M K Nguyen

T T Truong
email: truong@u-cergy.fr

Inversion of a new circular-arc Radon transform for Compton scattering tomography

Keywords: numbers: 00.00, 20.00, 42.10

A new circular arc Radon transform arising from the mathematical modeling of image formation in a new modality of Compton scattering tomography is introduced. We describe some of its properties and establish its analytic inverse formula. This result demonstrates the feasibility of image reconstruction from Compton scattered radiation in Compton scattering tomography. We also show that it belongs to a larger class of Radon transforms on algebraic curves, which remain invariant under a specific geometric inversion.

Introduction

Since the seminal work of J Radon [START_REF] Radon | Über die Bestimmung von Funktionnen durch ihre Integralwerte längs gewisser Mannigfaltikeiten Ber[END_REF], which finds numerous applications (Computed Tomography, Single Photon Emission Computed Tomography (SPECT) and Positron Emission Tomography (PET), etc.) many extensions of this integral transform have been widely discussed, in particular in the literature of imaging science. This is the case when results of measurements appear under the form of integrals of a physical quantity over lower dimensional manifolds. The relevant problem to solve is the recovery of the physical quantity of interest as a function in R 2 . The field of such problems is known in mathematics as integral geometry in the sense of I M Gel'fand [START_REF] Gelfand | Integral geometry and its relations to the theory of group representation Russian Math[END_REF] and in image processing as image reconstruction.

As the circle is the simplest non-trivial curve in the plane next to the straight line, it becomes the natural object on which a new Radon transform can be defined. However a circle has three parameters (two for its center and one for its radius). Therefore to reduce data redundancy, a constraint is usually imposed. A few examples of circular Radon transforms are known so far. Quinto [START_REF] Quinto | Radon transforms on curves in the plane[END_REF] has considered the case of all translations of a circle of fixed radius as well as circles centered on a circle and shown that they are invertible. This case is of interest for thermo-acoustic (and opto-acoustic) tomography [START_REF] Norton | Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution[END_REF][START_REF] Finch | Determining a function from its mean values over a family of spheres[END_REF]. Synthetic Aperture Radar -SAR (or SOund NAvigation and Ranging -SONAR) imaging makes use of circles of varying radius but centered on a straight line [START_REF] Redding | Inverting the circular Radon Transform[END_REF][START_REF] Sysoev | Unique recovery of a function integrable in a strip from its integral over circles centered on a fixed[END_REF][START_REF] Louis | Local Tomography Mathods in SONAR[END_REF]. Circular Radon transforms along paths that are not on the zero sets of harmonic polynomials were proved to be invertible in [START_REF] Agranovsky | Injectivity sets for the Radon transform over circles and complete systems of radial functions[END_REF]. Finally Compton scattering tomography, a two-dimensional imaging process based on scattered radiation, has one modality for which image formation is built on Radon transforms defined on a set of circles passing through a fixed point of the plane [START_REF] Norton | Compton scattering tomography[END_REF].

In this work we describe a recently proposed modality of Compton scattering tomography [START_REF] Nguyen | Nouvelle Tomographie Compton[END_REF], which is at the origin of a new Radon transform. It is defined on circular arcs having a chord of fixed length rotating around its middle point. Obviously such a condition is far more complicated that the two conditions previously met in imaging systems. Yet, it will be shown that its inversion can be achieved through analytic techniques established long ago by A M Cormack [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF].

Generally the choice of a family of circles is dictated by the physical mechanism of image formation. But there exists also other mechanisms which lead to Radon transforms on other types curves such as ellipses [START_REF] Mensah | Near-field ultrasound mammography[END_REF], parabolas [START_REF] Maeland | Focusing aspects of the parabolic Radon transform[END_REF], hyperbolas [START_REF] Lissianoi | On the inversion of the geodesic Radon transform on the hyperbolic plane[END_REF][START_REF] Maass | Wideband radar: the hyp-transform[END_REF] or V-line (or pair of half-lines from a vertex in the plane) [START_REF] Morvidone | On the V-line Radon transform and its imaging applications Preprint UCP[END_REF]. It is not known whether higher order algebraic curves could support more sophisticated image formation mechanisms.

From the mathematical point of view, it was A M Cormack who first succeeded to invert a large class of Radon transform on what he called α-curves, of which are straight lines, parabolas and one branch right-angle hyperbolas. He also showed that geometrically inverted α-curves, which are called by him β-curves support also invertible Radon transforms. The β-curves contains as special cases circles passing through a fixed point, cardioids and one-branch Bernoulli lemniscates. Palamodov has given the reconstruction of functions from the data of its integrals over half-circles with centers at the diameter of a half-disc H [START_REF] Paladomov | Reconstruction from limited data of arc means[END_REF]. But, as far as we are aware of, our circular arc Radon transform seems to be a new member of invertible Radon transforms in the plane.

Relations between these algebraic curve Radon transforms with the classical Radon have been routinely studied. In particular, the connection between parabolic and straightline Radon transforms has been found by Denecker et. al. [START_REF] Denecker | The general quadratic Radon transform Inverse[END_REF], whereas the relation between standard Radon transforms and SAR/SONAR transforms is discussed in [START_REF] Redding | Inverting the circular Radon Transform[END_REF].

In this paper, we shall be concerned essentially with the new Radon transform on arcs of circles and the derivation of its inverse. In section 2, we recall the principle of Compton scattering tomography as suggested more than thirty years back. We then review the modality proposed by S J Norton, who has shown that a possible implementation of Compton scattering tomography can be made with a circular Radon transform based on circles which pass through the radiation point source. We then introduce the new circular-arc Radon transform after showing how it occurs in a new modality of Compton scattering tomography. The merits of this circular-arc Radon transform cannot be claimed until the existence of its inverse transform is established. Section 3 is precisely devoted to the derivation of the inverse transform. This implies that image reconstruction is feasible with Compton scattered radiation in this modality of Compton scattering tomography. Finally section 4 discusses some possible extensions of this Radon transform on families of algebraic curves globally invariant under geometric inversion. Conclusion and perspectives are contained in the last section.

Compton scattering tomography (CST) and circular Radon Transforms

Compton scattering tomography (CST)

For more than forty years, transmitted penetrating radiation such as X-or gamma-rays have been routinely used to probe the hidden parts of matter and/or tissues [START_REF] Hussein | On the solution of the inverse problem of radiation scattering imaging[END_REF][START_REF] Arendtsz | Energy-spectral Compton scatter Imaging -Part 1: theory and mathematics[END_REF][START_REF] Guzzardi | A critical review of Compton imaging[END_REF]. The measurement of their attenuation along all possible linear paths in the plane form a set of Radon data, which, once fed into a chosen inversion formula provides the reconstruction of the probed medium. In this imaging modality radiation scatter acts as a nuisance by blurring images and it should be removed or at least be compensated.

However it was realized, in the earlier seventies, that the Compton effect which is the scattering of X-or gamma-photons with electric charges in matter may give rise to new challenging imaging modalities [START_REF] Gerl | Gamma-ray imaging exploiting the Compton effect[END_REF]. The idea is to register the outgoing scattered photons according to their energies in order to image the hidden part of objects of interest. Compton scattering tomography (CST) was then born. Several modalities have been proposed and tested [START_REF] Grangeat | La Tomographie[END_REF]. But so far none of them has emerged as sufficiently efficient to be of widespread use.

Extension of this concept to three-dimensional imaging exists in two modalities: Compton camera imaging [START_REF] Singh | An electronically collimated gamma camera for single photon emission computed tomography. Part 1: Theoretical considerations and design criteria[END_REF] and Compton scattered radiation imaging [START_REF] Nguyen | On an integral transform and its inverse in nuclear imaging Inverse[END_REF][START_REF] Nguyen | A novel inverse problem in γ-rays emission imaging Inverse Problems in[END_REF][START_REF] Nguyen | Radon transforms on a class of cones with fixed axis direction[END_REF][START_REF] Nguyen | On a novel approach to Compton scattered emission imaging[END_REF]. The first one is based on Radon transform on conical surfaces with swinging axis whereas the second one is modeled by Radon transform on cone surfaces with fixed axis direction. Both of them are illustrations of an astute use of Compton scattering for imaging hidden parts of objects.

Norton's CT

In 1984 S J Norton [START_REF] Norton | Compton scattering tomography[END_REF][START_REF] Wang | Analytic reconstruction of Compton scattering tomography[END_REF] worked out a CST modality which is based on a Radon transform on circles having a fixed common point. The functioning principle is given by figure 1. A point source S emits primary radiation towards an object, of which M is a running point. A point detector D moves along an Ox-axis and collects, at given energy E , scattered radiation from the object. The physics of Compton scattering demand that the registered radiation flux energy f at site D is due to the contribution of all scattering sites M lying on an arc of circle from S to D subtending an angle (π -ω), where ω is the scattering angle corresponding to the outgoing energy E , as given by the Compton formula, see for example [START_REF] Barrett | Radiological Imaging[END_REF]. Mathematically, f is essentially the Radon transform of the object electron density f (M) on such arc of circles, when radiation attenuation and photometric effects on radiation propagation are neglected. Image reconstruction is then feasible if an exact inversion formula for this type of circular Radon transform is available. This was done first by A M Cormack with a new technique of inversion, which turned out to work also for αand β-curves, see [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF]. Much later on, Norton came up with an alternative inversion formula [START_REF] Norton | Compton scattering tomography[END_REF].

Object

Review of the Cormack circular Radon transform inverse

To facilitate the reading of the next sections, we briefly recall the inversion approach to the Cormack circular Radon transform, introduced above. Consider a circle of diameter p, which goes through S the origin of coordinates, see figure 1. Let its center Ω be defined by -→ SΩ = (p/2)n, such that n makes an angle ϕ with respect to a reference direction Sx. A running point M on the circle is defined by --→ SM = r (γ) k, where cos γ = (k • n). The polar equation of the circle is clearly cos γ = r /p, with -π/2 < γ < π/2. The circle arc element can be easily evaluated as

d s = √ (d r ) 2 + (r d γ) 2 = r d γ 1 cos γ = p d γ. ( 1
)
An integrable function f (M) is represented, in this polar coordinate system by f (r, θ), where θ = γ + ϕ. Its circular Radon transform is given by the integral

f (p, ϕ) = ∫ Arc d s f (r, γ + ϕ)| r =p cos γ = p ∫ π/2 -π/2 d γ f (r, γ + ϕ)| r =p cos γ , (2)
This defining equation can be reformulated in terms of Fourier angular components of f and f

f (r, θ) = ∑ l f l (r ) e i l θ , with f l (r ) = 1 2π ∫ 2π 0 d θ e -i l θ f (r, θ) (3) f (p, ϕ) = ∑ l f l (p) e i l ϕ , with f l (p) = 1 2π ∫ 2π 0 d ϕ e -i l ϕ f (p, ϕ) as f l (p) = p ∫ π/2 -π/2 d γ f l (r )| r =p cos γ e i l γ . ( 4
)
From the polar equation of the circle we have

r d γ 1 cos γ = p d γ = - d r sin γ , ( 5
)
and taking into account the symmetry in γ, a final form, after going back to the variable r , change of integration bounds, and expressing all quantities in terms of r , arises as

f l (p) = 2 ∫ p 0 d r cos l ( cos -1 (r /p) ) √ 1 -(r /p) 2 f l (r ). ( 6
)
This is in fact one of the forms of the Tchebycheff transform [START_REF] Li | A new class of integral transforms[END_REF], called Cormack transform, by H H Barrett in [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]. The recovery of the circular component f l (r ) is done by integrating both sides of this equation on an appropriately chosen function of p. Thanks to a formula discovered by A M Cormack [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF] (which is in fact a special case of a general property of Gauss hypergeometric functions [START_REF] Bushman | Integrals of hypergeometric functions[END_REF]), the right-hand-side of the equation turns out to be the primitive of the product of f l (r ) with a known function of r . Thus f l (r ) can be extracted by a simple derivation. We shall meet the same inversion procedure for the circular arc Radon transform.

A new modality in CST

Recently we have suggested a new modality for Compton scattering tomography and presented some preliminary results on its performance [START_REF] Nguyen | Nouvelle Tomographie Compton[END_REF]. Figure 2 shows how this novel modality of Compton scattering tomography works. An emitting radiation point source S is placed at a distance 2p from a point detector D. The segment SD joining them rotates around its middle point O. At site D is collected the single-scattered radiation flux density from the scanned object for a given angular position of the line SD and at a given scattering energy E , (or equivalently at scattering angle ω). Thus, thanks to the physics of the Compton effect, the detected radiation flux density f (τ, φ) is the integral of the electron density f (M) on this class of circular arc, where τ = cot ω and φ is the rotation angle made by the mediator line of the segment SD with a fixed reference axis, see figure 2. Consequently this image formation leads to a novel class of Radon transform on a particular class of circular arcs, provided that radiation attenuation and photometric effects on radiation propagation are not taken into account.

Circular-Arc Radon (CAR) transform

The family of circular arcs, on which this new Radon transform is defined, subtend an angle (π -ω). Their radius is p/ sin ω. A running point M on the circular arc, is localized by its polar angle θ and its distance to the origin OM = r . The direction of OM makes an angle γ with the mediator line of SD, which itself makes an angle φ with a fixed polar direction, as shown in figure 2.

From the cosine identity for the triangle ΩOM

ΩM 2 = OM 2 + OΩ 2 -2OM OΩ cos ΩOM , (7) p
p 2 = r 2 + 2pr τ cos γ. ( 8
)
The r -positive root of eq. ( 8) describes the physically relevant circular arc

r = r (cos γ) = p ( √ 1 + τ 2 cos 2 γ -τ cos γ ) . ( 9
)
The second r -root describes the complementary arc below the SD line, which corresponds to a subtended angle ω, hence physically not relevant. As the product of the r -roots of eq. ( 8) is p 2 , these two arcs are inverse from each other in a geometric inversion transform of center O and module p. Consequently the whole circle is globally invariant with respect to this inversion and the locus of invariant points in this inversion is the circle (Γ) of radius p and center O. Thus the physical range of γ is -π/2 < γ < π/2, and p

( 1 + τ 2 -τ ) < r < p.
Outside this interval r = 0. Alternatively, the circular-arc equation may also be written as

cos γ = 1 2τ ( p r - r p ) , ( 10
)
displaying a reflection symmetry with respect to γ. In fact the symmetry is even greater, as the equation remains invariant under the simultaneous transformations

ω -→ (π -ω) and (r, p) -→ (1/r, 1/p). ( 11
)
Actually we do not need back scattering because by rotating the scanning equipment beyond φ > π, we can find ourselves in the same situation as exploring with an angle larger than π/2. This fact can be also seen in the inversion equation. Replacing ω by (π -ω) amounts to flip the sign of τ. This is also equivalent to take the equation

cos γ = 1 2τ ( p r + r p
) , instead of equation [START_REF] Norton | Compton scattering tomography[END_REF]. The circular-arc line element d s can be readily given as

d s = r d γ √ 1 + τ 2 1 + τ 2 cos 2 γ = d r 1 + τ 2 τ sin γ . ( 12
)
With this line element, the Circular-Arc Radon (CAR)transform of a function f (r, θ), is

f (τ, φ) = ∫ SD d s f (r, θ), ( 13
)
in which f is an integrable function having a compact support, which is assumed to be strictly inside the inversion circle (Γ). We now derive a new integral equation linking f l (τ) to f l (r ), the circular components of f (τ, φ) and f (r, θ). Since θ = (φ -γ) from figure 2 and accounting for the invariance of the integrand under γ ←→ -γ, eq. ( 13) takes the form

f l (τ) = 2 ∫ π/2 0 d γ r (cos γ) √ 1 + τ 2 1 + τ 2 cos 2 γ f l (r (cos γ)) cos l γ. ( 14
)
Now using

d γ r (cos γ) √ 1 + τ 2 cos 2 γ = d r τ sin γ = dr √ τ 2 -1 4 ( p r -r p ) 2 . ( 15
)
we change back to the r -variable in eq. ( 14) and compute the new r -integration bounds for γ = 0, r = p(

√ 1 + τ 2 -τ), (16)
for γ = π/2, r = p. (

Eq. ( 14) becomes finally

τ f l (τ) 1 + τ 2 = 2 ∫ p p( 1+τ 2 -τ) dr √ 1 -1 4τ 2 ( p r -r p ) 2 f l (r ) cos [ l cos -1 ( 1 2τ ( p r - r p ))] . ( 18
)
Compare this equation ( 18) to equation ( 6).

Inversion of the Circular-Arc-Radon Transform

Derivation

We are now in a position to put equation [START_REF] Paladomov | Reconstruction from limited data of arc means[END_REF] in the form of a Cormack's integral [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF], which lends itself to inversion using an integral identity [START_REF] Bushman | Integrals of hypergeometric functions[END_REF]. Let

q = 1 τ = tan ω, and s -1 = 1 2 ( p r - r p ) . ( 19
)
Since r > 0, we have r = p( 1 + s -2s -1 ), and

d r r = d s s 1 + s 2 . ( 20
)
For simplicity, let h l (r ) = r f l (r ). The new integration boundary values on s are for r = p(

√ 1 + τ 2 -τ), s = q (21) for r = p, s = ∞. ( 22
)
Moreover with τ

1 + τ 2 = 1 √ 1 + q 2 , ( 23
)
eq. ( 18) becomes

f l (1/q) √ 1 + q 2 = 2 ∫ ∞ q d s s 1 + s 2 h l (p( √ 1 + s -2 -s -1 )) cos l (cos -1 (q/s)) √ 1 -(q/s) 2 . ( 24
)
This equation ( 24) has now exactly the structure of equation (12a) of [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF] and becomes appropriate for Cormack's inversion technique. We follow Cormack's procedure by multiplying both sides of equation ( 24) by cosh

( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 , ( 25
)
and integrate over q from t to ∞. This yields

∫ ∞ t d q cosh ( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 f l (1/q) √ 1 + q 2 = 2 ∫ ∞ t d q cosh ( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 ∫ ∞ q d s s 1 + s 2 h l (p( √ 1 + s -2 -s -1 )) cos l (cos -1 (q/s)) √ 1 -(q/s) 2 . ( 26
)
Since f l (r ) is of compact support, we can rearrange the two-dimensional integration on the right-hand-side of eq. ( 26) into 2

∫ ∞ t d s s 1 + s 2 h l (p( √ 1 + s -2 -s -1 )) ∫ s t d q cosh ( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 cos l (cos -1 (q/s)) √ 1 -(q/s) 2 . ( 27
)
But in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF][START_REF] Bushman | Integrals of hypergeometric functions[END_REF], it is shown that

∫ s t d q cosh ( l cosh -1 ( q t )) q √ ( q t ) 2 -1 cos l (cos -1 (q/s)) √ 1 -(q/s) 2 = π 2 . ( 28
)
This result leads to the following form of the integral relation between circular components

∫ ∞ t d s s 1 + s 2 h l (p( √ 1 + s -2 -s -1 )) = 1 π ∫ ∞ t d q cosh ( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 f l (1/q) √ 1 + q 2 . ( 29
)
Extraction of f l is achieved by differentiation with respect to t on both sides of eq. ( 29)

-π t 1 + t 2 p( √ 1 + t -2 -t -1 ) f l (p( √ 1 + s -2 -s -1 )) =    d d t ∫ ∞ t d q cosh ( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 f l (1/q) √ 1 + q 2    t = 2pr (p 2 -r 2 ) . ( 30
)
It remains now to re-insert the original variable r in eq. [START_REF] Wang | Analytic reconstruction of Compton scattering tomography[END_REF]. Recalling that, by definition,

r = p( 1 + t -2 -t -1 ), we can work out t = 2pr (p 2 -r 2 )
, and 1

t 1 + t 2 = 2pr (p 2 + r 2 ) . (31)
Eq. ( 30) yields the circular component f l (r ) of f (r, θ)

f l (r ) = (-) (p 2 + r 2 ) 2πpr 2    d d t ∫ ∞ t d q cosh ( l cosh -1 ( q/t )) q √ ( q/t ) 2 -1 f l (1/q) √ 1 + q 2    t = 2pr (p 2 -r 2 ) . ( 32
)
A simple change of variables in the integration shows that equation ( 32) may be recast as

f l (r ) = (-) (p 2 + r 2 ) 2πpr 2 [ ∫ ∞ t d q cosh ( l cosh -1 ( q/t )) √ q 2 -t 2 d d q ( f l (1/q) √ 1 + q 2 )] t = 2pr (p 2 -r 2 ) . ( 33
)
Finally f (r, θ) is reconstructed through its Fourier expansion with the circular components f l (r ), as given by eq. [START_REF] Li | A new class of integral transforms[END_REF]. Note that the integration on q means that one has to collect data with scattering angle from π/2 to a certain value corresponding to t . This is the hole theorem as quoted in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF]. Finally this inversion structure suffers from data noise propagation for which a cure is proposed in [START_REF] Cormack | Radon's problem -Old and new[END_REF].

Advantages of the new CST modality

The analytical inverse formula (32) (or [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]) is the mathematical basis for a new image reconstruction method via the object electron density in this CST modality. This may be viewed as an alternative to image reconstruction via the object attenuation map by standard computed tomography (CT), because the attenuation property of matter may change in time as opposed to its electron density. It opens a new way for probing the inside of matter under working conditions which complement those of the existing CST modality. In fact in Norton's 1995 CST modality, the nature of the circular Radon transform dictates a scanning process on one side of the line source -detector. This is appropriate for large objects such as concrete walls, metal structures in ship building and the like. However there exists also a need for imaging smaller objects in non-destructive industrial testing as well as in medical imaging. For these objects, scanning should be rapid and restricted to an adjustable reduced volume in which residual stray radiation can be easily shielded. This is why the proposed CST modality, based on the circular-arc Radon transform, is more appropriate, in particular when a large number of objects need to be imaged in a row. Moreover, such a CST scanner can be built with the existing detector/detection technology, which has been shown to work with primary (or non-scattered radiation) as well as with scattered radiation. This would make it suitable for widespread use.

A generalization of the circular-arc Radon transform

Definition

In this section we examine the question of whether or not one can extend the idea of Cormack for finding more general curves on which a Radon transform can be defined so that it can be inverted by the same procedure. The essence of the idea is to come up with a similar functional form by assuming at first that relevant curves should be of the general type τ cos(aγ) = g (x a ), where x = p r , -π

2a < γ < π 2a and a ∈ R + . ( 34
)
Here g is an arbitrary function, p a characteristic length of the system and τ ∈ R + a parameter. By differentiation one can show that

r d γ x a g ′ (x a ) = d r τ sin (aγ) = d r √ τ 2 -g 2 (x a ) . ( 35
)
Hence the arc element is

d s = √ d r 2 + (r d γ) 2 = dr √ τ 2 -g 2 (x a ) + (x a g ′ (x a )) 2 τ 2 -g 2 (x a ) . ( 36
)
This allows to define the Radon transform of a function f (r, θ) as

f (τ, φ) = ∫ SD d s f (r (γ), γ + φ) = ∫ r 2 r 1 dr √ τ 2 -g 2 (x a ) + (x a g ′ (x a )) 2 τ 2 -g 2 (x a ) f (r, γ + φ), ( 37
)
where γ = a -1 cos -1 (g /τ). The integration limits are determined by g ((p/r 1 ) a ) = τ and by g ((p/r 2 ) a ) = 0. Now going to the circular components as before, the integral equation linking them is

f l (τ) = 2 ∫ r 2 r 1 dr √ τ 2 -g 2 (x a ) + (x a g ′ (x a )) 2 τ 2 -g 2 (x a ) f l (r ) cos(l γ)| γ=a -1 cos -1 (g /τ) . ( 38
)
Since g does not depend on τ, if we demand that

-g 2 (x a ) + (x a g ′ (x a )) 2 = C constant, (39)
then eq. ( 38) becomes

τ τ 2 +C f l (τ) = 2 ∫ r 2 r 1 dr cos(l /a cos -1 (g /τ)) √ 1 -(g (x a )/τ) 2 f l (r ). ( 40
)
This equation has the required form for inversion with the Cormack's method. So we must solve the differential equation (39). Putting z = (x a ) with x = p/r , it is easy to verify that the general solution takes the form of a linear fractional relation

g (z) = ( pz + q z ) with C = -4pq. ( 41
)
Conversely we can work out z (resp. r ) as a function of

g z = g 2p ± √ ( g 2p ) 2 - q p and r = p ( g 2p ± √ ( g 2p ) 2 - q p )1/a , ( 42
)
and compute d r in terms of g , and replace r by its expression in g .

Remark

A different type of generalization has been studied by Kurusa [START_REF] Kurusa | Support curves of invertible Radon transforms[END_REF]. It is defined on closed curves of R 2 having strictly convex distance functions.

Special cases

We observe that eq. ( 41) contains the two cases solved by A M Cormack when either p or q is zero, i.e.

cos aγ = ( p r
) a for q = 0, and cos aγ = ( r q

) a for p = 0 (43)
We now examine a non-trivial example of the previous generalization, for which the curve equation is simply

cos aγ = 1 2τ [ ( p r ) a - ( r p ) a ] , (44)
where a ∈ R + and -π/2a < γ < π/2a. Alternatively, we can solve for r ( r p

) a = ( √ 1 + τ 2 cos 2 aγ -τ cos aγ ) . (45)
For a = 1, the curve is a circular arc SD rotating around a point O, which is no longer the middle point of the chord SD. The corresponding differential d r is

d r = τ sin aγ √ 1 + τ 2 cos 2 aγ r (cos γ) d γ, ( 46
)
hence the curve element d s is

d s = r d γ √ 1 + τ 2 1 + τ 2 cos 2 aγ = d r 1 τ sin aγ . ( 47
)
The curvilinear-arc Radon transform is now given by the integral

f (τ, φ) = ∫ Arc d s f (r (cos γ), θ) = ∫ π/2a -π/2a r d γ √ 1 + τ 2 1 + τ 2 cos 2 aγ f (r (cos γ), γ + φ). ( 48
)
Going over circular components we find

f l (τ) = 2 ∫ π/2a -π/2a r (cos γ) d γ √ 1 + τ 2 1 + τ 2 cos 2 aγ f l (r (cos γ)) cos(l γ). ( 49
)
Now after going back to the r -variable, this integral equation becomes

τ f l (τ) 1 + τ 2 = 2 ∫ p p( 1+τ 2 -τ) 1/a dr √ 1 - ( 1 2τ [ ( p r ) a - ( r p ) a ]) 2 f l (r ) cos(l γ). ( 50
) with γ = 1 a cos -1 1 2τ [ ( p r ) a - ( r p ) a ] . (51)
Setting as before

q = 1 τ and 1 s = 1 2
[ ( p r ) a - ( r p ) a ] ⇒ d r = r a d s s 1 + s 2 , ( 52
)
we end up with the final equation

τ f l (τ) 1 + τ 2 = 2 ∫ ∞ q 1 a d s s 1 + s 2 1 √ 1 -(q/s) 2 h l (p( √ 1 + s -2 -(1/s)) 1/a ) cos ( l a cos -1 (q/s) ) , (53)
where h l (r ) = r f l (r ) as before. This last equation has again the precise structure of Cormack's equation [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF], thus invertible with the same technique.

In particular, for a = 1/2, 2, we have the equivalent inversion invariant αand βcurves of [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF].

• for a = 1/2 √ r p = ( √ 1 + τ 2 cos 2 (γ/2) -τ cos(γ/2) ) (54)
This curve is closed since -π < γ < π. It has the form of a one branch lemniscate.

• for a = 2, ( r p

) 2 = ( √ 1 + τ 2 cos 2 2γ -τ cos 2γ ) . ( 55
)
This is an arc located in a π/2-quadrant of the plane since -π/4 < γ < π/4.

Inclusion of attenuation and photometric effects in realistic working conditions

For a = 1, under realistic working conditions, traveling radiation is affected by medium attenuation and by dispersion due to photometric propagation effects. A standard way to take into account for these effects is to put the following factor

e -µ M S M S 2 × e -µ M D M D 2 , (56)
(where µ is the average linear attenuation coefficient, here assumed to be constant) in the integrand of the circular-arc Radon transform of eq. ( 13). The value of this additional factor can be evaluated by using triangular identities where

sin γ = √ 1 - ( 1 2τ
( p r - r p )) 2 > 0, (58)
We observe that attenuation brings up a term in sin γ, which has a non-separable dependence on τ. Thus inversion cannot be achieved in this case with the present method. However the inclusion of photometric effects, which yields terms of the form of a product of a function of r and a function of τ, will not spoiled the present mechanism of inversion. These terms have apparent divergences at r = p, which are in fact avoided since the class of functions of interest have their support strictly inside the inversion circle (Γ).

Conclusion and perspectives

A new circular arc Radon transform arising in the mathematical modeling of image formation of a new modality of Compton scatter tomography (CST) is introduced and shown to be invertible via the technique of A M Cormack. It is also proved that it is a member of a larger class of inversion invariant algebraic curve Radon transforms.

Interesting mathematical problems can be raised in this context, such as questions on range and injectivity in general, which has been studied in depth for previous classes of circular Radon transforms [9,[START_REF] Ambartsoumian | On the injectivity of the circular Radon transform[END_REF][START_REF] Ambartsoumian | A range description for the planar circular Radon transform[END_REF]. Another aspect is the development of numerical reconstruction algorithms based on an adapted filtered back-projection process, which remains to be constructed and exploited. These topics could be the subjects for future work which can include the study of an extension to higher dimensional spaces as fascinating new invertible spherical Radon transforms of potential use for imaging science.

Figure 1 .

 1 Figure 1. Norton's Compton scatter tomography

Figure 2 .

 2 Figure 2. Principle of the Circular-Arc-Radon transform

M S 2 = p 2 + r 2 -p 2 + r 2 - 2 + r 2 +)(p 2 -r 2 ) 2 + 2 + r 2 +( p 2 -r 2 ) 2 ( 1

 2222222222222221 2pr sin γ and M D 2 = p 2 + r 2 + 2pr sin γ. 2pr sin γ + √ p 2pr sin γ 4p 2 r 2 cos 2 γ = exp -µ ( √ p 2 + r 2 -2pr sin γ + √ p 2pr sin γ ) + τ -2 ) .


Romain Rieger

Ridha Hambli
email: ridha.hambli@univ-orleans.fr

Rachid Jennane

Modeling of biological doses and mechanical effects on bone transduction

published or not. The documents may come

Modeling of biological doses and mechanical effects on bone transduction

stimulation/inhibition of induced pathways stimulates osteogenic response in vivo. We then tested the model response in term of stimulus signal variation versus the biological factors doses to external mechanical stimuli. Despite the limitations of the model, it is consistent and has physiological bases. Biological inputs are histologically measurable.

This makes the model amenable to experimental verification.

Keywords

Introduction

It is well admitted that mechanical strain is one of the main stimulus triggering bone remodeling.

Bone adaptation to its environment is influenced by both mechanical and biological stimuli [START_REF] Feskanich | Calcium, vitamin D, milk consumption, and hip fractures: a prospective study among postmenopausal women[END_REF][START_REF] Goltzman | Interactions of PTH and PTHrP with the PTH/PTHrP Receptor and with Downstream Signaling Pathways: Exceptions That Provide the Rules[END_REF][START_REF] Heldring | Estrogen Receptors: How Do They Signal and What Are Their Targets[END_REF]. Fluid flow imposes a shear stress on osteocytes that appears to deform the cells [START_REF] Weinbaum | A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stresses[END_REF]. [START_REF] Bonewald | Osteocytes, mechanosensing and Wnt signaling[END_REF] rewieved a cascades of transduction events into the osteocyte as a response to mechanical loading. Important signaling molecules like Nitric Oxide (NO) and Prostaglandin E2 (PGE 2 ) influencing bone remodeling homeostasis have been shown to be produced in the osteocyte in response to fluid flow. [START_REF] Weinbaum | A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stresses[END_REF] were one of the first to propose a theoretical model to predict the fluid shear stress and streaming potential at the surface of osteocytic process in the lacunocanalicular porosity. [START_REF] Lemaire | Multiscale analysis of the coupled effects governing the movement of interstitial fluid in cortical bone[END_REF] enhanced the model of Zeng by including mass flux. [START_REF] Lemaire | Multiscale analysis of the coupled effects governing the movement of interstitial fluid in cortical bone[END_REF] developed a more complex model based on the idea that the motion of interstitial fluid is caused by a combination of mechanical strain, electro-osmotic and osmotic actions.

Few numerical studies modeling the load-induced fluid flow have been developed [START_REF] Goulet | Influence of cortical canal architecture on lacunocanalicular pore pressure and fluid flow[END_REF][START_REF] Gururaja | Modeling Deformation-Induced Fluid Flow in Cortical Bone's Canalicular-Lacunar System[END_REF][START_REF] Steck | Fluid flow through anisotropic, poroelastic bone models in the opposite direction to that through analogous models[END_REF][START_REF] Swan | Micromechanically Based Poroelastic Modeling of Fluid Flow in Haversian Bone[END_REF]. Some other theoretical and numerical studies have been performed considering the osteocyte's tissue strain amplification [START_REF] Han | Mechanotransduction and strain amplification in osteocyte cell processes[END_REF][START_REF] Rathbonivtch | Tissue strain amplification at the osteocyte lacuna: A microstructural finite element analysis[END_REF] or the integrin function on the mechanosensation process [START_REF] Wang | A model for the role of integrins in flow induced mechanotransduction in osteocytes[END_REF][START_REF] Weinbaum | Mechanotransduction and flow across the endothelial glycocalyx[END_REF]. Some others have investigated the effect of mechanical stimulus on bone remodeling without considering the metabolic factor effects [START_REF] Hambli | Multiscale methodology for bone remodelling simulation using coupled finite element and neural network computation[END_REF][START_REF] Hambli | Strain-damage coupled algorithm for cancellous bone mechano-regulation with spatial function influence[END_REF][START_REF] Mcnamara | Bone remodelling algorithms incorporating both strain and microdamage stimuli[END_REF][START_REF] Scott | A mechanistic model for internal bone remodeling exhibits different dynamic responses in disuse and overload[END_REF].

Recently

Despite of the large number of studies dealing with bone mechanotransduction, there is still a lack of mechanobiological models combining in a unified way the most observed mechanical and biological mechanisms during transduction phase on bone [START_REF] Rieger | A mechanobiological transduction model for bone remodelling[END_REF]. Such models are very useful to enhance bone remodeling predictions based on "biophysical" description rather than phenomenological ones. Following the transduction concept proposed by [START_REF] Oesterhelt | The Signal Transduction Cascade[END_REF]

Materials and methods

Concepts of bone transduction

A summary of the previous discussed components is presented in table 1. The integrin senses the amount of Ca and regulates PTH. In the same manner the integrin produces NO, PGE 2 and a mechanical stimulus due to fluid flow. All those components are organized into mechanical and biological normalized signals and then summed up into a full mechanobiological stimulus.

Table 1

Proposition of a transduction model

Mathematical description

The proposed model is based on the idea that the osteocyte is the operating center which generates the interaction between both mechanical stimulus and biological factors.

After combining both stimuli (mechanical and biological), the osteocyte triggers the signal to active BMUs to form or resorb bone.

For the signaling phase we consider the fluid flow velocity in the canaliculi due to mechanical pressure proposed by [START_REF] Lemaire | Multiscale analysis of the coupled effects governing the movement of interstitial fluid in cortical bone[END_REF]. They concluded that the fluid movement P V (equ. ( 1)) generated by pressure gradient is the most important driving effect in fluid flow since it accounts for 95%. The term ( )

1 D γ -
was added to Darcy's law to incorporate the bone matrix damage (D) where γ is a damage exponent that defines the sensitivity of the damage in the fluid velocity.

The biological signaling part is assumed to be triggered by Ca demands. The Ca level ( )

Ca

x is considered as an input in the model (equ.

(2) which leads to PTH release ( ) PTH x . To describe the mathematical relation between PTH release rate and level of Ca, we propose a fitted model obtained by experiments (Haden et al., 2000;Houillier P. et al., 2000) expressed by equation ( 3 (5-6)), from the integrin. The NO and PGE 2 rates growth [START_REF] Maldonado | Mathematical modeling and analysis of force induced bone growth[END_REF] depends on a fluid shear stress function P V f (equ. 7) applied to osteocytes. If NO evolution is purely stimulated by the fluid shear stress function, PGE 2 are also dependant of the NO evolution and then the coupling is expressed in equation ( 6). Osteocyte detects the concentration of Ca through PTH [START_REF] Langub | Parathyroid Hormone/Parathyroid Hormone-Related Peptide Type 1 Receptor in Human Bone[END_REF][START_REF] Teti | Do osteocytes contribute to bone mineral homeostasis? Osteocytic osteolysis revisited[END_REF] c x and considers an auto-regulation between these two types of cells. Indeed the autocrine and paracrine factors are phenomenological modeled as a rate coefficient by an exponent in the model. Insulin-like Growth Factor (IGF) (g22) and Transforming Growth factor Beta (TGF-β) (g11) are autocrine factor stimulating osteoblasts and osteoclasts respectively. Whereas for the stimulation of osteoblasts the paracrine factor is modeled by an exponent (g12) representing IGF and TGF-β actions. Regarding osteoclasts the paracrine stimulation takes effects trough the exponent (g21) modeling Receptor Activator of Nuclear factor κB Ligand (RANKL) and Osteoprotegerin (OPG) actions. [START_REF] Ryser | Mathematical Modeling of Spatio-Temporal Dynamics of a Single Bone Multicellular Unit[END_REF] developed a model describing the evolution of a single BMU taking into account dynamic biochemical factors such as RANKL and OPG driving osteoblasts and osteoclasts evolution. Moreover it describes temporal and spatial features of the digging hole and the movement of the BMUs across the bone surface. This model is biochemically more complex than the one from [START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF] and is more suited for visual bone remodeling evolution.

In order to investigate the potential in our approach to model the coupling effects on the transduction signal, the model of [START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF] has been retained here for its simplicity in the autocrine and paracrine factors description which allows us to focus on the effects of the transduction mechanism.

The integration and amplification phase ensures a normalized signals variations between 0% and 100% of their maximum expressed by the relation / (equ. 11-14). It enables to leave aside dimensional consideration of the variables since we combine mechanical dimension (MPa) and biological concentration (mM) [START_REF] Brazel | Dimensionless analysis of swelling of hydrophilic glassy polymers with subsequent drug release from relaxing structures[END_REF][START_REF] Coatanéa | Applying dimensionless indicators for the analysis of multiple constraints and compound objectives in conceptual design[END_REF][START_REF] Coatanéa | Applying dimensionless indicators for the analysis of multiple constraints and compound objectives in conceptual design[END_REF].

n M a x i i i x x x = ,
Finally, the mechanical and the biological signals are weighted and summed up (equ.

15-17) to generate a normalized stimulus function as a signal (varying between 0% and 100%) mediated by the biological factors (equ. 17). Table 2 Table 3

Results

We It is well known that NO and PGE 2 have an important role in bone remodeling regulation [START_REF] Brandi | Bidirectional regulation of osteoclast function by nitric oxide synthase isoforms[END_REF][START_REF] Pilbeam | Prostaglandins and Bone Metabolism[END_REF][START_REF] Wimalawansa | Skeletal Effects of Nitric Oxide: Novel Agent for Osteoporosis[END_REF]. One can notice that fluid flow gradually increases the production of NO and PGE 2 , which results in a higher stimulus. Although this stimulus can be increased or decreased at some specific magnitude Since this is an original work, there is no comparable study which allows us to calibrate the appropriate values of the weighting-factors. Thus we propose two different configurations of the stimulus equations which represent the possibility of the model to match with the large adaptability of any biological system. Figure 4 exhibits the stimulus' responses to different combinations of weighting-factors. In fact we propose in the first case to express the stimulus to be equally responsive to the mechanical and biological part. This is expressed by assigning a weighting-factors of 0.5 (filled solid curves). In the second case the stimulus is more influenced by the biological component which is expressed by weighting f Bio by 0.8 (open dashed curves). Moreover for the second case we consider the effect of NO and PGE 2 to be more influencing than the PTH, which is represented by weighting f MB by 0.8. Our model incorporates the effects of the bone matrix damage. Fluid velocity is obtained by Darcy's law and is weighted by the term ( 1) D γ -. The damage considered here is due to the presence of microcracks into the bone which leads to an alteration of the osteocyte's communication [START_REF] Burger | Mechanotransduction in bone-role of the lacunocanalicular network[END_REF][START_REF] Prendergast | Microdamage and Osteocyte-Lacuna Strain in Bone: A Microstructural Finite Element Analysis[END_REF]. Instead of an open/close communication relation we shall consider here the possibility of some degradation of the signals sent trough the lacunocanalicular network. So when the damage increases it reduces the magnitude of the mechanical signal sent to the osteocytes. The figure 5 depicts the influence of the level of damage in the mechanobiological stimulus for basal physiological evolution of NO and PGE 2 .

Figure 5 The disrupted communication between osteocytes increases as the damage on the lacunocanalicular network increases. This can be represented by a leakage of the bony fluid contained in the canaliculi. So the proportionality between the fluid flow and the deformation applied onto the bone is altered and the resulted signal does not match properly with the expected information. This non-proportionality is expressed by the modification of the stimulus' magnitude.

Discussion

The work presented here is an attempt to develop a dose dependent integrated transduction model incorporating both mechanical and biological reactions. The idea has arisen from the observation that actual transduction model are only based on mechanical stress neglecting the biological effects.

The current model predicts the influence of Ca concentration on the magnitude of the mechanobiological stimulus. The concentration of Ca can be a regulating factor for the stimulus increase. Low level of Ca induces an important release of PTH [START_REF] Houillier | Le récepteur du calcium : un rôle central dans le métabolisme calcique[END_REF] which stimulates bone resorption in order to deliver Ca from the bone to restore falling plasma Ca. This effect leads to a reduced level of the stimulus. On the opposite high concentration of Ca diminishes the release of PTH and as a consequence, it stimulates bone formation. This is consistent with the work of [START_REF] Chen | Strain differences in bone density and calcium metabolism between c3h/hej and c57bl/6j mice[END_REF] who have shown that there was a progressive increase in the PTH secretion as dietary Ca level decreased which would favor Ca mobilization from bone.

Experimental [START_REF] Bakker | The production of nitric oxide and prostaglandin E2 by primary bone cells is shear stress dependent[END_REF][START_REF] Klein-Nulend | Mechanobiology of bone tissue[END_REF][START_REF] Pitsillides | Mechanical strain-induced NO production by bone cells: a possible role in adaptive bone (re)modeling?[END_REF] and numerical [START_REF] Mcgarry | A comparison of strain and fluid shear stress in stimulating bone cell responses a computational and experimental study[END_REF] studies have shown NO and PGE 2 to be increased when fluid flow grows which confirms the results of the model for two reasons. Firstly we can observe the increase of NO and PGE 2 as the fluid flow magnifies. Secondly the stimulus is amplified as the amount of NO and PGE 2 increase. Based on the work of Maldonado (2006) the kinetics of NO and PGE 2 have been integrated in the present study in order to create a more precise stimulus taking into account the influence of NO and PGE 2 .

Other studies have proven the importance and the benefit of non-mechanical agent to enhance mechanical loading [START_REF] Jee | The benefit of combining non-mechanical agents with mechanical loading: a perspective based on the Utah Paradigm of Skeletal Physiology[END_REF]. This confirms the regulating role of biological messengers which inhibit the action of mechanical loading or improve bone adaptation responsiveness. Their action can be easily illustrated in some hormonal bone diseases by submitting patients onto drugs administration (mainly non-mechanical agents such as estrogen or growth hormone). The threshold values are then moved in order to improve mechanical loading effects.

Nevertheless, the model presents some limitations and simplifications. First, the mechanotransduction process is much more complex than the one described in this paper.

Vitamin D [START_REF] St-Arnaud | The direct role of vitamin D on bone homeostasis[END_REF][START_REF] Wasserman | Vitamin D and intestinal calcium transport : facts, speculations and hypotheses[END_REF], Ca homeostasis (Peterson and Riggs), age, sex, drugs treatments and nutrition should be taken under consideration in order to get an accurate model of transduction. Building such comprehensive model requires proceeding step by step and implies to develop a block diagram model in order to be easily updated.

Second, our model does not consider the pulsatile and the continuous injection of PTH. Many studies suggest an anabolic effect of PTH for pulsated injection [START_REF] Komarova | Mathematical Model of Paracrine Interactions between Osteoclasts and Osteoblasts Predicts Anabolic Action of Parathyroid Hormone on Bone[END_REF][START_REF] Kousteni | The cell biology of parathyroid hormone in osteoblasts[END_REF] and catabolic one for continuous injection. An extended version of the model could incorporate the pulsatile/continuous PTH dosing [START_REF] Potter | Response to continuous and pulsatile PTH dosing: A mathematical model for parathyroid hormone receptor kinetics[END_REF].

Experiments are required to further study the many parameters required for the model and the effects of biological factors dosing the mechanobiological stimulus. Further investigation of the molecular basis of mechanotransduction on bone physiology and disease is needed to confirm that the mechanobiological model can provide realistic predictions.

In spite of these limitations the model can predicts a plausible mechanobiological stimulus consistent with experimental, clinical and numerical observations. Moreover it is easily updatable regarding the block diagram architecture.

In conclusion two perspectives arise from this work. First of all we are going to link this work with a model of BMUs in order to define a complete bone remodeling algorithm [START_REF] Rieger | A mechanobiological transduction model for bone remodelling[END_REF]. Then the whole model will be implemented into finite element code in order to predict bone quality evolution for specific patients and to estimate the necessity of putting them under drugs medication.

Mechanical part

Ref.

Biological part

Ref.

Signal

(1 )

P P dP V K D dz γ = - - (1)
dx K f x X K f x X dt = - - - (8)

:

  Osteocytes; Shear stress; Ca-PTH; NO; PGE 2 Notations ocy x : Active sensing osteocytes number.

:

  Interstitial fluid velocity generated by pressure. dP dz : Pressure gradient in the canaliculi; z denotes the axial coordinate of the canaliculi.

  , Adachi et al., (2010) developed a bone remodeling finite element model based on a stimulus function expressed in term of fluid flow in the lacunocanalicular system. The limitations of this work it that the model neglects the coupling effects with the main biological mechanisms. No biological entities were considered and the bone adaptation is purely triggered by mechanical stimulus. Some studies have attempted to explore the effect of metabolic doses regarding bone remodeling or bone diseases in numerical models. Maldonado, (2006) has developed a formulation of the production of NO and PGE 2 by the osteocyte and a description of the evolution of the osteocyte cell population. The osteocyte rate growth was expressed as a function of the number of osteoblasts (some osteoblasts differentiate into osteocytes) and a fluid shear stress function (mechanical loading is involving on osteocyte viability).Recently, enhanced mathematical model was developed by[START_REF] Peterson | A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling[END_REF] to describe the bone and Calcium-Parathyroid Hormone (Ca-PTH) homeostasis.[START_REF] Komarova | Mathematical Model of Paracrine Interactions between Osteoclasts and Osteoblasts Predicts Anabolic Action of Parathyroid Hormone on Bone[END_REF] have developed a theoretical model describing the autocrine and paracrine factors which are involved in the Bone Multicellular Units (BMUs) autoregulation composed of osteoblast and osteoclast. In this work the authors studied the influence of pulsatile PTH therapy on those autocrine and paracrine factors to induce bone formation.[START_REF] Lemaire | Modeling the interactions between osteoblast and osteoclast activities in bone remodeling[END_REF] have developed a more dynamic BMUs interaction model and investigated the effects on bone remodeling of the different biological factor such as Osteoprotegerin (OPG) or Transforming Growth Factor β (TGF-β).

  Oesterhelt (2010), proposed a general framework to describe the transduction concept based on four steps. (i) Reception of the external signals as inputs (mechanical and biological). Signals can be stimulatory or inhibitory; (ii) Integration (combination) of the signals (iii); Amplification of the signals as outputs (iv); Signals sent to the specific target.

  Figure 1

  ) exhibiting a sigmoidale relation.The reception phase is expressed by the fluid velocity (equ. (4)) where P

  and responses to the fluid shear stress. Thus the Osteocyte population growth ( ) ocy x developed by Maldonado et al., (2006) has been enhanced and is expressed by equation (8). It is directly affected by the osteoblasts population ( ) b x and the osteocytes. 9, 10). It includes osteoblasts ( ) b x , osteoclasts ( )

  Mecha, Bio, MB, PTH, NO, PGE 2 : are mechanical and biological weight factors verifying

  have modeled and investigated the main important phenomena of bone transduction in a mechanical and biological point of view. The influence of metabolic doses on stimulus level has been explored in order to investigate their sensitivity on the mechanobiological stimulus. The following figures show the ability of the model to interpret mechanical effects due to bony fluid flow through the lacunocanalicular system and/or different biological effects due to the virtual injection or the basal production of different messengers such as NO and PGE 2 .If mechanical effect is the most influenced part in bone remodeling it is also highly dependent on the biological components considered as a regulating factor. Figure2shows the evolution of the stimulus versus of the Ca concentration for different magnitudes of the fluid flow.

Figure 2

 2 Figure2

Figure 3

 3 Figure 3

  of fluid flow by considering a virtual injection of continuous controlled dose (0.37 & 0.2 for the square curve; 1.6 & 0.2 for the circle curve; 1.6 & 0.85 for the triangle curve respectively for NO and PGE 2 [pM]) in a specific area. By doing so the model is able to magnify or to reduce NO and PGE 2 effects in the integral domain of influence. The model is considered as dose dependent because it enables the stimulus to be adjusted by the amount of NO and PGE 2 injected or auto regulated.

Figure 4 For

 4 Figure 4

  Adachi, T., Kameo, Y., Hojo, M., 2010. Trabecular bone remodelling simulation considering osteocytic response to fluid-induced shear stress. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, 2669-2682, doi:10.1098/rsta.2010.0073.

Figure 1 :

 1 Figure 1: Block diagram describing the events cascade for bone stimulus with coupled mechanical and biological effects. Dashed arrows represent the mechanical effects and dotted ones represent the biological effects. Solid arrows denote biological reaction due to mechanical stimulation represented by NO & PGE 2 production by the osteocyte. Transduction phases (Signal, Reception, Integration & Amplification, Target) are based on Oesterhelt's concept (2010).

Figure 2 :

 2 Figure 2: Mechanobiological stimulus versus calcium concentration for different levels of fluid flow on basal physiological evolution of NO and PGE 2 .

Figure 3 :

 3 Figure 3: Predicted stimulus versus fluid velocity for different levels of NO [pM] and PGE 2 [pM].

Figure 4 :

 4 Figure 4: Predicted stimulus versus calcium concentration at constant fluid flow for different levels of NO [pM], PGE 2 [pM] and different configurations of the stimulus Ψ.

Figure 5 :

 5 Figure 5: Predicted mechanobiological stimulus at different levels of canaliculus damages for basal physiological evolution of NO [pM] and PGE 2 [pM].

Table 1 : Mechanical and biologic components used in the model.

 1

		Signal	Reception	Integration &	Target
				Amplification
	Mechanics	Fluid flow,	Production	fMecha,fNO,fPGE2
		Shear stress	of NO & PGE 2		BMUs
	Biology	Ca	PTH	fPTH,fBio
			regulation

Table 2 : Model's equations

 2

 [START_REF] Maldonado | Mathematical modeling and analysis of force induced bone growth[END_REF][START_REF] Maldonado | Mathematical modeling and analysis of force induced bone growth[END_REF][START_REF] Calc | [END_REF]based on [START_REF] Calc | [END_REF] based on [START_REF] Maldonado | Mathematical modeling and analysis of force induced bone growth[END_REF][START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF]-Parfitt 1977 Osteoblasts 1.e0 0.e0

Ref. value Prod. rate Elimination rate Paracrine factor Autocrine factor [START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF]-Parfitt 1977[START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF][START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF][START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF] [START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF]-Parfitt 1977[START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF][START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF][START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF][START_REF] Komarova | Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling[END_REF]


MD Zoltán Vokó
email: voko@caesar.elte.hu

Sophie Brouwer

PhD Jacobus Lubsen

PhD Nicolas Danchin

FESC Jan-Erik Otterstad

Peter H J M Dunselman

PhD Bridget-Anne Kirwan

PhD Sophie De Brouwer

FESC Peter Hjm Dunselman

Long-term impact of secondary preventive treatments in patients with stable angina

Keywords: angina pectoris, cardiovascular agents, coronary artery bypass, coronary disease, percutaneous coronary intervention, secondary prevention, G-estimation

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

INTRODUCTION

Several drug classes, such as anti-platelet agents, beta-blockers, and statins have been shown in clinical trials to reduce morbidity and mortality in patients with coronary heart disease. In the recent CORONA trial, statins surprisingly did not show benefit the sub-group of patients with coronary heart disease and heart failure (HF); this trial demonstrated the need for caution when applying results in a broad population to sub-groups of that population [START_REF] Kjekshus | Rosuvastatin in older patients with systolic heart failure[END_REF]. Few large and long-term trials or data bases exist that allow the evaluation of preventive therapy in the sub-group of patients with coronary heart disease and angina pectoris. However, the ACTION trial was one such trial [START_REF] Poole-Wilson | Coronary disease Trial Investigating Outcome with Nifedipine gastrointestinal therapeutic system investigators. Effect of long-acting nifedipine on mortality and cardiovascular morbidity in patients with stable angina requiring treatment (ACTION trial): randomised controlled trial[END_REF]; the primary purpose of ACTION was to assess the effects of nifedipine GITS in patients with stable symptomatic angina over a mean follow-up of five years [START_REF] Poole-Wilson | Coronary disease Trial Investigating Outcome with Nifedipine gastrointestinal therapeutic system investigators. Effect of long-acting nifedipine on mortality and cardiovascular morbidity in patients with stable angina requiring treatment (ACTION trial): randomised controlled trial[END_REF][START_REF] Lubsen | Design and current status of ACTION: A Coronary disease Trial Investigating Outcome with Nifedipine GITS. Gastro-Intestinal Therapeutic System[END_REF]. Patients were randomly assigned to treatment with nifedipine GITS or placebo in addition to the other medications prescribed as clinically indicated. In this paper we describe the pattern of prescribed medications, as recommended in guidelines for patients with established coronary heart disease and the effects on mortality and major cardiovascular events of these cardiovascular medications and coronary interventions [START_REF]Management of stable angina pectoris. Recommendations of the Task Force of the European Society of Cardiology[END_REF][START_REF] Dalal | Recent developments in secondary prevention and cardiac rehabilitation after acute myocardial infarction[END_REF][START_REF] Fox | Guidelines on the management of stable angina pectoris: executive summary: the Task Force on the Management of Stable Angina Pectoris of the European Society of Cardiology[END_REF]. We adjusted for prognostic factors and considered both the relationship between outcome and treatment at baseline, and treatment during follow-up.

METHODS

The ACTION trial was carried out in accordance with the Declaration of Helsinki and ICH guidelines for good clinical practice (GCP). To protect patient safety, predefined interim analyses were performed by an independent safety monitoring committee that had access to the medication code. These did not lead to early termination. ACTION was a multinational, randomized, placebo-controlled trial comparing the effects of nifedipine GITS versus placebo on clinical outcome in patients aged at least 35 years old with stable symptomatic coronary artery disease who required treatment with anti-anginal drugs [START_REF] Poole-Wilson | Coronary disease Trial Investigating Outcome with Nifedipine gastrointestinal therapeutic system investigators. Effect of long-acting nifedipine on mortality and cardiovascular morbidity in patients with stable angina requiring treatment (ACTION trial): randomised controlled trial[END_REF][START_REF] Lubsen | Design and current status of ACTION: A Coronary disease Trial Investigating Outcome with Nifedipine GITS. Gastro-Intestinal Therapeutic System[END_REF]. The inclusion criteria were angina pectoris that had been stable for at least one month, and the need for oral or transdermal treatment either to treat or prevent anginal attacks. Patients had to have a history of myocardial infarction (MI), or angiographically proven coronary artery disease, or a positive exercise test or perfusion defect. Major exclusion criteria included left ventricular ejection fraction below 40% and clinically significant symptoms or signs of HF.

Nifedipine GITS (target dose 60 mg once daily) or matching placebo was added to the conventional anti-anginal medication regimen. Physicians were encouraged to optimise risk factor modification and to treat symptomatic angina with appropriate additional medications. Standard baseline assessments included echocardiography, vital signs, electrocardiography, laboratory tests and a full medical history. Blood pressure was recorded with a standard sphygmomanometer in the sitting position after 5 min of rest. ECG abnormalities were coded locally by the treating physician, using a standard coding form. Baseline coronary angiography was not required but clinically significant lesions identified in those past coronary angiograms that were done were documented as described elsewhere [START_REF] Kirwan | Treatment of angina pectoris: associations with symptom severity[END_REF]. After randomisation, patients were seen in the out-patient clinic at two weeks, six weeks and six months after randomisation; and from then onwards every six months. Between visits, patients were contacted by telephone.

ACTION was completed as planned [START_REF] Poole-Wilson | Coronary disease Trial Investigating Outcome with Nifedipine gastrointestinal therapeutic system investigators. Effect of long-acting nifedipine on mortality and cardiovascular morbidity in patients with stable angina requiring treatment (ACTION trial): randomised controlled trial[END_REF]. Between November 1996 and December 1998, 7665 patients were started on study medication (3825 nifedipine GITS, 3840 placebo), and are the basis of this report. The mean duration of follow-up until death or end of study was the same for both treatments (4.94 years) and followup was 97.3% complete.

Assessment of clinical events during follow-up

Any untoward experience that occurred during the trial was reported as a (serious) adverse event. This included major diagnostic and intervention procedures. According to procedures and pre-defined criteria described elsewhere [START_REF] Kirwan | A Coronary disease Trial Investigating Outcome with Nifedipine GITS) investigators. Diagnostic criteria and adjudication process both determine published event-rates: The ACTION trial experience[END_REF], the Critical Events Committee (CEC) independently determined the occurrence of acute MI, refractory angina, new overt HF, debilitating stroke, and peripheral revascularisation. Each patient could have several clinical events. The CEC determined the order of occurrence.

For the purpose of the present report, we defined chest pain requiring hospitalisation as any chest pain or worsening angina reported by the investigator for which the patient was hospitalised, and included in the definition refractory angina as determined by the CEC. We defined any stroke as either debilitating stroke determined by the CEC or stroke diagnosed by the investigator.

Statistical analyses

The treatments considered in this analysis were beta blockers, calcium antagonists To estimate the occurrence of death and of the composite of death, MI, stroke and HF, we determined the rates of these events in two different ways. Considering treatment at baseline only, we took rates as the number of patients with event by treatment at baseline, divided by the total person-time of follow-up 'at risk' of event by treatment at baseline based on 'intention-to-treat', i.e. ignoring treatment changes during follow-up. Considering actual treatment during follow-up, we took rates as the number of patients with event by the treatment actually used at the time of event, divided by the total person-time of follow-up 'at risk' that the treatment concerned was actually used. In the latter rate calculations, each patient could contribute to follow-up time 'at risk' both as a user and as a non-user of a particular class of drug, based on start and stop dates for the treatment concerned.

(
In rate calculations for CABG and PCI respectively, we considered that the patient concerned was treated from the date of the first procedure onwards, and ignored repeat coronary revascularisation procedures.

We estimated the effect of treatments on all cause mortality and on the composite of death, MI, stroke and HF. Firstly, to estimate the effects of the treatments given at baseline (including a history of CABG or PCI) on clinical events, we used multivariate Cox regression to estimate hazard ratios comparing treatment with no treatment adjusted for covariates (aHR) and their 95% confidence intervals Secondly, we performed G-estimation to estimate the effect of each treatment prescribed during follow-up as a time-dependent covariate based on start and stop dates for drugs, and for CABG and PCI, the date when the first procedure was performed, respectively [START_REF] Mark | Estimating the causal effect of smoking cessation in the presence of confounding factors using a rank preserving structural failure time model[END_REF]. We used the application stgest (STATA10.1) to perform the analysis [START_REF] Sterne | G-estimation of causal effects, allowing for time-varying confounding[END_REF]. Besides the baseline confounders, we adjusted for blood pressure, angina requiring hospitalisation, and total cholesterol level in a time dependent manner. When death was considered as outcome, additional adjustment was made for MI, HF, and stroke in a time-dependent manner. The parameter estimated by Gestimation is the causal survival time ratio. To estimate hazard ratios, we assumed that survival times followed a Weibull distribution and fitted Weibull regression models with the same covariates as the corresponding models in the G-estimation procedures. We have taken the estimates of the shape parameters from the Weibull models to calculate hazard ratios from the causal survival time ratios [START_REF] Mark | Estimating the causal effect of smoking cessation in the presence of confounding factors using a rank preserving structural failure time model[END_REF]. We used application gesttowb (STATA10.1) to perform the analysis [START_REF] Sterne | G-estimation of causal effects, allowing for time-varying confounding[END_REF].

Both in the analyses for treatment at baseline and in the time-dependent analyses for treatment during follow-up, the use of vitamin K antagonists was adjusted for atrial fibrillation or irregular pulse at baseline and in a time-dependent manner.

Finally, in G-estimation procedures, we estimated the independent effect of each treatment assessed adjusted for the same covariates as in the previous analysis and for the other treatments.

RESULTS

Baseline patient profile

The ACTION trial was completed as planned and 7665 patients were started on study medication [START_REF] Poole-Wilson | Coronary disease Trial Investigating Outcome with Nifedipine gastrointestinal therapeutic system investigators. Effect of long-acting nifedipine on mortality and cardiovascular morbidity in patients with stable angina requiring treatment (ACTION trial): randomised controlled trial[END_REF]. 79% of patients (n=6084) were male and 46% (n=3548) were 65 years or older. A history of MI was present in 51% (n=3898) and 92% (n=7070) reported anginal attacks. 7479 (98%) had complete information on all covariates considered in Cox regression analyses and are the basis for the remainder of this report.

Cardiovascular drug use

The prescription pattern of beta blockers, lipid-lowering drugs, ACEIs or ARBs and aspirin or another anti-platelet at baseline is given in Table 1. A vitamin K antagonist was prescribed at baseline for 305 patients (4%).

During follow-up, patients assigned nifedipine GITS or placebo were prescribed beta blocking, lipid-lowering drugs and aspirin or other anti-platelet drugs to a similar degree (c.f. Table 1). Patients assigned nifedipine used an ACEI or ARB significantly less frequently than those assigned placebo (mean 27% as opposed to 33% of follow-up time, p<0.001). All four drugs were used during a mean of 14% of follow-up time by patients assigned nifedipine as opposed to 18% of patients assigned placebo (p<0.001).

page 10 of 28

The impact of cardiovascular drugs use on mortality

As shown in Table 2, only lipid-lowering drugs and aspirin or other anti-platelet drugs were associated with a significantly (95% confidence interval of aHR does not cover unity) lower death rate adjusted for other covariates when comparing users with nonusers at baseline (aHR 0.78 and 0.75 respectively). Relative to non-users, the use of vitamin K antagonists was associated with a significantly increased rate of death (aHR 1.38) in an analysis that was also adjusted for atrial fibrillation or irregular pulse.

Time-dependent analyses comparing use during follow-up with non-use with adjustment for other covariates time-dependently are shown in Tables 2 and3 in addition to analyses comparing treatments at baseline. When drug use during followup was taken into account in this manner, lipid-lowering drugs, ACEIs or ARBs, aspirin or other anti-platelet drugs, and calcium antagonists significantly reduced the rate of death (aHRs 0.54, 0.82, 0.49, and 0.93 respectively, c.f. Table 2). Use of vitamin K increased mortality (aHR 1.19, CI 1.02, 1.37, c.f. Table 2).

The impact of cardiovascular drugs use on mortality, MI, stroke or HF

As shown in Table 3, for the outcome death (all cause), MI, stroke or HF, use of beta blockers at baseline was associated with a significantly increased rate of event (aHR 1.15) while lipid-lowering drugs was associated with a significantly reduced rate (aHR 0.84), relative to non-users. Vitamin K antagonists had no effect on this outcome.

Drugs that reduced mortality in the time-dependent covariate also significantly reduced the rate of mortality (all cause), MI, stroke or HF (c.f. Table 3). Lipid lowering drugs had the largest effect (aHR 0.73, CI 0.70, 0.78, c.f. Table 3). Vitamin K antagonists were not significantly associated with the combined outcome. Betapage 11 of 28 blockers use on the other hand increased the risk of death, MI, stroke or HF (aHR 1.07).

Coronary interventions and outcome

CABG had been performed before inclusion in 1749 (23%) patients, and was performed during follow-up in a further 584 (10% of those not having baseline history and included in this analysis). CABG was not associated with the rate of death or the combined rate of death, MI, stroke or HF as a baseline covariate. In the timedependent analysis, it did not reduce the rate of the combined event significantly, but it reduced mortality by 9% (c.f. Tables 2 and3).

PCI had been performed before inclusion in 1956 (26%) patients, and was performed during follow-up in a further 455 (8% of those not having baseline history and included in this analysis). Patients with history of PCI at baseline had a significantly reduced rate of death (aHR 0.67) but no significant reduction of the rate of any death, MI, stroke or HF (c.f. Tables 2 and3). PCI as a time-dependent covariate significantly reduced the rate of death (aHR 0.74) and the rate of any death, MI, stroke or HF (aHR 0.92).

Combined effect of cardiovascular treatments

The conditionally independent effects of cardiovascular treatments during follow-up adjusted for relevant covariates at baseline and time-dependently are shown in Table 4. Use of an ACEI or ARB, calcium antagonist, lipid-lowering drugs, aspirin or another anti-platelet drug, and Vitamin K antagonist had significant conditionally independent preventive effect both on mortality and on any death, MI, stroke or HF.

The rate of death and of the combined outcome of any death, MI, stroke or HF was page 12 of 28 higher in those on a beta-blocker but for death alone, this was not statistically significantly.

CABG had a statistically significant independent effect on mortality (aHR 0.84) but did not have an independent effect on the combined outcome. PCI was associated with a significant reduction of mortality and any death, MI, stroke or HF.

DISCUSSION

The selection criteria of the ACTION study allowed for the inclusion of a broad spectrum of patients with symptomatic stable angina requiring anti-anginal treatment and who did not have clinical signs of heart failure. Hence the present population is likely to be representative of stable angina patients seen in everyday clinical practice.

Patients included in ACTION were slightly older and more often male than patients investigated in two recent large cohorts managed in general practice [START_REF] Filippi | Management of patients with suspected angina, but without known myocardial infarction: a cross-sectional survey[END_REF][START_REF] Hemingway | Incidence and prognostic implications of stable angina pectoris among women and men[END_REF].

The main finding of the present analyses is that three of the four therapies currently recommended for secondary prevention in patients with coronary heart disease, i.e. lipid-lowering drugs, ACEIs or ARBs and anti-platelet drugs all improve outcome [START_REF] Fox | Guidelines on the management of stable angina pectoris: executive summary: the Task Force on the Management of Stable Angina Pectoris of the European Society of Cardiology[END_REF][START_REF] Gibbons | American College of Cardiology; American Heart Association Task Force on practice guidelines (Committee on the Management of Patients With Chronic Stable Angina). ACC/AHA 2002 guideline update for the management of patients with chronic stable angina--summary article: a report of the American College of Cardiology/American Heart Association Task Force on practice guidelines (Committee on the Management of Patients With Chronic Stable Angina)[END_REF]. These treatments significantly reduced mortality and the rate of any death, MI, stroke or HF in adjusted time-dependent analyses that took drug use during follow-up into account. Only lipid-lowering and anti-platelet drugs significantly reduced mortality and only lipid lowering drugs reduced the rate of any death, MI, stroke or HF when comparing treatment at baseline with no treatment. We attribute this to dilution caused by treatment changes during follow-up and our results underline the importance of taking such changes into account by time-dependent analyses. In time-dependent analysis of actual drug use, adjustment for previous health status is needed, because these are related to the indication of treatments. On page 13 of 28 the other hand, adjustment for future health status should be avoided because they are intermediates in the causal pathways (i.e. they predict the outcome).

Conventional multivariate regression methods (like Cox-regression) are not, but Gestimation procedure is a suitable method to achieve this [START_REF] Mark | Estimating the causal effect of smoking cessation in the presence of confounding factors using a rank preserving structural failure time model[END_REF].

Lipid-lowering drugs, ACEIs or ARBs and anti-platelet drugs have all been

shown to be effective in a broad spectrum of patients with coronary heart disease, or with risk factors for its development [START_REF] Gibbons | American College of Cardiology; American Heart Association Task Force on practice guidelines (Committee on the Management of Patients With Chronic Stable Angina). ACC/AHA 2002 guideline update for the management of patients with chronic stable angina--summary article: a report of the American College of Cardiology/American Heart Association Task Force on practice guidelines (Committee on the Management of Patients With Chronic Stable Angina)[END_REF]. In addition, the present results confirm that secondary preventive agents have conditionally independent effects.

In a cohort of 3031 patients studied between 2002 and 2003, Daly et al. found that beta blockade, statins and anti-platelet therapy were prescribed as monotherapy, dual or triple therapy in 18%, 36% and 35% of patients respectively [START_REF] Daly | The impact of guideline compliant medical therapy on clinical outcome in patients with stable angina: findings from the Euro Heart Survey of stable angina[END_REF]. In the present study (Table 1), we found that the proportion of patients prescribed these therapies as mono, dual and triple therapy were 11% (n=814), 38% (n=2910) and 50% (n=3853) respectively so patients were treated as intensively with triple therapy as patients discharged after acute MI in a French registry [START_REF] Danchin | Impact of combined secondary prevention therapy after myocardial infarction: data from a nationwide French registry[END_REF]. When the present study was performed (between 1996 to 2003), beta blockers were prescribed more frequently than ACEIs or ARBs, drug classes not then considered as components of combination therapy [START_REF] Danchin | Impact of combined secondary prevention therapy after myocardial infarction: data from a nationwide French registry[END_REF]. Indeed, 64% of patients were prescribed a beta blocker and 42% an ACEI in the Daly et al. cohort as opposed to 80% and 23% respectively in this cohort (c.f. Table 1) [START_REF] Daly | The impact of guideline compliant medical therapy on clinical outcome in patients with stable angina: findings from the Euro Heart Survey of stable angina[END_REF]. In the most recent guidelines for secondary prevention it is recommended to consider the use of ACEIs in all patients and ARBs in patients who are ACEI intolerant [START_REF] Fox | Guidelines on the management of stable angina pectoris: executive summary: the Task Force on the Management of Stable Angina Pectoris of the European Society of Cardiology[END_REF][START_REF] Gibbons | American College of Cardiology; American Heart Association Task Force on practice guidelines (Committee on the Management of Patients With Chronic Stable Angina). ACC/AHA 2002 guideline update for the management of patients with chronic stable angina--summary article: a report of the American College of Cardiology/American Heart Association Task Force on practice guidelines (Committee on the Management of Patients With Chronic Stable Angina)[END_REF].

The only randomised comparison in the ACTION trial was between nifedipine GITS and placebo. To remove confounding when assessing effects of nonrandomised treatments as much as possible, we used G-estimation to adjust for time-varying confounders. However, confounding by indication can never be absolutely controlled for in non-randomised comparisons [START_REF] Miettinen | The need for randomization in the study of intended effects[END_REF]. Adjusted comparisons for non-randomised treatments cannot replace appropriately designed randomised trials. Nonetheless, we believe that adjusted analyses of observational data are relevant to confirm that effects predicted based on results of trials can be achieved in clinical practice.

Currently, calcium antagonists are not generally recommended for secondary prevention in patients with coronary heart disease although their usefulness in treating anginal symptoms and lowering blood pressure is widely accepted. The ACTION study did not show an effect of nifedipine GITS on mortality or on the combination of death, MI, stroke or HF in a conventional intention-to-treat analyses;

nifedipine did reduce the incidence of HF, any stroke, the need for coronary angiography and the need for CABG [START_REF] Poole-Wilson | Coronary disease Trial Investigating Outcome with Nifedipine gastrointestinal therapeutic system investigators. Effect of long-acting nifedipine on mortality and cardiovascular morbidity in patients with stable angina requiring treatment (ACTION trial): randomised controlled trial[END_REF]. In our analysis, adjusted time-dependent analyses taking the actual use of a calcium antagonist during follow-up into consideration showed that calcium antagonists (which included active study treatment) reduced both mortality and the combination of death, MI, stroke or HF. We found that patients assigned placebo were more intensively treated with combination therapy including an ACEI or ARB than patients randomly assigned nifedipine GITS.

This may explain why the intention-to-treat analysis did not show significant effects of nifedipine on mortality and the composite endpoint while time-dependent analyses did. Intention-to-treat comparisons are unbiased if there is no effect, but may be diluted when the drug considered has effects relative to placebo on changes made to study or other medications for clinical reasons during follow-up. Furthermore, it needs to be emphasised that this analysis compared the rate of events in patients who actually were using any calcium antagonist (which might have been active double blind drug or an open-label calcium antagonist) with the rate of events in patients who were not using a calcium antagonist adjusted for other determinants. Such an analysis is different from comparing the rate of events in patients assigned active treatment with the rate of events in patients assigned placebo, and then disregarding whether the assigned drug was really used during a mean follow-up of almost five years.

Vitamin K antagonists increased mortality even in the time-dependent analysis. However, when its use was adjusted for the use of other drugs, it significantly reduced both mortality and the rate of any death, MI, stroke or HF. The WARIS II trial concluded that warfarin in combination with aspirin was superior to aspirin alone in reducing the incidence of death, non-fatal reinfarction or thromboembolic cerebral stroke, but was associated with a higher risk of bleeding [START_REF] Hurlen | Warfarin, aspirin, or both after myocardial infarction[END_REF]. Our results may reflect the independent benefit of Vitamin K antagonists.

Both CABG and PCI had a large positive effect on survival. PCI significantly reduced the rate of any death, MI, stroke or HF, too. These results came as a surprise as we are not aware of any randomised trial that has shown unequivocally that these procedures improve outcome. Recently, the 5-year results of the Clinical Outcomes Utilizing Revascularization and Aggressive Drug Evaluation (COURAGE) trial were reported [START_REF] Boden | 5 Optimal medical therapy with or without PCI for stable coronary disease[END_REF]. In this largest and most contemporary trial addressing the benefit of PCI in addition to optimal medical therapy in patients with stable coronary artery disease, there were no significant differences in mortality, MI, and stroke between patients with and without PCI. The only difference was that patients initially on medical therapy alone subsequently had more frequently a revascularisation procedure (33% vs. 21%). Patients randomised in the COURAGE trial [START_REF] Boden | 5 Optimal medical therapy with or without PCI for stable coronary disease[END_REF], however, represented only a very small proportion of those screened for eligibility (6.4%), raising the question of the relevance of the conclusions of the trial in the real world setting. Comparing outcomes between patients with and without PCI by a timedependent analysis in the present cohort may be biased because it involves events immediately after the procedure only in a minority of patients. Of all patients in the time-dependent analysis who had PCI, 1956 had a history of this procedure performed at least three months before inclusion while 455 had PCI during follow-up.

In addition, residual confounding by indication may be present after adjustment.

CONCLUSION

As far as we are aware, this is the first time that such an analysis has been performed to investigate the effectiveness of secondary preventive therapies as actually used in patients with a cardiovascular disease, on outcome. The ACTION trial enabled this analyses to be performed because of the detailed data collected during follow-up for each patient. In summary, this study confirms that ACEIs or ARBs, lipid-lowering and anti-platelet drugs as used in the everyday management of patients with stable angina have the secondary preventive effects claimed based on clinical trials. Calcium antagonists PCI, and to a smaller extent CABG although not proven in clinical trials, also seem to improve outcome in time-dependent nonexperimental comparisons.

FINANCIAL DISCLOSURE STATEMENT

The ACTION trial was funded by Bayer Healthcare AG. S de Brouwer, B-A Kirwan, page 25 of 28

  and J Lubsen were full-time employees of SOCAR Research SA, which managed the study. Z Vokó worked as consultant epidemiologist for SOCAR Research SA. PHJM 5 Dunselman was member of the Critical Events Committee and Nicolas Danchin was Chairman of this committee. JE Otterstad was member of the Steering Committee and one of the country coordinators. Members of committees were funded by SOCAR Research SA to attend meetings related to the trial. Z Vokó, N Danchin, JE Otterstad, PHJM Dunselman, J Lubsen have served as consultants to or received 10 travel expenses, or funding for research from other pharmaceutical companies.

5 left

 5 ventricular ejection fraction, serum creatinine at baseline and for blood pressure, angina requiring hospitalisation and total cholesterol level in a time dependent manner. Drug use treated as a time-dependent indicator variable that could change continuously and interventions as a time-dependent indicator variable that could change once at the time of the first intervention concerned. ‡ Adjusted in addition for atrial fibrillation or irregular pulse at baseline and in a time-dependent manner.

Table 1 .

 1 Use of (combinations of) secondary preventive treatments at baseline and during follow-up in 7665 patients with stable angina

	Beta-blocker	Lipid-lowering drug	ACEI or ARB	Aspirin or other anti-platelet	N (%) prescribed at baseline	Mean % of follow-up time prescribed Nifedipine Placebo	P
	None of these			67 (0.9%)	1.2%	1.2%	0.9
	X				147 (1.9%)	1.7%	1.8%	0.7
		X			88 (1.1%)	1.5%	1.1%	0.1
			X		21 (0.3%)	0.3%	0.4%	0.3
				X	357 (4.7%)	4.1%	3.9%	0.6
	Any one drug class			613 (8.0%)	7.6%	7.2%	0.5
	X	X			277 (3.6%)	3.8%	3.4%	0.3
	X		X		49 (0.6%)	0.7%	0.7%	1.0
	X			X	1357 (18%)	13%	12%	0.6
		X	X		36 (0.5%)	0.6%	0.7%	0.5
		X		X	571 (7.4%)	9.1%	7.6%	0.01
			X	X	137 (1.8%)	1.9%	2.0%	0.8
	Any combination of two drug classes		2427 (32%)	29%	27%	0.02
	X	X	X		84 (1.1%)	1.5%	1.9%	0.1
	X	X		X	3062 (40%)	39%	35%	<0.001
	X		X	X	332 (4.3%)	3.7%	4.8%	0.005
		X	X	X	289 (3.8%)	4.3%	5.1%	0.05
	Any combination of three drug classes	3767 (49%)	48%	47%	0.2
	All four drug classes			791 (10%)	14%	18%	<0.001
	Total				6099 (80%)	77%	78%	0.28
		Total			5198 (68%)	74%	73%	0.35
			Total		1739 (23%)	27%	33%	<0.001
				Total	6896 (90%)	89%	89%	0.96
	ACEI, angiotensin-converting enzyme inhibitor; ARB, angiotensin receptor blocker

Table 2 .

 2 Cardiovascular therapy and deathAdjusted for age, gender, history of MI, HF and stroke, frequency of angina, number of significant lesions on coronary angiogram, smoking, total cholesterol, blood pressure, diabetes mellitus, 5 left ventricular ejection fraction, serum creatinine at baseline and for blood pressure, angina requiring hospitalisation, MI, HF, stroke and total cholesterol level in a time dependent manner. Drug use treated as a time-dependent indicator variable that could change continuously and interventions as a time-dependent indicator variable that could change once at the time of the first intervention concerned. ‡ Adjusted in addition for atrial fibrillation or irregular pulse at baseline and in a time-dependent manner.

		Comparing treatment with no treatment		Comparing treatment with no treatment
			at baseline			during follow-up
		Treatment	No. of deaths	Adjusted* HR	% of total	No. of deaths	Survival time	Adjusted † HR
		at baseline	(rate/100 pyrs)	(95% CI)	f-up used	(rate/100 pyrs)	ratio † (95% CI)	(95% CI)
	Beta-blocker
	Yes	5950 (80%)	463 (1.57)	1.07 (0.87, 1.30)	77.7	377 (1.31)	0.99 (0.93, 1.08)	1.01 (0.91, 1.09)
	No	1529 (20%)	124 (1.66)	ref. cat.		210 (2.55)	ref. cat.	ref. cat.
	Lipid-lowering drug
	Yes	5074 (68%)	342 (1.35)	0.78 (0.66, 0.92)	74.5	274 (1.00)	1.60 (1.45,1.72)	0.54 (0.49, 0.62)
	No	2405 (32%)	245 (2.11)	ref. cat.		313 (3.32)		ref. cat.
	ACEI or ARB:
	Yes	1694 (23%)	142 (1.73)	0.99 (0.81, 1.20)	30.1	176 (1.58)	1.17 (1.09, 1.25)	0.82 (0.75, 0.89)
	No	5785 (77%)	445 (1.55)	ref. cat.		411 (1.59)	ref. cat.	ref. cat.
	Aspirin
	or other anti-platelet:
	Yes	6715 (90%)	495 (1.49)	0.75 (0.60, 0.94)	89.2	378 (1.15)	1.76 (1.67, 1.98)	0.49 (0.42, .0.53)
	No	764 (10%)	92 (2.51)	ref. cat.		209 (5.23)	ref. cat.	ref. cat.
	Vitamin K antagonist:
	Yes	294 (4%)	45 (3.15)	1.38 (1.00, 1.90) ‡	5.4	73 (3.66)	0.87 (0.77, 0.99) ‡ 1.19 (1.02, 1.37) ‡
	No	7185 (96%)	542 (1.53)	ref. cat.		514 (1.47)	ref. cat.	ref. cat.
	Calcium antagonist:
	Yes	3740 (50%)	305 (1.65)	1.09 (0.93, 1.28)	44.8	163 (0.98)	1.06 (1.02, 1.12)	0.93 (0.87, 0.98)
	No	3739 (50%)	282 (1.53)	ref. cat.		424 (2.08)	ref. cat.	ref. cat.

pyrs, person years; HR, hazard ratio; CI, confidence interval; ref. cat., reference category; ACEI, angiotensin-converting enzyme inhibitor; ARB, angiotensin receptor blocker; CABG, coronary artery bypass grafting; PCI, percutaneous coronary intervention * Adjusted for age, gender, history of MI, HF and stroke, frequency of angina, number of significant lesions on coronary angiogram, smoking, total cholesterol, blood pressure, diabetes mellitus, left ventricular ejection fraction, and serum creatinine at baseline. †

Table 3 .

 3 Cardiovascular therapy and first occurrence of death, MI, stroke or HF Adjusted for age, gender, history of MI, HF and stroke, frequency of angina, number of significant lesions on coronary angiogram, smoking, total cholesterol, blood pressure, diabetes mellitus, left ventricular ejection fraction, and serum creatinine at baseline. † Adjusted for age, gender, history of MI, HF and stroke, frequency of angina, number of significant lesions on coronary angiogram, smoking, total cholesterol, blood pressure, diabetes mellitus,

		Comparing treatment with no treatment		Comparing treatment with no treatment
			at baseline			during follow-up
		Treatment	No. of patients	Adjusted* HR	% of total	No. of patients	Survival time	Adjusted † HR
		at baseline	(rate/100 pyrs)	(95% CI)	f-up used	(rate/100 pyrs)	ratio † (95% CI)	(95% CI)
	Beta-blocker:
	Yes	5950 (80%)	962 (3.41)	1.15 (1.00, 1.33)	77.6	889 (3.24)	0.94 (0.89, 0.98) 1.07 (1.03, 1.14)
	No	1529 (20%)	234 (3.26)	ref. cat.		307 (3.87)	ref. cat.	ref. cat.
	Lipid-lowering drug:
	Yes	5074 (68%)	746 (3.07)	0.84 (0.74, 0.94)	74.4	742 (2.82)	1.33 (1.26-1.37)	0.73 (0.70; 0.78)
	No	2405 (32%)	450 (4.07)	ref. cat.		454 (5.01)	ref. cat.	ref. cat.
	ACEI or ARB:
	Yes	1694 (23%)	293 (3.74)	0.97 (0.85, 1.12)	29.1	384 (3.73)	1.13 (1.08, 1.19) 0.88 (0.83, 0.92)
	No	5785 (77%)	903 (3.28)	ref. cat.		812 (3.24)	ref. cat.	ref. cat.
	Aspirin
	or other anti-platelet:
	Yes	6715 (90%)	1056 (3.32)	0.99 (0.83, 1.18)	89.3	942 (2.98)	1.23 (1.10, 1.26) 0.80 (0.78, 0.90)
	No	764 (10%)	140 (3.97)	ref. cat.		254 (6.70)	ref. cat.	ref. cat.
	Vitamin K antagonist:
	Yes	294 (4%)	67 (4.90)	1.01 (0.78, 1.30) ‡	5.1	103 (5.77)	1.04 (0.90, 1.12) 0.96 (0.88, 1.12) ‡
	No	7185 (96%)	1129 (3.32)	ref. cat.		1093 (3.26)	ref. cat.	ref. cat.
	Calcium antagonist:
	Yes	3740 (50%)	587 (3.31)	0.95 (0.85, 1.07)	45.2	448 (2.80)	1.05 (1.01, 1.09) 0.95 (0.91, 0.99)
	No	3739 (50%)	609 (3.45)	ref. cat.		748 (3.86)	ref. cat.	ref. cat.

pyrs, person years; HR, hazard ratio; CI, confidence interval; ref. cat., reference category; ACEI, angiotensin-converting enzyme inhibitor; ARB, angiotensin receptor blocker; CABG, coronary artery bypass grafting; PCI, percutaneous coronary intervention *

ACKNOWLEDGEMENTS

The contribution of investigators, committee members and other study personnel as listed elsewhere is gratefully acknowledged [2], as is the support of Bayer Healthcare AG. Bayer Healthcare AG had no role in the design and conduct of the study; or in the preparation, review and approval of this manuscript.

page 27 of 28 page 28 of 28 † Adjusted in addition for atrial fibrillation or irregular pulse at baseline and in a time-dependent manner.