Fedor S Stonyakin

Evgeniya A Vorontsova
email: vorontsovaea@gmail.com

Mohammad S Alkousa
email: mohammad.alkousa@phystech.edu

New Version of Mirror Prox for Variational Inequalities with Adaptation to Inexactness ⋆

Keywords: Variational Inequality, Mirror Prox, Inexactness, Adaptation, Non-Smooth Operator, Lipschitz-continuous Operator, Matrix Game

Some adaptive analogue of the Mirror Prox method for variational inequalities is proposed. In this work we consider the adaptation not only to the value of the Lipschitz constant, but also to the magnitude of the oracle error. This approach, in particular, allows us to prove a complexity near O 1 ε log 2 1 ε for variational inequalities for a special class of monotone bounded operators. This estimate is optimal for variational inequalities with monotone Lipschitz-continuous operators. However, there exists some error, which may be insignificant. The results of experiments on the comparison of the proposed approach with some known analogues are presented. Also, we discuss the results of the experiments for matrix games in the case of using non-Euclidean proximal setup.

Introduction

Variational inequalities (VI) and saddle point problems often arise in a variety of important applications [START_REF] Facchinei | Finite-Dimensional Variational Inequality and Complementarity Problems[END_REF]. For solving such problems a lot of algorithmic schemes are known (see e.g. [START_REF] Facchinei | Finite-Dimensional Variational Inequality and Complementarity Problems[END_REF][START_REF] Antipin | Dynamics and variational inequalities[END_REF][START_REF] Chambolle | A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging[END_REF][START_REF] Nemirovski | Prox-method with rate of convergence O(1/T ) for variational inequalities with Lipschitz continuous monotone operators and smooth convexconcave saddle point problems[END_REF]). The Mirror Prox method proposed by A. S. Nemirovski [START_REF] Nemirovski | Prox-method with rate of convergence O(1/T ) for variational inequalities with Lipschitz continuous monotone operators and smooth convexconcave saddle point problems[END_REF] is currently one of the most popular of such methods. This method goes back to the well-known extragradient method proposed by G. M. Korpelevich in [START_REF] Korpelevich | Extragradient method for finding saddle points and other problems[END_REF]. At the same time, unlike the standard extragradient method, Mirror Prox allows to effectively solve problems with non-Euclidean norms, as well as with the Holder-continuous operators:

g(x) -g(y) * L ν x -y ν ∀ x, y ∈ Q for some ν ∈ [0; 1], (1)
all notations are explained in Sect. 2 below. Recently, a universal analogue of the A. S. Nemirovski method [START_REF] Nemirovski | Prox-method with rate of convergence O(1/T ) for variational inequalities with Lipschitz continuous monotone operators and smooth convexconcave saddle point problems[END_REF] was proposed in [START_REF] Dvurechensky | Generalized Mirror Prox: Solving variational inequalities with monotone operator, inexact oracle, and unknown Hölder parameters[END_REF][START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF]. The universality is understood as an adaptive adjustment to the optimal smoothness level ν in [START_REF] Facchinei | Finite-Dimensional Variational Inequality and Complementarity Problems[END_REF], as well as the constant value L ν > 0. Note that universal gradient method for unconstrained convex optimization problems was proposed by Yu.E. Nesterov [START_REF] Nesterov | Universal gradient methods for convex optimization problems[END_REF] (see also Sect. 5 of the textbook [START_REF] Gasnikov | Modern Numerical Optimization Methods, Universal Gradient Descent[END_REF]). And it is possible to observe the convergence rate of the proposed method, which is typical for the smooth case ν = 1 (Lipschitz-continuous operators), for some problems with bounded operators (L 0 < +∞ and L ν = +∞ for all ν > 0).

This paper is devoted to the modification of Mirror Prox method [START_REF] Dvurechensky | Generalized Mirror Prox: Solving variational inequalities with monotone operator, inexact oracle, and unknown Hölder parameters[END_REF][START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF] for the following analogue of the Lipschitz condition for the operator g with constant L > 0 g(y) -g(x), y -z LV (y, x) + LV (y, z) + δ y -z ∀x, y, z ∈ Q,

where δ > 0 is a fixed value and V (y, x) is the Bregman divergence (see Sect. 2 below).

We propose an analogue of аdaptive Mirror Prox method [START_REF] Dvurechensky | Generalized Mirror Prox: Solving variational inequalities with monotone operator, inexact oracle, and unknown Hölder parameters[END_REF][START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF]. At the same time, we consider adaptive tuning both for the value of the parameters L and δ. One of the features of the proposed method which are important for applications is the possibility for the value of δ to reflect the inexactness of operator g. In addition, the value of δ can indicate the degree of discontinuity of the operator g. Adaptive tuning to its value can approximate the convergence rate for variational inequalities with bounded operators (ν = 0) to the convergence rate for variational inequalities with Lipshitz-continuous operators (ν = 1). Effects of this approach can be observed for the universal method but without a theoretical justification for the convergence rate O 1 ε for non-smooth operators [START_REF] Dvurechensky | Generalized Mirror Prox: Solving variational inequalities with monotone operator, inexact oracle, and unknown Hölder parameters[END_REF]. This means that the proposed approach in this paper is an alternative to the universal method.

The contribution of the paper can be summarized as follows:

-An analogue of the Mirror Prox method for variational inequalities with a monotone Lipschitz-continuous operator, which allows for adaptive tuning to the value of the Lipschitz constant L, as well as the limit value of the error δ of the specifying operator g, is proposed.

-The applicability of the proposed method to a certain class of variational inequalities with bounded operators (ν = 0) is discussed. The rate of convergence O 1 ε of this method is proved with some finite error associated with the nonsmoothness of the operator. Thus, some alternative to the universal method has been proposed, but with a clearer theoretical rationale for acceleration.

-The results of numerical experiments for finding the equilibrium in a bilinear matrix game (or VI with Lipschitz-continuous operator) with a bounded error in the definition of the operator are given. A comparison of the quality of the calculated solution is given depending on the number of iterations for the adaptive Mirror Prox method from [START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF] and the method proposed in this paper with an adaptive setting for the magnitude of the error.

-The results of numerical experiments for a variational inequality with a bounded (ν = 0) operator (related to the Fermat-Torricelli-Steiner problem) are presented. These results show that the method due to the proposed adaptation of the non-smoothness error can converge much faster than the optimal lower estimate O 1 ε 2 for the corresponding class of problems.

Problem Statement and Some Examples

Let (E, • ) be a normed finite-dimensional vector space and E * be the conjugate space of E with the norm:

||y|| * = max x { y, x , ||x|| ≤ 1},
where y, x is the value of the continuous linear functional y at x ∈ E.

Let Q ⊂ E be a (simple) closed convex set, d : Q → R be a distance generating function (d.g.f.), which is continuously differentiable and 1-strongly convex with respect to the norm • and assume that min

x∈Q d(x) = d(0). For all x, y ∈ Q ⊂ E we consider the corresponding Bregman divergence V (x, y) = d(x) -d(y) -∇d(y), x -y . (3)
Let g : Q → E * be a continuous operator. We consider the problem of finding a solution to a variational inequality of the form

g(x * ), x * -x 0 ∀ x ∈ Q. ( 4
)
Under the assumption of the monotony of the operator g, i.e.

g(x) -g(y), x -y ≥ 0 ∀ x, y ∈ Q, the inequality (4) is equivalent to the following weak variational inequality

g(x), x * -x 0 ∀ x ∈ Q. (5)
Assume that the operator g satisfies the condition [START_REF] Antipin | Dynamics and variational inequalities[END_REF]. In this section, we show some examples of problems for which a condition of the form (2) naturally arises. First of all, this is due to the inexactness of the oracle for the operator of a variational inequality. But also the value of δ y -z can describe the degree of discontinuity of the operator g (i.e. using considered approach, one can propose an approach to the solution of some VI's with bounded operators).

Example 1. Let g : Q → R n be a Lipschitz-continuous operator with constant L > 0, i.e. g(x) -g(y) * L x -y ∀ x, y ∈ Q.

However, suppose that the exact value of the operator g is not available, and only an approximate value of g(x), i.e. g(x), is known:

g(x) -g(x) * δ 2 ∀ x ∈ Q.
Then for each x, y, z ∈ Q we have:

| g(y)-g(x), y-z -g(y)-g(x), y-z | = | g(y)-g(y), y-z + g(x)-g(x), y-z | g(y)-g(y) * • y -z + g(x)-g(x) * • y -z δ 2 + δ 2 y -z = δ y -z .
Therefore,

g(y)-g(x), y-z g(y)-g(x), y-z +δ y-z g(y)-g(x) * • y-z +δ y-z L y -x • y -z + δ y -z L 2 y -x 2 + L 2 y -z 2 + δ y -z LV (y, x) + LV (y, z) + δ.
Example 2. Note that the term δ y -z in (2) can describe non-smoothness for the operator g along any fixed vector segment {ty + (1 -t)x} 0≤t≤1 . In general (if you combine all possible vector segments), on the domain of non-smoothness points there can be an infinite number. For example, assume that for some subset Q 0 ⊂ Q the function f is differentiable at all points of Q \ Q 0 and that for an arbitrary x ∈ Q 0 there exists a finite subdifferential ∂f (x) in the sense of convex analysis.

For fixed x, y ∈ Q with t ∈ [0; 1] we denote y t := (1 -t)x + ty.

Definition 1. ( [START_REF] Stonyakin | Some analogue of quadratic interpolation for a special class of nonsmooth functionals and its application to the adaptive mirror descent method[END_REF]) Fix δ > 0 and L > 0. We say that the convex function

f : Q → R (Q ⊂ R n ) has (δ, L)-Lipschitz subgradient (f ∈ C 1,1 L,δ (Q))
, if: (i) for arbitrary x, y ∈ Q the function f is differentiable at all points of the set {y t } 0 t 1 , with the exception of the sequence (possibly finite)

{y t k } ∞ k=1 : t 1 < t 2 < t 3 < . . . and lim k→∞ t k = 1; (6)
(ii) for a sequence of points from (6) there exist finite subdifferentials {∂f

(y t k )} ∞ k=1 and diam ∂f (y t k ) =: δ k > 0, where +∞ k=1 δ k =: δ < +∞, (7)
and diam ∂f (x) = max{ y -z * | y, z ∈ ∂f (x)}; (iii) if for x, y ∈ Q the function f is differentiable at each point y t , t ∈ (0; 1), then the following inequality holds:

min ∂f (x)∈∂f (x), ∂f (y)∈∂f (y) ∂f (x) -∂f (y) * L x -y . (8)
Indeed, the property of (δ, L)-Lipschitzness for each subgradient g

(x) = ∂f (x) means that g(y) -g(x) * L y -x + δ. (9)
To prove [START_REF] Gasnikov | Modern Numerical Optimization Methods, Universal Gradient Descent[END_REF], it suffices to split the segment {y t } 0 t 1 into the intervals of smoothness and take into account the boundedness of diameters of the subdifferentials at non-smoothness points of f . The inequality [START_REF] Gasnikov | Modern Numerical Optimization Methods, Universal Gradient Descent[END_REF] means that

g(y) -g(x), y -z g(y) -g(x) * • y -z L y -x • y -z + δ y -z L 2 ( y -x 2 + y -z 2 ) + δ y -z LV (y, x) + LV (z, y) + δ y -z .
Let us give a concrete example of a non-smooth functional with a (δ, L)-Lipschitz subgradient with an arbitrarily large Lipschitz constant.

Example 3. We fix some k > 0, the value δ > 0 and consider the piecewise linear function

f : [0; 1] → R (here Q = [0; 1] ⊂ R) defined as follows f (x) =      kx ; 0 x 1 2 , k + n i=1 δ 2 i x - n i=1 δ 2 i 1 -1 2 i ; 1 -1 2 n < x 1 -1 2 n+1 , lim x→+1 f (x) ; x = 1. (10)
In this case,

Q 0 = {1-1 2 n } ∞ n=1 , ∂f (q n ) = k + n-1 i=1 δ 2 i ; k + n i=1 δ 2 i with n > 1, ∂f (q 1 ) = k; k + δ 2 (here q n = 1 -1 2 n with n = 1, 2, 3, . . .). It is clear that ∂f (q n ) = δ
2 n , which is true for the entered value δ > 0. Moreover, on the intervals (0; q 1 ), (q n ; q n+1 ) the function f has a Lipschitz-continuous gradient with the constant L = 0. Therefore, for the function f from [START_REF] Stonyakin | Some analogue of quadratic interpolation for a special class of nonsmooth functionals and its application to the adaptive mirror descent method[END_REF], we find that f ∈ C 1,1 0,δ (Q). Any functional with a finite set of non-smooth points along an arbitrary segment will satisfy the proposed Lipschitz condition for the subgradient. Obviously, this condition holds for each objective function with finite points of nonsmoothnes on each vector segment [x; y]. Thus, it is possible to apply this technique to problems of minimization for sum distances to several balls in Hilbert spaces [START_REF] Mordukhovich | Applications of variational analysis to a generalized Fermat-Torricelli problem[END_REF]. Such an objective function, obviously, will not be differentiable in the usual sense at the points of the boundaries of the balls of which there are infinitely many. Note that among points of each vector segment [x; y] such an objective function have finite points of non-smoothness. However, the considered Lipschitz condition for a special choice of subgradient holds for some functions with infinitely many points of non-smoothness (e.g. for maximum of linear functions).

Adaptive Method for Variational Inequalities with Adaptation to Inexactness

In this section, we introduce a new version of the Mirror Prox method for variational inequalities (see [START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF]), which we call Mirror Prox with Adaptation to Inexactness (MPAI). In this version, which is listed as Algorithm 1 below, we consider the adaptation not only to the level of operator smoothness, but also to the magnitude of the oracle error, which may allow to receive complexity near O 1 ε for VI with bounded operators, i.e. the optimal complexity for VI with Lipshitz-continuous operators.

We evaluate the solution quality of the problem (4), produced by Algorithm 1, by using the Bregman divergence (3).

Algorithm 1 Mirror Prox with Adaptation to Inexactness (MPAI).

Input:

x 0 = arg min x∈Q d(x), L 0 , δ 0 . 1: N := N + 1; L N+1 := L N 2 ; δ N+1 := δ N 2 . 2: Calculate y N+1 := arg min x∈Q { g(x N ), x -x N + L N+1 V (x, x N )}, (11)
x N+1 := arg min

x∈Q { g(y N+1 ), x -x N + L N+1 V (x, x N )}. ( 12
) 3: If g(y N+1 ) -g(x N ), y N+1 -x N+1 ≤ L N+1 V (y N+1 , x N )+ ( 13
)
+L N+1 V (x N+1 , y N+1 ) + δ N+1 y N+1 -x N+1 ,
then go to the next iteration (item 1). 4: Else increase L N+1 and δ N+1 by two times and go to item 2.

Theorem 1. After N iterations of Algorithm 1, the following estimate holds:

N -1 k=0 1 L k+1 g(y k+1 ), y k+1 -x V (x, x 0 )-V (x, x N )+ N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 .
Proof. One can directly check the following inequalities:

∇ x V (x, x k ) x=x k+1 , x -x k+1 = V (x, x k ) -V (x, x k+1 ) -V (x k+1 , x k ), (14)
∇ x V (x, x k ) x=y k+1 , x -y k+1 = V (x, x k ) -V (x, y k+1 ) -V (y k+1 , x k ). ( 15
)
Further, for each x ∈ Q and k = 0, N -1:

∇ x g(x k ), x -x k + L k+1 V (x, x k ) x=y k+1 , x -y k+1 0, ∇ x g(y k+1 ), x -x k + L k+1 V (x, x k ) x=x k+1 , x -x k+1 0.
Thus,

g(y k+1 ), x k+1 -x L k+1 V (x, x k ) -L k+1 V (x, x k+1 ) -L k+1 V (x k+1 , x k ) and g(x k ), y k+1 -x L k+1 V (x, x k ) -L k+1 V (x, y k+1 ) -L k+1 V (y k+1 , x k ).
Taking into account (13), we have for each k = 0, N -1:

g(y k+1 ), y k+1 -x = g(y k+1 ), x k+1 -x + g(x k ), y k+1 -x k+1 + + g(y k+1 ) -g(x k ), y k+1 -x k+1 L k+1 V (x, x k ) -L k+1 V (x, x k+1 ) -L k+1 V (x k+1 , x k ) + L k+1 V (x k+1 , x k )- -L k+1 V (x k+1 , y k+1 ) -L k+1 V (y k+1 , x k ) + L k+1 V (y k+1 , x k )+ +L k+1 V (x k+1 , y k+1 ) + δ k+1 y k+1 -x k+1 , i.e. 1 L k+1 g(y k+1 ), y k+1 -x V (x, x k ) -V (x, x k+1 ) + δ k+1 L k+1 y k+1 -x k+1 .
(16) After summing ( 16) by k = 0, N -1, we have

N -1 k=0 1 L k+1 g(y k+1 ), y k+1 -x V (x, x 0 )-V (x, x N )+ N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 .
Let us denote

S N = N -1 k=0 1 L k+1 , y = 1 S N N -1 k=0 y k+1 L k+1 and R 2 = max x∈Q V (x, x 0 ).
Theorem 2. For monotone operator g after N iterations of Algorithm 1, the following estimate holds:

max x∈Q g(x), y -x R 2 S N + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 . ( 17
)
Assume that for fixed ε

N -1 k=0 1 L k+1 R 2 ε . ( 18
)
Then the following inequality holds:

max x∈Q g(x), y -x ε + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 . ( 19
)
If L 0 2L, then inequality (18) holds at no more than

N = 2LR 2 ε iterations of Algorithm 1.
Proof. By monotony of g we have for each k = 0, 1, ...: g(x), y k+1 -x = g(y k+1 ), y k+1 -x + g(x)-g(y k+1 ), y k+1 -x g(y k+1 ), y k+1 -x , so the inequality

1 S N max x∈Q N -1 k=0 1 L k+1 g(y k+1 ), y k+1 -x R 2 S N + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 ε + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 (20)
can be replaced by

max x∈Q g(x), y -x R 2 S N + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 ε + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 . ( 21
)
Remark 1. Due to adaptive choice of parameters L k+1 and δ k+1 at each iteration of Algorithm 1 the expression

R 2 S N + 1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1
in (21) may be small enough even in the case of L = +∞ or δ = +∞ in (2).

Remark 2. Clearly, for each k, we have δ k C L δ (C L = max 1, 2L L 0 ) and:

1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 C L δ max k=0,N -1 y k+1 -x k+1 .
This means that the value associated with the error in the specifying operator g is bounded on the set Q of a finite diameter. Remark 3. If g ≡ 0, then the condition L 0 2L can be satisfied by choosing

L 0 := g(x) -g(y) * x -y at g(x) = g(y).
Remark 4. Note that the estimate of the number of iterations N = 2LR 2 ε with accuracy to a numerical factor is optimal for variational inequalities with a Lipschitz-continuous operator [START_REF] Nemirovsky | Information-based complexity of linear operator equations[END_REF]. Note that the evaluation of the inexactness of the value of the operator, as we see from the previous remark, is bounded and does not accumulate. Note that similarly Remark 4 in [START_REF] Stonyakin | An adaptive analog of Nesterov's method for variational inequalities with a strongly monotone operator[END_REF] the total number of attempts to solve [START_REF] Mordukhovich | Applications of variational analysis to a generalized Fermat-Torricelli problem[END_REF] and ( 12) is bounded by 4N + max log 2 2L L 0 , log 2 2δ δ 0 . Remark 5. It was pointed earlier that δ can describe the degree of non-smoothness of the operator g. Let us show how Algorithm 1 can be slightly modified in order to provably obtain an approximate complexity estimate

O 1 ε 2 log 2 1 ε
to achieve the quality of the solution y:

max x∈Q g(x), y -x ε
in the case of a bounded operator g (generally speaking, non-smooth). Suppose that at each iteration L L k+1 2L (this can always be achieved with a constant number of calculations in item 2 of the listing of Algorithm 1).

We propose such a procedure for some positive integer p: repeat the operations item 2 of Algorithm 1, p times, each time

L k+1 := 2 • L k+1 with the same δ k+1 2δ (22)
(for each iteration of the algorithm 1, k = 0, 1, 2, . . . , N -1). A procedure of type (22) will be stopped if one of the following two inequalities holds:

δ k+1 y k+1 -x k+1 ε 2 , (23)
or

g(y k+1 ) -g(x k ), y k+1 -x k+1 2 p-1 L y k+1 -x k 2 + y k+1 -x k+1 2 .
(24) Note that the procedure (22) involves updating x k+1 and y k+1 during repetitions. Let us estimate p (the number of times required for executing (23) or (24) to repeat a procedure of type ( 22)).

It is clear that ∀ x k , x k+1 , y k+1 ∈ Q it is true that

g(y k+1 ) -g(x k ), y k+1 -x k+1 L 2 y k+1 -x k 2 + L 2 y k+1 -x k+1 2 +δ y k+1 -x k+1 .
Moreover, δ k+1 2δ. If (23) is not true, then y k+1 -x k+1 > ε 4δ and the inequality (24) is obviously satisfied at

2 p > 1 + 16δ 2 Lε , (25)
since in that case

2 p -1 2 L y k+1 -x k+1 2 > δ y k+1 -x k+1 .
So, after repeating the p procedures of type (22) at each of the N iterations of the method, the following inequality holds:

max x∈Q g(x), y -x R 2 S N + ε 2 .
Further, In this section, to show the advantages of the proposed Algorithm 1, we consider some numerical experiments for the saddle point problem (and the corresponding VI), which corresponds to the convex programming problem for some analogues of the Fermat-Torricelli-Steiner problem with functional constraints. Note that the objective functions are non-smooth and the corresponding operators of the variational inequality of the problem under consideration are bounded (ν = 0). However, experimentally, due to adaptation, we can observe an estimate of the complexity inherent in the case of Lipshitz-continuous operators of VI. All experiments in this section were implemented in Python 3.4, on a computer equipped with Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz, 1992 Mhz, 4 Core(s), 8 Logical Processor(s). RAM of the computer is 8GB.

S N = N -1 k=0 1 L k+1 N 2 p+1 L . Therefore, R 2 S N 2 p+1 LR 2 N ε 2 at N 2 p+2 LR 2 ε , whence, taking into account (25) N 4LR 2 ε + 64δ 2 R 2 ε 2 . ( 26
)
For a given set of N points {A k = (a 1k , a 2k , . . . , a nk ); k = 1, N }, that represent the centers of the balls ω k with radii r k , in the n-dimensional Euclidean space R n , we need to find such a point X = (x 1 , x 2 , . . . , x n ) of the objective function [START_REF] Mordukhovich | Applications of variational analysis to a generalized Fermat-Torricelli problem[END_REF]

f (x) := N k=1 d(X, A k ), (27)
where

d(X, A k ) = XA k -r k , if |XA k | r k 0, otherwise,
would take the minimal value on the set Q, which is given by several functional constraints:

ϕ 1 (x) = α 11 x 2 1 + α 12 x 2 2 + . . . + α 1n x 2 n -1, ϕ 2 (x) = α 21 x 2 1 + α 22 x 2 2 + . . . + α 2n x 2 n -1, . . . ϕ m (x) = α m1 x 2 1 + α m2 x 2 2 + . . . + α mn x 2 n -1, (28)
where the coefficients α 11 , α

    ,
in which one element of each row is an integer belonging to the interval (1; 10), and the remaining elements of the row are equal to 1.

To solve such a problem, we can consider a saddle point problem min

x max λ L(x, λ),
where

L(x, λ) = f (x) + m p=1 λ p ϕ p (x), -→ λ = (λ 1 , λ 2 , . . . , λ m ).
Consider the corresponding variational inequality:

G(x * , -→ λ * ), (x * , -→ λ * ) -(x, -→ λ ) 0 ∀(x, -→ λ ) ∈ B ⊂ R n+m ,
where

B = (x, -→ λ ) | n k=1 x 2 k + m p=1 λ 2 p 1 , G(x, λ) =   ∇f (x) + m p=1 λ p ∇ϕ p (x), -ϕ 1 (x), -ϕ 2 (x), . . . , -ϕ m (x)   .
We give an example for n = 100, m = 20, N = 5, initial approximation

(x 0 , λ 0 ) = 1 √ m + n , 1 √ m + n . . . , 1 √ m + n ∈ R n+m ,
and δ 0 = 1 20 . The coordinates of the points A k are chosen in such a way that A k ∈ [1; 2]. We choose the standard Euclidean proximal setup as a proxfunction.

Note that the centers of the balls were chosen with the norm in the interval [1; 2], and the radii of the balls are 1. Therefore, in a single ball with the center at zero there will be points of the boundary of the balls in which the objective function ( 27) and the operator of the corresponding variational inequality will be bounded. At the same time, it can be shown that the diameter of the subdifferential at such points will be at least 1. It means that theoretically δ can be at least 1. However, experimentally, we can see significantly better solution quality (see the column "General estimate" in Table 1).

The results of the work of Algorithm 1, for objective function ( 27) are represented in Table 1 below. It is known ( [START_REF] Nemirovsky | Information-based complexity of linear operator equations[END_REF][START_REF] Nemirovsky | Problem Complexity and Method Efficiency in Optimization[END_REF]) that for variational inequalities with bounded operators, the theoretical estimate of the complexity (the convergence rate) O 1 ε 2 is theoretically optimal. However, experimentally we see from Table 1 that, for example, an accuracy of 0.1051 is achieved in 17 iterations, and a 10-fold greater accuracy of 0.0106 is achieved in 25 iterations and approximately in the same time. If the method worked without adaptation and strictly according to optimal lower bounds for the specified class of problems, then the increase in costs could be approximately 100 times. Thus, due to the adaptability of the proposed method, we observe a convergence rate close to O ln 1 ε . Now for a given set of N points {A k = (a 1k , a 2k , . . . , a nk ); k = 1, N } in ndimensional Euclidean space R n we need to find such a point x = (x 1 , x 2 , . . . , x n ), that the objective function

f (x) := N k=1 (x 1 -a 1k ) 2 + . . . + (x n -a nk ) 2 = N k=1 x -A k 2 (29)
would take the minimal value on the set Q, which is defined by the previous constraints (28). The coordinates of the points A k for k = 1, N , are chosen as the rows of the matrix A ∈ R N ×n . The entries of the matrix A are random integers in the closed interval [-10; 10], which are drawn from the discrete uniform distribution.

The results of the work of Algorithm 1, for objective function (29) and for some different values of n, m and N , are presented in Table 2 below. These results demonstrate the number of iterations produced by Algorithm 1 to reach the solution of the problem, the quality of the solution "General estimate", which is the right side of inequality [START_REF] Stonyakin | Gradient Methods for Problems with Inexact Model of the Objective[END_REF], and the running time of the algorithm in seconds. As we see from Table 2 we also observe convergence rate close to O ln 1 ε , due to the adaptability of the proposed method.

Remark 6. Now we take all previous parameters but with points A k (k = 1, N ) in the unit ball. The results of Algorithm 1, for objective function (29) and for some different values of n, m and N , with constraints (28), are presented in Table 3 below.

In this case, since the points A k are chosen in the unit ball, the operator of the variational inequality is bounded. The results of experiments in Table 3 show that the rate of convergence of the proposed method is close to O 1 ε , which is significantly better than the optimal one, which is O 1 ε 2 , for non-smooth convex optimization problems and bounded variational inequalities [START_REF] Nemirovsky | Problem Complexity and Method Efficiency in Optimization[END_REF].

where

x = (x 1 , x 2 , . . . , x n ) ∈ R n , y = (y 1 , y 2 , . . . , y n ) ∈ R m , ∆ n is a standard simplex in R n , i.e. ∆ n = {x ∈ R n | x ≥ 0, n i=1 x i = 1}, ∆ m is a standard simplex in R m ,
A is the payoff matrix for the y player. In all experiments we use payoff distributions centered at zero. Consider the following operator

g(u) = ∇ x (x T Ay) -∇ y (x T Ay) = A T y -Ax , u = (x, y) ∈ Q ≡ ∆ n × ∆ m . (31)
The operator g(u) from ( 31) is monotone on Q, and with this operator the VI (5) has the same solution as the saddle point problem (30). So, Mirror Prox methods could be used for solving it. In all experiments with matrix games we use the entropy prox-function

d(x) = n i=1
x i ln x i in Bregman divergence (3). Entropy prox for matrix games on simplex is the best option (see [START_REF] Nesterov | Smooth minimization of non-smooth functions[END_REF]). First of all, we calculate experimental numbers of iterations for adaptive proximal method for VI [START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF] and compare these numbers with theoretical estimation N = C ε , for some C > 0 [START_REF] Nemirovski | Prox-method with rate of convergence O(1/T ) for variational inequalities with Lipschitz continuous monotone operators and smooth convexconcave saddle point problems[END_REF]. For that kind of experiments we run simulations on two classes of random matrix games: 10 × 10 and 100 × 100 normally distributed payoff matrices. For the first setting we create 50 games at random and calculate average experimental number of iterations over all games. Fig. 1 shows the results for different ε. The experimental results are better than theoretical estimation in all cases. In the second part of experiments with matrix games we compare the proposed Algorithm MPAI with adaptive Mirror Prox method from [START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF]. In this part of experiments we modified problem (30) by adding inexactness (a bounded by δ) random noise) to the operator g of VI [START_REF] Korpelevich | Extragradient method for finding saddle points and other problems[END_REF]. Fig. 23show the results of these experiments.

For comparison we calculate the specific values that determine the degree of influence of the inexactness on the final estimate of the decisions' accuracy. So, not the whole error estimations are compared, but only improved by our approach part of the error estimations.

For Algorithm 1 this specific value is equal to (see (19)):

1 S N N -1 k=0 δ k+1 L k+1 y k+1 -x k+1 (32)
and for adaptive Mirror Prox [START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF] method we can estimate the analogous value in the following way:

1 S N N -1 k=0 δ L k+1 y k+1 -x k+1 .
According to the scheme of the proofs of Theorems 1 and 2, we can obtain the analogous value for non-adaptive Mirror Prox with constant step 1/L:

1 N N -1 k=0 δ y k+1 -x k+1 . ( 33
)
It is worth mentioning that estimate (32) should be less than (33) because of adaptive reduction δ k+1 < δ.

We may see that the accumulated error for proposed MPAI method is smaller than the error for adaptive Mirror Prox method from [START_REF] Gasnikov | Adaptive proximal method for variational inequalities[END_REF].

Conclusion

The paper proposes an analogue of the Mirror Prox method for variational inequalities with adaptive tuning not only for the constant L ν , but also for the magnitude of the operator's error. Moreover, such an error can set the degree of discontinuity of the operator. It is proved that the proposed method converges with the optimal estimate of complexity O 1 ε for the Lipschitz-continuous operator and the magnitude of the error is limited. It is important that the result applies to a certain class of variational inequalities with bounded (generally, discontinuous) operators (ν = 0). The paper also presents the results of experiments that demonstrate the ability to work with the estimate of the complexity close to O ln 1 ε even for a problem with bounded operator (ν = 0) and the experimental comparison between adaptive Mirror Prox and the proposed algorithm for a special application (matrix games with inexactness). We also note that all result of this paper is applicable for VI with relatively smooth operators (with the exception of Remark 5). More precisely, the prox-function and Bregman divergence in (2) may not be strongly convex (for convex optimization problems this situation was studied e.g. in [START_REF] Haihao | Relatively-Smooth Convex Optimization by First-Order Methods, and Applications[END_REF][START_REF] Stonyakin | Gradient Methods for Problems with Inexact Model of the Objective[END_REF]).

Fig. 1 :

 1 Fig. 1: The first kind of experiments with matrix games. Dependence between experimental and theoretical number of iterations for different ε. (a) 10 × 10 matrix A from (30), (b) 100 × 100 matrix A from (30).

Fig. 2 :

 2 Fig. 2: The second kind of experiments with matrix games. The logarithmic scale on the Error-axis. (a) ε = 1/100, δ = 1/300, (b) ε = 1/1000, δ = 1/6000.

Fig. 3 :

 3 Fig. 3: The second kind of experiments with huge-scale matrix games. The logarithmic scale on the Error-axis. 1000 × 1000 matrix A from (30).

  12 , . . . , α mn are represented by the matrix     α 11 α 13 . . . α 1n α 21 α 23 . . . α 2n . . . . . . . . . . . . . . . . . α m1 α m3 . . . α mn

Table 1 :

 1 The results of Algorithm 1 for objective function (27).

	Iterations General estimate Time (sec.)
	17	0.1051	0.264
	19	0.0527	0.291
	21	0.0266	0.315
	22	0.0212	0.342
	23	0.0177	0.354
	24	0.0133	0.364
	25	0.0106	0.380
	26	0.0082	0.427
	27	0.0063	0.467
	28	0.0048	0.423
	29	0.0044	0.443

Table 2 :

 2 The results of Algorithm 1 for objective function (29).

		n = 600, m = 400, N = 25		n = 1000, m = 500, N = 50
	Iteration General estimate Time (sec.) Iteration General estimate Time (sec.)
	22	0.122	67.955	19	0.1343	252.151
	23	0.061	70.587	20	0.0672	252.673
	24	0.0305	75.107	21	0.0336	266.636
	25	0.0153	72.917	22	0.0168	279.883
	26	0.0076	76.686	23	0.0084	293.866

Table 3 :

 3 The results of Algorithm 1 for objective function (29), with points A k in the unit ball.

		n = 100, m = 50, N = 25			n = 200, m = 100, N = 50
	Iteration General estimate Time (sec.) Iteration General estimate Time (sec.)
	318	0.2539	35.805		684	0.2522	441.744
	468	0.1702	52.809	1016	0.1688	645.082
	618	0.1279	71.484	1350	0.1267	857.185
	768	0.1026	87.851	1682	0.1015	1049.885
	918	0.0857	103.686	2016	0.0846	1305.006
	1068	0.0736	123.056	2349	0.0727	1534.333
	1218	0.0645	141.044	2683	0.0637	1753.489
	1368	0.0575	153.877	3014	0.0567	2026.402
	1518	0.0518	173.538	3348	0.0511	2210.362
	1668	0.0472	190.714	3680	0.0465	2301.327
	1818	0.0434	208.966	4014	0.0427	2611.942
	1968	0.0401	228.198	4346	0.0394	2970.324
	2118	0.0373	243.970	4678	0.0367	3153.905
	2268	0.0349	253.112	5012	0.0343	3387.451
	2426	0.0323	266.583	5346	0.0322	3619.831
	5 Numerical Experiments for Matrix Games with
	Inexactness
	We continue our experiments with computing a Nash equilibrium of a matrix
	game. For that purpose one should solve the saddle point problem
			min x ∈ ∆n	max y ∈ ∆m	x T Ay,

⋆ The research of F. Stonyakin in Sections 2, 3 and numerical experiments in Table 1 were supported by Russian Science Foundation (project 18-71-00048). The research of E. Vorontsova was supported by Russian Foundation for Basic Research (project number 18-29-03071 mk and project number 18-31-20005 mol-a-ved).


Dag Sehlin

Stina Syvänen
email: stina.syvanen@pubcare.uu.se

Gérard N Bischof

Delphine Boche

Hennig Boecker

Karl Peter Bohn

Alexander Drzezga

Kathrin Giehl

Michel Goedert

Jochen Hammes

Oskar Hansson

Karl Herholz

Günter Höglinger

Frank Jessen

Thomas Klockgether

Eckhard Mandelkow

Eva-Maria Mandelkow

Andreas Maurer

Bernd Neumaier

Agneta Nordberg

Özgur Onur

Kathrin Reetz

Axel Rominger

James Rowe

Osama Sabri

Anja Schneider

Antonio Strafella

Thilo Van Eimeren

Neil Vasdev

Engineered antibodies: new possibilities for brain PET?

Keywords: Transferrin receptor1 (TfR1)-mediated transcytosis, Alzheimer's disease (AD), Amyloid-β (Aβ), Antibody, Bloodbrain barrier (BBB), Positron emission tomography (PET)

Almost 50 million people worldwide are affected by Alzheimer's disease (AD), the most common neurodegenerative disorder. Development of disease-modifying therapies would benefit from reliable, non-invasive positron emission tomography (PET) biomarkers for early diagnosis, monitoring of disease progression, and assessment of therapeutic effects. Traditionally, PET ligands have been based on small molecules that, with the right properties, can penetrate the blood-brain barrier (BBB) and visualize targets in the brain. Recently a new class of PET ligands based on antibodies have emerged, mainly in applications related to cancer. While antibodies have advantages such as high specificity and affinity, their passage across the BBB is limited. Thus, to be used as brain PET ligands, antibodies need to be modified for active transport into the brain. Here, we review the development of radioligands based on antibodies for visualization of intrabrain targets. We focus on antibodies modified into a bispecific format, with the capacity to undergo transferrin receptor 1 (TfR1)-mediated transcytosis to enter the brain and access pathological proteins, e.g. amyloid-beta. A number of such antibody ligands have been developed, displaying differences in brain uptake, pharmacokinetics, and ability to bind and visualize the target in the brain of transgenic mice. Potential pathological changes related to neurodegeneration, e.g. misfolded proteins and neuroinflammation, are suggested as future targets for this novel type of radioligand. Challenges are also discussed, such as the temporal match of radionuclide half-life with the ligand's pharmacokinetic profile and translation to human use. In conclusion, brain PET imaging using bispecific antibodies, modified for receptor-mediated transcytosis across the BBB, is a promising method for specifically visualizing molecules in the brain that are difficult to target with traditional small molecule ligands.

Introduction

Positron emission tomography (PET) is a non-invasive, quantitative, functional imaging method. Clinically, PET is used to aid diagnosis, especially in cancer, where the radioactive glucose analogue [ 18 F]FDG is used to localize primary tumours and metastases. PET has also become an important tool for diagnosis of brain disorders, since naturally it is difficult to obtain biosamples from the brain. Further, PET is an attractive method in translational research and drug development, as the same experiments can be performed in vivo in both animals and humans, and it allows for repeated investigations in one subject.

The main hurdle for the delivery of drugs (and radioligands) to the brain, irrespective of their size, is the blood-brain barrier (BBB), comprising tightly connected endothelial cells. Traditionally, PET radioligands for the central nervous system (CNS) have been based on small "drug-like" molecules preferably labelled with clinically compatible positron-emitting radionuclides such as carbon-11 ( 11 C) or fluorine-18 ( 18 F). Radioligands for brain imaging have to be fairly lipophilic to be able to pass through the BBB into the brain parenchyma. Unfortunately, increased lipophilicity also increases nonspecific distribution into the lipophilic brain tissue. This may lead to a poor specific-to-nonspecific PET signal. Further, and especially relevant in proteopathies such as Alzheimer's disease (AD) and Parkinson's disease (PD), it is unlikely that small-molecule radioligands could discriminate between different aggregation forms of a protein or proteins with similar fibrillary structures. Thus, in line with the shift in therapeutic focus from small-molecule drugs to biologics, antibodies or fragments thereof could turn out to be a completely novel class of neuroPET radioligands and could be used for highly specific PET imaging in the CNS, including imaging of target proteins for which radioligands are lacking today.

Antibody transport across the blood-brain barrier

Radioligands based on antibodies or other proteins have already been introduced for peripheral targets related to cancer diagnostics and theranostics, including some applications in clinical use as well [START_REF] Niemeijer | Whole body PD-1 and PD-L1 positron emission tomography in patients with non-small-cell lung cancer[END_REF][START_REF] Jauw | Immuno-positron emission tomography with Zirconium-89-labeled monoclonal antibodies in oncology: what can we learn from initial clinical trials?[END_REF]. However, antibodies are large molecules, displaying highly restrictive BBB transcytosis. It has been reported that only 0.1% of peripherally administered antibody reaches the brain [START_REF] Bard | Peripherally administered antibodies against amyloid beta-peptide enter the central nervous system and reduce pathology in a mouse model of Alzheimer disease[END_REF][START_REF] Poduslo | Macromolecular permeability across the blood-nerve and blood-brain barriers[END_REF], and it has even been questioned whether antibodies penetrate the brain parenchyma at all, or whether antibody concentrations measured in the brain rather reflect transport from the blood into the cerebrospinal fluid (CSF) [START_REF] Pardridge | Re-engineering therapeutic antibodies for Alzheimer's disease as blood-brain barrier penetrating bi-specific antibodies[END_REF]. Thus, antibodies and other proteins will most likely have to be specifically engineered for facilitated transport across the BBB to enable their use as PET radioligands within the CNS.

Carrier-mediated transporters at the BBB have been described for essential compounds such as glucose and amino acids, while insulin and transferrin (Tf) are transported into the brain with receptor-mediated transcytosis (RMT). Especially the transferrin receptor (TfR) has been used to increase transport of antibody-based therapeutics across the BBB. This has in recent years proven to be a successful strategy in several preclinical studies [START_REF] Niewoehner | Increased brain penetration and potency of a therapeutic antibody using a monovalent molecular shuttle[END_REF][START_REF] Syvanen | Efficient clearance of Aβ protofibrils in AβPPtransgenic mice treated with a brain-penetrating bifunctional antibody[END_REF][START_REF] Yu | Boosting brain uptake of a therapeutic antibody by reducing its affinity for a transcytosis target[END_REF]. In early clinical studies, the insulin receptor was utilized for boosting the brain delivery of recombinant lysosomal enzyme α-L-iduronidase, aimed at treating patients with the metabolic disorder mucopolysaccharidosis (MPS) type I [START_REF] Giugliani | Neurocognitive and somatic stabilization in pediatric patients with severe mucopolysaccharidosis type I after 52 weeks of intravenous brain-penetrating insulin receptor antibodyiduronidase fusion protein (valanafusp alpha): an open label phase 1-2 trial[END_REF]. Other reported strategies for increasing brain delivery of antibodies include various binders targeting the endothelial cell surface protein CD98 [START_REF] Zuchero | Discovery of novel blood-brain barrier targets to enhance brain uptake of therapeutic antibodies[END_REF] and the singledomain antibody fragment FC5, targeting the transmembrane protein 50A (TMEM50A) [START_REF] Muruganandam | Selection of phage-displayed llama singledomain antibodies that transmigrate across human blood-brain barrier endothelium[END_REF][START_REF] Webster | Brain penetration, target engagement, and disposition of the blood-brain barrier-crossing bispecific antibody antagonist of metabotropic glutamate receptor type 1[END_REF]. The common feature of all the reported studies is that a binder for the selected BBB receptor (insulin receptor, TfR, CD98, etc.) is fused, either by recombinant expression or chemical conjugation, to the therapeutic antibody/protein. The generated bispecific protein or antibody is then expected to bind its BBB receptor and subsequently, after successful delivery across the BBB, interact with its primary intrabrain target (Fig. 1). Several initiatives are ongoing to find proteins expressed at the BBB that could act as brain shuttles for therapeutic antibodies or other biologics [START_REF] Stanimirovic | Emerging technologies for delivery of biotherapeutics and gene therapy across the bloodbrain barrier[END_REF][START_REF] Pardridge | Delivery of biologics across the blood-brain barrier with molecular Trojan horse technology[END_REF]. This molecular "Trojan horse" strategy, in addition to enhancing the concentration of therapeutic biologics in the brain, can be applied to generate bispecific radioligands. This review will focus on TfR binding bispecific radioligands.

The transferrin receptor

The TfR is found in two different isoforms: TfR1 and TfR2. TfR1 (also known as cluster of differentiation 71, or CD71) is expressed in brain endothelial cells, hepatocytes, and erythrocyte precursors, especially in bone marrow, lung, and other rapidly dividing cells. TfR2 is only expressed in hepatocytes, enterocytes of the small intestine, and erythroid cells. Both forms are expressed as transmembrane glycoproteins composed of two disulfide-linked monomers joined by two disulfide bonds. Each monomer binds one holo-transferrin molecule, creating an iron-Tf-TfR complex which enters the cell by endocytosis. In the endosome, the lower pH of around 5.5 will cause Tf to release its iron ions, which can subsequently be used by the cell. The TfR-Tf complex will then be recycled to the cell surface.

A number of antibodies have been generated against TfR1, among them the monoclonal mouse antibody OX-26 [START_REF] Jefferies | Transferrin receptor on endothelium of brain capillaries[END_REF] that binds to the rat TfR1, and rat antibody 8D3 [START_REF] Kissel | Immunohistochemical localization of the murine transferrin receptor (TfR) on blood-tissue barriers using a novel anti-TfR monoclonal antibody[END_REF] that binds to the murine TfR1. Initially, OX-26 and 8D3 were developed for immunohistochemical visualisation of brain capillaries. However, it was later discovered that the antibodies were also found in high concentrations in the brain after in vivo systemic Fig. 1 Transferrin receptor (TfR)-mediated transcytosis of a bispecific antibody. The bispecific antibody binds to TfR at the luminal side of the BBB. TfR transports the bispecific antibody inside an endosome across the BBB. The bispecific antibody is released on the abluminal side of the BBB and can then bind to its target (e.g. Aβ, reactive astrocytes, microglia) in the brain parenchyma administration, and further, that when fused to a protein cargo, they were able to carry its cargo across the BBB [START_REF] Jefferies | Transferrin receptor on endothelium of brain capillaries[END_REF][START_REF] Friden | Anti-transferrin receptor antibody and antibodydrug conjugates cross the blood-brain barrier[END_REF]18]. despite binding to TfR1 [START_REF] Sade | A human blood-brain barrier transcytosis assay reveals antibody transcytosis influenced by pH-dependent receptor binding[END_REF]. It is debated which factors govern the ability to induce TfR1 transcytosis. Some studies have suggested that low/moderate affinity promotes transcytosis, while high affinity reduces release from the receptor and leads to lysosomal degradation [START_REF] Yu | Boosting brain uptake of a therapeutic antibody by reducing its affinity for a transcytosis target[END_REF]. Other studies have suggested that the binding valency to the TfR1 is of importance, monovalent binding being more efficient than bivalent binding, which may cause TfR clustering at the cell surface and intracellular degradation [START_REF] Niewoehner | Increased brain penetration and potency of a therapeutic antibody using a monovalent molecular shuttle[END_REF]. These theories are not necessarily conflicting, as monovalent binding in general results in lower affinity (avidity). Yet another feature that may govern transcytosis efficiency is pH-dependent affinity to the receptor. Ideally, the antibody should have moderate TfR affinity at neutral pH for optimal engagement at the cell surface, and low affinity at low pH for release in the acidic endosome and further transport across the cell, into the brain. While it has not been systematically studied and reported, the location of the TfR binding epitope may also be of importance for an antibody's ability to undergo TfR-mediated transcytosis.

Alzheimer's disease and amyloid-β

Insoluble plaques of amyloid-beta (Aβ) in the brain of AD patients can today be visualized with PET using the thioflavin-T-derived small-molecule radioligand [ 11 C]PIB (Pittsburgh compound B) that was developed by the Uppsala University Hospital PET Centre in collaboration with Pittsburgh University in the early 2000s [START_REF] Klunk | Imaging brain amyloid in Alzheimer's disease with Pittsburgh compound-B[END_REF]. The introduction of amyloid imaging has been an important improvement in clinical diagnosis of AD, especially in cases when the cause of dementia is unclear [START_REF] Ossenkoppele | Prevalence of amyloid PET positivity in dementia syndromes: a meta-analysis[END_REF]. However, [ 11 C]PIB, and all later developed analogues, bind to the beta-sheet structure of insoluble plaques and give rise to a fairly static signal that does not correlate well with disease severity during the clinical stages of AD [START_REF] Rodriguez-Vieitez | Diverging longitudinal changes in astrocytosis and amyloid PET in autosomal dominant Alzheimer's disease[END_REF][START_REF] Engler | Two-year follow-up of amyloid deposition in patients with Alzheimer's disease[END_REF]. Further, since [ 11 C]PIB does not bind to diffuse plaques [START_REF] Scholl | Low PiB PET retention in presence of pathologic CSF biomarkers in Arctic APP mutation carriers[END_REF], a form of Aβ deposits that dominates AD pathology in about 5% of AD patients, there is a substantial risk for a false-negative diagnosis with [ 11 C]PIB. In addition, it appears that soluble aggregates of Aβ, e.g. protofibrils/oligomers formed before fibrils (Fig. 2), are involved in the pathological process that leads to neuronal degeneration, and further, that levels of soluble aggregates correlate better than insoluble plaques with disease severity [START_REF] Esparza | Soluble amyloid-beta aggregates from human Alzheimer's disease brains[END_REF][START_REF] Esparza | Amyloid-beta oligomerization in Alzheimer dementia versus high-pathology controls[END_REF][START_REF] Walsh | Naturally secreted oligomers of amyloid beta protein potently inhibit hippocampal long-term potentiation in vivo[END_REF]. Thus, efforts today are devoted towards therapeutic targeting of soluble Aβ, including protofibrils/ oligomers [START_REF] Lannfelt | Perspectives on future Alzheimer therapies: amyloid-beta protofibrils -a new target for immunotherapy with BAN2401 in Alzheimer's disease[END_REF], and the development of PET ligands that can visualize these Aβ species is therefore also of great interest.

Antibody-based radioligands for imaging of Aβ

One attractive feature of antibodies is that they can be generated to be specific, or selective, for certain aggregation forms or modifications (truncation, phosphorylation, etc.) of a protein. A few attempts to develop radioligands based on antibodies, mainly targeting Aβ, without a specific BBB shuttle moiety have been described in the literature, although some have been conjugated to nonspecific BBB modifiers like polyethylene glycol (PEG) [START_REF] Fissers | Synthesis and evaluation of a Zr-89-labeled monoclonal antibody for immuno-PET imaging of amyloid-beta deposition in the brain[END_REF][START_REF] Magnusson | Specific uptake of an amyloid-beta protofibrilbinding antibody-tracer in AβPP transgenic mouse brain[END_REF][START_REF] Mclean | Positron emission tomography imaging of fibrillar parenchymal and vascular amyloid-beta in TgCRND8 mice[END_REF][START_REF] Mclean | Anti-amyloid-beta-mediated positron emission tomography imaging in Alzheimer's disease mouse brains[END_REF][START_REF] Syvanen | Cationization increases brain distribution of an amyloid-beta protofibril selective F(ab')2 fragment[END_REF]. While few of the published articles show PET images or whole brain section ex vivo autoradiography, most of them report elevated brain concentrations of Aβ antibodies in transgenic versus wild-type (WT) mice. However, the total brain concentration of radiolabelled antibodies is low or animals are not perfused to exclude the radioactivity in blood, which makes it difficult to determine antibody concentrations in the brain parenchyma [START_REF] Magnusson | Specific uptake of an amyloid-beta protofibrilbinding antibody-tracer in AβPP transgenic mouse brain[END_REF][START_REF] Mclean | Positron emission tomography imaging of fibrillar parenchymal and vascular amyloid-beta in TgCRND8 mice[END_REF][START_REF] Mclean | Anti-amyloid-beta-mediated positron emission tomography imaging in Alzheimer's disease mouse brains[END_REF]. Fissers and co-workers include images of 89 Zr-labelled JRF/ AβN/25, a monoclonal antibody directed against full-length Aβ [START_REF] Fissers | Synthesis and evaluation of a Zr-89-labeled monoclonal antibody for immuno-PET imaging of amyloid-beta deposition in the brain[END_REF]. The radioligand showed in vitro stability and high affinity to Aβ, but ex vivo autoradiography showed that the radioligand mainly accumulated in local high-intensity deposits. This binding pattern has also been observed for other Aβ-binding antibodies such as 3D6 and mAb158 [START_REF] Syvanen | Efficient clearance of Aβ protofibrils in AβPPtransgenic mice treated with a brain-penetrating bifunctional antibody[END_REF][START_REF] Gustafsson | Blood-brain barrier integrity in a mouse model of Alzheimer's disease with or without acute 3D6 immunotherapy[END_REF]. It is likely that the high-intensity deposits represent radioligand binding to Aβ aggregates in the brain vasculature, i.e. cerebral amyloid angiopathy (CAA). To enable PET imaging of intrabrain Aβ, it is important that the radioligand is able to access the whole brain parenchyma, as illustrated in Fig. 3. This is most efficiently achieved by transport across the BBB in the whole brain capillary network, since diffusion from potential CAA deposits or areas of broken BBB into the deeper brain parenchyma is likely to be too slow [START_REF] Pardridge | CSF, blood-brain barrier, and brain drug delivery[END_REF][START_REF] Pizzo | Intrathecal antibody distribution in the rat brain: surface diffusion, perivascular transport and osmotic enhancement of delivery[END_REF].

The first bispecific antibody-based PET study that took advantage of TfR1 as a brain shuttle utilized the complete 8D3 antibody chemically fused to a F(ab') 2 fragment of an Aβ protofibril-binding antibody, mAb158 (Fig. 5a). Hence, the TfR1 binding was bivalent and led to a 15-fold increase in brain concentrations, measured 4 h after administration, compared with the F(ab') 2 fragment of the original antibody [START_REF] Sehlin | Antibody-based PET imaging of amyloid beta in mouse models of Alzheimer's disease[END_REF]. The same study also showed that 8D3 was equally efficient in shuttling a F(ab') 2 fragment of an antibody without any intrabrain target, and that the BBB transport took place both in WT animals and in two different models of AD. Thus, the study showed that BBB transcytosis per se was not influenced by subsequent binding (or no binding) in the brain parenchyma. The bispecific antibody was radiolabelled with 124 I, and PET images obtained 3 days post-administration, when unbound radioligand had been cleared, showed an increasing signal intensity with age (i.e. with increasing Aβ pathology) in the two AD animal models, while brains of WT mice were devoid of signal regardless of age (Fig. 4) [START_REF] Sehlin | Antibody-based PET imaging of amyloid beta in mouse models of Alzheimer's disease[END_REF][START_REF] Sehlin | Pharmacokinetics, biodistribution and brain retention of a bispecific antibody-based PET radioligand for imaging of amyloid-beta[END_REF].

To introduce monovalent TfR binding, fragments of 8D3 have been recombinantly fused with the primary antibody (Fig. 5). RmAb158-scFv8D3 (Fig. 5b) resembles the original full-sized mAb158 with an intact Fc domain [START_REF] Hultqvist | Bivalent brain shuttle increases antibody uptake by monovalent binding to the transferrin receptor[END_REF], with two scFv8D3 fragments attached to each of the IgG light chains. The short linkers between scFv8D3 and the light chain hinder simultaneous binding to TfR1 with both scFv8D3, while the two copies of scFv8D3 increase the chance of binding. Further, since RmAb158-scFv8D3 is a symmetrical antibody, it is easier to produce than asymmetrical, bispecific full-sized IgG-like antibodies that have been described in the literature [START_REF] Niewoehner | Increased brain penetration and potency of a therapeutic antibody using a monovalent molecular shuttle[END_REF][START_REF] Yu | Boosting brain uptake of a therapeutic antibody by reducing its affinity for a transcytosis target[END_REF]. PET imaging showed that [ 124 I]RmAb158-scfv8D3 reached considerably higher brain concentrations, resulting in higher-contrast images compared with [ 124 I]8D3-F(ab') 2 -h158 (Fig. 5a), which binds bivalently to TfR [START_REF] Hultqvist | Bivalent brain shuttle increases antibody uptake by monovalent binding to the transferrin receptor[END_REF]. In subsequent studies, smaller bispecific constructs have been used to promote faster clearance from blood, i.e. a feature that is desired for PET radioligands to minimize the radioactivity in the vascular volume of the brain. These studies showed that a tribody, based on two scFv158 and a Fab-8D3 [START_REF] Syvanen | A bispecific Tribody PET radioligand for visualization of amyloid-beta protofibrils -a new concept for neuroimaging[END_REF], with a size of approximately 110 kDa (Fig. 5c), and a tandem-scFv based on an scFv3D6 fused to an scFv8D3 [START_REF] Fang | High detection sensitivity with antibody-based PET radioligand for amyloid beta in brain[END_REF], with a size of approximately 58 kDa (Fig. 5d), were more rapidly cleared from blood than the larger 210 kDa [ 124 I]RmAb158-scfv8D3, and allowed for imaging of Aβ pathology at an earlier time point after administration (Fig. 5e).

Antibody vs PIB PET

PET imaging with bispecific antibodies has been performed primarily in two AD mouse models, tg-ArcSwe and tg-Swe [START_REF] Sehlin | Antibody-based PET imaging of amyloid beta in mouse models of Alzheimer's disease[END_REF][START_REF] Sehlin | Pharmacokinetics, biodistribution and brain retention of a bispecific antibody-based PET radioligand for imaging of amyloid-beta[END_REF][START_REF] Fang | High detection sensitivity with antibody-based PET radioligand for amyloid beta in brain[END_REF], expressing human APP with both the Arctic (E693G) and Swedish (KM670/671NL) amyloid precursor protein (APP) mutations or the Swedish mutation alone. Tg-ArcSwe is characterized by early formation of soluble Aβ aggregates and subsequent formation of dense core plaques, closely mimicking plaques formed in the human brain, while tg-Swe has a late-onset pathology, rapid progression, and less dense plaques [START_REF] Philipson | Animal models of amyloid-beta-related pathologies in Alzheimer's disease[END_REF]. PET which was readily detectable only in aged mice (18 months) [START_REF] Sehlin | Antibody-based PET imaging of amyloid beta in mouse models of Alzheimer's disease[END_REF] (Fig. 6a). When quantified as standardized uptake value ratio (SUVR), using cerebellum as reference region, [ 124 I]8D3-F(ab') 2 -h158 showed earlier and better discrimination between transgenic and WT mice compared with [ 11 C]PIB (Fig. 6b) [START_REF] Sehlin | Antibody-based PET imaging of amyloid beta in mouse models of Alzheimer's disease[END_REF][START_REF] Fang | High detection sensitivity with antibody-based PET radioligand for amyloid beta in brain[END_REF]. Interestingly, tg-ArcSwe mice, showing more dense plaque pathology, had higher [ 11 C]PIB retention than Swe mice, which instead gave a higher signal with the antibody ligand. This is in line with the two ligands' binding characteristics, i.e. d Tandem-scFv composed of scFv3D6 fused via a polypeptide linker to scFv8D3 (Mw ~58 kDa). e Blood elimination curves of recombinant antibody ligands (b), (c), and (d). The dashed red line represents an approximate antibody blood concentration threshold below which PET imaging is feasible. From the intersection of each antibody's blood curve with the threshold line, there is a projection to a time window where AD transgenic mice can at the earliest be discriminated from wild-type mice accumulation between the different models, including low accumulation in two models based on the Swedish mutation: APPSwe and APP/PS1dE9 [START_REF] Brendel | Cross-sectional comparison of small animal [18F]florbetaben amyloid-PET between transgenic AD mouse models[END_REF]. Only one model, PS2APP, showed SUVRs above 1.1 at the age of 15 months. Further, a clinical study demonstrated that AD patients with mainly diffuse pathology were [ 11 C]PIB negative [START_REF] Scholl | Low PiB PET retention in presence of pathologic CSF biomarkers in Arctic APP mutation carriers[END_REF]. Taken together this implies that antibody-based PET, but not [ 11 C]PIB, is able to visualize and quantify early formed and diffuse Aβ assemblies.

Additional evidence for the sensitivity of antibody PET is provided by a study where 10-month-old tg-ArcSwe mice were treated for 3 months with a BACE-1 inhibitor to decrease Aβ production. PET imaging with the recombinant [ 124 I]RmAb158-scfv8D3 readily detected reduced Aβ levels in treated compared to non-treated animals, at an age when [ 11 C]PIB can hardly detect Aβ [START_REF] Meier | Antibody-based in vivo PET imaging detects amyloid-beta reduction in Alzheimer transgenic mice after BACE-1 inhibition[END_REF].

Other pathological changes in need of novel radioligands α-Synuclein

Similar to AD, protein misfolding and aggregation is also a pathological feature in PD. In PD, the presynaptic protein αsyn initially forms oligomers and later insoluble aggregates. There are presently no PET ligands available for imaging of either soluble or insoluble α-syn, and several research programmes are aimed at developing small-molecule PET ligands for α-syn [START_REF] Eberling | Alpha-synuclein imaging: a critical need for Parkinson's disease research[END_REF]. The lack of in vivo biomarkers for α-syn is a major limitation for the development of disease-modifying treatments for PD, and the Michael J. Fox Foundation (MJFF) recently announced a $2 million prize to the first team to develop a viable selective α-syn PET radioligand. A number of initiatives to develop a radioligand for α-syn are ongoing, and most of these are based on small molecules. Although some promising compounds have been presented, one major hurdle is the cross-reactivity and binding to Aβ. Even if the affinity of a specific compound is much higher for α-syn than Aβ, the abundance and availability of Aβ may be higher. Thus, it might be difficult to differentiate between an α-syncontaining brain and an Aβ-containing brain, even if Aβ levels are low. This may be especially a problem when different protein pathologies coexist, which is often the case in older individuals. In this respect, highly specific antibodies for αsyn may be a new option for developing radioligands truly specific for α-syn. One potential challenge is that the majority of aggregated α-syn in the brain appears to be intracellular, and thus an additional barrier has to be conquered by the bispecific antibody in order to reach its primary target.

Microglia and astrocytes

PET imaging of activated microglia and reactive astrocytes has been used as an indication of neuroinflammation. For example, a number of PET radioligands have been developed for the 18-kDa translocator protein (TSPO), which is highly expressed on activated microglia. However, quantitative interpretation of the PET signal with the second-and thirdgeneration TSPO PET radioligands is confounded by large interindividual variability in binding affinity due to a genetic polymorphism leading to a trimodal distribution, reflecting high-affinity binders (HABs), low-affinity binders (LABs), and mixed-affinity binders (MABs) [START_REF] Vivash | Imaging microglial activation with TSPO PET: lighting up neurologic diseases?[END_REF]. In addition, TSPO is also expressed on astrocytes, and hence the TSPO ligands b Quantification of PET images from (a) expressed as standardized uptake value ratio (SUVR) of two regions of interest, cortex (ctx) and hippocampus (hc), using cerebellum as reference region. Significant differences were observed for all regions and ages except for hippocampus in the 12-month-old animals are not specific for microglia. Reactive astrocytes, also indicative of neuroinflammation, can be imaged with the PET radioligand deuterium-L-depreneyl ([ 11 C]DED). [ 11 C]DED binds to monoamine oxidase-B, primarily found in activated astrocytes, and although studies indicate that the radioligand indeed visualizes astrocytosis, its binding differs from that of other astrocytic markers often used in immunohistochemical analysis in post-mortem neuropathological studies, such as glial fibrillary acidic protein (GFAP) [START_REF] Rodriguez-Vieitez | Astrocytosis precedes amyloid plaque deposition in Alzheimer APPswe transgenic mouse brain: a correlative positron emission tomography and in vitro imaging study[END_REF][START_REF] Olsen | Astroglial responses to amyloid-Beta progression in a mouse model of Alzheimer's disease[END_REF]. Attempts have been made to image GFAP with antibody fragments [START_REF] Li | Cell-penetrating anti-GFAP VHH and corresponding fluorescent fusion protein VHH-GFP spontaneously cross the blood-brain barrier and specifically recognize astrocytes: application to brain imaging[END_REF], and myriad well-characterized antibodies for proteins expressed on microglia and astrocytes have been described in the literature. Hence, the possibility of engineering these into bispecific brain-penetrating radioligands is tempting and could potentially allow for "in vivo immunohistochemistry" of classical ex vivo immunohistochemical targets.

Challenges

TfR transport capacity

One central paradigm of PET is the use of tracer doses that do not elicit a pharmacological response or occupy a significant fraction of potential binding sites. The use of doses above true tracer doses may have an impact on the PET signal, and hence the interpretation of the study. It is therefore important to estimate the capacity of the TfR. For example, a study using [ 125 I]RmAb158-scfv8D3 showed that, compared with unmodified [ 125 I]RmAb158, the transport of bispecific radioligand into the brain was increased almost 100-fold at tracer doses (0.05 mg/kg), while a tenfold increase was observed at a dose of 10 mg/kg [START_REF] Hultqvist | Bivalent brain shuttle increases antibody uptake by monovalent binding to the transferrin receptor[END_REF]. The blood pharmacokinetics were linear, i.e. the half-life in blood was the same for the tracer and the pharmacological dose, and did therefore not contribute to the changed BBB transcytosis efficacy. Moreover, the study showed that co-administration of fullsized 8D3, also at a dose of 10 mg/kg, even more efficiently inhibited transcytosis by reducing the brain uptake of [ 1 2 5 I]RmAb158-scfv8D3 to threefold more than [ 125 I]RmAb158. The lower inhibition capacities of RmAb158-scfv8D3 compared with 8D3 can most likely be explained by the former compound's monovalent TfR binding.

To further understand the TfR transcytosis efficiency in relation to TfR occupation, brain uptake was investigated at different intravenously administered doses of RmAb158-scfv8D3. Doses up to 1 mg/kg had no impact on BBB transport, resulting in brain antibody concentrations of around 1.5% of the injected dose per gram brain tissue at 2 h postinjection. At higher doses, TfR seemed to be saturated, and the brain uptake was reduced in a dose-dependent manner (Fig. 7). While this may be a concern for therapeutic applications, all PET studies described here were conducted at antibody doses below 0.5 mg/kg body weight, calculated as IgG equivalents, i.e. below the limit of TfR saturation.

Radiolabelling

Protein-and antibody-based PET radioligands have traditionally been radiolabelled with radionuclides such as iodine-124 ( 124 I; half-life 4.2 days), zirconium-89 ( 89 Zr; half-life 3.3 days), or gallium-68 ( 68 Ga; half-life 68 min). Except for 124 I, these radionuclides require the attachment of a chelator on the antibody/protein backbone before the introduction of the radionuclide. The instability of the chelators was initially a major challenge, but new and more stable versions that may be better suited for clinical use have been introduced [START_REF] Van Dongen | PET imaging with radiolabeled antibodies and tyrosine kinase inhibitors: immuno-PET and TKI-PET[END_REF]. On the other hand, 124 I has not been a preferred alternative due to its accumulation in the thyroid, resulting in high local exposure. Further, the above-mentioned radionuclides are not ideal for clinical use due to low fraction of positron decay (26% for 124 I and 23% for 89 Zr-only this fraction of radioactivity is detected by PET). Also, the high energy of 124 I positrons, which allows them to travel a longer distance in the tissue before electron annihilation, causes low-resolution PET images. In contrast, the low-energy positrons from 18 F decay generate PET images with high resolution.

For small-molecule radioligands used clinically, fluorine-18 ( 18 F; half-life 110 min) is the first choice for radiolabelling. However, radiochemistry methods for introducing 18 F on Fig. 7 Dose vs brain uptake of RmAb158-scFv8D3. Brain antibody uptake (% of injected dose per gram brain tissue) was measured 2 h post-injection of RmAb158-scFv8D3 at doses ranging from 0.25 mg/kg body weight to a high therapeutic dose of 10 mg/kg body weight. While antibody doses relevant to PET imaging (below 0.5 mg/kg) had no impact on brain delivery, doses above 1 mg/kg seemed to saturate the TfR transport mechanism, resulting in reduced brain uptake antibody-based ligands have been lacking. With the increased interest in protein-based radioligands, increasing efforts have also been observed when it comes to 18 F radiolabelling. Different strategies involving" click-chemistry", e.g. Diels-Alder reaction, have been described [START_REF] Wang | Conformationally strained trans-cyclooctene (sTCO) enables the rapid construction of (18)F-PET probes via tetrazine ligation[END_REF]; the radiolabelling is somewhat similar to the above-mentioned chelator methods, i.e. a two-step process. First, the antibody/protein is modified by introducing a small-molecule group that in a second step can react with another small molecule that carries the 18 F. Some methods for direct radiolabelling of amino acids, e.g. amine groups, with 18 F have also been described [START_REF] O L B E R G D E , E T A L . O N E | s t e p r a d i o s y n t h e s i s o f 6[END_REF][START_REF] Basuli | Fast indirect fluorine-18 labeling of protein/peptide using the useful 6-fluoronicotinic acid-2,3,5,6-tetrafluorophenyl prosthetic group: a method comparable to direct fluorination[END_REF].

Pharmacokinetics

The radioactivity measured with PET in the brain (or any tissue of interest) comprises radioactivity in the tissue itself as well as radioactivity in the blood pool of the tissue. For example, around 3% of the brain volume is blood [START_REF] Todd | Microwave fixation for the determination of cerebral blood volume in rats[END_REF][START_REF] Watabe | CBF/CBV maps in normal volunteers studied with (15)O PET: a possible index of cerebral perfusion pressure[END_REF]. Thus, PET radioligands should be cleared from the blood fairly rapidly to minimize the radioactivity contribution from the blood pool of the tissue. This is especially important for the brain and for neuroPET radioligands with limited brain distribution. Full-sized antibodies are generally associated with long systemic half-life, while a more rapid elimination can be achieved by modifying antibodies into fragments (Fab, F(ab') 2 , scFv). However, even fragments may display halflives that are not compatible with clinically preferred radionuclides 11 C and 18 F. In addition, specific and nonspecific binding to peripheral targets may contribute to the observed halflife. For example, binding to soluble circulating and erythrocyte-expressed TfR1 may either prolong or shorten the circulation time. Although there are limited published data, it appears that smaller size and lower TfR1 affinity leads to faster elimination from blood [START_REF] Syvanen | A bispecific Tribody PET radioligand for visualization of amyloid-beta protofibrils -a new concept for neuroimaging[END_REF][START_REF] Fang | High detection sensitivity with antibody-based PET radioligand for amyloid beta in brain[END_REF]. Hence, studying the systemic pharmacokinetics of antibody-based ligands is essential before deciding on a labelling strategy. Moreover, to achieve a high signal-to-noise ratio, unbound ligand must also be eliminated from the brain within a time frame that matches the half-life of the radionuclide. Knowledge about clearance rates and routes of bispecific antibodies from the brain is sparse, and more research will be required to elucidate whether brain clearance is passive or mediated by active transport mechanisms, and whether it is size-dependent.

Translation

Translation from preclinical imaging to clinical imaging is challenging from many perspectives. In addition to dosimetry and pharmacokinetics that may differ between different species, the actual target under investigation may also be different although its biological "purpose" may be the same in different species. In PET, species differences have been observed for both neuroreceptors and transporters at the BBB [START_REF] Syvanen | Species differences in blood-brain barrier transport of three positron emission tomography radioligands with emphasis on P-glycoprotein transport[END_REF][START_REF] Pike | PET radiotracers: crossing the blood-brain barrier and surviving metabolism[END_REF]. The differences include different density of the target, different function and different amino acid composition. Limited knowledge about species-related differences in BBB receptors that mediate transcytosis, e.g. the TfR1 and the insulin receptor, makes it difficult to predict the brain delivery in humans based on preclinical data. The murine TfR1 and the human TfR1 differ in the amino acid sequence at the domain where the 8D3 antibody binds, resulting in no binding of 8D3 to the human TfR. However, antibodies binding human TfR1 at the same domain as 8D3 have been generated and shown to successfully shuttle biologics across the in vitro human BBB and in vivo in monkeys [START_REF] Yu | Therapeutic bispecific antibodies cross the blood-brain barrier in nonhuman primates[END_REF][START_REF] Pardridge | Blood-brain barrier transport, plasma pharmacokinetics, and neuropathology following chronic treatment of the rhesus monkey with a brain penetrating humanized monoclonal antibody against the human transferrin receptor[END_REF]. Attempts to generate speciesindependent TfR1 antibodies, which would aid in translation, have been described, but with limited success. One exception is perhaps the generation of variable new antigen receptors (VNARs) that appear to bind both mTfR1 and hTfR1 [START_REF] Stocki | Combined in vitro/in vivo methods for selecting VNARs that shuttle large therapeutic molecules across the blood brain barrier[END_REF]. Published data on the capacity of the VNARs to shuttle antibody cargos across the BBB is limited, however, and it appears that the VNARS may be less efficient for BBB transport of full-sized antibodies compared with smaller fragments. This is not the case for 8D3. In summary, it appears that TfR1 can be used in both mice and men, but may require species-specific TfR1 binders. The expression of TfR1 at the murine and human BBB has been reported to be similar [START_REF] Pardridge | Re-engineering therapeutic antibodies for Alzheimer's disease as blood-brain barrier penetrating bi-specific antibodies[END_REF].

Very little data on translatability and species differences have been reported for CD98 and the TMEM50A binder FC5, although they are claimed to be species-independent [START_REF] Zuchero | Discovery of novel blood-brain barrier targets to enhance brain uptake of therapeutic antibodies[END_REF][START_REF] Muruganandam | Selection of phage-displayed llama singledomain antibodies that transmigrate across human blood-brain barrier endothelium[END_REF][START_REF] Webster | Brain penetration, target engagement, and disposition of the blood-brain barrier-crossing bispecific antibody antagonist of metabotropic glutamate receptor type 1[END_REF].

Discussion

The development of antibody-based PET ligands for brain disorders has so far not been feasible, as techniques to facilitate large protein delivery to the brain have been lacking. Still, antibodies have many benefits, particularly their ability to bind specifically to their target, which is advantageous for obtaining PET images without background noise caused by nonspecific binding. Thus, the development of protein engineering strategies to increase antibody concentrations in the brain may enable a completely new class of radioligands with no or very low nonspecific binding.

The high affinity and specificity of antibodies are attractive, especially when specific aggregation forms of a protein are of interest. This is the case for many misfolded proteins, e.g. Aβ and α-syn, for which protein aggregates representing intermediate stages in the aggregation pathway is believed to be more toxic and dynamic than the insoluble end state deposits. In addition, numerous antibodies have been described for inflammation markers on astrocytes and microglia. Generation of bispecific antibodies able to pass the BBB based on these well-characterized and frequently used antibodies for immunohistochemical analyses could likely lead to novel in vivo imaging biomarkers.

Although already shown to be successful in the preclinical setting, the translation into clinical use will require the development of new radiochemistry for incorporation of more clinically suitable radionuclides such as 18 F. A number of radiochemical strategies have been described, but large-scale synthesis and reproducibility has to be improved. However, labelling antibodies with 18 F, which has a half-life of less than 2 h, will also require fast clearance of the ligand from blood and of unbound ligand from the brain. Further research on the pharmacokinetics of bispecific antibody ligands of different formats is needed to optimize these parameters for clinical development.

Another challenge is finding species-independent BBB shuttles, or validated shuttles in higher species. This has to some extent been accomplished for TfR1, although no species-independent binders have been proven to be as efficient as current mTfR1 binders. Several large-scale projects aiming to discover novel shuttles beyond TfR binders are ongoing.

In conclusion, we have already entered the era of biologics, both for the periphery and for the CNS, and it is likely that antibody-or protein-based radioligands will also become an important class of PET radioligands. A number of preclinical studies have shown the feasibility of antibody-based imaging of Aβ pathology in the brain, and although some hurdles remain, this novel class of tracers is likely to enter clinical development within the next few years.

  imaging with [ 124 I]8D3-F(ab') 2 -h158 and [ 11 C]PIB revealed an intense PET signal with the antibody-based ligand, but a fairly low signal with [ 11 C]PIB,

Fig. 2

 2 Fig. 2 Protein misfolding causes aggregation of monomeric Aβ into oligomers and protofibrils, which ultimately form insoluble fibrils, i.e. the main constituent of Aβ plaques which are characteristic for AD. Of the soluble species, oligomers/protofibrils are most likely the toxic forms

  [ 11 C]PIB requires dense core plaques, while [ 124 I]8D3-F(ab') 2 -h158 binds to diffuse assemblies including oligomers and protofibrils. Quantification of Aβ levels in brain tissue from the previously scanned animals showed that the [ 124 I]8D3-F(ab') 2 -h158 PET signal correlated closely with protofibril levels, while no correlation was observed with total Aβ levels (corresponding to plaque load). The need for dense core plaques for imaging of Aβ with [ 11 C]PIB and analogues has also been shown in other studies. Snellman and co-workers demonstrated increasing [ 11 C]PIB signal with age in APP23, a model with compact Aβ assemblies, while no such trend and very low brain concentrations were found in Tg2576 mice, a model expressing only the Swedish mutation and thus characterized by the absence of dense plaques [43]. Brendel and co-workers conducted a cross-sectional study with [ 18 F]florbetaben in four different AD mouse models and showed large variation in radioligand

Fig. 4 Fig. 5

 45 Fig. 4 Sagittal PET images obtained at 3 days after administration of the bispecific radioligand [ 124 I]8D3-F(ab') 2 -h158 in two mouse models of AD (ArcSwe and Swe) and wild-type (WT) mice at different ages

Fig. 6

 6 Fig. 6 Comparison of antibody and PIB PET imaging. a Sagittal view of PET images obtained from 12-and 18-month-old tg-ArcSwe mice 3 days post-injection of [ 124 I]8D3-F(ab') 2 -h158 (upper panel) in comparison with mice scanned 40-60 min after injection of [ 11 C]PIB (lower panel). b Quantification of PET images from (a) expressed as standardized

Acknowledgements We would like to thank Tobias Gustavsson for the generation of the autoradiography and immunohistochemistry images shown in Fig. 3. This work was supported by grants from the Swedish Research Council (2017-02413, 2018-02715), Swedish Innovation Agency, the Swedish Alzheimer's Foundation and the Swedish Brain Foundation.

Faculty of the Multimodal Imaging in Neurodegeneration Cologne (MINC) symposium

Bénédicte Ballanger, Lyon Neuroscience Research Center, Lyon, France, EU Henryk Barthel, University Hospital Leipzig, University of Leipzig,

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