Ruben Blasco-Garcia

Conchita Martinez-Perez

Luis Paris

Rubén Blasco-García

Conchita Martínez-Pérez

Poly-freeness of even Artin groups of FC type

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

One of the families of groups where the interaction between algebraic and geometric techniques has been most fruitful is the family of Artin groups, also known as Artin-Tits groups or generalized braid groups. There are very few works dealing with all the Artin groups, and the theory mainly consists on the study of more or less extended subfamilies. The subfamily of right-angled Artin groups is one of the most studied, in particular because of they resounding applications in homology of groups by Bestvina-Brady [START_REF] Bestvina | Morse theory and finiteness properties for groups[END_REF] and in low dimensional topology by Haglund-Wise [START_REF] Haglund | Special cube complexes[END_REF] and Agol [START_REF] Agol | The virtual Haken conjecture, With an appendix by Agol, Daniel Groves, and Jason Manning[END_REF]. We refer to Charney [START_REF] Charney | An introduction to right-angled Artin groups[END_REF] for an introductory survey on these groups.

The main goal of the present paper is to prove that some known structural property of right-angled Artin groups also holds for a larger family of groups (see Theorem 3.17). The property is the "poly-freeness" and the family is that of even Artin groups of FC type.

A group G is called poly-free if there exists a tower of subgroups

1 = G 0 G 1 • • • G N = G
for which each quotient G i+1 /G i is a free group. Recall that a group is locally indicable if every non-trivial finitely generated subgroup admits an epimorphism onto Z, and a group is right orderable if it admits a total order invariant under right multiplication. Poly-free groups are locally indicable, and locally indicable groups are right orderable (see Rhemtulla-Rolfsen [START_REF] Rhemtulla | Local indicability in ordered groups: braids and elementary amenable groups[END_REF]). The fact that right-angled Artin groups are poly-free was independently proved by Duchamp-Krob [START_REF] Duchamp | Free partially commutative structures[END_REF], Howie [START_REF] Howie | Bestvina-Brady groups and the plus construction[END_REF] and Hermiller-Šunić [START_REF] Hermiller | Poly-free constructions for right-angled Artin groups[END_REF].

Artin groups of FC type were introduced by Charney-Davis [START_REF] Charney | The K(π, 1)-problem for hyperplane complements associated to infinite reflection groups[END_REF] in their study of the K(π, 1) conjecture for Artin groups. They are particularly interesting because there are CAT(0) cubical complexes on which they act. This family contains both, the rightangled Artin groups, and the Artin groups of spherical type (those corresponding to finite Coxeter groups). An Artin group is called even if it is defined by relations of the form (st) k = (ts) k , k ≥ 1. There are some papers in the literature dealing with even Coxeter groups (see for example [START_REF] Antolín | L Ciobanu Geodesic growth in right-angled and even Coxeter groups European[END_REF]) but as far as we know, there is no other paper dealing with even Artin groups (although they are mentioned in [START_REF] Bartolo | Quasi-projectivity, Artin-Tits groups, and pencil maps[END_REF]). However, we think that even Artin groups deserve to be studied because they have remarkable properties. One of them is that such a group retracts onto any parabolic subgroup (see Section 2).

Our proof is partially inspired by the one of Duchamp-Krob [START_REF] Duchamp | Free partially commutative structures[END_REF] for right-angled Artin groups in the sense that, as them, we define a suitable semi-direct product of an Artin group based on a proper subgraph with a free group, and we show that this semi-direct product is isomorphic to the original Artin group. However, in our case, this approach is more complicated, essentially because we have to deal with relations that are also more complicated.

In Section 4 we prove that even Artin groups of FC type are residually finite. This result is a more or less direct consequence of Boler-Evans [START_REF] Boler | The free product of residually finite groups amalgamated along retracts is residually finite[END_REF], but it deserves to be pointed out since it increases the list of Artin groups shown to be residually finite. Actually, this list is quite short. It contains the spherical type Artin groups (since they are linear), the right-angled Artin groups (since they are residually nilpotent), and few other examples. In particular, it is not known whether all Artin groups of FC type are residually finite.

Preliminaries

Artin groups

Let S be a finite set. A Coxeter matrix over S is a square matrix M = (m s,t ) s,t∈S indexed by the elements of S, with coefficients in N ∪ {∞}, and satisfying m s,s = 1 for all s ∈ S and m s,t = m t,s ≥ 2 for all s, t ∈ S, s = t. We will represent such a Coxeter matrix M by a labelled graph Γ, whose set of vertices is S, and where two distinct vertices s, t ∈ S are linked by an edge labelled with m s,t if m s,t = ∞. We will also often use the notation V(Γ) to denote the set of vertices of Γ (that is, V(Γ) = S).

Remark The labelled graph Γ defined above is not the Coxeter graph of M as defined in Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF]. It is another fairly common way to represent a Coxeter matrix. For T ⊂ S, we denote by A T (resp. W T ) the subgroup of A (resp. W ) generated by T , and by Γ T the full subgraph of Γ spanned by T . Here we mean that each edge of Γ T is labelled with the same number as its corresponding edge of Γ. By Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF], the group W T is the Coxeter group of Γ T , and, by van der Lek [START_REF] Van Der Lek | The homotopy type of complex hyperplane complements[END_REF], A T is the Artin group of Γ T . The group A T (resp. W T ) is called a standard parabolic subgroup of A (resp. of W ).

As pointed out in the introduction, the theory of Artin groups consists in the study of more or less extended families. The one concerned by the present paper is the family of even Artin groups of FC type. These are defined as follows. We say that A = A Γ is of spherical type if its associated Coxeter group, W Γ , is finite. A subset T of S is called free of infinity if m s,t = ∞ for all s, t ∈ T . We say that A is of FC type if A T is of spherical type for every free of infinity subset T of S. Finally, we say that A is even if m s,t is even for all distinct s, t ∈ S (here, ∞ is even).

Remark As we said in the introduction, we think that even Artin groups deserve special attention since they have particularly interesting properties as the following ones. Assume that A is even.

(1) Let s, t ∈ S, s = t. If we set m s,t = 2k s,t , then the Artin relation Π(s, t : m s,t ) = Π(t, s : m s,t ) becomes (st) ks,t = (ts) ks,t . This form of relation is less innocuous than it seems (see Subsection 2.3 for example).

(2) Let T be a subset of S. Then the inclusion map A T ֒→ A always admits a retraction π T : A → A T which sends s to s if s ∈ T , and sends s to 1 if s ∈ T .

Britton's lemma

Let G be a group generated by a finite set S. We denote by (S ∪ S -1 ) * the free monoid over S ∪ S -1 , that is, the set of words over S ∪ S -1 , and we denote by (S ∪ S -1 ) * → G, w → w, the map that sends a word to the element of G that it represents. Recall that a set of normal forms for G is a subset N of (S ∪ S -1 ) * such that the map N → G, w → w, is a one-to-one correspondence.

Let G be a group with two subgroups A, B ≤ G, and let ϕ : A → B be an isomorphism.

A useful consequence of Britton's lemma yields a set of normal forms for the HNNextension G * ϕ = G, t | t -1 at = ϕ(a), a ∈ A in terms of a set N of normal forms for G and sets of representatives of the cosets of A and B in G (see Lyndon-Schupp [START_REF] R C Lyndon | Combinatorial group theory[END_REF]). Explicitly, choose a set T A of representatives of the left cosets of A in G containing 1, and a set T B of representatives of the left cosets of B in G also containing 1.

Proposition 2.1 (Britton's normal forms) Let Ñ be the set of words of the form

w 0 t ε 1 w 1 • • • t εm w m
, where m ≥ 0, ε i ∈ {±1} and w i ∈ N for all i, such that:

(a) wi ∈ T A if ε i = -1, for i ≥ 1, (b) wi ∈ T B if ε i = 1, for i ≥ 1,
(c) there is no subword of the form

t ε t -ε .
Then Ñ is a set of normal forms for the HNN-extension G * ϕ .

Variations of the even Artin-type relations

For a, b in a group G, we denote by b a = a -1 ba the conjugate of b by a. The aim of this subsection is to illustrate how the Artin relations in even Artin groups can be expressed in terms of conjugates. This observation will be a key point in our proof of Theorem 3.17.

Let s, t be two generators of an Artin group A. The relation st = ts can be expressed as t s = t. Analogously, from tsts = stst, we obtain s -1 tst = tsts -1 , and therefore t s -1 = t -1 t s t and t s 2 = t s t(t s ) -1 . We deduce that t s -1 , t s 2 ∈ t, t s . Doing the same thing with the relation tststs = ststst one easily gets

t s -1 = t -1 (t s ) -1 t s 2 t s t and t s 3 = t s 2 t s t (t s ) -1 (t s 2 ) -1 .
This can be extended to any even Artin-type relation (ts

) k = (st) k giving (2-1) t s -1 = t -1 (t s ) -1 • • • (t s k-2 ) -1 t s k-1 t s k-2 • • • t s t , t s k = t s k-1 • • • t s t (t s ) -1 • • • (t s k-1 ) -1 .
As a consequence we see that t s i ∈ t, t s , t s 2 , . . . , t s k-1 for any integer i.

Poly-freeness of even Artin groups of FC type

In this section we prove that even Artin groups of FC type are poly-free (Theorem 3.17). We begin with a characterization of these groups in terms of their defining graphs.

Lemma 3.1 Let A Γ be an even Artin group. Then A Γ is of FC type if and only if every triangular subgraph of Γ has at least two edges labelled with 2.

Proof From the classification of finite Coxeter groups (see Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF], for example) follows that an even Artin group A Γ is of spherical type if and only if Γ is complete and for each vertex there is at most one edge with label greater than 2 involving it. Suppose that A Γ is of FC type. Let Ω be a triangular subgraph of Γ. Then A Ω is even and of spherical type, hence, by the above, Ω has at least two edges labelled with 2. Suppose now that every triangular subgraph of Γ has at least two edges labelled with 2. Let Ω be a complete subgraph of Γ. Then Ω is even and every triangular subgraph of Ω has at least two edges labelled with 2, hence, by the above, A Ω is of spherical type. So, A Γ is of FC type.

The proof of Theorem 3.17 is based on the following.

Proposition 3.2 Assume that A Γ is even and of FC type. Then there is a free group F such that A Γ = F ⋊ A 1 , where A 1 is an even Artin group of FC type based on a proper subgraph of Γ.

We proceed now with some notations needed for the proof of Proposition 3.2. Fix some vertex z of Γ. Recall that the link of z in Γ is the full subgraph lk(z, Γ) of Γ with vertex set V(lk(z, Γ)) = {s ∈ S | s = z and m s,z = ∞}. As ever, we see lk(z, Γ) as a labelled graph, where the labels are the same as in the original graph Γ. We set L = lk(z, Γ), and we denote by Γ 1 the full subgraph of Γ spanned by S \ {z}. We denote by A 1 and A L the subgroups of A Γ generated by V(Γ 1 ) and V(L), respectively. Recall from Section 2 that A 1 and A L are the Artin groups associated with the graphs Γ 1 and L, respectively (so this notation is consistent).

As pointed out in Section 2, since A Γ is even, the inclusion map A 1 ֒→ A has a retraction π 1 : A Γ → A 1 which sends z to 1 and sends s to s if s = z. Similarly, the inclusion map

A L ֒→ A 1 has a retraction π L : A 1 → A L which sends s to s if s ∈ V(L),
and sends s to 1 if s ∈ V(L). It follows that A Γ and A 1 split as semi-direct products

A Γ = Ker(π 1 ) ⋊ A 1 and A 1 = Ker(π L ) ⋊ A L .
For s ∈ V(L) we denote by k s the integer such that m z,s = 2k s . Lemma 3.1 implies the following statement. This will help us to describe A L as an iterated HNN extension.

Lemma 3.3 Let s, t be two linked vertices of L.

(1) Either k s = 1, or k t = 1.

(

) If k s > 1, then m s,t = 2. 2
Let L 1 be the full subgraph of L spanned by the vertices s ∈ V(L) such that k s = 1. Lemma 3.3 implies that L\L 1 is totally disconnected. We set V(L\L 1 ) = {x 1 , . . . , x n }. Again, from Lemma 3.3, we deduce that, for each i ∈ {1, . . . , n}, if the vertex x i is linked to some vertex s ∈ V(L 1 ), then the label of the edge between x i and s must be 2. For each i ∈ {1, . . . , n} we set S i = lk(x i , L), and we denote by X i the full subgraph of L spanned by {x 1 , . . . , x i } ∪ V(L 1 ) and X 0 = L 1 . Note that S i is a subgraph of L 1 and, therefore, is a subgraph of X i . The subgraphs of Γ that we have defined so far are sitting as follows inside Γ

S i ⊆ L 1 = X 0 ⊆ X 1 ⊆ . . . ⊆ X n = L ⊆ Γ 1 ⊆ Γ
where i ∈ {1, . . . , n} The defining map of each of the HNN extensions will be the identity in the subgroup generated by the vertices commuting with x i , that is,

ϕ i = Id : A Si → A Si .
Then, writing down the associated presentation, we see that A Xi = (A X i-1 ) * ϕi with stable letter x i . So, we get the following.

Lemma 3.4 We have A L = ((A L 1 * ϕ 1 ) * ϕ 2 • • • ) * ϕn .
Now, fix a set N 1 of normal forms for A L 1 (for example, the set of shortlex geodesic words with respect to some ordering in the standard generating system). We want to use Britton's lemma to obtain a set of normal forms for A L in terms of N 1 . To do so, first, for each i ∈ {1, . . . , n}, we need to determine a set of representatives of the right cosets of A Si in A L 1 . The natural way to do it is as follows. Consider the projection map π Si :

A L 1 → A Si which sends s ∈ V(L) to s if s ∈ V(S i ) and sends s to 1 otherwise. Observe that A L 1 = A Si ⋉ Ker(π Si ). Then, Ker(π Si ) is a well-defined set of representatives of the right cosets of A Si in A L 1 .
In our next result we will use this set of representatives together with Britton's lemma to construct a set N L of normal forms for A L . More precisely, N L denotes the set of words of the form

w 0 x ε 1 α 1 w 1 • • • x εm αm w m
, where w j ∈ N 1 for all j ∈ {0, 1, . . . , m}, α j ∈ {1, . . . , n}, wj ∈ Ker(π Sα j ) and ε j ∈ {±1} for all j ∈ {1, . . . , m}, and there is no subword of the form x ε α x -ε α with α ∈ {1, . . . , n}. Lemma 3.5 The set N L is a set of normal forms for A L .

Proof For i ∈ {0, 1, . . . , n}, we denote by N L,i the set of words of the form

(3-1) w 0 x ε 1 α 1 w 1 • • • x εm αm w m
, where w j ∈ N 1 for all j ∈ {0, 1, . . . , m}, α j ∈ {1, . . . , i}, wj ∈ Ker(π Sα j ) and ε j ∈ {±1} for all j ∈ {1, . . . , m}, and there is no subword of the form x ε α x -ε α with α ∈ {1, . . . , i}. We prove by induction on i that N L,i is a set of normal forms for A Xi . Since A L = A Xn , this will prove the lemma.

The case i = 0 is true by definition since A L 1 = A X 0 and N L,0 = N 1 . So, we can assume that i ≥ 1 plus the inductive hypothesis. Recall that A Xi = (A X i-1 ) * ϕi , where ϕ i is the identity map on A Si . By induction, N L,i-1 is a set of normal forms for A X i-1 . We want to apply Proposition 2.1, so we also need a set T i of representatives of the right cosets of A Si in A X i-1 . Since A L 1 = A Si ⋉ Ker(π Si ), we see that we may take as T i the set of elements of A X i-1 whose normal forms, written as in Equation 3-1, satisfy w0 ∈ Ker(π Si ). Now, take g ∈ A Xi and use Proposition 2.1 with the set N L,i-1 of normal forms and the set T i of representatives to write a uniquely determined expression for g. The set of these expressions is clearly N L,i .

Given g ∈ A L , we denote by n(g) the normal form of g in N L .

The following sets will be crucial in our argument. The set T 0 is the set of g ∈ A L such that n(g) does not begin with s -1 or s ks for any s ∈ V(L). In other words, T 0 denotes the set of elements in A L such that n(g) has w 0 = 1, ε 1 = 1, and no consecutive sequence of x α 1 's of length k xα 1 at the beginning. On the other hand, we set T = T 0 Ker(π L ). The reason why these sets are so important to us is that T will serve as index set for a free basis for the free group F of Proposition 3.2, that is, F = F(B) is the free group with basis a set B = {b g | g ∈ T} in one-to-one correspondence with T . We will also consider a smaller auxiliary free group F 0 = F(B 0 ) with basis B 0 = {b h | h ∈ T 0 }. Observe that, since T 0 ⊆ A L , we have that any g ∈ T can be written in a unique way as g = hu with h ∈ T 0 and u Ker(π L ).

At this point, we can explain the basic idea of the proof of Proposition 3.2. We will define an action of A 1 on F and form the corresponding semi-direct product. Then we will construct an explicit isomorphism between this semi-direct product and the original group A Γ that takes F onto Ker(π 1 ). More explicitly, this isomorphism will be an extension of the homomorphism ϕ : F → Ker(π 1 ) which sends b g to z g for all g ∈ T .

We proceed now to define the action of A 1 on F . This action should mimic the obvious conjugation action of A 1 on Ker(π 1 ). As a first step, we start defining an action of A L on F 0 . This will be later extended to the desired action of A 1 on F .

Given h ∈ T 0 , we denote by supp(h) the set of x i ∈ V(L \ L 1 ) which appear in the normal form n(h). We will need the following technical result. Lemma 3.6 Let s ∈ V(L) and h ∈ T 0 .

(1) If s ∈ V(L 1 ) and s ∈ V(S i ) for every x i ∈ supp(h) (including the case h = 1), then hs ∈ T 0 and hs -1 ∈ T 0 , but shs -1 , s -1 hs ∈ T 0 .

(

) If s = x ∈ V(L \ L 1 ) 2
and h = x kx-1 , then hs ∈ T 0 and hs -1 ∈ T 0 .

(

) If s = x ∈ V(L \ L 1 ) 3
and h = 1, then hs ∈ T 0 and hs -1 ∈ T 0 .

(4) We have hs, hs -1 ∈ T 0 in all the other cases.

Proof Since h ∈ T 0 , h has a normal form

x ε 1 α 1 w 1 • • • x εm
αm w m , with each w j the normal form of an element in Ker(π Sα j ). In the case when s = x ∈ V(L \ L 1 ), multiplying this expression by x on the right, we get, after a possible cancellation x -1 x, a normal form. So, hx ∈ T 0 if and only if h = x kx-1 . Similarly, we have hx -1 ∈ T 0 if and only if h = 1. Now, suppose that s ∈ V(L 1 ). Assume that s ∈ S α j for j = j 0 + 1, . . . , m, but s ∈ S α j 0 . Then the normal form for hs is

x ε 1 α 1 w 1 • • • w j 0 -1 x ε j 0 α j 0 w ′ j 0 • • • x ε m-1 α m-1 w ′ m-1 x εm αm w ′ m
, where w ′ j 0 is the normal form for wj 0 s, and, for j ∈ {j 0 + 1, . . . , m}, w ′ j is the normal form for s -1 wj s. Hence, hs ∈ T 0 . Similarly, hs -1 ∈ T 0 . On the contrary, assume that s ∈ S α j for all those x j appearing in the normal form for h. Then the normal form for hs is

sx ε 1 α 1 w ′ 1 • • • x εm αm w ′ m ,
where w ′ j is the normal form for s -1 wj s for all j ∈ {1, . . . , m}. This form begins with s, hence hs ∈ T 0 . However, the normal form for s -1 hs is obtained from the above form by removing the s at the beginning, hence s -1 hs ∈ T 0 . Similarly, hs -1 ∈ T 0 and shs -1 ∈ T 0 .

Hidden in the proof of Lemma 3.6 is the fact that, if g 1 , g 2 ∈ A L 1 and h ∈ T 0 , then g 1 hg 2 ∈ A L 1 T 0 . Moreover, by Lemma 3.5, every element of A L 1 T 0 is uniquely written in the form gh with g ∈ A L 1 and h ∈ T 0 . In this case we set u(gh) = h. So, by Lemma 3.6, if s ∈ V(L 1 ) and h ∈ T 0 , then u(hs) = s -1 hs if s ∈ V(S i ) for every x i ∈ supp(h), and u(hs) = hs otherwise. If s = x ∈ V(L \ L 1 ), then u(hx) is not defined if h = x kx-1 , and u(hx) = hx otherwise.

We turn now to define the action of A L on F 0 . We start with the action of the generators. Let s ∈ V(L). For h ∈ T 0 we set

b h * s = b u(hs) if hs ∈ A L 1 T 0 , b x kx -1 • • • b x b 1 b -1 x • • • b -1 x kx -1 if s = x ∈ V(L \ L 1 ) and h = x kx-1 . Then we extend the map B → F 0 , b h → b h * s, to a homomorphism F 0 → F 0 , f → f * s. Lemma 3.7
The above defined homomorphism * s : F 0 → F 0 is an automorphism.

Proof The result will follow if we show that the map has an inverse. Our candidate to inverse will be the map * s -1 :

F 0 → F 0 defined by b h * s -1 =    b u(hs -1 ) if hs -1 ∈ A L 1 T 0 , b -1 1 b -1 x • • • b -1 x kx-2 b x kx-1 b x kx-2 • • • b x b 1 if s = x ∈ V(L \ L 1 ) and h = 1 .
Assume first that either s ∈ V(L 1 ), or h ∈ {1, x kx -1 } with s = x ∈ V(L \ L 1 ). In this case we only have to check that u(u(hs)s -1 ) = h = u(u(hs -1 )s). Observe that the condition s ∈ V(L 1 ) and s ∈ S i for every x i ∈ supp(h) is equivalent to s ∈ V(L 1 ) and s ∈ S i for every x i ∈ supp(u(hs)), thus, if that condition holds, we have u(hs) = s -1 hs and u(u(hs)s -1 ) = u((s -1 hs)s -1 ) = ss -1 hss -1 = h. If the condition fails, then u(hs) = hs and u(u(hs)s -1 ) = u((hs)s -1 ) = hss -1 = h. Analogously, one checks that h = u(u(hs -1 )s). Now, we assume that s = x ∈ V(L \ L 1 ) and either h =

1 or h = x kx-1 . If h = x kx-1 , then (b h * x) * x -1 = (b x kx -1 • • • b x b 1 b -1 x • • • b -1 x kx-1 ) * x -1 = b x kx -2 • • • b 1 (b 1 * x -1 ) b -1 1 • • • b -1 x kx-2 = b x kx -2 • • • b 1 b -1 1 b -1 x • • • b -1 x kx -2 b x kx -1 b x kx -2 • • • b x b 1 b -1 1 • • • b -1 x kx -2 = b x kx -1 = b h . On the other hand, (b h * x -1 ) * x = b x kx-2 * x = b x kx -1 = b h .
The case when h = 1 is analogous.

To show that this action, defined just for the generators of A L , yields an action of A L on F 0 , we need to check that it preserves the Artin relations. We do it in the next two lemmas.

Lemma 3.8 Let s, t ∈ V(L) such that m s,t = 2. Then (g * s) * t = (g * t) * s for all g ∈ F 0 .
Proof Since s and t are linked in V(L), Lemma 3.1 implies that at least one of the vertices s, t lies in V(L 1 ). Without loss of generality we may assume s ∈ V(L 1 ), i.e.,

k s = 1. Let h ∈ T 0 . Then b h * s = b u(hs) .
Assume first that either ht ∈ T 0 , or t ∈ S i for every x i ∈ supp(h). In this last case, we have t ∈ S i for any x i ∈ supp(u(hs)). Therefore

(b h * s) * t = b u(hs) * t = b u(u(hs)t) and (b h * t) * s = b u(ht) * s = b u(u(ht)s) .
Depending on whether hs lies in T 0 or not we have u(hs) = hs or u(hs) = s -1 hs, and the same for t. So, we have four cases to consider. If hs, ht ∈ T 0 , then hst = hts ∈ T 0 and u(u(hs

)t) = u((hs)t) = hst = hts = u((ht)s) = u(u(ht)s) .
If hs ∈ T 0 and ht ∈ T 0 , then hst ∈ T 0 but t -1 hts ∈ T 0 , hence

u(u(hs)t) = u((hs)t) = t -1 hst = t -1 hts = u((t -1 ht)s) = u(u(ht)s) . Lemma 3.11 Let g ∈ A L 1 T 0 , and let n(g) = w 0 x ε 1 α 1 w 1 • • •
x εn αn w n be its normal form. Then any prefix of n(g) also represents an element in A L 1 T 0 .

Proof The only case where it is not obvious is when the prefix is of the form

w 0 x ε 1 α 1 w 1 • • • x ε j α j u j with u j a prefix of w j . Let h be the element of A L represented by w 0 x ε 1 α 1 w 1 • • • x ε j
α j u j , and let h ′ be the element represented by

w 0 x ε 1 α 1 w 1 • • • x ε j α j . It is clear that h ′ ∈ A L 1 T 0 . Moreover, h = h ′ ūj
, hence, as pointed out after the proof of Lemma 3.6, we have h ∈ A L 1 T 0 . Lemma 3.12 For every g

∈ A L 1 T 0 we have b 1 * g = b u(g) . Proof Set again n(g) = w 0 x ε 1 α 1 w 1 • • • x εn αn w n .
Let vs ε be a prefix of n(g) with s a vertex and ε ∈ {±1}. Note that Lemma 3.11 implies that vs ε and v represent elements in A L 1 T 0 . We are going to prove that, if b 1 * v = b u(v) , then also b 1 * vs ε = b u(vs ε ) . Note that this will imply the result. As vs ε lies in A L 1 T 0 , the element u(vs ε ) is well-defined. Observe that u(u(v)s ε ) is also well-defined. We have b

1 * (vs ε ) = b u(v) * s ε = b u(u(v)s ε ) . So, we only need to show that u(u(v)s ε ) = u(vs ε ). Set v = qh with h ∈ T 0 and q ∈ A L 1 . Then u(v) = h, thus, by Lemma 3.6, u(u(v)s ε ) = u(hs ε ) = u(qhs ε ) = u(vs ε ).
Our next objective is to extend the action of A L on F 0 to an action of A 1 on F . Recall that T = T 0 Ker(π L ) and that any h ∈ T can be written in a unique way as h = h 0 u with h 0 ∈ T 0 and u ∈ Ker(π L ). Taking this into account we set b h = b h 0 u = b h 0 • u. We extend this notation to any element ω = b εi hi ∈ F 0 by setting ω • u = b εi hiu . Now, let g ∈ A 1 and h ∈ T . We write h = h 0 u with h 0 ∈ T 0 and u ∈ Ker(π L ). So, with the previous notation, we have b h = b h 0 • u. Then we set

b h * g = (b h 0 * π L (g)) • (π L (g) -1 ug) .
We can also write this action as follows. Let ω = b εi hi ∈ F 0 and let u ∈ Ker(π L ). Then

(3-2) (ω • u) * g = (ω * π L (g)) • (π L (g) -1 ug) .
Lemma 3. [START_REF] Hermiller | Poly-free constructions for right-angled Artin groups[END_REF] The above defined map

F × A 1 → F , (ω, g) → ω * g, is a well-defined right-action of A 1 on F .
Proof The lemma is essentially a consequence of the fact that the action of A L on F 0 is well-defined. Let g 1 , g 2 ∈ A 1 and let b h = b h 0 • u ∈ B, where h 0 ∈ T 0 and u ∈ Ker(π L ). Then, using Equation 3-2,

(b h * g 1 ) * g 2 = ((b h 0 • u) * g 1 ) * g 2 = (b h 0 * π L (g 1 )) • (π L (g 1 ) -1 ug 1 ) * g 2 = ((b h 0 * π L (g 1 )) * π L (g 2 )) • (π L (g 2 ) -1 (π L (g 1 ) -1 ug 1 )g 2 ) = (b h 0 * π L (g 1 g 2 )) • (π L (g 1 g 2 ) -1 ug 1 g 2 ) = b h * g 1 g 2 .
Recall the homomorphism ϕ : F → Ker(π 1 ) that sends b h to z h for all h ∈ T . We consider the semi-direct product G = A 1 ⋉ F associated with the above action, and we turn to define an extension of ϕ to G.

Lemma 3. [START_REF] Howie | Bestvina-Brady groups and the plus construction[END_REF] The map G → A, (g, ω) → g ϕ(ω), is a well-defined homomorphism.

Proof We have to check that, for all g 1 , g 2 ∈ A 1 and all ω 1 , ω 2 ∈ F , we have

ϕ(g 1 , ω 1 )ϕ(g 2 , ω 2 ) = g 1 g 2 ϕ(ω 1 ) g 2 ϕ(ω 2 ) = ϕ(g 1 g 2 , (ω 1 * g 2 )ω 2 ) = g 1 g 2 ϕ((ω 1 * g 2 )ω 2 ) .
Since the restriction of ϕ to F is a group homomorphism, this is equivalent to show that ϕ(ω 1 ) g 2 = ϕ(ω 1 * g 2 ). It is enough to prove this for the group generators. So, we can assume that ω 1 = b h for some h ∈ T , and that g 2 = s for some vertex s ∈ V(Γ), s = z. We have ϕ(b h ) s = s -1 z h s = s -1 h -1 zhs, and we need to check that this is equal to ϕ(b h * s). To see it we set h = h 0 u, with h 0 ∈ T 0 and u ∈ Ker(π L ), so that b So, from now on, we will assume that s ∈ V(L). Then π L (s) = s, thus b s h = (b h 0 * s) • (s -1 us), and therefore

h = b h 0 • u. Observe first that, if s ∈ V(L), then π L (s) = 1,
ϕ(b h * s) = ϕ(b h 0 * s) s -1 us = s -1 ϕ(b h 0 * s) s -1 u s .
So, we only have to prove that ϕ(b h 0 * s) = z h 0 s . We distinguish three different cases.

If h 0 s ∈ T 0 , then b h 0 * s = b h 0 s , thus ϕ(b h 0 * s) = ϕ(b h 0 s ) = z h 0 s . If h 0 s ∈ T 0 , s ∈ V(L 1 ) and s ∈ S i for every x i ∈ supp(h 0 ), then b h 0 * s = b s -1 h 0 s , thus ϕ(b h 0 * s) = ϕ(b s -1 h 0 s ) = z s -1 h 0 s = z h 0 s . Finally, if s = x ∈ V(L \ L 1 ) and h 0 = x kx-1 , then b h 0 * x = b x kx -1 • • • b x b 1 b -1 x • • • b -1 x kx -1 , thus ϕ(b h 0 * x) = ϕ(b x kx -1 • • • b x b 1 b -1 x • • • b -1 x kx-1 ) = z x kx-1 • • • z x z (z x ) -1 • • • (z x kx -1 ) -1 = z x kx = z h 0 x .
Now, we want to define the inverse map of ϕ. We do it by giving the images of the Artin generators of A, that is, the vertices of Γ.

Lemma 3.15 There is a well-defined homomorphism ψ : A → G that sends s to s for all s ∈ V(Γ) \ {z}, and sends z to b 1 .

Proof We have to check that the Artin relations are preserved by ψ . Note that it suffices to check it for the Artin relations that involve z and some s ∈ V(Γ) \ {z}. If s ∈ V(L), then there is nothing to check because, in that case, s and z are not linked in Γ, hence there is no relation between them. If s ∈ V(L), then we can rewrite the Artin relation as

z s ks = z s ks-1 • • • z s z (z s ) -1 • • • (z s ks-1 ) -1 .
We include here the case s ∈ V(L 1 ), where we have k s = 1 and the above formula is z s = z. Applying ψ to the left hand side of this equation we get Proof We have already seen that both maps are group homomorphisms. We claim that they are inverses of each other. This will prove the result. Let s ∈ V(Γ), s = z. We have (ϕ • ψ)(s) = ϕ(s) = s and (ψ • ϕ)(s) = ψ(s) = s. Also (ϕ • ψ)(z) = ϕ(b 1 ) = z and (ψ • ϕ)(b 1 ) = ψ(z) = b 1 . Moreover, Lemma 3.12 implies that b 1 and the Artin generators of A 1 generate the whole group G, so ψ • ϕ is the identity of G. Similarly, ϕ • ψ is the identity of A. Now, we obtain immediately our main result. Theorem 3.17 Every even Artin group of FC type is poly-free.

ψ(z s ks ) = s -ks b 1 s ks = s -1 b s ks-1 s = b s ks-1 • • • b s b 1 b -1 s • • • b -1 s ks-1 ,
Proof By Proposition 3.16, A Γ ≃ G = F ⋊ A 1 . By induction we may assume that A 1 is poly-free, thus A is also poly-free.

Residually finiteness

In this section we will show that even Artin groups of FC type are residually finite. Recall that a group G is said to be residually finite if, for every g ∈ G \ {1}, there is a normal subgroup of finite index in G not containing g. It is well-known that being residually finite is not closed under short exact sequences, in the sense that, if N is a normal subgroup of G and both N and G/N are residually finite, then one cannot deduce the same for G itself. However, the situation changes if we work under some extra hypothesis. For example, a direct product of residually finite groups is residually finite. This can be generalized to the following result of Boler-Evans [START_REF] Boler | The free product of residually finite groups amalgamated along retracts is residually finite[END_REF] that will be crucial in our argument. Proof We argue by induction on the number of vertices of Γ. Assume first that Γ is complete. Then, by definition, A Γ is of spherical type, hence, by Cohen-Wales [START_REF] A M Cohen | Linearity of Artin groups of finite type[END_REF] and Digne [START_REF] Digne | On the linearity of Artin braid groups[END_REF], A Γ is linear, and therefore A Γ is residually finite. Assume that Γ is not complete. Then we can choose two distinct vertices s, t ∈ S = V(Γ) such that m s,t = ∞. Set X = S \ {s}, Y = S \ {t} and Z = S \ {s, t}. From the presentation of A follows that A = A X * A Z A Y . Moreover, since A is even, the inclusion map A Z ֒→ A X has a retraction π X,Z : A X → A Z which sends r to r for all r ∈ Z and sends t to 1, hence A X = Ker(π X,Z )⋊ A Z . Similarly, the inclusion map A Z ֒→ A Y has a retraction π Y,Z : A Y → A Z , hence A Y = Ker(π Y,Z ) ⋊ A Z . By the inductive hypothesis, A X and A Y are residually finite, hence, by Theorem 4.1, A is also residually finite.

1 2

 1 If a, b are two letters and m is an integer ≥ 2, we set Π(a, b : m) = (ab) m 2 if m is even, and Π(a, b : m) = (ab) ma if m is odd. In other words, Π(a, b : m) denotes the word aba • • • of length m. The Artin group of Γ, A = A Γ , is defined by the presentation A = S | Π(s, t : m s,t ) = Π(t, s : m s,t ) for all s, t ∈ S, s = t and m s,t = ∞ . The Coxeter group of Γ, W = W Γ , is the quotient of A by the relations s 2 = 1, s ∈ S.

  thus b h * s = b h 0 • (us) = b h 0 us = b hs , and therefore ϕ(b h * s) = ϕ(b hs ) = z hs = ϕ(b h ) s .

  which is exactly what we get applying ψ to the right hand side. Lemma 3.14 and Lemma 3.15 show part of the following result. Proposition 3.16 The maps ϕ : G → A and ψ : A → G are well-defined group isomorphisms.

Theorem 4 . 1 (

 41 Boler-Evans[START_REF] Boler | The free product of residually finite groups amalgamated along retracts is residually finite[END_REF]) Let G 1 , G 2 be residually finite groups, and letL ≤ G 1 , G 2 such that both split as G i = H i ⋊ L. Then the amalgamated free product G = G 1 * L G 2 is residually finite.Using this we get the following.

Theorem 4 . 2

 42 Every even Artin group of FC type is residually finite.

Acknowledgements

We thank Yago Antolín for suggesting us the main problem studied in this paper. The first two named authors were partially supported by Gobierno de Aragón, European Regional Development Funds and MTM2015-67781-P (MINECO/FEDER). The first named author was moreover supported by the Departamento de Industria e Innovación del Gobierno de Aragón and Fondo Social Europeo Phd grant

Similarly, if hs ∈ T 0 and ht ∈ T 0 , then u(u(hs)t) = u(u(ht)s). If hs, ht ∈ T 0 , then s -1 hst ∈ T 0 and t -1 hts ∈ T 0 , thus

We are left with the case where t = y ∈ V(L \ L 1 ) and h = y ky-1 . Then, since s and y are linked, for every α ∈ {0, 1, . . . , k y -1} we have u(y α s) = s -1 y α s = y α , thus

If w is a word over {s, t}, where s, t ∈ V(L), and if h ∈ T 0 we define b h * w by induction on the length of w by setting

Proof Note that, since s and t are linked and the edge between them is labelled with 2k > 2, Lemma 3.1 implies that s, t ∈ V(L 1 ). Take h ∈ T 0 . Then, in a similar way as in the proof of Lemma 3.8, we have hs ∈ T 0 if and only if u(hstst • • • t)s ∈ T 0 and this is also equivalent to u(htsts • • • t)s ∈ T 0 . The same thing happens for t. So, we may distinguish essentially the same cases as in the first part of the proof of Lemma 3.8 and get the following. If hs, ht ∈ T 0 , then

If hs ∈ T 0 and ht ∈ T 0 , then

Similarly, if hs ∈ T 0 and ht ∈ T 0 , then (b h ) * (st

All the previous discussion implies the following.

Lemma 3.10 The mappings * s, s ∈ V(L), yield a well-defined right-action F 0 ×A L → F 0 , (u, g) → u * g.

Moreover, this action behaves as one might expect. To show this, we will need the following technical lemma.


Roch Mollero

Jakob A Hauser

Xavier Pennec

Manasi Datar

Hervé Delingette

Alexander Jones

Nicholas Ayache

Tobias Heimann

Maxime Sermesant

Longitudinal Parameter Estimation in 3D Electromechanical Models: Application to Cardiovascular Changes in Digestion

Computer models of the heart are of increasing interest for clinical applications due to their discriminative and predictive abilities. However the number of simulation parameters in these models can be high and expert knowledge is required to properly design studies involving these models, and analyse the results. In particular it is important to know how the parameters vary in various clinical or physiological settings. In this paper we build a data-driven model of cardiovascular parameter evolution during digestion, from a clinical study involving more than 80 patients. We first present a method for longitudinal parameter estimation in 3D cardiac models, which we apply to 21 patient-specific hearts geometries at two instants of the study, for 6 parameters (two fixed and four time-varying parameters). From these personalised hearts, we then extract and validate a law which links the changes of cardiac output and heart rate under constant arterial pressure to the evolution of these parameters, thus enabling the fast simulation of hearts during digestion for future patients.

Introduction

The main function of the heart is to create the necessary blood flow through the cardiovascular system, so that the oxygen supply of all the organs meets their needs. When an organ or a part of the body needs more energy (such as the muscles during exercise, or the digestive system during digestion), the heart rate and the blood flow increase because the overall demand in oxygen is higher.

The main changes in the cardiac function leading to an increase of the cardiac output are an increased heart rate, a decreased action potential duration and an increased contractility (positive inotropy). When the cardiac output increase is small (such as digestion or a mild exercice), the systolic pressure usually rochmollero@hotmail.com increases but the diastolic pressure is constant, the latter being a consequence of the dilation of the arteries which lowers the arterial resistance [START_REF] Laughlin | Cardiovascular response to exercise[END_REF]. Those qualitative changes are well-known, but are rarely quantified in the context of 3D cardiac electromechanical models, in part because most studies only involve personalisations on a single beat only (see [START_REF] Chabiniok | Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics[END_REF] for a complete review).

A clinical study was performed in [START_REF] Hauser | Comprehensive assessment of the global and regional vascular responses to food ingestion in humans using novel rapid mri[END_REF] to assess the cardiovascular response to a food stress protocol, involving the ingestion of a high-energy meal after fasting for 12h. From the data of this study, we propose a consistent estimation of patient-specific 3D cardiac electromechanical models at two different instants of the protocol (pre-ingestion and t+1h). We first calibrate both the biomechanical parameters which are constant in time (such as the myocardial fibre directions) and time-varying (such as the arterial resistance) from the pre-ingestion measurements and heart motion extracted from the MRI. Then, we re-estimate values of the time-varying parameters (contractility and haemodynamics parameters) to reproduce changes in cardiac output and blood pressure at the second instant.

From these personalised simulations, we analyse the trends of the estimated parameters in relation to known physiological changes during mild exercise [START_REF] Otsuki | Contribution of systemic arterial compliance and systemic vascular resistance to effective arterial elastance changes during exercise in humans[END_REF][START_REF] Albert | The response of the peripheral venous pressure to exercise in congestive heart failure[END_REF]. Finally, we build a law of evolution of the biomechanical parameters which leads to arbitrary changes of both the simulated cardiac output and stroke volume, while maintaining the same mean and diastolic pressure. The good accuracy of this law, which we validate with cross-validation over the 21 patients, then opens the door to the fast simulation of hearts during digestion in future patients.

Clinical Study and Data

More than 80 patients participated to a clinical study to assess the cardiovascular response after the ingestion of a high-energy (1635 kcal), high-fat (142g) meal after fasting for 12h, following the stress protocol in [START_REF] Hauser | Comprehensive assessment of the global and regional vascular responses to food ingestion in humans using novel rapid mri[END_REF]. Informed consent was obtained from the subjects and the protocol was approved by the local Research Ethics Committee. An objective of the study was to analyze the evolution of blood flow toward the various organs of the body. In particular, a short axis cardiac cine MRI sequence was acquired before the ingestion, as well as measurements of the stroke volume, systolic, diastolic and mean cuff pressures at several time points within 1h of the ingestion of the meal. Two instants are considered in particular: T 1 which is before the meal ingestion, and the latest measurement time T 2 around 1h after ingestion, which also corresponds to the peak of the increased cardiac activity.

Overall (see Table 1), an increase of both the Heart Rate (HR) and the Cardiac Output (CO) of around 17% was observed. There were no significative changes in the values of the Systolic, Diastolic and Mean cuff pressure (SP, DP, MP) during the 1h process of digestion (beyond the intra-patient variability of the measurements). Finally the Stroke Volume (SV) was constant on average but the measurement showed a high inter-patient variability of the evolution (11%).

Additionally, we tracked the boundaries of the endocardium over the entire cine MRI sequence acquired at T 1 , then extracted from this sequence a point at the apex of the left ventricle and one at the top of the left ventricle septum. This was used to calculate the Septal Shortening (SS) as the maximal shortening of the distance between these two points during the cycle. It has an average value of -17% and a standard deviation of 3.7% across the population.

3 Patient-Specific Cardiac Modelling

3D Electromechanical Cardiac Model

We performed 3D cardiac modelling for 21 of these patients. A high-resolution biventricular tetrahedral mesh of the patient's heart morphology was extracted as in [START_REF] Mollero | Propagation of myocardial fibre architecture uncertainty on electromechanical model parameter estimation: A case study[END_REF] from the pre-ingestion MRI at T 1 , made of around 15 000 nodes. On this mesh, a myocardial fibre direction can be defined at each node of the mesh (see Fig 1a), by varying the elevation angles of the fibre across the myocardial wall from α 1 on the epicardium to α 2 degrees on the endocardium. In this paper, α 2 is set at the default value of 90 o and α 1 is a variable parameters in our experiments. The depolarization times across the myocardium were computed with the Multi-front Eikonal method [START_REF] Sermesant | An anisotropic multi-front fast marching method for realtime simulation of cardiac electrophysiology[END_REF]. The APD is set from the Heart Rate with classical values of the restitution curve and default values of conductivities are used as in [START_REF] Pernod | A multi-front eikonal model of cardiac electrophysiology for interactive simulation of radio-frequency ablation[END_REF]. Myocardial forces are computed based on the Bestel-Clement-Sorine model as detailed in [START_REF] Chapelle | Energy-preserving muscle tissue model: formulation and compatible discretizations[END_REF]. It models the forces as the combination of an active contraction force in the direction of the fibre, in parallel with a passive anisotropic hyperelasticity driven by the Mooney-Rivlin strain energy. In this paper, we only consider two main parameters of the model: the Maximal Contractility σ and the Passive Stiffness c 1 . Finally for the haemodynamics, the pressure in the cardiac chambers are described by global values, and the mechanical equations are coupled with a circulation model implementing the 4 phases of the cardiac cycle [START_REF] Marchesseau | Simulation of patient-specific cardiac models for therapy planning[END_REF].

In particular the pressure of the aortic artery P ar (cardiac after-load) is modeled with a 4-parameter Windkessel model [START_REF] Westerhof | The arterial windkessel[END_REF], which describes the evolution of arterial blood pressure with the second-order equation of an electric circuit (see Fig 1b). The blood inertia is modeled by the inductance L, the arterial compliance by a capacity C and the proximal and distal (peripheral) resistances respectively by a resistance Z C and R (see Fig 1b). Finally, the venous pressure P ve models the mean pressure in the venous system. In the following, Z C and L are fixed at a default value (see [START_REF] Marchesseau | Simulation of patient-specific cardiac models for therapy planning[END_REF]) while C, R and P ve are variable parameters.

Longitudinal Parameter Estimation

After building the heart mesh geometry, parameter estimation is the next step in order to have model simulations which reproduce the available data. Considering a set of simulated quantities called the "outputs" O (such as the Stroke Volume or the Mean Pressure for example), and a set of model parameters P , it consists in finding adequate values x of the parameters such that the output values O(x) in the 3D model simulation fit the "target values" O available in the data. This is done by performing an optimization of the parameter values x in order to minimize a distance S(x, O) = ||O(x) -O|| S between the simulated values O(x) and the target values O (normalised to compare quantities with different units).

For each patient, we have here measurements of different varying quantities at the two instants T 1 and T 2 (such as the stroke volume and the heart rate), so we need to estimate different values for some cardiac model parameters (in particular the haemodynamic parameters) at these two instants. On the other hand, during the time-scale of the study (1h on average), some parameters of the cardiac model can be considered constant. This is the case of the Epicardial Fibre Elevation Angle α 1 for example, or the cardiac stiffness c 1 . In order to have consistent sets of estimated parameters at these two different instants, we need to use the same values for these parameters at these two instants.

To that end, we perform a two-step parameter estimation. First, we estimate values of both the fixed and varying parameters from the data at T 1 . Then we reuse the estimated values of the fixed parameters for T 2 and estimate new values for the varying parameters only, from the data at T 2 . As summarized in Table ,   we then have two distinct Parameter Estimation problems: the estimation of 6 parameters values in order to fit 4 target output values at T 1 (with the heart rate of the simulations set to its value at T 1 ). Then the estimation of 4 parameters values in order to fit 3 target output values at T 2 (with the heart rate at T 2 ). 2: Estimated Parameters and Target Outputs in the parameter estimations at T1 and T2. Constant parameters whose values are reused for the estimation at T2 are outlined in bold. The heart rate in the simulations for the estimation at T1 (resp T2) correspond to the measured value at T1 (resp T2).

The optimisation was performed with an extended version of the framework described in [START_REF] Mollero | A multiscale cardiac model for fast personalisation and exploitation[END_REF]: the main algorithm is the CMA-ES genetic algorithm, which asks at each iteration for the score of a high number of 3D simulations. Instead of actually computing all these 3D simulations, we only compute a few within the parameter space (2N + 1 where N is the number of estimated parameters). Then we build a "low-fidelity" surrogate model [START_REF] Peherstorfer | Survey of multifidelity methods in uncertainty propagation, inference, and optimization[END_REF] from these simulations which allows to approximate the outputs of the 3D simulations for many successive iterations of the algorithm, without performing all the 3D simulations. This robust and efficient "multifidelity optimization" allows a very fast exploration of large parameter sets with a low computational cost. In particular for the two problems at T 1 and T 2 , we performed the optimization for the 21 patients simultaneously and the convergence was reached in around two days.

Exploitation of Estimated Parameters

Analysis of Parameter Trends in the Population

Across the 21 patients and the two estimations, the average fit error on the target ouput values are 1.9 mL for the Stroke Volume, 1% for the Septal Shortening, and 0.1 mmHg for both the mean and diastolic pressures, with few outliers. As a consequence of this step, we now have a population of 21 personalised patient hearts at two instants. For each parameter, we report in Table 3 the mean and standard deviation of its estimated values at T 1 across the 21 patients, as well as the mean of its evolution ∆ between the instants T 1 and T 2 (difference between the values estimated at T 2 and T 1 ). The first remark is that on average, the parameters R which models the arterial peripheral resistance decreases by 14%. This was expected and corresponds to findings in [START_REF] Hauser | Comprehensive assessment of the global and regional vascular responses to food ingestion in humans using novel rapid mri[END_REF]. In a clinical setting the peripheral resistance is indeed computed as the ratio between the blood flow and the blood pressure, and a similar relationship can be derived in the model: as shown in Fig 2a the ratio (MP-P ve )/CO is almost exactly equal to the peripheral resistance R in our simulations. Across the population, since the cardiac output CO increases by around 17% but the pressures are constant, the peripheral resistance has to decrease by a close number (14.2% here) on average.

c1 (kPa) α1 ( • ) σ (MPa) Pve (mmHg) R (MPa.m 3 .s) C (MPa -1 .m -3 ) Mean 54.2e
We then notice both an average increase of the venous pressure P ve and decrease of the arterial compliance C. These two trends can be explained as to compensate the decrease of the resistance and avoid a drop in the mean blood pressure. Indeed, in the model, a decrease of R leads to a decrease of the "characteristic time" τ = RC at which the blood pressure decreases from the systolic pressure to the "asymptotic pressure" P ve . A decrease of R only leads then to a decrease of the mean pressure. On the other hand, a decrease of C leads to an increase of the "pulse pressure" (difference between systolic and diastolic pressure) since C links an increase of arterial volume to an increase of arterial pressure with the formula C∆P = ∆V (a less compliant aorta has a higher pulse pressure for the same stroke volume). This contributes to the increase of the mean pressure (see Fig 2b), and it is also the case of the increase of P ve . Interestingly, we can note that these two trends (decrease of the arterial compliance and increase of venous pressure) in parameters correspond to actual cardiovascular phenomena which are commonly observed during exercise [START_REF] Otsuki | Contribution of systemic arterial compliance and systemic vascular resistance to effective arterial elastance changes during exercise in humans[END_REF][START_REF] Albert | The response of the peripheral venous pressure to exercise in congestive heart failure[END_REF].

Finally, we can also observe a high correlation between changes in the Maximal Contractility σ and changes in the ejected volume, as shown in Fig 2c. This is also a known phenomenon in cardiac dynamics, in particular at the core of the Starling Effect.

Parameter Evolution Law

From this data and the estimated parameters, we then build a law f which, from a given simulation, gives variations of the electromechanical parameters σ, P ve , R and C which leads to a new simulation with prescribed changes in heart period (HP) and stroke volume (SV) while having same mean and diastolic pressures: f (∆HP, ∆SV ) = (∆σ, ∆P ve , ∆R, ∆C) This is done by computing a multivariate regression between the changes (in %) in Heart Rate and Stroke Volume and the changes in the estimated parameters values at the two instants T 1 and T 2 , for the 21 patients. We report in Table 4 the coefficients of this multivariate regression: The predicted variations of parameters with the variations of the heart period (∆HP) are consistent with the mean variations across the population described earlier. Interestingly with the coefficients of the second row (∆SV), we can also note how the parameters have to change for an increase in Stroke Volume only with constant pressures.

We finally tested the accuracy of this law with a leave-one-out approach: for each patient, we computed the regression f from the data and estimated parameters of all the others patients. Then we changed the baseline parameters (at T 1 ) of this patient with the parameters predicted from f , and simulated the Pressure and Stroke Volume values at T 2 . The obtained results were accurate: on average, the target Stroke Volume at T 2 was predicted within 1.9 mL and the mean absolute variations in Diastolic and Mean Pressure were within 2.1mmHG, which is beyond the variability of both the intra-patient and population variabilities.

In this manuscript we performed a consistent longitudinal estimation of cardiac model parameters for 21 patient-specific hearts at two different instants within a 1h time span, from clinical data. This was done through two successive parameter estimation problems: we first estimated 6 parameters to fit the simulated Stroke Volume, the Septal Shortening and the Mean and Diastolic Pressures to their values at the first instant. Then we reused the estimated values of the fixed parameters at this step and performed a second estimation of 4 parameters to fit values of Stroke Volume and Pressures at the second instant. This was done in parallel for the 21 hearts in around two days and a maximum of 150 simulations of the 3D model per patient.

From those personalised hearts, we identified relationships between the estimated parameters and the simulated pressure and volume outputs, and linked their evolution between these two instants to classical physiological phenomena. Then we extracted a law which computes changes of electromechanical parameters from changes of stroke volume and heart rate with constant pressure. This law allows in particular to easily simulate the changes observed between the two instants without having to perform the parameter estimation step at the second instant. This was evaluated in a leave-one-out test and showed that it can predict accurately changes in the model parameters.

A first direct continuation of this work would be to quantify (from further data) to what extent this law holds for changes of cardiac outputs which are more important (digestion can be seen as a 'mild' exercise and it is known for example that blood pressure rises during more intense exercises). Finally, for future patients, it could also be interesting to evaluate to what extent the changes in both the Stroke Volume and the Heart Rate can be predicted, and use our law to simulate the predicted heartbeats.

Fig 1a :

 1a Fig 1a: 3D heart geometry with myocardial fiber direction Fig 1b: Schema and rheological model and of the windkessel model (figure from [7])

Fig

  Fig 2a: (MP-Pve)/CO as a function of R

Table 1 :

 1 Statistics of the measurements and their evolution ∆ between T1 and T2 (in percentage of the value at T1). Systolic, Diastolic and Mean cuff Pressure (SP, DP, MP), Stroke Volume (SV), Cardiac Output (CO) and Heart Rate (HR).

		SP (mmHg) DP (mmHg) MP (mmHg) SV (mL) CO (L/min) HR (bpm)
	Mean	117.13	60.95	84.16	92.11	6.17	67.65
	Std.	9.99	6.45	5.98	19.69	1.34	10.25
	Mean ∆ (%)	-	-	-	-0.10	17.58	17.76
	Std. ∆ (%)	-	-	-	11.43	17.74	13.19

Table 3 :

 3 Statistics of the estimated parameters and of the difference ∆ between estimated parameters at T1 and T2

			1 -58.7	82.6	48.3	47.4	6.23e -3
	Std.	27.7e 1 2.94	34.0	12.9	17.2	1.98e -3
	Mean ∆	-	-	-1.52%	6.93%	-14.2%	-7.30%

Table 4 :

 4 Coefficient of the multivariate regression f

	∆ σ ∆ Pve ∆ R ∆ C
	∆ HP -0.02 -0.15 1.20 0.51
	∆ SV 3.05 0.52 -1.04 1.19

Ackowledgements This work has been partially funded by the EU FP7-funded project MD-Paedigree (Grant Agreement 600932) and contributes to the objectives of the ERC advanced grant MedYMA (2011-291080).