we consider only the planar component of velocity i.e. ∣Vh∣= ∣Vx,Vy,0∣, ∣vi

Y,0∣

and vertical velocity Vz. To reduce notational confusion, the ∣.∣around the induced horizontal velocity and horizontal induced velocity will be dropped. The total airflow velocity at the blade element is U(r,ψ) ∈R3 and U(r,ψ) ∈{D}. The transverse scalar velocity at a blade element Uh(r,ψ) ∈R which is the magnitude of the planar

(32)

Normalising or non-dimensionalising by dividing by the tip velocity of the rotor ϖR, the following are obtained

(34)

The total or resultant velocity at the blade element is

Assumption 3.2 We Assume That U2(R,Ψ) ≅U2

h(r,ψ) for the quadrotor used in this report. This is a valid assumption as the quadrotor which weighs under 2kg, experiments have shown that for its rotor of radius 10in with ϖ ≥5000RPM, the velocity of the vehicle is bounded and in this case ∣V ∣≤5m/s. This Uz(r,ψ) = 5m/s. Hence Uh(r,ψ) > 10Uz(r,ψ) from which it is easily seen that U2(r,ψ) ≅U2 h(r,ψ).

Aerodynamic Forces Acting On Blade Elements

The aerodynamic forces acting on a blade element shown in Figure 9 are defined as Lift dL(r,ψ) ∈R is the force generated by the blade element perpendicular to the direction of the resultant airflow U(r,ψ).

Drag dD(r,ψ) ∈R is the force generated by the blade element that is parallel to the direction of the resultant airflow. The elemental lift and drag forces on a blade element expressed in the TPP {D} are defined by

(36)

where Cl(r,ψ) is the lift coefficient, Cd(r,ψ) the drag coefficient and c(r) is the chord length at a section radius r from the hub. The coefficients Cl(r,ψ) and Cd(r,ψ) are expressed respectively as

(37)

Cd(r,ψ) = Cd0 + KCl(r,ψ)2,K > 0.

For A 3 −D Wing Planform, The Constant K =

πARe, where AR is the aspect ratio of the wing and e is the Oswald span efficiency. The AR for helicopter blades is usually large > 10 and e = 0.8 for an elliptical lift distribution. From Figure 9, the blade element angle of attack α(r,ψ) which is the angle between the mean chord line of the aerofoil and the direction of motion of the blade or airflow is defined as

(39)

where θ(r) is the blade pitch and φ(r,ψ) is the relative inflow angle at the blade section defined by

(40)

for which we make the following assumption. Assumption 3.3 The relative inflow angle ∣φ(r,ψ)∣≤10○is small such that cosφ(r,ψ) ≅1 and sinφ(r,ψ) ≅ φ(r,ψ) for all r and ψ.

(42)

Remark 7 The most ideal or optimum rotor geometry is one that maintains a constant angle of attack and constant induced velocity across the entirety of the blade [6, pg. 54]. To produce a constant angle of attack for purely axial flights, the optimum rotor makes use of a hyperbolic pitch geometry from the hub to the tip of the rotor.

In addition, using a hyperbolic geometry for the chord ensures constant spanwise induced velocity. In [15, 16], the authors designed a slight variation of such an optimum rotor for the ANU-X4 flyer. It should be noted that there are physical limitations on the rotor around the hub such that limr→0 c(r),θ(r) →R+ or c(r),θ(r) have physical values. In addition the 20 to 30% rotor around the hub is curved inwards to prevent large horizontal forces, torque and practicallity of manufacture. The blade element chord and pitch at a distance r from the hub for the ideal/optimum rotor are defined by the following hyperbolic functions

R/R,

where ctip ad θtip are the tip chord and pitch respectively, r ∈[0,R] is the distance from the rotor hub and R is the rotor radius. The theoretical development of models for T,H and P using the ideal blade geometry with aerofoils for which along the span and every azimuth angle ψ, Cl(r,ψ) and Cd(r,ψ) are defined by (37) and (38) respectively are carried out in Section 6.

From the elemental forces defined by (35) and (36) are the forces along the Ð→e 1,Ð→e 2 plane and the Ð→e 3 of the tip path plane {D}. These horizontal and vertical forces defined in {D} are given by

(43)

dFz(r,ψ) = dL(r,ψ)cosφ(r,ψ) −dD(r,ψ)sinφ(r,ψ).

(44)

It should be noted that Fx represents the magnitude of the force along the x−y plane or plane containing Ð→e 1,Ð→e 2. Furthermore we define the following elemental forces along the rotor hub or body fixed frame {B}. In-plane H force dH(r,ψ) ∈R is the resultant elemental force generated on the plane of Ð→e 1,Ð→e 2 of {B} that opposes the motion along the Ð→e 1,Ð→e 2 plane of {B}.

Thrust dT(r,ψ) ∈R is the resultant elemental force generated along Ð→e 3 of {B}. In the next sections, derivations of these elemental forces and associated elemental torque and their respective sums in {B} will be presented in accordance with Figure 10 which shows the rotor reference frame ({C}) with the horizontal and vertical forces that are tilted from {B} by the flapping angle β(ψ). The derivations are carried out in the body fixed frame of the rotor ({B}) or the rotor hub or shaft for different blade geometries and aerodynamic characteristics of the aerofoil section and rotor aspect ratio AR.

Figure 10: Blade element forces in {B} and {D}. The figure shows the horizontal and vertical forces in the TPP {D} rotated by β(ψ) from the rotor hub/body fixed frame {B}.

Nduced Power Factor Κ

The induced power factor κ is a factor that accounts for the additional power/energy dissipated due to wake rotation, tip loss and non-uniform flow that is not modelled by momentum theory. These power lost effects are more significant for small rotors such as quadrotor blades with high disc loading and low power efficiency ( T

P ) That

are generally less efficient than helicopters. It changes with changing aerodynamic conditions around a rotor and increases with increasing tip loss and decreases with increasing rotor efficiency. These aerodynamic losses only apply to the induced power component of power. The induced power factor is closely related to but not the same as the figure of merit. Unlike the figure of merit used in the analysis of full scale helicopters, κ does not model profile power losses. This κ contains κe described in Section 2.5 which is incorporated to account for the use of the assumption of uniform and constant induced inflow velocities.

The induced power factor can be better explained in terms of disc loading DL. For helicopters with low disc loading i.e. large rotor areas relative to the thrust they produce, they have a high thrust coefficient CT and high

Power Efficiency ( T

P ) than quadrotors. Furthermore, the following also apply for a helicopter 1. Given the disc area and high torque engines for helicopter rotors implies that they require less RPM compared to quadrotors. For example the blades on the quadrotor under study require RPM ≈5000 to hover while those on normal helicopters require significantly less.

2. The induced velocity vi, is lower for helicopters compared to quadrotors at hover. Hence, to maintain the low angle of attack, helicopter blades have low collective pitch at hover. The higher CT is as a result of higher rotor radius as CT is proportional to the cube of the radius.

3. The high rotor efficiency at hover is as a result of the high thrust generated by large area with small induced velocity where in hover the total airflow through the rotor is vi z and therefore low required power.

4. The low disc loading on helicopter blades implies that they are under less back pressure than quadrotor blades. 5. Because the thrust is very high with less back pressure, axial relative wind only slightly affects the back pressure and the thrust produced at a given power. It should be noted that the thrust and power are very high thus there is no significant change in the efficiency i.e.

P Of The Rotor With Axial Wind. Note However

that given the low RPM of the rotors implies that small changes in thrust results in observable changes in

T .

So in the presence of an updraft, there is only a slight increase in rotor efficiency despite an increase in CT . The dominant effect is the additional work done by the rotors as a result of increased swirl in the wake as well as additional tip losses in the generation of tip vortices. Thus increasing CT as a result of an updraft causes additional losses with little changes in rotor efficiency and therefore corresponds to a moderate increase in κ.

For high disc loading rotors with low thrust coefficients CT such as quadrotors, 1. The angle of attack of such rotors is very high which corresponds to higher blade pitch angle in static free air or at hover. It should be noted that in such aerodynamic condition, given the small rotor disc area implies that to produce thrust requires higher induced velocity vi

Z Which Corresponds To Higher Power And Therefore

lower efficiency than do helicopter rotors. 2. With an updraft, the thrust produced increases, hence CT and a decrease in total velocity through the rotor. This leads to an increase in rotor efficiency. The relative high disc loading, low thrust and power requirements including high ϖ compared to helicopters implies that adding or removing power into the system will have a significant effect on the rotor efficiency. This can be illustrated mathematically by using the rotor efficiency equations (for e.g. (15)) and recognising the low profile power requirement for quadrotor blades compared to helicopter blades.

Hence for quadrotor rotor blades an updraft increases CT slightly (due to high ϖ), increases efficiency significantly due to low thrust and power and a negligible change in the already high tip loss. Hence overall for high disc loading low CT quadrotor blades, κ decreases with increasing CT .

With these intuitions and with reference to Figure 11, we propose the following general model relating κ to CT

(45)

It should be noted that the model illustrated in Figure 11 is supported by Figure 3.18 [9, pg. 105] although Leishman only considers low disc loading helicopter rotor blades. In the region of operation of quadrotors (CT <<

−4), The Dominant Part Of The Model Is D1 1

CT . For helicopters with large CT , the dominant part is d2CT . Given that the dominant part of the induced power factor model (45) for quadrotors (CT << 10−4) is d1 1

T And

if ¯CT is an operating point of the rotor, then it can be shown algebraically that

(Ct −¯Ct ),

where the constants const are some arbitrary constants. Hence one can approximate the κ model for quadrotors

(46)

where β0 > 0 and β1 < 0 is a large negative constant. With this linear model, there is a significant reduction in the computational requirement for κ when implementing on computationally constrained embedded electronic speed controllers used on quadrotors. In the derivations carried out in the sequel, the induced velocity components of the power contains κ. To reduce the number of variables during the development of the models, we make the following remark.

Remark 8 To reduce the many variables in the torque/power derivations, we will use λ in the power models and later on account for the effects mentioned in the final model by replacing it with κλi + λz or κvi

Z −Vz. This Also

applies to the contribution of the horizontal force i.e. κµi + µh or κvi h + Vz. Blade Element Theory for Classical Rotor Geometry (Constant chord and

Pitch) And Infinite Aspect Ratio

In this section, we apply the elemental forces obtained in Section 3 to model T,H and τ hence power P of the entire rotor blade in {B}. The blade geometry used in the analysis is the simplest geometry which consists of a constant chord c(r) = c, constant pitch θ(r) = θ0 and a blade of infinite aspect ratio (AR) of length R. In addition,

Κ And Ct Model

Figure 11: An illustration of the induced power factor κ and thrust coefficient CT for low and high disc loading rotor blades used on quadrotors and helicopters respectively. the rotor aerofoil considered has zero-lift angle of attack i.e. Cl0 = 0 and a linear lift slope. This is the rotor geometry and aerofoil characteristics considered in the analysis contained in the helicopter literature [6, 9, 21].

Though this rotor is far from the rotors used on quadrotors, it is however a good starting point for modelling. The final models obtained are simplified to obtain lumped parameter models that can be used for Robotic applications. With AR = ∞implies K = 0 and Cl0 = 0, the lift coefficient (37) and drag coefficient (38) become

(47)

which implies that with Assumption 3.2, (35) becomes

Rotor Thrust

Classical helicopter theory for this is covered in [6, pg. 96-98] and [21, pg. 58-60]. From the definitions given in Section 3.6, an expression for the thrust modulus dT(r,ψ) or the elemental thrust can be obtained. To do this, consider again Figure 10 and resolving forces in the Ð→e 3 direction of {B} for a blade element, dT(r,ψ) = dFz(r,ψ)cosβ(ψ) + dFx(r,ψ)sinβ(ψ).

Substituting For Dfx And Dfz,

dT(r,ψ) = [dL(r,ψ)cosφ(r,ψ) + dD(r,ψ)sinφ(r,ψ)]cosβ(ψ) + [dL(r,ψ)sinφ(r,ψ) + dD(r,ψ)cosφ(r,ψ)]sinβ(ψ). Realising that dD(r,ψ)sinφ(r,ψ)cosβ(ψ) and dD(r,ψ)cosφ(r,ψ)sinβ(ψ) both consist of two small terms that can be neglected i.e.

Dd(R,Ψ)Sinφ(R,Ψ)Cosβ(Ψ) ≅0,

dD(r,ψ)cosφ(r,ψ)sinβ(ψ) ≅0.

Hence,

dT(r,ψ) = dL(r,ψ)cosφ(r,ψ)cosβ(r,ψ) + dL(r,ψ)sinφ(r,ψ)sinβ(ψ). In addition, dL(r,ψ)sinφ(r,ψ)sinβ(ψ) consists of two small terms hence its effect is also negligible i.e. dL(r,ψ)sinφ(r,ψ)sinβ(ψ) ≅0.

= 1

2ρClα (θ(r)Uh(r,ψ)2 −Uz(r,ψ)Uh(r,ψ))c(r)dr.

(51)

Substituting for Uz(r,ψ) and Uh(r,ψ) using their normalised forms (33) and (34) and from (16), (51) becomes

R)

+ µ2 sin2 ψ)]c(r)drdψ. Given that we are using a constant chord c(r) = c and pitch θ(r) = θ0,

Θ0 (1 + 3

2µ2)]. Consider now the φ component of T and substituting for β(ψ) and dβ(ψ)

Ρcclαϖ2R2 (Λ + R

R (a1 sinψ −b1 cosψ) + µ(a0 −a1 cosψ −b1 sinψ)cosψ)( r

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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