Paola Boito
email: paola.boito@unilim.fr

Olivier Ruatta
email: olivier.ruatta@unilim.fr

Generalized companion matrix for approximate GCD

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f (x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]/(f ). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f, g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.

Introduction

The approximate polynomial greatest common divisor (denoted as AGCD) is a central object of symbolic-numeric computation. The main difficulty of the problem comes from the fact that is no universal notion of AGCD. One can find different approaches and different notions for AGCD. We will not give a review of all the existing work on this subject, but we will recall one of the most popular approaches to show how our work brings a different point of view on the problem.

The main approach to the computation of an AGCD consists in considering two univariate polynomials whose coefficients are known with uncertainty. This uncertainty can be the result of the fact that the polynomials have floating point coefficients coming from previous computation (and so are subject to round-off errors). The most frequently adopted formulation is related to semi-algebraic optimization : given f and g two approximate polynomials, find two polynomials f and g such that ff and gg are small (lower than a given tolerance for instance) and such that the degree of gcd(f, g) is maximal. That is, one looks for the most singular system close to the input ( f , g). An ε-gcd is obtained if the conditions ff < ε and gg < ε are satisfied. One can try to compute the tolerance on the perturbation of the input polynomial thanks to direct computation (for instance from a jump on singular values of particular matrix for instance). This last approach has received a great interest following the work of Zeng using Sylvester like matrices ( [START_REF] Zeng | The approximate GCD of inexact polynomials Part I: a univariate algorithm[END_REF]).

Here, we consider a slightly different problem. One of the polynomials, say f , is known exactly (it is the result of an exact model) and the second one, say g, is an approximate polynomial (result of measures or previous approximation for instance). This case occurs in applications such as model checking (to compare results of an exact model and measures). There are many other instances of such a problem, such as simplification of fractions when one of the polynomial is known exactly but the other one is not.

We give a example of such a situation. When modeling an electromagnetic filter, one might want to parametrize its behavior with respect to the frequency. But one may need to do so even if there are singularities and to do so one may use Padé approximations of the electromagnetic signal at each point as a function of the frequency. In some cases of interest, one can know all the singularities and so compute an exact polynomial called characteristic. Padé approximations are computed independently for each point by a numerical process and denominators may have a non trivial gcd with the "characteristic" polynomial. The denominators are not known exactly. So, in order to identify unwanted common factors in denominators one has to compute approximate gcds between an exact and non exact polynomials.

This AGCD problem can also be interpreted as an optimization problem. Given f exactly and g approximately, compute a polynomial g close to g such that g has a maximal degree gcd with f . Our approach takes advantage of the asymmetry of the problem and of the structure of the quotient algebra C[x]/(f (x)) (more accurately, of the displacement rank of the multiplication operator in this algebra). So, we address the following problem :

Problem 1 Let f (x) ∈ C[x]
a given polynomial and g(x) another polynomial. Find g(x) close to g(x) (in a sense that will be explained) such that f (x) and g(x) have a gcd of maximal degree.

This may be also an interesting approach when one has two polynomials, one known with high confidence and another with worse accuracy. This approach may take advantage of this asymmetry which would not be possible for classical framework based on Sylvester or Bézout matrices.

In this paper, we propose an approach and an algorithm to address this problem. The proposed algorithm is "fast" since the exponent of its complexity is better than the classical linear algebra exponent in the degree of the input polynomials.

Organisation of the paper: The second section is devoted to some basic result on algebra needed after, the third section gives an algebraic method for gcd based on linear algebra, the fourth section recalls the Barnett formula allowing to compute the multiplication matrix without division, the fifth gives the displacement rank structure of the multiplication matrix, the sixth describes the final algorithm and experiments before finishing with conclusions and perpectives.

Euclidian structure and quotient algebra

In this section, we recall basic algebraic results allowing to understand the principle of our approach. All material in this section can be found (even in the non reduced case and in the multivariate setting) in [START_REF] Mourrain | Relations between roots and coefficients, interpolation and application to system solving[END_REF].

Assume that K is an algebraically closed field (here we think about C).

Let f (x) and g(x) ∈ K[x] and assume that f

(x) = f d * d i=1 (x -ζ i )
and that

ζ i = ζ j for all i = j in {1, . . . , d}. Let A = K[x]/(f ) and π : K[x] -→ A be the natural projection. For i ∈ {1, . . . , d}, we define L i (x) = j =i (x-ζj ) j =i
(ζi-ζj ) , the i th Lagrange polynomial associated to {ζ 1 , . . . , ζ d }. Clearly, since deg(L i ) < deg(f ) we have π(L i ) = L i , for all i ∈ {1, . . . , d}. Let A * = Hom K (A, K) be the usual dual space of A. For all i ∈ {1, . . . , d}, we define 1 ζi : A -→ K by 1 ζi (p) = p(ζ i ) for all p ∈ A. The following lemma is obvious form the definition of the polynomials L i that for i and j ∈ {1, . . . , d}, we have g(ζ i )L i (x). In other word, divide g by f is equivalent to evaluate g at the roots of f . The general philosophy of this last proposition will allows us to make a lot of proof in a very simple way. For example, it is very easy to see the different operation in A using this representation. Let g and h be to elements in A, then we have g

L i (ζ j ) = 1 if i = j 0 else .
+ h = d i=1 (g(ζ i ) + h(ζ i )) * L i (x) and g * h = d i=0 (g(ζ i ) * h(ζ i ))L i (x) in A.
This allows us to avoid the use of the section σ. In fact, the Lagrange polynomials L 1 , . . . , L d reveal a deeper structure on the algebra A : The polynomials

L 1 , . . . , L d are the idempotents of A, i.e. L i * L j = L i if i = j 0 else .
Thanks to this description of the quotient algebra, it is easy to derive algorithms for both polynomial solving and gcd computation even though the problems are of very different nature.

Remark that we have expressed everything in the monomial basis since it is the most widely used basis to express polynomials but we could use other bases. A particular basis is the Chebyshev basis where all results are exactly the same since it is a graduated basis.

An algebraic algorithm for gcd computation

To first give an idea on how to exploit the section above in order to design algorithm for gcd, We recall a classical method for polynomial solving (see [START_REF]On two iterative methods for approximating roots of a polynomial[END_REF] for instance). Proofs are given for the sake of completeness and because very similar ideas will lead us to the AGCD computation.

Roots via eigenvalues

Let f (x) = d i=0 f i x i ∈ C[x
] be a polynomial of degree d. Then we consider the matrix of the multiplication by x in C[x]/(f ). Its matrix in the monomial basis 1, . . . , x d-1 is the following:

Frob(f ) = 1 x x 2 • • • x d-1        1 x x 2
. . .

x d-1                0 0 0 • • • -f0 f d 1 0 0 • • • -f1 f d 0 1 0 • • • -f2 f d . . . . . . . . . . . . . . . 0 0 0 • • • - f d-1 f d        
well known as the Froebenius companion matrix associated to f .

Proposition 1 Let f (x) ∈ C[x]be polynomial of degree d with d distinct roots Z(f ) = {z 1 , . . . , z d }, then the eigenvalues of Frob(f ) are the roots of f (x), i.e. Spec(Frob(f )) = {z 1 , . . . , z d }.
Proof It follows directly from the fact that Frob(f ) is the matrix of the multiplication by x in C[x]/(f ). But here we propose to give a direct proof by induction. In fact, we prove by induction that the characteristic polynomial of Frob(f ) is f (x) itself (up to a sign and a scalar factor 1/f d ), i.e.:

-

x 1 0 • • • -f0 f d 0 -x 1 • • • -f1 f d . . . . . . . . . . . . . . . 0 0 0 • • • -x - f d-1 f d = -f (x).
since we have:

-x 1 0 • • • -f0 f d 0 -x 1 • • • -f1 f d . . . . . . . . . . . . . . . 0 0 0 • • • -x - f d-1 f d = -x * -x 1 0 • • • -f1 f d . . . . . . . . . . . . . . . 0 0 0 • • • -x -f d-1 f d - f 0 f d = -(x * f (x)+ f 0 f d )
and by assumption f (x) = f (x)-f0

f d * x
and finally we have the wanted result. Remark that this proof allows to avoid the condition that all the roots of f (x) are distinct.

Then, to compute the roots of f (x) one can compute the eigenvalues of is Froebenius companion matrix. This is the object of the method proposed (reintroduced) by Edelman and Murakami [START_REF] Edelman | Polynomial roots from companion matrix eigenvalues[END_REF] and revisited by Fortune [START_REF] Fortune | An iterated eigenvalue algorithm for approximating roots of univariate polynomials[END_REF] and many others trying to use the displacement structure of the companion matrix. In fact, often, the author realized that the monomial basis of the quotient algebra is not the most suitable one and proposed to express the matrix of the same linear application but in other basis. In the case of the Chebyshev basis this algorithm was already known by Barnett [2] and Cardinal later [START_REF]On two iterative methods for approximating roots of a polynomial[END_REF].

In the next section, we will also take advantage of the structure of the quotient algebra to design an algorithm for gcd computation mainly using linear algebra (eigenvalues are used in theory and never computed).

Structure of quotient and gcd

Let f (x) and g(x) ∈ K[x] such that they are both monic. As above, we denote A = K[x]/(f ) and d = deg(f ). We denote denote {ζ 1 , . . . , ζ d } the set of roots of f (x) and we assume that f (x) is squarefree, i.e. ζ i = ζ j if i = j. We define

M g : A -→ A h -→ π(gh) where π(p) ∈ A denote the remainder of p(x) ∈ K[x]
by division by f (x). We denote M g the matrix of M g in the monomial basis 1, x, . . . , x d-1 of A but other bases can be used. A matrix representing the map M g is called an extended companion matrix.

Proposition 2

The eigenvalues of M g are {g(ζ 1 ), . . . , g(ζ d )}.

Proof It is a direct corollary of the proposition ?? since if we write the matrix of this linear map in the Lagrange basis associated to {ζ 1 , . . . ,

ζ d } is    g(ζ 1 ) • • • 0 . . . . . . . . . 0 • • • g(ζ d )   
and gives the wanted result.

Trivially, we have:

Corollary 1 We have corank(M g ) = deg(f ) -rank(M g ) = deg(gcd(f, g)).
The column of index i of M g is the column vector of the coefficients of x i-1 * g(x).

Let p 1 , . . . , p l be a basis of Ker(M g ) and let P 1 (x), . . . , P l (x) be the corresponding polynomials. First remark that Ann A (g) = {P (x) ∈ A|P (x) * g(x) = 0} is an ideal of A.

Lemma 1 The ideal Ann

A (g) is a principal ideal. Proof Let us define s(x) = ζ∈Z(f )\(Z(f )∩Z(g)) (x -ζ). For all h ∈ Ann A (g) it is clear that Z(h) ⊃ Z(f )\(Z(f ) ∩ Z(g)) and then s divide h. Furthermore s ∈ Ann A (g) since in the Lagrange basis s(x) * g(x) = d i=1 s(ζ i ) * g(ζ i )L i (x) = 0.
This shows that Ann A (g) = (s).

To compute s(x), we built the matrix with columns formed by p 1 , . . . , p l and we make a triangulation operating only on the columns. This way we obtain the polynomial of minimal degree linear combination of P 1 (x), . . . , P l (x) and it is easily seen that this s(x) up to a multiplicative scalar factor.

Lemma 2

The first column of a column echelon form of the matrix K g built from a basis of Ker(M g ) is the generator of Ann A (g), i.e. it is the vector of the coefficients of s(x) up to a scalar multiplication.

Proof Since the columns of a column echelon form of the matrix K g are linearly independent, they form a basis of Ann A (g) as K-vector space. So s(x) is a linear combination of the polynomials associated to those columns. The polynomial associated to the column echelon form of K g have all different degree (because it is an echelon form) and so s(x) is a linear combination of those polynomial. Because s(x) as the lowest degree possible, it is a scalar multiple of the polynomial associated to the first column.

Proposition 3 f (x) ∧ g(x) = f (x) s(x) .
Proof By construction, we have s(x) * g(x) = 0 mod f (x) and so s(x) divide f (x). We also have gcd( f (x) s(x) , g(x)) = gcd(f (x), g(x)) since the roots of f (x) s(x) are the root of f (x) where g(x) vanishes. Since deg( f (x) s(x) ) = deg(gcd(f (x), g(x)) we have the wanted result.

In all this section, we did not care if the polynomials are known in monomial or Chebyshev basis for instance. In fact, in order to have an algebraic algorithm, we only need to be able to perform euclidian division and this is always the case if the polynomial basis is graduated (as for monomial, Chebyshev, most of the orthogonal bases).

Bezoutian and Barnett's formula

A classical matricial formulation of resultant is given by the Bézout matrix. In this part, we recall the construction of the Bézout matrix and a special factorization of the multiplication matrix expressed in the monomial basis. This factorization is called Barnett formula (see [START_REF] Barnett | Polynomials and linear control systems[END_REF]). The Barnett's formula allows to build the classical extended companion matrix without using euclidian division and only stable numerical computations. Furthermore, this factorization reveals that the extended companion matrix has a special rank structure and we will use this fact later to design a fast algorithm to compute AGCD.

Definition 1 Let f and g ∈ C[x] of degree m and n respectively (with n m),

we denote Θ f,g (x, y) = f (x)g(y)-f (y)g(x)
x-y = i,j θ i,j x i y j = m-1 j=0 κ f,g,j (x)y j . The Bézout matrix associated with f and g is B f,g = θ j,j i,j∈{0,...,m-1} .

Remark that since Θ f,g (x, y) = Θ f,g (y, x) the matrix B f,g is symmetric. The polynomials κ f,g,j (x) are univariate polynomials of degree at most m -1. One particular case of interest is when f = 1. In this case the Bézout matrix has a Hankel structure, i.e. θ i,j = θ i-1,j+1 . In this case we denote H g,i (x) = κ 1,g,i (x) for i ∈ {0, . . . , m -1} which are called the Horner polynomials.

Proposition 4 Let i ∈ {0, . . . , m -1}, the polynomial H g,i (x) = c 1,m-i + • • • + c 1,m x i has degree i and since they have different degree, they form a basis of

C[x]/(g). Furthermore, Θ 1,g (x, y) = m-1 i=0 H g,m-i (x)y i .
Corollary 2 The matrix B 1,g is the basis conversion from the Horner basis H 0 , . . . , M m-1 to the monomial basis 1, x, . . . , x n-1 of C[x]/(g). This leads us to the following theorem, known as Barnett formula (see [START_REF] Barnett | Polynomials and linear control systems[END_REF]):

Theorem 1 Let M f be the multiplication matrix associated to f in C[x]/(g) in the monomial basis, we have:

M f = B f,g B -1 1,g . Proof We have Θ f,g (x, y) = f (x) g(y)-g(x)
x-y +g(x) f (x)-f (y)

x-y and so f (x) g(x)-g(y)

x-y ≡ Θ f,g (x, y) in C[x, y]/(g(x)). So, for each i ∈ {0, . . . , m -1}, we have Θ f,g,i (x) ≡ f (x)Θ 1,g,i (x). This last equality means that B f,g is the matrix of the multiplication by f (x) in C[x]/(g). The result follows directly from this fact. The Barnett's formula reveals the rank structure of the multiplication matrix. Furthermore, this formula is already known if we choose Chebyshev basis instead of monomial basis to express the polynomials and the matrices have exactly the same nature.

Structured matrices and asymptotically fast algorithms

In this section, we briefly recall some basics on displacement structured matrices and related algorithms.

Displacement structure

Given an integer n and a complex number ϑ with |ϑ| = 1, define the circulant matrix

Z ϑ n =         0 ϑ 1 0 1 . . . . . . . . . 1 0         ∈ C n×n .
Next, define the Toeplitz-like displacement operator as the linear operator

∇ T : C m×n -→ C m×n ∇ T (A) = Z 1 m A -AZ ϑ n . A matrix A ∈ C m×n
is said to be Toeplitz-like if ∇ T (A) is a small rank matrix (where "small" means small with respect to the matrix size). The number α = rank(∇(A)) is called the displacement rank of A. If A is Toeplitz-like, then there exist (non-unique) displacement generators G ∈ C m×α and H ∈ C α×n such that ∇(A) = GH.

Toeplitz matrices and their inverses are examples of Toeplitz-like matrices. Another useful example is the multiplication matrix M f , which has Toeplitz-like displacement rank equal to 2, regardless of its size. A similar definition holds for Cauchy-like structure; here the relevant displacement operator is

∇ C : C m×n -→ C m×n ∇ C (A) = D 1 A -AD 2 ,
where D 1 and D 2 are diagonal matrices of appropriate size with disjoint spectra. See [START_REF] Kailath | Displacement Structure: Theory and Applications[END_REF] for a detailed description of displacement structure.

Fast solution of displacement structured linear systems

Gaussian elimination with partial pivoting (GEPP) is a well-known and reliable algorithm that computes the solution of a linear system. Its arithmetic complexity for an n × n matrix is asymptotically O(n 3 ). But if the system matrix exhibits displacement structure, it is possible to apply a variant of GEPP with complexity O(n 2 ). The main idea consists in operating on displacement generators rather than on the whole matrix; see [START_REF] Gohberg | Fast Gaussian elimination with partial pivoting for matrices with displacement structure[END_REF] for details. Strictly speaking, the GKO algorithm performs GEPP (or, equivalently, computes the PLU factorization) for Cauchy-like matrices. However, several authors have pointed out (see [START_REF] Gohberg | Fast Gaussian elimination with partial pivoting for matrices with displacement structure[END_REF], [START_REF] Heinig | Inversion of generalized Cauchy matrices and other classes of structured matrices[END_REF], [START_REF] Pan | On computations with dense structured matrices[END_REF]) that Toeplitz-like matrices can be stably and cheaply transformed into Cauchy-like matrices; the same is true for displacement generators.

Consider, for instance, the case ϑ = 1 and let A be an n×n Toeplitz-like matrix with generators G and H. Denote by D 0 the matrix diag(1, e πi/n , . . . , e (n-1)πi/n ) and let F be the Fourier matrix of size n × n. Then the matrix F AD -1 0 F H is Cauchy-like, of the same displacement rank as A, with respect to the displacement operator defined by D 1 = diag(1, e 2πi/n , . . . , e 2πi(n-1)/n ) and D 2 = diag(e πi/n , e 3πi/n , . . . , e (2n-1)πi/n ). Its Cauchy-like generators can be computed

as Ĝ = F G and H H = F D 0 H H .
Generalization to the case of m × n rectangular matrices is possible. In this case, the parameter ϑ should be chosen so that the spectra of D 1 and D 2 are well separated (see [START_REF] Aricò | A fast solver for linear systems with displacement structure[END_REF] and [START_REF] Bini | A fast algorithm for approximate polynomial gcd based on structured matrix computations[END_REF]).

We also point out that the GKO algorithm can be adapted to pivoting techniques other than partial pivoting ( [START_REF] Gu | Stable and Efficient Algorithms for Structured Systems of Linear Equations[END_REF], [START_REF] Stewart | Stable Pivoting for the Fast Factorization of Cauchy-Like Matrices[END_REF]). This is especially useful in case of instability due to internal growth of generator entries. A Matlab implementation of the GKO algorithm that takes into account several pivoting strategies is found in the package DRSolve described in [START_REF] Aricò | A fast solver for linear systems with displacement structure[END_REF]. In our implementation, we the pivoting strategy proposed in [START_REF] Gu | Stable and Efficient Algorithms for Structured Systems of Linear Equations[END_REF].

A structured approach to AGCD computation

We propose here an algorithm that exploits the algebraic and displacement structure of the multiplication matrix to compute the AGCD of two given polynomials with real coefficients (as defined in section 1).

Rank estimation

It has been pointed out in Section 3 that the rank deficiency of the multiplication matrix equals the AGCD degree. Here we use the structured pivoted LU decomposition to estimate the approximate rank of the multiplication matrix. Recall that M g has a Toeplitz-like structure with displacement rank 2; it can then be transformed into a Cauchy-like matrix Mg as described in Section 5.2. Fast pivoted Gauss elimination yields a factorization Mg = P 1 LU P 2 , where L is a square, nonsingular, lower triangular matrix with diagonal entries equal to 1, U is upper triangular and P 1,2 are permutation matrices. Inspection of the diagonal entries (or of the row norms) of U allows to estimate the approximate rank of Mg and, therefore, of M g .

Minimization of a quadratic functional

Let us suppose that:

• the polynomial f (x) = n j=0 f j x j is exactly known,
• the polynomial g(x) = m j=0 g j x j is approximately known and may be perturbed, so that we consider its coefficients as variables,

• the AGCD degree is known.

Then we can reformulate the problem of AGCD computation as the minimization of a quadratic functional. Indeed, recall that the cofactor v(x) with respect to f (x) is defined by the "shortest" vector (i.e., the vector with the maximum number of trailing zeros) that belongs to the null space of M g . We assume v(x) to be monic; we denote its degree as k and we have

M g v = M g •             v 0 . . . v k-1 1 0 . . . 0             =                0 . . . . . . . . . . . . . . . 0                .
Also observe that the entries of M g are linear functions of the coefficients of g(x). Then the equation M g v = 0 can be rewritten as F (g, v)=0, where the functional F is defined as

F : C m+1 × C k -→ R + F (g, v) = M g v 2 2 .
For a preliminary study of the problem, we have chosen to solve the equation F (g, v)=0 by means of Newton's method, applied so as to exploit structure. Denote by z = [g 0 , . . . , g m , v 0 , . . . , v k-1 ] T the vector of unknowns; then each Newton step has the form

z (j+1) = z (j) -J(g (j) , v (j) ) † M g (j) v (j) .
In particular, notice that the Jacobian matrix associated with F is an n × (m + k + 1) Toeplitz-like matrix of displacement rank 3. This property allows to compute a solution of the linear system J(g (j) , v (j) )y = M g (j) v (j) in a fast way; therefore, the arithmetic complexity of each iteration is quadratic w.r.t. the degree of the input polynomials.

We propose in the future to take into consideration other optimization methods in the quasi-Newton family, such as BFGS.

Computation of displacement generators

In order to perform fast factorization of the multiplication matrix M g , we need to compute Toeplitz-like displacement generators. It turns out that the range of ∇(M g ) is spanned by the first and last column of the displaced matrix, and the columns of indices from 2 to n -1 are multiples of the first one. Therefore, it suffices to compute a few rows and columns of M g in order to obtain displacement generators. This can be done in a fast and stable way by using Barnett's formula. If we denote as e j the jth vector of the canonical basis of C n , then the computation of the j-th column of M g can be seen as M g (:, j) = B(f, g) • B(1, f ) -1 e j , that is, it consists in solving a triangular Hankel linear system and computing a matrix-vector product. For row computation, recall that the Bezoutian is a symmetric matrix; we have analogously:

M g (j, :) = e T j • B(f, g) • B(1, f ) -1 = B(1, f ) -1 B(f, g)e T j T ,
so that the computation of a row of M g amounts to performing a matrix-vector product and solving a Hankel triangular system. A similar approach holds for computation of displacement generators of the Jacobian matrix J(g, v) associated with the functional F (g, v).

Description of the algorithm

Input: coefficients of polynomials f (x) and g(x). Output: a perturbed polynomial g(x) such that f and g have a nontrivial common factor.

1. Estimate the approximate rank k of M g by computing a fast pivoted LU decomposition of the associated Cauchy-like matrix.

2. Again by using fast LU, compute a vector v = [v 0 , v 1 , . . . ., v k-1 , 1, 0, . . . ., 0] T in the approximate null space of M g .

3.

Apply structured Newton with initial guess (g, v) and compute polynomials g and ṽ such that f and g have a common factor of degree deg fk and ṽ is the monic cofactor for f .

Numerical experiments and computational issues

We have written a preliminary implementation of the proposed method in Matlab (available at the URL http://www.unilim.fr/pages perso/paola.boito/MMgcd.m).

The results of a few numerical experiments are shown below. The polynomials f and g are monic and have random coefficients uniformly distributed over [-1, 1]. They have an exact GCD of prescribed degree. A perturbation is then added to g. The perturbation vector has random entries uniformly distributed over [-η, η] and its norm is of the order of magnitude of η. We show:

• the residual F (g, ṽ),

• the 2-norm distance between the exact and the computed cofactor v,

• the 2-norm distance between the exact and the computed perturbed polynomial g (which is expected to be roughly of the same order of magnitude as η).

In the following 2.54e-4 There are several issues in our approach that deserve further investigation. Let us mention in particular:

• The choice of a threshold (or a more refined technique) for estimating approximate rank.

• Normalization of polynomials: here we mostly work with monic polynomials, but other normalizations may be considered.

• The structured implementation of the optimization step (minimizing F (g, v)).

We have used for now a heuristic structured version of the Gauss-Newton algorithm. Observe that each step of classical Gauss-Newton applied to our problem has the form z (j+1) = z (j)y (j) , where z (j) is the vector containing the coefficients of the j-th iterate polynomials g (j) and v (j) , and y (j) is the least-norm solution to the underdetermined system J(g (j) , v (j) )y (j) = M g (j) v (j) . Computing this least-norm solution in a structured and fast way is a difficult point that will require more work. Our implementation gives a solution which is not, in general, the least-norm one, even though it is typically quite close. Further work will also include a study of other possible optimization methods that lend themselves well to a structured approach.

Conclusions

We have proposed and implemented a fast structured matrix-based approach to a variant of the AGCD problem, namely, the problem of computing an approximate greatest common divisor of two univariate polynomials, one of which is known to be exact. To our knowledge, this variant has been so far neglected in the existing literature. It may be also interesting when one polynomial is known with high accuracy and the other is not.

Our approach is based on the structure of the multiplication matrix and on the subsequent reformulation of the problem as the minimization of a suitably defined functional. Our choice of the multiplication matrix M g over other resultant matrices (e.g., Sylvester, Bézout...) is motivated by

• the smaller size of M g , with respect e.g. to the Sylvester matrix,

• the strong link between the null space of M g and the gcd, and in particular the fact that the null space of M g immediately yields a gcd cofactor,

• the displacement structure of M g ,

• the possibility of computing selected rows and columns of M g in a stable and cheap way, thanks to Barnett's formula. This is, however, a preliminary study. Further work will include generalizations of the proposed problem and a more thorough analysis of the optimization part of the algorithm. Furthermore, this approach can be generalized in several intersting way:

• using better bases then the monomial one,

• it can be extended to some multivariate setting to compute the co-factor of a polynomial g in C[x 1 , . . . , x n ]/(f 1 , . . . , f n ) when f 1 , . . . , f n define a complete intersection since Barnett formula still holds,

• to compute the AGCD of f with g 1 , . . . , g k where f is known with accuracy but g 1 , . . . , g k are inaccurate, one can take g as a linear combination of g 1 , . . . , g k with our method and succeed with a high probability.

  This implies that the set {L 1 , . . . , L d } is a basis of A. A well known fact is that the set {1 ζ1 , . . . , 1 ζ d } form a basis A * dual of the basis {L 1 , . . . , L d } of A.

	As a corollary, we have the Lagrange interpolation formula : Each p ∈ A can
	d
	be written p(x) =	1 ζi (p) * L i (x). A funny consequence is that if we choose
	i=1
	{L 1 , . . . , L d } as a basis of A, for all g ∈ K[x], the remainder π(g) of the euclidian
	division of g by f is given by (g(ζ 1 ), . . . , g(ζ d )) in the basis {L 1 , . . . , L d }, i.e.
	d
	r =
	i=1

  table we have taken η =1e-5.

	n, m, deg f F (g, ṽ)	v -ṽ 2	g -g 2
	8, 7, 3	1.02e-15 1.19e-15 1.40e-5
	15, 14, 5	1.51e-15 2.26e-15 1.35e-4
	22, 22, 7	2.07e-13 1.40e-13 2.20e-4
	36, 36, 11	1.19e-12 5.07e-14 0.0012
	Here are results for η =1e-8:
	n, m, deg f	F (g, ṽ)	v -ṽ 2	g -g 2
	8, 7, 3	5.49e-15 1.63e-15	5.85e-8
	28, 27, 13 7.90e-14 8.98e-14	6.50e-7
	38, 37, 13 4.88e-12 4.26e-12	2.30e-5
	58, 57, 23 2.03e-12 4.40e-12


PhD Roza Ghotbi

MD Buster Mannheimer
email: buster.mannheimer@sodersjukhuset.se

PhD Eleni Aklillu

MD, PhD Akira Suda

PhD Leif Bertilsson

MD, PhD Erik Eliasson

Mannheimer

MD, PhD Urban Ösby

Carriers of the 142T>G gene variant are predisposed to reduced olanzapine exposure-an impact similar to male gender or smoking in schizophrenic patients

Keywords: Antipsychotics, therapeutic failure, drug metabolism, pharmacogenetics, polymorphism

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Schizophrenia is in most cases a chronic disease that requires continuous, effective pharmacological intervention. However, in the recent CATIE trials it was found that three out of four patients with schizophrenia discontinue treatment with antipsychotic drugs within 1.5 years from start of treatment. [1] One major reason for drug discontinuation is lack of therapeutic effect. This may be explained by several contributing factors such as poor compliance, adverse effects of poly-pharmacy, illicit drug abuse or more simply; subtherapeutic exposure to the antipsychotic drug. Clearly, it may be difficult to distinguish between all possible explanations.

In the western world, olanzapine is one of the most commonly used antipsychotic drugs.

Therapeutic drug monitoring (TDM) of olanzapine has been advocated to circumvent the potential problem with high inter-patient variability in olanzapine exposure at standard dosage, which relates to almost 10-fold differences in clearance. [2] However, even the intra-individual variability is considerable, according to a recent report [3], which means that single individual analyses of the steady state level are not fully reliable as guide for individual dose adjustments.

In addition, a target interval for plasma olanzapine has not yet been established and is still a matter of controversy. [4,21] However, initial studies have indicated that 12 h post-dose plasma levels of olanzapine at 74 nmol/L (23 ng/mL) or higher increase the likelihood for therapeutic effect. [4] In fact, it was reported that only 25% of patients below this level of exposure actually responded to treatment, which suggests that patients may be at risk of subtherapeutic exposure at standard dosage. In conclusion, identification of factors that predispose patients to low exposure should help to individualize treatment. In the present study on patients with schizophrenia and other psychotic diseases, we aimed to investigate the relative importance of gender, smoking status and specific genetic variants of UGT1A4, CYP1A2, and MDR1, on olanzapine plasma levels. The rational for the focus on these polymorphic genes is the documented role of uridine diphosphate glucuronosyltransferase (UGT) 1A4 [5,6] and cytochrome P450 (CYP) 1A2 [7] in olanzapine metabolism and clearance. A potential role for the MDR1 gene product P-glycoprotein, a CNS drug efflux pump, was implicated in a recent report that described a relationship between MDR1 gene variants and treatment response to olanzapine in female schizophrenic patients. [8] F o r P e e r R e v i e w

Material and methods

Study subjects

The present study focused on patients with schizophrenia or other psychoses, recruited from psychosis-specialized psychiatric outpatient departments mainly in Stockholm County. These departments are responsible for treatment of all outpatients with long-term psychotic disorders in their respective catchments area. The aim of the study was to recruit all patients from a number of such departments, and therefore this olanzapine-treated sub-sample is likely to represent psychosis treatment with olanzapine in the total catchments areas. In a total sample of 775 patients, 121 (15.6%) were prescribed olanzapine. After the patients had given their informed consent, information such as gender, age, smoking habits, dosage of antipsychotic drugs, time between dosage and blood sampling and co-medication was documented. DNA was extracted from venous blood samples, which also was used for isolation of plasma for drug concentration analysis. Ethical permission was obtained from the Stockholm Regional Ethics Committee #04-447/4.

Human genomic DNA samples

DNA was extracted with conventional methods from venous blood samples drawn from the patients by the Department of Forensic Medicine in Linköping, Sweden.

SNP genotyping

Genotyping of CYP1A2 -2467delT, MDR1 2677G>T and 3435C>T and UGT1A4 142T>G (L48V), was performed using Polymerase Chain Reaction-Restriction Fragment Length Polymorphism (PCR-RFLP) methods. The amplified PCR products were digested over night with respective restriction enzymes indicated in Supplementary table 1 and the fragments were analyzed on 2-3% agarose gels. Genotyping for UGT1A4 L48V polymorphism was performed using a forward primer (5´-GTTGGGCCCATAACGAAAGGCAGTT-3´) from Ehmer et al and a newly designed UGT1A4 specific reverse primer (5-'GCTCCACACAACACCTATGAAG-3´) to amplify a 576 bp fragment spanning the polymorphic site. [9] The amplified PCR products were digested over night with STU I. Wild type allele was cut into 324-bp and 152-bp fragments while the mutant allele was not. We confirmed the genotyping result by direct sequencing in both directions using ABI PRISM™ BigDye terminator cycle sequencing ready reaction kit V3.1 (Applied Biosystems) and analyzed on an ABI Prism 377 DNA sequencer.

CYP1A2 polymorphisms -729 C>T and -163C>A were genotyped by Minisequencing using taq arrays, as described earlier. [10] PCR primers and Minisequencing primers with 5´ Tag sequences covering the SNP site were designed using the Autoprimer (http://www.autoprimer.com) (Beckman Coulter) software. The primers were obtained from Integrated DNA Technologies (IDT Inc., Coralville, IA, USA). PCR and Minisequencing primers are listed in Supplementary table 2. PCR fragments spanning the SNPs were amplified in individual reactions by multiplex PCR (10 ng/µl DNA) containing forward and reverse primers for both SNPs. To remove remaining dNTPs and primers, the PCR products (7.2 µl) were treated with 1 U/µl shrimp alkaline phosphatase and 0.48 U/µl exonuclease I (USB Coorporation, Cleveland, OH, USA). The oligonucleotides complementary to the taq sequence attached to the 5´of the minisequencing primer were immobilized on the CodeLink Activated Slides (GE Healthcare, Uppsala, Sweden) in duplicate spots in each sub array. The minisequencing reaction was run in the presence of fluorescently labeled ddNTP (0.1 µM Texas Red-ddATP, Tamra-ddCTP, R110-ddGTP and 0.15 µM Cy5-ddUTP (Perkin Elmer Life Sciences, Boston, MA, USA)) and 0.065 U of KlenThermase™ DNA polymerase (GeneCraft, Germany). After annealing of the extension products to the immobilized complementary tag

Determination of olanzapine plasma concentration

The plasma olanzapine concentration was measured by a HPLC method developed in the Karolinska University Hospital Therapeutic Drug Monitoring laboratory. 500 µl of olanzapine standard (160 nmol/L) was added to a triplet of 14 ml polypropene tubes; a blank plasma sample, a control, and a sample in duplicates. To this solution, 50 µl (750 ng/ml) of the internal standard methyl-olanzapine, 500 µl of 1M glycine buffer, pH 9.8, and 3 ml of diisopropyl ether (DIPE) was added and vortexed. The mixture was extracted for 10 minutes with a rotator at a low speed and centrifuged for 10 minutes at 3000 rpm. The water phase was frozen in a water bath at -15°C and the top phase was transferred to a new polypropene tube where 75 µl 25 mmol/L aqueous acetic acid was added and additionally extracted for 5 min with rotator and centrifuged for 5 min at 3000 rpm. The DIPE phase was discarded and any remaining DIPE was removed under vacuum. The samples were injected into a Waters XTerra RP 18 3, 3 µm, 100x3,0 mm column, pre-warmed to 40°C, coupled to a Dionex HPLC P 580 HPG system. Three µl of the samples was injected into the column with a flow rate of 0.6 ml/min and were monitored at 275 nm. The concentration of olanzapine was determined by taking the olanzapine over the olanzapine standard based on the ratio against the internal standard. The range of measurement was 20-2000 nmol/L (6.2ng/l -625ng/L).

The accuracy was 93-101% in the range 160-2000 nmol/L. The HPLC analyses were approved in general when the within-day coefficient (CV) of the control sample was less than 15% and when the CV of the samples in duplicate were less than 15%, however for the present analyses, CV was close to 5%, only.

Statistical analysis

Haplotype analysis and estimation of expected haplotype frequency from the raw genotype data was done using population genetic software program, ARLEQUIN 2.000. [11] A multiple regression analysis was performed to investigate the combined effect of patient characteristics and genotypes/haplotypes on daily dose-corrected plasma concentrations of olanzapine as the dependent variable. Correction of plasma concentrations with regards to daily dose was based on a linear relationship between given dose and achieved plasma concentrations, as clearly demonstrated in. [12] Unfortunately the time between last dose and sampling was not standardized. We therefore included a time variable 0.97716 Hours between last dose and sampling assuming a 50% decrease in plasma concentration of olanzapine for each 30 hours (the reported average half-life of olanzapine).

Results

Patients

Out of a total of 121 patients treated with olanzapine, the great majority suffered from schizophrenia or schizoaffective syndrome (Table 1). There were 77 men (64%) and 44 women (36%). Fifty patients were smokers (41%) and 71 non-smokers (59%). The mean age was 47 years (range18-81, SD 12), and the mean body weight was 87 kg (range 49-138, SD 18). The women were slightly older that the men (50 versus 45 years). Smoking was somewhat less common among women (36%) as compared to the male population (44%).

There were 118 patients with information on daily dose (range 2.5-40 mg/day). Average daily doses of olanzapine are given in Table 1. The most commonly used dose of olanzapine was 10 mg daily (fig. 1) but the mean dose in the study population was 12.2 mg daily. There was no significant difference in the doses used for different diagnoses, nor in relation to gender or smoking status (Table 1). The average mean time of sampling after the last dose intake was 11.8 hours (range 0.5-24h SD 4.7). Including olanzapine, the patients used a mean of 3.1 (SD 2.19) prescribed drugs, none of which were known to interact with the pharmacokinetics of olanzapine.

Olanzapine plasma concentrations

Of the 118 patients with recorded daily dose of olanzapine, plasma concentration-to-dose ratios were not obtained in 11 patients because of analytical interference (n=6) or failure to detect olanzapine in the HPLC assay (n=5). In 8 patients (including the patient with missing DNA and including one of the patients with unknown daily dose) olanzapine concentrations were below 20nmol/L (6.2ng/mL) and were assigned a reference value of 10 nmol/L (3.1 ng/mL). There was a wide inter-patient variation in olanzapine plasma levels, with concentrations ranging between 10 and 646 nmol/L (202 ng/mL). Forty seven percent (52/110) of the patients had a plasma level below 74 nmol/L (23 ng/mL), the lower limit of the proposed therapeutic interval, i.e. an increased number of non-responders were found below this limit in a material of 198 schizophrenic patients. [14] The corresponding range in olanzapine C/D ratios was 0.5-47 nmol/L/mg/day (0.16-15 ng/mL/mg/day); however this great variability was to some extent explained by different time points of sampling in relation to last dose. There was a significant difference in the mean C/D between smokers and nonsmokers (mean difference: 3.56 nmol/L/mg/day (1.11 ng/mL/mg/day) lower in smokers, 95% CI of the mean difference 1.27, 5.85; p<0.003) and regardless of smoking habits, men (range 0.5-27 nmol/L/mg/day (0.16-8.4 ng/mL/mg/day)) had significantly lower C/D than the females (range 1.0-47 nmol/L/mg/day (mean difference: 3.4 nmol/L/mg/day (1.1ng/mL/mg/day) lower in males, 95% CI of the mean difference 1.05, 5.75; p<0.006).

Genotype/haplotype

Out of all 121 patients, DNA samples were missing from nine. The genotype of CYP1A2, MDR1 and UGT1A4 variants in 112 patients are given in Table 2. The observed genotype frequencies did not show a significant deviation from Hardy-Weinberg equilibrium. The most frequent SNP in CYP1A2 was -163C>A followed by -2467delT with frequencies of 76.8% and 9.8%, respectively. The SNP CYP1A2 -729C>T was not detected in this population. The frequencies of SNPs MDR1 3435C>T and 2677G>T were 58.5% and 45.5%, respectively, and SNP UGT1A4 142T>G with a frequency of 17.9%. The CYP1A2 and MDR1 haplotypes were constructed using the raw genotype data using population genetic software, ARLEQUIN 2.000. Three known haplotypes, CYP1A2*1A, CYP1A2*1F and CYP1A2*1D, were found in frequencies of 21.9%, 68.3% and 1.3%, respectively (Table 3). Since CYP1A2 -3860G>A, -739T>G and -3113A>G SNPs were not included, the remaining 8.3% of the haplotype could not be determined. MDR1 haplotypes were estimated by using two SNPs, 3435C>T and 2677G>T. The most frequent haplotype was TT followed by CG in frequencies of 41.2% and 37.2% respectively (Table 3).

The association of olanzapine plasma concentration with individual SNPs and haplotypes

The prevalence of each genotype together with the corresponding mean C/D is presented in Table 2. Using ANOVA, the individual genotypes and haplotypes were compared with mean C/D. None of the genotypes or haplotypes unadjusted for covariates showed any significant association with olanzapine plasma concentrations. To be able to control for other factors that have a possible effect on the results, a multiple regression analysis was performed with the combined effect of patient characteristics and genotypes/haplotypes on dose-corrected plasma concentrations of olanzapine (Table 4). Figure 2 shows the values of plasma olanzapine C/D predicted by the model, plotted against the measured concentrations. Male gender, UGT1A4 142T>G genotype and smoking, did each explain a significant proportion of the variation independently and predicted a decrease in dose normalized plasma concentration of olanzapine of 35%, 25% and 21% respectively. Overall, the full regression model (Table 4) accounted for 25% of inter-individual variability in olanzapine C/D, as indicated by the R 2 value (fig. 2).

No additional contribution to the olanzapine C/D variability was apparent from the different CYP1A2 and MDR1 genotypes/haplotypes. The regression model predicted a 5.1 fold higher olanzapine plasma C/D for a non-smoking female patient, without the UGT1A4 142T>G SNP, 13 with last dose intake 12 hours prior to sampling (14.0 nmol/mg/day), compared to a smoking man heterozygous for UGT1A4 142T>G (2.75 nmol/mg/day).

To further investigate the clinical impact of the UGT1A4 142T>G genotype for olanzapine exposure, the proportion of patients below the lower limit of the therapeutic interval (74 nmol/L (23 ng/mL)) was compared between homozygous carriers of the 142T-variant. i.e. the most common genotype, or patients being heterozygous for 142G. A larger proportion of patients that carry the UGT1A4 142G variant had a plasma concentration of olanzapine below 74 nmol/L (55%) as compared to 142T homozygous cases (42%), but this difference was not significant (p=0.22).

We also investigated the possible interaction between gender, UGT1A4 142T>G and smoking respectively, with time of sampling, and between UGT1A4 142T>G and smoking.

Neither of the multiplicative terms explained a significant proportion of the variability. To further investigate the interaction of CYP1A2*1F haplotypes and smoking, crude C/D plasma concentrations among smokers were compared. The plasma concentrations in individuals with 0, 1 and 2 CYP1A2*1F haplotypes were 8.19 (SD:5.82 ), 6.26 (SD:3.92) and 6.78 (SD:4.89) nmol/L/mg/day respectively. The difference was not significant (p=0.688).

Discussion

The present study suggests that the UGT1A4 142T>G SNP is associated with lower average plasma concentrations of olanzapine.

Initially we performed genotyping for UGT1A4 142T>G using primers described by Nozawa M et al. [15] However we found no clear result to differentiate the different genotypes. Later we performed sequencing of the PCR product and the result indicated that the primers coamplified UGT1A3 and UGT1A5 together with UGT1A4. After multiple alignments of the UGT1A families, we designed UGT1A4 specific primers for genotyping and the result were confirmed by direct sequencing. The impact of the UGT1A4 polymorphism was significant after adjusting for other factors and the 142T>G allele was associated with a similar reduction in olanzapine exposure as smoking or male gender. Applying the defined algorithm (Table 4) to literature-based mean morning plasma levels of 140 nmol/L (46 ng/mL) at steady state after mean daily doses of 16 mg [3], corresponding to 8.75 nmol/L/mg/day, and assuming this represents the exposure in UGT1A4 142TT subjects mainly ('wild type'); the corresponding exposure in subjects genotyped as heterozygous for UGT1A4 142T>G is predicted to [8.75 minus 3.46] = 5.29 nmol/L/mg/day and 85 nmol/L (30 ng/mL) in plasma 12h after last dose at repeated intake of 16 mg/d. In other words, patients carrying one allele of UGT1A4 142G (population frequency of around 15-20%), treated with standard dosages of olanzapine are at risk of sub-therapeutic exposure, according to the cut-offs defined in earlier work. [4,14] It follows that olanzapine exposure in subjects homozygous for the UGT1A4 142T>G SNP would be much lower. However, this needs to be confirmed in separate pharmacokinetic studies on genotyped individuals; i.e. under controlled experimental conditions. We are currently planning to run such a study after screening of several hundred volunteers for the UGT1A4 142T>G SNP, in order to include homozygous 142GG carriers. Confirmatory data from such studies on a significant impact of the UGT1A4 polymorphism would unavoidably lead to the question whether prospective genotyping of patients should be considered, for an early detection of subjects at risk of sub-therapeutic olanzapine exposure.

The UGT1A4 142T>G SNP causes a leucine to valine amino acid substitution (L48V), and there are contradictory reports in the literature how this affects enzyme activity. Ehmer et al.

showed the almost complete absence of glucuronidation of steroid substrates by the UGT1A4 valine-48 enzyme. [9] However, a study including Japanese healthy volunteers showed that the UGT1A4 142T>G genetic variant may be associated with either an increase or decrease in glucuronidation activity, depending on substrate. Thus, the UGT1A4 L48V amino acid substitution decreased the intrinsic clearance (V max /K m ) for tigogenin while the intrinsic clearance for clozapine, another atypical antipsychotic with very close structural similarity to olanzapine, was twice as high in the UGT1A4 valine-48 enzyme variant compared to that of the wild type leucine-48 enzyme. [16] In light of these data, it is tempting to speculate that the UGT1A4 L48V amino acid substitution involves ethylene-group-dependent sterical changes in the substrate-binding region of the UGT1A4 enzyme.

The present data did not show any effect of the CYP1A2*1F haplotype on olanzapine exposure.

Interestingly, previous work has suggested the impact of this haplotype to be dependent on smoking, and that the extent of CYP1A2 gene expression and enzyme induction by smoking would be more pronounced in carriers of this haplotype. [17,18] However, we did not observe a corresponding positive relationship between the CYP1A2*1F haplotype and smoking in the present material. This might be explained by the fact that UGT1A4 probably plays a greater role in the metabolism of olanzapine than CYP1A2. [5] It is also of interest that UGT1A4 seems to be regulated through the aryl hydrocarbon receptor [19], in similarity with the CYP1A2 gene. As yet, there are no published studies that describe the influence of smoking on UGT1A4 gene expression or enzyme activity, and we failed to detect a difference between smokers and non-smokers regarding the impact of the UGT1A4 polymorphism on olanzapine plasma levels.

We lacked information regarding coffee intake which may be of some relevance for CYP1A2 metabolism since both caffeine and olanzapine are metabolised by this enzyme, and in theory coffee intake may contribute to inter-individual variability in olanzapine exposure. Coffee consumption has been suggested also to induce CYP1A2 [20], but this needs to be concluded from studies on intra-individual changes before and after repeated coffee intake since baseline differences in CYP1A2 activity may predispose to differences in coffee intake behaviour. In any case, the potential bias is likely to be non-differential and should therefore not have affected the results on the impact of UGT1A4 genotype on olanzapine levels.

Using multiple regression analysis, we showed that the male gender is associated with a 35% decrease in the daily dose corrected plasma concentration of olanzapine compared to females, after adjustment for other factors. Adjusting for body weight did not lower this effect, which is concordant with previous studies. Perry et al. reported that male patients required 15.8 mg/day and women 8.6 mg per day, to reach the therapeutic concentration of olanzapine, and Gex-Fabry et al. found that female gender was associated with an 11% increase in plasma concentration after controlling for other factors. [4,21] A possible explanation may be the undocumented use of oral contraceptives which appear to inhibit CYP1A2 activity in vivo. [17] Unfortunately we did not in the present study specifically ask the patients for information regarding contraceptives. To address this issue indirectly, we included age as a dichotomized variable separating patients with an age above 40 years, who may use contraceptives to a lesser extent, from patients 40 years or less. However, this variable did not explain a significant proportion of the variation, supporting the observed gender-specific difference in olanzapine metabolism. It is of interest to note that dose levels in our material did neither differ significantly between men and women, nor between smokers and non-smokers. Apparently, the established influence of these factors on olanzapine exposure did not translate into different dose levels in the clinic, despite being mentioned in drug label. Whether this is a matter of clinical neglect or indicates only a limited clinical importance of such differences for treatment outcome is not clear at this stage.

The regression model was based on an exponential association between the time passed between last dose and olanzapine exposure. As the interaction terms between the sample time and the factors shown to predict olanzapine exposure were not significant, with the exception to that of UGT1A4 142T>G SNP, we chose to keep this association fixed instead of dependent on the characteristics of each individual. This may, from a strict pharmacokinetic point of view be theoretically inappropriate, but appeared as the best approach due to the limited number of patients which resulted in non-significant interaction terms. One non-smoking man, wild type for UGT1A4 142T>G, with a sample that was taken 3.5 hours after the last dose, was identified as an outlier with an exceptionally high C/D olanzapine value of 47 nmol/mg/day (Figure 2). If this case was excluded from the analysis, also the time between last dose and sample became statistically non-significant. Also the predictive value of UGT1A4 142T>G for C/D olanzapine became weaker with a beta coefficient that decreased from -3.46 to -2.19, but did however remain significant (p=0.042). In 8 patients the olanzapine concentration was found to be <20nmol/L, i.e. below the level of more exact quantification. These samples were given a value of 10 nmol/L. By this approach we aimed to avoid the bias of excluding the 18 nmol/L. Calculations without these 8 patients did not change the results with regard to the effect of gender nor UGT1A4 142T>G. The effect of smoking was however not statistically significant (p=0.086). When scrutinizing the excluded patients it was evident that a disproportional large part of them (n=5) were smokers. Whether these patients had not taken their medication, misused their recommended dose or if they are ultra-rapid metabolizers remain unclear. In this respect, it should be commented that patient compliance with measurable or at least detectable intake of prescribed drug in > 90% of subjects may be considered surprisingly high. If it were assumed that patient compliance is independent of genotype, a lower compliance would significantly reduce the power to detect the genotypeconcentration relationship described above.

Conclusion

The results of the present study, based on multiple regression analysis, indicate that carriers of the UGT1A4 142T>G SNP have reduced exposure to olanzapine. The validity of this finding clearly needs to be confirmed in other similar naturalistic settings and by pharmacokinetic studies in genotyped healthy volunteers after oral administration of olanzapine. Also, studies of relapses and discontinuation of olanzapine treatment are needed to assess the clinical effects of the variation in olanzapine levels. In addition, the enzyme variants encoded by the UGT1A4 142T and 142G variants needs to be characterised in vitro with regards to potential catalytic properties. Such studies are now underway. We were also able to confirm that men and smokers are exposed to lower levels of the drug. Whether or not these factors influence the risk of therapeutic failure remains to be established. However, it became clear from the present study that factors already known to reduce olanzapine exposure such as smoking and male gender, was not translated into treatment with higher daily doses. We propose that treatment response to olanzapine in Results: In addition to time of sampling from last dose intake, the results indicate that interpatient differences in olanzapine exposure was explained by the following individual factors in order of relative impact (a) male gender (b) carrier of the UGT1A4 142T>G SNP and (c) smoking. Each of these predicted a decrease in daily dose-corrected plasma concentrations by 35%, 25% and 21% and respectively. In contrast, age, body weight, MDR1 or CYP1A2 haplotype did not have a significant impact.

Conclusions: At 12h after dose intake, the regression model predicted a 5.1-fold higher olanzapine plasma level in a non-smoking female patient, without the UGT1A4 142T>G SNP, compared to a smoking man treated with the same dose, but heterozygous for UGT1A4 142T>G. Whether these combined genetic and environmental factors influence the risk of therapeutic failure remains to be established.

Keywords:

Antipsychotics, therapeutic failure, drug metabolism, pharmacogenetics, polymorphism

Introduction

Schizophrenia is in most cases a chronic disease that requires continuous, effective pharmacological intervention. However, in the recent CATIE trials it was found that three out of four patients with schizophrenia discontinue treatment with antipsychotic drugs within 1.5 years from start of treatment. [1] One major reason for drug discontinuation is lack of therapeutic effect. This may be explained by several contributing factors such as poor compliance, adverse effects of poly-pharmacy, illicit drug abuse or more simply; subtherapeutic exposure to the antipsychotic drug. Clearly, it may be difficult to distinguish between all possible explanations.

In the western world, olanzapine is one of the most commonly used antipsychotic drugs.

Therapeutic drug monitoring (TDM) of olanzapine has been advocated to circumvent the potential problem with high inter-patient variability in olanzapine exposure at standard dosage, which relates to almost 10-fold differences in clearance. [2] However, even the intra-individual variability is considerable, according to a recent report [3], which means that single individual analyses of the steady state level are not fully reliable as guide for individual dose adjustments.

In addition, a target interval for plasma olanzapine has not yet been established and is still a matter of controversy. [4,21] However, initial studies have indicated that 12 h post-dose plasma levels of olanzapine at 74 nmol/L (23 ng/mL) or higher increase the likelihood for therapeutic effect. [4] In fact, it was reported that only 25% of patients below this level of exposure actually responded to treatment, which suggests that patients may be at risk of subtherapeutic exposure at standard dosage. In conclusion, identification of factors that predispose patients to low exposure should help to individualize treatment. In the present study on patients with schizophrenia and other psychotic diseases, we aimed to investigate the relative importance of gender, smoking status and specific genetic variants of UGT1A4, CYP1A2, and MDR1, on olanzapine plasma levels. The rational for the focus on these polymorphic genes is the documented role of uridine diphosphate glucuronosyltransferase (UGT) 1A4 [5,6] and cytochrome P450 (CYP) 1A2 [7] in olanzapine metabolism and clearance. A potential role for the MDR1 gene product P-glycoprotein, a CNS drug efflux pump, was implicated in a recent report that described a relationship between MDR1 gene variants and treatment response to olanzapine in female schizophrenic patients. [

Material and methods

Study subjects

The present study focused on patients with schizophrenia or other psychoses, recruited from psychosis-specialized psychiatric outpatient departments mainly in Stockholm County. These departments are responsible for treatment of all outpatients with long-term psychotic disorders in their respective catchments area. The aim of the study was to recruit all patients from a number of such departments, and therefore this olanzapine-treated sub-sample is likely to represent psychosis treatment with olanzapine in the total catchments areas. In a total sample of 775 patients, 121 (15.6%) were prescribed olanzapine. After the patients had given their informed consent, information such as gender, age, smoking habits, dosage of antipsychotic drugs, time between dosage and blood sampling and co-medication was documented. DNA was extracted from venous blood samples, which also was used for isolation of plasma for drug concentration analysis. Ethical permission was obtained from the Stockholm Regional Ethics Committee #04-447/4.

Human genomic DNA samples

DNA was extracted with conventional methods from venous blood samples drawn from the patients by the Department of Forensic Medicine in Linköping, Sweden.

SNP genotyping

Genotyping of CYP1A2 -2467delT, MDR1 2677G>T and 3435C>T and UGT1A4 142T>G (L48V), was performed using Polymerase Chain Reaction-Restriction Fragment Length Polymorphism (PCR-RFLP) methods. The amplified PCR products were digested over night with respective restriction enzymes indicated in Supplementary table 1 and the fragments were

Determination of olanzapine plasma concentration

The plasma olanzapine concentration was measured by a HPLC method developed in the Karolinska University Hospital Therapeutic Drug Monitoring laboratory. 500 µl of olanzapine standard (160 nmol/L) was added to a triplet of 14 ml polypropene tubes; a blank plasma sample, a control, and a sample in duplicates. To this solution, 50 µl (750 ng/ml) of the internal standard methyl-olanzapine, 500 µl of 1M glycine buffer, pH 9.8, and 3 ml of diisopropyl ether (DIPE) was added and vortexed. The mixture was extracted for 10 minutes with a rotator at a low speed and centrifuged for 10 minutes at 3000 rpm. The water phase was frozen in a water bath at -15°C and the top phase was transferred to a new polypropene tube where 75 µl 25 mmol/L aqueous acetic acid was added and additionally extracted for 5 min with rotator and centrifuged for 5 min at 3000 rpm. The DIPE phase was discarded and any remaining DIPE was removed under vacuum. The samples were injected into a Waters XTerra RP 18 3, 3 µm, 100x3,0 mm column, pre-warmed to 40°C, coupled to a Dionex HPLC P 580 HPG system. Three µl of the samples was injected into the column with a flow rate of 0.6 ml/min and were monitored at 275 nm. The concentration of olanzapine was determined by taking the olanzapine over the olanzapine standard based on the ratio against the internal standard. The range of measurement was 20-2000 nmol/L (6.2ng/l -625ng/L).

The accuracy was 93-101% in the range 160-2000 nmol/L. The HPLC analyses were approved in general when the within-day coefficient (CV) of the control sample was less than 15% and when the CV of the samples in duplicate were less than 15%, however for the present analyses, CV was close to 5%, only.

Statistical analysis

Haplotype analysis and estimation of expected haplotype frequency from the raw genotype data was done using population genetic software program, ARLEQUIN 2.000. [11] A multiple regression analysis was performed to investigate the combined effect of patient characteristics and genotypes/haplotypes on daily dose-corrected plasma concentrations of olanzapine as the dependent variable. Correction of plasma concentrations with regards to daily dose was based on a linear relationship between given dose and achieved plasma concentrations, as clearly demonstrated in. [12] Unfortunately the time between last dose and sampling was not standardized. We therefore included a time variable 0.97716 Hours between last dose and sampling assuming a 50% decrease in plasma concentration of olanzapine for each 30 hours (the reported average half-life of olanzapine). [13] Possible independent variables tested included gender, smoking, age, weight, time between last dose and sampling, co-medication with one of the following potent CYP2D6 inhibitors (there was no documented use of CYP1A2 inhibitors in the study population); paroxetine, fluoxetine, perphenazine or levomepromazine, polymorphisms in genes coding for CYP1A2, UGT1A4 and P-glycoprotein,

Results

Patients

Out of a total of 121 patients treated with olanzapine, the great majority suffered from schizophrenia or schizoaffective syndrome (Table 1). There were 77 men (64%) and 44 women (36%). Fifty patients were smokers (41%) and 71 non-smokers (59%). The mean age was 47 years (range18-81, SD 12), and the mean body weight was 87 kg (range 49-138, SD 18). The women were slightly older that the men (50 versus 45 years). Smoking was somewhat less common among women (36%) as compared to the male population (44%).

There were 118 patients with information on daily dose (range 2.5-40 mg/day). Average daily doses of olanzapine are given in Table 1. The most commonly used dose of olanzapine was 10 mg daily (fig. 1) but the mean dose in the study population was 12.2 mg daily. There was no significant difference in the doses used for different diagnoses, nor in relation to gender or smoking status (Table 1). The average mean time of sampling after the last dose intake was 11.8 hours (range 0.5-24h SD 4.7). Including olanzapine, the patients used a mean of 3.1 (SD 2.19) prescribed drugs, none of which were known to interact with the pharmacokinetics of olanzapine.

Olanzapine plasma concentrations

Of the 118 patients with recorded daily dose of olanzapine, plasma concentration-to-dose ratios were not obtained in 11 patients because of analytical interference (n=6) or failure to detect olanzapine in the HPLC assay (n=5). In 8 patients (including the patient with missing DNA and including one of the patients with unknown daily dose) olanzapine concentrations were below 20nmol/L (6.2ng/mL) and were assigned a reference value of 10 nmol/L (3.1 ng/mL). There was a wide inter-patient variation in olanzapine plasma levels, with concentrations ranging between 10 and 646 nmol/L (202 ng/mL). Forty seven percent (52/110) of the patients had a plasma level below 74 nmol/L (23 ng/mL), the lower limit of the proposed therapeutic interval, i.e. an increased number of non-responders were found below this limit in a material of 198 schizophrenic patients. [14] The corresponding range in olanzapine C/D ratios was 0.5-47 nmol/L/mg/day (0.16-15 ng/mL/mg/day); however this great variability was to some extent explained by different time points of sampling in relation to last dose. There was a significant difference in the mean C/D between smokers and non-smokers (mean difference: 3.56 nmol/L/mg/day (1.11 ng/mL/mg/day) lower in smokers, 95% CI of the mean difference 1.27, 5.85; p<0.003) and regardless of smoking habits, men (range 0.5-27 nmol/L/mg/day (0.16-8.4 ng/mL/mg/day)) had significantly lower C/D than the females (range 1.0-47 nmol/L/mg/day (mean difference: 3.4 nmol/L/mg/day (1.1ng/mL/mg/day) lower in males, 95% CI of the mean difference 1.05, 5.75; p<0.006).

Genotype/haplotype

Out of all 121 patients, DNA samples were missing from nine. The genotype of CYP1A2, MDR1 and UGT1A4 variants in 112 patients are given in Table 2. The observed genotype frequencies did not show a significant deviation from Hardy-Weinberg equilibrium. The most frequent SNP in CYP1A2 was -163C>A followed by -2467delT with frequencies of 76.8% and 9.8%, respectively. The SNP CYP1A2 -729C>T was not detected in this population. The frequencies of SNPs MDR1 3435C>T and 2677G>T were 58.5% and 45.5%, respectively, and SNP UGT1A4 142T>G with a frequency of 17.9%. The CYP1A2 and MDR1 haplotypes were constructed using the raw genotype data using population genetic software, ARLEQUIN 2.000. Three known haplotypes, CYP1A2*1A, CYP1A2*1F and CYP1A2*1D, were found in frequencies of 21.9%, 68.3% and 1.3%, respectively (Table 3). Since CYP1A2 -3860G>A, -739T>G and -3113A>G SNPs were not included, the remaining 8.3% of the haplotype could not be determined. MDR1 haplotypes were estimated by using two SNPs, 3435C>T and 2677G>T. The most frequent haplotype was TT followed by CG in frequencies of 41.2% and 37.2% respectively (Table 3).

The association of olanzapine plasma concentration with individual SNPs and haplotypes

The prevalence of each genotype together with the corresponding mean C/D is presented in Table 2. Using ANOVA, the individual genotypes and haplotypes were compared with mean C/D. None of the genotypes or haplotypes unadjusted for covariates showed any significant association with olanzapine plasma concentrations. To be able to control for other factors that have a possible effect on the results, a multiple regression analysis was performed with the combined effect of patient characteristics and genotypes/haplotypes on dose-corrected plasma concentrations of olanzapine (Table 4). Figure 2 shows the values of plasma olanzapine C/D predicted by the model, plotted against the measured concentrations. Male gender, UGT1A4 142T>G genotype and smoking, did each explain a significant proportion of the variation independently and predicted a decrease in dose normalized plasma concentration of olanzapine of 35%, 25% and 21% respectively. Overall, the full regression model (Table 4) with last dose intake 12 hours prior to sampling (14.0 nmol/mg/day), compared to a smoking man heterozygous for UGT1A4 142T>G (2.75 nmol/mg/day).

accounted
To further investigate the clinical impact of the UGT1A4 142T>G genotype for olanzapine exposure, the proportion of patients below the lower limit of the therapeutic interval (74 nmol/L (23 ng/mL)) was compared between homozygous carriers of the 142T-variant. i.e. the most common genotype, or patients being heterozygous for 142G. A larger proportion of patients that carry the UGT1A4 142G variant had a plasma concentration of olanzapine below 74 nmol/L (55%) as compared to 142T homozygous cases (42%), but this difference was not significant (p=0.22).

We also investigated the possible interaction between gender, UGT1A4 142T>G and smoking respectively, with time of sampling, and between UGT1A4 142T>G and smoking. Neither of the multiplicative terms explained a significant proportion of the variability. To further investigate the interaction of CYP1A2*1F haplotypes and smoking, crude C/D plasma concentrations among smokers were compared. The plasma concentrations in individuals with 0, 1 and 2 CYP1A2*1F haplotypes were 8.19 (SD:5.82 ), 6.26 (SD:3.92) and 6.78 (SD:4.89) nmol/L/mg/day respectively. The difference was not significant (p=0.688).

Discussion

The present study suggests that the UGT1A4 142T>G SNP is associated with lower average plasma concentrations of olanzapine.

Initially we performed genotyping for UGT1A4 142T>G using primers described by Nozawa M et al. [15] However we found no clear result to differentiate the different genotypes. Later we performed sequencing of the PCR product and the result indicated that the primers coamplified UGT1A3 and UGT1A5 together with UGT1A4. After multiple alignments of the UGT1A families, we designed UGT1A4 specific primers for genotyping and the result were confirmed by direct sequencing. The impact of the UGT1A4 polymorphism was significant after adjusting for other factors and the 142T>G allele was associated with a similar reduction in olanzapine exposure as smoking or male gender. Applying the defined algorithm (Table 4) to literature-based mean morning plasma levels of 140 nmol/L (46 ng/mL) at steady state after mean daily doses of 16 mg [3], corresponding to 8.75 nmol/L/mg/day, and assuming this represents the exposure in UGT1A4 142TT subjects mainly ('wild type'); the corresponding exposure in subjects genotyped as heterozygous for UGT1A4 142T>G is predicted to [8.75 minus 3.46] = 5.29 nmol/L/mg/day and 85 nmol/L (30 ng/mL) in plasma 12h after last dose at repeated intake of 16 mg/d. In other words, patients carrying one allele of UGT1A4 142G (population frequency of around 15-20%), treated with standard dosages of olanzapine are at risk of sub-therapeutic exposure, according to the cut-offs defined in earlier work. [4,14] It follows that olanzapine exposure in subjects homozygous for the UGT1A4 142T>G SNP would be much lower. However, this needs to be confirmed in separate pharmacokinetic studies on genotyped individuals; i.e. under controlled experimental conditions. We are currently planning to run such a study after screening of several hundred volunteers for the UGT1A4 142T>G SNP, in order to include homozygous 142GG carriers. Confirmatory data showed the almost complete absence of glucuronidation of steroid substrates by the UGT1A4 valine-48 enzyme. [9] However, a study including Japanese healthy volunteers showed that the UGT1A4 142T>G genetic variant may be associated with either an increase or decrease in glucuronidation activity, depending on substrate. Thus, the UGT1A4 L48V amino acid substitution decreased the intrinsic clearance (V max /K m ) for tigogenin while the intrinsic clearance for clozapine, another atypical antipsychotic with very close structural similarity to olanzapine, was twice as high in the UGT1A4 valine-48 enzyme variant compared to that of the wild type leucine-48 enzyme. [16] In light of these data, it is tempting to speculate that the UGT1A4 L48V amino acid substitution involves ethylene-group-dependent sterical changes in the substrate-binding region of the UGT1A4 enzyme.

The present data did not show any effect of the CYP1A2*1F haplotype on olanzapine exposure.

Interestingly, previous work has suggested the impact of this haplotype to be dependent on smoking, and that the extent of CYP1A2 gene expression and enzyme induction by smoking would be more pronounced in carriers of this haplotype. [17,18] However, we did not observe a corresponding positive relationship between the CYP1A2*1F haplotype and smoking in the present material. This might be explained by the fact that UGT1A4 probably plays a greater role in the metabolism of olanzapine than CYP1A2. [5] It is also of interest that UGT1A4 seems to be regulated through the aryl hydrocarbon receptor [19], in similarity with the CYP1A2 gene. As yet, there are no published studies that describe the influence of smoking on UGT1A4 gene expression or enzyme activity, and we failed to detect a difference between smokers and non-smokers regarding the impact of the UGT1A4 polymorphism on olanzapine plasma levels.

We lacked information regarding coffee intake which may be of some relevance for CYP1A2 metabolism since both caffeine and olanzapine are metabolised by this enzyme, and in theory coffee intake may contribute to inter-individual variability in olanzapine exposure. Coffee consumption has been suggested also to induce CYP1A2 [20], but this needs to be concluded from studies on intra-individual changes before and after repeated coffee intake since baseline differences in CYP1A2 activity may predispose to differences in coffee intake behaviour. In any case, the potential bias is likely to be non-differential and should therefore not have affected the results on the impact of UGT1A4 genotype on olanzapine levels.

Using multiple regression analysis, we showed that the male gender is associated with a 35% decrease in the daily dose corrected plasma concentration of olanzapine compared to females, after adjustment for other factors. Adjusting for body weight did not lower this effect, which is concordant with previous studies. Perry et al. reported that male patients required 15.8 mg/day and women 8.6 mg per day, to reach the therapeutic concentration of olanzapine, and Gex-Fabry et al. found that female gender was associated with an 11% increase in plasma concentration after controlling for other factors. [4,21] A possible explanation may be the undocumented use of oral contraceptives which appear to inhibit CYP1A2 activity in vivo. [17] Unfortunately we did not in the present study specifically ask the patients for information regarding contraceptives. To address this issue indirectly, we included age as a dichotomized variable separating patients with an age above 40 years, who may use contraceptives to a lesser extent, from patients 40 years or less. However, this variable did not explain a significant proportion of the variation, supporting the observed gender-specific difference in olanzapine metabolism. It is of interest to note that dose levels in our material did neither differ significantly between men and women, nor between smokers and non-smokers. Apparently, the established influence of these factors on olanzapine exposure did not translate into different dose levels in the clinic, despite being mentioned in drug label. Whether this is a matter of clinical neglect or indicates only a limited clinical importance of such differences for treatment outcome is not clear at this stage.

The regression model was based on an exponential association between the time passed between last dose and olanzapine exposure. As the interaction terms between the sample time and the factors shown to predict olanzapine exposure were not significant, with the exception to that of UGT1A4 142T>G SNP, we chose to keep this association fixed instead of dependent on the characteristics of each individual. This may, from a strict pharmacokinetic point of view be theoretically inappropriate, but appeared as the best approach due to the limited number of patients which resulted in non-significant interaction terms. One non-smoking man, wild type for UGT1A4 142T>G, with a sample that was taken 3.5 hours after the last dose, was identified as an outlier with an exceptionally high C/D olanzapine value of 47 nmol/mg/day (Figure 2). If this case was excluded from the analysis, also the time between last dose and sample became statistically non-significant. Also the predictive value of UGT1A4 142T>G for C/D olanzapine became weaker with a beta coefficient that decreased from -3.46 to -2.19, but did however remain significant (p=0.042). In 8 patients the olanzapine concentration was found to be <20nmol/L, i.e. below the level of more exact quantification. These samples were given a value of 10 nmol/L. By this approach we aimed to avoid the bias of excluding the 18 effect of gender nor UGT1A4 142T>G. The effect of smoking was however not statistically significant (p=0.086). When scrutinizing the excluded patients it was evident that a disproportional large part of them (n=5) were smokers. Whether these patients had not taken their medication, misused their recommended dose or if they are ultra-rapid metabolizers remain unclear. In this respect, it should be commented that patient compliance with measurable or at least detectable intake of prescribed drug in > 90% of subjects may be considered surprisingly high. If it were assumed that patient compliance is independent of genotype, a lower compliance would significantly reduce the power to detect the genotypeconcentration relationship described above.

Conclusion

The results of the present study, based on multiple regression analysis, indicate that carriers of the UGT1A4 142T>G SNP have reduced exposure to olanzapine. The validity of this finding clearly needs to be confirmed in other similar naturalistic settings and by pharmacokinetic studies in genotyped healthy volunteers after oral administration of olanzapine. Also, studies of relapses and discontinuation of olanzapine treatment are needed to assess the clinical effects of the variation in olanzapine levels. In addition, the enzyme variants encoded by the UGT1A4 142T and 142G variants needs to be characterised in vitro with regards to potential catalytic properties. Such studies are now underway. We were also able to confirm that men and smokers are exposed to lower levels of the drug. Whether or not these factors influence the risk of therapeutic failure remains to be established. However, it became clear from the present study that factors already known to reduce olanzapine exposure such as smoking and male gender, was not translated into treatment with higher daily doses. We propose that treatment response to olanzapine in psychosis might improve if both genetic and environmental factors are taken into account by individualized dosage.

  1.5-2 h, the slides were washed with 2x SSC and 0.1% SDS twice for 5 min at 42°C and twice with 0.2 x SSC for 1 min at room temperature. Fluorescence signals from the microarrays were measured using a ScanArray® Express instrument (Perkin Elmer Life Sciences, Boston, MA, USA). The laser power was kept constant at 80% and the photo multiplier tube gain adjusted to obtain equal signal levels from the reaction control spots in all four-laser channels. The excitation lasers were Blue Argon 488 nm for R110; Green HeNe 543.8 nm for Tamra; Yellow HeNe 594 nm for Texas Red and Red HeNe 632.8 nm for Cy5.The signal intensities were determined with the QuantArray analysis 3.1 software (Perkin-Elmer Life Sciences, Boston, Ma, USA). The SNP genotypes were assigned using the SNPsnapper software v3.0.0.191 (http://www.bioinfo.helsinki.fi/SNPsnapper/) based on scatter plots with the logarithm of the sum of both fluorescence signals (S Allele1 +S Allele2 ) on the vertical axis and the fluorescence signal fractions [S Allele1 /(S Allele1 +S Allele2 )] on the horizontal axis.

[ 13 ]

 13 Possible independent variables tested included gender, smoking, age, weight, time between last dose and sampling, co-medication with one of the following potent CYP2D6 inhibitors (there was no documented use of CYP1A2 inhibitors in the study population); paroxetine, fluoxetine, perphenazine or levomepromazine, polymorphisms in genes coding for and P-glycoprotein, and CYP1A2 and MDR1 haplotypes identified in the study population, using forward stepwise regression. The validity of the final model was tested using conventional residual analysis. The calculations were performed in SPSS software version 16.

  8 patients with the lowest daily dose adjusted plasma concentrations. The use of other reference values (e.g. 1 or 20 nmol/L) led to almost identical results to those based on 10

  if both genetic and environmental factors are taken into account by individualized dosage.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

  To study the impact of UGT1A4, CYP1A2, and MDR1 genetic variants on olanzapine plasma levels was investigated in patients with schizophrenia, in relation to other individual factors such as gender, smoking status, body weight and age.Methods: One hundred and twenty-one patients were recruited from psychosis-specialized outpatient departments in Stockholm County. Olanzapine plasma concentrations were analyzed by HPLC. Genotyping was carried out by PCR-RFLP or Minisequencing and haplotypes were analyzed by the use of computer software on population genetics. Multiple regression analysis was performed to investigate the combined effect of patient characteristics and genotypes/haplotypes on daily dose-corrected plasma concentrations of olanzapine.

6 analyzed on 2 -

 62 3% agarose gels. Genotyping for UGT1A4 L48V polymorphism was performed using a forward primer (5´-GTTGGGCCCATAACGAAAGGCAGTT-3´) from Ehmer et al and a newly designed UGT1A4 specific reverse primer (5-'GCTCCACACAACACCTATGAAG-3´) to amplify a 576 bp fragment spanning the polymorphic site.[9] The amplified PCR products were digested over night with STU I. Wild type allele was cut into 324-bp and 152-bp fragments while the mutant allele was not. We confirmed the genotyping result by direct sequencing in both directions using ABI PRISM™ BigDye terminator cycle sequencing ready reaction kit V3.1 (Applied Biosystems) and analyzed on an ABI Prism 377 DNA sequencer.CYP1A2 polymorphisms -729 C>T and -163C>A were genotyped by Minisequencing using taq arrays, as described earlier.[10] PCR primers and Minisequencing primers with 5´ Tag sequences covering the SNP site were designed using the Autoprimer (http://www.autoprimer.com) (Beckman Coulter) software. The primers were obtained from Integrated DNA Technologies (IDT Inc., Coralville, IA, USA). PCR and Minisequencing primers are listed in Supplementary table 2. PCR fragments spanning the SNPs were amplified in individual reactions by multiplex PCR (10 ng/µl DNA) containing forward and reverse primers for both SNPs. To remove remaining dNTPs and primers, the PCR products (7.2 µl) were treated with 1 U/µl shrimp alkaline phosphatase and 0.48 U/µl exonuclease I (USB Coorporation, Cleveland, OH, USA). The oligonucleotides complementary to the taq sequence attached to the 5´of the minisequencing primer were immobilized on the CodeLink Activated Slides (GE Healthcare, Uppsala, Sweden) in duplicate spots in each sub array. The minisequencing reaction was run in the presence of fluorescently labeled ddNTP (0.1 µM Texas Red-ddATP, Tamra-ddCTP, R110-ddGTP and 0.15 µM Cy5-ddUTP (Perkin Elmer Life Sciences, Boston, MA, USA)) and 0.065 U of KlenThermase™ DNA polymerase (GeneCraft, Germany). After annealing of the extension products to the immobilized complementary tag 1.5-2 h, the slides were washed with 2x SSC and 0.1% SDS twice for 5 min at 42°C and twice with 0.2 x SSC for 1 min at room temperature. Fluorescence signals from the microarrays were measured using a ScanArray® Express instrument (Perkin Elmer Life Sciences, Boston, MA, USA). The laser power was kept constant at 80% and the photo multiplier tube gain adjusted to obtain equal signal levels from the reaction control spots in all four-laser channels. The excitation lasers were Blue Argon 488 nm for R110; Green HeNe 543.8 nm for Tamra; Yellow HeNe 594 nm for Texas Red and Red HeNe 632.8 nm for Cy5. The signal intensities were determined with the QuantArray analysis 3.1 software (Perkin-Elmer Life Sciences, Boston, Ma, USA). The SNP genotypes were assigned using the SNPsnapper software v3.0.0.191 (http://www.bioinfo.helsinki.fi/SNPsnapper/) based on scatter plots with the logarithm of the sum of both fluorescence signals (S Allele1 +S Allele2 ) on the vertical axis and the fluorescence signal fractions [S Allele1 /(S Allele1 +S Allele2 )] on the horizontal axis.

9 and

 9 CYP1A2 and MDR1 haplotypes identified in the study population, using forward stepwise regression. The validity of the final model was tested using conventional residual analysis. The calculations were performed in SPSS software version 16.

  for 25% of inter-individual variability in olanzapine C/D, as indicated by the R 2 value (fig. 2). No additional contribution to the olanzapine C/D variability was apparent from the different CYP1A2 and MDR1 genotypes/haplotypes. The regression model predicted a 5.1 fold higher olanzapine plasma C/D for a non-smoking female patient, without the UGT1A4 142T>G SNP,

  on a significant impact of the UGT1A4 polymorphism would unavoidably lead to the question whether prospective genotyping of patients should be considered, for an early detection of subjects at risk of sub-therapeutic olanzapine exposure.The UGT1A4 142T>G SNP causes a leucine to valine amino acid substitution (L48V), and there are contradictory reports in the literature how this affects enzyme activity. Ehmer et al.

8

  patients with the lowest daily dose adjusted plasma concentrations. The use of other reference values (e.g. 1 or 20 nmol/L) led to almost identical results to those based on 10 nmol/L. Calculations without these 8 patients did not change the results with regard to the

Table 1 . Summary of diagnoses and the mean daily doses* and plasma concentrations of olanzapine in the study population

 1

	Diagnosis**	Schizophrenia	Other psychosis	Other diagnosis
		or
		schizoaffective	(psychosis NUD +	(including bipolar)
		syndrome	delusional disorder)
	N total	75	27	14
	mg, daily dose (mean ± SD)	(13 ± 7.6)	(11 ± 5.6)	(11 ± 7.0)
	nmol/L, plasma concentration,	(115 ± 112)	(77 ± 53)	(78 ±55)
	(mean ± SD)
	N males	46	17	11
	mg, daily dose (mean ± SD)	(13 ± 7.5)	(11 ± 5.0)	(8.4 ± 4.4)
	nmol/L, plasma concentration,	(104 ± 112)	(69 ± 42)	(58 ± 35)
	(mean ± SD)
	N females	29	10	3
	mg, daily dose (mean ± SD)	(12 ± 7.9)	(11 ± 6.7)	(18 ± 10)
	nmol/L, plasma concentration,	(135 ± 111)	(91 ± 70)	(131 ± 71)
	(mean ± SD)
	N non-smokers	43	16	9
	mg, daily dose (mean ± SD)	(12 ± 8.1)	(12 ± 6.0)	(11 ± 8.3)
	nmol/L, plasma concentration,	(139 ± 132)	(95 ± 56)	(87 ± 60)
	(mean ± SD)
	N smokers	32	11	5
	mg, daily dose (mean ± SD)	(13 ± 7.0)	(9.3 ± 4.5)	(9 ± 4.2)
	nmol/L, plasma concentration,	(86 ± 71)	(50 ± 34)	(53 ± 35)
	(mean ± SD)
	* Information on daily doses was available in 118 patients and missing in 3
	** Information on diagnosis was available in 119 patients and missing in 2
	***Information on plasma concentrations of olanzapine was available in 110 patients and
	missing in 11

Table 2 . Genotype frequencies of CYP1A2, MDR1 and UGT1A4 polymorphisms investigated and corresponding mean concentration-to-dose rations (C/D) of olanzapine

 2

	Gene and SNP Genotype Number of patients	Mean C/D (SD)
			(n=112)	(n=102)
	CYP1A2
	-2467 delT	TT	92	8.8 (6.4)
		T/delT	18	9.6 (6.0)
		delT/delT 2	-
	-729C>T	CC	112	8.9 (6.3)
		CT	0	-
		TT	0	-
	-163C>A	CC	7	11.0 (9.1)
		CA	38	9.0 (7.6)
		AA	67	8.3 (4.7)
	UGT1A4
	142T>G	TT	83	9.6 (6.9)
		TG	29	7.0 (3.6)
		GG	0
	MDR1
	3435C>T	CC	21	8.2 (10.3)
		CT	51	8.8 (4.8)
		TT	40	9.3 (5.5)
	2677G>T	GG	32	9.2 (8.5)
		GT	58	8.2 (5.1)
		TT	22	10.7 (5.3)

Table 3 . Haplotype frequencies of CYP1A2 and MDR1 in 112 schizophrenic patients.

 3

		Haplotype	Observed
				frequencies
	CYP1A2
	CYP1A2*1A		0.219
	CYP1A2*1F (-163C>A)	0.683
	CYP1A2*1D (-2467delT)	0.013
	MDR1
	3435C>T 2677G>T
	CG	C	G	0.372
	CT	C	T	0.043
	TG	T	G	0.173
	TT	T	T	0.412

Table 4 . Inter-individual variability factors for plasma concentration of olanzapine per milligram oral daily dose, in the study population 1 .

 4

	Factors affecting olanzapine plasma			p-value
		Expected values Std. Error
	concentration (nmol/L)/(mg/day)
	Male gender	-4.91	1.3	<0.001
	UGT1A4 142T>G	-3.46	1.4	0.019
	Smoking	-2.93	1.3	0.022

1

The final linear model was as follows: Daily dose corrected plasma concentration of olanzapine = 1.29 + (16.83* 0.97716 Hours between last dose and sampling ) + (-4.91* Male gender) + (-3.46* UGT1A4 142T>G) + (-2.93* Smoking). The predicted dose corrected olanzapine concentration in (nmol/L)/(mg/day) in a non-smoking female, wild type for UGT1A4 142T>G genotype that had taken her last dose 12 hours prior to the sampling = 1.29 + (16.83* 0.97716 12 ) + (-4.91* 0) + (-3.46* 0) + (-2.93* 0) = 14.0.

CARRIERS OF THE UGT1A4 142T>G GENE VARIANT ARE PREDISPOSED TO REDUCED OLANZAPINE EXPOSURE -AN IMPACT SIMILAR TO MALE GENDER OR SMOKING IN SCHIZOPHRENIC PATIENTS Roza

  Table 4 and explained 25% of the inter-individual variability in plasma levels corrected for differences in daily dose. Ghotbi, PhD* 1 , Buster Mannheimer, MD, PhD* , Eleni Aklillu, PhD 1 , Akira Suda, MD, PhD 1, § , Leif Bertilsson, PhD 1 , Erik Eliasson, MD, PhD 1 # and Urban Ösby, MD, PhD 3 # * Both authors contributed equally and share first authorship. # Both authors contributed equally and share last authorship. § Visiting scientist from Yokohama City University School of Medicine, Department of Psychiatry. Karolinska Institutet, Stockholm, Sweden; 1 Department of Laboratory Medicine, Division of Clinical Pharmacology, Karolinska University Hospital, Huddinge 2 Department of Internal Medicine, Södersjukhuset 3 Department of Clinical Neuroscience, R&D Unit, Department of Psychiatry, Danderyd University Hospital

	Figure 1. Ghotbi, Mannheimer et al. 2009
	Figure 2. Ghotbi, Mannheimer et al. 2009
	F o r F o r F o r F o r
	P P P P
	e e e e
	Address for correspondence: Dr Buster Mannheimer Tel: +4686163478	e r e r e r e r
	R e R e R e R e e-mail: buster.mannheimer@sodersjukhuset.se Fax: +4686163566 Mobile: +46733260444
		v i e v i e v i e v i e
		w w w w

Table 1 . Summary of diagnoses and the mean daily doses* and plasma concentrations of olanzapine in the study population

 1 Information on daily doses was available in 118 patients and missing in 3 ** Information on diagnosis was available in 119 patients and missing in 2 ***Information on plasma concentrations of olanzapine was available in 110 patients and missing in 11

	Diagnosis**	Schizophrenia	Other psychosis	Other diagnosis
		or
		schizoaffective	(psychosis NUD +	(including bipolar)
		syndrome	delusional disorder)
	N total	75	27	14
	mg, daily dose (mean ± SD)	(13 ± 7.6)	(11 ± 5.6)	(11 ± 7.0)
	nmol/L, plasma concentration,	(115 ± 112)	(77 ± 53)	(78 ±55)
	(mean ± SD)
	N males	46	17	11
	mg, daily dose (mean ± SD)	(13 ± 7.5)	(11 ± 5.0)	(8.4 ± 4.4)
	nmol/L, plasma concentration,	(104 ± 112)	(69 ± 42)	(58 ± 35)
	(mean ± SD)
	N females	29	10	3
	mg, daily dose (mean ± SD)	(12 ± 7.9)	(11 ± 6.7)	(18 ± 10)
	nmol/L, plasma concentration,	(135 ± 111)	(91 ± 70)	(131 ± 71)
	(mean ± SD)
	N non-smokers	43	16	9
	mg, daily dose (mean ± SD)	(12 ± 8.1)	(12 ± 6.0)	(11 ± 8.3)
	nmol/L, plasma concentration,	(139 ± 132)	(95 ± 56)	(87 ± 60)
	(mean ± SD)
	N smokers	32	11	5
	mg, daily dose (mean ± SD)	(13 ± 7.0)	(9.3 ± 4.5)	(9 ± 4.2)
	nmol/L, plasma concentration,	(86 ± 71)	(50 ± 34)	(53 ± 35)
	(mean ± SD)

*

Table 2 . Genotype frequencies of CYP1A2, MDR1 and UGT1A4 polymorphisms investigated and corresponding mean concentration-to-dose rations (C/D) of olanzapine

 2

	Gene and SNP Genotype Number of patients	Mean C/D (SD)
			(n=112)	(n=102)
	CYP1A2
	-2467 delT	TT	92	8.8 (6.4)
		T/delT	18	9.6 (6.0)
		delT/delT 2	-
	-729C>T	CC	112	8.9 (6.3)
		CT	0	-
		TT	0	-
	-163C>A	CC	7	11.0 (9.1)
		CA	38	9.0 (7.6)
		AA	67	8.3 (4.7)
	UGT1A4
	142T>G	TT	83	9.6 (6.9)
		TG	29	7.0 (3.6)
		GG	0
	MDR1
	3435C>T	CC	21	8.2 (10.3)
		CT	51	8.8 (4.8)
		TT	40	9.3 (5.5)
	2677G>T	GG	32	9.2 (8.5)
		GT	58	8.2 (5.1)
		TT	22	10.7 (5.3)

Table 3 . Haplotype frequencies of CYP1A2 and MDR1 in 112 schizophrenic patients.

 3

		Haplotype	Observed
				frequencies
	CYP1A2
	CYP1A2*1A		0.219
	CYP1A2*1F (-163C>A)	0.683
	CYP1A2*1D (-2467delT)	0.013
	MDR1
	3435C>T 2677G>T
	CG	C	G	0.372
	CT	C	T	0.043
	TG	T	G	0.173
	TT	T	T	0.412

Table 4 . Inter-individual variability factors for plasma concentration of olanzapine per milligram oral daily dose, in the study population 1 .

 4 The final linear model was as follows: Daily dose corrected plasma concentration of olanzapine = 1.29 + (16.83* 0.97716 Hours between last dose and sampling ) + (-4.91* Male gender) + (-3.46* UGT1A4 142T>G) + (-2.93* Smoking). The predicted dose corrected olanzapine concentration in (nmol/L)/(mg/day) in a non-smoking female, wild type for UGT1A4 142T>G genotype that had taken her last dose 12 hours prior to the sampling = 1.29 + (16.83* 0.97716 12 ) + (-4.91* 0) + (-3.46* 0) + (-2.93* 0) = 14.0.Aricle title: CARRIERS OF THE UGT1A4 142T>G GENE VARIANT ARE PREDISPOSED TO REDUCED OLANZAPINE EXPOSURE -AN IMPACT SIMILAR TO MALE GENDER OR SMOKING IN SCHIZOPHRENIC PATIENTS

	Factors affecting olanzapine plasma			p-value
		Expected values Std. Error
	concentration (nmol/L)/(mg/day)
	Male gender	-4.91	1.3	<0.001
	UGT1A4 142T>G	-3.46	1.4	0.019
	Smoking	-2.93	1.3	0.022

1

table 2 . PCR primers and Minisequencing primers used for the genotyping of CYP1A2 729T>G and -163C>A.

 2

	SNP	PCR primer (5´-3´)	Minisequencing primer with tag sequence
	-729T>C	fo-AAGCTAGTGGGGACAGAAAGA	re-ATTGACCAAACTGCGGTGCGAGTCAA
			GAGCTGGGTAGCAAAGCCC
		re-TTAAAAATGGCTTAGTCCAAACTG
	-163C>A	fo-AATCTTGAGGCTCCT TTCCA	fo-ATTAACTCGACTGCCGCGTGTGCTCAA
			AGGGTGAGCTCTGTGGGC
		re-AGCTGGATACCAGAAAGACTAAGC	re-AATACGCTGAATAGAGCCCTTAAGCTCC
			ATCTACCATGCGTCCTG

European Journal of Clinical Pharmacology

Acknowledgements

The study was supported by the Swedish Research Council, Medicine (grants 3902 and 10909), by the Stockholm County Council (ALF grants 20060420, 20060100 and 20080022), and by grants from Department of Drug Management and Informatics, Stockholm County Council, and from Söderström-Königska Hospital. We thank Professor Martin Schalling for important contributions in planning and setting up the study of metabolic risks in psychosis. We also thank Lili Milani for excellent assistance in the development of CYP1A2 Minisequencing technique and Carina Schmidt for highly qualified administration of patient samples. Drs Sören Akselson, Deanne Mannelid and Helena Ring are acknowledged for recruiting most of the patients. Acknowledgements The study was supported by the Swedish Research Council, Medicine (grants 3902 and 10909), by the Stockholm County Council (ALF grants 20060420, 20060100 and 20080022), and by grants from Department of Drug Management and Informatics, Stockholm County Council, and from Söderström-Königska Hospital. We thank Professor Martin Schalling for important contributions in planning and setting up the study of metabolic risks in psychosis. We also thank Lili Milani for excellent assistance in the development of CYP1A2 Minisequencing technique and Carina Schmidt for highly qualified administration of patient samples. Drs Sören Akselson, Deanne Mannelid and Helena Ring are acknowledged for recruiting most of the patients.

Legends to figures:

 4and explained 25% of the inter-individual variability in plasma levels corrected for differences in daily dose. [19] [19]