Local convergence of the Levenberg-Marquardt method under H\"{o}lder metric subregularity

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under H\"{o}lder metric subregularity of the function defining the equation and H\"older continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the \L{}ojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the \L{}ojasiewicz gradient inequality assumption.Comment: 30 pages, 10 figure

Structural conserved moiety splitting of a stoichiometric matrix

Characterising biochemical reaction network structure in mathematical terms enables the inference of functional biochemical consequences from network structure with existing mathematical techniques and spurs the development of new mathematics that exploits the peculiarities of biochemical network structure. The structure of a biochemical network may be specified by reaction stoichiometry, that is, the relative quantities of each molecule produced and consumed in each reaction of the network. A biochemical network may also be specified at a higher level of resolution in terms of the internal structure of each molecule and how molecular structures are transformed by each reaction in a network. The stoichiometry for a set of reactions can be compiled into a stoichiometric matrix N is an element of Z(mxn), where each row corresponds to a molecule and each column corresponds to a reaction. We demonstrate that a stoichiometric matrix may be split into the sum of m - rank(N) moiety transition matrices, each of which corresponds to a subnetwork accessible to a structurally identifiable conserved moiety. The existence of this moiety matrix splitting is a property that distinguishes a stoichiometric matrix from an arbitrary rectangular matrix. (C) 2020 Elsevier Ltd. All rights reserved.Analytical BioScience

Effects of Inhalation Flow Rate on Particle Deposition and Flow Structure in a Model of Tracheobronchial Airway.

Due to the prevalence of respiratory diseases, effective drug delivery to the lungs is important for researchers. The main objective of current study is investigating the transfer and deposition of micron-sized particles (1-10 ?m) as well as airflow structure at different respiratory flow rates (i.e. 30, 60, and 90 L/min) in a realistic airway model according to the CT images of a 48-year-old healthy female. Computational fluid dynamics (CFD) is used for simulation of particle transport and deposition in an airway model that includes mouth-Throat zone, trachea region, and bronchial airways up to the fourth generation and the results were compared with available data in the literature. To investigate airflow structure, velocity contours with streamlines at different regions are obtained. Deposition fraction (DF) is used to present the results of particle deposition pattern. The results show that mouth-Throat region and trachea filters out largest inhaled aerosols, which lead to highest particle deposition fractions for these regions. In addition, increasing the inhalation flow rate, increases turbulence level and particles inertia which result in higher deposition fractions

Effect of laryngeal jet on dry powder inhaler aerosol deposition: A numerical simulation.

Although pulmonary drug delivery has been deeply investigated, the effect of the laryngeal jet on particle deposition during drug delivery with dry powder inhalers (DPI) has not been evaluated profoundly. In this study, the flow structure and particle deposition pattern of a DPI in two airway models, one with mouth-throat region including the larynx and the other one without it, are numerically investigated. The results revealed that the laryngeal jet has a considerable effect on particle deposition. The presence of laryngeal jet leads to 4-fold and 2-fold higher deposition efficiencies for inlet flow rates of 30 and 90 L/min, respectively

Flow structure and particle deposition analyses for optimization of a pressurized metered dose inhaler (pMDI) in a model of tracheobronchial airway.

Inhalation therapy plays an important role in management or treatment of respiratory diseases such asthma and chronic obstructive pulmonary diseases (COPDs). For decades, pressurized metered dose inhalers (pMDIs) have been the most popular and prescribed drug delivery devices for inhalation therapy. The main objectives of the present computational work are to study flow structure inside a pMDI, as well as transport and deposition of micron-sized particles in a model of human tracheobronchial airways and their dependence on inhalation air flow rate and characteristic pMDI parameters. The upper airway geometry, which includes the extrathoracic region, trachea, and bronchial airways up to the fourth generation in some branches, was constructed based on computed tomography (CT) images of an adult healthy female. Computational fluid dynamics (CFD) simulation was employed using the k-ω model with low-Reynolds number (LRN) corrections to accomplish the objectives. The deposition results of the present study were verified with the in vitro deposition data of our previous investigation on pulmonary drug delivery using a hollow replica of the same airway geometry as used for CFD modeling. It was found that the flow structure inside the pMDI and extrathoracic region strongly depends on inhalation flow rate and geometry of the inhaler. In addition, regional aerosol deposition patterns were investigated at four inhalation flow rates between 30 and 120 L/min and for 60 L/min yielding highest deposition fractions of 24.4% and 3.1% for the extrathoracic region (EX) and the trachea, respectively. It was also revealed that particle deposition was larger in the right branches of the bronchial airways (right lung) than the left branches (left lung) for all of the considered cases. Also, optimization of spray characteristics showed that the optimum values for initial spray velocity, spray cone angle and spray duration were 100 m/s, 10∘ and 0.1 sec, respectively. Moreover, spray cone angle, more than any other of the investigated pMDI parameters can change the deposition pattern of inhaled particles in the airway model. In conclusion, the present investigation provides a validated CFD model for particle deposition and new insights into the relevance of flow structure for deposition of pMDI-emitted pharmaceutical aerosols in the upper respiratory tract