J Chenot

D Melodelima

W A N'djin

Rémi Souchon

M Rivoire

J Y Chapelon

Intra-operative Ultrasound hand-held Strain imaging for the Visualization of Ablations Produced in the liver with a Toroidal HIFU Transducer: First In Vivo Results

Keywords: strain imaging, real-time, ultrasound, liver, HIFU, Elastography

The use of hand-held ultrasound strain imaging for intra-operative real-time visualization of HIFU ablations produced in the liver by a toroidal transducer was investigated. A linear 12 MHz ultrasound imaging probe was used to obtain radiofrequency signals. Using a fast crosscorrelation algorithm, strain images were calculated and displayed at 60 frames/s, allowing the use of hand-held strain imaging intra-operatively. Fourteen HIFU lesions were produced in 4 pigs. Intra-operative strain imaging of HIFU ablations in the liver was feasible owing to the high frame rate. The correlation between dimensions measured on gross pathology and dimensions measured on B-mode images and on strain images were R = 0.72 and R = 0.94 respectively. The contrast between ablated and non-ablated tissue was significantly higher (p<0.05) in the strain images (22 dB) than in the B-mode images (9 dB). Strain images allowed equivalent or improved definition of ablated regions when compared with B-mode images. Real-time intra-operative hand-held strain imaging seems to be a promising complement to conventional B-Mode imaging for the guidance of HIFU ablations produced in the liver during an open procedure. These results support that hand-held strain imaging outperforms conventional B-mode ultrasound and could potentially be used for assessment of thermal therapies.

INTRODUCTION

Imaging elastic properties of tissue for diagnosis of disease or for thermal treatment guidance is gaining attention since it is non-invasive and can provide new information regarding tissues. Among these techniques, elastography (strain imaging) allows the estimation of deformations in the medium when stress is applied [START_REF] Ophir | Elastography: a quantitative method for imaging the elasticity of biological tissues[END_REF]. The principle of strain imaging using ultrasound is the calculation of time delay between pre-and post-compression radiofrequency signals (RF), using a sliding window. Assuming a constant speed of sound in tissue, it is possible to estimate displacement. The strain image, called an elastogram, is obtained from the gradient of the displacement. Using this approach, many methods have been proposed for imaging the mechanical properties of tissues, and these may differ in a variety of ways, including the method used to stress the tissue, the nature of this stress (e.g. whether controlled stress or controlled displacement, and whether constant or time-varying), the property measured to describe the resulting tissue response, and the method used to measure this property [START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF]. In several applications, the load or displacement is applied externally to the surface of the tissue as in quasi-static compression [START_REF] Doyley | A freehand elastographic imaging approach for clinical breast imaging: system development and performance evaluation[END_REF][START_REF] Ophir | Elastography: a quantitative method for imaging the elasticity of biological tissues[END_REF][START_REF] Plewes | Visualization and quantification of breast cancer biomechanical properties with magnetic resonance elastography[END_REF] or in dynamic vibration excitation [START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF][START_REF] Taylor | Three-dimensional sonoelastography: principles and practices[END_REF]. Internal physiologic motion has also been utilized, for cardiac [START_REF] Varghese | Ultrasonic imaging of myocardial strain using cardiac elastography[END_REF], liver [START_REF] Kolen | Characterization of cardiovascular liver motion for the eventual application of elasticity imaging to the liver in vivo[END_REF]) and arterial strain imaging [START_REF] Arnett | Arterial stiffness: a new cardiovascular risk factor?[END_REF]. Acoustic radiation force has also been used to generate an internal load, non-invasively [START_REF] Bercoff | Supersonic shear imaging: a new technique for soft tissue elasticity mapping[END_REF][START_REF] Fatemi | Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission[END_REF], Melodelima et al. 2007a[START_REF] Melodelima | 64-element intraluminal ultrasound cylindrical phased array for transesophageal thermal ablation under fast MR temperature mapping: an ex vivo study[END_REF][START_REF] Nightingale | Observations of tissue response to acoustic radiation force: opportunities for imaging[END_REF].

Given previous demonstrations of stiffness contrast of thermal lesions created by radiofrequency ablation [START_REF] Techavipoo | Monitoring stiffness changes in lesions after radiofrequency ablation at different temperatures and durations of ablation[END_REF][START_REF] Fahey | In vivo guidance and assessment of liver radio-frequency ablation with acoustic radiation force elastography[END_REF][START_REF] Varghese | Elastographic imaging of thermal lesions in the liver in vivo following radiofrequency ablation: preliminary results[END_REF], methods of imaging tissue elasticity may also hold promise for guiding and assessing ablation procedures, particularly for High Intensity Focused Ultrasound (HIFU) treatments (Kallel et al. 1999[START_REF] Souchon | Visualisation of HIFU lesions using elastography of the human prostate in vivo: preliminary results[END_REF]. HIFU treatments are already guided by ultrasound scanner for numerous applications [START_REF] Illing | Visually directed high-intensity focused ultrasound for organ-confined prostate cancer: A proposed standard for the conduct of therapy[END_REF][START_REF] Pichardo | New integrated imaging high intensity focused ultrasound probe for transrectal prostate cancer treatment[END_REF], Wu et al. 2004). Therefore, strain imaging could provide additional information to conventional B-mode imaging since elastograms can now be calculated in real-time [START_REF] Pesavento | New real-time strain imaging concepts using diagnostic ultrasound[END_REF][START_REF] Zahiri-Azar | Motion estimation in ultrasound images using time domain cross correlation with prior estimates[END_REF][START_REF] Zhu | A modified block matching method for real-time freehand strain imaging[END_REF]. In a previous study (Melodelima et al. 2007b) we have described the use of a toroidal high intensity focused ultrasound (HIFU) device for treating colorectal liver metastases. The selected approach for the toroidal HIFU treatment is surgical laparotomy. The main advantage of performing the HIFU treatment during an open procedure is to make it possible to associate HIFU with hepatic resection. Indeed, hepatic resection, eventually combined with radiofrequency ablation (RFA) or cryosurgery ablation (CSA), is the gold standard in the treatment of colorectal liver metastases [START_REF] Antoniou | Meta-analysis of clinical outcome after first and second liver resection for colorectal metastases[END_REF][START_REF] Pawlik | Combined resection and radiofrequency ablation for advanced hepatic malignancies: results in 172 patients[END_REF][START_REF] Rivoire | Combination of neoadjuvant chemotherapy with cryotherapy and surgical resection for the treatment of unresectable liver metastases from colorectal carcinoma[END_REF]. Compared with RFA or CSA, HIFU, could represent a promising alternative for treating colorectal liver metastases. The principal interest in HIFU lies in the possibility of treating metastases for which the use of other destruction techniques is contraindicated, notably for metastases located near a large vascular structure [START_REF] Parmentier | High-intensity focused ultrasound ablation for the treatment of colorectal liver metastases during an open procedure: study on the pig[END_REF]). This approach allows for treatment to all regions of the liver without penetrating the hepatic capsule. Furthermore, such an intra-operative approach ensures the protection of surrounding organs and eliminates the risk of lesions in other organs. Using this device, it was previously reported that the demarcation between ablated and non-ablated tissues was apparent using B-Mode images (Melodelima et al. 2009b).

In the current study, we hypothesize that ultrasound hand-held strain imaging may be useful in enhancing the guidance of ablations produced using this toroidal HIFU transducer during an open procedure for the treatment of liver metastases. Strain imaging may be useful during several stages of HIFU procedures, mainly intraprocedure guidance (demarcation between ablated and non-ablated tissues as well as detection of tumors not discovered on preoperative imaging) and postprocedure outcome assessment. Ultrasound hand-held strain imaging for guiding HIFU treatments in the liver during an open procedure has never been investigated. If successful, this imaging modality could also be used in the future to provide visualization of the ablated region for other thermal techniques like radiofrequency or cryosurgery.

Experiments were conducted in vivo in porcine livers. Taking advantage of the open procedure approach that allows ultrasound imaging at high frequency (12 MHz), strain images were produced by manually and directly stressing the liver with an ultrasonic imaging probe.

A fast algorithm allows calculating and displaying strain images at the frame rate of the ultrasound scanner. Strain imaging was implemented on a modified ultrasound scanner, allowing for combined real-time sonography/strain imaging. We present a feasibility study on use of the technique of combined hand-held sonography/strain imaging to detect HIFU ablations intra-operatively.

MATERIALS

A. Animals

Experiments were conducted in vivo in pig liver. There is no established liver tumor model in pigs due to the absence of porcine liver tumor cell lines. However, the pig is an ideal animal to study the different treatment options available for liver tumors because of its size and similar physiology to humans. Therefore, in this study, trials were conducted on normal liver tissues in 4 female Landrace pigs, 12-14 weeks old, with an average weight of 27.6 ± 1.5 kg

B. HIFU probe

The HIFU device was composed of a driving equipment similar to that previously reported [START_REF] Melodelima | 64-element intraluminal ultrasound cylindrical phased array for transesophageal thermal ablation under fast MR temperature mapping: an ex vivo study[END_REF]) and a sterilized treatment probe also previously described (Melodelima et al. 2009b). Briefly, the ultrasound fields were generated using eight ultrasound emitters operating at a frequency of 3 MHz and distributed according to a toroidal geometry with a diameter of 70 mm and a radius of curvature of 70 mm. The use of a 3 MHz HIFU transducer is not typical for abdominal applications, which usually use lower frequencies to allow for application through the skin/subcutaneous fat. However the use of the device during an open procedure allows the use of a higher frequency. Using numerical simulations, the choice of 3 MHz was judged to be the best compromise between heat deposition and no cavitation effects.

Due to the toroidal shape of the transducer, the focal zone was slightly conical. The focal zone observed in the focal plane was a ring. The -6 dB beam axial focal length of the produced focal zone was 28.8 mm. The -6 dB outer diameters at the two extremities of the produced conical focal zone were 19.5 and 18.0 mm, respectively. These pressure fields were described in detail in a previous study (Melodelima et al. 2007b). The ablation zone (conical in shape) induced in 40 seconds has a diameter of 2 cm, a major axis of 2.5 cm, and was placed at 7 cm from the emitting surface to enable treatment of deep regions in the liver. A 7.5 MHz ultrasound imaging probe (Vermon, Tours, France) was placed in the centre of the device and connected to a B-K HAWK 2102 EXL scanner (B-K Medical, Herlev, Denmark) to guide the treatment. The HIFU probe was brought into contact with the liver using a sterile ultrasound

METHODS

A. Treatment and imaging procedure

Twenty-four hours before treatment, the food intake of the animal was restricted with free access to water. Premedication was performed 30 minutes before anesthesia using an intramuscular injection of ketamine (Kétalar®, Parke-Davis, Courbevoie, France) at a dosage of 20 mg/kg. A catheter (Willy Rush, Waiblingen, Germany) was placed in an auricular vein.

A dose of 5 mg/kg of Propofol (Diprivan®, AstraZeneca, Rueil Malmaison, France) was administered just before intubation. Ventilation was performed with a Servo 900B® type respirator (Siemens Life Support Service, Solna, Sweden) at 10 L/min and at a frequency of 18 to 20 respiratory movements per minute with a 50% air/oxygen mix. Anesthesia was maintained using a slow intravenous injection of Propofol (20mg/kg/h) and sufentanil (10µg/kg/h; Sufenta®, Jassen-Cilag, Berchem, Belgique) in a continuous perfusion of physiological saline solution. Hydration was provided by an isotonic perfusion of physiological saline solution at 9%. The animals were monitored continuously by evaluating hemodynamic status and blood oxygen saturation.

A 25 cm median laparotomy was performed from the xyphoid process. Pads were placed below and on the sides of the liver and were utilized in order to protect adjacent organs from possible unwanted lesions. The HIFU probe was held by hand at the surface of the liver. The region to be treated was located using the 7.5 MHz integrated ultrasound imaging probe. Each HIFU exposure was performed during apnea to avoid liver movement. Apnea periods always began 5 seconds before sonication and lasted 45 seconds. The purpose of this maneuver was to limit the movements of the liver, which could have been deleterious to treatment quality.

Fourteen HIFU ablations were performed using an underrated acoustic power of 50 watts, and the exposure duration was 40 seconds. The exposure conditions are derived from preliminary in vivo studies (Melodelima et al. 2009b). At the end of the treatment the laparotomy was closed in two planes. After the treatment, post-operative analgesia was administered to each animal by a fentanyl patch at a 100-μg dosage every three days (Durogesic ®, Janssen-Cilag, Berchem, Belgium).

B. Strain images acquisition

Imaging during the surgical intervention would of course have been of great interest.

However, for this preliminary study, the imaging procedure was approved by our local institutional review board to be performed at the end of a protocol approved for the above described HIFU treatment. Therefore, strain images were acquired 21 days after the HIFU treatment, during a second open procedure. One advantage is that it allows a direct comparison between the images and gross pathology since lesions were examined just after the imaging procedure.

The animals were prepared as described above. After laparotomy, the 12 MHz ultrasound probe was housed in a sterile polyurethane envelope (Civco, Kaloma, Iowa, USA) and used to acquire strain images. Strain was induced manually by pushing and pulling the imaging probe at the surface of the liver. The amplitude and speed of the displacement were adapted by the operator as a function of the strain results visualized in real-time. The liver was precompressed manually by about 1 -3 mm to ensure good contact between the imaging probe and the organ. During the strain imaging procedure, the amplitude and speed of the axial displacement were adapted by the operator as a function of the strain images visualized in real-time. Axial displacements of less than 0.5 mm were applied around this initial position of the imaging probe to produce strain images that were judged satisfactory. The thickness of the livers used in this study was between 2 and 4 cm. Therefore the applied stress represents a maximal strain of 1% using the maximal scan depth of 5 cm used in this study. This value was a compromise between the stress that can be applied at the surface of the liver without damaging the organ or adjacent viscera and the quality of the resulting strain images.

Ultrasound data was captured continuously and strain images were processed and displayed in real-time. Displacement estimation was performed using a time-domain cross-correlation algorithm similar to that described by [START_REF] Zahiri-Azar | Motion estimation in ultrasound images using time domain cross correlation with prior estimates[END_REF]. Strain was estimated using the staggered strain estimator [START_REF] Srinivasan | Elastographic imaging using staggered strain estimates[END_REF]. The displacement estimation (tracking) window length was set to 0.6 mm (approximately 4 pulse lengths) and the size of the strain estimation kernel ("staggering kernel") was increased until the subjective visibility of strain at the ablation location was maximized. A staggering kernel length of 0.9 mm was deemed satisfactory. The window shift was chosen as small as possible (0.3 mm) while still allowing the images to be produced at a speed of 60 frames per second (fps). The same values were used for processing all data sets. Sonograms, correlation, displacements and strains fields were displayed simultaneously on the screen at a rate of 60 fps. Care was taken by visual observation of the real-time images during all acquisitions to minimize the lateral and elevational displacements (caused by transducer motion), thus optimizing correlation between pre and post compression RF images.

When strain images were deemed satisfactory, the acquisition system was paused and a cineloop containing the last 200 consecutive RF images was saved (corresponding to a recording time of 3.7 seconds). The diameter and the depth of all ablations were measured on strain images and B-mode images.

After the acquisition of the strain images, the pigs were euthanized by a single intravenous 0.3 ml/kg injection containing embutramide, mebezonium and tetracaine (T61®, Intervet, Beaucouze, France). A total hepatectomy was then done in order to visually inspect the ultrasound effects. The lesions were cut along the imaging axis. The diameter and the depth of all ablations were measured so that their dimensions were defined by a boundary (defined as a lighter color) outlining the ablated region. It is known that the edge of the red tissue is really the boundary between the normal tissue and that treated by thermal ablation. This narrow transition from treated to normal tissue was previously described and confirmed by histological observations to be approximately 200 μm (Melodelima et al. 2009a[START_REF] Vykhodtseva | Induction of apoptosis in vivo in the rabbit brain with focused ultrasound and Optison[END_REF]. Dimensions were measured using a caliper leading to an uncertainly of around 1 mm. The examiner measuring the dimensions of the ablations on gross pathology was different from the examiner measuring dimensions on B-Mode imaging and strains images.

The two examiners were blinded to the results of each other.

C. Data analysis

Data are presented as mean ± standard deviation [min -max]. Pearson's correlation was used to determine the relationship between the dimensions of the lesions measured on gross pathology, strain images and sonograms. The quality of strain images was assessed by computing the contrast-to-noise ratio (CNRe) and the signal-to-noise ratio (SNRe), which were defined as follows [START_REF] Chaturvedi | Testing the limitations of 2-D companding for strain imaging using phantoms[END_REF]): Contrast calculations were also performed using regions of interest (ROIs) that were selected both inside the ablated zone and in adjacent untreated tissues. The reason for choosing a location near the ablated zone for the untreated ROI is that visual contrast demarcation is generally made by the human eye for region located close to each other. ROI location was chosen by the consensus of two observers after examination of B-Mode and strain image data. ROIs within ablations were chosen to be as large as possible while confidently remaining within ablation boundaries. For each data set, the ROIs used for B-mode and strain image contrast calculations were identical. Contrast was calculated using equation 3:

         T A C   log * 20 (3)
Where A  and T  are the mean B-mode or strain pixel magnitude inside the ablated and the untreated tissues, respectively. In the B-mode images, the HIFU ablations were seen as a hypoechoic region with a central hyperechoic zone. Therefore, three different contrast values were computed: (i) the contrast between untreated liver and the entire ablation zone, (ii) the contrast between the untreated liver and the central part of the ablation (hyperechoic zone) and (iii) the contrast between the untreated liver and the edges of the ablation (hypoechoic zone). Contrasts were compared using Student's t test. The significance level for all tests was fixed at p < 0.05.

RESULTS

Fourteen HIFU ablations were observed on sonograms and strain images. The mean diameter of ablations measured on gross pathology was 11.4 ± 3.3 mm [6.0 -18.0] with a mean depth of 15.6 ± 6.2 mm [8.0 -30.0]. The uncertainly of the measurement was around 1mm because of the graduation of the caliper. The correlation between the dimensions measured on strain images and those measured on gross pathology was R=0.94. The correlation between the dimensions measured on B-mode images and those measured on gross pathology was R=0.72.

This value is consistent with our previous experiments with B-mode monitoring only (Melodelima et al. 2009b) where a coefficient R=0.67 was found. The correlation between the dimensions measured on strain images and on sonograms was 0.77. Figure 2 In two strain images, it was possible to observe the presence of a blood vessel (figure 5(a)).

During the compression cycles, the veins were flattened and then dilated. They were associated with decorrelation and significant displacement estimation error. This phenomenon could be observed before and after an ablation [START_REF] Fan | Sixth International Conferenceon the Ultrasonic Measurement and Imaging of Tissue Elasticity© 68[END_REF]). These random displacement estimates induced very visible zones (artefactual strain) in strain images. This particularity can be utilized, for example, to verify the presence of blood vessels before a HIFU ablation is created. In addition, two abscesses caused by ablations on biliary tracts were also imaged by hand-held strain imaging (figure 5(b)).

In two cases, strain images allowed more accurate evaluation of the ablation dimensions. In these two cases, the HIFU ablation was inhomogenous. For example, Figure 6 In that particular case, the strain images showed better correspondence with the shape observed on gross pathology.

DISCUSSION

Intra-operative ultrasound hand-held strain imaging of HIFU ablation in the liver was feasible.

Combined hand-held sonography/strain imaging is a straightforward imaging protocol that allows for the strengths of both modalities to be used in a highly complementary manner.

Real-time hand-held sonography/strain imaging was particularly well-suited for assessing intra-operative HIFU ablation and good correspondence was found between measurements from pathology and strain imaging. Thus, this approach may provide additional relevant information during HIFU treatments in the liver during an open procedure without sacrificing the cost, convenience and portability benefits of conventional ultrasound monitoring. In this preliminary study, the imaging procedure could only be performed 21 days after the HIFU treatment. Nevertheless, we envision that the method will be feasible during or immediately after ablation. For this purpose, future work will be conducted with the ultrasound imaging probe integrated in the HIFU device.

Although the hand-held method used does not allow compressions to be as reproducible as the ones produced using motorized arms, the real-time display allowed the operator to visualize the effects of probe positioning and to adjust the strength and speed of axial compression in real-time. In addition, the operator could simultaneously visualize the B-mode image, the displacement, the correlation and the strain images.

The key elements to successful scanning were fast frame rate acquisition (60 fps), and realtime visual feedback of B-mode and strain images that guide the probe positioning and compression direction [START_REF] Chandrasekhar | Elastographic image quality vs. tissue motion in vivo[END_REF][START_REF] Hall | In vivo real-time freehand palpation imaging[END_REF]. More than the overlap, the frame rate is the most important factor for the quality of strain images. In our initial experiments (unpublished) we used a slower system running at 8 strain images per second.

This frame rate was too slow for hand-held strain imaging in the liver due to high sensitivity to motion artifacts and also to axial displacements that were produced too rapidly. The threshold to have acceptable strain images seems to be at least of 20-30 images/s [START_REF] Chandrasekhar | Elastographic image quality vs. tissue motion in vivo[END_REF][START_REF] Hall | In vivo real-time freehand palpation imaging[END_REF] but further studies are required to thoroughly evaluate this limit. In this study, a frame rate of 60 fps allowed the operator to see the effect of axial compressions immediately and to adjust the induced displacement to improve the quality of the resulting strain images. Additionally, high frame rate minimized the deleterious effects of movements (breathing and swiping of the probe) on strain images quality. At 60 fps, the effect respiratory motions were negligible and the resulting strain images were due to the stress applied by the probe at the surface of the liver. Faster movements, such as the ones caused by cardiac activity were localized around veins. The induced strain was about 5% and was localized 1 mm around the vessel. This has not compromise the strain imaging of HIFU ablations.

During compression of a non-homogenous medium, the different mediums are displaced in a non-uniform manner, leading to a phenomenon of a border effect between the mediums. This effect is visible as a discontinuity in the strain images. It allows the operator to visualize the ablations even if the elasticity of the ablation is close to the elasticity of the untreated liver.

Indeed, in this study, there was one case were the HIFU ablation contained two zones of different elasticity. One was close to the untreated tissues (Figure 6). Based on a discontinuity visible in the corresponding strain images, it was possible to estimate correctly the dimensions of the ablation. At the opposite, it was not possible to determine the dimension of this ablation on B-Mode images. However this is a very specific and illustrating case and no definitive conclusions can be drawn from this example. In some cases, biliary ducts were ablated by the HIFU exposure, creating abscesses. The results of our study show that the elasticity of abscesses was comparable to that of the liver causing the CNRe of these strain images to be very low. Nevertheless, it was possible to image abscesses due to the edge effect previously described. The displacement images were continuous within the liver and within the abscess, with a characteristic discontinuity at the frontier. The contour of the abscess was more rigid compared with the liver, allowing the abscesses to appear on strain images.

In the past, in vitro experiments conducted by other investigators have shown that tissue damage induced by high-intensity focused ultrasound in excised canine liver specimens reveals high correlation (R 2 =0.93) between elastographic and gross pathological measurements [START_REF] Righetti | Elastographic characterization of HIFU-induced lesions in canine livers[END_REF]. Our results show that high correlation (R=0.94, i.e. R 2 =0.88) can be obtained as well but under in vivo conditions. The sonographic contrast between the untreated liver and the zones treated by HIFU allowed satisfactory identification of the ablations boundary in sonograms (R=0.72). Measurements performed on strain images showed a superior agreement with pathology (R=0.94).

In two cases, the sizes of ablations were larger than the others (Fig. 2). Previous studies shown that such dimensions (about 30mm) are usual (Melodelima et al. 2009b) and created in deeper liver regions. The correlation coefficients obtained when these two points are excluded were R=0.87 for strain images and R=0.80 for B-Mode images. Hence the inclusion/exclusion of these particular points did not significantly affect correlation. Regarding correlation results it is noteworthy that the dispersion is less important for the measurements made using strain images. This can be explain by the fact that lesion boundary are visible more precisely on strain images.

Despite the limited number of samples, this study demonstrated that in some cases strain images can allow visualization of the treated zone with better accuracy, and particularly with a more homogenous contrast. Therefore, with combined hand-held sonography/strain imaging, HIFU ablation could be guided with ultrasound, eventually using real-time strain imaging when increased contrast or further information about the treated region are required.

This method would likely increase operator confidence in assessing the ablation zone, compared to the use of B-Mode images alone. This combined approach seems promising and has also been suggested for the guidance and assessment of liver radiofrequency ablation with acoustic radiation force elastography (ARFI) [START_REF] Fahey | In vivo guidance and assessment of liver radio-frequency ablation with acoustic radiation force elastography[END_REF].

Due to the thickness of porcine liver (2 -4 cm), ablations produced with this device generally extend from the surface to the bottom of the liver as previously described (Melodelima et al. 2009a). As a consequence, most HIFU lesions imaged in this study were directly in contact with the imaging probe. Human liver is thicker and it is expect that HIFU lesions will be deeper, most often behind a superficial untreated area. In such circumstances the HIFU lesions would be totally surrounded by soft untreated tissues. This should reinforce the contrast of the ablated area (Doyley et al. 2000). Hence we hypothesize that under these conditions, the quality of the imaging results would be similar or better. However, future clinical studies will need to be conducted to confirm the validity of this hypothesis.

Another limitation in this study is the relatively low SNRe compared with in vitro studies assessing relative stiffness of lesions created using high intensity focused ultrasound (HIFU) in the liver [START_REF] Kallel | The feasibility of elastographic visualization of HIFU-induced thermal lesions in soft tissues. Imageguided high-intensity focused ultrasound[END_REF]. This was expected because of unwanted displacements encountered in vivo. In addition to the above mentioned breathing movements, the surface of the liver was not flat and the probe can slightly slide inducing lateral displacement during the applied axial compression.

In order to propose a therapeutic method that uses HIFU and adapted to the treatment of colorectal liver metastases the treatment was performed during an open procedure. This choice allowed associating the HIFU destruction technique with hepatic resection, which appears to be the best current strategy in curative treatment methods for liver metastases.

Moreover, general anesthesia makes it possible to use apnea in order to create precise HIFU ablation in the liver. The intra-operative approach has also been suggested for radiofrequency ablation (RFA) and cryosurgery (CSA). Therefore the imaging results presented here could also be used for RFA or CSA. Nevertheless, extracorporeal use of HIFU is possible as has been clinically demonstrated for other disease [START_REF] Illing | The safety and feasibility of extracorporeal high-intensity focused ultrasound (HIFU) for the treatment of liver and kidney tumours in a Western population[END_REF][START_REF] Kennedy | High-intensity focused ultrasound for the treatment of liver tumours[END_REF]) and future work will concentrate on the potential of high frame rate for hand-held extracorporeal elastography.

In conclusion, we have presented initial results from an in vivo study investigating the performance of combined hand-held sonography/strain imaging in real-time for use with an intra-operative toroidal HIFU device developed for the treatment of liver metastases in conjunction with surgery. Initial results demonstrate the feasibility of this application. Realtime hand-held strain imaging allowed accurate measurement of the lesion dimensions, and was a useful complement to conventional sonograms. Further investigation is required to determine the full potential of intra-operative ultrasound strain imaging for visualizing tumor masses and the zones of the tumors that are effectively treated. Our results suggest that strain imaging may also provide useful information for the guidance of other thermal-based procedures (RF ablation and cryotherapy)

[

  min. 26.2max. 29.5]. The experiments were conducted at the Institute of Experimental Surgery of the Léon Bérard Center, after local Institutional Review Board approval. These experiments conformed to the requirements of the local Office of Animal Experimentation and were in accordance with the legal conditions of the National Commission on Animal Experimentation. The animals were kept on-site seven days before the start of the experiments.

  cooling and coupling liquid (Ablasonic®, Edap-Technomed, Vaulx en Velin, France) contained in a sterile polyurethane envelope (Civco, Kaloma, Iowa, USA). The software developed in our laboratory allows the user to target the HIFU treatment zone from a conventional B-mode image. The user interface allowed the visualization of the region to be coagulated by ultrasound exposure. The position of the HIFU focal region was superimposed on the ultrasound image. In this manner, the user can precisely position the treatment zone within the hepatic tissue. C. Elastography system Strain images were obtained using a linear 12 MHz ultrasound probe (Model 8805, B-K Medical, Herlev, Denmark) connected to a B-K ultrasound scanner (Hawk 2102 EXL, B-K Medical, Herley, Denmark). The scanner was modified in order to allow acquisition of RF lines by a computer (Dell Precision 690 Workstation with 2 quad-core Intel Xeon E5345 processors) via an analogue/digital converter (GaGe Octopus CS8327) having a sampling frequency of 40 MHz. A schematic diagram of the experimental apparatus is shown in figure 1.



  represent the mean strain of the ablated zone and in the untreated tissues respectively, and 2 A  and 2 T  represent the variances in strain in the ablated zone and untreated tissues respectively. represent the mean strain and the standard deviation in the ablated zone.

  (a) and figure 2(b) shows the correlation between the dimensions of the ablations measured on strain images, B-mode images and the actual dimensions measured on gross pathology.The mean CNRe of the strain images was 11.4 ± 3.1 [7.7 -18.2] with a mean SNRe of 2.8 ± 1.1 [1.4 -4.7]. The mean contrast between the treated zone and the untreated zone was 22.0 ± 10.3 dB[9.1 -40.6] in the strain images versus 9.3 ± 5.5 dB [0.1 -20.0] in B-mode images (when the entire ablation appearing in B-mode images was taken into account). On B-mode images the mean contrast between the treated zone and the untreated zone was 3.5 ± 3.0 dB [0.0 -10.0] when only the hyperechoic zone of the lesion was taken into account, and 14.4 ± 5.5 dB[1.6 -21.3] when only the hypoechoic zone of the lesion was taken into account. All these results are summarized in table 1. Relationships between contrasts measured on strain images and those measured on sonograms are presented in figure3. The contrast of the entire lesion observed on strain images was significantly higher when compared to the different contrast observed on sonograms (p<0.05 in all cases).

Fig. 4a shows

  Fig.4ashows an example of a single ablation produced in 40 seconds using the toroidal device and imaged using conventional B-Mode imaging. Figure4(b) shows the corresponding strain image. In figure4(a), the ablation appears as a hypoechoic cone containing a conical hyperechoic zone. These features are typical of HIFU ablations created in the liver with the toroidal device(Melodelima et al. 2009b). The strain images display the treated zones with a more homogenous appearance (figure 4(b)) and corresponding well to the actual dimensions of the ablation (figure 4(c)).

  figure 6(c) -shows cavities and dark zones in the treated area. In the corresponding B-mode image (figure 6(a)), only half of the HIFU ablation was visible. The sonogram shows two hypo and hyper echogenic zones with a slight shadowing behind, but does not match the dimensions measured on gross pathology. In the strain image (fig. 6(b)), it is possible to distinguish two regions: a cone formed by the dark zone supported by a triangular point. In

Figures captionsFigure 1 :

 1 Figures captions

Figure 2 :

 2 Figure 2: Relation between sizes measured (A) on pathology and on strain images and (B) on pathology and sonograms. Significant correlation was found in both cases.

Figure 3 :

 3 Figure 3: Comparison using Student tests of the contrasts measured on sonograms and strain images. Raw data and mean ± standard deviation are shown. The contrast of the entire lesion observed on strain images was significantly higher when compared to the contrast of the entire ablation observed on sonograms (p<0.05).

Figure 4 :

 4 Figure 4: Single ablation produced in 40 seconds using the toroidal HIFU device. (A) Ablation imaged using B-mode imaging (full arrow = 15 mm, doted arrow = 17 mm, contrast = 5.6 dB), (B) corresponding strain image (full arrow = 15 mm, doted arrow = 18 mm, contrast = 22.5 dB, CNRe = 9.3, SNRe = 2.1) and (C) corresponding gross pathology (full arrow = 16 mm, doted arrow = 18 mm).

Figure 5 :

 5 Figure 5: Strain images have allowed the visualization of veins (in circle) close to the ablated zone (A) and of abscesses (B), black and white representing low (0%) and high (5%) strain, respectively in both images.

Figure 6 :

 6 Figure 6: In that particular case, only a portion of the HIFU ablation was visible on B-mode images (full arrow = 15 mm, doted arrow = 10 mm, contrast = 1.9 dB) (A). The corresponding strain image (B) (full arrow = 14 mm, doted arrow = 21 mm, contrast = 9.1 dB, CNRe = 1.9, SNRe = 1.8) is more closely related to (C) the actual ablation observed on gross pathology (full arrow = 15 mm, doted arrow = 20 mm) where the HIFU ablation has different appearances, corresponding to a zone where the activity of cavitation and/or ebullition may have been significant (darker area and cavities at the top of the image).

  Figure 3.

  Figure 5.

Table 1 :

 1 CNRe, SNRe, contrast and dimension measured on elastograms, sonograms and pathology.

	Mean Standard deviation

ACKNOWLEDGMENT

The authors wish to thank the staff of the laboratory for experimental surgery for their aid in the animal study. This work was supported by funding from the Cancéropôle Lyon Auvergne Rhône Alpes (PDC 2006.4.8), EDAP TMS (N° EDAP 88513 B45) and by NIH/NIBIB Program Project Grant P01-EB-02105 to the University of Texas Health Science Center at Houston.


Mohamed Achibi
email: mohamed.achibi@snecma.fr

Michel Broniatowski
email: michel.broniatowski@upmc.fr

BIVARIATE COX MODEL AND COPULAS

Keywords: . 2000 Mathematics Subject Classification. 62H05;62N05 Cox models, positive quadrant dependence, archimedean copula, extreme value copulas, asymmetric logistic copula, frailty models

Bivariate Cox model and copulas

Introduction

The aim of this paper is to present a new description for bivariate dependence. It extends the proportional hazard (PH) model and is relevant in various fields of biostatisctics and industry. Denote z an environmental covariate and consider two positive random variables X and Y with absolutely continuous survival functions (1.1) F z (x) = P(X > x; z) and

(1.2) G z (y) = P(Y > y; z)

under the covariate z. The joint distribution of (X, Y ) is modelled through (1.1) and (1.2) and through the conditional s.d.f

(1.3) G z x (y) = P(Y > y|X > x; z). The precise setting of our model is as follows: we assume that the survival function of X depends on z through a PH model, i.e

(1.4) λ z X (x) := -

d dx log F z (x) = λ 0 X (x)Φ(z)
for some positive function z → Φ(z) and some baseline hazard λ 0 X (x) corresponding to z = 0. Denote

(M1) λ z X (x) = λ 0 X (x)Φ(z) λ z Y |X>x (y) = λ 0 Y |X>x (
y)Ψ(z) Other models for bivariate dependence have been defined, for example [START_REF] Clayton | A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence[END_REF][START_REF] Clayton | Multivariate generalizations of the proportional hazards model[END_REF][START_REF] Oakes | Bivariate survival models induced by frailties[END_REF]. In [START_REF] Clayton | A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence[END_REF] where joint lifetimes of sons and fathers are considered, a Cox model is used to handle the role of the covariate upon the margins, and the association between the margins is described through the fact that the local association measure θ (x, y) := λ (y| X = x) λ (y| X > x)

is independent upon x and y and upon z. This index expresses the influence of parental history of a given disease upon the incidence in the offspring. X is the lifetime of the father, while Y describes that of the son. It is assumed that association arises because the two members of a given pair share some common influence and not because one lifetime influences the other. In fact these models are of frailty type, and the margins are independent conditionally upon the covariate. More globally the standard model building strategy is to have marginal survival functions and a copula for their dependency; see f.i. [START_REF] Choi | Accelerated life regression modelling of dependent bivariate Time-to-Event data[END_REF] or [START_REF] Saïd | Score tests for independence in semiparametric competing risks models[END_REF]. In [START_REF] Lindström | Familial concordance in cancer survival: a swedish population-based study[END_REF] a study on the familial concordance in cancer survival based on a Swedish population showed that cancer specific survival in parents predicts survival for the same cancer in their children. The risk in dying in children in relation to parental survival was modelled by use of two PH models; first parental survival was modelled and next survival risk in children in relation to parental survival was assessed; thus in this case the lifetime of the father influences the lifetime of the son whatever the level of the covariate (here z represents the different cancer sites), however in relation with it. In this case Model (1.6) seems to be natural. Setting x = 0 in (1.5) shows that Y follows a PH model so that (1.6) is PH on both components. The copula of the couple of r.v's (X, Y ) for a given z can be written in terms of the values of Φ(z) and Ψ(z), as seen further. The unusual feature of this model is positive in a number of cases, since z acts on the margins, but also specifically on their dependence. The claim that the goal of copula modelling is to distinguish the parameters for the dependency from those associated to the marginal models cannot be considered as a general principle. Other authors have considered cases when the association of the margins is specifically related to their distributions; see f.i. [START_REF] Gupta | Reliability studies of bivariate distributions with exponential conditionals[END_REF]. In conjunction with (1.6) it is worth noting that direct approaches based on regression type models cannot satisfy our purpose. Indeed consider for example a model defined through

X = r(z, U) Y = s(z, V )
with r(z, .) and s(z, .) strictly increasing for all z. Then following [START_REF] Nelsen | An introduction to copulas[END_REF], Theorem 2.4.3, (X, Y ) has the same copula as (U, V ) for all z, which implies that the covariate z plays no role in the dependency of X and Y . Introducing a new model imposes to determine its range of applicability; it will be shown that (1.6) is adapted for positive quadrant dependence (PQD) between the margins, a concept which is recalled in Section 2. Stating that the margins follow a PH (typically Cox) model can be checked using standard tools (see f.i. [START_REF] Grambsch | Proportional hazards tests and diagnostics based on weighted residuals[END_REF]. PQD property can be tested through a Kolmogorov-Smirnov test (see f.i. [START_REF] Scaillet | A Kolmogorov-Smirnov type test for positive quadrant dependence[END_REF]. Let us now show the main results which we present in connection with model (1.6):

(1) The T P 2 class of sdf's is a subclass of the Positive Quadrant Dependence (PQD) sdf class and is stable under the model which is properly defined when the hazard baseline H 0 is T P 2 . This class appears quite naturally as the one under which the model is properly defined, and it is appropriate for the description of positive dependence between its margins. Definitions of T P 2 and PQD properties are given in Section 2.

(2) Since the T P 2 property of a multivariate sdf refers only to its copula, model (1.6) describes the changes of the baseline copula induced by the covariate. Also this implies that the model is valid independently of marginal distributions. Only the structure of dependence is involved in the domain of validity of the model. (3) Two main classes of copulas are stable under the model namely: when the baseline bivariate copula is in such a class, so is the copula for all value of the covariate z. The class of extreme values copulas (evc) enjoys this property. The class of extended archimedean copulas is also stable under the model. This class results as a special by-product of a technique intended to produce asymmetric copulas due to Genest et al.; see [START_REF] Frees | Understanding relationships using copulas[END_REF] and [START_REF] Liebscher | Construction of asymmetric multivariate copulas[END_REF]. The so-called class of logistic asymmetric copulas (see [START_REF] Tawn | Bivariate extreme value theory: models and estimation[END_REF], which is a simple extension of the Gumbel family of copulas, enjoys an important role in the present model.

It is stable under the model and admits a simple parametrization. The covariate z acts in an adaptive way when the value of the covariate is changed. It is the only bivariate distribution in the class of frailty models which enjoys such properties in the model. This paper is organized as follows. In Section 2 we briefly recall the necessary background from bivariate dependence. Section 3 describes the model. In Section 4 we focus on the asymmetric Gumbel class of copulas, which is the natural parametric setting of our model; we also provide some connection with bivariate frailty models. All proofs are deferred to the Appendix.

Some useful facts in bivariate dependence

Let X and Y be two random variables (r.v) with joint sdf H, with margins F and G. All dependence properties of X and Y are captured through the survival copula C which is a cdf defined on [0, 1] × [0, 1] through

C(u, v) = H(F ← (u), G ← (v))
where u and v belong to [0, 1] and where F ← (t) := sup{x : F (x) ≥ t}. It is easily checked that C is indeed a copula. The definition of a copula is given in [START_REF] Nelsen | An introduction to copulas[END_REF], definition 2.2.2. We will make use of the following definition and notation.

Definition 1 (Min-id property). A bivariate cdf H is min-infinitely divisible (min-id) if for all positive γ, H γ is a sdf.

Assume that H is min-id and let V = (X, Y ) be a random vector with sdf H. Then for all n in N, H 1/n is a sdf. Further let (X i , Y i ) , i = 1, . . . , n be n copies independent and identically distributed with sdf H 1/n . It holds

V d = (min i X i , min i Y i ).
Definition 2 (PQD property). X and Y are positively quadrant dependent (PQD) iff, for all (x, y) in R 2 , P(X > x, Y > y) ≥ P(X > x)P(Y > y); in this case we also say that H is PQD.

Definition 3 (T P 2 property). A mapping φ from R 2 onto R is totally positive of order 2 (T P 2 ) if φ(x, y) ≥ 0 for all (x, y) in R 2 and φ(x 1 , y 1 ) φ(x 1 , y 2 ) φ(x 2 , y 1 ) φ(x 2 , y 2 ) = φ(x 1 , y 1 )φ(x 2 , y 2 )φ(x 1 , y 2 )φ(x 2 , y 1 ) ≥ 0, for all x 1 < x 2 and y 1 < y 2 .

Remark 1. When φ is C 2 , then φ is T P 2 iff (2.7) ∂φ ∂x (x, y) ∂φ ∂y (x, y) ≤ ∂ 2 φ ∂x∂y (x, y)φ(x, y)
The proof is given in [START_REF] Resnick | Extreme values, regular variation, and point processes[END_REF], p.254.

We also recall the following results.

Theorem 1 [START_REF] Joe | Multivariate models and dependence concepts[END_REF], Theorem 2.3). If H is a T P 2 sdf, then H is P QD.

Theorem 2 [START_REF] Joe | Multivariate models and dependence concepts[END_REF], Theorem 2.6). Let H be a cdf, then H is min-id iff H is T P 2 .

The relation between properties of the s.d.f's and their copulas is captured in the following result.

Lemma 1. Let H be a sdf with copula C. Then H is T P 2 iff C is T P 2 .

Proof. By Remark 1,

H T P 2 ⇒ ∂H ∂x (x, y) ∂H ∂y (x, y) ≤ ∂ 2 H ∂x∂y (x, y) × H(x, y). Furthermore, H(x, y) = C(F (x), G(y)). Therefore, ∂H ∂x (x, y) = - ∂C ∂u (u, G(y)) u=F (x) × f (x) ∂H ∂y (x, y) = - ∂C ∂v (F (x), v) v=G(y) × g(y) ∂ 2 H ∂x∂y (x, y) = ∂ 2 C ∂u∂v (u, v) u=F (x) v=G(y) × f (x) × g(y)
Hence,

∂C ∂u (u, G(y)) u=F (x) × ∂C ∂v (F (x), v) v=G(y) ≤ ∂ 2 C ∂u∂v (u, v) u=F (x) v=G(y) × C(F (x), G(y))
Definition 4 (Archimedean copula). An Archimedean copula is a function

C from [0, 1] 2 to [0, 1] given by C(u, v) = ϕ [-1] (ϕ(u) + ϕ(v))
, where ϕ is a continuous strictly decreasing convex function from [0, 1] to [0, ∞] such that ϕ(1) = 0, and where ϕ [-1] denotes the "pseudo-inverse" of ϕ, namely

ϕ [-1] (t) = ϕ -1 (t) for t in [0, ϕ(0)] 0 for t ≥ ϕ(0) When ϕ(0) = ∞, ϕ is said to be strict and ϕ [-1] ≡ ϕ -1 . ϕ is called a generator.
The class of so called extreme value copulas (evc) is important in this model, although not related here with the theory of bivariate extremes; therefore we define an extreme value copula through the basic Pickands representation, without further reference to the theory of bivariate extremes.

Theorem 3 (Pickands Theorem). C is an extreme value copula iff there exists a convex function A defined on [0, 1], which satisfies

A(0) = A(1) = 1 and max(t, 1 -t) ≤ A(t) ≤ 1 such that (2.8) C(u, v) = exp log(uv)A log v log uv .
The function A is referred to as the dependence function or Pickands function of the copula C.

Introducing covariates in dependence models

3.1. Description of the model. Not all baseline survival d.f's H 0 defines a model, so that λ z X and λ z Y |X>x are the marginal and conditional specific cause hazards for some bivariate sdf H z with margins F z and G z under a given covariate z. We conclude from the first equation of (1.6) that F

z (x) = F 0 (x) Φ(z)
. By the second equation in (1.6),

plugging x = 0, we get G z (y) = G 0 (y) Ψ(z)
. The model is defined when z holds if

F 0 (x) Φ(z) H 0 Y |X>x (y) Ψ(z) defines a sdf. Notice that (3.9) H z (x, y) = H 0 (x, y) Ψ(z) F 0 (x) Φ(z)-Ψ(z)
which is indeed a sdf when Φ(z) ≥ Ψ(z) > 0 and H 0 (x, y)

Ψ(z)
is a sdf.

Also not all bivariate survival d.f's H 0 are such that for all positive γ, H 0 γ is a sdf. Min-infinite divisibility of the baseline hazard seems to be a natural assumption here. Assume therefore that:

(H) H 0 is min-infinitely divisible
By Theorem 2 and Lemma 1, (H) holds iff C H 0 is T P 2 . We have the following result.

Proposition 1. When (H) holds then H z defined in (3.9) is a sdf for all z such that

Φ(z) ≥ Ψ(z) > 0.
Let us consider the case when 0 < Φ(z) ≤ Ψ(z). Analogously with (1.6) the model may then be written

(3.10) (M2) λ z Y (y) = λ 0 Y (y)Ψ(z) λ z
X|Y >y (x) = λ 0 X|Y >y (x)Φ(z) permuting the role of X and Y . In a similar way to the above we have that

H z (x, y) = H 0 (x, y) Φ(z) G 0 (y) Ψ(z)-Φ(z)
is a proper sdf. To summarize the above arguments we state:

Let the model be defined by

(1.6) if Φ(z) ≥ Ψ(z) and by (3.10) if Φ(z) < Ψ(z). Call (M) the model defined through (M) := (M1)1 Φ(z)≥Ψ(z) + (M2)1 Φ(z)<Ψ(z) .
This model is well defined, even if Φ(z) and Ψ(z) are not ordered uniformly on the covariate z (which can be multivariate); the functions Φ and Ψ can be easily estimated through the data, since they characterize the marginal PH models in (M). Suppose that X and Y are fitted to the same scale under the baseline, namely F 0 (t) = G 0 (t) for all t.

Then Φ(z) ≥ Ψ(z) implies F z (t) ≤ G z (t) for all t, stretching the fact that X becomes stochastically smaller than Y under the stress parameter z.

Identifiability of (M) holds; assume for example that Φ(z) ≥ Ψ(z). Then Φ(z) and Ψ(z) are defined uniquely. Indeed, assume

H z (x, y) = H 0 (x, y) Ψ(z) F 0 (x) Φ(z)-Ψ(z) = H 0 (x, y) Ψ ′ (z) F 0 (x) Φ ′ (z)-Ψ ′ (z)
, for all x, y. Then taking logarithms yields Φ(z) = Φ ′ (z) and Ψ(z) = Ψ ′ (z).

When (H) holds then for all z , H z is a sdf and

H z (x, y) = 1 Φ(z)≥Ψ(z) H 0 (x, y) Ψ(z) F 0 (x) Φ(z)-Ψ(z) (3.11) +1 Φ(z)<Ψ(z) H 0 (x, y) Φ(z) G 0 (y) Ψ(z)-Φ(z)
.

Min-infinite divisibility of the baseline will also make any H z min-infinitely divisible, showing that this class is stable under (M). Indeed for any positive γ,

H z (x, y) γ = 1 Φ(z)≥Ψ(z) H 0 (x, y) γΨ(z) F 0 (x) γ[Φ(z)-Ψ(z)] +1 Φ(z)<Ψ(z) H 0 (x, y) γΦ(z) G 0 (y) γ[Ψ(z)-Φ(z)]
,

which still is a sdf. By Theorem 2 and Lemma 1 min-infinite divisibility is not a property of the cdf but of its copula. Formula (3.11) can be written for copulas through

C H z (u, v) = 1 Φ(z)≥Ψ(z) u Φ(z)-Ψ(z) Φ(z) C H 0 u 1 Φ(z) , v 1 Ψ(z) Ψ(z) (3.12) +1 Φ(z)<Ψ(z) v Ψ(z)-Φ(z) Ψ(z) C H 0 u 1 Φ(z) , v 1 Ψ(z) Φ(z)
.

(H) is only a sufficient condition for the model to be defined. The following example illustrates this fact.

Example 1. Let C H 0 (u, v) = uv exp {-θ log u log v}, with θ ∈ (0; 1]. This is the Gumbel-Barnett family (see Nelsen, 2006, p.119). By (2.7) it is easy to check that C H 0 is not T P 2 . Using (3.12) and assuming Φ(z) ≥ Ψ(z) we obtain

C H z (u, v) = uv exp - θ Φ(z) log u log v
which still is a Gumbel-Barnett copula when θ Φ(z) belongs to (0; 1]. This example shows that (H) is indeed the only acceptable condition for existence. Otherwise the baseline hazard H 0 defines a model only for specific values of the covariate. This motivates our interest in good classes of min-infinitely divisible copulas which we intend to regress on the covariate z.

3.2.

Stability properties of the model. We introduce two classes of copulas which are stable under (M).

3.2.1. Extended archimedean copulas. Among all possible types of bivariate dependence which can be described through the present bivariate Cox model, there exists a class of copulas which contains the archimedean copulas and which enjoys peculiar stability properties.

From now on, denote

(3.13) α(z) = min Ψ(z) Φ(z) , 1 and (3.14) β(z) = min Φ(z) Ψ(z) , 1 .
Let ϕ 0 be a generator and C ϕ 0 be the archimedean copula with generator ϕ 0 . We assume that C ϕ 0 is T P 2 . When H z is defined, set C H z its copula.

Proposition 2. Let H 0 be a sdf with T P 2 copula C ϕ 0 . Then, H z is defined for all z in the domain of Φ and Ψ. Further

(3.15) C H z (u, v) = u 1-α(z) v 1-β(z) ϕ -1 z ϕ z u α(z) + ϕ z v β(z) with (3.16) ϕ z (t) = ϕ 0 t 1 Φ(z)α(z) = ϕ 0 t 1 Ψ(z)β(z)
.

More generally we have, denoting Π(u, v) = uv Proposition 3. Assume that

(3.17) C H 0 (u, v) = Π(u 1-κ , v 1-η )C ϕ 0 (u κ , v η ), 0 ≤ κ, η ≤ 1 then (3.18) C H z (u, v) = Π(u 1-α(z)κ , v 1-β(z)η )C ϕz (u α(z)κ , v β(z)η )
where C ϕz denotes the archimedean copula with generator ϕ z defined in (3.16).

Extreme values copulas.

We show that the class of evc's also enjoys stability properties, as seen in the present Section. Extreme values copulas are T P 2 [START_REF] Hürlimann | Hutchinson-Lai's conjecture for bivariate extreme value copulas[END_REF].

Proposition 4. When H 0 has an evc C H 0 with Pickands function A then, denoting C H z the copula of H z and using (3.12), we have

(3.19) C H z (u, v) = exp log(uv)B z log v log uv with (3.20) B z (s) = 1 -W (z)K(z) -sW (z)[1 -K(z)] + W (z)[(1 -s)K(z) + s]A s K(z)(1 -s) + s where K(z) = Ψ(z) Φ(z) and W (z) = min 1 K(z) , 1 = β(z).
Remark 2. This basic result shows that C H z is an evc with Pickands function B z . Proposition 4 shows that the class of evc's is stable under (M). Although the copula of H 0 is an evc, this does not imply in any respect that its marginals should be extreme value sdf 's.

From (3.20) we deduce the transition formula which links B z ′ to B z for two different values of the covariate. It holds Proposition 5. Under (M) let H 0 has an evc. Then with the above notation, for all z, z ′ , (3.21) Proposition 5 proves that the transition from z to z ′ is independent of the baseline dependence function. Formula (3.21) can be seen as a kind of expression of the proportional hazard property, which links two hazard rates independently on the baseline. When the covariate acts equally on X and Y , i.e. Φ(z) = Ψ(z) for all z, then B z (s) = A(s) for all values of s as seen in Proposition 4. Thus, the copula of H z equals that of the baseline H 0 ; the dependency structure of X and Y should not be altered through (M). Only the marginal distributions of X and Y in this case reflect the role of the covariate.

B z ′ (s) = 1- α(z ′ ) α(z) + α(z ′ ) α(z) - β(z ′ ) β(z) s+ (1 -s) α(z ′ ) α(z) + s β(z ′ ) β(z) B z s β(z ′ ) β(z) (1 -s) α(z ′ ) α(z) + s β(z ′ ) β(z)
We propose some illustration. We use Φ(z) = e αz , with α = 1.5 and Ψ(z) = e βz , with β = 2. Figure 3.1 illustrates formula (3.15). We represent the change of the density of C H z with z. The archimedean copula is the Clayton copula whose generator is defined by ϕ(t) = t -θ -1. We take θ = 3. In this figure the model tends rapidely to independence since the density of the copula tends to 1 as z increases. Figure 3.2 illustrates the transition formula (3.20). The baseline copula is the Gumbel copula with θ = 3. The Pickands function of the Gumbel copula is

A(t) = t θ + (1 -t) θ

Asymmetric logistic models of dependence

This section deals with specific parametric models for dependence which are stable under (M). We consider model (M) specialized in the case when the copula of H 0 is a Gumbel copula. The margins of H 0 can be any. It is a simple parametrized model of copulas, which is an evc on one hand, and which models frailty bivariate dependence, being hence an archimedean copula. Indeed it is the only copula satisfying jointly these two properties (see [START_REF] Nelsen | An introduction to copulas[END_REF], Theorem 4.5.2; [START_REF] Genest | A characterization of Gumbel's family of extreme value distributions[END_REF], statement A). The Gumbel copula writes

C(u, v):=exp -(-log u) θ + (-log v) θ 1/θ with θ ≥ 1. The dependence function of this copula is (4.22) A(s) = s θ + (1 -s) θ 1/θ .
Assume that H 0 has an evc with dependence function A. When the covariate z acts, the dependence function B z defined through Proposition 4 determines the asymmetric logistic copula. This copula has three parameters α(z), β(z) and θ. Recall from (3.13), (3.14) and proposition 4 that

α(z) = min Ψ(z) Φ(z) , 1 and
β(z) = min Φ(z) Ψ(z) , 1 . It holds (4.23) B z (s) = 1 -α(z) + [α(z) -β(z)] s + α(z) θ (1 -s) θ + β(z) θ s θ 1/θ .
When α(z) = β(z) (which implies that they equal 1), i.e. when z acts equally on X and Y, B z (s) = A(s) for all s. The copula of H

z is (4.24) C H z (u, v) = Π(u 1-α(z) , v 1-β(z) )C(u α(z) , v β(z) ).
where Π(u, v) := uv is the product copula (see Nelsen, 2006, p.11). The dependence function B z is an asymmetric form of the Gumbel dependence function A defined in (4.22). This is the asymmetric logistic model in [START_REF] Tawn | Bivariate extreme value theory: models and estimation[END_REF] when the margins are standard exponential. As developped by Khoudraji (see Khoudraji, 1995, chap 4) and Genest et al. in Frees and Valdez (2007), Proposition 3, given two dependence functions A 1 and A 2 , two constants κ and η with 0 < κ, η < 1 the function defined through B(s) := (κs + ηs) A 1 κs κs + ηs + (κs + ηs) A 2 κs κs + ηs where s denotes 1s, is the dependence function of the extreme value copula defined by Genest et al. define this procedure as a technique to generate asymmetric copulas. The class of copulas defined in (4.24) has been introduced by Genest et al (see their Proposition 2 in [START_REF] Frees | Understanding relationships using copulas[END_REF]).

C A 1 (u 1-κ , v 1-η )C A 2 (u κ , v η ).
We now analyze this class of copulas in terms of frailty models. The Gumbel copula is associated with a frailty model of order 1, namely

C(u, v) = Λ -1 (Λ(u) + Λ(v))
where Λ -1 (s) = ∞ 0 e -sw dM 1/θ (w) = e -s 1/θ , θ > 1, is the Laplace transform of the positive stable law M 1/θ on R + with tail heaviness index 1/θ, location parameter 0, scale parameter 1 and skewness parameter 0 (see [START_REF] Ravishanker | Multivariate survival models with a mixture of positive stable frailties[END_REF] and the example 5 in [START_REF] Oakes | Bivariate survival models induced by frailties[END_REF]). Denote W a positive random variable with cdf M 1/θ . A bivariate sdf H 0 with Gumbel copula C writes

H 0 (x, y) = ∞ 0 F (x)G(y) w dM 1/θ (w),
for some sdf F and G. Therefore H 0 is a frailty bivariate sdf, with stable frailty measure and margins

∞ 0 F (x)
w dM 1/θ (w) and

∞ 0 G(y) w dM 1/θ (w).
Let U 1 and U 2 be two independent r.v's, both independent of W . We assume that U 1 and U 2 have a positive stable law M 1 and M 2 with tail heaviness index 1/θ. The r.v U 1 (resp. U 2 ) has shape parameter 1 α(z) -1 (resp. 1 β(z) -1). Both have location and skewness parameters 0. Define S i := U i + W, i = 1, 2 (see [START_REF] Drouet | Construction de copules asymétriques avec dépendance extrême. Application à des données océanométéorologiques[END_REF]). Denote ϕ -1 i (s) the Laplace transforms of the distribution of S i . Denote further ψ -1 i (s) the Laplace transform of the distribution of U i . Let M denote the probability measure of (S 1 , S 2 ). For arbitrary sdf H 1 and H 2 define the bivariate sdf

H(x, y) := H 1 (x) s 1 H 2 (y) s 2 dM(s 1 , s 2 )
which we call a frailty model of order 2 since it implies a bivariate latent variable. Frailty models of order two have been considered in [START_REF] Marshall | Families of multivariate distributions[END_REF] (see their formula (2.2)). The marginals of H are

F (x) = ∞ 0 H 1 (x) s 1 dM S 1 (s 1 ) and G(y) = ∞ 0 H 2 (y) s 2 dM S 2 (s 2 ).
We prove that the copula of H is (4.24). Indeed

H(x, y) = H 1 (x) u 1 dM 1 (u 1 ) H 2 (y) u 2 dM 2 (u 2 ) H 1 (x)H 2 (y) w dM 1/θ (w).
Introducing the Laplace transforms defined above and rewriting the marginals

F (x) = ϕ -1 1 -log H 1 (x) and G(y) = ϕ -1

2

log H 2 (y) we obtain the following expression for the copula of H

(4.25) C(u, v) = ψ -1 1 (ϕ 1 (u)) ψ -1 2 (ϕ 2 (v)) Λ -1 (ϕ 1 (u) + ϕ 2 (v))
, since H(x, y) = C(F (x), G(y)). Substituting ψ i and ϕ i , i = 1, 2 by their expressions in the above expression, noting that ϕ -1 i = ψ -1 i Λ -1 , (4.25) coincides with (4.24). We now prove that for an adequate choice of H 1 and H 2 the bivariate sdf H has same marginals as H z . Indeed let

H 1 (x) : = exp --min (Φ(z), Ψ(z)) log F 0 (x) θ H 2 (y) : = exp --min (Φ(z), Ψ(z)) log G 0 (y) θ which yields F (t) = F z (t) and G(t) = G z (t) for all t.
Therefore H and H z coincide. We have proved Proposition 6. When H 0 is a frailty bivariate sdf with Gumbel copula, then for all z, H z is a frailty sdf of order 2 with asymmetric logistic copula given in (4.24).

More generally we have

Proposition 7. The class of second order frailty models with asymmetric logistic copula is stable under (M).

Proof. Let A denote the dependence function of an asymmetric logistic copula.

A(s)

:= 1 -κ + (κ -η) s + κ θ (1 -s) θ + η θ s θ 1/θ .
By (3.20) it holds

B z (s) = 1 -κ ′ + (κ ′ -η ′ ) s + κ ′θ (1 -s) θ + η ′θ s θ 1/θ with κ ′ = α(z)κ η ′ = β(z)η.
These new parameters are in (0, 1) , as are κ and η. We have proved that the class of sdf with asymmetric logistic copula is stable under (M). Any sdf with such a copula is necessarily a frailty sdf of order 2. Indeed this follows from (4.25) which enables identifying the frailty measure M of H z as the joint distribution of (S 1 , S 2 ) as defined here above.

Remark 3. It can be seen that the only sdf which are frailty of order 2 with evc are precisely the frailty models with asymmetric logistic copula

C(u, v) = Π(u 1-κ , v 1-η )C θ (u κ , v η )
with 0 < η, κ < 1, and where C θ is the standard Gumbel copula with parameter θ ≥ 1.

Simulation results

5.1. Stability of the estimate under the model. For a given copula C θ in a parametric family we simulated N independent couples (X i , Y i ), i = 1, ..., N, with joint distribution function C θ . Estimation of θ was performed using plug-in technique, leading θ. We repeated the procedure 1000 times. We compared C z θ with C z θ for various z, given known functions Φ and Ψ. Here Φ(z) = e αz with α = 1.5 and Ψ(z) = e βz with β = 2. The figures hereunder show the mean relative error

C z θ (u,v)-C z θ (u,v) C z θ (u,v)
dC z θ (u, v) with respect to z, together with the 95% confidence interval. This indicator is obtained on a grid of 100 points in

[0, 1] × [0, 1]. Figure 5.3 pertains to the Clayton copula C θ (u, v) = (u -θ + v -θ -1) -1 θ with θ = 3. The estimate θ is defined through θ = 2τ
1-τ where (5.26) τ = number of concordant pairsnumber of discordant pairs N 2

is the empirical estimate of the Kendall's tau (see Nelsen, 2006, p.158). Figure 5.4 pertains to the Gumbel copula

C θ (u, v) = exp[-{(-log u) θ + (-log v) θ } 1 θ ] with θ = 3. The estimate θ is defined through θ = 1
1-τ . It appears from those curves that a good estimate in the reference zone (z = 0) propagates accordingly to other zones (indexed by z), without deteriorating the estimation accuracy. These facts also hold for very small values of N; obviously the larger N, the better the accuracy, for all z.

Remark 4.

(1) These simulations are closely related to some industrial application where the environmental variable z has range [0, 0.3]. Obviously α (resp. β) is fitted accordingly around 1.5 (resp. 2), as in these simulations. Changing the scale of z modified the value of α and β. This entails that no comparison can be performed with higher values of z on these graphs, keeping in mind the underlying applied statistical context.

(2) In these simulations the paramater θ of the copula is estimated using the baseline group. The estimation of the copula under z is obtained through a transformation of the copula under z = 0. Taking into account all the pooled data for all z values introduces complex estimation procedures. Relative error pertaining to Gumbel copula 5.2. A case study through simulation. In this section, z is bivariate, z = (z 1 , z 2 ) with z i ∈ {0; 1}, i = 1, 2. We consider model (M) with Weibull marginals. In the reference zone, z ′ = (0, 0), the marginals are W (2, 12000) for X and W (1.5, 8000) for Y . The copula in the reference zone is Clayton (archimedean) or Gumbel (which is both archimedean and e.v.c). The function Φ (resp Ψ) is exp(α ′ z) (resp exp(β ′ z)) where α ′ = (0.1, 0.06) (resp β ′ = (0.07, 0.25)). We simulated N = 200 couples with Gumbel copula or with Clayton copula under z ′ = (0, 0), 100 values under z ′ = (1, 0) and 100 values under z ′ = (0, 1). Simulations was performed using Khoudraji's algorithm (see [START_REF] Khoudraji | Contributions à l'études des copules et à la modélisation de valeurs extrêmes bivariées[END_REF]. We estimated α and β using the whole sample (400 couples) through Cox's partial likelihood estimation method (implemented in R via the coxph procedure); therefore we estimated α and β independently by working separately with the X i 's and the Y i 's. The copula parameter θ under z ′ = (0, 0) was estimated through the plug-in of the empirical Kendall's tau. We used formula (3.15) or formula (4.24) in order to obtain an estimated copula Ĉz under z ′ = (0, 0) (with 3 parameters instead of 1) by replacing α(z), β(z) in (3.13) and (3.14) and θ by their estimators; the estimator of θ is unchanged and has been obtained under z ′ = (0, 0). On the other hand, these same formulas provide the theoretical copula C z under z. We performed the relative accuracy of the estimation scheme through

N = 10 z ⌠ ⌡ ⌠ ⌡ |C(u, v) -C ^(u, v)| C(u, v) dC(u, v)
N = 100 z ⌠ ⌡ ⌠ ⌡ |C(u, v) -C ^(u, v)| C(u, v) dC(u, v)
N = 1000 z ⌠ ⌡ ⌠ ⌡ |C(u, v) -C ^(u, v)| C(u, v) dC(u, v)
N = 10000 z ⌠ ⌡ ⌠ ⌡ |C(u, v) -C ^(u, v)| C(u, v) dC(u, v) Mean 95% confidence interval
N = 10000 z ⌠ ⌡ ⌠ ⌡ |C(u, v) -C ^(u, v)| C(u, v) dC(u, v) Mean 95% confidence interval
C z (u,v)-Ĉz (u,v) C z (u,v)
dC z (u, v). The procedure is repeated 1000 times. The results are given in Table 1. We give the average of the uniform relative error over the 1000 simulations (bold character) and its 95-percent confidence interval. The estimators of Φ and Ψ however are of mean accuracy, which nevertheless does not deteriorate the quality of the estimators of the copula, in all cases which we considered. In Table 1 the indicator mixes both propagation and estimation errors. The relative error w.r.t the theoretical propagated copula is very small in the case of the most commonly used baseline copulas. Measured in terms of Spearman's rho, the relative error does not deteriorate significantly either, as seen in Table 2. 5.3. Propagation of misspecification errors. 5.3.1. Assuming Φ and Ψ known. We simulated N = 200 couples of r.v's with Clayton distribution function C θ with parameter θ = 3. Here Φ and Ψ are as in Section 5.1. We estimated the Kendall's tau through the classical non parametric estimate (5.26). We used a misspecified model assuming that the data have been generated under a Ali-Mikhail-Haq copula G θ (Nelsen, 2006, Table 4.1 p.116) with parameter θ = 2 3-τ . As seen in Nelsen ( 2006), both copulas present common features. Misspecification may therefore occur as presented here. For various z we used formula (3.15) to define both the true copula C z θ and the misspecified estimated copula G z θ . The misspecification error is defined through

z ′ = (0, 0) z ′ = (1, 0) z ′ = (0, 1)
z ′ = (0, 0) z ′ = (1, 0) z ′ = (0, 1)
Err(z) = C z θ (u,v)-G z θ (u,v) C z θ (u,v)
dC z θ (u, v). We simulated N = 200 couples of r.v's with Clayton distribution function C θ with parameter θ = 3. z is bivariate and the functions Φ and Ψ are as in Section 5.2. We estimated Φ and Ψ through partial likelihood and we used a misspecified model assuming that the data have been generated under a Ali-Mikhail-Haq copula G θ. The results in Table 3 show that the misspecification error is of order 17% and keeps stable through the propagation. At the contrary Table 1 shows that the error under the true model is much smaller and propagates with great accuracy. This enlights the need for a good specification in this model. 4 the relative error of the Spearman's rho is presented, when a Clayton baseline copula is substituted by an AMH. The error deteriorates while propagating along z.

⌠ ⌡ ⌠ ⌡ |C(u, v) -G ^(u, v)| C(u, v) dC(u, v) Mean 95% confidence interval
z ′ = (0, 0) z ′ = (1, 0) z ′ = (0, 1)
z ′ = (0, 0) z ′ = (1, 0) z ′ = (0, 1)

Conclusion

In this paper a new model for bivariate dependence is proposed. The margins follow a PH model; the covariate acts both on the margins and on their dependence function, handled through the copula. Such model is in contrast with other approaches in which the role of the covariate is restricted to its influence on the margins, the copula deserving a separate study. Applications of the present model fit in "father and sons" paradigm as well as in various industrial contexts. Classical families of copulas (archimedean or extreme value) are shown to be stable under the action of the covariate, in the sens that the model acts in a transitive way in those classes. In parametric classes this model provides explicit transformation of the parameter of the copula w.r.t the covariate. This model is adequate for the case of positive dependence between the margins for all the values of the covariate.

We now prove that C H z in (3.15) is T P 2 . It is readily checked that the product of two T P 2 functions is T P 2 . Now (u, v) → u a v b is T P 2 for all 0 ≤ a, b ≤ 1 and (u, v) → C ϕz (u a , v b ) is T P 2 for all 0 ≤ a, b ≤ 1, since ϕ z satisfies ϕ ′ z (t) + tϕ ′′ z (t) ≥ 0. We conclude that C H z is T P 2 as a product of two T P 2 functions. B.2. Proof of Proposition 3. C H z is (3.18) through simple calculations. That (u, v) → C H z (u, v) is T P 2 is proved as in Proposition 2. In the above display it holds

A s K(z ′ )(1 -s) + s = A     s (1-s) K(z ′ ) K(z) +s K(z) 1 - s (1-s) K(z ′ ) K(z) +s + s (1-s) K(z ′ ) K(z) +s     .
The RHS in this latter expression can be written as a function of B z s (1-s)

K(z ′ ) K(z) +s
. Some calculus yields (D.30)

B z ′ (s) = 1 -W (z ′ )K(z ′ ) -sW (z ′ )[K(z) -K(z ′ )] -[1 -W (z)K(z)] W (z ′ ) W (z) (1 -s) K(z ′ ) K(z) + s + W (z ′ ) W (z) (1 -s) K(z ′ ) K(z) + s B z s
(1s) K(z ′ ) K(z) + s which is (3.21).

  function of Y depends on z through the PH model (1.5) λ z Y |X>x (y) = λ 0 Y |X>x (y)Ψ (z) for some positive function z → Ψ(z) and some baseline hazard λ 0 Y |X>x . Denote (1.6)

Figure 3 . 1 .

 31 Figure 3.1. Illustration of (3.18) with the Clayton copula density for the baseline (z = 0)

1θ.Figure 3 . 2 .

 32 Figure 3.2. Illustration of formula (3.20)

Figure

  Figure 5.3. Relative error pertaining to Clayton copula

Figure

  Figure 5.4. Relative error pertaining to Gumbel copula

Figure 5 . 5 .

 55 Figure 5.5. Relative error pertaining to the misspecification when Φ and Ψ are known

  ) = (1s)[1 -K(z)] + [(1s)K(z) + s] A s (1s)K(z) + s = 1 -K(z)s[1 -K(z)] + [(1s)K(z) + s] A s (1s)K(z) + s If K(z) > 1 ĈH z (u, v) ) -Φ(z) Ψ(z) + (1s) + s Φ(z) Ψ(z) A s (1s) Ψ(z) Φ(z) + s = exp ln(uv)B z 2 ln v ln uvas sought. It remains to prove that B z (s) is a convex function. Some calculus yields∂ 2 B z ∂s 2 (s) = W (z)K 2 (z) [(1s)K(z) + s] Proof of Proposition 5 Write B z ′ (s) = 1-W (z ′ )K(z ′ )-sW (z ′ )[1-K(z ′ )]+W (z ′ )[(1-s)K(z ′ )+s]A s K(z ′ )(1s) + s .

Table 1 .

 1 Relative error for different copulas

	Clayton (θ = 3)	0.93% [0.88%,0.97%] [1.95%,2.12%] [2.49%,2.71%] 2.03% 2.60%
	Gumbel (θ = 3)	0.81% [0.77%,0.85%] [1.49%,1.62%] [1.93%,2.11%] 1.55% 2.02%

Table 2 .

 2 Relative error on Spearman's rho

	Clayton	3.49% [3.32%,3.66%] [5.44%,6.02%] [11.08%,12.23%] 5.73% 11.65%
	Gumbel	2.35% [2.24%,2.46%] [3.82%,4.22%] [8.11%,8.89%] 4.01% 8.50%

Table 3 .

 3 Relative error Err(z) pertaining to the misspecification when Φ and Ψ are unknown

	Clayton -AMH	17.38% [17.36%,17.40%] [17.54%,17.67%] [14.65%,14.80%] 17.61% 14.72%

Table 4 .

 4 Relative error on Spearman's rhoIn Table

	Clayton -AMH	53.39% [53.31%,53.46%] [54.61%,55.09%] [54.83%,55.75%] 54.85% 55.29%

Appendix A. Proof of Proposition 1

Trivially we show that lim

x j →+∞ H z (x 1 , x 2 ) = 0, j = 1, 2; and lim

Now we will prove that H z satisfies the rectangle inequality [START_REF] Joe | Multivariate models and dependence concepts[END_REF], p.11) which we recall here: for all (a 1 , a 2 ), (b

We denote r(t) = H 0 (t, b 2 )

is a 2-increasing function), then by Lemma 2.1.3 in [START_REF] Nelsen | An introduction to copulas[END_REF], the function r is nondecreasing. Therefore

.

and

is a decreasing function of its second argument) to obtain

Appendix B. Proof of Proposition 2 and 3

. Using (3.12) some calculus yields (3.15). We prove that ϕ z (t) = ϕ 0 t

is also a generator for all values of t in [0, 1].

(1) ϕ z (1) = ϕ 0 (1) = 0 (2) ϕ z (t) is strictly decreasing in t, since t 1 min(Φ(z),Ψ(z)) is increasing in t and ϕ 0 (t) is strictly decreasing.

(3) It holds ϕ ′ 0 (t) + tϕ ′′ 0 (t) ≥ 0 since since C H 0 is T P 2 . Let us prove that ϕ ′′ z (t) ≥ 0. We have,

We prove that B z is a dependence function. It holds

We prove the upper and lower bounds for B z .

Upper bound. Using A(s) ≤ 1 for all s in[0, 1]

We prove that the RHS in the above display is larger than both s and 1s.