Particle Swarm Optimization Robot

Abstract

The random drift particle swarm optimization (RDPSO) algorithm, inspired by the free electron model in metal conductors placed in an external electric field, is presented, systematically analyzed and empirically studied in this paper. The free electron model considers that electrons have both a thermal and a drift motion in a conductor that is placed in an external electric field. The motivation of the RDPSO algorithm is described first, and the velocity equation of the particle is designed by simulating the thermal motion as well as the drift motion of the electrons, both of which lead the electrons to a location with minimum potential energy in the external electric field. Then, a comprehensive analysis of the algorithm is made, in order to provide a deep insight into how the RDPSO algorithm works. It involves a theoretical analysis and the simulation of the stochastic dynamical behavior of a single particle in the RDPSO algorithm. The search behavior of the algorithm itself is also investigated in detail, by analyzing the interaction between the particles. Some variants of the RDPSO algorithm are proposed by incorporating different random velocity components with different neighborhood topologies. Finally, empirical studies on the RDPSO algorithm are performed by using a set of benchmark functions from the CEC2005 benchmark suite. Based on the theoretical analysis of the particle’s behavior, two methods of controlling the algorithmic parameters are employed, followed by an experimental analysis on how to select the parameter values, in order to obtain a good overall performance of the RDPSO algorithm and its variants in real-world applications. A further performance comparison between the RDPSO algorithms and other variants of PSO is made to prove the

Ntroduction

Particle swarm optimization (PSO) is a population-based optimization method attributed to be originally developed by Kennedy and Eberhart (Kennedy and Eberhart, 1995; Eberhart and Kennedy, 1995). It is widely known that PSO is rooted in two paradigms (Kennedy and Eberhart, 1995). One obvious root is its ties with artificial life in general, and bird flocking, fish schooling, and swarm theory in particular. The other root is associated with evolutionary algorithms (EAs), such as genetic algorithms (GAs) and evolutionary programming (EP). However, unlike EAs, PSO has no evolution operators similar to crossover and selection.

PSO optimizes a problem by iteratively improving a population of candidate solutions with respect to an objective (fitness) function. The candidate solutions, called particles, move through the problem space according to simple mathematical formulae describing the particles’ position and velocity. The movement of each particle is influenced by its own experiences, and is also guided towards the current best known position.

During the last decade, PSO has gained increasing popularity due to its effectiveness in performing difficult optimization tasks. The reason why PSO is attractive is that it gets better solutions, in a faster and cheaper way compared to other methods, whereas has fewer parameters to adjust. It has been successfully used in many research and application areas. An extensive survey of PSO applications can be found in (Poli, 2007; 2008).

To gain insights into how the algorithm works, some researchers have theoretically analyzed the PSO algorithm. These analyses mainly aimed for the behavior of the individual particle in the PSO algorithm, which is essential to the understanding of the search mechanism of the algorithm and to the parameter selection (Kennedy, 1998; Ozcan, and Mohan, 1999; Clerc and Kennedy, 2002; van den Bergh, 2002; Eberhart and Shi, 1998; Trelea, 2003; Emara, and Fattah, 2004; Gavi and Passino, 2003; Kadirkamanathan, et al, 2006; Jiang, 2007; Solis and Wets, 1981). For example, Kennedy analysed a simplified particle behavior and demonstrated different particle trajectories for a range of design choices (Kennedy, 1998).

Clerc and Kennedy undertook the first formal analysis of the particle trajectory and its stability properties (Clerc and Kennedy, 2002). As for the algorithm itself, Van den Bergh proved that the canonical PSO is not a global search algorithm (van den Bergh, 2002), even not a local one, by using the convergence criterion provided by Solis and Wets (Solis and Wets, 1981).

In addition to the analyses mentioned above, there has been a considerable amount of work performed in improving the original version of the PSO through empirical studies. The original PSO proposed in (Kennedy and Eberhart, 1995) appeared to have weak local search ability, due to the slow convergence speed of the particles. It is universally known that the tradeoff between the local search (exploitation) and the global search (exploration) is vital for the performance of the algorithm. As such, the original PSO needs to accelerate the convergence speed of the particles in order to achieve a better balance between exploitation and exploration. The work in this area, first carried out by Shi and Eberhart, involves introducing an inertia weight into the update equation for velocities (Shi and Eberhart, 1998). Clerc proposed another acceleration method by adding a constriction factor in the velocity update equation, in order to release the restriction on the particle’s velocity during the convergence history (Clerc, 1999). The acceleration techniques were shown to work well, and the above two variants of PSO have laid the foundation for further enhancement of the PSO algorithm.

In the original PSO, the PSO with inertia weight (PSO-In) and the PSO with constriction factor (PSO-Co), the search of the particles is guided by the global best position and their personal best positions. In these versions of PSO, all particles are neighbors of each other so that their neighborhood topology is known as the global best topology or the global best model. Although the algorithm with this model is able to efficiently obtain the best approximate solutions for many problems, it is more prone to encounter suboptimal point, it would mislead the other particles to move towards that point. In other words, other promising search areas might be missed. This had led to the investigation of other neighborhood topologies known as the local best models, first studied by Eberhart and Kennedy (1995) and subsequently in depth by 2004; Parrott and Li, 2006; Bratton and Kennedy, 2007; Kennedy and Mendes, 2002; van den Bergh and Engelbrecht, 2004; Lane et al., 2004; Li, 2004). The objective there was to find other possible topologies to improve the performance of the PSO algorithm.

In PSO, the particle essentially follows a semi-deterministic trajectory defined by a velocity update formula with two random acceleration coefficients. This is a semi-deterministic search, which restricts the search domain of each particle and may weaken the global search ability of the algorithm, particularly at the later stage of the search process. In view of this limitation, some researchers have proposed several probabilistic PSO algorithms, which simulate the particle trajectories by direct sampling, using a random number generator, or from a distribution of practical interests (Kennedy, 2003; 2004; Sun, et al., 2012; Krohling, 2004; Secrest and Lamon, 2003; Richer and Blackwell, 2006; Kennedy, 2006). The Bare Bones PSO (BBPSO) family is a typical class of probabilistic PSO algorithms (Kennedy, 2003). In BBPSO, each particle does not have a velocity vector, but its new position is sampled “around” a supposedly good one, according to a certain probability distribution, such as the Gaussian distribution in the original version (Kennedy, 2003). Several other new BBPSO variants used other distributions which seem to generate better results (Kennedy, 2004; 2006).

Researchers also turned to hybrid algorithms that incorporate other search methods into the PSO algorithm, for the purpose of playing to the advantages of different optimization algorithms (Angeline, 1998; Løvbjerg et al., 2001; Zhang and Xie, 2003; Devicharan and Mohan, 2004; Chen et al., 2007; Settles and Soule, 2005; Higashi and Iba, 2003; Pant et al., 2008). Angeline undertook the first work in this area by introducing a tournament selection into the PSO, based on the particle’s current fitness, so that the properties that make some solutions superior were transferred directly to some of the less effective particles (Angeline, 1998). Besides, some researchers introduced various efficient strategies into the PSO in order to enhance the search ability of the algorithm (Riget and Vesterstroem, 2002; Lovbjerg and Krink, 2002; Xie et al., 2002; Krink et al., 2002; Liang et al., 2006). For instance, Liang et al. proposed a PSO with a novel learning strategy, in which all other particles' historical best information is used to update a particle's velocity. It was shown that this strategy can diversify the swarm to avoid premature convergence (Liang et al., 2006).

In this paper, based on a random drift model, we present a new version of PSO, which is called the random drift particle swarm optimization (RDPSO). This PSO variant is inspired by the free electron model in metal conductors in an external electric field. The model considers that each electron in a conductor, which is situated in an external electric field, has both a thermal motion as well as a drift motion (Omar, 1993). The drift motion is caused by the electric field and is the directional movement of the electron in the opposite direction to the electric field. On the other hand, the thermal motion is random in essence, and it exists even in the absence of an external electric field. The two motions together bring the electron into a location with minimum potential energy, which is analogous to the process of searching for the optimal solution to an optimization problem. Our motivation of designing the RDPSO algorithm, based on this model, was to improve the search ability of the PSO algorithm by only modifying the update equation of the particle’s velocity, instead of by revising the algorithm based on the update equation of the canonical PSO, which would probably increase the complexity of the algorithm and its computational cost.

The basics of the original concept of the random drift model for PSO were sketched in our previous work (Sun et al., 2010). In the initial limited version of the algorithm, the velocity of the particle’s drift motion is simply expressed by the summation of the cognition part and the social part in the velocity update equation of the original PSO, which is not consistent with the physical meaning of the random drift model.

This paper is to propose a more concise form for the drift velocity, which is more in line with the physical meaning of the model, and a novel strategy for determining the random velocity, and thus a new version of the RDPSO algorithm. In order to gain an in-depth understanding of how the RDPSO works, we make comprehensive theoretical analyses of the behavior of the individual particle in the RDPSO and the search behavior of the algorithm. Four variants of the RDPSO are proposed based on different random velocities and neighborhood topologies. Comprehensive empirical studies on the RDPSO algorithm by using the CEC2005 benchmark suite are performed to verify the effectiveness of the algorithm.

To this end, the paper firstly describes the principle of the RDPSO and analyzes the behavior of the single particle in the RDPSO. The motivations of the RDPSO from a trajectory analysis point of view together with the free electron model are formulated in detail, and the random drift model for the RDPSO is presented. Based on this model, the velocity of the particle is assumed to be the superimposition of the random velocity component and the drift velocity component, which reflect the global search as well as the local search of the particle, respectively. The mathematical expressions of the two velocity components and, subsequently, the update equation for the particle’s velocity are given. After that, the conditions for the particle’s position to be probabilistically bounded are theoretically derived, and are later verified by stochastic simulations on the particle’s behavior.

Then, the search mechanism of the RDPSO algorithm is investigated. The effects of the thermal and drift motions on the particle’s search behavior are analyzed. The drift velocity component leads the particle to move toward its personal best position and the global best position as well, and thus it essentially implements the local search of the particle. The random velocity component makes the particle more volatile and its position is pulled or pushed away from the global best position, reflecting the global search of the particle. The interactions between the particles in the swarm are also analyzed in order to show that the RDPSO algorithm may provide a good balance between the global and the local search. Next, four RDPSO variants are proposed based on the combination of the two topologies (i.e., the global best model, and the ring neighborhood topology for the local best model) with two strategies for the random velocity components (i.e., one that uses the mean best position to determine the random velocity component, and the other one that employs randomly selected personal best position to compute the random velocity component).

Finally, empirical studies on the RDPSO algorithm are undertaken by using the CEC2005 benchmark suite. The issues of the parameter control and selection with respect to the thermal and drift coefficients of the algorithm are addressed by testing the algorithm with different parameter settings on three benchmarks.

Then, the parameter settings that are identified to result in good algorithmic performance are further tested and compared on the first twelve functions of the CEC2005 benchmark suite. For each RDPSO variant, the parameter settings that yield good overall algorithmic performance are found out. The RDPSO variants with the identified parameter configurations and some other PSO variants are tested by all the twenty five problems of the benchmark suite in order to make a thorough performance comparison and verify the The remainder of the paper is organized as follows. Section 2 gives a brief introduction to the PSO algorithm. Section 3 presents the motivation, the procedure and the analyses of the RDPSO algorithm as well as its variants. Empirical studies on the parameter selection for the RDPSO algorithm and the performance comparison are provided in Section 4. Finally, the paper is concluded in Section 5.

Particle Swarm Optimization

In a PSO with M individuals, each individual is treated as a volume-less particle in the N-dimensional space, with the current position vector and the velocity vector of particle i at the nth iteration represented as

=

, respectively. The particle moves according to the

C And

2c are known as the acceleration coefficients. The vector

=

is the best previous position (the position giving the best objective function value or fitness value) of particle i, called the personal best (pbest) position, and the vector

Is

the position of the best particle among all the particles in the population and called the global best (gbest) position. Without loss of generality, we consider the following minimization problem:

F

is an objective function and S is the feasible space. Accordingly,

Ir , Are Sequences

of two different random numbers distributed uniformly on (0, 1), which is denoted by

−

. The original PSO algorithm with equation (1) appears to have a weak local search ability. It should be noted that the tradeoff between the local search (exploitation) and the global search (exploration) is vital for the performance of the algorithm. Therefore, the original PSO needs to accelerate the convergence speed of the particles in order to achieve a better balance between exploitation and exploration. Work in this area, first carried out by Shi and Eberhart , involves introducing an inertia weight into equation (1), and the

, (5)

where w is the inertia weight. The PSO algorithm with equation (5) replacing equation (1) is known as the PSO with inertia weight (PSO-In). The inertia weight w can be a positive value chosen according to experience or from a linear or nonlinear function of the iteration number. When w is 1, the PSO-In is equivalent to the original PSO. The values of c1 and c2 in equation (5) are generally set to be 2 as originally recommended by Kennedy and Eberhart (1995), which implies that the ‘social’ and ‘cognition’ parts have the same influence on the velocity update.

Clerc (1999) proposed another acceleration method by adding a constriction factor in the velocity update equation (1) in order to ensure the convergence of the PSO without imposing any restriction on velocities, as given below.

, (6)

where the constant χ is known as the constriction factor and is determined by

(7)

This version of PSO is known as the PSO with constriction factor (PSO-Co). It was shown by Clerc and Kennedy (Clerc and Kennedy, 2002) that the swarm shows stable convergence if

, The

approach is very similar to the concept of the inertia weight with

Lerc And

Kennedy recommended a value of 4.1 for the sum of c1 and c2, which leads to

And C1=C2=2.05

(Clerc and Kennedy, 2002). These two versions of the PSO algorithm collectively referred to as the canonical PSO algorithms accelerate the convergence speed of the swarm effectively and have better performance than the original PSO in general. They have laid the foundation for further enhancement of the PSO. There are many other versions of the PSO algorithm as have been mentioned in Section 1, but most of them are based on these two versions.

3. Random Drift Particle Swarm Optimization (RDPSO)

The Motivation And Procedure Of Rdpso

In (Clerc and Kennedy, 2002), the trajectory analysis demonstrated that the convergence of the whole particle swarm may be achieved if each particle converges to its local focus,

(8)

In fact, as the particles are converging to their own local attractors, their current positions, pbest positions, local focuses and the gbest position are all converging to one point. This way, the canonical PSO algorithm

N

ip , is a random point uniformly distributed within the hyper-rectangle with

N

G being the two ends of its diagonal, the particle’s directional movement towards

Ip , Makes

the particle search around this hyper-rectangle and improves its fitness value locally. Hence, this directional movement essentially reflects the local search of the particle. In equation (1), (5) or (6), there are three parts on the right side. The last two ones are known as the ‘cognition’ part and the ‘social’ part, the superimposition of which results in the directional motion of the particle toward

Ip , . The First Part On The

right side of each equation is the ‘inertia part’, which may lead the particle to fly away from

N

and provide necessary momentum for the particle to search globally in the search space. The ‘inertia part’ is deterministic and reflects the global search of the particle. The motivation of the proposed RDPSO algorithm comes from the above trajectory analysis of the canonical PSO and the free electron model in metal conductors placed in an external electric field (Omar, 1993). According to this model, the movement of an electron is the superimposition of the thermal motion, which appears to be a random movement, and the drift motion (i.e., the directional motion) caused by the electric field. That is, the velocity of the electron can be expressed by

, Where Vr And Vd

are called the random velocity and the drift velocity, respectively. The random motion (i.e., the thermal motion) exists even in the absence of the external electric field, while the drift motion is a directional movement in the opposite direction of the external electric field. The overall physical effect of the electron’s movement is that the electron careens towards the location of the minimum potential energy. In a non-convex-shaped metal conductor in an external electric field, there may be many locations of local minimum potential energies, which the drift motion generated by the electric force may drive the electron to.

If the electron only had the drift motion, it might stick into a point of local minimum potential energy, just as a local optimization method converges to a local minimum of an optimization problem. The thermal motion can make the electron more volatile and, consequently, helps the electron to escape the trap of local minimum potential energy, just as a certain random search strategy is introduced into the local search technique to lead the algorithm to search globally. Therefore, the movement of the electron is a process of minimizing its potential energy. The goal of this process is essentially to find out the minimum solution of the minimization problem, with the position of the electron represented as a candidate solution and the potential energy function as the objective function of the problem.

Inspired by the above facts, we assume that the particle in the RDPSO behaves like an electron moving in a metal conductor in an external electric field. The movement of the particle is thus the superposition of the thermal and the drift motions, which implement the global search and the local search of the particle, respectively. The trajectory analysis, as described in the first paragraph of this subsection, indicates that, in the canonical PSO, the particle’s directional movement toward its local attractor

Ip , Reflects The Local

search of the particle. In the proposed RDPSO, the drift motion of the particle is also defined as the

N

ip , , which is the main inheritance of the RDPSO from the canonical PSO. However, in the RDPSO, the ‘inertia part’ in the velocity equation of the canonical PSO is replaced by the random velocity component. This is the main difference between the RDPSO and the canonical PSO. The thermal motion in the RDPSO is far different from the ‘inertia’ movement in the canonical PSO, although both of them have an identical functionality, namely, to implement the global search ability of the particle.

From a physical perspective, the particle in the canonical PSO is in a mechanical movement and its velocity and acceleration can be depicted by a set of deterministic dynamics equations. Consequently, it is natural that there should be an ‘inertia part’ in the velocity equation to reflect the change in the particle’s velocity with time. On the contrary, the thermal motion of the particle in the RDPSO is random in nature and can not be described by the dynamics equations used for mechanical movements. The only way of describing the thermal motion in statistical physics is by providing the probability distribution function of the particle’s velocity or momentum. It is unnecessary and impossible to depict the exact change of the particle’s velocity with time. Therefore, there is no ‘inertia part’ in the velocity equation of the RDPSO algorithm anymore.

From an algorithm design point of view, the role of the ‘inertia part’ in the global search is assumed by the random velocity component in the RDPSO, so that there is no need of an ‘inertia part’ in the RDPSO. Therefore, the velocity of the particle in the RDPSO algorithm has two components, i.e., the thermal component and the drift component. Mathematically, the velocity of particle i in the jth dimension can be

, + Are The Random

velocity component and the drift velocity component, respectively. A further assumption is that the value of the random velocity component

R

, + essentially follows a normal distribution (i.e., Gaussian distribution) whose probability density function is given by

Σ

is the standard deviation of the distribution. Using stochastic simulation, we can express

Φ

is a random number with a standard normal distribution, i.e.,

=

is known as the mean best (mbest) position defined by the mean of the pbest

Α

is an algorithmic parameter called the thermal coefficient.

I

, + , its role is to achieve the local search of the particle. As has been mentioned above, the directional movement toward

I

, + in (Sun et al., 2010) is just the combination of the ‘cognitive part’ and the ‘social part’

+

is a random number, its effect is to make the movement of the particle randomized, which is not consistent with the free electron model. Therefore, in this paper we modify it to be

Β

is a deterministic constant and is another algorithmic parameter called the drift coefficient. Equation (14) has a clear physical meaning that it reflects the particle’s directional movement towards

N

ip , . In Theorem A1 in the Appendix, it is proven that, if there is only drift motion and, i.e.,

, + In Equation (14) Can Indeed

guarantee the particle’s directional movement toward

N

ip , in oscillation and thus the sampling space of

N

G , where points with better fitness values may exist. As such, when we select the value of β for real application of the RDPSO algorithm, it may be desirable to set

For Good Local Search

ability of the particles. With the above specification, a novel set of update equations can be obtained for the particle of the

(16)

The procedure of the algorithm is outlined below. Like in the canonical PSO, the value of

Begin

Initialize the current positions and velocities of all the particles randomly; Set the personal best position of each particle to be its current position;

Ynamical Behavior Of The Rdpso Particle

An analysis of the behavior of an individual particle in the RDPSO is very essential to understanding how the RDPSO algorithm works and how to select the algorithmic parameters. Since the particle’s velocity is the superimposition of the thermal velocity and the drift velocity, the conditions for the particle’s position to converge or to be bounded are far more complex than those given in subsection 3.1 when only the drift motion exists. In this subsection, we undertake theoretical and empirical studies on the stochastic dynamical behavior of the particle in the RDPSO. Since each dimension of the particle’s position is updated independently, we only need to consider a single particle in the one-dimensional space without loss of generality. As such, equations (15) and (16) can be simplified as

N

V denote the current position and the velocity of the particle, respectively, and the local focus of the particle and the mean best position are denoted by p and C , which are treated as

Φ

is a sequence of independent identically distributed random variables with

N

ϕ is symmetrical with respect to the ordinate, equation (17) has the

N

in equations (17) and (19) are the same. Based on equations (19) and (18), several theorems on the dynamical behavior of a single particle in RDPSO are proved in the Appendix. As shown by Theorem A2, the particle’s behavior is related to the convergence of

Λ

, namely, the values of α and β satisfy the following

N

ρ is probabilistically bounded and, thus, the position of the particle is probabilistically bounded too. In inequality (20), the value of Δ is an improper integral which is undefined at

By Dirichlet Test, This

improper integral is convergent if both α and β are two finite numbers (Courant, 1989). Inequality (20) does not provide any explicit constraint relation between α and β due to the difficulty in calculating the improper integral in the inequality. A sufficient condition for

Ρ

) is derived in Theorems A4. It says that if the values of α and β are subject to

Ρ

converges to zero, which consequently ensures the probabilistic boundedness of the particle’s position as shown. Figure 1 visualizes some simulation results on the stochastic behaviour of the particle by using different values of α and β , with C fixed at

=

. Figures 1 (a) to (c) show the results with α and β satisfying constraint (21). It can be observed that the particle’s position oscillated around p and C, implying that the position is probabilistically bounded in these cases. Figures 1 (d) to (i) show that the particle’s position is probabilistically bounded in some cases when α and β do not satisfy constraint (21). This verifies that constraint (21) is a sufficient condition for

N −

reached 700 and stopped changing after a certain number of iterations, as shown in Figures 1 (j) to (o). In such cases, the value of

Reaches The Maximum

positive value that the computer can identify, so that it can be considered to have diverged to infinity. Constraint (21) is of practical significance to the application of the RDPSO algorithm, although it does

Δ

. In practice, the values of α and β can generally be selected within the intervals given by (21), for a satisfactory algorithmic performance when the algorithm is applied is undertaken by using a set of benchmark functions from the CEC2005 benchmark suite.

(O)

Figure 1 The figure visualizes the simulation results for the behavior of the particle at different values of α and β . Figures (a) to (c) show that when the values of α and β are selected within the intervals

, The

particle’s position is probabilistically bounded. Figures (d) to (i) show that the particle’s position may be also probabilitcally bounded at some values of α and β not satisfying constraint (21). Figures (j) to (o) show some cases that when α and β do not satisfy constraint (21),

The Rdpso’S Search Behavior

In the above analysis, it is assumed that each particle in the RDPSO updates its velocity and position independently, with the mean best position C and the local focus p being treated as independent probabilistically bounded random variables, and thus it is revealed that the behavior of the particle is related

N

ρ . However, the actual situation is more complex when the RDPSO algorithm is running in a real-world landscape. During the search process of the RDPSO algorithm,

Ip , , Which Can Not Be Treated As

independent random variables anymore, but are relevant to the other particles. As for

, It Is The Mean Of The

pbest positions of all the particles, moving with the variation of each pbest position. The local focus

Ip , , Is A

random point associated with the pbest position of particle i (

G That Rotates

among the pbest positions of the member particles according to their fitness values. In contrast to

N

C averages the changes of all the pbest positions.

Φ

Generally, the pbest positions of all the particles converge to a single point when the RDPSO algorithm is performing an optimization task, which implies that

As Mentioned In The Proof Of

Theorem A1. Referring to equations (A7) to (A10), we can infer that if and only if

P

. That means the current positions and the pbest positions of all the particles converge

Δ

. It can also be found from Theorems A2 and A3 that, when

, The Particle’S

position is probabilistically bounded and oscillates around but does not converge to

Δ

, it is shown by Theorems A2 and A3 that the particle’s current position diverges and the explosion of the whole particle swarm happens.

Iϕ

In practical applications, it is always expected that the particle swarm in the RDPSO algorithm can converge to a single point, like that in the canonical PSO. Essentially, there are two movement trends, i.e. the random motion and the drift motion, for each particle in the RDPSO, as has been described in the motivation of the algorithm. These two motions reflect the global search and the local search, respectively. The drift

I

, + in the velocity update equation (15), draws the particle towards the local focus and makes the particle search in the vicinity of the gbest position and its pbest position so that the particle’s current and pbest positions can constantly come close to the gbest position. On the other hand, the

R

, + results in a random motion, leading the particle to be so volatile that its current position may reach a point far from the gbest position and its pbest position. This component can certainly provide the particle a global search ability, which, in the canonical PSO algorithm, is given by the velocity at the last iteration, i.e.

Iv

, + . Nevertheless, an important characteristic distinguishing the RDPSO from other randomized PSO methods is that the random component of the particle’s velocity uses an adaptive standard deviation for its distribution, i.e.

Α

. Such a random component makes the random motion of the particle have a certain orientation. The effect of

R

, + is to pull or push the particle away from the gbest

N

C as shown by Figure 2, not only to displace the particle randomly as the mutation operation

On The Direction Of The Particle’S Motion Is

opposite to that in Figure 2(a). Generally speaking, the longer the distance

I

, + will be away from the gbest position. If the particle’s position is close to the gbest position, the random component can help the particle escape the gbest position easily, when the gbest position is stuck into a local optimal solution. As far as the whole particle swarm is concerned, the overall effect is that the RDPSO has a better balance between the global search and the local search, as illustrated below.

N

C is shifted toward the lagged particles and thus far from the particles clustering

N

G . The particles are pulled or pushed away from the neighbourhood of

G And Would Search The Landscape

globally. In the RDPSO method, the swarm could not gather around the gbest position without waiting for the lagged particles. Figure 3 depicts the concept where the pbest positions of several particles, known as the lagged particles, are located far away from the rest of the particles and the gbest position

G , While The Rest

of the particles are nearer to the global best position, with their pbest positions located within a neighbourhood of the gbest position. The mbest position

Would Be Shifted Towards The Lagged Particles

and be located outside the neighbourhood. When the lagged particles are chasing after their colleagues, that

N

G slowly. The current positions of the particles within the neighbourhood would be pulled or pushed outside the neighbourhood by

N

C is careening toward the neighbourhood, the exploration scope of the particle is becoming narrower. After the lagged particles move into the neighbourhood of the gbest position,

Also Enter The

neighbourhood and the particles would perform the same search process based on a smaller neighbourhood of the gbest position. In the canonical PSO, each particle converges to the gbest position independently and has less opportunity to escape from the neighbourhood of the gbest position. When the speed of the particle is small, it is impossible for the particles within the neighbourhood to jump out of the neighbourhood. As a result, these particles would perform local search around the gbest position and only the lagged particles could search globally. Evident from the above analysis, the RDPSO algorithm generally has a better balance between exploration and exploitation than the canonical PSO.

Agged Particles

Moreover, different from mutation operations that play minor roles in some variants of PSO and evolutionary algorithms, the random motion has an equally important role as the drift motion in the RDPSO.

N

C , the RDPSO achieves a good balance between the local and global searches during the search process. By the influences of both

And Their Local Focuses, The

particles in the RDPSO have two movement trends, convergence and divergence, but the overall effect is their convergence to a common point of all the particles if

The Convergence Rate Of The Algorithm

depends on the values of α and β , which can be tuned to balance the local and global search, when the algorithm is used for a practical problem.

Ariants Of Rdpso

In order to investigate the RDPSO in depth, some variants of the algorithm are proposed in this paper. Two methods are used for determining the random component of the velocity. One employs equation (13) for this component and the other replaces the mbest position in (13) by the pbest position of a randomly selected particle in the population at each iteration. For convenience, we denote the randomly selected pbest

N

C′ . For each particle, the probability for its pbest position to be selected as

, The Current Position Of Each Particle At Each

iteration shows to be more volatile than that of the particle with equation (13), which diversifies the particle swarm and in turn enhances the global search ability of the algorithm. In addition to the global best model, the local best model is also examined for the RDPSO. The ring topology is a widely used neighborhood topology for the local best model (Li, 2010), in which each particle connects exactly to two neighbors. The standard PSO (SPSO) in (Bratton and Kennedy, 2007) is defined by the integration of the PSO-Co with the ring topology. Although there are various neighborhood topologies, we chose the ring topology for the RDPSO with the local best model. Thus, the combination of the two topologies with the two strategies for the random velocity component produces the four resulting RDPSO

Variations:

RDPSO-Gbest: The RDPSO algorithm with the global best model and the random velocity component described by equation (13). RDPSO-Gbest-RP: The RDPSO algorithm using the global best model and employing a randomly selected pbest position to determine the random velocity component.

RDPSO-Lbest: The RDPSO algorithm with the ring neighborhood topology and the random velocity component in (13), where, however, the mbest position is the mean of the pbest positions of the neighbors of each particle and the particle itself, instead of the mean of the pbest positions of all the particles in the population.

RDPSO-Lbest-RP: The RDPSO algorithm using the ring neighborhood topology and employing the pbest position of a particle randomly selected from the neighbors of each particle and the particle itself.

Benchmark Problems

The previous analysis of the RDPSO provides us with a deep insight into the mechanism of the algorithm. However, it is not sufficient to evaluate the effectiveness of the algorithm without comparing it twenty five functions from the CEC2005 benchmark suite (Suganthan, 2005) were employed for this purpose. Functions F1 to F5 are unimodal, functions F6 to F12 are multi-modal, F13 and F14 are two expanded functions, and F15 to F25 are hybrid functions. The dimension N was chosen as 30 for each of these functions. The mathematical expressions and properties of the functions are described in detail in (Suganthan, 2005). The codes in Matlab, C and Java for the functions could be found at http://www.ntu.edu.sg/home/EPNSugan/.

4.2. Empirical Studies on the Parameter Selection of the RDPSO Variants Parameter selection is the major concern when a stochastic optimization algorithm is being employed to solve a given problem. For the RDPSO, the algorithmic parameters include the population size, the maximum number of iterations, the thermal coefficient α and the drift coefficient β . Like in the canonical PSO, the population size in the RDPSO is recommended to be set from 20 to 100. The selection of the maximum number of iterations depends on the problem to be solved. In the canonical PSO, the acceleration coefficients and the inertia weight (or the constriction factor) have been studied extensively and in depth since these parameters are very important for the convergence of the algorithm. For the RDPSO algorithm, α and β play the same roles as the inertia weight and the acceleration coefficients for the canonical PSO.

In Section 3, it was shown that it is sufficient to set α and β according to (21), such that

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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