Salvador Mascarenhas

Review of Mercier and Sperber's The Enigma of Reason (preprint of article to appear in Teorema)

Mercier and Sperber argue very convincingly that the traditional intellectualist approach is altogether inadequate to explain the workings of deliberate reasoning about inferences. However, they indict an entire field of inquiry and whole classes of theories of human inferential behavior on this charge, when in fact many of these theories were primarily designed to account for intuitive, non-deliberative, first-order inference making. This tension can be resolved in a constructive way that can propel the field forward into an appropriately ambitious psychology of reasoning: non-interactionist theories of reasoning need to scour their extant empirical coverage for aspects of phenomena that might best be seen as instances of reasoning about reasons, and address those within frameworks that provide satisfactory answers to the challenges raised by Mercier and Sperber. In the process, one's non-interactionist theory of reasoning itself can only become clearer, and will possibly acquire greater explanatory adequacy.

The enigma of reasons explained

In their (2017) book "The Enigma of Reason," Mercier and Sperber (M&S) present a view of reason and a program for its study that addresses a number of old and new challenges. Center stage is a careful discussion of the failures and failings of the traditional approach to reasoning, dubbed the intellectualist view by the authors, which holds that the purpose of the human capacity for reasoning is to track truth and to allow for good decision making. Mercier and Sperber show that this view is ill equipped to provide an explanatory account of human reason given what we know about the failures and successes in how humans deploy and evaluate reasons. If reason aims at truth, why are we prey to so many biases, why do we confabulate reasons so promptly, why do we make so many mistakes when reasoning by ourselves? And what of the successes of reason, how does the goal of truth tracking explain that, in social contexts, our capacity for using and evaluating reasons in argumentation so often leads us to good or goodenough decisions, or allows for coordinated action on subjects of great complexity? In response to these challenges, and a host of others, Mercier and Sperber propose an interactionist view of reason: the purpose of this human faculty is to produce reasons to justify oneself, and to produce arguments to convince others.

In what follows, I argue that Mercier and Sperber's work presents a compelling view of a particular phenomenon that can be called "reason" or "reasoning," namely the production and evaluation of reasons. But the psychology of reasoning has at least for the past several decades been about far more than reasons, and M&S's sweeping criticism of this work is not entirely justified. In fact, the traditional approach is a valid way to investigate another human phenomenon that can intelligibly be called "reason" and "reasoning." And even if it is not, the interactionist approach fares no better as a foundation for the study of this other kind of reasoning. I conclude that, once we evaluate the theories in the field under this light, M&S's approach is an important advance toward a full understanding of human reasoning, it carves out a chunk of the phenomenon and shows that it is significantly different from the rest, and it lays the foundations for how to understand it. But this work does not offer a broad indictment of the research on reasoning that precedes it.

Inferences, reasons, and inferences about reasons

In my view, Mercier and Sperber's critical outlook on extant theories of reasoning is more negative than what is warranted. Mercier and Sperber are happy to grant that what they call the intellectualist approach has produced a wealth of empirical results on failures and successes of reasoning. After all, a carefully mapped empirical landscape can be of great use to science, even if its cartographers were theoretically misguided.

More importantly though, extant approaches have produced a wealth of models of those successes and failures of reasoning. The empirical scope, theoretic insight, and formal rigor of those models vary greatly, from spectacularly sophisticated to modest on all three fronts. But before we scratch the past fifty years of modeling proposals about human reasoning, it is worth checking that all we're left with after salvaging the empirical discoveries is indeed bathwater.

First and higher order inferences

Mercier and Sperber point out that, while reason might well be an exclusively human faculty, its super-category inference making certainly is not. Indeed, humans and other animals make inferences all the time, say about what to eat, whom to mate with, when to attack, when to flee. These inferences seem to be the product of specialized modules rather than of some general-purpose faculty.1 I will call these kinds of inferences firstorder inferences. They contrast with reason as M&S propose we see it: a domain-general faculty that operates on representations of inferences to produce reasons connected to those inferences. Reasons in turn are themselves representational entities that explain and justify inferences (and presumably all sorts of other things, like desires and actions).

In this discussion, it will be useful to call the product of reason in M&S's sense higherorder inferences. One does this not without some abuse of terminology, for they are inferences about representations of inferences. But there is a gain in perspicuity: this terminology fulfills M&S's recommendation not to see the reason module as something qualitatively distinct from the lower inferential modules, while pinpointing the difference in levels between the two kinds of inferences. Both first-order inferences and higherorder inferences are the result of modules that deliver intuitive inferences, leading us to expect that they should share some psychological signatures. But we also expect them to differ in interesting ways due to the fact that the principal objects they operate on are of different kinds: precepts and knowledge of the world vs. representations of inferences.

Mercier and Sperber's central criticism of the state of the art in the study of reasoning can be articulated as follows. Most if not all extant theories of reasoning hold, be it overtly or tacitly, that the functional aim of the higher-order inference module is to track truth closely and to achieve better decisions. Yet this idea is at odds with over fifty years of research showing the fallibility of this higher-order inference module at guaranteeing truth tracking and optimal decision making. When humans introspect about reasons, they often end up with arguments that violate fundamental laws of logic. Moreover, humans confabulate reasons that often have little or nothing to do with the actual causes of their lower-level inferential behavior. This is baffling from a view that holds that the capacity to draw higher-order inferences is about truth tracking and good decision making, but it becomes natural and clear once one moves to the idea of social interaction as the functional aim of higher-order inferences.

Stated in this way, I am persuaded by the arguments put forth in the book in favor of this thesis.

The object of study of reasoning

The issue is that it is by no means obvious just how many (and which ones) of the extant theories of reasoning are about higher-order inferences. In fact, it is clear that many of them, including the most well-known ones, have instead focused on particular subclasses of first-order inferences. Consider the famous bat-and-the-ball problem, part of the cognitive-reflection test [START_REF] Frederick | Cognitive reflection and decision making[END_REF].

A bat and a ball cost $1.10 together. The bat costs $1.00 more than the ball. How much does the ball cost?

Many people answer that the ball costs $0.10, when in fact $0.05 is the only solution to this simple system of equations. This is a striking result, but what exactly makes this problem and others like it informative and insightful from a theoretical standpoint? Sure enough, interesting things happen when we tell participants in this experiment that in fact ten cents is not the right answer. Subjects might engage in what looks like mathematical reasoning: well the bat is one-ten and it's one dollar more than the ball, one-ten minus one equals ten cents. Then, with a bit of prodding, and in particular when confronted with the fact that 1.1 + 0.1 = 1.2, subjects will often see the light and correct their answer. There are good questions about higher-order inferences to be asked here, and M&S's approach to reason offers a promising framework in which to pursue those questions. But there is also something else going on.

You may not have uttered the incorrect response the first time you were presented with this problem. But almost certainly you had the experience of hearing an insistent voice in your head whispering "stop thinking, this is very easy, the answer is ten cents." As far as you can tell, this voice addressed you prior to any reasoning about the reasons why the answer provided by the voice was right. This first-order inference is also a worthy object of study. Presumably it comes from one of those intuitive modules that deliver low-level inferences, and perhaps M&S are right that to call that "reasoning" is not the most vernacular or conceptually tidy use of the English word. What is clear is that a good chunk of the field has been focusing on precisely these inferences, has been happily calling them "reasoning," and has been developing complex models to describe their workings, the best of which have strong predictive power or theoretic insight.

Two examples: the conjunction fallacy and the Wason selection task

The conjunction fallacy [START_REF] Tversky | Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment[END_REF] offers an even more instructive example, because we have a large array of competing theories of it. 2 At around 85% rate of conjunctive responses in the original studies, the conjunction fallacy looks a lot like the bat-and-the-ball problem, in that participants feel a strong attraction toward the purportedly fallacious answer. 3 As far as I know, there is good reason to think that the intuitive pull of the conjunctive option precedes any inferences experimental subjects may make about the reasons they have for picking that option. Accordingly, Tversky and Kahneman's account of the phenomenon in terms of representativeness is an account about an entirely unconscious and automatic process that does not require deliberation. Specifically, subjects substitute a question about representativeness (typicality in this case) for a question about probabilities. Notice that nothing in this account is in any way 2 I agree with M&S that it is suspicious that after more than 50 years of research on reasoning we've achieved so little in way of consensus, a good example of which is the conjunction fallacy. But I believe that the right diagnosis involves much more than just a confusion about first order and higher-order inferences. In my view, the lack of systematic and appropriately sophisticated research on the interplay between interpretation and reasoning has introduced a host of confounds into the field, which we are only now addressing with the appropriate linguistic tools (Mascarenhas 2014).

3 I say "purportedly" as a nod to the tradition in the psychology of reasoning, behavioral economics, and more recently linguistics of seeking absolving interpretations of the conjunction fallacy that dispel it as an experimental artifact. See for example the work of [START_REF] Hertwig | The conjunction fallacy revisited: How intelligent inferences look like reasoning errors[END_REF] or [START_REF] Dulany | Conversational implicature, conscious representation, and the conjunction fallacy[END_REF].

about attaching reasons to representations of inferences. Tversky and Kahneman's story is about the under-the-hood process that delivers the intuitive and attractive first-order response, irrespective of what experimental subjects might want to tell us about the reasons they think they had for picking it. Now, M&S can of course say that Tversky and Kahneman were looking at something far less interesting than what M&S call reason. But it makes little sense to discard their theory simply on the grounds that if it is not about higher-order inferences then a theory cannot use the English word "reason" and its morphological cousins when it tells us what it is meant to be about.

Consider now the Wason selection task [START_REF] Wason | Reasoning about a rule[END_REF]) as an example where first-order inferences play a more dubious role. Unlike the conjunction fallacy, typical answers to the Wason selection task in its original formulation cover an appreciably wider spectrum, with the most popular answer (turn around antecedent-verifying and consequent-verifying cards, modulo negations) corresponding to about half of responses. Using an old fashioned but often useful tool, introspection suggests to me that I have no immediate pull toward any answers in the Wason selection task. In fact, I feel puzzlement and frustration at the question, and then start engaging in overt deliberation. The fallacious response, it seems to me, comes alongside the higher-order reasoning I engage in.

The usefulness of the old ways

Mercier and Sperber argue convincingly that the functional aim of what they call "reason" and "reasoning" is not to arrive at good decisions given a particular notion of what is ecologically relevant, but rather to serve as a tool for social interaction. 4 But the old-fashioned dogma is a perfectly plausible hypothesis for the functional aim of the low-level intuitive inferential modules whose existence M&S happily grant.

Indeed, once we look at the arguments against the intellectualist approach through this lens, it becomes clear that they are nowhere near as strong when applied exclusively to first-order, intuitive inferences. Interactionist considerations that are compelling explanations of logically puzzling behavior in argumentation are of little use in the realm of low-level inferences. How is, say, the availability heuristic of [START_REF] Tversky | Judgment under uncertainty: Heuristics and biases[END_REF] explained in terms of its usefulness in social interaction? 5 The traditional view that these kinds of processes aim at truth tracking and good decision making certainly still has important challenges to answer, but it is clear that those challenges are not dispelled by an interactionist approach. 6Other elements of M&S's arguments in favor of interactionism are less easy to apply to first-order inferences. Higher-order reasoning improves in dialogical contexts, and better decisions can be achieved in such contexts, under certain conditions. But have we reason to think that first-order intuitions themselves improve in dialogical situations? If you solve the bat-and-the-ball problem in conversation with your friends you are more likely to find the solution, but does something deep happen to the little voice in your head that whispers "ten cents"? Is it silenced in a way that is interestingly predicated on the dialogical activity? These questions deserve study of course, but it is as of now entirely plausible to think that they are answered in the negative.

Synthesis

Mercier and Sperber's bleak outlook on state-of-the-art research on reasoning is justified only insofar as the state of the art is meant to be about what M&S call "reason" and "reasoning." For example, work from the heuristics and biases paradigm on the conjunction fallacy is very plausibly not about M&S's "reason" and "reasoning" (higher-order inferences), but about intuitive (first order) inferences. As far as I can tell, M&S do not provide arguments impugning research on intuitive inferences from a non-interactionist perspective, and consequently a sizable portion of the existing work on reasoning, conceived in this strict way, is in principle perfectly compatible with M&S's framework. It could well be that the first-order inferential module responsible for the attractiveness of the conjunctive option in the conjunction fallacy is indeed aiming at a judgment of typicality, as Tversky and Kahneman argue. And this substitution of questions is a plausible adaptation under the functional goal of delivering good-enough decisions within actionable time frames. I propose we do not think of M&S's work as an alternative to the past 50 years of theoretical research on reasoning. Instead, let us see it as an important and overdue piece of scholarship that zooms out of low-level accounts of particular classes of fallacies to offer a framework in which every extant theory of reasoning should now situate itself. In particular, the lack of clarity in the field so far between first and higher-order inferences is real and has certainly been pernicious. Thanks to M&S's work, we now see that, despite all the properties the two kinds of inferences share, a unified account in terms of their functional aims is not forthcoming: higher-order inferences most plausibly exist for social interaction, first-order inferences for decision making.

It is interesting in this connection to consider early work on dual-process approaches to reasoning. In particular, [START_REF] Wason | Dual processes in reasoning?[END_REF] outline a framework for the study of reasoning that recognizes two kinds of processes: (1) "the processes underlying the and Chater (2007) argue that in fact our intuitive responses are far more rational than they seem. Philipp Koralus and I toss another brand of fuel into the flame in our work on the erotetics of reasoning [START_REF] Koralus | The erotetic theory of reasoning: Bridges between formal semantics and the psychology of deductive inference[END_REF]: the contents of humans' propositional thoughts are structured in singular or multiple alternatives, a way of recruiting attention in reasoning; but representing multiple alternatives is costly, so we sometimes hastily discard alternatives that evidence suggests are less relevant, producing illusory inferences.

reasoning performance," and (2) "introspective accounts of [reasoning] performance."7 These two levels correspond quite neatly to the distinction M&S and I make between first order and higher-order inferences. Moreover, there is good reason to think that Wason and Evans consider both processes to be legitimate objects of study for a psychology of reasoning. After all, the authors argue there and elsewhere for the existence of a matching bias in the Wason selection task, whereby reasoners solve the task partly by looking for cards that match the cards mentioned in the conditional sentence. Since they give precisely this matching bias as an example of a type 1 process, we can only conclude that their account of the Wason selection task involves an account in terms of first-order inferences. Thus, the view that I am suggesting here, where both processes are worthy of study and need to be incorporated into a larger view of reasoning, is by no means novel.

Importantly, this perspective has concrete consequences for how the field should proceed. For one thing, we need to find ways of operationalizing the question of whether a particular class of inferential behavior falls under first-order or higher-order reasoning. If one's theory makes crucial appeal say to rational analysis in purely non-interactionist settings, one must either focus on first-order reasoning or respond satisfactorily to the challenges raised in M&S's work. These questions will be extremely complex in many cases. In the Wason selection task for example, it may well be that we find a conspiracy of low-level processes such as matching bias and high-level reasoning about reasons, which will lead us to error in non-dialogical situations.

We also need to investigate the properties of these low-level inferential modules. Are there several such modules? Do they ever pull in different directions and if so how are disagreements resolved? Is a unified account of their properties at all possible? Since they do not depend on human-specific meta-representational abilities, we expect to find some of these low-level inferential modules in our closest relatives in the animal kingdom. Can those investigations lead to a better evolutionary account of this component of human reasoning?

In conclusion, M&S's thought-provoking book can be seen as offering a broader, conceptually tidier, and more ambitious framework for thinking about human reasoning than the field has seen so far. But there is no good argument in it to reject non-interactionist approaches as a matter of principle, they are in fact very plausibly the right way to approach first-order inferences. A true synthesis of the two lines of research will require work from interactionists and intellectualists alike, but the potential for advance in the field is impressive. Who knows just to what extent otherwise excellent theories of firstorder inferences have been led astray by trying to account for instances of higher-order reasoning in the same fashion?

Like M&S, I use the term "module" to mean roughly "cognitive function."

There is to my mind an unresolved tension between this position and the authors' observations about epistemic vigilance. Reasoning exists for social interaction, yet this is less transparent in the case of hearers than in the case of speakers. Addressees of argumentative discourse have a strong interest in exercising epistemic vigilance, but it is unclear whether and how M&S expect this to fall from the general view of reasoning as aiming at social interactions.

To use the availability heuristic is to answer questions about the frequency of an event in terms of how easily past instances of that event present themselves to one's mind.

Moreover, understanding the truth-tracking failures of a system that aims at truth tracking has always been a crucial goal of the traditional perspective. The heuristics and biases program tells us that we evolved imperfect but approximative strategies meant to give us responses within actionable time frames. Oaksford

Mercier and Sperber discuss Wason and Evans's work in chapter 2, but they do so from their perspective that only the introspective accounts of performance are "reasoning proper." This is precisely the unwarranted move I criticize in this review.

* The author's work is funded by Agence Nationale de Recherche's grants ANR-17-EURE-0017 and ANR-18-CE28-0008.


domain: phys spi

text:
Clement Le Touze
email: clement.letouze@onera.fr

Nicolas Rutard
email: nicolas.rutard@onera.fr

NUMERICAL METHODS FOR DIFFUSE INTERFACE MULTIFLUID MODELS

Keywords: Multiphase Flows, Diffuse Interface Multifluid Models, Numerical Methods

We describe some numerical methods developed in ONERA's Cedre platform to solve diffuse interface multifluid models with a view to realistic industrial applications. These methods are illustrated on test cases such as a shock-droplet interaction case.

INTRODUCTION

Multiphase flows are involved in many engineering applications and a large variety of scientific fields. For example, they are encountered in the aerospace industry within all kinds of reaction engines involving liquid reactants injected in the combustion chamber. The use of numerical simulations for design and optimization purposes is increasingly developing, and it is therefore crucial to develop more and more reliable and efficient numerical platforms. To simulate complex multiphase flows in realistic industrial applications, diffuse interface multifluid models are the best suited. In this family, the most general model is the 7-equation model [1,2] allowing the fluids to be locally in full desequilibrium. From this model, it is possible to derive a hierarchy of models by successive relaxations considering infinitely small relaxation times. This gives rise to some well-known models such as the 4-equation model [3,4], the 5-equation model [5], or the 6-equation model with immediate pressure relaxation which has gained much attention in the past decade [6][7][8][9].

The implementation of multifluid models in Cedre [10], the ONERA's CFD platform dedicated to industrial multiphysics applications (aerodynamics and energetics, heat and radiative transfers, turbulent combustion, two-phase flows), is a long-standing objective for ONERA. The challenge is twofold: first to equip these models with the physical submodels required to achieve a sufficient level of precision, which means e.g. including a proper modelling for fluid thermodynamics and phase change, turbulence or combustion. Second, to develop efficient, robust and accurate numerical methods to deal with the challenging conditions involved: complex geometries and general unstructured meshes, strong gradients of physical quantities, stiff relaxation and source terms, thermodynamics of phase change, interaction with turbulence and combustion, wide range of Mach numbers and large variety of spatial and temporal scales... This paper focuses on the second topic regarding the numerical methods. A compressive technique to reduce the numerical diffusion of the interface is implemented in the framework of multislope MUSCL methods. In order to deal with the whole range of Mach numbers, low-Mach corrections are also implemented in HLLC Riemann solvers. Using the 4-equation diffuse interface model in Cedre, these methods are illustrated on different test cases such as a shock-droplet interaction case. 1 Clément Le Touze and Nicolas Rutard

THE 4-EQUATION DIFFUSE INTERFACE MODEL 2.1 Governing equations

The 4-equation diffuse interface model is very similar to the compressible Navier-Stokes equations for a multi-species homogeneous mixture, also referred to as the homogeneous relaxation model when mass transfer between phases is considered [11]. The system of conservation equations can be written in vector form as [3]:

∂ t Q + ∇. (f C + f D + f σ ) = S , (1)
where Q = (ρY j,k ρu ρE) t is the vector of the conservative variables, comprising species partial densities (j is the index for species within the fluid k), momentum and total energy of the mixture, and ρ is the density of the whole mixture. From Q, one can derive a set of primitive variables U = (Y j,k u P T ) t , i.e. the species mass fractions with respect to the mixture, plus the mixture mean velocity, pressure and temperature. The convective fluxes f C are that of the standard compressible Euler equations, the diffusive fluxes f D may include laminar and turbulent diffusion (either in a RANS or LES context), f σ stands for surface tension (CSS method), while S comprises the source terms to be taken into account depending on the configuration (e.g. body forces, chemical reactions or coupling with another solver). The thermodynamic closure of the system is based on the assumption of thermal and mechanical equilibrium between phases [11], together with dedicated equations of state for each phase (see section 2.2 below). As a result, the liquid phase volume fraction α l is not an independent transported variable of system ( 1), but can be straightforwardly obtained from the liquid mass fraction Y l and the densities of pure fluids ρ g and ρ l as

α l = ρY l /ρ l = ρ g Y l / [ρ g Y l + ρ l (1 -Y l )].

Equations of state

A fluid made up of N g gaseous species will be modeled most of the time as an ideal gas mixture, leading to

ρ = P RT Ng i=1 Y i /W i -1 , (2)
with R the ideal gas constant and W i the molar mass of a given gaseous species i. For a fluid made up of one liquid species, our usual choice is to use a linearized version of the Isentropic Stiffened Gas EOS (ISG, also referred to as the Tait's law). Starting from the ISG law:

P + P ∞ P 0 + P ∞ = ρ ρ 0 γ ⇔ ρ = ρ 0 [1 + γβ 0 (P -P 0 )] 1/γ , (3)
where P ∞ is the Stiffened Gas constant of the liquid, β 0 = [γ(P 0 + P ∞ )] -1 the isothermal compressibility at a given reference state (ρ 0 , P 0 ) and γ the isentropic exponent. Then, the linearization of this law under the assumption β 0 (P -P 0 ) ≪ 1 (see [12]) leads to:

ρ = ρ 0 [1 + β 0 (P -P 0 )] ⇔ P = P 0 + c 2 0 (ρ -ρ 0 ) , (4)
where c 0 = (β 0 ρ 0 ) -1/2 is the constant sound speed of the liquid. This is a barotropic (isothermal) linearized equation of state, which is accurate in the limit of small variations around the reference state (ρ 0 , P 0 ), and restricted to the pure liquid area of the phase diagram.

The numerical discretization in Cedre is based on the finite volume method on general unstructured meshes. In this paper, we will focus on the discretization of convective fluxes, and in particular on the two following topics: a compressive limitation technique in the framework of the multislope MUSCL method to reduce the numerical diffusion of the interface in multifluid simulations, and low-Mach corrections for HLLC-type Riemann solvers.

The multislope MUSCL method for general unstructured meshes

To briefly introduce the multislope MUSCL method for general unstructured meshes (see further details and a visual representation on figure 1 in [13]), let us consider a domain Ω made up of general polyhedral elements. For each element

K i ∈ Ω, V i = {K j |K i ∩ K j = S ij
} is the set of its neighbor cells sharing a common face S ij . For each face S ij ∈ V i , we seek two pairs of two (in 2D) or three (in 3D) vertex-neighbor cells of K i as close as possible to the axis connecting B i and M ij (the barycenters of K i and S ij respectively), so that we can compute backward and forward points H - ij and H + ij . Then second-order consistent approximations of any reconstructed variable W can be computed at these points using barycentric interpolations, namely W H - ij and W H + ij . The procedure is then very close to the classical MUSCL approach in one dimension. Both a backward and a forward slope are computed, and then used to compute the interpolation W ij at the face barycenter:

p - ij = W i -W H - ij ∥B i H - ij ∥ , p + ij = W H + ij -W i ∥B i H + ij ∥ , W ij = W i + p + ij φ r ij , η - ij , η + ij ∥B i M ij ∥ , (5)
where φ is a non-linear limiter function whose purpose is to provide the second-order accuracy in smooth regions while preserving the monotonicity near discontinuities. For the scheme stability (a proof of which is provided for the linear advection equation in [13]), this limiter function has to be expressed in terms of the slope ratio

r ij = p - ij /p + ij but also in terms of geometrical parameters η - ij = ∥B i H - ij ∥/∥B i M ij ∥ and η + ij = ∥B i H + ij ∥/∥B i M ij ∥
. This is the standard procedure for the interpolation of a given variable W . But in practice, when solving a particular system of equations, we also need to specify the set of variables on which to perform the interpolations. This is not a trivial issue and may have an impact on accuracy and robustness. For instance, it is well-known than interpolating primitive rather than conservative variables is a safer choice. Another issue of the same kind is how to interpolate vectorial variables, such as the velocity (this topic is addressed in another contribution to the 2022 ECCOMAS conference [START_REF] Tételin | ECCOMAS 2022[END_REF]). The result is that we may have to adopt a limitation procedure adapted to each variable and / or depending on the flow region. This is typically how the compressive limitation technique operates.

Compressive limitation in the multislope MUSCL framework

In spite of their advantages to deal with compressible multiphase flows, diffuse interface models also have the drawback to entail excessive numerical diffusion of interfaces when using classical second-order schemes. This is why so-called compressive limitation techniques have been proposed in the literature [START_REF] Chiapolino | [END_REF][16][17]. In this paper, we present such a technique adapted to the multislope MUSCL method introduced in [13]. Inspired by [17], in which a cell-based interface sensor ξ i was defined in a monoslope context using the volume fraction gradient, we introduce in the multislope context a face-based interface sensor ξ ij as:

∀K i ∈ Ω, ∀S ij ∈ V i , ξ ij = true if ||B i H + ij ||p + ij = α H + ij -α i > ϵ α false else , (6)
where ϵ α is a threshold (typical value ϵ α = 10 -3 ), and with α i and α H + ij the volume fractions at points B i and H + ij . Then we compute slope compression coefficients for each cell:

γ i = max 0, min γ min i , γ max i , γ min i = α i -min (α i , ϵ) max α i -α min i , ϵ , γ max i = max (α i , 1 -ϵ) -α i max α max i -α i , ϵ , (7)
where ϵ ≈ 10 -20 and α min i , α max i are the minimum and maximum values of the volume fraction in the vertex-based neighborhood of the cell K i :

α min i = min α i , min j∈V i α min ij α max i = max α i , max j∈V i α max ij with: α min ij = min α H - ij , α H + ij , α H - ji , α H + ji α max ij = max α H - ij , α H + ij , α H - ji , α H + ji . (8)
This enables us to perform the slope compression as follows:

∀K i ∈ Ω, ∀S ij ∈ V i | ξ ij = true, p - ij = γ i p - ij , p + ij = γ i p + ij , (9)
where p - ij and p + ij are the compressed slopes for the volume fraction. The next step afterwards is to adapt the limitation procedure for every face, depending on whether it belongs to the interface or not. For the faces localized outside the interface (ξ ij = false), we just interpolate all the usual variables with our favorite standard limiter. For the faces localized inside the interface on the other hand (ξ ij = true), the procedure is as follows: first interpolate the volume fractions α ij with a hyper-compressive limiter (see below), then interpolate the other variables with a standard limiter, and finally update the variables required to compute the numerical fluxes from the interpolated variables. The final step is to derive a hyper-compressive limiter dedicated to the interpolation of the volume fraction within the interface. To do so, we use a so-called Overbee limiter adapted to the multislope formalism:

φ(r ij , η - ij , η + ij ) = max 0, min η - ij r ij , η + ij . ( 10
)
This limiter matches the boundaries of the stability zone without ensuring the second-order consistency property (ϕ(r) = 1). As a result, it tends to transform the solutions that vary linearly into discontinuities, which is precisely the behavior we seek here in order to maintain a discontinuous profile for the volume fraction as long as possible. With this limiter, interpolations of the volume fractions can be recast as:

α ij = α i + α H + ij -α i max 0, min 1, (α i -α H - ij )/(α H + ij -α i ) , (11)
that is to say as a convex combination of α i and α H + ij , thereby preserving the domain of definition of the volume fraction.

The effect of this compressive limitation technique is clearly shown in figure 1, which represents a classical sloshing test case under gravity performed with the 4-equation model. It can be seen that the liquid-gas interface undergoes a very strong numerical diffusion with classical second-order schemes, whereas the interface is much more preserved thanks to the limitation technique.

Low-Mach number corrections in HLLC Riemann solvers

An important feature of multiphase flows in energetics is that they may involve a wide scale of Mach numbers in a same application, ranging from low-Mach number (or even zero) for example in steady liquid regions, to supersonic or even hypersonic areas. For these applications, the use of incompressible models is therefore impossible, while the use of compressible models can be a challenge due to their well-known inaccuracy in the low-Mach number limit [18]. This may result in excessive numerical diffusion of material interfaces moving at low speed, but also in strong pressure oscillations, especially when a weakly compressible liquid phase is involved. Much work has been done in the literature to deal with this low-Mach issue in compressible multifluid models. Starting from a reformulation of HLLC Riemann solvers proposed in an unpublished ONERA report by B. Courbet, we introduce here low-Mach corrections as additional terms to the standard upwinding, thus simplifying the implementation of an all-Mach HLLC scheme in a CFD code. Considering the Euler equations without loss of generality, the classical HLLC Riemann solver is rewritten as:

F HLLC = 1 2 (F L + F R ) + 1 4 (σ R + σ L )(F L -F R ) + 1 4 (σ R -σ L )D, ( 12
)
where D is the subsonic upwinding (D = 0 in the supersonic case) that reads:

D = (F * L -F L ) + (F * R -F R ) + σ M (F * L -F * R ) = D ρY D ρu D ρE t , (13)
with F L , F * L , F * R and F R the physical fluxes considered in the different regions of the Riemann problem, and σ L , σ R , σ M the sign of the three waves speeds S L , S R , S M . The subsonic upwinding can be split into convective vs. pressure force terms, i.e. D = D c + D p with:

D p = . . . + (f R -f L )   0 n S M   , f L = ρ L (u n,L -S L )(S M -u n,L ) and f R = ρ R (S R -u n,R )(S M -u n,R
) standing for pressure forces (with u n,L and u n,R the left and right normal velocities), while n is the unit vector normal to the face. The term in the box is actually the one to blame for the over dissipation and inaccuracies at low-Mach. In order to compensate for it, we add a contribution D sup to the standard upwind term D:

D sup = (1 -β) [ρ * L S L (u n,L -S M ) -ρ * R S R (S M -u n,R )]   0 n S M   ,
where ρ * L and ρ * R are the densities in the left and right starred region of the Riemann problem. The important part is β, a function of the Mach number at least, and whose role is to tend to unity outside of the low-Mach regime to recover the classical HLLC scheme. Most of low-Mach corrections proposed in the literature can be recast in this way, with particular expressions for the function β. Here we employ an ad hoc formulation proposed by D. Scherrer at ONERA and depending on the left and right Mach numbers, sound speeds, densities and pressures:

β = min 1, M L + M R + |P L -P R | ρ L c 2 L + ρ R c 2 R + ϵ + 10 -2 , ϵ = 10 -10 . ( 14
)
Figure 2 shows the contribution of this low-Mach correction on a simulation performed with the 4-equation model (equipped with a k -ω RANS turbulence model) during the internship project of W. Haegeman. It consists of a high-velocity gas jet impacting a free surface of liquid water initially at rest. Without the low-Mach correction, spurious pressure fluctuations of high amplitude are triggered just below the liquid surface as soon as the pressure wave generated by the gas jet impacts on it (even before the impact of the jet itself), which eventually lead to the failure of the computation. In this very low-Mach liquid zone, the misbehavior of the HLLC Riemann solver is more than a dissipation issue, it is primarily a robustness issue. With the low-Mach correction detailed above, these pressure shoots are avoided and the simulation can be carried out until convergence.

FOCUS ON A SHOCK-DROPLET INTERACTION TEST-CASE

Description of the case

Similar test cases have been widely studied in the literature (see [19][20][21] for example) and aim at reproducing the deformation and breakup of a water droplet (or water column [19]) exposed to a supersonic flow. For the validation of the numerical strategy presented in this paper, both experimental and numerical studies are used as reference [21]. This test case consists in a spherical water droplet of diameter ∅ d = 1.135 mm located in ambiant air in which propagates a planar shock wave. In order to reduce computational cost, a 2D axisymmetrical geometry is simulated. The numerical domain and the initial state of the simulation are described in figure 3. Red values represent input data extracted from the reference [21], while other data are computed mostly thanks to Rankine-Hugoniot jump conditions. The numerical mesh consists of two zones of quadrangles with two different characteristic sizes. The first zone corresponds to the lower part of the domain and extends up to 3∅ d from the symmetry axis. In this zone, a uniform cell characteristic size of 90 cells per initial droplet diameter is used. Due to the high Weber ("catastrophic") regime considered in this application (We ∼ 1.2e5), such refinement is clearly not sufficient to capture all final liquid fragments but is considered to be adapted for a macroscopic analysis of the early droplet breakup dynamics. In the second zone, which corresponds to the rest of the domain, the mesh is progressively coarsened Clément Le Touze and Nicolas Rutard in vertical direction down to a cell characteristic size of 30 cells per initial droplet diameter at the upper boundary condition in order to reduce computational cost. Besides, viscous stresses, heat diffusion and capillary effects are neglected since such mechanisms do not participate to the early droplet breakup [21]. Finally, the left and right boundary conditions are supersonic air inflow (P = 2.122 334 × 10 6 Pa, T = 1336 K and V = 1160 m.s -1 ) and outflow respectively, while the up and down boundary conditions correspond to a wall and the symmetry axis. In the following section, the initial droplet diameter ∅ d and Rayleigh time

Quantity

t c = ∅ d V 1 ρ d
ρ 1 are used to get dimensionless space and time quantities, with ρ 1 and V 1 the upstream air density and velocity, and ρ d the water droplet density.

Results with the 4-equation model

Figure 4 represents the liquid volume fraction and numerical Schlieren fields at different times after the shock impacts the droplet. In this figure, the strongest air density gradients extracted from the numerical simulation of reference [21] (red dots) are used to visualize bow and wake shocks positions as well as parts of the liquid-gas interface. This first comparison tends to demonstrate the capability of the 4-equation model to reproduce the displacement and deformation of bow and wake shocks as well as the droplet breakup dynamics. To demonstrate that the numerical methods presented in section 3 play a major role in the reproduction of the shock-droplet interaction, the same analysis is performed in figure 5 for simulations carried out either without compressive limitation (left) or without low-Mach number correction (right). Without compressive simulation, an excessive diffusion of the interface is noticed at the droplet stagnation point, leading to a significant error on the bow shock location. Regarding the low-Mach number correction, some may think that no effect should be observed in such a test case due to the overall supersonic nature of the flow. However, as its name suggests, the droplet stagnation point is associated with a very low Mach number. At this particular location, the HLLC scheme thus leads to an excessive diffusion of the solution, which results once again in a significant error on the bow shock location. This demonstrates that both compressive limitation and low-Mach number correction participate to enhance the precision of such multifluid simulations. Further analyses are performed hereafter to pursue the validation of the multifluid strategy in Cedre. First, displacements of the droplet center of mass (DCM) and the droplet stagnation point (DSP) are measured. As mentioned in the literature, the DCM displacement can be used to calculate the drag coefficient of the deforming droplet, which is an important feature to model the liquid acceleration [19], while the DSP displacement is a key feature to evaluate the bow shock position in time. The DCM coordinates X DCM can be calculated in numerical simulations as X DCM = Ω α l ρ l XdV Ω α l ρ l dV , where the integrated volume Ω is the entire computational domain, X corresponds to each computational cell coordinates and dV is their volume. As shown in figure 6, a good agreement is found between reference and Cedre results as far as it concerns DCM displacement. Regarding DSP, measurements are much more uncertain with a diffuse interface method than with the Volume Of Fluid method used in reference [21]. Indeed, diffuse interface models do not involve any interface reconstruction, leading to an ambiguity in the exact interface location and thus in the DSP position. Therefore, two measurements of the DSP displacement are performed and displayed on left part of figure 6 with two different threshold liquid volume fractions α th viz. 0 and 0.1 (any mesh cell with a liquid volume fraction α l ≥ α th is considered part of the liquid structures). According to figure 6, the threshold value 0.1 seems to be a better tracer of the interface since it leads to a perfect match with the reference DSP. The time evolution of bow shock position along the symmetry axis plotted on the middle part of figure 6 tends to confirm this conclusion. Indeed, as mentioned before, the DSP and bow shock displacements are strongly related. Finally, right part of figure 6 shows the evolution of the dimensionless distance between bow and wake shocks at 2.5∅ d from the symmetry axis. A good agreement is found with numerical results of reference, with however slight differences whose magnitude is the same as between experimental and numerical results of reference. This last result participates to demonstrate the good reproduction of the macroscopic droplet breakup dynamics since the position of the wake shock strongly depends on the formation of vortices in the droplet's wake [19]. All these results show that thanks to the numerical methods presented in section 3, the 4-equation model is able to reproduce complex breakup mechanisms in an environment experiencing a wide range of Mach numbers, from locally sub-to supersonic flows.

TOWARDS THE USE OF A TWO-TEMPERATURE MODEL IN CEDRE

Being a mirror of the compressible Navier-Stokes equations, the 4-equation model can be easily implemented in advanced industrial CFD codes like Cedre, which makes it attractive. Indeed, these codes more easily include the whole complexity required for industrial applica- tions (e.g. unstructured meshes and advanced numerical methods, general thermodynamics or comprehensive physical modelling) than specialized codes. Nevertheless, the use of more refined multifluid models, giving locally access e.g. to the temperature and the velocity of each fluid, as well as to some geometrical variables describing the interface topology at the subgrid scale level, seems unavoidable if we want to progress towards more predictive multifluid simulations for industrial applications. This is why we are interested in bringing a more accurate multifluid model, namely the 6-equation model with instantaneous pressure relaxation, to industrial maturity. This model has been first introduced by [6], and then revisited by some authors [7][8][9]. In this model, each fluid is locally described with its own temperature. This entails a more robust thermodynamic closure than the 4-equation model when strong temperature gradients between fluids are involved [22], and offers a more solid basis to model heat transfers and phase changes. The version of this model that has been implememented in Cedre is the one derived during the PhD of P. Cordesse [9], carried out at ONERA in collaboration with M. Massot's team at CMAP -Ecole Polytechnique. The comprehensive version of this model includes transported equations on subscale geometrical variables (surface density area, mean curvature) derived from the principle of least action, accounting in particular for capillary effects at the subscale level. Without considering here the geometrical variables equations for the sake of simplicity, this system reads (for the case of two fluids, k = 1, 2):

∂ t Q + ∇ • (f C + f D + f σ ) + S D + K∇ • u = R ϵ p + S , (15)
where Q = (α k ρ k y j,k ρu ρE α k ρ k ϵ k α 1 ) t is the vector of conserved or transported variables, comprising species partial densities of each fluid (j is the index for species within the fluid k), momentum and total energy of the whole mixture, plus internal energies and volume fractions of each fluid. From Q, one can derive a set of primitive variables, e.g. U = (u α k y j,k P k T k ) t , i.e. the mixture mean velocity, plus for each fluid the volume fraction, the mass fractions of its species (with respect to the fluid), the pressure and the temperature. As each fluid is described with its own T k and P k (before applying the instantaneous pressure relaxation), this makes the thermodynamic closure independant for each fluid:

ϵ k = ϵ EOS k (P k , T k , y j,k ) , P k = P EOS k (ρ k , ϵ k , y j,k ) , T k = T EOS k (ρ k , ϵ k , y j,k ) , etc. (16)
It is therefore possible to use any equation of state for each fluid (see section 2.2). A mixture pressure is defined as P = α k P k , and becomes the unique pressure of the fluids as soon as the pressure relaxation is applied. Moreover, f C and K∇ • u are respectively the conservative and non conservative parts of convective fluxes, f D and S D comprise the diffusive terms (see [START_REF] Schmidmayer | 46th AIAA Fluid Dynamics Conference[END_REF]), f σ stands for surface tension, R ϵ p is the instantaneous pressure relaxation term (ϵ p → 0), while S gathers the source terms, e.g. body forces such as gravity. Under the assumption of instantaneous pressure relaxation, the solutions of the 6-equation model converge towards that of the classical 5-equation model [5]. Even though the equation on the total energy of the mixture is redundant with the transport equations on internal energies, it is still solved numerically in order to ensure the conservation of the total energy. Indeed, the latter is not guaranteed due to the numerical discretization of the non conservative terms in the internal energies equations [6].

The numerical discretization of this 6-equation model in Cedre is made in the same way as for the 4-equation model. In particular, both numerical methods described in this paper (compressive limitation and low-Mach corrections) have been easily adapted to the 6-equation model. However, while the 4-equation model in Cedre has reached the level of maturity required for numerical simulations on industrial configurations (it has already been used for many complex applications in various fields, such as liquid atomization in jet or rocket engines, supercritical combustion, phase change in heat exchangers, spray / acoustics interaction with space launch vehicles, etc.), the simulations with the 6-equation model have not yet achieved the same level of maturity in Cedre. The main difficulty currently occurs when shocks interact with the diffuse interface, like e.g. in the shock-droplet interaction test case presented in section 4. The numerical discretization of the non conservative terms in the volume fraction and internal energy equations may lead to small discrepancies when updating the fluid densities in the shock layer, resulting in strong discrepancies when updating the fluid pressures due to the weak compressibility of the liquid. This occurs after the convective step and before the pressure relaxation procedure. To ensure the robustness of the computation, especially when dealing with general EOS, it may be necessary to use clipping patches, but then at the expense of a detrimental loss of conservativity. These difficulties seem to be kwown in the literature, but the potential solutions such as the artificial heat exchange mentioned by [6] are not compatible with practical simulations on an industrial scale. We currently envision different ways to remedy this problem, e.g. using an alternative equation of state for the liquid phase, less sensitive to the pressure and density discrepancies. Otherwise it will be necessary to further study the numerical discretization methods employed, or maybe consider an alternative version of the model such as the 6-equation model of [7] based on the total energies of the fluids.

Apart from this shock / interface interaction issue, satisfactory simulations with the 6equation model have been performed on several cases, such as those presented in section 3 with the 4-equation model. In particular, challenging simulations of high-temperature and high-speed gas jets impacting water surfaces have been conducted with the 6-equation model, exhibiting good accuracy and robustness, even better than with the 4-equation model due to the strong temperature gradients involved. When these numerical robustness issues are solved, our main focus will be on the formulation of advanced physical submodels for the 6-equation model (subscale geometrical variables, general thermodynamics and phase change, turbulence and combustion...).

Figure 1 :

 1 Figure 1: Effect of the compressive limitation technique on a water tank sloshing test case under gravity with the 4-equation model. Left: volume fraction field just after the initial time. Middle: final solution with a standard second-order scheme. Right: final solution with the compressive method.

Figure 2 :

 2 Figure 2: Top: solution at t = 0.01 s without low-Mach correction, a few time before the computation fails due to spurious pressure fluctuations. Bottom: converged solution at t = 3 s obtained with the low-Mach correction (slightly zoomed out). From left to right: density, Mach number and pressure fields.

Figure 3 :

 3 Figure 3: Geometry and initialization data for the simulation of shock-droplet interaction.

Figure 4 :

 4 Figure 4: Liquid volume fraction and Schlieren fields at different times. Red dots: strongest air density gradients from reference simulation [21].

Figure 5 :

 5 Figure 5: Liquid volume fraction and Schlieren fields after droplet breakup (t = 2.98t c ). Red dots: strongest air density gradients from reference simulation [21]. Left: without compressive limitation. Right: without low-Mach number correction.

Figure 6 :

 6 Figure 6: Left: DCM and DSP displacements along the shock propagation axis. Middle: bow shock position along the symmetry axis.Right: distance between bow and wake shocks at 2.5∅ d from the symmetry axis. Reference results: see[21]


Pour Le Diplôme D'état De Docteur En

Médecine Spécialité

Par M Mazen

Kallel Thèse

Soutenue Publiquement

Président Du

M Le Pr

Emmanuel Gay

Stéphan Chabardès

M Le Dr Éric

Seigneuret M Le Dr

Valérie Fraix

ASSESSING THE USE OF DIRECTIONAL LEADS IN PARKINSON'S DISEASE PATIENTS DEEP BRAIN STIMULATION: CLINICAL EFFICACY AND VOLUME OF TISSUE ACTIVATED -TARGET OVERLAPS IN COMPARISON TO OMNIDIRECTIONAL LEADS THÈSE

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Résumé

Objectif : Les électrodes directionnelles sont utilisées depuis quelques années pour la stimulation cérébrale profonde (SCP) et sont particulièrement employées dans la SCP du noyau sous-thalamique (NST) dans la maladie de Parkinson (MP). Dans cette étude, nous avons cherché à comparer deux cohortes de patients implantés avec des électrodes omnidirectionnelles et directionnelles en ce qui concerne l'efficacité clinique et à évaluer la corrélation avec l'intersection volume de tissu activé -cible (VTA-cible).

Matériel et méthodes : Au total, 60 patients ont été inclus. 27 patients avec électrodes directionnelles ont été comparés à 33 patients avec électrodes omnidirectionnelles. Les scores MDS UPDRS III, la levodopa equivalent daily dose (LEDD) et les volumes d'intersection du VTA avec le NST moteur et avec le volume du sweetspot d'amélioration motrice ont été comparés 12 mois après la chirurgie.

Résultats : Il y a une réduction LEDD significativement plus élevée dans la cohorte des électrodes directionnelles de plus de 8 % (51,3 % de réduction contre 42,7 % de réduction, p = 0,042) par rapport au groupe omnidirectionnel, avec des scores moteurs MDS UPDRS III similaires à 12 mois. Les patients avec électrodes omnidirectionnelles avaient un volume de chevauchement VTA-NST moteur significativement supérieur à celui des patients avec électrodes directionnelles (32,01 mm 3 contre 20,38 mm 3 , p = 0,0226), mais la proportion de VTA dans le NST moteur ou dans le sweetspot d'amélioration motrice globale ne différait pas statistiquement entre les deux groupes. Chez les patients avec électrodes directionnelles, la réduction de la LEDD était corrélée au chevauchement du VTA avec le sweetspot d'amélioration motrice globale (R = 0,36, p = 0,036), ce qui n'était pas le cas pour les patients avec électrodes omnidirectionnelles (R = 0,11, p = 0,276). 41 % des patients implantés avec des électrodes directionnelles avaient un réglage de stimulation directionnelle à 12 mois. Dans l'analyse du sous-groupe de patients avec électrodes directionnelles, il n'y avait pas de différence significative dans les scores MDS UPDRS III, la réduction des LEDD, les chevauchements du VTA avec le NST moteur ou avec le sweetspot de l'amélioration motrice globale entre les patients stimulés de manière omnidirectionnelle et directionnelle à 12 mois.

Conclusion :

Par rapport aux électrodes omnidirectionnelles, les électrodes directionnelles permettent, avec des volumes d'intersections VTA-cibles moins importants, d'obtenir un score UPDRS III comparable avec une meilleure réduction de la dose quotidienne d'équivalent de dopa à 12 mois de la SCP du NST chez les patients atteints de la maladie de Parkinson.

Table of Contents

Introduction

Neurosurgical treatment of Parkinson's disease patients mainly began with stereotactic thalamotomy. The latter was a lesioning technique that was used on the ventral intermediate nucleus (Vim) of the thalamus and was very effective on essential tremor patients and on tremor symptoms of PD patients 1 . However, it presented major drawbacks mainly related to its irreversible nature and neurological complications when performed bilaterally that led to the development of deep brain stimulation (DBS). High frequency DBS for Parkinson's disease (PD) started with Benabid et al. [START_REF] Benabid | Chronic VIM thalamic stimulation in Parkinson's disease, essential tremor and extra-pyramidal dyskinesias[END_REF][START_REF] Benabid | Chronic electrical stimulation of the ventralis intermedius nucleus of the thalamus as a treatment of movement disorders[END_REF] as a reversible strategy for stereotactic thalamotomy. Deep brain stimulation has proven overtime to be a safe and reliable procedure [START_REF] Mendes Martins | Surgical complications of deep brain stimulation: clinical experience of 184 cases[END_REF] . The target for DBS then progressively extended to subthalamic nucleus [START_REF] Benabid | Acute and long-term effects of subthalamic nucleus stimulation in Parkinson's disease[END_REF] (STN) mainly in Europe, because it allowed control over tremor but over akinesia and rigidity as well [START_REF] Fraix | Subthalamic nucleus stimulation in tremor dominant parkinsonian patients with previous thalamic surgery[END_REF] . Since then, deep brain stimulation of the subthalamic nucleus remains a well -established treatment for Parkinson's disease 6 (PD). DBS of the STN significantly improves the MDS UPDRS III score [START_REF] Chou | The MDS-UPDRS tracks motor and non-motor improvement due to subthalamic nucleus deep brain stimulation in Parkinson disease[END_REF] and the quality of life of patients [START_REF] Floden | Predicting quality of life outcomes after subthalamic nucleus deep brain stimulation[END_REF][START_REF] Lagrange | Bilateral subthalamic nucleus stimulation improves health-related quality of life in PD[END_REF][START_REF] Fraix | Clinical and economic results of bilateral subthalamic nucleus stimulation in Parkinson's disease[END_REF] even on the long term [START_REF] Bove | Long-term Outcomes (15 Years) After Subthalamic Nucleus Deep Brain Stimulation in Patients With Parkinson Disease[END_REF] .

Many targets were reported in the literature including the thalamus, globus pallidus internus (GPI), STN, pedunculopontine nucleus and pre-lemniscal radiation [START_REF] Temel | Targets for deep brain stimulation in Parkinson's disease[END_REF] but the STN remains one the preferred targets [START_REF] Benabid | Deep brain stimulation of the subthalamic nucleus for the treatment of Parkinson's disease[END_REF] . It successfully reduces medication dosage intake more than GPi targeting [START_REF] Liu | Meta-analysis comparing deep brain stimulation of the globus pallidus and subthalamic nucleus to treat advanced Parkinson disease[END_REF] .

The STN is a small ovoid shaped biconvex nucleus that is oriented ventrally and medially and measures about 9 mm in length and 6 mm in width, but these values vary greatly according to the studies and whether the measures are made on anatomic specimens or on MRI [START_REF] Mavridis | Anatomy of the Human Subthalamic Nucleus: A Combined Morphometric Study[END_REF][START_REF] Daniluk | Assessment of the variability in the anatomical position and size of the subthalamic nucleus among patients with advanced Parkinson's disease using magnetic resonance imaging[END_REF] . The STN is moreover divided into three parts according to its connectivity patterns: limbic, associative and sensorimotor [START_REF] Emmi | Anatomy and Connectivity of the Subthalamic Nucleus in Humans and Non-human Primates[END_REF] . The latter part is often cited as DBS target for PD patients and in general the dorsolateral part of the STN and the surrounding white matter seem to be the optimal ''sweetspot'' for targeting [START_REF] De Roquemaurel | Stimulation Sweet Spot in Subthalamic Deep Brain Stimulation -Myth or Reality? A Critical Review of Literature[END_REF] . Having such a small target requires great precision and accuracy during lead implantation and can be quite a challenge.

The main goal of precise targeting is to maximize beneficial effects of the stimulation while at the same time avoid the possible side effects. In fact, stimulation can induce side effects in correlation with the STN surrounding anatomy: muscle contraction if the internal capsule is stimulated laterally and posteriorly, oculomotor movements if stimulated medially, cognitive, and behavioral changes if stimulated anteriorly and akinesia if stimulated inferiorly in relation with substantia nigra [START_REF] Umemura | Complications of subthalamic nucleus stimulation in Parkinson's disease[END_REF] .

Actual targeting methods include lead implantation within the STN with an error margin of up to 3 mm [START_REF] Guo | Surgical targeting accuracy analysis of six methods for subthalamic nucleus deep brain stimulation[END_REF] . Considering the small size of the STN [START_REF] Emmi | Anatomy and Connectivity of the Subthalamic Nucleus in Humans and Non-human Primates[END_REF] , a deviation of a few millimeters of the lead during implantation can put the contacts close to the adjacent structures around the nucleus. Historically, ring contacts leads were implanted, from single contact leads to nowadays quadripolar leads. The contacts in this type of electrodes activated the tissue around it forming a sphere without surrounding tissue discrimination. This can rapidly decrease the side-effect threshold and substantially decrease the DBS efficacy. Thus, current steering has emerged as a promising solution to this issue [START_REF] Schüpbach | Directional leads for deep brain stimulation: Opportunities and challenges[END_REF] . A directional lead can steer the current towards a particular direction, which changes the shape of the classic spheric volume of tissue activated (VTA). Theoretically this should allow for better dorsolateral STN targeting while avoiding activation of surrounding tissues responsible for DBS side-effects [START_REF] Masuda | Surgical Strategy for Directional Deep Brain Stimulation[END_REF] . It should compensate for lead mispositioning by correctly orienting VTAs [START_REF] Kramme | Potentials and Limitations of Directional Deep Brain Stimulation: A Simulation Approach[END_REF] .

Studies regarding directional leads efficacy have demonstrated an increase in therapeutic window in individual patients implanted with directional electrodes [START_REF] Steigerwald | Directional deep brain stimulation of the subthalamic nucleus: A pilot study using a novel neurostimulation device[END_REF][START_REF] Dembek | Directional DBS increases side-effect thresholds-A prospective, doubleblind trial[END_REF][START_REF] Schnitzler | Directional Deep Brain Stimulation for Parkinson's Disease: Results of an International Crossover Study With Randomized, Double-Blind Primary Endpoint[END_REF] , with an increase in side-effect threshold. However, very few studies tried to directly compare a group of patients implanted with omnidirectional leads with a second group implanted with directional leads which therefore has a ''choice'' of directional stimulation. Only one study had such configuration and tried to compare such two groups when it comes to motor and medication intake [START_REF] Pintér | Antiparkinsonian Drug Reduction After Directional Versus Omnidirectional Bilateral Subthalamic Deep Brain Stimulation[END_REF] . It has shown a superior decrease in LEDD in the directional leads group.

To our knowledge, there has been no direct comparison between omnidirectional and directional leads with correlation to the anatomical post-operative lead localization and their VTAs. We therefore aimed to assess this aspect in our study.

Materials and methods

Patients' cohort

All patients implanted with DBS electrodes for PD (Parkinson disease) from February 2017 until September 2020 were screened. Among 90 PD patients eligible for inclusion, 60 patients were selected for the study. Reasons for exclusion were missing ON or OFF medication UPDRS III score at 12 months (22 patients), absence of reliable LEDD for certain dopaminergic agonists (2 patients), infection (2 patients), one sided implantation (2 patients), per-operative hemorrhage (1 patient) and death (1 patient).

33 patients were implanted with omnidirectional electrodes and 27 patients with directional electrodes. Implanted omnidirectional leads were quadrapolar Medtronic 3389 (Figure 1), except for one patient who had Boston scientific Vercise standard linear 8 contact leads. All implanted directional leads were Boston Scientific Vercise Cartesia (Figure 2).

For each patient we identified the age, sex, disease starting year, MDS UPDRS III score OFF and ON medication (MED) pre-operatively and at 12 months, LEDD pre-operatively and at 12 months, date of surgery, chosen contacts and stimulation parameters post-operatively and at 12 months. All MDS UPDRS III scores measured at 12 months were with stimulation ON.

LEDD calculations were performed using the Equidopa 27 online tool. Pre-operative imaging was performed with a 3T MRI and included a whole brain 3D T1 and a basal ganglia focused coronal T2 sequences. Post-operative imaging was performed using a CT scan with slices' thickness inferior to 1 mm.

Image processing

All calculations were performed using the Lead-DBS Toolbox (https://www.lead-dbs.org/) [START_REF] Horn | Lead-DBS v2: Towards a comprehensive pipeline for deep brain stimulation imaging[END_REF] on MATLAB R2022a. MRI pre-operative acquisitions were co-registered linearly using SPM12 [START_REF]Statistical parametric mapping: the analysis of funtional brain images[END_REF] and CT postoperative acquisition were linearly co-registered to pre-operative T1 MRI using two-stage linear registration (rigid followed by affine) as implemented in Advanced Normalization Tools 32 (ANTs).

Normalization

Pre and post-operative acquisitions were spatially normalized into Montreal Neurological Institute (MNI) space (2009b, nonlinear, asymmetric) using the symmetric diffeomorphic registration approach (SyN) as implemented in ANTs 32-34 . This step was followed by a ''subcortical refine'' step, if necessary, as implemented in Lead-DBS software.

Post-operative brainshift [START_REF] Horn | Lead-DBS: a toolbox for deep brain stimulation electrode localizations and visualizations[END_REF] was corrected as implemented in Lead-DBS software.

Lead Localization

DBS leads were automatically pre-localized in native and template space using PaCER 36 algorithm. When this method failed TRAC/CORE [START_REF] Horn | Lead-DBS: a toolbox for deep brain stimulation electrode localizations and visualizations[END_REF] algorithm was used (6 patients). All electrodes were individually visualized and manual corrections were performed when necessary. Directional electrodes orientation was automatically detected in Lead-DBS according to DiODe v2 [START_REF] Dembek | DiODe v2: Unambiguous and Fully-Automated Detection of Directional DBS Lead Orientation[END_REF] .

STN subcortical atlases and VTA calculations

VTAs were calculated using a finite element method approach as implemented in Lead-DBS [START_REF] Horn | Connectivity Predicts deep brain stimulation outcome in Parkinson disease[END_REF] .

Subcortical atlases used for VTA-target overlap calculations were DISTAL Minimal [START_REF] Horn | Lead-DBS v2: Towards a comprehensive pipeline for deep brain stimulation imaging[END_REF][START_REF] Nowacki | Accuracy of different three-dimensional subcortical human brain atlases for DBS -lead localisation[END_REF] for motor STN and STN Sweetspots [START_REF] Dembek | Probabilistic sweet spots predict motor outcome for deep brain stimulation in Parkinson disease[END_REF] for ''overall motor mean map gradient''. The latter is a probabilistic sweet spot predicting motor outcome.

TEED calculations (Total electrical energy delivered)

The TEEDs were calculated for voltage-based leads (Medtronic) according to Koss et al. formula [START_REF] Koss | Calculating total electrical energy delivered by deep brain stimulation systems[END_REF] TEED (uJ) = voltage(V)² * impulse width (S) * frequency (Hz) / impedance (Ohm) * 10 6For current-based leads TEEDs were calculated using a derived formula where voltage was replaced with current according to Ohm's law as described by Hui et al [START_REF] Hui | Assessing the effect of current steering on the total electrical energy delivered and ambulation in Parkinson's disease[END_REF] :

TEED (uJ) = intensity(A)² * impulse width (S) * frequency (Hz) * impedance (Ohm) * 10 6

Since impedance was not available for all patients, all TEEDs were calculated by assuming that the impedance was equal for both groups and a standard value of 1000 ohms was used.

Statistics

Statistical analysis was performed using McNemar standard test for qualitative variables and Student t-test for quantitative variables.

A linear regression model (Pearson coefficient) was used to correlate VTA-target overlap to different variables.

Results :

Patients' pre-operative characteristics Patients' pre-operative characteristics were summarized in Table 1. Patients implanted with omnidirectional leads did not differ from patients implanted with directional leads when it comes to mean age at surgery (62.21 vs 61.11, p = 0.58), mean disease age (10.55 vs 10.37, p = 0.86) and mean pre-operative MDS UPDRS III score OFF MED (46.45 vs 45.74, p = 0.87) nor ON MED (16.5 vs 16.26, p = 0.93). However directional leads patients had a significantly higher LEDD intake pre-operatively than omnidirectional leads patients (1575.45 mg vs 1303.28 mg, p= 0.044). This might be due to a selection bias in our center; clinically worse patients were possibly preferably implanted with directional leads. There was a slightly superior male predominance in directional leads implanted patients (74% vs 58%).

Omnidirectional leads patients vs directional leads patients comparison MDS UPDRS III motor score

All MDS UPDRS III score values are measured with ongoing stimulation (STIM ON). The patients implanted with directional electrodes had similar UPDRS motor score as patients implanted with omnidirectional electrodes (Figure 3). There was no significant difference preoperatively OFF and ON medication (respectively OFF med 46. 4

LEDD intake pre-operatively versus at 12 months

Considering that directional leads should theoretically be able to steer the current ''better'' towards the stimulation sweet spot, we can hypothesize that the LEDD intake decrease should be greater in these patients.

At 12 months omnidirectional and directional leads patients had similar LEDDs (748 mg vs 768 mg, p = 0.85), even though pre-operative LEDD in directional patients were significantly higher (1575 mg vs 1303 mg, p = 0.044 for a two tailed t-test). The daily LED decrease was therefore significantly higher in directional electrodes patients by more than 8% (-808 mg or 51.3% reduction vs -556 mg or 42.7 % reduction, p = 0.042 for a one tailed t-test) (Figure 4).

VTA-target overlap at 12 months

Both omnidirectional and directional leads patients' groups showed a positive correlation between the LEDD intake reduction and the VTA-motor STN overlap volume (R = 0.46, p = 0.002 vs R = 0.36, p = 0.034). In other words, the more motor STN volume is within the VTA, the more levodopa consumption is further reduced (Figure 5). When using the overall motor improvement mean map, LEDD decrease was significantly positively correlated with VTAmotor mean map sweetspot intersection in directional leads patients (R = 0.36, p = 0.036) however we did not find this correlation with omnidirectional leads (R = 0.11, p = 0.278).

Omnidirectional leads patients had a significantly superior VTA-motor STN overlap volume than directional leads patients (32.01 mm 3 vs 20.38 mm 3 , p = 0.0226). This is also applicable to the overall motor improvement mean map (p = 9.04 x 10 -07 ). However, there was no significant difference (p = 0.10) between the two groups when it comes to the normalized VTA-motor STN intersection volume (VTA -motor STN intersection/VTA) (p = 0.10) nor when it comes to normalized VTA-overall motor improvement mean map (p = 0.72), which suggests that both groups had comparable proportion of VTA volumes within motor STN and overall motor improvement mean map (Figure 6).

TEED at 12 months

Surprisingly the total electrical energy delivered with the stimulation settings at 12 months in omnidirectional leads patients was roughly similar to the energy delivered in directional leads patients (111.5 uJ vs 110.6 uJ, p = 0.93). Using current steering leads did not reduce current consumption as expected (Figure 7).

Stimulation configuration during follow-up

Of the 60 patients included in the study, 27 were implanted with directional electrodes. The stimulation setting was noted post-operatively, at 3 months and at 12 months (Figure 8). Postoperatively, only 4 out of 27 patients had a directional stimulation setting (15%). However, this proportion significantly increased at 3 months (33%, p = 0.01) and at 12 months (41%, p = 0.02). There was no statistically significant difference between the proportion of directionally stimulated patients at 3 months vs 12 months (p = 0.23).

MDS UPDRS III motor score

All MDS UPDRS III score values are measured with ongoing stimulation (STIM ON). Within the patients implanted with directional electrodes, there was no significant difference in UPDRS motor score pre and post-operatively at 12 months when comparing the patients with an omnidirectional setting and those with a directional setting (at 12 months respectively: OFF med: 26.4 vs 28.9 p = 0.51, ON med: 15.6 vs 17.9 p = 0.39) (Figure 9).

LEDD intake pre-operatively versus at 12 months

There was no significant difference in daily LED between directional electrodes patients stimulated omnidirectionally and directionally pre-operatively (1586 mg vs 1560 mg, p = 0.89) and at 12 months (742 mg vs 805 mg, p = 0.73). Daily LED decrease was therefore similar (Figure 10).

VTA-target intersection at 12 months

No visually detectable pattern was found during reconstruction of omnidirectionally and directionally stimulated leads (Figure 13 and14). There was no statistically significant difference in VTA-motor STN overlap between patients stimulated omnidirectionally or directionally at 12 months (p = 0.7436) (Figure 15). There was a tendency to a superior VTA-overall motor improvement mean map intersection in directionally stimulated patients, without statistical significance (p = 0.0992).

Figure 15 VTA-target intersection volumes (mm 3 ). In blue: omnidirectionally stimulated patients, in red: directionally stimulated patients.

Discussion

To our knowledge this is the largest study that assesses the correlation between clinical findings and anatomical localization of the implanted DBS leads with a group comparison between omnidirectional and directional leads.

We did not find any significant difference in MDS UPDRS III scores at 12 months between directional and omnidirectional leads implanted in PD patients which is consistent with another study 26 in the literature.

In our study, directional leads patients had a significantly higher LEDD intake pre-operatively than omnidirectional leads patients (1575.45 mg vs 1303.28, p= 0.044). This is probably due to a selection bias in our center; clinically worse patients were preferably implanted with directional leads. Even though these patients had more LEDD intake pre-operatively, they had comparable LEDDs at 12 months to those of the omnidirectional leads' patients. In fact, the LEDD intake decrease was significantly higher using a one-tailed t-test in directional leads patients compared to omnidirectional leads, by 252 mg (about 8.7% more dose reduction). This LEDD intake decrease is consistent with results in D.Pinter et al study [START_REF] Pintér | Antiparkinsonian Drug Reduction After Directional Versus Omnidirectional Bilateral Subthalamic Deep Brain Stimulation[END_REF] that found a 13% drug reduction in directional leads patients compared to omnidirectional leads patients.

Furthermore, we demonstrated that the LEDD intake decrease was directly correlated to the volume of the intersection between the volume of tissue activated and motor STN in both omnidirectional and directional leads. However, when using the overall motor improvement mean map sweetspot as described in Dembek et al article [START_REF] Dembek | Probabilistic sweet spots predict motor outcome for deep brain stimulation in Parkinson disease[END_REF] , only directional leads patients had a significant positive correlation between LEDD decrease and VTA overlap. This might suggest that current steering manages to better target the necessary volume for best LEDD decrease effects in the overall motor improvement mean map. Further investigation with a larger cohort of patients may be needed.

Omnidirectional leads had a larger VTA-motor STN overlap and VTA-overall motor improvement map overlap than directional leads, which is probably due to larger absolute VTAs in omnidirectional leads. When normalizing these values to the VTA volumes, we found no significant difference between the two groups, which suggests that even with a smaller VTA-target overlap, directional leads had the same proportion of VTA volume within target. Directional leads, with smaller overlaps manage to obtain comparable UPDRS III score with better LEDD dosage reduction.

Another study [START_REF] Nguyen | Analysis of patient-specific stimulation with segmented leads in the subthalamic nucleus[END_REF] demonstrated that in individual patients and at therapeutic currents, the VTAs with best directional stimulation were about half the VTAs with omnidirectional stimulation. This could explain some of our VTA overlaps results. Current steering probably reduces the absolute volume of activated tissue. Using directional leads decreased the VTAs but with probable better sweetspot overlap, which might explain comparable UPDRS III score and better LEDD intake decrease than omnidirectional leads. However, the total electrical energy delivered did not differ between the two groups. Directional leads offered no advantage when it comes to current consumption, and thus battery life should not be affected.

Patients with directional leads preferred a directional stimulation type at 12 months in 41% of the cases. This number is consistent with other studies in the literature and notably the PROGRESS randomized cross-over study [START_REF] Schnitzler | Directional Deep Brain Stimulation for Parkinson's Disease: Results of an International Crossover Study With Randomized, Double-Blind Primary Endpoint[END_REF] that reported a preference for directional stimulation in 52.8% of patients and 58.5% of clinicians . In this study authors found that this choice is because of the increase of the therapeutic window in those individual patients; however, the MDS UPDRS III score and quality of life of patients did not improve.

In directional leads patients' group, the MDS UPRS III score and LEDD intake at 12 months did not vary between patients stimulated directionally an patients stimulated omnidirectionally. This suggests that at 12 months each of these patients are ''well' in their stimulation setting whether it is omnidirectional or directional. Stimulation setting and the choice of directionality at 12 months are logically the best stimulation setting for the patients.

Active contacts were significantly closer to nearest motor STN voxel in directionally stimulated patients than in omnidirectionally stimulated patients, but only by about 0.3 mm which is much smaller than the lead thickness of 1.3 mm or the contacts spacing of 0.5 mm. On the other hand, VTA overlaps with targets were comparable between the two groups, which goes well with the fact that every patient has an optimal stimulation corresponding to his anatomy and lead localization. A slight superiority for patients stimulated directionally might exist in VTA overlap with overall motor improvement sweetspot, but this should be confirmed in further studies with larger cohorts.

Directionality might finally not ''correct'' mispositioned leads, but rather potentialize LEDD intake reduction, which might improve the quality of life of patients. Directional leads should therefore be used when available.

Limitations

DiODe v2 37 automatic lead orientation algorithm used in Lead DBS presents a risk of 180 degrees marker flip, which can affect the VTA orientation and thus overlap with the target volume for patients stimulated directionally. However, authors claim a 100% of correct orientation using the center of mass method, which we preferably used.

We could not perform contact per contact side-effect analysis and correlate the directionality choice with VTA overlap with side effects cluster due to the retrospective nature of this study.

For practical reasons impedances used were around 1000 ohms, however some studies suggest that impedance might be higher in directional leads [START_REF] Pintér | Antiparkinsonian Drug Reduction After Directional Versus Omnidirectional Bilateral Subthalamic Deep Brain Stimulation[END_REF][START_REF] Eleopra | Brain impedance variation of directional leads implanted in subthalamic nuclei of Parkinsonian patients[END_REF] . Taking this into account and based on the total electrical energy delivered formulas, directional leads TEEDs could be even higher than what we found this study. Therefore, directional leads TEEDs are at least comparable to omnidirectional leads TEEDs, but could not be inferior, and this only confirms that there is no gain in using directional leads for current consumption reduction. A Julien plus qu'un chef, qu'un co-interne ou qu'un ami, tu es toujours là pour me pousser en avant. Toujours partant pour prendre une pizza avec toi en garde.

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  Abbreviations ...................................................................................................................... Introduction ........................................................................................................................ Materials and methods ....................................................................................................... Patients' cohort ............................................................................................................... Image processing ............................................................................................................. Normalization .................................................................................................................. Lead Localization ............................................................................................................. STN subcortical atlases and VTA calculations ................................................................... TEED calculations (Total electrical energy delivered) ....................................................... Statistics .......................................................................................................................... Results : .............................................................................................................................. Patients' pre-operative characteristics............................................................................. Omnidirectional leads patients vs directional leads patients comparison ........................ MDS UPDRS III motor score .......................................................................................... LEDD intake pre-operatively versus at 12 months ........................................................ VTA-target overlap at 12 months ................................................................................. TEED at 12 months ....................................................................................................... Directional leads patients' subgroup analysis................................................................... Stimulation configuration during follow-up .................................................................. MDS UPDRS III motor score .......................................................................................... LEDD intake pre-operatively versus at 12 months ........................................................ Distance from target .................................................................................................... VTA-target intersection at 12 months .......................................................................... Discussion ........................................................................................................................... Limitations ....................................................................................................................... Conclusion: ......................................................................................................................... Bibliography ........................................................................................................................ Remerciements ...................................................................................................................

Figure 1

 1 Figure 1Medtronic 3389 omnidirectional lead characteristics according to Kahan et al.[START_REF] Kahan | The Safety of Using Body-Transmit MRI in Patients with Implanted Deep Brain Stimulation Devices[END_REF]

Figure 2

 2 Figure 2 Boston scientific Vercise Cartesia directional lead characteristics according to Boston Scientific 29 .

Figure 3

 3 Figure 3 Omnidirectional and directional leads patients had similar MDS UPDRS III scores pre-operatively and at 12 months OFF and ON MED.

Figure 4

 4 Figure 4 LEDD (mg) pre-operatively and at 12 months.

Figure 5 Figure 6 A

 56 Figure 5 Correlation between LEDD decrease and intersection (mm 3 ) of volume of tissue activated with motor STN or overall motor mean map sweetspot. Figures on the left with the blue dots represent omnidirectional leads patients and figures on the right with the green dots directional leads patients.

Figure 7 12 months

 712 Figure 7 Total electrical energy delivered (uJ) at 12 months was similar in omnidirectional leads (111.5 uJ) and directional leads patients (110.6 uJ) with a p-value = 0.93.

Figure 8

 8 Figure 8 Proportion of directionally stimulated patients post-operatively, at 3 months and at 12 months.

Figure 9

 9 Figure9 Omnidirectionally and directionally stimulated patients had similar MDS UPDRS III scores pre-operatively and at 12 months OFF and ON MED.

Figure 10 Figure 12

 1012 Figure 10 LEDD pre-operatively and at 12 months in omnidirectionally and directionally stimulated patients.

Figure 13

 13 Figure 13 Omnidirectionally stimulated patients leads position (red leads) with a VTA example model.

Figure 14

 14 Figure 14 Directionally stimulated patients leads positions (green leads) with a VTA example model.

Table 1

 1 vs 45.8 p = 0.89, ON med: Patients' pre-operative characteristics are summarized in the above table. There was a slightly superior male predominance in directional leads implanted patients (74% vs 58%). Directional leads patients had a significantly higher preoperative LEDD intake than omnidirectional leads patients (1575.45 mg vs 1303.28 mg, p-value = 0.044). There was no significant difference in all other pre-operative parameters.16.7 vs 16.1 p = 0.83) nor post-operatively at 12 months OFF and ON medication (respectively OFF med: 27.2 vs 27.6 p = 0.91, ON med: 17.8 vs 16.6 p = 0.59).

	Patients' characteristics	Omnidirectional	leads	Directional leads	p-value
		patients		patients
	Number of patients	33		27
	Mean age at DBS (years)	62.21		61.11	0.58
	Sex	19 M (58%) 14 F (42%)	20 M (74%) 7 F
				(26%)
	Mean disease age at DBS	10.55		10.37	0.86
	(years)
	Mean pre-op MDS UPDRS	46.45		45.74	0.87
	III score OFF MED
	Mean pre-op MDS UPDRS	16.50		16.26	0.93
	III score ON MED
	Pre-op LEDD (mg)	1303.28		1575.45	0.044

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Remerciements

Distance from target

Using Lead DBS 3D visualization tool, active contacts of patients stimulated omnidirectionally and directionally can be spatially represented (Figure 11). No visually detectable trend or pattern was found, active contacts from both groups seemed to be homogenously distributed along the STN. Interestingly directionally stimulated patients had a mean distance between lead contacts and the nearest motor STN voxel that was significantly smaller than omnidirectionally stimulated patients by almost 0.3 mm (0.99 mm vs 1.31 mm, p = 0.01) (Figure 12). Omnidirectionally stimulated patients leads were in globality further away from motor STN than directionally stimulated patients leads. Active contacts were also significantly closer to nearest motor STN voxel in directionally stimulated patients than in omnidirectionally stimulated patients (0.59 mm vs 0.96 mm, p = 0.02).

Conclusion:

When compared to omnidirectional leads, directional leads manage, with smaller VTA-target overlaps, to obtain comparable UPDRS III score with better LEDD dosage reduction in STN DBS for PD patients.