G Merle

J.-M Roussel
email: jean-marc.roussel@lurpa.ens-cachan.fr

J.-J Lesage

Algebraic determination of the structure function of Dynamic Fault Trees

Keywords: Dynamic fault tree, algebraic modeling, structure function

This paper presents an algebraic framework allowing to algebraically model dynamic gates and determine the structure function of any Dynamic Fault Tree (DFT). This structure function can then be exploited to perform both the qualitative and quantitative analysis of DFTs directly, even though this latter aspect is not detailed in this paper. We illustrate our approach on a DFT example from the literature.

Introduction

The structure function of a Static Fault Tree (SFT) -a fault tree (FT) which only contains gates OR, AND, and K-out-of-N -is a Boolean function which represents the failure of the top event (TE) according to the failure of the basic events (BEs) of the FT. This algebraic model is classically used to perform both the qualitative and quantitative analysis of SFTs directly. For complex systems, these analyses are most often performed thanks to BDDbased methods [START_REF] Dugan | Developing a low-cost highquality software tool for dynamic fault-tree analysis[END_REF][START_REF] Reay | A fault tree analysis strategy using binary decision diagrams[END_REF] or other combinatorial techniques [START_REF] Bobbio | Parametric fault trees with dynamic gates and repair boxes[END_REF][START_REF] Ortmeier | Safety analysis of the height control system for the Elbtunnel[END_REF].

The introduction of dynamic gates -gates PAND, FDEP, and Spare -in FTs has changed the nature of the relation between the TE and the BEs. In a Dynamic Fault Tree (DFT), the failure of the TE depends not only on the failure of the BEs but also on the order of occurrence of these failures. As this last aspect is not taken into account in the Boolean model of failures (which only expresses whether a BE has occurred or not), a classical Boolean function cannot represent the dynamic relations between the TE and the BEs that exist in a DFT.

In a previous article, we presented the basics of an algebraic framework allowing to algebraically model dynamic gates PAND and FDEP, and determine the structure function of any Dynamic Fault Tree (DFT) containing these gates [START_REF] Merle | Probabilistic algebraic analysis of fault trees with priority dynamic gates and repeated events[END_REF]. In this paper, we extend our previous work to Spare gates in order to be able to determine the structure function of any DFT. This structure function is based on a specific algebraic model of failures which allows to take into account the order of occurrence of failures. As this algebraic model is an extension of the Boolean model used for SFTs, all the results previously obtained for SFTs are preserved.

This paper is organised as follows. The most common approaches used to perform the analysis of DFTs are presented in Section 2. The algebraic framework that we introduce to model DFTs is detailed in Section 3, and the algebraic model of dynamic gates which can be determined from it is presented in Section 4. This algebraic model allows to determine the canonical form of the structure function of any DFT, as shown in Section 5, and our approach is illustrated on a DFT example in Section 6. Finally, we show how the qualitative analysis of DFTs can be performed directly from the canonical form of the structure function in Section 7.

State of the art

Several approaches have been used to avoid the problem of the determination of the structure function of DFTs. These approaches can be either modular or global.

Global approaches consist in solving the whole DFT directly, whereas modular approaches consist in:

• dividing the DFT into independent static and dynamic subtrees (or modules) prior to analysis: if a substree contains static gates only, it is considered as static; if a subtree contains at least one dynamic gate, it is considered as dynamic;

• solving the modules separately; and

• combining the results of the various modules to get the overall result for the entire tree.

Various methods exist to analyze the static and dynamic modules of DFTs. On the one hand, solving static modules can be done by using Binary Decision Diagrams (BDDs), other combinatorial techniques, or even some DFT Analysis models such as Markov Chains. On the other hand, solving dynamic modules is generally done using Input/Output Interactive Markov Chains [START_REF] Boudali | Dynamic fault tree analysis through input/output interactive Markov chains[END_REF][START_REF] Boudali | A compositional semantics for dynamic fault tree in terms of interactive Markov chains[END_REF][START_REF] Boudali | A rigorous, compositional, and extensible framework for dynamic fault tree analysis[END_REF], Stochastic Petri Nets (SPN) [START_REF] Bobbio | Parametric fault trees with dynamic gates and repair boxes[END_REF][START_REF] Raiteri | The conversion of dynamic fault trees to stochastic petri nets, as a case of graph transformation[END_REF], or Temporal Bayesian Networks [START_REF] Boudali | A discrete-time Bayesian network reliability modeling and analysis framework[END_REF][START_REF] Boudali | A continuous-time Bayesian network reliability modeling, and analysis framework[END_REF][START_REF] Montani | DBNet, a tool to convert dynamic fault trees into dynamic Bayesian networks[END_REF]. On the one hand, Markov Chains provide the cut sequences of the (sub)tree, which are the failure sequences that lead to the states of the Markov Chain in which the TE of the (sub)tree fails. They also provide the failure probability of the TE of the (sub)tree by solving the set of differential equations which is equivalent to the Markov Chain. On the other hand, the reachability graph of SPNs provides the cut sequences of the (sub)tree, and the failure probability of the TE can be computed after converting the SPN into its corresponding Markov Chain. However, in both cases, the failure of the components of the system is most often modeled by exponential time-tofailure distributions. Temporal Bayesian Networks allow to address this limit by allowing to consider other distributions. However, Bayesian Networks only allow to perform the quantitative analysis of the dynamic (sub)tree, and the inference algorithms used limit the distributions considered to Gaussian distributions [START_REF] Lauritzen | Stable local computation with conditional Gaussian distributions[END_REF] and mixtures of truncated exponentials [START_REF] Moral | Mixtures of truncated exponentials in hybrid Bayesian networks[END_REF].

An analytic approach was introduced in [START_REF] Walker | Qualitative temporal analysis: towards a full implementation of the fault tree handbook[END_REF] to analyze DFTs by modelling the dynamic gate PAND and by determining simplification theorems. The authors focus on three temporal gates: gates PAND, Simultaneous-AND (SAND), and Priority-OR (POR). Gate SAND was created to address the ambiguity encountered in the definition of gate PAND regarding the simultaneity of input events, whereas gate POR was created from the definition of the Exclusive-OR gate found in [START_REF] Vesely | Fault tree handbook[END_REF]. Each event of the FT is assigned a sequence value which allows to know the order in which events occur. The authors propose an extension of truth tables, denoted as Temporal Truth Tables and based on these sequence values, to prove theorems allowing to simplify the FTs which contain the three temporal gates considered. Nevertheless, this approach allows to perform the qualitative analysis of FTs, only, and the only dynamic gate considered is gate PAND.

We have not found in the literature any attempt to provide an algebraic model for all dynamic gates allowing to determine the structure function of a DFT explicitly as it is currently the case for SFTs. The goal of the algebraic determination of the structure function of DFTs is to be able to perform their analysis directly whatever the distribution considered for basic events. We present such an algebraic framework in Section 3.

Algebraic framework for the modeling of Dynamic Fault Trees

Temporal model of non-repairable events

The structure function of SFTs is based on a Boolean model of events, and of basic events in particular. With this simple model, the only aspect which is taken into account is the presence or absence of failure. However, this Boolean model cannot render the order of occurrence of events which is necessary for the modeling of dynamic gates. To count on the temporal aspect of events, we consider the top event, the intermediate events, and the basic events as temporal functions, which are piecewise rightcontinuous on R + ∪ {+∞}, whose range is B = {0,1}. In accordance with [START_REF] Vesely | Fault tree handbook[END_REF], we consider all events as nonrepairable, each of them being perfectly defined by its unique date of occurrence-noted d(a) for an event a. A generic timing diagram of such an event a is given in Fig. 1. In this paper, we denote by E nr the set of these temporal functions, which corresponds to the set of non-repairable events.

With this temporal model, non-repairable events can be ordered according to their date of occurrence. This characteristics is the cornerstone of our approach, and it is used to model each of the operators which are needed to model both the static and dynamic gates of FTs.

However, specific attention must be paid to the simultaneity of events when modeling the order of occurrence of events, as it is explained below.

Simultaneity in DFTs

In a FT, simultaneity among events may arise in two ways. Independent basic events can occur simultaneously if they have a discrete probability distribution with a nonnull probability mass exactly at the same time. Because the failure probability distributions are usually considered as continuous functions with infinite support, the simultaneous occurrence has null probability, and can be neglected. A second case of simultaneity may arise at any level of a FT when there are repeated basic events. FTs with repeated events represent the most powerful combinatorial model in dependability [START_REF] Malhotra | Power-hierarchy among dependability model types[END_REF], and require ad hoc analysis techniques. Nevertheless, the presence of repeated events across modules of dynamic gates has not yet been explored in its full generality. In [START_REF] Yuge | Quantitative analysis of a fault tree with priority AND gates[END_REF], repeated events are allowed, but the paper does not provide any algorithm to derive the list of the cut sequences.

Let us consider the DFT in Fig. 2, in which event A is a repeated basic event.

If basic events A, B, C, and D occur according to se- quences [B,C,A],[C,B,A],or [D,A], intermediate events G and H occur simultaneously at the same time as A occurs. This example shows that intermediate nodes of a FT can occur simultaneously because of the presence of repeated basic events. The simultaneity problem has been briefly addressed in [START_REF] Boudali | Dynamic fault tree analysis through input/output interactive Markov chains[END_REF], and has been solved by resorting to the concept of "non-determinism", a concept that is not easy to accept in engineering practice because many engineers believe that the behavior of technical systems, and in particular control systems, must necessarily be deterministic. We assert that a choice must be made regarding the semantics of simultaneous events, and dynamic gates. For instance, in the case of simultaneous events in input to a PAND gate, two choices are possible (Fig. 2):

• if the order relation is considered strictly, when intermediate events G and H occur simultaneously, TE1 does not occur, and gate PAND would then be considered as being "non-inclusive"; and

• if the order relation is not considered strictly, when intermediate events G and H occur simultaneously, TE1 occurs at the same time as G or H, and gate PAND would then be considered as being "inclusive".

Both interpretations of the order relation can be taken into account, and algebraically modeled.

Boolean operators

Any elements of E nr can be composed thanks to a rewriting of classical Boolean operators. The temporal definition of Boolean operators OR and AND, based on the dates of occurrence of a and b (which are denoted by d(a) and d(b), respectively), is

d(a + b) = ⎧ ⎪ ⎨ ⎪ ⎩ d(a) if d (a) < d (b) d(a) if d (a) = d (b) d(b) if d (a) > d (b) d(a • b) = ⎧ ⎪ ⎨ ⎪ ⎩ d(b) if d (a) < d (b) d(a) if d (a) = d (b) d(a) if d (a) > d (b)
Indeed, a + b occurs as soon as a or b occurs, and a • b occurs as soon as a and b have occurred. It can be noted that operators OR and AND are commutative. The identity elements of operators OR and AND in E nr are denoted by ⊥, and , respectively, to which these dates can be assigned:

d(⊥) = +∞, d( ) = 0
⊥ is the never-occurring event whereas is the alwaysoccurring event.

Temporal operators

To model the order of occurrence of events, we introduce an operator non-inclusive BEFORE (BF, with symbol ¡), and an operator SIMULTANEOUS (SM, with symbol ), whose formal definitions, based on the dates of occurrence of a and b, are

d(a ¡ b) = ⎧ ⎪ ⎨ ⎪ ⎩ d(a) if d (a) < d (b) +∞ if d (a) = d (b) +∞ if d (a) > d (b) d(a b) = ⎧ ⎪ ⎨ ⎪ ⎩ +∞ if d (a) < d (b) d(a) if d (a) = d (b) +∞ if d (a) > d (b)
The result of the composition of two events a and b by operators BF and SM is illustrated by the timing diagrams in Figs. 3 and4, respectively, in three cases: Case 1:

d (a) < d (b), Case 2: d (a) = d (b), Case 3: d (a) > d (b).
Based on the previous two operators, we can introduce a non-strict or INCLUSIVE BEFORE (IBF, with symbol whose definition, based on the dates of occurrence of a and b, is

¢) operator a ¢ b = a ¡ b + a b (1)
d(a ¢ b) = ⎧ ⎪ ⎨ ⎪ ⎩ d(a) if d (a) < d (b) d(a) if d (a) = d (b) +∞ if d (a) > d (b)
The result of the composition of two events a and b by operator IBF is illustrated by the timing diagrams in Fig. 5 in three cases: Case 1:

d (a) < d (b), Case 2: d (a) = d (b), Case 3: d (a) > d (b).
According to these timing diagrams, and to (1), a ¢ b occurs in two cases: when a occurs strictly before b, Case 1 (which corresponds to a ¡ b); and when a occurs at the same time as b, Case 2 (which corresponds to a b).

Algebraic structure of E nr

Static Fault Tree Analysis is mainly based on the structure function of SFTs, which is determined and simplified thanks to a substructure of the Boolean algebra of Boolean values: the Abelian dioid {{0, 1}, +, •, 0, 1} 1 .

We have demonstrated in [START_REF]Algebraic modelling of dynamic fault trees, contribution to qualitative and quantitative analysis[END_REF] that by providing a new temporal definition of events (Section 3.1) and by rewriting the Boolean operators + and • (Section 3.3), the framework obtained still has an Abelian dioid structure. The properties that are commonly used for the simplification of SFTs can hence still be applied with our model, and their structure functions can be determined as usual. In particular, operators OR and AND satisfy the four additional following theorems, which are theorems which hold on Boolean algebras:

a + (b • c) = (a + b) • (a + c) ( 2 ) a + = (3) a + (a • b) = a (4) a • (a + b) = a ( 5
)
(E nr , +, •) thus has an algebraic structure which allows to express gates OR, AND, and K-out-of-N, and to determine the structure function of SFTs as it is commonly done by using the classical Boolean algebra of Boolean variables.

Temporal operators non-inclusive BEFORE and Inclusive BEFORE satisfy several theorems which are useful for calculation, some of which are presented in Appendix A. The proofs of these theorems can be found in [START_REF] Merle | Algebraic framework for the modelling of priority dynamic fault trees[END_REF]. Furthermore, the temporal operators introduced in this section allow to determine an algebraic model of dynamic gates, as shown in Section 4.

Algebraic model of dynamic gates

Based on the algebraic framework introduced in Section 3, the algebraic model of dynamic gates is now going to be developed. The temporal operators introduced in Section 3.4 allow to take into account and algebraically model both a strict and a non-strict order relation. However, a non-strict inclusive interpretation of dynamic gates seems more coherent with the designers' expectations. For this reason, in the remainder of this paper, we define an algebraic model of dynamic gates by means of operator IBF (¢), only, even though it is easy to define an algebraic model of dynamic gates by means of operator BF (¡) as well.

Furthermore, in accordance with [START_REF] Merle | Probabilistic algebraic analysis of fault trees with priority dynamic gates and repeated events[END_REF], we assume that basic events are statistically independent, and have a continuous failure time distribution, so that they cannot occur simultaneously. Hence, for any two basic events a and b with the above characteristics, the following relation holds:

a b = ⊥ (6)
The algebraic models of gates PAND and FDEP are presented in Sections 4.1 and 4.2, respectively. The algebraic model of Spare gates is presented in an increasing order of complexity: Spare gates with 2 and 3 input events are studied in Sections 4.3 and 4.4, respectively. Besides, we consider that there is only one type of Spare gate which is the Warm Spare gate and that Cold and Hot Spare gates [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF] are particular cases of Warm Spare gates. Both of them are studied in Section 4.6.

Algebraic model of gate PAND

According to [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF], gate PAND is defined in Fig. 6. An algebraic model of gate PAND can hence be determined as

Q = (A • B) • (A ¢ B)
This model can be simplified, thanks to the theorems of Appendix A, as follows 2 :

Q = (A • B) • (A ¢ B) Q (A.30) = B • (A ¢ B)

Algebraic model of gate FDEP

According to [START_REF] Boudali | Dynamic fault tree analysis through input/output interactive Markov chains[END_REF][START_REF] Dugan | Dynamic fault-tree models for fault-tolerant computer systems[END_REF][START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF], the FDEP gate -Functional Dependency gate -is a dynamic gate comprised of a trigger input event -either a basic event or the output of another gate of the tree -and a set of dependent basic events. Fig. 7 provides a pictorial depiction of an FDEP gate with two dependent basic events A and B, T representing the trigger event. When the trigger event occurs, the dependent basic events are forced to occur.

In Fig. 7, basic events A and B can fail by themselves, or can be forced to fail by the trigger event T . In accordance with [START_REF] Boudali | A continuous-time Bayesian network reliability modeling, and analysis framework[END_REF], we choose to denote the global failure of basic events A, and B by the substituted variables A T , and B T to explicitly indicate the effect of trigger T : basic event A fails (A T ) if it is forced to fail by the trigger event (T ) or if it fails by itself before the trigger event fails (A ¢ T ). The algebraic model of gate FDEP thus is

A T = T + (A ¢ T) B T = T + (B ¢ T)
This model can be simplified, thanks to the theorems of Appendix A, as follows: Thanks to this simplification, it appears that the behavior of dynamic gate FDEP is equivalent to the behavior of static gate OR, as it has been suggested by some authors [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF].

A T = T + (A ¢ T) (A.29) = A + T B T = T + (B ¢ T) (A.

Algebraic model of Spare gates with two input events

In this section, we completely detail the algebraic model of Spare gates with two input events in the main configurations which may be encountered in DFTs. The different cases are treated in an increasing order of complexity, from a single Spare gate in Section 4.3.1 to two Spare gates sharing a spare event in Section 4.3.2, and even to the generalization to n Spare gates sharing a spare event in Section 4.3.3.

Algebraic model of a single Spare gate

Let us consider a Spare gate with two input eventsthe primary event A and one spare event B -as shown in Fig. 8.

As stated in [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF], the output Q of the gate occurs when the primary and all spares have failed, so when A and B have failed, in this case. A and B are basic events and cannot fail simultaneously -A B = ⊥ -so Q will occur if A and B fail according to sequences [A, B] or [B, A]. It is important to note that in sequence [A, B], B fails while in its active mode (denoted as B a ), whereas in sequence [B, A], B fails while in its dormant mode (denoted as B d ). It is essential to distinguish both failure modes by using two different variables, for quantitative analysis purposes. Indeed, B does not have the same failure distribution when it fails during its dormant mode (B ≡ B d ) or during its active mode (B ≡ B a ). As we aim at making possible the quantitative analysis of DFTs from their structure function, this structure function must hence provide sufficient information to know whether spare events are in their dormant or active mode.

The algebraic model of gate Spare can hence be expressed as Furthermore, as B cannot be both in an active state and in a dormant state, we have

Q = B a • (A ¡ B a ) + A • (B d ¡ A)
B d • B a = ⊥

Algebraic model of two Spare gates sharing a spare event

Let us now consider two Spare gates with two input events -with primary events A and B -sharing a spare event C, as shown in Fig. 9.

If we focus on the Spare gate on the left side, Q1 will occur as soon as A and C have failed -as stated in Section 4.3.1 -or if A fails and C is made unavailable because B has failed before A. As a consequence, the algebraic model of this Spare gate is

Q1 = C a • (A ¡ C a ) + A • (C d ¡ A) + A • (B ¡ A) C d • C a = ⊥
The algebraic expression for the Spare gate on the right side can be determined in the same way by symmetry. Consequently, the final algebraic model of any of two Spare gates sharing a spare event is

⎧ ⎨ ⎩ Q1 = C a • (A ¡ C a ) + A • (C d ¡ A) + A • (B ¡ A) Q2 = C a • (B ¡ C a ) + B • (C d ¡ B) + B • (A ¡ B) C d • C a = ⊥

Algebraic model of n

Spare gates sharing a spare event Let us consider n Spare gates with 1 output event Q i and two input events: a primary event P i -i ∈ {1, • • • , n} -and a spare event S.

If we focus on the first Spare gate, Q 1 will occur as soon as P 1 and S have failed -as stated in Section 4.3.1 -or if P 1 fails and S is made unavailable because the primary event of any of the other Spare gates has failed before P 1 . As a consequence, the algebraic model of the first Spare gate is

⎧ ⎨ ⎩ Q 1 = S a • (P 1 ¡ S a ) + P 1 • (S d ¡ P 1 ) + i =1 P 1 • (P i ¡ P 1 ) S d • S a = ⊥
The algebraic expression for Q i , i ∈ {1, . . . , n}, can be determined in the same way by symmetry. Consequently, the final algebraic model of any of n Spare gates sharing a spare event is

⎧ ⎨ ⎩ Q i = S a • (P i ¡ S a ) + P i • (S d ¡ P i ) + j =i P i • (P j ¡ P i ) S d • S a = ⊥

Algebraic model of a single Spare gate with three input events

Let us consider a Spare gate with three input eventsthe primary event A and two spare events B and C -as shown in Fig. 11.

As stated in [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF], the output Q of the gate occurs when the primary and all spares have failed, so when A, B, and C have failed. A, B, and C are basic events and cannot fail simultaneously so Q will occur if A, B, and C fail according to sequences [A,B,C],[A,C,B],[B,A,C],[B,C,A],[C,A,B],or [C,B,A]. It is important to note that, when the quantitative analysis will be performed from the structure function, B and C will not have the same distribution function in the six sequences. For instance, in sequence

Q = C a • (A ¡ B a ) • (B a ¡ C a ) +B a • (A ¡ C d ) • (C d ¡ B a ) +C a • (B d ¡ A) • (A ¡ C a ) +A • (B d ¡ C d ) • (C d ¡ A) +B a • (C d ¡ A) • (A ¡ B a ) +A • (C d ¡ B d ) • (B d ¡ A)
Figure 10: A benchmark from [START_REF] Zhu | A benchmark for quantitative fault tree reliability analysis[END_REF] with four Spare gates with three input events sharing two spare events.

As B and C cannot be both in an active state and in a dormant state, we have

B d • B a = ⊥ C d • C a = ⊥
The algebraic model of many Spare gates with many common spare events can be deduced from these models by considering the same approach. It will not be detailed here though.

General case of n Spare gates with m input events sharing p ≤ (m -1) spare events

In the general case, the algebraic model of n Spare gates with m input events sharing p ≤ (m-1) spare events is complex. However, a few information can be provided regarding its complexity.

The algebraic model of each Spare gate can be divided into two parts:

• A first part which describes the failure of the gate 'by itself', i.e. without taking into account the influence of the other Spare gates. In the case of two Spare gates sharing a spare event described in Section 4.3.2, this first part, respectively, is

C a •(A¡C a )+A•(C d ¡ A) and C a • (B ¡ C a ) + B • (C d ¡ B) for Q1 and Q2;
it can be noted that this first part does not depend on the main components of the other gates.

• A second part which describes the influence of the other Spare gates on the Spare gate considered. In To illustrate this complexity, if we consider the case of n Spare gates with three input events sharing two spare events, the algebraic model of each Spare gate contains two parts:

• the first part contains 3! terms • the second part contains:

-6(n -1) + (n -1)! terms if there are three Spare gates or more (n ≥ 3) -6(n -1) terms if there are less than three Spare gates (n < 3)

If we consider Fig. 10 which shows a benchmark from [START_REF] Zhu | A benchmark for quantitative fault tree reliability analysis[END_REF] containing four Spare gates with three input events sharing two spare events (n = 4), the algebraic model of each Spare gate includes 30 terms:

• six terms which correspond to the first part, and which represent the 3! possible failure sequences of the inputs of the Spare gate

• 24 = 6 × (4 -1) + (4 -1)! terms which correspond to the second part, and hence to the influence of the three other gates. These 24 terms are divided as follows:

six terms which correspond to the 6 order-3 sequences in which the spare events V MS1 and V MS2 fail first six terms which correspond to the 6 order-3 sequences in which the spare events V MS1 and V MS2 fail second six terms which correspond to the 6 order-3 sequences in which the spare events V MS1 and V MS2 fail last six terms which correspond to the 3! order-3 sequences in which the failures of the main components only are sufficient to engender the failure of the gate

Specific case of Cold and Hot Spare events

The algebraic models presented in Sections 4.3 and 4.4 are models of Spare gates in the general case of Warm Spare events. These algebraic models can be simplified in the specific cases of Cold and Hot Spare events:

• if a spare event S is a Cold Spare event, it cannot fail while in a dormant state, so S d will never occur and any expression containing S d in the algebraic models can be removed;

• if a spare event S is a Hot Spare event, it will have the same distribution function when in an active and in a dormant state, so S a ≡ S d ≡ S and the algebraic models can be simplified.

Thanks to the algebraic model of all dynamic gates, we are now going to show how the structure function of any DFT can be determined and simplified to a canonical form.

Determination of the canonical form of the structure function of Dynamic Fault Trees

The algebraic models of dynamic gates presented in Section 4 allow to determine the structure function of any DFT. It can be interesting to manipulate this structure function to obtain a sum-of-product canonical form since such a canonical form of the structure function can be much useful to perform both the qualitative and the quantitative analysis of the DFT.

On the one hand, each product term of this canonical form is an algebraic expression which engenders the occurrence of the TE of the DFT and which holds for a given number of failure sequences. The cut sequences of each product term, and hence of the whole DFT, can hence be determined from this canonical form. On the other hand, the failure probability of the TE can be determined from this canonical form thanks to the standard inclusion-exclusion formula [START_REF] Trivedi | Probability & statistics with reliability, queueing & computer science applications[END_REF].

Given a DFT with n basic events {b i , i ∈ (1, ..., n)}, the structure function for the TE becomes an expression containing at most the n basic events, and operators +, •, ¡, , and ¢. In [START_REF] Merle | Probabilistic algebraic analysis of fault trees with priority dynamic gates and repeated events[END_REF], we showed that the structure function of any DFT with gates PAND and FDEP could be developed and simplified, thanks to the theorems presented in Appendix A, to arrive to a standardized sum-of-product canonical form where each product term contains operator •, and ordered pairs of variables linked by operator ¡ only, as in [START_REF] Raiteri | The conversion of dynamic fault trees to stochastic petri nets, as a case of graph transformation[END_REF]:

T E = b i • (b j ¡ b k ) , j / ∈ {i, k} (7)
However, in this paper, we extend the work presented in [START_REF] Merle | Probabilistic algebraic analysis of fault trees with priority dynamic gates and repeated events[END_REF] to Spare gates, and the algebraic model of Spare gates defined in Sections 4.3 to 4.6 involves the same temporal operator that is used to model gates PAND and FDEP, so the expression [START_REF] Raiteri | The conversion of dynamic fault trees to stochastic petri nets, as a case of graph transformation[END_REF] still holds in the case of a DFT with Spare gates.

In this canonical form of the structure function, each product term b i • (b j ¡ b k ) is not a single cut sequence, but an algebraic expression providing a sufficient condition on the order of basic event failures that leads to the TE which may contain more than one cut sequence, and actually is a cut sequence set (CSS).

If we suppose that there are n product terms in ( 7), the canonical form can be rewritten in the compact form

T E = n i=1 CSS i (8)
Nevertheless, a CSS may be included in one or more CSSs. CSS i is included in one of the CSS j if it satisfies the criterion [START_REF] Rauzy | Mathematical foundations of minimal cutsets[END_REF] CSS i •

j =i CSS j = CSS i (9)
If CSS i is included in one of the CSS j , it is redundant, and can be removed from the structure function [START_REF] Dugan | Dynamic fault-tree models for fault-tolerant computer systems[END_REF]. Iterative application of the criterion (9) removes all the redundant CSSs, and returns the minimal set S min of non-redundant CSSs.

If S min contains (m ≤ n) cut sequence sets, the minimal canonical form of the structure function can be expressed as [START_REF] Lauritzen | Stable local computation with conditional Gaussian distributions[END_REF] This minimal canonical form of the structure function can be exploited to perform the qualitative analysis -as it will be shown through an example in the next section -and the quantitative analysis of any DFT (even though this latter aspect will not be detailed in this paper).

T E = m i=1 CSS i , CSS i ∈ S min

Determination of the structure function of a DFT example

We propose to determine the structure function of a DFT example extracted from [START_REF] Boudali | A discrete-time Bayesian network reliability modeling and analysis framework[END_REF] which is depicted in Fig. 12. This DFT models the failure of a cardiac assist system (HCAS) which is divided into four modules: Trigger, CPU unit, motor section, and pumps. The Trigger consists of a crossbar switch (CS) and a system supervisor (SS). The failure of either CS or SS triggers the failure of both CPUs. The CPU unit is a Warm Spare, which has a primary P and a spare unit B having a dormancy of 0.5. For the motor section to function, either MOTOR or MOTORC need to be working. The pumps unit is comprised of two Cold Spares, each having a primary pump (PUMP 1 and PUMP 2), and sharing a common spare pump (Backup PUMP). In order for the pumps unit to fail, all three pumps need to fail and CSP 1 needs to fail before (or at the same time as) CSP 2, i.e. PAND gate.

The structure function of this DFT could be determined without any further simplification. However, in an educational purpose, we propose to divide this DFT into three subtrees whose structure functions will be successively determined. These three subtrees are as follows:

• subtree 1, which corresponds to the failure of the CPU unit: this subtree contains one OR gate, one FDEP gate, and one Warm Spare gate, and is hence dynamic;

• subtree 2, which corresponds to the failure of the motor section: this subtree contains a single AND gate and is hence static;

• subtree 3, which corresponds to the failure of the pumps unit: this subtree contains one PAND gate and two Cold Spare gates, and is hence dynamic.

If we denote by T E 1 , T E 2 , and T E 3 the top events of these three subtrees, the structure function of the DFT in Fig. 12 can be expressed as

T E = T E 1 + T E 2 + T E 3
The structure function of each one of these three subtrees can now be determined thanks to the algebraic model of dynamic gates presented in Section 4.

Determination of the structure function for T E 2

Subtree 2 is static since it contains a single AND gate. Its structure function can thus be determined directly as

T E 2 = MOT OR • MOT ORC
T E 1 = (P ¡ B a ) • (B a ¡ (CS + SS)) +(P ¡ (CS + SS)) • ((CS + SS) ¡ B a ) +(B d ¡ P ) • (P ¡ (CS + SS)) +(B d ¡ (CS + SS)) • ((CS + SS) ¡ P ) +((CS + SS) ¡ B d ) • ((CS + SS) ¡ P ) +((CS + SS) ¡ P ) • ((CS + SS) ¡ B a )
Some of the terms of this structure function can be grouped to obtain the final simplified following form:

T E 1 = CS + SS + P • (B d ¡ P ) + B a • (P ¡ B a )
6.4. Determination of the structure function of the whole DFT in Fig. 12 The structure function of the DFT in Fig. 12 can finally be determined as

T E = CS + SS + MOT OR • MOT ORC +P • (B d ¡ P ) + B a • (P ¡ B a ) +BP • (P 2 ¡ P 1) • (P 1 ¡ BP ) +P 2 • (P 1 ¡ BP ) • (BP ¡ P 2)
This structure function can be used to perform the qualitative analysis of the DFT in Fig. 12 directly, as explained in Section 7.

Qualitative analysis of DFTs based on the structure function

The canonical form of the structure function of the DFT in Fig. 12, which was determined previously, is a sum-of-product form whose each product term can provide minimal cut sets or minimal cut sequences. Two cases may happen:

• if a product term does not contain the temporal operator BF (¡), it is static and provides minimal cut sets for the DFT;

• if a product term contains the temporal operator BF (¡), it is dynamic and provides minimal cut sequences for the DFT. In some cases, a set of minimal cut sequences may represent all the possible sequences which correspond to a minimal cut set and can hence be reduced to this minimal cut set.

The structure function of the DFT in Fig. 12 contains seven terms. On the one hand, three terms do not contain the temporal operator BF (¡). They are static and can hence provide three minimal cut sets for the DFT:

CS, SS, (MOT OR • MOT ORC)

On the other hand, four terms contain the temporal operator BF (¡). They are dynamic and can hence provide the minimal cut sequences of the DFT: In this case, it can be noted that the two minimal cut sequences [B d , P ] and [P, B a ] are logically equivalent to the single minimal cut set P • B. However, the minimal cut set does not render the two states of the basic event B which will be needed to perform the quantitative analysis of the DFT. This is the reason why these two minimal cut sequences were not reduced to the equivalent minimal cut set B • P .

This canonical form of the structure function thus provides a hybrid result for the qualitative analysis of DFTs by allowing to determine both minimal cut sets and minimal cut sequences. Furthermore, as it was shown above, a set of minimal cut sequences may sometimes be equivalent to a single minimal cut set. Two cases may happen:

• if these minimal cut sequences contain spare events, they must not be reduced to their equivalent minimal cut set since the knowledge of the state in which spare events fail will be needed to perform the quantitative analysis of the DFT;

• if these minimal cut sequences do not contain spare events, they can be reduced to their equivalent minimal cut set. Indeed, even though both results are equivalent, minimal cut sets represent a more concise -and hence more useful -result to the practitioner than the corresponding set of minimal cut sequences.

Conclusion

In this paper, we presented an algebraic framework allowing to determine the structure function of any DFT, as it is commonly the case for SFTs. Furthermore, we showed that this structure function can be simplified to a canonical form for any DFT. Starting from this canonical form of the structure function, the qualitative analysis of the DFT can be performed directly by determining both the minimal cut sets and sequences of the DFT.

Regarding the quantitative analysis of DFTs, which has not been developed in this paper, it can also be performed from the structure function of DFTs thanks to appropriate probabilistic models of all dynamic gates (PAND, FDEP, and Spare). We have already determined such probabilistic models, which do not depend on the failure distribution considered for basic events, but they could not be presented in this paper.

Even though cut sequences can be extracted quite easily from the structure function, ongoing work is currently addressed to the systematic determination of the minimal set of minimal cut sequences. Besides, the work presented in this paper allowed to propose a formal background for the determination of the structure function of DFTs, and future work will be dedicated to the elaboration of efficient algorithms allowing to automatically perform the calculation of this structure function and the qualitative analysis of DFTs.

Figure 1 :

 1 Figure 1: A non-repairable event.

Figure 2 :

 2 Figure 2: An example of DFT with one repeated basic event.

Figure 3 :

 3 Figure 3: Timing diagrams of operator non-inclusive BEFORE (BF).

Figure 4 :

 4 Figure 4: Timing diagrams of operator SIMULTANEOUS (SM).

Figure 5 :

 5 Figure 5: Timing diagrams of operator INCLUSIVE BEFORE (IBF).

Figure 6 :

 6 Figure 6: Definition of gate PAND from [20].

Figure 7 :

 7 Figure 7: An FDEP gate with two dependent basic events A and B.

2

  In the equation below, the notation (A.30) = indicates that the expression B •(A¢B) is obtained from the expression (A•B)•(A¢B) by applying theorem (A.30) from the Appendix. This notation will be used in the remainder of this paper.

Figure 8 :

 8 Figure 8: A single Spare gate with one primary event A and one spare event B.

Figure 9 :

 9 Figure 9: Two Spare gates sharing a spare event C.

Figure 11 :

 11 Figure 11: A single Spare gate with one primary event A and two spare events B and C.

  [A, B, C], both B and C fail during their active mode (denoted by B a and C a ), whereas in sequence [B, C, A], both B and C fail during their dormant mode (denoted by B d and C d ). The algebraic model of gate Spare can hence be expressed as

  the case of two Spare gates sharing a spare event described in Section 4.3.2, this second part, respectively, is A • (B ¡ A) and B • (A ¡ B) for Q1 and Q2.

Figure 12 :

 12 Figure 12: The HCAS Dynamic Fault Tree from [2].

6. 2 .

 2 Determination of the structure function for T E 3 Subtree 3 is dynamic since it contains gates PAND and Cold Spare. The algebraic model of gate PAND presented The additional term P T B T can first be determined since T = CS + SS: P T B T = (P + CS + SS) (B + CS + SS) = ((CS + SS) ¡ B) • ((CS + SS) ¡ P ) = ((CS + SS) ¡ B d ) • ((CS + SS) ¡ P ) + ((CS + SS) ¡ P ) • ((CS + SS) ¡ B a ) Consequently, T E 1 = (B a +CS +SS)•((P +CS +SS)¡(B a +CS +SS)) + (P + CS + SS) • ((B d + CS + SS) ¡ (P + CS + SS)) + ((CS + SS) ¡ B d ) • ((CS + SS) ¡ P ) + ((CS + SS) ¡ P ) • ((CS + SS) ¡ B a ) Finally, this structure function can be developed thanks to the use of theorems (A.4) and (A.8), and simplified to the following form:

[

  B d , P ] , [P, B a ] , [P 2, P 1, BP ] , [P 1, BP, P 2] The minimal cut sets and sequences of the DFT in Fig. 12 can then be determined as CS, SS, (MOT OR • MOT ORC), [B d , P ] , [P, B a ] , [P 2, P 1, BP ] , [P 1, BP, P 2]

{{0, 1}, +, •, 0, 1} is an Abelian dioid because + and • are commutative, idempotent, and, respectively, allow 0 and 1 as their identity element, • is distributive over +, and • allows 0 as an absorbing element.

in Section 4.1 allows to express T E 3 as

According to the algebraic model of two Spare gates sharing a spare event presented in Section 4.3.2, CSP 1 = BP • (P 1 ¡ BP ) + P 1 • (P 2 ¡ P 1) CSP 2 = BP • (P 2 ¡ BP ) + P 2 • (P 1 ¡ P 2)

where BP denotes the active state BP a of the spare pump since it cannot fail while in a dormant state, and P 1 and P 2 denote P UMP 1 and P UMP 2, respectively, for the sake of clarity.

The following result will be exploited to determine the structure function:

CSP 1 ¢ CSP 2 can now be expressed as

= (BP ¡ P 2) + (BP ¡ P 1) + BP • P 1 • (P 2 ¡ P 1) On the other hand,

(1),( 6)

Since BP ≡ BP a cannot fail before P 1 and P 2 as it can only fail in its active mode,

Determination of the structure function for T E 1

Subtree 1 is dynamic since it contains gates FDEP and Warm Spare. The model of the Spare gate presented in Section 4.3.1 is valid when the input events of the Spare gate are independent basic events which can consequently not occur simultaneously. However, in the case of subtree 1, basic events P and B are basic events which have a common cause failure represented by the Trigger, and they can hence occur simultaneously when the trigger occurs. Nevertheless, this particular aspect can be taken into account in our model by introducing an additional term related to the simultaneous occurrence of P T and B T -P T B Tin the algebraic model of the Spare gate. T E 1 can hence first be expressed as

where B a and B d denote the active and dormant state of the spare unit B, according to the algebraic model of the Warm Spare gate presented in Section 4.3.1. As explained in Section 4.2, the substituted variables B aT , B dT , B T , and P T explicitly indicate the effect of trigger T and denote the global failure of basic events B and P . Thus we have:

Development and simplification theorems

The temporal operators non-inclusive BEFORE and Inclusive BEFORE introduced in Section 3.4 satisfy the following theorems (their proofs can be found in [START_REF] Merle | Algebraic framework for the modelling of priority dynamic fault trees[END_REF]), for any non-repairable events a, b, and c. These theorems will allow to calculate and simplify the structure function of DFTs.

Appendix A.1. Theorems satisfied by operator non-inclusive BEFORE Operator non-inclusive BEFORE satisfies the following theorems, for all a, b, c ∈ E nr :

Appendix A.2. Theorems satisfied by operator Inclusive BEFORE Operator Inclusive BEFORE satisfies the following theorems, for all a, b, c ∈ E nr : a ¢ a = a (A.17)


Guillaume Ploussard

Stéphane Terry

Pascale Maillé

Yves Allory

Nanor Sirab

Laurence Kheuang

Pascale Soyeux

Nathalie Nicolaiew

Estelle Coppolani

Bernard Paule

St Phane Terry

Pascale Maill

Laurent J Salomon

St Phane Culine

Ralph Buttyan

Francis Vacherot

Alexandre De

La Taille

Alexandre De La

Class III beta

Keywords: Aged, Antineoplastic Agents, therapeutic use, Cell Line, Tumor, Cell Survival, drug effects, genetics, Drug Resistance, Neoplasm, genetics, Humans, Immunoblotting, Immunohistochemistry, Male, Middle Aged, Multivariate Analysis, Neoplasm Invasiveness, Orchiectomy, Prognosis, Prostate, drug effects, metabolism, surgery, Prostatectomy, Prostatic Neoplasms, drug therapy, metabolism, pathology, RNA Interference, Survival Analysis, Taxoids, therapeutic use, Tubulin, biosynthesis, genetics, Tumor Markers, Biological, biosynthesis, genetics class III -tubulin β, prostate cancer, predictive value, docetaxel, therapeutic resistance, prognosis

INTRODUCTION

Prostate cancer (PCa) is the most common solid malignancy and the second leading cause of death attributable to cancer in men ( ). 1

Despite the widespread use of PSA testing to screen for early stage PCa, men continue to be diagnosed with locally advanced or metastatic disease, and approximately 30 of newly diagnosed patients treated with curative intent will eventually relapse during follow-up ( ).

% 2

Treatments at this advanced stage usually include androgen deprivation therapy to deplete systemic androgens in patients, thus reducing the levels of a known PCa growth factor. This treatment is most often only transiently effective because prostate tumor cells can progress to a seeming androgen-independent growth phase now diagnosed as castration resistant prostate cancer (CRPC) ( ). Once at this stage, 3 docetaxel-based chemotherapy has been shown to improve though the survival advantage is relatively limited ( , ). At this time, several 4 5 molecular markers have been proposed to have dependent or independent utility for PCa patient prognostic assessments ( , ) but these 6 7 latter biomarkers remain unproven. Here, we discuss our efforts to evaluate the potential utility of class III -tubulin for purposes of β prognostication of PCa patient response to early, localized therapy or to late-stage, taxane-based chemotherapy.

Taxanes constitute an important group of chemotherapeutic agents that specifically target the -tubulin subunit of microtubules. By preclinical and clinical data now suggests that increased expression of one particular -tubulin isoform, class III -tubulin ( III-tubulin),

β β β
confers cancer cell resistance to taxanes ( ) and that III-tubulin resistance to taxanes is clinically relevant for human lung, breast, 9 -11 β and ovarian cancers ( ). For these human malignancies, high tumor cell expression of III-tubulin was associated with significantly 12 β poorer survival rates in patients treated with taxane-based chemotherapy ( ). For lung cancer, one study suggested that tumor cell 13 -17 β III-tubulin expression was a prognostic factor for men who did not receive adjuvant chemotherapy ( ) [START_REF] Seve | Class III beta-tubulin expression and benefit from adjuvant cisplatin/vinorelbine chemotherapy in operable non-small cell lung cancer: analysis of NCIC JBR. 10[END_REF] For PCa, despite evidence from in vitro studies showing that -tubulin isotype expression was altered in paclitaxel-resistant cells, there β has been only little informative clinical data evaluating the prognostic or predictive value of III-tubulin expression in patient tumor cells ( β , ). Previously, we showed that expression of III-tubulin was increased in castration-resistant prostate cancers and that expression of 19 20 β this tubulin isoform might have a role in progression to CRPC ( ). The aim of the present study was to determine whether III-tubulin 21

MATERIALS AND METHODS

Cell culture

Human prostate cancer cell lines LNCaP (clone FGC), 22Rv1, and DU145 were obtained from American Type Culture Collection (Manassas, VA, USA). Cells were maintained in RPMI 1640 supplemented with 10 FBS and penicillin/streptomycin. For this study low % passage cells were used (<20 passages). The LNCaP-AI variant (passage 16 to 20) was derived from LNCaP cultures maintained more than 18 months growth in androgen depleted medium (phenol red-free RPMI supplemented with 10 charcoal-stripped FBS). LNCaP and % 22Rv1 were authenticated through cell morphology monitoring, response to androgen treatment, and expression of Androgen Receptor which harbor mutations in these lines ( ). DU145 was authenticated by lack of AR expression, expression of mutant TP53 ( ), and 22 -24 [START_REF] Van Bokhoven | Molecular characterization of human prostate carcinoma cell lines[END_REF] assessment of the invasive behavior using Boyden chamber assays.

Western blot analysis

Protein lysates were prepared in the RIPA buffer (radioimmunoprecipitation assay lysis buffer) supplemented with protease inhibitor cocktail (Roche Diagnostics, Basel, Switzerland) and phosphatase inhibitors (25mmol/L orthovanadate and 50 mmol/L NaF) (Sigma-Aldrich). The total protein concentration of the soluble extract was determined using the Bicinchoninic Acid Kit (Sigma-Aldrich).

Each protein sample (30 g) was resolved to SDS PAGE, transferred onto a polyvinylidene difluoride membrane (Millipore, Molsheim,

μ -

France) and incubated with a monoclonal antibody against III-tubulin (1 : 10 000; clone TUJ1; Covance, Emeryville, CA) or -actin (1 :

β β
16 000; AC-15; Sigma-Aldrich). Primary antibodies against the other -tubulin isotypes were from Abcam (San Francisco, CA). Immune β complexes were visualised by enhanced chemiluminescence detection (ECL plus kit, GE Healthcare, Little Chalfont, UK).

cDNA synthesis and real-time PCR

Quantitative PCR was carried out using SYBR Green dye on an Applied Biosystems 7000 Real Time PCR system (Applied Biosystems, Foster City, CA, USA). The conditions for RT PCR have been described previously ( ). The amount of III-tubulin mRNA -25 β levels relative to the housekeeping gene Ribosomal Protein, large, P0 (RPLP0) was determined on the basis of the comparative threshold cycle CT method (2 ). The primer sequences for III-tubulin and RPLP0 have been described previously ( , ).

CT -Δ Δ β 25 26

Construction of bIII-tubulin expression vector and Generation of stable bIII-tubulin overexpressing cells

The ORF encoding bIII-tubulin was purchased from Invitrogen (Ultimate ORF clone, clone ID IOH3755, NM 00608. s instructions. Cells were selected in geneticin (G418, 400 g/mL) for 3 weeks. Resistant colonies were isolated and allowed to grow as μ monoclonal population. Clones expressing the different levels of bIII-tubulin were selected, as determined by western blot analysis, for further studies.

siRNA transfection

Small interfering RNA (siRNA) against TUBB3, and control NonTargeting siRNA were obtained from Invitrogen, Inc. Three specific Stealth RNAi sequences were tested: TUBB3HSS115886 (5 -GACAUCUCUUCAGGCCUGACAAUUU-3 ), TUBB3HSS115887 (5

′ ′ ′
-GCAUCAUGAACACCUUCAGCGUCGU-3 ), and TUBB3HSS173592 (5 -CAGCUGGAGCGGAUCAGCGUCUACU-3 ). The non Subconfluent human prostate cells were transfected with siRNA by using Lipofectamine 2000 (Invitrogen) following the manufacturer s instructions. 72 hours after the transfection, the efficacy of the siRNA knock down was assessed by qRT-PCR and by immunoblotting.

'

The optimal amount of siRNA used for transfection was determined as being 50 nmol/L and the best siRNA sequence allowing to reduce more than 70 of bIII-tubulin expression was identified as the sequence TUBB3HSS115887.

%

Docetaxel dose response curve

To assess the effect of III-tubulin overexpression on chemoresistance, 1x10 cells were seeded in 96-well microtiter plates. The next To assess the effect of the combination treatment of III-tubulin silencing plus docetaxel, 22Rv1 cells were transfected with 50 nM of β stealth siRNA against III-tubulin or control vector as described above and then treated with docetaxel at various concentrations for three β days. Cell viability was then determined by MTT assay.

The cell survival curve was presented as the percentage of surviving cells versus the concentration of docetaxel. The IC values were 50 defined as the concentration of drug required for 50 cell survival, and were calculated using a logarithmic regression. Results were % expressed as means SEM. Each assay was done in triplicate and was repeated on three separate experiments.

±

Patients and Tissue Samples

Written informed consent was obtained. The study included : progressive increase in PSA level after androgen blockade. Hormonal castration had to be biologically confirmed. Chemotherapy was combined with low-dose orally administered prednisone.

(i)
Formalin-fixed, paraffin-embedded specimens were obtained from the Department of Pathology at the Henri Mondor Hospital, Cr teil, é France. Immunostaining was performed: (i) on the initial prostatic biopsies of 37 patients who had received first-line docetaxel-based chemotherapy; and ( ) on tissue microarrays (TMA) for the HNPC patient s cohort. When considering TMAs, for each PCa case, 4 7 ' replicate cores (diameter 0.6 mm) were obtained from cancer foci and four additional cores were also taken from nonneoplastic areas as previously described 25 .

[ ]

Immunohistochemistry (Supplementary Method)

Immunostaining was done on 5-m tissue sections mounted on silane-coated slides. III-tubulin protein expression was evaluated μ β using a monoclonal antibody specific for the III-tubulin isotype (1:500; clone TUJ1; Covance, Emeryville, CA).

β A numerical score was assigned for the epithelial cells of each specimen. Samples with no stained tumour cells were scored as 0. A score of 1, 2, or 3 was assigned to samples with weak, moderate, or strong staining, respectively, independently of the proportion of stained tumour cells. The proportion of immunostained tumour cells was also assessed. For TMAs analysis, the mean of staining in the 4 neoplastic cores was considered. Only cytoplasmic staining was taken into account. All of the slides were independently evaluated by 3 observers (GP, ST, YA). Observers were blinded to the patients adjuvant treatment, final pathologic assessment, and outcome. Interrater

Statistical analysis

Null expression (0 of stained cells) and weak to strong expression (score 1 to 3, and 5 of stained cells) were grouped separately as

% ≥ %
dichotomic variables for statistical analysis.

The Student-t test was used for continuous data. The Mann-Whitney test and the Kruskal-Wallis test were used when data were not normally distributed. Qualitative data were tested using a Chi-square test or Fisher s test as appropriate. The Gleason score was ' dichotomized according to the definition of high-risk prostate cancer as follows: Gleason score<8 versus 8 ( ). Survival curves were ≥ 2 generated by the Kaplan-Meier method and compared using the log-rank test. The Cox proportional hazards model was used to evaluate the independent value of III-tubulin expression among commonly used prognostic factors. Hazard ratios (HR) were presented with 95 β % CI.

(i) In the CRPC cohort, the starting point of the analysis was the first cycle of chemotherapy. The PSA Working Group criteria were used to evaluate PSA responses: we chose a decrease of >75 from the baseline PSA level as the criterion for PSA response ( ). The % 28 primary endpoint was overall survival (OS) defined as the time from the start of chemotherapy until death from any cause or last follow-up for censored patients. Time-to-progression was defined as the time between the first cycle of chemotherapy and an elevated PSA finding.

(7) In the HNPC cohort, Biochemical RFS was analyzed. The day of surgery was reported as the starting point of analysis. Recurrence was defined as the first detectable elevation of PSA above 0.20 ng/ml (at least two consecutive measurements).

A value of p<0.05 was considered statistically significant, and all p values were two-sided. SPSS 13.0 (Chicago, Illinois) software was used for statistical analyses.

RESULTS

Prognostic value of III-tubulin expression for prostate cancer recurrence in hormone-na ve PCa patients treated by radical β ï prostatectomy

We had previously reported that III-tubulin was expressed significantly lower in hormone na ve PCa (HNPC) as compared to β ï castration resistant PCa (CRPC). In our previous study, however, the small patient sample size in the HNPC group (n 74) prevented us = from deriving any statistically reliable association with other patient prognostic factors ( ). Here, we extended our assessment of 21

β III-tubulin expression to PCa-containing specimens obtained from 258 PCa patients that were treated by radical prostatectomy ( ). Table 1 We identified III-tubulin expression in 43 of the 258 specimens from patients with HNPC (16.7 ,

). Strong III-tubulin

β %

Increased expression of III-tubulin in prostate cancer cell lines in response to in vitro docetaxel treatment β

We previously reported that III-tubulin expression was increased in LNCaP cells, ( ) that were grown in androgen depleted ). Despite some decrease of II isotype was seen in III-tubulin transfected LNCaP cells ( III-tubulin clone 1), forced Fig. 2C β β β expression of TUBB3 did not seem to affect consistently the expression of the -tubulin isotypes. Interestingly, the III-tubulin

β β
overexpressing LNCaP cells had a neuroendocrine-like appearance ( ) suggesting that forced expression of Supplementary Fig. S3 β III-tubulin is associated with a trans-differentiation process that has been frequently described for these cells ( , ). To examine the 31 32 effects of III-tubulin over-expression, we measured the half-time inhibitory concentration (IC50) of docetaxel assessed by measuring cell β viability at 72 hrs after exposure. The cell viability assays showed that III-tubulin transfected LNCaP cells ( III-tubulin clone 1 and

β β β
III-tubulin clone 7) were significantly more resistant to docetaxel treatment than control vector-transfected LNCaP cells with a 6.6-fold increase in the IC (33.1 versus 5.0 nmol/L, p<0.001, ). A significant difference between clones and control cell survival was 50 Fig. 2D already achieved at 2 nmol/L docetaxel and persited up to 50 nmol/L docetaxel treatment (p<0.001). III-tubulin transfected LNCaP cells β were again poorly sensitive to docetaxel when higher doses were tested (Supplementary Fig. S4) Having established that III-tubulin has β severe implications for docetaxel resistance in LNCaP cells, we sought to determine if this could be due in part to an effect of III-tubulin β on cell proliferation. To this end, cell proliferation was assayed in the absence of drug. Cell doubling times for vector-transfected LNCaP and III-tubulin transfectants were estimated at 31h and 44h, respectively (supplementary Table 1). Two III-tubulin transfected DU145 β β clones were also selected for subsequent treatment including 1 clone expressing high levels of III-tubulin ( III-tubulin clone 7), and 1

β β
clone expressing moderate III-tubulin ( III-tubulin clone 5) at a level similar to that found in parental or control vector-transfected DU

β β
145 cells ( ). The cell viability assays showed that bIII-tubulin-transfected clone 7 was significantly more resistant to docetaxel Fig. 3A treatment compared to clone 5 or control vector-transfected DU 145 cells with a 2-fold increase in IC (11.1 versus 5.1 nmol/L, p<0.001, 50

). A significant difference in cell viability between clone 7 and control cells was already achieved at 5 nmol/L (p<0.001). There Fig. 3B were no noticeable morphological differences between the parental vector-transfected DU145 and either of the two sublines. Nor were there any evident differences in growth rates noted during standard passage and cell proliferation assays of the sublines compared to the vector-transfected or parental cells (supplementary Table1).

βIII-tubulin silencing increases sensitivity to docetaxel

To further ascertain a role for III-tubulin in chemoresistance to docetaxel, siRNAs directed against III-tubulin was used to ). siRNA mediated loss of III-tubulin did not appear to affect the other III-tubulin isotypes (supplementary Figure S5). An Fig. 3C β β assessment of the doubling times revealed that concomitantly with the loss of III-tubulin upon III-tubulin siRNA treatment, there was a

β β
trend towards an increase in cell growth (supplementary table 1). Docetaxel was then added to the growth medium of control cells or targeting siRNA-treated cells 72 hours after the transfection. Seventy hrs later, targeting siRNA-transfected 22Rv1 cells were found to be more sensitive to docetaxel treatment at all doses tested compared to the control 22Rv1 cells. with III-tubulin positive tumours than those with III-tubulin-negative hormone-na ve tumours (13.5 versus 41.6 months, p 0.019;

β β ï = Fig.
)

. Finally, baseline PSA levels measured prior to the first cycle of chemotherapy in these patients were also significantly related to OS. 4C Multivariate analysis taking III-tubulin expression, baseline PSA, age, and duration of androgen deprivation therapy into account showed β that III-tubulin expression (HR 2.93; p 0.037) and baseline PSA level (HR 4.09; p 0.012) were independent predictors of OS.

β = =

DISCUSSION

In prostate cancer, clinical data and histoprognostic parameters, separated or integrated into nomograms still fail at the individual level to accurately determine the risk of biochemical or clinical relapse after local treatment ( , ). At CRPC stage, docetaxel-based 33 34 chemotherapy has proven to have some effectiveness in terms of overall response rates and survival in CRPC patients ( , ). However, 4 5

any selection of patients likely to benefit most from this form of chemotherapy is difficult and is often based simply upon patient age or the presence of co-morbidities in individuals.

Numerous preclinical studies have reported that the selective overexpression of III-tubulin constitutes an important mechanism for prostatectomy. The advent of PSA testing has lead to a considerable stage migration with an increase of low-risk prostate cancers. In our HNPC cohort study, we included patients who underwent radical prostatectomy since 1988. This constitutes a selection bias due a more important proportion of high-risk prostate cancer before the PSA era. These discrepancies may limit the study of III-tubulin expression as β the number of high-risk prostate cancers decreases over time. However, we showed that the prognostic impact of III-tubulin remained β significant in organ-confined PCa and therefore, we posit that the III-tubulin is of additional value to well-established histoprognostic β parameters even in the assessment of presumed low-risk PCa.

We also observed here that the III-tubulin expression in AI PCa cell lines was increased in response to acute or chronic exposure to to debate. In some instances, we found that alterations of III-tubulin expression were associated with changes in cell morphology and/or β cell proliferation rate. Thus, there is reason to believe that these events might be key determinants of drug resistance. Additionally, evidence is accumulating that microtubules containing III-tubulin exhibit an aberrant dynamicity. These microtubules are less stable than β microtubules composed of other -tubulin isotypes ( , , , ) Since the primary effect of taxane is to bind microtubules thereby β 9 10 40 41

enhancing the microtubule polymerization and decreasing microtubule dynamicity, it has been suggested that III-tubulin-containing β microtubules are more prone to overcome the suppressive effects of taxanes on microtubule dynamics. Although it should be noted that all the above studies ( , , ) have focused on the assembly of purified microtubules in cell free systems and therefore remain 12 13 43 controversial given contradictory reports obtained in intact cells, and hence more biologically relevant, that III-tubulin did not β intrinsically affect microtubule dynamic, at least in some instances ( , ). Interestingly, in these last studies, the effects of paclitaxel on 38 42 microtubule dynamics were altered, suggesting a role for III-tubulin in drug-microtubule interactions. Consistent with this, Mozetti et al. Collectively, our results suggest that the functional and clinical biomarker aspects of III-tubulin expression are linked. Consistent β with our previous findings and other studies reporting an increase in the III-tubulin isotype as a result of anti-microtubule drug treatments β ( , , ), it seems likely that adjuvant treatments other than the current taxane-based chemotherapy regimen may be required for this 19 20 50 group of patients. More importantly, our results now support the idea that III-tubulin expression may be linked to multiple forms of PCa; β progression to recurrence in HNPC patients, progression to CRPC in hormone-treated patients and, finally, progression to docetaxel resistance in docetaxel-treated CRPC patients. In the area of cancer treatment, clinicians have to deal with two limitations: the difficulty to predict accurately the relapse after local treatment; and the ability to anticipate ineffective adjuvant or systemic therapy. To date, in prostate cancer, new molecular markers are needed to better define the subset of patients who are most likely to benefit from an adjuvant strategy after radical treatment, and from chemotherapy at CRPC stage. Our findings were consistent with the role of III-tubulin β expression as prognostic marker of biochemical recurrence at HNPC stage. Assessment of the III-tubulin expression in cancer tissue β therefore could be useful to identify and monitor prostate cancer at high risk for recurrence after radical prostatectomy. If longer follow-up and rates of specific mortality confirm these results, the III-tubulin expression might be interesting for predicting recurrence rates and for Ackowledgements: Western blot analysis showed that. The III-tubulin clone 5 expressed a similar level of III-tubulin compared with control vector-transfected β β cells, whereas the III-tubulin clone 7 showed a higher level of III-tubulin expression.

β

  targeting microtubule activity, taxanes can block cell mitosis and induce apoptosis, especially in tumor cells ( ). A growing body of β

β

  expression might have prognostic value for hormone-na ve PCa patients treated by surgery or for CRPC patients treated with taxane-based ï therapy (docetaxel). Here we evaluated clinical prostate cancer specimens for expression levels of III-tubulin by immunohistochemistry β and we manipulated III-tubulin expression in prostate cancer cell lines to determine the effects of this manipulation on in vitro β responsiveness to docetaxel.

5 %

 5 2). The bIII-tubulin ORF was provided in the Gateway entry vector pENTR 221. The bIII-tubulin expression was constructed by recombining the bIII-tubulin pENTR 221 into the destination vector pcDNA 3.2/V5-DEST (Invitrogen) via the LR reaction according to the manufacturer s ' instructions (Gateway LR clonase II enzyme mix, Invitrogen). The resulting vector was designated as pcDNA-TUBB3. LNCaP and DU145 cells were seeded at the density of 1.5x10 cells in 100-mm culture dishes in RPMI 1640 supplemented with 10 FBS. The next day, cells were transfected with 4 g of pcDNA-TUBB3 vector using Lipofectamine 2000 reagent (Invitrogen) following the manufacturer μ '

β 4 day- 2 , 5 -

 425 , cells were treated with docetaxel at growing concentrations for 72 hours. Cell viability was determined by 3-4,5-dimethylthiazol-2-yl [ ] diphenyl tetrazolium bromide (MTT) assay.

'

  Photomicrographs were taken using a Zeiss Axioplan2 microscope (Carl Zeiss, Le Pecq, France) from imaging platform (INSERM, U955, UPEC).

β 21 medium

 21 or in tumor xenografts from these cells after castration of the host mouse. In the present work, we tested whether docetaxel treatment could affect III-tubulin expression in two androgen-independent (AI) PCa cell lines; LNCaP-AI that expresses low levels ofβ βIII-tubulin or 22Rv1 that endogenously expresses higher levels of III-tubulin. LNCaP-AI variant cells were treated with docetaxel at 3 β nmol/L for up to 30 days. As shown in , using Western blots to assess the levels of expressed III-tubulin protein, expression Figure2Aβ was continuously elevated by exposure to docetaxel over the time course of the experiment. It is noteworthy that the class II -tubulin, an β isotype known to be predominantly expressed in the brain also increased along with total -tubulin under docetaxel exposure whereas no β β III-tubulin protein between control vector-transfected LNCaP cells and TUBB3-transfected clones was established by Western blot analysis (

  -tubulin expression in 22Rv1 cells. The targeting siRNA with the highest reduction of III-tubulin was β β selected for further study based on RT-PCR and Western-blot analysis for effects on III-tubulin mRNA and protein. III-tubulin mRNA β β levels were decreased by 70 using this siRNA and III-tubulin knockdown was confirmed by Western blot analysis after 3 and 6 days ( % β

β

  resistance to tubulin-binding agents in various cancer cell lines ( , ). Several clinical studies have shown that high levels of 19 35 -38 β III-tubulin expression in tumor cells are associated with low response rates and poorer survival in patients treated with taxane-based chemotherapies ( , ). Our findings in prostate cancer were consistent with the previous studies and highlight that the expression 13 -17 39 of III-tubulin is associated with non organ-confined disease, metastatic lymph nodes and PSA failure. Thus, tumor cell III-tubulin β β expression might characterize a general subclass of PCa patients with aggressive behavior and poor prognosis. This could give reason to consider adjuvant treatments for patients with III-tubulin positive tumors. β Characteristics of our HNPC patient s cohort differed slightly from characteristics of patients who actually undergo a radical '

β

  docetaxel. In line with our findings, Ranganathan et al. previously showed an increase in III-tubulin expression in response to paclitaxel β treatment of DU145 cells ( ). The exact mechanism by which increased expression of III-tubulin mediates drug resistance remains open 19 β

β 19 β 11 48 49 β

 191149 have reported that III-tubulin overexpression was the prominent mechanism of paclitaxel resistance in ovarian cancer patients, as β compared with other mechanisms of drug resistance such as overexpression of MDR-1 and point mutations in tubulin at the binding site of paclitaxel ( ). Although many studies have focused on paclitaxel, such characteristics might also be pertinent for docetaxel. Interestingly, 43recent reports further showed that III-tubulin confers resistance to microtubule-destabilizing agents such as vinorelbine in breast cancer β Class III -tubulin and prostate cancer β . In the setting of non-small cell lung cancer, III-tubulin was shown to impact significantly on the response to both tubulin 44 β targeting-agents and DNA-damaging agents ( ). The investigators proposed that III-tubulin may serve as a survival factor to rescue 45 β tumor cells from death signals triggered by chemotherapeutic agents. Therefore, it is conceivable that elevated expression of III-tubulin β can exert similar effects in other malignancies, including prostate cancer, and future work should explore this question. Another interesting open question regarding antimicrotubule drug resistance associated with changes in III-tubulin expression in β cancer cells is: what is the contribution of other isotypes in response to taxane-based regimens? Ranganathan et al. have previously noted that class IVb -tubulin was increased collaterally in III-tubulin-transfected DU145 cells in response to paclitaxel treatment ( ). In a β β 20 separate study, wherein paclitaxel-resistant DU145 variants were selected after chronic exposure to the drug, examination of the -tubulin β isotype composition revealed increased expression of III-tubulin but no changes were observed for IVb -tubulin. Instead, some increase β β in IVa -tubulin expression was noted ( ). Although this previous work in DU145 has suggested that III-tubulin expression isβ upregulated in response to paclitaxel treatment, our survey using docetaxel treatment of two androgen-independent AR-positive PCa cells, 22Rv1 and LNCaP-AI provides additional evidence that this situation might be relevant in clinical settings. These studies further indicate that expression of the class II tubulin can be altered in PCa cells and highlight a potential role for this isotype in the emergence of β docetaxel resistance. The ectopic overexpression or silencing of III-tubulin resulted in a seeming specific up-or downregulation of the β β II-tubulin expression, respectively. Surprisingly, under these conditions, we failed to find any significant changes in the expression of other -tubulin isotypes. These instances strengthened the role of III-tubulin in contributing to chemosensitvity and illustrate the importance of β β monitoring the -tubulin isotype profile to confirm the contribution of the target isotype in drug response studies. Although previous work β has suggested that manipulation of II-and IVb-tubulin expression did not cause changes in paclitaxel sensitivity of CHO and lung β β cancer cells ( , ), these cell lines differ from prostate cancer cells, and future work is warranted in order to determine the full 46 47 functional implications of each -tubulin isotype in the emergence of drug-resistance in prostate cancer cells. β Our findings in experimental prostate cancer cell lines were consistent with our clinical observations that III-tubulin expression in β prostate tumors correlated significantly with outcomes of docetaxel chemotherapy for CRPC patients. Patients expressing III-tubulin at β diagnosis had reduced survival. Thus it is tempting to speculate that the assessment of III-tubulin expression could determine which β CRPC patients might benefit from taxane-based chemotherapy. Studies exploring potential molecular markers in response to chemotherapy are currently limited by the difficulty in obtaining tissues from patients with CRPC. Many changes occurred during the evolution to CRPC, and cancer tissue at diagnosis does not always reflect accurately the cancer tissue in its more advanced stage. In fact, obtaining tumor tissue at CRPC stage, or before and during therapy, though rarely feasible in clinical practice, would certainly be the best way to investigate the impact of biomarker on response to therapy. Nevertheless, by examining III-tubulin expression in the tumors of β HNPC patients, which are easily accessible to tumor sampling, it seems possible to evaluate their response to docetaxel-based therapy once the disease has progressed to CRPC. If these findings are confirmed, the use of novel tubulin-targeted agents, such as epothilones, could be useful for III-tubulin-positive PCa patients ( , , ). This might also offer the opportunity for therapeutic intervention by an antiβ III-tubulin treatment in such patients.

β

  proposing adjuvant therapy.Functional overexpression or knockdown of III-tubulin modulates the prostate cancer cell lines sensitivity to docetaxel. The tissue β β III-tubulin expression status also has predictive value in terms of OS in patients receiving docetaxel-based chemotherapy for hormone-independent disease. These findings underline the importance of III-tubulin expression, which could be used in addition to β clinicopathologic characteristics to select patients for docetaxel-based chemotherapy. Prospective studies incorporating III-tubulin β immunohistochemistry are warranted to determine the clinical relevance of routine use of this assay.

Figure 1 Figure 2 B

 12 Figure 1 Prognostic value of III-tubulin expression in predicting the biochemical recurrence after radical prostatectomy in HNPC prostate cancer β patients . Representative staining for III-tubulin in HNPC TMAs: absence of staining (A), moderate staining (B), strong staining (C). A, B, C β Photomicrographs are taken at x20 objective magnification (A C) and x10 objective magnification; insets (A C). Scale Bar, 100 m. --μ D. Recurrence-free survival (RFS) curves stratified by the expression of III-tubulin (negativity versus positivity of immunostaining, log-rank β test; p 0.002). HNPC: hormone naive prostate cancer; TMA: tissue micro-array.=

Figure 3

 3 Figure 3Functional overexpression and knockdown of III-tubulin modulates the androgen-independent prostate cancer cells sensitivity to docetaxel β Differential levels of III-tubulin protein between control vector-transfected and III-tubulin transfected (clone 5 and 7) DU145 cells: A.β β

  Dose response curve assessing the effect of B. β III-tubulin overexpression in androgen-independent DU145 cells. Cell viability assays showed that clone 7 was significantly more resistant to docetaxel treatment relative to III-tubulin clone 5 and control vector-transfected DU 145 cells. Points, mean; bars, SEM. NT: non targeting. β Western blot confirmation of III-tubulin knockdown in 22Rv1 cells after siRNA transfection. The III-tubulin protein level was decreased C. β β 72 hours after transfection and lasted at least 6 days after a single transfection. Dose response curve assessing the effect of III-tubulin D. β silencing in androgen-independent 22Rv1 cells. Cell viability assays showed that III-tubulin siRNA-transfected 22Rv1 cells were β significantly more sensitive to docetaxel treatment relative to non targeting siRNA-transfected and non-transfected 22Rv1 cells. Points, mean; bars, SEM.

Figure 4

 4 Figure 4 Predictive value of III-tubulin expression for response to docetaxel in castration-resistant prostate cancer patients β Representative staining for III-tubulin in prostate biopsy cores: absence of staining (A), staining positivity (B). Photomicrographs are A, B. β taken at x20 objective magnification (A B), insets x5 objective magnification (AC). Scale Bar, 100 m. -μ Overall survival in C. docetaxel-treated patients: stratification by III-tubulin expression in prostatic tissue (negativity versus positivity of immunostaining, log-rank β test; p 0.019): expression of III-tubulin in prostate cancer tissue is significantly predictive for poorer survival in docetaxel-treated patients.

  β

  Two hundred and fifty eight patients who had undergone RP for localized PCa between November 1988 and May 2007.

	Demographics, clinical and biological data, pathologic parameters, and outcomes in terms of PSA recurrence and adjuvant treatment of
	258 patients with hormone-na ve PCa (HNPC) was collected prospectively in a database and reviewed in a retrospective manner. Clinical, ï
	biological, and pathologic parameters and follow-up data are listed in	Table 1	. No patient had received neoadjuvant therapy.
	(7) Thirty seven patients with CRPC stage and analyzable initial hormone-naive tissue who received docetaxel-based chemotherapy as
	first-line treatment between January 2002 and July 2008. Demographics, clinical and biological data, pathologic parameters, and outcomes
	in terms of PSA were collected prospectively in a database and reviewed in a retrospective manner. The mean age was 68.1 years and the
	mean PSA level was 135.0 ng/ml at PCa diagnosis. The median Gleason score was 8 at diagnosis. Bone metastases were present in 21
	patients at PCa diagnosis. Docetaxel-based chemotherapy was administered for clinical and/or biochemical progression in patients with
	CRPC after a mean duration of androgen-ablation therapy of 33.2 months. The patient s hormone-refractory status was defined as a '

Table 2

 2

	-	%	%	%

β immunostaining (score of 3.0) was detected in 6.2 of tumors. When III-tubulin expression was observed, the percentage of % β positively-stained tumor cells showed a wide range of variability (range: 5 100 ; mean 25.8 ; median 10 ). Representative examples of β cells adjacent to the tumor. Correlations of III-tubulin immunostaining with histoprognostic parameters are described in . We β Table 2

found that positive immunostaining was significantly associated with a Gleason score 8 (p 0.001; OR 3.17), a primary Gleason grade of ≥ = 4 or 5 (p 0.013; OR 2.28), a pT stage 3 (p 0.042; OR 1.97), an extraprostatic extension (p 0.028; OR 2.10) and positive lymph nodes (p = ≥ = = 0.034; OR 3.05). It is noteworthy that III-tubulin expression was also significantly associated with a high risk for biochemical = β recurrence (p 0.029: OR 2.15). The intensity of immunostaining was also correlated with the histoprognostic parameters. A strong = immunostaining, defined by a staining score of 3 was also markedly associated with a Gleason score 8 (p<0.001), extraprostatic extension ≥ (p<0.001), positive surgical margins (p 0.025), pT stage (p<0.001), and positive lymph nodes (p 0.043), when compared with a = = null-to-moderate staining (score 0, 1, or 2). The 3-year and 5-year recurrence free survival (RFS) was 84.5 and 75.4 , respectively, in % %

patients with no III-tubulin expression in the prostate tumor. By contrast, the 3-year and 5-year RFS was 72.4 and 59.4 , respectively,

  Of the 37 cases, 17 were positive for III-tubulin (9 with moderate and 8 with strong staining) whereas 20 were negative. Gleason score >7 at diagnosis; 29.4 of III-tubulin-negative cancers were graded 8 or more as compared to 63.2 of

			β
	Representative examples of staining are shown in	Fig. 4A and 4B	. In this cohort, III-tubulin expression was significantly correlated with
	%	β		%	β
	III-tubulin-positive cancers (p 0.043). PSA responses were observed in 52 of III-tubulin-negative patients compared with 35 of = % β %	β
	III-tubulin-positive patients (p 0.337). Patients with III-tubulin positive tumours experienced shorter time-to-progression. Median = β
	time-to-progression was 4.7 months in III-tubulin-positive patients compared with 9.8 months in III-tubulin-negative patients (Breslow β β
	p 0.149; log-rank p 0.522). Overall, the median survival time for the entire cohort was 25.3 months (95 CI: 16.5 34.2) and the observed = = % -
	cumulative probabilities at years 1, 2, and 3 were 79.5 , 57.8 , and 43.7 , respectively. Median OS was significantly shorter for patients % % %

Overall, III-tubulin siRNA-transfected β 22Rv1 cells were found to be 2-fold more sensitive to docetaxel (IC of 4.4 versus 8.5 nmol/L, p<0.001, ) than controls. These 50 Fig. 3D results indicate that III-tubulin silencing sensitizes AI PCa cells to docetaxel. β Predictive value of early III-tubulin expression in a docetaxel-treated patient cohort β Class III -tubulin and prostate cancer β Cancer Res . Author manuscript Page / 6 14 By examining the III-tubulin status in the initial prostatic biopsies of patients subsequently treated for CRPC disease with docetaxel β chemotherapy, we investigated the usefulness of III-tubulin expression as a potential predictive tumor biomarker for response to β docetaxel. β a

Table 2

 2 Association between III-tubulin expression and clinico-pathologic parameters.

β Patients

with no TUBB3 expression n 215 = Patients with TUBB3 expression n 43

  Clinical data, pathologic features, PSA failure: correlations with III-tubulin expression in HNPC samples (univariate analysis). HNPC: hormone naive prostate cancer; PSA: prostate specific antigen.

			=	Univariate analysis
				p	OR 95 CI [ % ]
	Age, No ( ): %
	<65 years	105 (48.8)	19 (44.2)	0.577
	>65 years	110 (51.2)	24 (55.8)
	Age, years (mean)	64.6	63.8	0.399
	Clinical stage, No ( ): %
	T1	151 (70.2)	25 (58.2)	0.068
	T2 or more	64 (29.8)	18 (41.9)
	PSA, No ( ): %
	<10 ng/ml	119 (55.3)	19 (44.2)	0.180
	>10 ng/ml	96 (44.7)	24 (55.8)
	PSA, ng/ml (mean)	13.1	19.5	0.066
	Gleason score, No ( ): %
	≤7	172 (80.0)	24 (55.8)	0.001	3.17 1.59 6.30 [ -	]
	≥8	43 (20.0)	19 (44.2)
	Primary Gleason grade 4 or 5, No ( ): %	72 (33.5)	23 (53.5)	0.013	2.28 1.18 4.43 [ -	]
	Prostate weight, No ( ): %
	<50 gr	88 (40.9)	22 (51.2)	0.264
	>50 gr	127 (59.1)	21 (48.9)
	Prostate weight, gr (mean)	55.9	54.2	0.724
	pT stage, No ( ): %
	pT2	131 (60.9)	19 (44.2)	0.042	1.97 1.02 3.82 [ -	]
	pT3-4	84 (39.1)	24 (55.8)
	Extracapsular extension, No ( ): %	74 (34.4)	22 (51.2)	0.028	2.10 1.08 4.09 [ -	]
	Positive surgical margins, No ( ): %	43 (20.0)	13 (30.2)	0.116
	Seminal vesicle invasion, No ( ): %	42 (19.5)	14 (32.6)	0.059
	Positive lymph nodes, No ( ): %	15 (7.0)	8 (18.6)	0.034	3.05 1.20 7.73 [ -	]
	Recurrence, No ( ): %	46 (21.4)	16 (37.2)	0.029	2.15 1.07 4.33 [ -	]
	Duration of follow-up, months (mean)	62.1	58.4	0.603

[START_REF] Galmarini | Class III beta-tubulin isotype predicts response in advanced breast cancer patients randomly treated either with single-agent doxorubicin or docetaxel[END_REF]

[START_REF] Galmarini | Class III beta-tubulin isotype predicts response in advanced breast cancer patients randomly treated either with single-agent doxorubicin or docetaxel[END_REF]

[START_REF] Galmarini | Class III beta-tubulin isotype predicts response in advanced breast cancer patients randomly treated either with single-agent doxorubicin or docetaxel[END_REF]

This work was supported by INSERM, the Conseil G n ral du Val de Marne , the Universit Paris-Est Cr teil Val de Marne , grants from ' é é ' ' é é '

the ARTP (to G.P, S.T, F.V), the Association pour la Recherche sur le Cancer (to G.P, F.V) and the FERCM (to G.P).

significant increase was seen for I and IV isotypes ( and ). To corroborate these findings, we set out to

Fig. 2A Supplementary Fig. S1 further characterize the expression of -tubulin isotypes in 22Rv1 under exposure to docetaxel at 15 nM for 72 hrs. The 22Rv1 PCa cell β line is derived from a primary tumor xenograft, CWR22, that relapsed during androgen ablation ( ). These cells express at least three 29

androgen receptor isoforms that appear to be constitutively active, likely promoting proliferation as well as the expression of multiple AR-dependent genes in a ligand-independent manner ( ). Consistent with earlier observations, upregulation of III-tubulin protein ( 23β Fig.

and

) as well as mRNA levels ( ) was evident in docetaxel-treated 22Rv1 cells.

2B

Supplementary Fig. S2A Supplementary Fig. S2B The II-and total protein levels were similarly increased in docetaxel-treated cells in contrast to that found for I and IV isotypes.

Together, these data demonstrate that both acute and chronic exposure of AI prostate cancer cells to docetaxel upregulates III-tubulin β along with a seeming general increase of total -tubulin.