Thi Huyen

Tram Nguyen

France Mentré

Micha Levi

Jing Yu

Jérémie Guedj

A pharmacokinetic -viral kinetic model describes the effect of alisporivir monotherapy or in combination with peg-IFN on hepatitis C virologic response

Alisporivir is a cyclophilin inhibitor with demonstrated in vitro and in vivo activity against hepatitis C virus (HCV). We estimated antiviral effectiveness of alisporivir alone or in combination with pegylated-Inteferon (peg-IFN) in 88 patients infected with different HCV genotypes treated for four weeks. The pharmacokinetics of both drugs were modeled and used as driving functions for the viral kinetic model.

Genotype was found to significantly affect pegylated-Inteferon effectiveness (ε= 86.3% and 99.1% in genotype-1/4 and genotype-2/3, respectively, p<10 -7 ) and infected cells loss rate (= 0.22 vs 0.39 day -1 in genotype-1/4 and genotype-2/3, respectively, p<10 -6 ). Alisporivir effectiveness was not significantly different across genotype and was high for doses ≥600 mg QD. We simulated virologic responses with other alisporivir dosing regimens in HCV genotype-2/3 patients using the model. Our predictions consistently matched the observed responses, demonstrating that this model could be a useful tool for anticipating virologic response and optimize alisporivir-based therapies.

Introduction

Chronic infection with hepatitis C virus (HCV) affects about 170 million people worldwide and is responsible for the majority of morbidity and mortality from liver diseases. [START_REF] Mohd Hanafiah | Global epidemiology of hepatitis C virus infection: New estimates of age-specific antibody to HCV seroprevalence[END_REF] The goal of treatment is to achieve a sustained virologic response (SVR), marker of viral eradication, assessed by the absence of detectable HCV RNA six months after treatment cessation. Over the last few years, a rapid development of direct acting antivirals (DAAs) has opened a new era for HCV therapy. The advent of the new therapies has dramatically increased SVR rate and decreased treatment duration. In recent phase 3 clinical trials including a protease or a polymerase inhibitor, SVR was higher than 90% following a 12 week-treatment. This holds the expectation that several one-in-a-pill (i.e., interferon-free) treatments with universal cure might be achieved. [START_REF] Flisiak | Emerging treatments for hepatitis C[END_REF][START_REF] Gentile | Efficacy and Safety of Sofosbuvir in Treatment of Chronic Hepatitis C: The Dawn of the a New Era[END_REF] However several challenges are still ahead. First DAAs have been optimized to treat HCV genotype-1 and have lower effectiveness against genotypes2/3, which account for 19 to 40% of HCV-infected patients. Second, high mutation rate of HCV of several DAA classes (i.e. protease and non-nucleoside polymerase inhibitors) requires therapies with a high barrier to resistance, which can be achieved by combining several antiviral agents. [START_REF] Flisiak | Emerging treatments for hepatitis C[END_REF][START_REF] Au | Novel therapeutic approaches for hepatitis C[END_REF] However therapy combination can be a particularly difficult endeavor and requires anticipating possible pharmacokineticpharmacodynamic (PK-PD) interaction between different agents.

Unlike DAAs that specifically block a viral protein, host targeting antivirals (HTA) block host proteins necessary for viral replication. [START_REF] Flisiak | Emerging treatments for hepatitis C[END_REF] Alisporivir, a second-generation cyclophilin inhibitor devoid of immunosuppressive activity, is the most advanced HTA. [START_REF] Flisiak | Emerging treatments for hepatitis C[END_REF][START_REF] Gallay | Profile of alisporivir and its potential in the treatment of hepatitis C[END_REF] Alisporivir acts primarily by blocking in a dose-dependent manner the interaction between cyclophilin A and the HCV NS5A protein. [START_REF] Gallay | Profile of alisporivir and its potential in the treatment of hepatitis C[END_REF] It has shown potent anti-HCV activity across various HCV genotypes in several in vitro and in vivo studies. 5- 7 Alisporivir was also found to have a high genetic barrier to resistance. [START_REF] Garcia-Rivera | Multiple mutations in hepatitis C virus NS5A domain II are required to confer a significant level of resistance to alisporivir[END_REF] In a phase 2a study (DEB-025-HCV-203, Euract number: 2006-002695-17), the virologic response to alisporivir monotherapy or in combination with pegylated interferon (peg-IFN) was studied in 90 patients infected with HCV genotype 1, 2, 3 or 4. [START_REF] Flisiak | The cyclophilin inhibitor Debio 025 combined with PEG IFNalpha2a significantly reduces viral load in treatment-naïve hepatitis C patients[END_REF] Alisporivir (200, 600, and 1000 mg given BID for one week then QD till the end of treatment) showed a dose-dependent antiviral effect in combination with peg-IFN, with mean viral decay of more than 4 and 5 log10IU/mL at week 4 in HCV genotype1 and genotype-2/3, respectively. [START_REF] Flisiak | The cyclophilin inhibitor Debio 025 combined with PEG IFNalpha2a significantly reduces viral load in treatment-naïve hepatitis C patients[END_REF] However, multiple factors involved (genotypes, monotherapy vs combination, alisporivir doses) made it difficult to anticipate the effect of different alisporivir dosing regimen on longer time period. Mathematical models for HCV kinetics have provided important insights into HCV life cycle as well as the effectiveness and the mechanisms of action of different anti-HCV agents . [START_REF] Guedj | A perspective on modelling hepatitis C virus infection[END_REF][START_REF] Chatterjee | Hepatitis C viral kinetics: the past, present, and future[END_REF][START_REF] Chatterjee | Mathematical modelling of HCV infection: what can it teach us in the era of direct-acting antiviral agents?[END_REF][START_REF] Guedj | Modeling shows that the NS5A inhibitor daclatasvir has two modes of action and yields a shorter estimate of the hepatitis C virus half-life[END_REF][START_REF] Guedj | Understanding silibinin's modes of action against HCV using viral kinetic modeling[END_REF] The understanding of virologic response can be further improved by incorporating PK information. [START_REF] Chatterjee | Hepatitis C viral kinetics: the past, present, and future[END_REF][START_REF] Shudo | Modeling HCV kinetics under therapy using PK and PD information[END_REF] However this approach became more complicated in drug combination as both PK and PD interaction can occur. Here we incorporated the PK of alisporivir and peg-IFN into a viral kinetic (VK) model to estimate the antiviral effectiveness of both agents and to tease out the effect of genotypes and alisporivir doses on the virologic response. Lastly we evaluated the robustness of our model by comparing the predictions of the model and virologic responses observed in a phase 2b study (VITAL-1, clinicaltrials.gov ID: NCT01215643), where alisporivir was given at different dosing regimens with response-adapted treatments.

Results

Data

The study population comprised 90 HCV patients, 57.8% were men and 66.7% were infected with HCV genotype-1. Baseline and patients characteristics are presented in Table S1. [START_REF] Flisiak | The cyclophilin inhibitor Debio 025 combined with PEG IFNalpha2a significantly reduces viral load in treatment-naïve hepatitis C patients[END_REF] This was a randomized, double-blind, placebo-controlled phase 2 study in treatment naïve HCV patients with compensated liver function. Patients were randomized into 5 treatment arms receiving peg-IFNα2a (180 μg/week) combined with placebo, 200, 600 or 1000 mg alisporivir or 1000 mg alisporivir monotherapy (Arm PEG, A, B, C, D, respectively). Alisporivir was administered as twice daily (BID) for one week, followed by once daily (QD) for three weeks. No lack of compliance was reported.

Alisporivir and peg-IFN plasma trough concentrations were determined on days 1, 8, 15, 22, 29, and 50 (21 days after the last dose). In addition, a 12-hour and 24-hour PK of alisporivir and peg-IFN was established on days 1 and 29 in a subset of patients. Plasma concentrations of alisporivir and peg-IFN were determined using validated LC/MS/MS and ELISA assays with the limit of quantification (LOQ) of 2.5 ng/mL and 0.17 ng/mL, respectively. [START_REF] Lindh | Dynamic tailoring of treatment durations improves efficiency of hepatitis C treatment with pegylated interferon and ribavirin[END_REF] HCV RNA levels (IU/mL) were determined by the Roche COBAS TaqMan, Roche laboratories, Basel, Switzerland with LOQ of 15 or 45 IU/mL and detection limits of 10 or 15 IU/mL.

The PK model of alisporivir was developed using 858 observations from 50 patients with rich design and 104 observations from 21 patients with sparse sampling. Seven post-dose concentrations recorded as trough concentrations were excluded from the analysis.

The PK model of peg-IFN was developed using 527 observations from 48 patients with rich design and 100 observations from 21 patients with sparse design.

The VK model was developed using on-and post-treatment data (521 viral loads from 88 patients). Two patients having no PK record for both alisporivir and peg-IFN were excluded. The percentages of viral load below quantification limits were 27.6%.

Alisporivir PK model

Alisporivir PK on day 29 was described well using a two-compartment model with delayed first-order absorption and Michaelis-Menten elimination (Table S2). However, this model over predicted alisporivir PK profile on day 1 (Figure S1). This discrepancy between day 1 and day 29 PK was described using a time-dependent bioavailability. The structural model is described as below:

( ) = + ( -)(1 -exp (-) = 0 ≤ - > = - - + - + = -
where F(t) is the bioavailability at time t. A, Q1, Q2 are the drug quantity at the absorption site, in central and peripheral compartment, respectively. As the bioavailability could not be identified using only oral data, we fixed F0 at the value found in monkeys (0.27).The relative bioavailability reached its maximal value in approximately 4 days (kf). The maximal relative bioavailability (Finf) was approximately 3-fold higher as compared to the base line value (F0). We also tested time-decreasing Vm or time-increasing Km models but their Bayesian Information Criterion (BIC), a criterion for model selection based on the log-likelihood penalized by number of parameters and observations (the lower the better), were much higher, compared to time-dependent bioavailability (Table S3).

However, alisporivir exposure following 1000 mg QD was higher at day 29 in the combination therapy as compared to monotherapy. In order to describe this higher exposure we tested a possible peg-IFN effect on alisporivir PK (peg-IFN effect on Vm, Km and Finf). A statistically significant effect of peg-IFN on reducing alisporivir elimination at the dose of 1000 mg QD was described by decreasing both Vm (55 vs 115 mg.h -1 , p<0.003) and Km (0.64 vs 2.44 mg.L -1 , p<0.00023). Peg-IFN effect on the Km and Vm led to 1.2-time increase in the median value (or 1.5-time increase in the mean value) of AUC on day 29. The parameter estimates for alisporivir PK were shown in Table 1.

Peg-IFN PK model

The PK of peg-IFN was described by one-compartment model with linear absorption and elimination.

alisporivir had no significant effect on peg-IFN PK (Table 1).

PK-VK model

The shrinkages of PK individual parameters (computed from ratio of variances) were higher than 50%

for some PK parameters of alisporivir such as ka (58%), k12 (77%), Vm (58%), Km (51%) and Finf (60%) as there were 21 out of 71 patients having only weekly assessed trough concentrations. However, using individual PK profiles predicted from the individual parameters, the PK-VK model fitted to the viral load data very well and was able to describe the fluctuation in virologic response due to the variation of concentrations over the dosing period (Figure 3).

HCV genotypes were found to significantly affect both the effectiveness of peg-IFN and the loss rate of infected cells but not the effectiveness of alisporivir (Table 1). The average effectiveness of peg-IFN in the seven days following the last dose was higher in genotype-2/3 than in genotype-1/4

patients with median values of 99.1% and 86.3% , respectively (p<10 -8 ). Likewise the loss rate of infected cells, , was higher in HCV genotype-2/3 than in genotype-1/4 patients (0.39 day -1 vs 0.22 day -

1

, respectively, p<10 -6 ). Alisporivir effectiveness reached the highest values at the end of BID dosing during week 1 with a median value of 71.3%, 95.2%, 97.6% and 95.7% in arms A, B, C and D (alisporivir 200 mg, 600 mg, 1000 mg with peg-IFN and alisporivir 1000 mg monotherapy), respectively and then dropped when switching to QD dosing with a median effectiveness at the last day of treatment of 50.5%, 89.4%, 97.1% and 93.6% in arm A, B, C and D, respectively. Higher alisporivir effectiveness in combination than in monotherapy (median ε=97.6% vs 95.7% at day 8 and 97.1% vs 93.6% at day 29, respectively) was explained by higher exposure of alisporivir due to PK interaction between the two drugs. Given in combination, alisporivir and peg-IFN acted additively and yielded to high effectiveness, with mean effectiveness at day 29 in HCV genotype-2/3 patients equal to 98.5%, 99.3% and 99.5% in arm A, B and C, respectively as compared to 92.0%, 96.1% and 99.3% in genotype-1/4 patients in arm A, B and C, respectively. The VK in different treatment arms and in two genotype populations was illustrated in Figure 4.

External validation

We used the PK-VK model to simulate virologic responses and compared them to the results from the VITAL-1 study (see Methods). The observed responses were included within the 95% prediction interval in the large majority of cases evaluated (Table 2). Likewise a good match was also obtained for later data points (after week 6) where treatment regimen depended on the initial virologic response.

Discussion

Models of HCV kinetics have become valuable tools to estimate antiviral effectiveness and optimize therapy. [START_REF] Guedj | A perspective on modelling hepatitis C virus infection[END_REF][START_REF] Mihm | Review article: predicting response in hepatitis C virus therapy[END_REF][START_REF] Nguyen | Influence of a priori Information, Designs, and Undetectable Data on Individual Parameters Estimation and Prediction of Hepatitis C Treatment Outcome[END_REF][START_REF] Sypsa | Modelling of viral dynamics in hepatitis B and hepatitis C clinical trials[END_REF] Initially developed for peg-IFN/ribavirin they have been recently applied to various classes of DAA agents. [START_REF] Guedj | A perspective on modelling hepatitis C virus infection[END_REF][START_REF] Chatterjee | Mathematical modelling of HCV infection: what can it teach us in the era of direct-acting antiviral agents?[END_REF] However most of the results published considered constant treatment effectiveness over time and/or empirical function to describe varying-effectiveness models. [START_REF] Chatterjee | Hepatitis C viral kinetics: the past, present, and future[END_REF][START_REF] Chatterjee | Mathematical modelling of HCV infection: what can it teach us in the era of direct-acting antiviral agents?[END_REF][START_REF] Shudo | Modeling HCV kinetics under therapy using PK and PD information[END_REF][START_REF] Laouénan | Using pharmacokinetic and viral kinetic modeling to estimate the antiviral effectiveness of telaprevir, boceprevir and Peg-IFN during triple therapy in treatment-experienced HCV infected cirrhotic patients (ANRS CO20-CUPIC)[END_REF] In fact there have been only a few attempts to include PK into VK models, even less in drug combinations. [START_REF] Guedj | Modeling viral kinetics and treatment outcome during alisporivir interferonfree treatment in HCV genotype 2/3 patients[END_REF] This, indeed, considerably complicates the analysis as interactions are susceptible to occur both at the PK and PD levels. Here we characterized the VK obtained in a phase 2a study using both alisporivir and peg-IFN PK information.

Alisporivir PK model was developed via several steps. First a structural model using day 29 data was developed. A two-compartment model with Michaelis-Menten elimination was the best to describe these data, consistent with the nonlinearity in alisporivir PK reported in previous studies. [START_REF] Flisiak | The cyclophilin inhibitor Debio-025 shows potent anti-hepatitis C effect in patients coinfected with hepatitis C and human immunodeficiency virus[END_REF][START_REF] Lindh | Dynamic tailoring of treatment durations improves efficiency of hepatitis C treatment with pegylated interferon and ribavirin[END_REF] This model (based on day 29 data) systematically over-estimated the concentrations of alisporivir after the first dose on day 1. This overestimation could be due to a time-varying PK or to non-linearity in absorption and/or distribution and/or elimination that was not yet properly handled. To correctly identify the cause for this overestimation and to build a more mechanistic PK model for alisporivir, further data such as IV data and more detailed PK profiles during treatment would be required. We modeled the discrepancy of PK between day 1 and 29, using an empirical model allowing PK parameters (Vm, Km and bioavailability) to change with time. The model assuming time-dependent bioavailability was the best statistical model to describe the full PK profile. Here we chose to model the change in exposure, which may be the consequences of changes in bioavailability and/or distribution volume, using a timevarying bioavailability. The relevance of this model can be explained by the fact that as alisporivir concentration increases, the transporters involved in the absorption and tissue distribution of alisporivir (P-gp, MRP2, OATP1B1, OATP1B3, NTCP, BSEP) could be inhibited. [START_REF] Crabbé | An evaluation of the cyclophilin inhibitor Debio 025 and its potential as a treatment for chronic hepatitis C[END_REF] This would increase alisporivir plasma concentration (by increasing bioavailability and reducing the uptake to the liver and other distribution sites) and hence, increase bioavailability.

The effect of peg-IFN on alisporivir PK was investigated by adding peg-IFN as a covariate on alisporivir PK parameters. Peg-IFN decreases both Vm and Km. Indeed, IFN was shown to affect the activity and the expression of the cytochrome P450 system [START_REF] Lee | CYP-mediated therapeutic protein-drug interactions: clinical findings, proposed mechanisms and regulatory implications[END_REF][START_REF] Aitken | Gene-specific effects of inflammatory cytokines on cytochrome P450 2C, 2B6 and 3A4 mRNA levels in human hepatocytes[END_REF][START_REF] Islam | Differential Effect of IFNα-2b on the Cytochrome P450 Enzyme System: A Potential Basis of IFN Toxicity and Its Modulation by Other Drugs[END_REF][START_REF] Flaman | The effect of interferon-α on the expression of cytochrome P450 3A4 in human hepatoma cells[END_REF][START_REF] Chen | Effect of interferon-α2b on the expression of various drug-metabolizing enzymes and transporters in co-cultures of freshly prepared human primary hepatocytes[END_REF] and of several drug transporters such as P-gp, MRP2, MRP3, BCRP, BSEP in human hepatocytes. [START_REF] Chen | Effect of interferon-α2b on the expression of various drug-metabolizing enzymes and transporters in co-cultures of freshly prepared human primary hepatocytes[END_REF][START_REF] Vee | Regulation of drug transporter mRNA expression by interferon-γ in primary human hepatocytes[END_REF][START_REF] Fernandez | Influence of the proinflammatory cytokines on P-glycoprotein expression and functionality[END_REF] These proteins are all involved in the elimination of alisporivir, via metabolism (mainly by CYP3A4) or excretion via the bile as parent drug.

Although the mechanism for the PK interaction between alisporivir and peg-IFN remains unclear, it could be in part explained by the inhibition effect of peg-IFN on the CYP450 system, possibly on CYP3A, and on drug transporters such as P-gp. This hypothesis is also supported by the case of telaprevir, another substrate of CYP3A and P-gp [START_REF] Weiss | Telaprevir is a substrate and moderate inhibitor of P-glycoprotein, a strong inductor of ABCG2, but not an activator of PXR in vitro[END_REF] , where plasma exposure tends to increase in presence of peg-IFN. [START_REF] Garg | Telaprevir: pharmacokinetics and drug interactions[END_REF][START_REF] Forestier | Antiviral activity of telaprevir (VX-950) and peginterferon alfa-2a in patients with hepatitis C[END_REF] However, this interaction was evaluated using only an alisporivir dose of 1000 mg and additional monotherapy dosing cohorts are needed to evaluate whether this effect is universal or specific to this dosing regimen. Also, in this analysis, we did not account for the fluctuation in peg-IFN exposure that may influence the drug-drug interaction.

Peg-IFN PK is well described with a one-compartment model. The parameters obtained were close to what had been reported during peg-IFN/ribavirin therapy. [START_REF] Bruno | Viral dynamics and pharmacokinetics of peginterferon alpha-2a and peginterferon alpha-2b in naive patients with chronic hepatitis c: a randomized, controlled study[END_REF][START_REF] Harris | Pegylation: a novel process for modifying pharmacokinetics[END_REF][START_REF] Dahari | Pharmacodynamics of PEG-IFN-alpha-2a in HIV/HCV co-infected patients: implications for treatment outcomes[END_REF] We used estimated individual parameters to predict individual PK profiles of both drugs and these predictions were incorporated into a VK model to fit on-and post-treatment viral load data. A more rigorous approach could be to simultaneously fit PK and VK. However the sequential approach has been shown to lead to limited bias while being much less time-consuming than simultaneous approach. [START_REF] Zhang | Simultaneous vs. sequential analysis for population PK/PD data I: best-case performance[END_REF] To examine if the model can explain well the VK in each treatment arm, we used plots of prediction discrepancies versus time. Prediction discrepancies are simulation-based evaluation metrics, similar to visual predictive check (VPC) but unlike VPC, the heterogeneity in covariate values are handled in their computation [START_REF] Comets | Computing normalised prediction distribution errors to evaluate nonlinear mixed-effect models: the npde add-on package for R[END_REF][START_REF] Nguyen | Extension of NPDE for evaluation of nonlinear mixed effect models in presence of data below the quantification limit with applications to HIV dynamic model[END_REF][START_REF] Mentré | Prediction discrepancies for the evaluation of nonlinear mixedeffects models[END_REF] . Therefore, we can use one prediction discrepancies plot for all data from all genotypes instead of splitting data into genotype-1/4 and 2/3 groups (with 12 and 6 patients, respectively) to make two VPC plots for each treatment arm. Assuming a Bliss independence model for the combined effect, we obtained satisfactory prediction discrepancies plots for all treatment arms (Figure 5), which indicated that this model without PD interaction was sufficient to characterize the combined effect. Although a slightly synergistic effect was observed in vitro, [START_REF] Paeshuyse | The non-immunosuppressive cyclosporin DEBIO-025 is a potent inhibitor of hepatitis C virus replication in vitro[END_REF] the data of this study did not allow ruling out any significant synergy.

With this PK-VK model, the improved virologic response observed in HCV genotype-2/3 patients receiving dual-therapies was attributed to a larger loss rate of infected cells ( = 0.39 day -1 vs 0.22 day -1 in genotype-2/3 and genotype-1/4, respectively, p=10 -6 ) and a lower IC50peg-IFN than in genotype-1/4 patients (0.146 µg.L -1 vs 2.26 µg.L -1 , corresponding to 99.1% vs 86.3% for peg-IFN, p=10 -

8

). These results are consistent with previous studies comparing VK parameters between HCV genotype-2/3 and genotype-1/4 patients with peg-IFN/ribavirin. [START_REF] Neumann | Differences in viral dynamics between genotypes 1 and 2 of hepatitis C virus[END_REF][START_REF] Medeiros-Filho | Differences in viral kinetics between genotypes 1 and 3 of hepatitis C virus and between cirrhotic and non-cirrhotic patients during antiviral therapy[END_REF] Our analysis revealed that the loss rate of infected cells in the alisporivir monotherapy arm was also larger in genotype-2/3 patients (=0.41 in genotype-2/3 vs 0.23 day -1 in genotype-1/4 patients, respectively, p<10 -11 ), supporting that larger loss rate of infected cells was probably inherent to HCV genotypes and independent from the drug. In contrast, we could not show any significant effect of HCV genotypes on alisporivir effectiveness; however this effect was associated with a large confidence interval. Further studies with an appropriate design allowing better estimation of antiviral effectiveness, i.e., with frequent sampling in the first 3 days of treatment, [START_REF] Guedj | Design evaluation and optimization for models of hepatitis C viral dynamics[END_REF] are needed to correctly evaluate this effect. Altogether our analysis suggested that the larger response observed in HCV genotype-2/3 receiving alisporivir monotherapy was due to an intrinsically larger loss rate of infected cells and not to a differential effect of alisporivir in blocking virus production.

Next, in order to validate our model on an independent dataset, we used our model to predict the early virologic response (until week 12) with the design of the VITAL-1 study, where alisporivir was given in combination with ribavirin or peg-IFN for 24 weeks. To do this, two main simplifications were made. First, the effect of ribavirin was neglected, consistent with the hypothesis that ribavirin primarily acts against drug-resistant virus and has no or low effect on drug-sensitive virus. [START_REF] Rotman | Effect of ribavirin on viral kinetics and liver gene expression in chronic hepatitis C[END_REF][START_REF] Pawlotsky | Antiviral action of ribavirin in chronic hepatitis C[END_REF] Second, our model was built on short term data until day 29 where no resistance-related viral breakthrough was observed. Therefore we excluded from our analysis patients treated with alisporivir monotherapy without ribavirin where significant prevalence of non-PK related suboptimal response (18 among 83 patients, corresponding to 21.5%) [START_REF] Flisiak | The cyclophilin inhibitor Debio-025 shows potent anti-hepatitis C effect in patients coinfected with hepatitis C and human immunodeficiency virus[END_REF] was reported which requires to incorporate other mechanisms such as resistance.

Therefore three dosing regimens were considered, two with alisporivir/ribavirin (600 or 800 mg QD) and one with alisporivir/peg-IFN (600 mg QD). The prediction was complicated by the responseadapted treatment strategy. Interestingly, our predicted proportions of patients with undetectable virus matched very well the observed results in all groups, including the IFN-free regimen. A small discrepancy between predictions and observations was noticed at week 6, where we slightly overestimated the proportion of patients with undetectable HCV RNA in the group receiving alisporivir 800 mg without IFN (57.4% vs 45.0%). Nevertheless, the observed response was probably artefactually low, as hinted by the fact that it was 5 points less than that observed with alisporivir 600 mg without IFN. Our model also predicted slightly lower responses at week 12 for these two IFN-free arms (~93% vs 98% observed in VITAL-1) and this under-prediction could be explained by the fact that ribavirin could help to improve the second phase decline in patients whose responses to peg-IFN are low while in the simulation, we ignored this effect of ribavirin on add-on peg-IFN treatment.

Given the robustness of our early predictions, we extrapolated the proportion of patients considered as cured after 24 weeks of treatment, using a theoretical cure boundary. This cure boundary is a theoretical threshold for viral eradication, defined as less than one viral particle or infected cell remaining in 15L of extracellular fluid [START_REF] Dixit | Modelling how ribavirin improves interferon response rates in hepatitis C virus infection[END_REF] . It was shown to provide good predictions for SVR in previous studies. [START_REF] Dixit | Modelling how ribavirin improves interferon response rates in hepatitis C virus infection[END_REF][START_REF] Snoeck | A comprehensive hepatitis C viral kinetic model explaining cure[END_REF] Here as well our predictions nicely matched reported results (Table 2). Our simulations also suggested that treatment duration could be optimized using early virologic response.

In conclusion, we introduced a PK-VK model for drug combination. This model could well describe VK observed with different dosing groups of alisporivir given alone or with peg-IFN. Moreover this model provided good predictions for both early and long-term response of another study. Overall our approach demonstrates the use of mathematical modeling to anticipate treatment outcome in the coming era of oral combinations therapy for chronic HCV infections.

Methods

Alisporivir PK structural model

We built the structural model using frequent sampling PK data at day 29 (steady-state) in patients receiving alisporivir/peg-IFN . For this, we tested one-, two-, three-compartment models, linear and Michaelis-Menten elimination, linear absorption with or without a latent time. We then included day 1 data and trough concentrations obtained in this subset of patients to characterize the change in PK over time. This was achieved by allowing the elimination parameters and the apparent bioavailability to change over time:

( ) = - - (exp(-) -1)
where (t) is value of the concerning parameter  at time t, 0 is the value at time 0, inf is the maximum/minimum value achieved at infinity and k is the changing rate.

Peg-IFN PK structural model

A one-compartment model was fitted to all the available data. [START_REF] Bruno | Viral dynamics and pharmacokinetics of peginterferon alpha-2a and peginterferon alpha-2b in naive patients with chronic hepatitis c: a randomized, controlled study[END_REF][START_REF] Harris | Pegylation: a novel process for modifying pharmacokinetics[END_REF][START_REF] Dahari | Pharmacodynamics of PEG-IFN-alpha-2a in HIV/HCV co-infected patients: implications for treatment outcomes[END_REF]

PK interaction

Once the PK structural models of both alisporivir and peg-IFN were determined, the impact of alisporivir or peg-IFN on the other drug was evaluated by including alisporivir (yes/no), peg-IFN (yes/no) as a covariate.

PK individual parameters

After PK models for both drugs were obtained, individual parameters were retrieved as the Empirical Bayes Estimates (EBEs). The PK predictions obtained with these EBE and dosing information were used as driving functions in VK model.

Viral kinetic modeling

Let f the function describing the structural model, the statistical model for the observation Oij (drug concentrations or log10 viral loads) of subjects i at time tij is:

= , +
wherei is the vector of parameters of subject i and eij is the residual error.

The individual parameters i are supposed to follow log-normal distribution:

= exp( × ) exp ( )
or logit-normal distribution (for the bioavailability parameter):

= + (1 -) exp(- × ) exp(-)
where µ is the fixed effects, representing population values, cov is the covariate effect, COVi is the binary covariate (treatment combination or genotype 1/4) and i is the random effect, supposed to follow N(0, ²).

The residual error eij were assumed to follow N(0, ²ij), where ²ij can be additive ( = 0) , proportional ( = 0) or combined model:

= + ,
We supposed a combined and an additive model for PK and VK error model, respectively.

Parameter estimation

Parameters were estimated using the SAEM algorithm implemented in MONOLIX 4.2.2 to handle data below LOQ. Models were compared using BIC values. Covariate was selected using stepwise method, based on p-value of likelihood ratio test. The likelihood was obtained using important sampling method.

Only random effects correlations with a Pearson's coefficient higher than 0.7 were tested. Standard goodness-of-fit plots, visual predictive checks, prediction discrepancies [START_REF] Comets | Computing normalised prediction distribution errors to evaluate nonlinear mixed-effect models: the npde add-on package for R[END_REF][START_REF] Nguyen | Extension of NPDE for evaluation of nonlinear mixed effect models in presence of data below the quantification limit with applications to HIV dynamic model[END_REF][START_REF] Mentré | Prediction discrepancies for the evaluation of nonlinear mixedeffects models[END_REF] were used for model evaluation.

External validation

We evaluated the predictive ability of the PKVK model by comparing model predictions with the virologic responses obtained in patients genotype 2/3 in three alisporivir-containing arms of the VITAL-1 study. [START_REF] Gallay | Profile of alisporivir and its potential in the treatment of hepatitis C[END_REF] All the three groups received a loading dose of 600 mg alisporivir BID for 1 week, followed by alisporivir 600 mg QD + Ribavirin (ribavirin) (arm VITAL-1b, N=84), alisporivir 800 mg QD + ribavirin (arm VITAL-1c, N=94) and alisporivir 600 mg QD + peg-IFN (arm VITAL-1d, N=39).

Patients with RVR, i.e., HCV RNA below LOQ (<25 IU/mL) at treatment week 4, continued the same treatment until week 24, while non-RVR patients switched to alisporivir 600 mg QD plus peg-IFN/ribavirin from week 6 to week 24. Of note, the VITAL-1 study also comprised a group treated with alisporivir monotherapy. But a large proportion of patients developed non-PK related suboptimal response which requires other mechanism (e.g. resistance) to be incorporated. [START_REF] Flisiak | The cyclophilin inhibitor Debio-025 shows potent anti-hepatitis C effect in patients coinfected with hepatitis C and human immunodeficiency virus[END_REF] Thus this arm was excluded from our simulation.

We used the PK-VK model (including residual errors) to simulate the virologic responses in the three alisporivir-containing arms of VITAL-1, considering the adaptive treatment and the number of patients retained in per protocol analysis. We assumed 100% treatment adherence and considered that ribavirin could prevent the growth of resistant viruses but had no effect on drug sensitive viruses [START_REF] Rotman | Effect of ribavirin on viral kinetics and liver gene expression in chronic hepatitis C[END_REF] , i.e., we simulated alisporivir/ribavirin co-treatment using only the PKVK model of alisporivir. For each simulated patient, viral eradication (i.e., SVR) was considered attained if the model predicted less than 1 virion or infected cells in the whole extracellular fluid volume, i.e., 6.7x10 -5 copy(cell)/mL, during treatment period. 46 Observed data are presented in blue; data below the quantification limit are presented in red.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. Spaghettiplots of drugs concentration over time.

Figure 5 .

 5 Figure 5. Prediction discrepancies for the PK-VK model for different treatment arms. Dashed lines in blue represent 10 th and 90 th observed percentiles, dashed line in red represents the 50 th observed percentile. Blue and pink shaded areas are the 95% prediction interval of the corresponding percentiles.

Figure S1 .

 S1 Figure S1. Median of observed data and median predicted by the structural model for the last dose, obtained from 1000 simulations, for the last dose (first row) and for the first dose (last row). This model over-predicted concentration obtained at the first dose of alisporivir

Figure S5 .

 S5 Figure S5. VPC for peg-IFN PK model for all study period (left pattern), day 1 (middle pattern) and the last day of treatment (right pattern). Dashed lines in blue represent 10 th and 90 th percentiles of the observations; dashed line in red represents the 50 th percentile of the observations. Blue and pink shaded areas are the 95% prediction interval of the corresponding percentiles.

Figure S6 .

 S6 Figure S6. VPC for the PK-VK model for all data (left pattern), for genotype-1/4 patients (middle pattern) and for genotype-2/3 patients (right pattern). Dashed lines in blue represent 10 th and 90 th percentiles of the observations; dashed line in red represents the 50 th percentile of the observations. Blue and pink shaded areas are the 95% prediction interval of the corresponding percentiles. Data below the quantification limit were presented as red closed circle.

Table legends Table 1 .

 legends1 Parameter estimates for PK models of alisporivir, peg-IFN and for the PK-VK model

	Parameters	Fixed effects	Interindividual variability
		Estimates (RSE %)  (%) (RSE %)
	PK parameters of alisporivir
	Tlag (h)	0.453 (1)	5.1 (20)
	ka (h -1 )	2.77 (15)	46.8 (25)
	k12 (h -1 )	0.256 (9)	26.6 (31)
	k21 (h -1 )	0.0419 (12)	54.6 (16)
	V (L)	294 (8)	36.1 (14)
	Vm (Peg-IFN = 1) (mg.h -1 )	115 (24)	25.9 (41)
	Vm (Peg-IFN = 0) (mg.h -1 )	55 (24)
	Km (Peg-IFN = 1) (mg.L -1 )	2.44 (33)	39.3 (50)
	Km (Peg-IFN = 0) (mg.L -1 )	0.64 (11)
	F0	0.27	0 (-)
	Finf	0.853	97.6 (21)
	kF (h -1 )	0.0341	0 (-)
	cor(Vm,Km)	-	0.26 (241)
	a (µg/L)	6.6 (11)
	p (%)	34.1 (4)
	PK parameters of peg-IFN
	ka (h -1 )	0.0392 (12)	69.7 (13)
	V (L)	12 (5)	51.4 (11)
	k (h -1 )	0.00603 (7)	27 (14)
	a (µg/L)	0.243 (26)
	p (%)	30.0 (5)
	VK parameters
	VL0 (IU.mL -1 )	1.14 x 10 6 (20)	166 (9)
	c (day -1 )	6 (-)	0 (-)
	 in GT-1/4 (day -1 )	0.217 (8)	43.1 (9)
	 in GT-2/3 (day -1 )	0.393 (9)
	 (mL.virion -1 .day -1 )	5.6 x 10 -7 (2)	171 (13)
	IC50alv (µg.L -1 )	43.4 (38)	276 (10)
	IC50peg in GT-1/4 (µg.L -1 )	2.26 (43)	201 (13)
	IC50peg in GT-2/3 (µg.L -1 )	0.146 (54)
	a (log10IU/mL)	0.366 (4)

Table 2 .

 2 (1) Observed virologic response in VITAL-1 study; (2) Predicted virologic responses obtained by simulation using PK-VK population parameters (including residual errors for VK data). The response-adapted therapy and the number of patients remained in per protocol analysis were taken into account in simulation. Simulation was conducted assuming no ribavirin effect on virologic response and 100% adherence to treatment.

				(1)
		Observed % of patients below LOQ at	Observed SVR (%)
		Week 2	Week 4	Week 6	Week 12
	VITAL-1b* (N=77)	22.0	37.0	50.0	99.0		91.0
	VITAL-1c* (N = 80)	20.0	42.0	45.0	98.0		91.0
	VITAL-1d* (N=35)	62.0	85.0	90.0	96.0		91.0
				(2)
		Predicted % of patients below LOQ** at	Predicted SVR for a 24-week treatment** (%)
		Week 2	Week 4	Week 6	Week 12	Based on viral load	Based on infected cells

nd and

rd column, respectively. Observed data are presented as circle; data below the quantification limit are presented as star symbols.

Acknowledgements

The authors would like to thank Nikolai V. Naoumov for kindly providing the data. Comments and inputs given by Novartis HCV clinical team have been a great help in improving the manuscript. I would also like to thank Hervé Le Nagard and François Cohen for letting me with use the Fisher Machine for the simulation.

The following VK model was fitted to viral load data:

where T, I and V represent target cells, infected cells and free virus, respectively. We assumed a constant liver volume (i.e., T+I) during the study period. Infected hepatocytes die at rate δ and release virions at a rate p per cell. Virions infect target cells with infection rate  and are cleared from serum with rate c.

Anti-viral drugs are supposed to act mainly by blocking virion production, p, with effectiveness ε.

Because c cannot be estimated when there is no viral load data during the first week of treatment, we fixed c = 6 day -1 as done previously. [START_REF] Guedj | Hepatitis C viral kinetics with the nucleoside polymerase inhibitor mericitabine (RG7128)[END_REF] Next the concentration effect relationship was modeled using an Emax model:

where IC50ALV and IC50peg-IFN are concentrations of alisporivir and peg-IFN, respectively, needed to achieve an effectiveness of 50%. CALV(t) and Cpeg-IFN(t) are the drug concentrations predicted at time t.

Lastly the total effectiveness of alisporivir and peg-IFN was modeled using a Bliss independence equation to describe the combined effectiveness of two drugs having independent mechanisms of action [START_REF] Chou | Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies[END_REF] :

The effect of HCV genotypes was studied by including the covariate genotype1/4 (yes/no) on all VK parameters and treatment effectiveness.

Individual mean effectiveness

VK individual parameters were retrieved as the Empirical Bayes Estimates (EBEs). Then, the individual average effectiveness over a dosing interval [ti, tj] is defined by:

Study Highlights

What is the current knowledge on the topic?

Alisporivir is a host-targeting agent against HCV currently in phase 3 of clinical development. In a phase 2 dose-ranging study, the combination of alisporivir/peg-IFN for 28 days led to a rapid virologic decline for various HCV genotypes.

What question does this study address?

We built a pharmacokinetic/viral kinetic model to estimate antiviral effectiveness of alisporivir and peg-IFN and predicted the virologic responses with other therapeutic strategies.

What this study adds to our knowledge?

This pharmacokinetic/viral kinetic model accounted for PK/PD interactions in drug combination in HCV. Alisporivir is highly effective at doses higher than 600 mg QD and acts additively with peg-IFN without evidence of synergism. Alisporivir effectiveness was not found to be significantly different across genotypes. Faster virologic response in genotype-2/3 patients was due to a more elevated loss rate of infected cells and higher sensitivity to peg-IFN.

How this might change clinical pharmacology and therapeutics?

This study showed that model from early viral kinetics could be useful to predict treatment outcome and anticipate future clinical studies.

Conflict of Interest


Maya Khalifé

Astrid Decoene

Filipa Caetano

Ludovic De Rochefort

Emmanuel Durand

Dima Rodríguez

Estimating absolute aortic pressure using MRI and a one-dimensional model

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

To date, blood pressure (BP) is one of the most useful clinical indicator of cardiovascular disease. Hypertension, more specically, is a leading predictor of death in atherosclerosis diseases worldwide Cohn et al. [START_REF] Cohn | Surrogate markers for cardiovascular disease functional markers[END_REF], Collaboration [START_REF] Collaboration | Age-specic relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF]. Therefore, measuring BP is of great interest for diagnosis and risk prevention of cardiovascular events. An elevated pressure gives information about the aortic state, the presence of atherosclerotic plaques, stenosis, calcication or aneurisms. In a clinical routine, a sphygmomanometer is used to measure systolic and diastolic brachial pressure. However, due to reexion in the distal arteries, the aortic pressure waveform is altered while traveling through the vascular system. Thus distortion of the wave shape as well as systolic amplication occur on the systolic pressure measured in the brachial artery O'rourke et al. [START_REF] O'rourke | Pressure Wave Transmission along The Human Aorta: >Changes with Age and in Arterial Degenerative Disease[END_REF], Park and Guntheroth [START_REF] Park | Direct Blood Pressure Measurements in Brachial and Femoral Arteries in Children[END_REF], Salvi [START_REF] Salvi | Pulse Waves: How Vascular Hemodynamics Aects Blood Pressure[END_REF]. Although models and transfer functions to link brachial BP to aortic pressure exist Chen et al. [START_REF] Chen | Validation of Carotid Artery Tonometry as a Means of Estimating Augmentation Index of Ascending Aortic Pressure[END_REF][START_REF] Chen | Estimation of Central Aortic Pressure Waveform by Mathematical Transformation of Radial Tonometry Pressure: Validation of Generalized Transfer Function[END_REF], Liang [START_REF] Liang | Numerical validation of a suprasystolic brachial cu-based method for estimating aortic pressure[END_REF],

wave reection in the arterial system makes it dicult to reproduce the wave contour with great delity from such methods. Until now, the gold-standard of aortic pressure measurement is catheterization, which is invasive and not repeatable in a routine procedure Murgo et al. [START_REF] Murgo | Aortic input impedance in normal man: relationship to pressure wave forms[END_REF], Skinner and Adams [START_REF] Skinner | Outpatient cardiac catheterisation[END_REF]. In recent years, imaging techniques have been used to assess pressure gradients from velocity or acceleration maps and its combination with uid mechanics equations has been exploited in order to measure the BP non-invasively. Doppler Ultrasound (US) used to measure blood velocity in the arteries was associated to the standard simplied Bernoulli equation in order to determine pressure dierences between two measurement sites. This latter technique is reported to be userdependent and error-prone due to the wave angle of incidence Zananiri et al. [START_REF] Zananiri | A Comparative-Study of Velocity-measurements in Major Blood-vessels using Magnetic-Resonance-Imaging and Doppler Ultrasound[END_REF], Muhler et al. [START_REF] Muhler | Evaluation of Aortic Coarctation after Surgical Repair -Role of Magnetic-Resonance-Imaging and Doppler Ultrasound[END_REF], hence its accuracy in determining the maximum velocity in the artery is debatable. Also, extending US to other situations is limited by the inapplicability of Bernoulli equation to unsteady ows Yang et al. [START_REF] Yang | Computation of ow pressure elds from magnetic resonance velocity mapping[END_REF].

Phase-Contrast (PC) MRI allows accurate encoding of the blood velocity in the arteries in the three directions; hence, it has been largely used for non-invasive pressure estimation. Some authors computed pressure dierences using Poisson equation Yang et al. [START_REF] Yang | Computation of ow pressure elds from magnetic resonance velocity mapping[END_REF], others integrated the Navier-Stokes (NS) equations using MRI velocity maps Tyszka et al. [START_REF] Tyszka | Estimation of relative cardiovascular pressures using time-resolved three-dimensional phase contrast MRI[END_REF], Thompson and McVeigh [44], Bock et al. [START_REF] Bock | In vivo noninvasive 4D pressure dierence mapping in the human aorta: phantom comparison and application in healthy volunteers and patients[END_REF], Ebbers et al. [START_REF] Ebbers | Estimation of relative cardiovascular pressures using time-resolved threedimensional phase contrast MRI[END_REF] or acceleration maps to avoid computational errors arising from velocity derivation Buyens et al. [START_REF] Buyens | Calculation of left ventricle relative pressure distribution in MRI using acceleration data[END_REF]. These methods compute a pressure gradient, and to estimate an absolute pressure, require a zero-pressure reference point which has to be measured with a catheter, or user-dened in a gross assumption. Consequently, these methods are not an alternative to catheter measurements, which remain more accurate. In this work, we propose a non-invasive technique to extract absolute pressure in a straight artery from MR velocimetry using a biomechanical one-dimensional (1D) model as proposed by Formaggia et al. Formaggia et al. [18]. Although a three-dimensional (3D) model gives a more complete and realistic reproduction of the aortic ow, 1D models are able to describe the non-linear ow behavior in larger elastic vessels Hughes and Lubliner [START_REF] Hughes | On the one-dimensional theory of blood ow in the larger vessels[END_REF]. As these models are reasonably accurate, they are widely used for aortic ow simulations. Their accuracy has been assessed by comparison with experimental data acquired in a tube phantom Bessems et al. [START_REF] Bessems | Experimental validation of a time-domain-based wave propagation model of blood ow in viscoelastic vessels[END_REF], in a distributed arterial model Olufsen et al. [START_REF] Olufsen | Numerical simulation and experimental validation of blood ow in arteries with structured-tree outow conditions[END_REF], Alastruey et al. [START_REF] Alastruey | Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements[END_REF] and in vivo Reymond et al. [START_REF] Reymond | Validation of a one-dimensional model of the systemic arterial tree[END_REF], Alastruey et al. [START_REF] Alastruey | Modelling pulse wave propagation in the rabbit systemic circulation to assess the eects of altered nitric oxide synthesis[END_REF]. Furthermore, the 1D model relies on the establishment of a pressure law consisting of a relation between pressure and vessel section area. The pressure laws used in 1D model equations are determined experimentally with invasive measurements or estimated using complex algorithms. Additionally, these pressure laws are complicated and involve the determination of multiple parameters. Here, we propose a pressure law based on the aortic compliance which reects arterial elasticity and can be determined non-invasively with MRI. Using this pressure law, we coupled the 1D model with realistic boundary conditions measured by MRI to estimate absolute pressure in the aortic segment. The derived model was tested on a straight compliant phantom and computed pressure was compared to experimental pressure measurements recorded simultaneously with the MRI acquisition. The model was also tested on a real-sized compliant aortic phantom Then, the model is used to estimate BP on healthy volunteers.

Methods

The 1D-model, studied in Formaggia et al. [START_REF] Formaggia | One-dimensional models for blood ow in arteries[END_REF], is a reduced model describing blood ow in arteries and its interaction with wall motion. The artery is considered as a cylindrical compliant tube of length L and radius R (R L).

The model derivation approach consists of integrating the NS equations on a generic section S. Some simplifying assumptions are made:

• the model assumes axisymmetry

• the wall displacement is supposed to solely be in the radial direction

• pressure is assumed to be uniform in each section

• the axial velocity u z is predominant.

For large arteries such as the aorta, it is a safe assumption to consider a at velocity prole for the boundary layer is very thin compared to the vessel radius Olufsen et al. [START_REF] Olufsen | Numerical simulation and experimental validation of blood ow in arteries with structured-tree outow conditions[END_REF].

The main variables of the problem are (Figure 1):

• axial section area A A(t, z) = ˆS(t,z) dσ (1)
• mean ow Q Q(t, z) = ˆS u z dσ (2)
• blood pressure P (t, z),

where dσ denotes the area element. Their evolution is described by the momentum conservation and the mass conservation equations, while considering a constant viscosity:

∂A ∂t + ∂Q ∂z = 0 (3)
∂Q ∂t + ∂ ∂z Q 2 A + A ρ ∂P ∂z + K r Q A = 0 (4)
K r is the friction coecient; for a at prole in blood ow problems,

K r = 22πν,
where ν is the vessel wall kinematic viscosityFormaggia and Veneziani [START_REF] Formaggia | Reduced and multiscale models for the human cardiovascular system[END_REF]. ρ is the blood density.

Pressure law To close the system, a relation between the section area A and the pressure P is dened. This pressure law depends on section area A 0 (z) = πR 2 0 (z) at time t = 0 and on a set of parameters β = (β 0 , β 1 , ...β n ) related to the vessel wall physical and mechanical properties. P ext is the external pressure exerted by the environment of the vessel whereas P is the intravascular pressure. The pressure law should meet these conditions:

• ∂P ∂A > 0

• when A = A 0 , P = P ext .

Some pressure laws are proposed in the literature to link the pressure behavior to the section area. A frequently used relation between A and P is stated as follows Quarteroni and Formaggia [START_REF] Quarteroni | Mathematical modelling and numerical simulation of the cardiovascular system[END_REF], Alastruey [START_REF] Alastruey | Numerical modelling of pulse wave propagation in the cardiovascular system: development, validation and clinical applications[END_REF]:

P -P ext = β 0 √ A - √ A 0 A 0 ( 5
)
β 0 = √ πh0E
1-ξ 2 using Young's modulus E and the vessel thickness h 0 and Poisson's ratio ξ = 0.5 for an incompressible material deformed elastically at small strains.

A more general law proposed in Hayashi et al. [START_REF] Hayashi | Stiness and elastic behavior of human intracranial and extracranial arteries[END_REF], Smith and Hunter [START_REF] Smith | Generation of an anatomically based geometric coronary model[END_REF] is written as:

P -P ext = β 0 A A 0 β1 -1 . (6)
The parameters β 0 and β 1 values can be obtained either by tting experimental pressure versus section measurements Smith [START_REF] Smith | A computational study of the interaction between coronary blood ow and myocardial mechanics[END_REF], or by solving an inverse problem with a 3D-model solution Martin et al. [START_REF] Martin | Parameter identication for a one-dimensional blood ow model[END_REF], Dumas [START_REF] Dumas | Inverse problems for blood ow simulation[END_REF]. Hence, these laws cannot be determined non-invasively, and need knowledge about the vessel properties. Additionally, they seem too complex to determine during a clinical application. Finding a simple non-invasive way to determine a pressure law is of great interest, consequently, we turned to the compliance. Indeed, in physiological conditions, the aorta section deformation is commonly assumed to be linked to the intravascular pressure by the aortic compliance Langewouters et al. [START_REF] Langewouters | The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model[END_REF].

Aortic compliance establishes a linear relation between the pressure and section area; it represents the arterial wall ability to deform in response to a pressure variation Conrad [START_REF] Conrad | Flow relationships in collapsible tubes[END_REF]. Also, it is clinically used and can be estimated non-invasively by measuring the pulse wave velocity (PWV) in MRI. Hence, it provides a simple and non-invasive pressure law that can be applied in vivo and, as it includes compliance changes, is patient-specic.

In fact, compliance is given by the ratio of section variation to pressure variation :

C = dA dP t (7)
where P t = P -P ext is the transmural pressure. The compliance is considered as a local constant on an arterial segment. By integrating equation ( 7) and knowing that when P t = 0, i.e. P = P ext , A = A 0 , we write A = CP t + A 0 where A 0 is the section area at the equilibrium state.

P -P ext = A 0 C A A 0 -1 (8)
In the limit of small displacements, this pressure law is equivalent to the commonly used linear law (( 5)) for

β 0 = 2A 3/2 0 C , (9)
In fact, by linearizing (( 5)) (see also Alastruey et al. [START_REF] Alastruey | Physical determining factors of the arterial pulse waveform: theoretical analysis and calculation using the 1-D formulation[END_REF]), we have:

β 0 √ A - √ A 0 A 0 = β 0 2A 3/2 0 (A -A 0 ) + O(A -A 0 ) 2

Numerical approximation

System (3)-( 4) is written in the conservative form :

∂U ∂t + ∂F(U) ∂z = S(U) (10)
where

• U =    A Q    • F =    Q Q A 2 + C 1   , C 1 (A) = ´A 0 c 2 1 (τ ) dτ and c 1 = c 1 (A) = A ρ ∂P ∂A , • S =    0 -K r Q A    Since A > 0, the matrix DF possesses two real distinct eigenvalues λ ± = Q A ± c 1
and system (10) is a hyperbolic system of partial dierential equations Formaggia and Veneziani [START_REF] Formaggia | Reduced and multiscale models for the human cardiovascular system[END_REF]. The blood ow is assumed to be sub-critical, thus λ -< 0, and λ ± have opposite signs.

Following Formaggia and Veneziani [START_REF] Formaggia | Reduced and multiscale models for the human cardiovascular system[END_REF], system [START_REF] Buyens | Calculation of left ventricle relative pressure distribution in MRI using acceleration data[END_REF] is written in diagonal form, which is more suitable to understand the solution behavior and for numerical purpose:

∂W ∂t + Λ ∂W ∂z = L • S (11)
where

W = (W 1 , W 2 ) are the characteristic variables, Λ =    λ + 0 0 λ -    and L =    T + T -    is a matrix of left eigenvalues of DF such that L • DF = Λ • L.
In the case of the law (8), we have

c 1 = A ρC , W 1 = Q A + 2 A ρC , W 2 = Q A -2 A ρC . (12)
Inversely, the primitive set of variables (A, Q) is written in terms of the diagonal variables (W 1 , W 2 ) as

A = ρC W 1 -W 2 4 2 , Q = A W 1 + W 2 2 . ( 13
)
For system [START_REF] Buyens | Calculation of left ventricle relative pressure distribution in MRI using acceleration data[END_REF] approximation, a nite volume scheme is dened by a grid over

[0, L] × [0, T ],
with uniform mesh-spacing ∆z and time step ∆t n . The cells are

given by [z i , z i+1 [×[t n , t n+1 [, where z i = i∆z, i = 0, . . . , M, with M ∆z = L, and t n = n∆t n , n = 0, . . . , N, with N ∆t N = T. We seek U n i approximating

1 ∆z ´zi+1 z i U (x, t n )dz.
The numerical scheme is a classical nite volume scheme of Rusanov type for hyperbolic problems:

1 ∆t n (U n+1 i -U n i ) + 1 ∆z (F n i+1/2 -F n i-1/2 ) -S n i = 0, (14)
F i+ 1 2 = 1 2 F (U n i ) + F (U n i+1 ) -λ i+ 1 2 (U n+1 i -U n i ) with λ i+ 1 2 = max{λ -(U n i ), λ + (U n i+1
)}, and where Since ows involved in this model are sub-critical, the numerical scheme [START_REF] Collaboration | Age-specic relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF] requires one condition at each boundary. An admissible condition imposes the incoming characteristic and allows the wave corresponding to the outgoing characteristic to leave the domain. As W 2 remains constant along the characteristic curve dened by

S n i =    0 -K r Q n i A n i   .
dz dt = λ -(U (z, t)), if we approximate λ -(U (z, t)) by its numer- ical value λ -(U (z, t n )) in [t n , t n+1 [, we consider that W 2 (z, t n+1 ) W 2 (z -{λ -} n 0 ∆t n , t n ), thus W n+1 2,in = ∆z + λ∆t ∆z W n 2,-1 - λ∆t ∆z W n 2,0
where λ = λ n -0 and ∆t = ∆t n . Similarly, we impose

W n+1 1,out = ∆z -λ∆t ∆z W 1 n M +1 + λ∆t ∆z W 1 n M where λ = {λ + } n M .

Boundary conditions

We used MRI measured boundary conditions on the inlet of the tube phantom or the arterial segment. Flow measured on the entrance was imposed as a condition on the rst mesh. We extrapolated, as explained above, the inlet value of W 2 , and A was computed by using relations [START_REF] Chen | Validation of Carotid Artery Tonometry as a Means of Estimating Augmentation Index of Ascending Aortic Pressure[END_REF] and [START_REF] Cohn | Surrogate markers for cardiovascular disease functional markers[END_REF]. The outlet Frank Frank [START_REF] Frank | Die grundform des arteriellen pulses[END_REF] by removing pressure and ow oscillations Stergiopulos et al.

[43], Alastruey [START_REF] Alastruey | Numerical modelling of pulse wave propagation in the cardiovascular system: development, validation and clinical applications[END_REF]. The rst resistance Z c is the proximal aorta characteristic impedance, the compliance C in parallel to the resistance R 2 simulate respectively the volume compliance and the resistance of the vascular network found downstream the abdominal aorta (Figure 2).

The inlet ow of the R model, which is the 1D-model outlet ow (Q in-0D = Q out-1D ), is:

Q in-0D = P in-0D -P C Z c (15)
As for the RC-model:

C dP C dt - P out-0D -P C R 2 -Q in-0D = 0 (16)
P out-0D is the pressure at the 0D-model exit. We compute P C at each time step by explicitly discretizing equation ( 16):

P N +1 C = f (P N C , Q N in-0D , P out-0D )
then use the relation at the 1D-model exit:

Q N +1 in-0D = P N +1 in-0D -P N +1 C Z c MRI experiments
A straight exible silicone tube was used as a phantom to validate the 1D-model. The MRI setup (Figure 3), described in a previous work Khalife et al. [START_REF] Khalife | In vitro validation of non-invasive aortic compliance measurements using MRI[END_REF], consisted of the phantom connected to a non-magnetic CardioFlow 5000MR pump (Shelley Automation, Toronto, Canada) that can generate a programmable ow pattern. We used a blood mimicking uid (BMF, 62% water and 38% glycerol) with similar magnetic and physical properties as blood. It has a Newtonian behavior which is a safe assumption for blood circulating in large arteries Ottesen et al. [START_REF] Ottesen | Applied Mathematical Models in Human Physiology[END_REF]. Two MRI-compatible optical pressure sensors (Opsens, Québec, Canada) were introduced into the phantom. These sensors allow simultaneous pressure measurements used as validation for the simulation.

To test the 1D-model application on a curved segment and estimate the error, the tube phantom was replaced with an aortic phantom (Elastrat, Geneva, Switzerland) comprising an ascending aorta branch (AAo) and a descending aorta branch (DAo). Pressure sensors were introduced at the AAo entrance and at the DAo exit.

A sinusoidal waveform of amplitude 150mL.s -1 was applied in the tube phantom setup. In the aorta phantom setup, a ow wave similar to the one measured at the left ventricle exit was programmed on the pump control system with an amplitude of 250mL.s -1 and a period of 0.8s (75bpm).

Experiments were performed on a 1.5T MRI system (Achieva; Philips, Best, the Netherlands) using SENSE Flex-L coils. Phase-Contrast (PC) ow-encoded gradient-echo multiphase sequences were performed to assess through-plane velocity on two slices, one situated at the phantom entrance with the rst pressure sensor and the second at the phantom exit, with the second pressure transducer.

Scan parameters: spatial resolution: 1 × 1 × 8mm 3 , echo time/repetition time TE/TR = 5.75/9.59ms, velocity encoding V enc = 80cm.s -1 for a owrate of 150mL.s -1 and V enc = 100cm.s -1 for a owrate of 250mL.s -1 , 40 time frames per cardiac cycle with ECG-triggering on the pump signal.

For the in vivo feasibility of the 1D-model, 3 healthy subjects (male, ages 27, 30 and 33) were included in our study after having lled out a written consent form. All measurements were performed on a 1.5T MRI system with SENSE cardiac coil. PC ow-encoded sequence covering the thoracic aorta was acquired during breath-hold and ECG-gated gradient echo sequence (spatial resolution:

1.1 × 1.1 × 8mm 3 , TE/TR = 3.0/5.1 ms, V enc = 180cm.s -1 , 40 time frames per cardiac cycle). Data were acquired in an axial plane to measure through-plane velocity in the descending and the abdominal aorta (AbAo). A coronal plane of the aorta was acquired to measure the distance between velocity encoding sites.

Arm pressure was measured before and after MRI acquisitions and used as a reference for the 1D-model pressure simulations.

Compliance measurements MRI images were processed using an in-house code. Aorta section was segmented semi-automatically and visually assessed by the operator on the magnitude image then ow waveforms were computed from velocity data provided by the corresponding phase images. The inlet ow is thus used as the model input condition.

MRI non-invasive measurements were used to determine phantoms and aortic compliance through PWV. The method, described in a previous work Khalife et al. [START_REF] Khalife | In vitro validation of non-invasive aortic compliance measurements using MRI[END_REF], is based on the transit time (TT) method commonly used for the PWV estimationMurgo et al. [START_REF] Murgo | Aortic input impedance in normal man: relationship to pressure wave forms[END_REF], Avolio et al. [START_REF] Avolio | Eects of Aging on Changing Arterial Compliance and Left-Ventricular Load in a Northern Chinese Urban-Community[END_REF] (Figure 4).

For each experiment, the linear compliance is determined from PWV based on the Bramwell-Hill theory: C = A ρ(P W V ) 2 Bramwell and Hill [START_REF] Bramwell | Velocity of transmission of the pulse-wave: and elasticity of arteries[END_REF], Vulliémoz et al. [START_REF] Vulliémoz | Estimation of local aortic elastic properties with MRI[END_REF], where A is the vessel lumen area and ρ the uid density.

Simulations

Simulations were performed on an in-house program using Matlab (the Mathworks, Natick, USA) by setting the domain geometry, uid properties and MRI measured compliance for the pressure law. FlowQ, section area A and pressure P are computed on the whole domain as functions of time and space, then compared to measured data on the corresponding nodes.

The values of the input parameters were the following:

• Tube phantom : L = 40cm, C = 0.0012 cm 2 .mmHg -1 . The RCR-model components were estimated from measurements: Z c = 1.24 • 10 8 Pa.s.m -3

was computed using the approach proposed in Mitchell et al. [START_REF] Mitchell | Pulsatile Hemodynamics in congestive heart failure[END_REF], R 2 = 1.63 • 10 8 Pa.s.m -3 , C per = 0.288 cm 3 .mmHg -1 and P out-0D = 0.

• Aortic phantom:

L = 21.5cm, C = 0.051cm 2 .mmHg -1 , Z c = 2.63 • 10 6 Pa.s.m -3 , R 2 = 1.07 • 10 8 Pa.s.m -3 , C per = 0.137 cm 3 .mmHg -1 .
• In vivo : the RCR parameters at the outlet were taken from Alastruey et al. [START_REF] Alastruey | Lumped parameter outow models for 1-D blood ow simulations: Eect on pulse waves and parameter estimation[END_REF] because of the absence of invasive measurements. The character-

istic impedance is Z 0 = ρ c0 A0 . The peripheral resistance is R 2 = R T -Z c
where R T = 1.89 • 10 8 Pa.s.m -3 is the complete vascular system resistance.

The simulations were performed on a 100 meshgrid on a scope of 10 periods, with an average simulation time of 328 s. A and Q are computed at each time step,∆t = 10 -4 s. To evaluate results, the root mean square error in % (RMS)

was calculated between computed and measured pressure waves.

Furthermore, to evaluate the equivalence between the pressure law ( 5) and the compliance pressure law [START_REF] Bock | In vivo noninvasive 4D pressure dierence mapping in the human aorta: phantom comparison and application in healthy volunteers and patients[END_REF] proposed in this work, two simulations were carried out with the same ow conditions, but with dierent pressure laws. The ow waveform and the model parameters used were those of a physiological scenario as proposed in Alastruey et al. [START_REF] Alastruey | Lumped parameter outow models for 1-D blood ow simulations: Eect on pulse waves and parameter estimation[END_REF]: for the compliance pressure law.

A 0 = π • 10 -4 m 2 , β 0 = 4

Results

The results of the equivalence test between the pressure law ( 5) and the compliance pressure law (8) are depicted in gure 5. Based on these curves, we can say that the two pressure laws provide nearly identical results which means that the β 0 pressure law agree well with the linear pressure law with C = 2A 3/2 0 β0 , as in equation [START_REF] Bramwell | Velocity of transmission of the pulse-wave: and elasticity of arteries[END_REF]. The RMS between both pressure curves is 6.2% . Hence, even if the pressure to section area relation is not perfectly linear in the range of physiological pressure, the error committed by considering it linear is negligible.

In phantom experiments, simulated pressure was compared to pressure sensor measurements.

The inlet ow and the simulated pressure within the tube phantom are shown in gure 6 for the sinusoidal ow waveform of amplitude 150mL.s -1 .

Computed pressure curves show good agreement with the measured pressure in the tube phantom. RMS error computed between both is 8.15% and conrms the agreement of the model results with measurements.

Furthermore, the simulated pressure at the DAo outlet in the aortic phantom was compared to pressure sensor measurements acquired at the same location (Figure 7b). RMS between both pressure curves is 4.97%. Velocity measurement noise caused oscillations on the MR measured ow which were reported on the simulated pressure. This is due to the relatively low velocity measured in the descending branch. Thus, the ow was ltered before simulations (Figure 7a).

Beyond this, pressure curves simulated by the 1D-model on a straight segment formed by the DAo and the AbAo in healthy volunteers are plotted in gure 8. Their contour is in good agreement with pressure curves reported in vivo in the literature Schnabel et al. [START_REF] Schnabel | A technic of vascular catheterization with small plastic catheters: its utilization to measure the arterial pulse wave velocity in man[END_REF], Murgo et al. [START_REF] Murgo | Aortic input impedance in normal man: relationship to pressure wave forms[END_REF]. Also, according to the reection principle described in Salvi [START_REF] Salvi | Pulse Waves: How Vascular Hemodynamics Aects Blood Pressure[END_REF], the systolic pressure increases along the arterial tree and consequently the brachial systolic pressure overestimates the aortic systolic pressure up to 20mmHg Levick [START_REF] Levick | An introduction to cardiovascular physiology[END_REF]. Thus in gure 8, the simulated systolic pressure in the aorta is smaller than the systolic pressure measured on each volunteer. Mean arterial pressure (MAP) computed on the simulated and measured pressure is added in gure 8.

Discussion

In this work, we showed that by combining a 1D aorta model to non-invasive MRI velocity measurements, we were able to estimate aortic pressure. Flow MRI had commonly been used to derive pressure dierences, so estimating a local pressure without having to set a null pressure point is a step forward in cardiovascular diagnosis.

The MR-measured ow was used as input for the 1D-model and a coupled 0D-model was used as an output condition. The pressure law, which relies on invasive measurements in formal studies, is replaced by a non-invasive pressure law based on the aortic compliance. This compliance was measured from MRI assessed PWV. The model simulations were successfully validated on an MRI setup with a straight phantom, an aortic arch phantom and in vivo.

At rst, the simulations were applied on a simple straight tube to respect the model conditions and simplications. Pressure values obtained with simulations were in good agreement with measurements and showed satisfactory results as in previous studies Alastruey et al. [START_REF] Alastruey | Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements[END_REF]. Then, simulations were applied on an aortic phantom and results showed a negligible error when compared to pressure measurements, meaning that, in the physiological range of blood velocity, the 1D-model coupled with MRI ow measurements and subject-specic compliance pressure law would be enough to describe the aortic ow behavior. Results in vivo showed that it can estimate realistic pressure curves in the DAo, in agreement with the brachial pressure measured in the clinical routine. A bias was observed between the two brachial measurements carried out before and after the MRI exam due to the subject changing state going from agitated prior to the exam to rested state after the exam was completed. Another explanation could be the dierence in the subjects position (seated vs. supine) during measurements. Consequently, the measured arm pressures might have been dierent

than what the simulations would yield. This 1D-model, which is frequently used to model the aorta, is based on strong assumptions, considering only radial and axisymmetric wall deformations, constant pressure on a section along the tube axis, thus only depending on time and on the axial coordinate z, dominance of axial velocity, absence of turbulence and neglecting the vessel wall inhomogeneities. Nevertheless, the model serves its initial role of estimating an absolute pressure in a simple and fast non-invasive manner, thus it is widely used in large artery modeling for its fair reproduction of the physiological conguration. Even if the model is one-dimensional and must be applied to a straight segment, its application to a curved aortic phantom did not greatly aect the agreement of simulated pressure and measured pressure at the phantom outlet. However, the eect of an inertance in the 0D-model represented by an inductor in the electrical circuit could be considered. The inertance element as proposed in Segers et al. [START_REF] Segers | Three-and four-element Windkessel models: Assessment of their tting performance in a large cohort of healthy middle-aged individuals[END_REF] could absorb the oscillations which appear on the pressure curve.

The compliance based pressure law is simple, non-invasive and widely anchored in clinical practice and its association to the 1D-model may be more advantageous than the previously used pressure laws reported here. The model showed satisfactory results and good reproduction of invasive measurements recorded on a tube phantom. Additionally, we have shown that this simplied linear pressure law is in good agreement with the √ A law proposed in the literature.

Some improvements could be made to the 1D-model in the future. The model could be made more patient-specic by adapting the 0D-model parameters to the patient at the outlet boundary condition. Since these parameters are taken from the literature and are not patient-specic, previous work Willemet et al. [START_REF] Willemet | Validation of a 1D patient-specic model of the arterial hemodynamics in bypassed lowerlimbs: Simulations against in vivo measurements[END_REF] has proposed to measure peripheral resistance by introducing pressure catheters in the femoral artery. However, this is a disadvantage to a method which oers to be strictly non-invasive. Further studies are needed to propose a non-invasive way to measure the RCR parameters. Also, the model supposes a constant compliance along the artery which is a non-realistic condition when studying aortic disease or stenosis. This problem will be addressed in future studies.

In the future, it would be interesting to compare the pressure estimation found in our model with catheter measurements in an animal experiment. Also, more in vivo experiments would allow assessing the reliability and the repeatability of such estimation techniques in the aorta and to study pressure changes in patients suering from cardiovascular diseases or under various stress conditions.

Figure 1: The 1D model simplied geometry. It assumes that the artery is a straight cylinder of length L with a circular cross section A(t, z) that deforms in respect to the radial vector. The TT method for PWV measurement. The TT, which is the delay ∆t that a waveform needs to travel a distance L, is used to compute the wave velocity: P W V = L ∆t . Here, ∆t was measured between the feet of ow waveforms extracted at two sections: tangent lines are obtained by the linear regression of the upslope segment on each waveform.

6.4 6.5 6.6 6.7 6.8 6.9

6.4 6.5 6.6 6.7 6.8 6.9 (dotted line). Simulations are performed with a pulsed ventricular owrate and parameters given in Alastruey et al. [START_REF] Alastruey | Lumped parameter outow models for 1-D blood ow simulations: Eect on pulse waves and parameter estimation[END_REF]. Plotted results are such as: (a) ow waveform, (b) mean velocity and (c) pressure waveform simulated at three sites of the artery, at the inlet (0m), in the middle (0,2m) and at the exit (0,4m). Figure 7: (a) Flow mesured at the entrance of the aorta phantom (solid line). Due to the motion of the phantom and the low velocity amplitude, oscillations are visible at the measurements curves and are reported in the 1D model simulation results. Hence, the tted measured ow (dashed line) is used for pressure calculation. (b) Simulated vs. measured pressure at the exit of the aorta phantom with a ventricular ow of amplitude 250 mL.s -1 . Boundary conditions used are the ow at the entrance of the ascending aorta (AAo) and given by the 1D-0D coupled model at the outlet of the phantom.

  Time- space ∆t n must verify the well-known stability condition 2 max i |λ i+ 1 2 |∆t n ≤ ∆z. Numerical boundary conditions are prescribed, thus U n in = U n -1 and U n out = U n M +1 are given to compute U n+1 0 and U n+1 M at z = 0 and z = L respectively.

  boundary condition is obtained by coupling the 1D model with a 0D model that consists of a system of dierential equations linking pressure to ow at the 1D model exit. In fact, as shown in Alastruey et al.[START_REF] Alastruey | Lumped parameter outow models for 1-D blood ow simulations: Eect on pulse waves and parameter estimation[END_REF], the behavior happening beyond the 1D-modeled arteries can be represented by a 0D lumped-parameters model, also known as the Windkessel model, that simulates hemodynamics by electrical circuit analogy. Pressure gradient is represented by a potential difference, blood ow by the electrical current and hydraulic impedance by an electrical impedance. Hydraulic impedance combines friction loss, arterial wall elasticity and blood ow inertia, which are modeled by a resistance, a capacitor and an inductance respectively. A three-element (RCR) Windkessel model is a good compromise accounting for wall compliance and resistance in the peripheral network as well as proximal aortic impedance Wetterer[START_REF] Wetterer | Flow and pressure in the arterial system, their hemodynamic relationship, and the principles of their measurement[END_REF], Westerhof et al.[START_REF] Westerhof | Analog studies of the human systemic arterial tree[END_REF]. It is a combination of an R-model and an RC-model that improves considerably the behavior of the original two-element model proposed by Otto

3 √

 3 πhE for the β 0 pressure law, with E = 0.4•10 6 Pa, h = 1.5•10 -3 m, C = 7.85•10 -9 m 3 /Pa,

Figure 2 :

 2 Figure 2: Coupling the 1D model outlet with the 0D model: a three-element Windkessel model (RCR) forms the outlet boudary condition of the 1D model. (Q in-0D , P in-0D ) = (Q out-1D , P out-1D ) and (Q out-0D , P out-0D ) are ow and pressure at the inlet and the outlet of the RCR model respectively. To solve the RCR model, two sub-models, RC and R, are considered and then combined. P C is the pressure across C at the exit of the RC model and at the entrance of the R model.

Figure 3 :Figure 4 :

 34 Figure 3: Experimental MRI setup with the MR-compatible pump and the aorta phantom. Optical pressure sensors are introduced in two locations on the phantom.

Figure 5 :

 5 Figure 5: Results of the 1D model using pressure laws with β 0 in equation (5) (straight line) vs. with compliance C = 2A 3/2 0 β 0

Figure 6 :

 6 Figure 6: Pressure curves in the tube phantom with a sinusoidal ow waveform of amplitude 150 mL.s -1 . Boundary conditions used are the measured ow at the inlet (a) and given by the 1D-0D coupled model at the outlet. (b) Simulated pressure curves at the sensors locations are compared to pressure measurements.

(Psyst -P diast ).

Conict of interest statement

The authors declare that there are no conicts of interest.