Rapid Expansion Method Research Articles

Assessing the Efficacy and Stability of Rapid Maxillary Expansion and the Delaire Appliance in Skeletal Class III Malocclusion: A Narrative Review

Background: A Class III skeletal malocclusion is a very complex condition in orthodontics practice, and its solution is often carried out orthopedically. The most used orthopedic devices include rapid maxillary expansion (RME) and the Delaire Appliances to treat not only maxillary deficiency but also to correct the position of the upper jaw. Even if these interventions become sustainable in the short term, their long-term stability remains questionable. Aim: reveal a pretty complicated relationship between orthopaedic and orthodontic surgeries and their influences on the volume of the upper respiratory tract and craniofacial correction. Methods: We searched multiple online databases including; Web of Science, Scopus, Google Scholar, PubMed, and Cochrane Library. We used the following search strategy; "Rapid maxillary expansion" OR RME AND ("sleep apnea" OR "sleep-disordered breathing" OR "sleep-related breathing disorders" OR "sleep apnea-hypopnea syndrome" OR SAHS) AND (children OR pediatric) AND ("orthodontic treatment" OR "orthopaedic treatment" OR "maxillary expansion") AND (oximetry OR polysomnography). Results: RME and the Delaire appliance were seen to be promising in the short term with an outcome of correction of maxillary transverse deficiency and skeletal relationships. Nevertheless, the issue of the long-term stability was spotted, including with regard to growth. But nevertheless both interventions exhibited some really valuable additions in general treatment of skeletal class III malocclusion. Scientific Novelty: This literature review synthesizes and summarizes the current knowledge on the effectiveness and durability of RME and the Delaire splint concerning skeletal class III malocclusion, presenting perspectives on their short-term impact and long-term performance. With its emphasis on poorly understood issues and new research, it greatly assists in updating our knowledge about orthopedic treatments. Conclusion: In a relatively short period of time, the rapid maxillary expansion method as well as the Delaire appliance can produce positive outcomes in terms of correcting the retroposition in cranial base class III malocclusion. Although, long lasting stability issues remain a worry and make the case for continued research and clinical monitoring for achieving better treatment outcomes and evidence-based practices in orthodontics.

A numerical scheme based on the Taylor expansion and Lie product formula for the second‐order acoustic wave equation and its application in seismic migration

AbstractWe have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration.

Indications for treatment of rapid palatal expansion devices based on cone-beam computed tomography

Currently, many orthodontists often prescribe the method of rapid palatal expansion (RPE) to patients, however, there are no clear criteria for indications for the use of the method based on cone-beam computer tomography (CBCT) in the available literature.Aim. Determination of indications for the use of the technique of rapid palatal expansion based on the data of the CBCT.Materials and methods. A retrospective analysis of control and diagnostic models and CBCT of 40 patients with diagnosed narrowing of the upper jaw was carried out. All the subjects were divided into two groups: the first was patients who were treated with RPE devices. The second is patients whose orthodontic treatment used other methods (mainly expanding removable plate apparatus or partial bracket system). The analysis of control and diagnostic models according to the Pona method and the determination of palate narrowing by CBCT were carried out. Comparison of the two groups by a quantitative indicator having a normal distribution, provided that the variances were equal, was performed using the Student's t-test, with unequal variances – Welch's t-test. The comparison of the two groups by a quantitative indicator, the distribution of which differed from the normal one, was carried out using the Mann-Whitney U-test.Results. Analysis of measurements in the area of premolars and molars by the Pona method in both groups revealed no statistically significant differences (p = 0.093, p = 0.137, respectively). However, when analyzing the indicators obtained by measuring the width of the palate at the level of the deepest points of the bone base in the projection of the Pon line, it showed the presence of statistically significant differences (p < 0.001). The following values were obtained: the average width of the palate at the molar level in patients using the RPE method was 24± 2 mm, in the second group – 30± 2 mm. At the premolar level, the average width of the palate in patients treated with the RPE method is 20 mm, in group 2–22 mm.Conclusions. With a narrowing of the palate less than 26 mm at the level of the molars and less than 20 mm in the premolar region, the use of methods of rapid palatal expansion can be shown. Further development of the methodology and indications for treatment with rapid palatal dilation devices is required.

A new approach of CO2 separation by applying rapid expansion of supercritical CO2 rich natural gas

The previous methods for CO2 separation from CO2 rich natural gas led to expensive production costs. This work was implemented to overcome the problems utilizing a new approach economically. The cooling and rapid expansion processes were integrated for the CO2 separation from CO2 rich natural gas on the supercritical condition. The experimental apparatus was newly constructed to perform the experiments, and the results were simulated using a various equation of state. The result reveals that the inlet temperature of supercritical expansion diminished the outlet temperature and the gas condensed easily. The simulation indicated that the 70% CO2 in natural gas was condensed easier than 45% CO2. We found that the outlet temperature of −42 °C and the vapor fraction of 0.69 was attained at the CO2 composition of 70%. Besides, the pressure drop change influences the vapor fraction at various CO2 compositions. The vapor fraction under supercritical diminished significantly compared with the non-supercritical condition. The expansion coefficient determined utilizing the equation of state escalates by the enhancement of expansion inlet temperature based on CO2 composition in natural gas. The acid gas equation of state was the perfect equation to estimate the expansion coefficient with the absolute average error of 4.83%. This work suggests that the CO2 separation from CO2 rich natural gas with the cooling and rapid expansion method promotes the new approach to overcome the disadvantages of previous methods.

3D-organoid culture supports differentiation of human CAR+ iPSCs into highly functional CAR Tcells.

Unlimited generation of chimeric antigen receptor (CAR) Tcells from human-induced pluripotent stem cells (iPSCs) is an attractive approach for "off-the-shelf" CAR Tcell immunotherapy. Approaches to efficiently differentiate iPSCs into canonical αβ Tcell lineages, while maintaining CAR expression and functionality, however, have been challenging. We report that iPSCs reprogramed from CD62L+ naive and memory Tcells followed by CD19-CAR engineering and 3D-organoid system differentiation confers products with conventional CD8αβ-positive CAR Tcell characteristics. Expanded iPSC CD19-CAR Tcells showed comparable antigen-specific activation, degranulation, cytotoxicity, and cytokine secretion compared with conventional CD19-CAR Tcells and maintained homogeneous expression of the TCR derived from the initial clone. iPSC CD19-CAR Tcells also mediated potent antitumor activity invivo, prolonging survival of mice with CD19+ human tumor xenografts. Our study establishes feasible methodologies to generate highly functional CAR Tcells from iPSCs to support the development of "off-the-shelf" manufacturing strategies.

БЫСТРОЕ НЕБНЫЕ РАСШИРЕНИЕ И ЕГО ОСОБЕННОСТИ

Rapid Maxillary expansion or palatal expansion as it is sometimes called, occupies unique niche in dentofacial therapy. Rapid Maxillary expansion (RME) is a skeletal type of expansion that involves the separation of the mid-palatal suture and movement of the maxillary shelves away from each other. RME effects the maxillary complex, palatal vaults, maxillary anterior and posterior teeth, adjacent periodontal structures to bring about an expansion in the maxillary arch. Morphogenesis and anatomical features of the upper jaw determine the choice of the method of rapid palatal expansion with narrowing of the upper dentition, as an effective method for eliminating congenital deformities of the maxillofacial region. The majority of dental transverse measurements changed significantly as a result of RME. The maturity of the maxillofacial structures determines the timing and degree of success of rapid palatal dilatation treatment.

Study of Dispersions of Carbon Nanotubes Modified by the Method of Rapid Expansion of Supercritical Suspensions.

The effectiveness of carbon nanotubes (CNT) deagglomeration by rapid expansion of supercritical suspensions (RESS) in nitrogen and carbon dioxide fluids was studied in this work. Two different mechanisms of deagglomeration were proposed for these two fluids at various temperature and pressure conditions. Ultrasound attenuation spectroscopy was applied as an express method of determining median diameter and aspect ratio of CNTs. At least twofold reduction of the diameter was shown for CNT bundles processed by RESS technique. Aspect ratio of processed CNTs, calculated from acoustic attenuation spectra, increased to 340. These results were in a good agreement with atomic force microscopy data.

Comparison of expansion-based explicit time-integration schemes for acoustic wave propagation

We have developed a von Neumann stability and dispersion analysis of two time-integration techniques in the framework of Fourier pseudospectral (PS) discretizations of the second-order wave equation. The first technique is a rapid expansion method (REM) that uses Chebyshev matrix polynomials to approximate the continuous solution operator of the discrete wave equation. The second technique is a Lax-Wendroff method (LWM) that replaces time derivatives in the Taylor expansion of the solution wavefield with their equivalent spatial PS differentiations. In both time-integration schemes, each expansion term [Formula: see text] results in an extra application of the spatial differentiation operator; thus, both methods are similar in terms of their implementation and the freedom to arbitrarily increase accuracy by using more expansion terms. Nevertheless, their limiting Courant-Friedrichs-Lewy stability number [Formula: see text] and dispersion inaccuracies behave differently as [Formula: see text] varies. We establish the [Formula: see text] bounds for both methods in cases of practical use, [Formula: see text], and we confirm the results by numerical simulations. For both schemes, we explore the dispersion dependence on modeling parameters [Formula: see text] and [Formula: see text] on the wavenumber domain, through a new error metric. This norm weights errors by the source spectrum to adequately measure the accuracy differences. Then, we compare the theoretical computational costs of LWM and REM simulations to attain the same accuracy target by using the efficiency metric [Formula: see text]. In particular, we find optimal [Formula: see text] pairs that ensure a certain accuracy at a minimal computational cost. We also extend our dispersion analysis to heterogeneous media and find the LWM accuracy to be significantly better for representative [Formula: see text] values. Moreover, we perform 2D wave simulations on the SEG/EAGE Salt Model, in which larger REM inaccuracies are clearly observed on waveform comparisons in the range [Formula: see text].

Application of perfectly matched layers in 3D transient controlled-source electromagnetic modeling by the rapid expansion method

I have developed an extension of the rapid expansion method (REM) for 3D time-domain controlled-source electromagnetic modeling that includes perfectly matched layers (PMLs) as the absorbing boundary. The REM solves the time-domain electric field by a weighted summation of the Chebyshev polynomials. The results are free of temporal dispersion and accurate to the Nyquist frequency, yet the domain of Chebyshev polynomials lacks an accurate absorbing boundary. I find that by introducing a fictitious magnetic field in the Chebyshev domain, the recursion of the Chebyshev polynomials obeys a discrete coupled wave equation, which shares a similarity with the propagation of EM waves in a lossless medium. The time and frequency components in the Chebyshev domain are derived based on the eigenvalues of the propagation matrix, and the PML theory designed for EM waves can be extended to the Chebyshev domain in a straightforward way. Numerical tests against analytical solution and spectral methods show an excellent agreement after PML solves the boundary problem in the Chebyshev domain, which demonstrates the accuracy of the REM algorithm and the usefulness of the PML absorbing boundary.

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

ABSTRACTWe derive a governing second‐order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite‐difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme, numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections from the model boundaries.

Time evolution of the electric field using the rapid expansion method with pseudospectral evaluation of spatial derivatives — Part 2: Inclusion of anisotropy and the earth-air interface

We have evaluated an extension of the 3D 3C rapid-expansion method for time-domain electromagnetic modeling with pseudospectral evaluation of spatial derivatives that includes anisotropic conductivity and the earth-air interface. We found that transverse isotropic anisotropy can be included in the modeling with the manipulation of the conductivity tensor. We model the response of the earth-air interface by using the upward continuation of the time derivative of the magnetic induction. All three electric field components are collocated with the electric conductivities and are modeled independently and simultaneously with spectral accuracy in space and time. Numerical tests against the analytical solution for a half-space and spectral methods for 1D anisotropic layered earth models demonstrate the excellent accuracy of this approach.

Time evolution of the electric field using the rapid expansion method with pseudospectral evaluation of spatial derivatives — Part 1

We have developed a method for modeling transient electromagnetic data acquired on land or at sea. The 3D 3C time response of the electric wavefield is simulated by considering the spatial and temporal responses separately and using numerical methods best suited for each. We have used the rapid expansion method to develop the 3C electric wavefield time response from the spatial responses found using a pseudospectral method. The results are free of numerical dispersion and accurate to the Nyquist frequency in time and space. The resulting diffusive field is a weighted sum of Chebyshev polynomials that exhibit a wave-like character and are very sensitive to small perturbations in the medium. The method is intrinsically parallel, which leads to computational efficiency. Numerical results are compared favorably with the analytic response and 1D methods even though the computations are innately 3D.

S2-14 SESSION 2: POSTERIOR VAULT DISTRACTION QUANTITATIVE OUTCOMES IN POSTERIOR AND TRANSVERSE CRANIAL VAULT DISTRACTION FOR CRANIOSYNOSTOSIS

Introduction: CVD osteogenesis has been shown to establish greater cranial volume overall than conventional remodeling. Quantitative measurement with systematic evaluation of outcomes is vital for the comparison of surgical outcomes across international institutions. Computed tomography (CT) or magnetic resonance imaging (MRI) in addition to image analysis software has been increasingly used in recent times to predict ICV changes after cranial vault surgery. The primary study aim was to quantify ICV increase in CVD and secondarily to gauge acceptability of using a simple modality to measure ICV. Methods: A retrospective review of a prospectively maintained database of posterior and transverse cranial vault distraction procedures at the National Paediatric Craniofacial Centre in Temple Street Children’s University Hospital from May 2010 to September 2016 was performed. ICV was calculated using Osirix® Version 3.3 from post-operative CT-scans. Cephalometric analysis was also performed using lateral and anteroposterior radiographic views of the cranium. Data was analyzed with GraphPad Prism®. Results: 21 patients (9 PCVD and 13 TransCVD), underwent successful cranial vault distraction. Patients in the PCVD group achieved a mean ICV increase of 2947mm, 32.5% (range: 12.8% to 80.2%, p<0.0001).95% CI 2135 to 3758mm3. In the TransCVD group, an average ICV increase of 1912.96,15.8% (range: 4.4% to 24.2%, p<0.0001) was achieved. 95% CI 1507 to 2320 mm3. Cephalic index values moved towards normal range, average change of 10.1% (range: 2.04% to 19.83%, p<0.0552) in the PVD group and 6.8% in the TransCVD (range: 2.7% to 14.8%, p<0.0001) respectively. Conclusion: This institutional review of outcomes post cranial vault distraction for syndromic and non-syndromic synostosis contributes to the existing literature and confirms the satisfactory reliable results afforded by distraction surgery along with a favorable morbidity profile. Quantitative analysis of cranial vault expansion by distraction has shown a reproducible increase in total ICV and is an effective method for rapid expansion of ICV with normalization of head morphology. We found Osirix® used with CT scans an acceptable modality that does not compromise accuracy and confer excellent interactive user-software interface.

Up/down acoustic wavefield decomposition using a single propagation and its application in reverse time migration

The separation of up- and downgoing wavefields is an important technique in the processing of multicomponent recorded data, propagating wavefields, and reverse time migration (RTM). Most of the previous methods for separating up/down propagating wavefields can be grouped according to their implementation strategy: a requirement to save time steps to perform Fourier transform over time or construction of the analytical wavefield through a solution of the wave equation twice (one for the source and another for the Hilbert-transformed source), in which both strategies have a high computational cost. For computing the analytical wavefield, we are proposing an alternative method based on the first-order partial equation in time and by just solving the wave equation once. Our strategy improves the computation of wavefield separation, and it can bring the causal imaging condition into practice. For time extrapolation, we are using the rapid expansion method to compute the wavefield and its first-order time derivative and then we can compute the analytical wavefield. By computing the analytical wavefield, we can, therefore, separate the wavefield into up- and downgoing components for each time step in an explicit way. Applications to synthetic models indicate that our method allows performing the wavefield decomposition similarly to the conventional method, as well as a potential application for the 3D case. For RTM applications, we can now use the causal imaging condition for several synthetic examples. Acoustic RTM up/down decomposition demonstrates that it can successfully remove the low-frequency noise, which is common in the typical crosscorrelation imaging condition, and it is usually removed by applying a Laplacian filter. Moreover, our method is efficient in terms of computational time when compared to RTM using an analytical wavefield computed by two propagations, and it is a little more costly than conventional RTM using the crosscorrelation imaging condition.

Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion

We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

Modeling the wave propagation in viscoacoustic media: An efficient spectral approach in time and space domain

We present an efficient and accurate modeling approach for wave propagation in anelastic media, based on a fractional spatial differential operator. The problem is solved with the Fourier pseudo-spectral method in the spatial domain and the REM (rapid expansion method) in the time domain, which, unlike the finite-difference and pseudo-spectral methods, offers spectral accuracy. To show the accuracy of the scheme, an analytical solution in a homogeneous anelastic medium is computed and compared with the numerical solution. We present an example of wave propagation at a reservoir scale and show the efficiency of the algorithm against the conventional finite-difference scheme. The new method, being spectral in the time and space simultaneously, offers a highly accurate and efficient solution for wave propagation in attenuating media.

Solubility measurement and preparation of nanoparticles of an anticancer drug (Letrozole) using rapid expansion of supercritical solutions with solid cosolvent (RESS-SC)

A method of rapid expansion of supercritical solutions with solid cosolvent (RESS-SC) was applied, for the first time, to precipitate nanoparticles of Letrozole (LTZ). Solubility of LTZ, as a poorly water-soluble drug, was measured under various conditions in supercritical CO2 with/without menthol as cosolvent. The solubility of LTZ in ternary system (with solid cosolvent) was increased 7.1 times as large as that of binary system (without solid cosolvent). Then, influences of temperature, pressure, solid cosolvent and spray distance on the size and morphology of the particles were investigated by Taguchi technique. The analysis of variance results showed that, temperature and pressure had the most and least significant impacts on the size-reduction of nanoparticles, respectively. The particles were characterized using SEM, DLS, XRD, FTIR and DSC. Experimental observations revealed that the LTZ particles were nanosized from the original average size of 30μm to the smallest average size of 19nm.

Screening of conformational polymorphism of ibuprofen in supercritical CO2

The populations of conformers of ibuprofen in supercritical CO2 have been calculated for two isochors in the temperature range of 35–90°C, using the combination of data of IR spectroscopy, quantum chemical calculations and molecular dynamics simulation (metadynamics approach). The results of 2D cross-correlation analysis of IR spectra confirm the presence of two spectral contributions associated with two conformers. The population of the first conformer (Conf. I) is found to be dominating within the investigated temperature range. Moreover, at the temperature of 60°C, for both studied isochors, the population of the first conformer reaches the maximum. The assumption about the correlation between the populations of conformers in supercritical media and population of polymorphs, which form during crystallization from certain initial state parameters, has been formulated and checked using analysis of micronized ibuprofen powder obtained by the method of rapid expansion of supercritical solution (RESS). The analysis of the microcrystalline ibuprofen, obtained by RESS, was performed by Differential Scanning Calorimetry (DSC) and X-Ray Powder Diffraction (XRPD). Both methods lead to estimation of the polymorphs populations in accordance with the conformer populations at corresponding state parameters.

Third-order symplectic integration method with inverse time dispersion transform for long-term simulation

The symplectic integration method is popular in high-accuracy numerical simulations when discretizing temporal derivatives; however, it still suffers from time-dispersion error when the temporal interval is coarse, especially for long-term simulations and large-scale models. We employ the inverse time dispersion transform (ITDT) to the third-order symplectic integration method to reduce the time-dispersion error. First, we adopt the pseudospectral algorithm for the spatial discretization and the third-order symplectic integration method for the temporal discretization. Then, we apply the ITDT to eliminate time-dispersion error from the synthetic data. As a post-processing method, the ITDT can be easily cascaded in traditional numerical simulations. We implement the ITDT in one typical exiting third-order symplectic scheme and compare its performances with the performances of the conventional second-order scheme and the rapid expansion method. Theoretical analyses and numerical experiments show that the ITDT can significantly reduce the time-dispersion error, especially for long travel times. The implementation of the ITDT requires some additional computations on correcting the time-dispersion error, but it allows us to use the maximum temporal interval under stability conditions; thus, its final computational efficiency would be higher than that of the traditional symplectic integration method for long-term simulations. With the aid of the ITDT, we can obtain much more accurate simulation results but with a lower computational cost. We use the inverse time dispersion transform to the 3rd-order symplectic integration.It can greatly reduce the time-dispersion error, especially for long travel times.It is superior to the conventional 2nd- and high-order finite-difference methods.It allows the maximum temporal interval under stability conditions thus is efficient.We can obtain more accurate simulation results but with a lower computational cost.

Time-domain multiscale full-waveform inversion using the rapid expansion method and efficient step-length estimation

Full-waveform inversion (FWI) is rapidly becoming a standard tool for high-resolution velocity estimation. However, the application of this method is usually limited to low frequencies due to the high computational cost of the wavefield propagation and the inversion scheme. To mitigate this problem, we have developed a rapid expansion method (REM) for numerical wavefield extrapolation inside the FWI workflow. This method approximated the partial time derivative of the wave equation using Chebyshev polynomials instead of the conventional finite-difference (FD) approximation. This allowed the REM to accurately propagate wavefields with bigger time steps, thus improving the computational efficiency of FWI. We have compared high-frequency FWI results using REM and the traditional FD approximation of the time derivative to illustrate the ability of REM to remain stable and free of numerical dispersion noise even with large grid and time samplings. In addition, we studied a step-length estimation scheme with the objective of avoiding an extra forward wavefield propagation during the line search at each FWI iteration. In this scheme, we estimated the step-length value based on prior iterations and validated this value using the Wolfe conditions. If the value was accepted, then the forward modeling was availed in the next iteration and no extra propagation was necessary. We tried seven different step-length estimation methods, from which we highlighted the adaptive Barzilai-Borwein method when combined with the steepest-descent inversion scheme, and the unitary step length for the L-BFGS algorithm. Through synthetic numerical results, we showed how this scheme could achieve convergence, while keeping the number of extra forward modelings way below the number of FWI iterations.