Fahima Nekka

The modified box-counting method: Analysis of some characteristic parameters

Abstract One of the most popular ways of estimating fractal dimension (Df) is the box-counting method (BCM). The major problem with this method is that it leads to results with a very high percentage of error. The modified box-counting method (MBCM) was developed as a methodic procedure to set sequence and range. The procedure eliminates two problems of the computerized BCM, the border effect and noninteger values of e. The MBCM is a new, powerful tool, very simple to use, allowing accurate estimation of Df.

A model of growing vascular structures

Increasing attention is being paid to the configuration and development of vascular structures and their possible correlations with physiological events. The study of angiogenesis in normal and pathological states as well as in embryo and adult has provided new insights into the mechanism of vessel growth and organization of the vasculature. Various mathematical branching models have been developed. These constructions are mainly geometrical and only involve a branching phenomenon. We propose the use of a deterministic non-linear model based on physiological laws and hydrodynamics. Growth, branching and anastomosis, the three actual main events occurring in vascular growth, are included in this model. Space growth, including cells and vessels, is defined by a decreasing transformation. Space density and the length of new sprouts are controlled by a set of parameters. The conditions on these parameters are well established, which allows the production of realistic patterns.

Assessing drug distribution in tissues expressing P-glycoprotein through physiologically based pharmacokinetic modeling: model structure and parameters determination

BackgroundThe expression and activity of P-glycoproteins due to genetic or environmental factors may have a significant impact on drug disposition, drug effectiveness or drug toxicity. Hence, characterization of drug disposition over a wide range of conditions of these membrane transporters activities is required to better characterize drug pharmacokinetics and pharmacodynamics. This work aims to improve our understanding of the impact of P-gp activity modulation on tissue distribution of P-gp substrate.MethodsA PBPK model was developed in order to examine activity and expression of P-gp transporters in mouse brain and heart. Drug distribution in these tissues was first represented by a well-stirred (WS) model and then refined by a mechanistic transport-based (MTB) model that includes P-gp mediated transport of the drug. To estimate transport-related parameters, we developed an original three-step procedure that allowed extrapolation of in vitro measurements of drug permeability to the in vivo situation. The model simulations were compared to a limited set of data in order to assess the model ability to reproduce the important information of drug distributions in the considered tissues.ResultsThis PBPK model brings insights into the mechanism of drug distribution in non eliminating tissues expressing P-gp. The MTB model accounts for the main transport mechanisms involved in drug distribution in heart and brain. It points out to the protective role of P-gp at the blood-brain barrier and represents thus a noticeable improvement over the WS model.ConclusionBeing built prior to in vivo data, this approach brings an interesting alternative to fitting procedures, and could be adapted to different drugs and transporters.The physiological based model is novel and unique and brought effective information on drug transporters.

Distinct effects of amlodipine treatment on vascular elastocalcinosis and stiffness in a rat model of isolated systolic hypertension

Objective Medial elastocalcinosis (MEC) contributes to the development of large artery stiffness and isolated systolic hypertension. Since endothelin receptor antagonists can prevent and regress elastocalcinosis, our aim was to determine whether amlodipine, a calcium channel blocker that inhibits endothelin signaling, could likewise influence MEC, or reduce pressure mainly through its vasorelaxing properties. Methods Control male Wistar rats were compared with rats receiving warfarin (20 mg/kg per day) with vitamin K1 (15 mg/kg per day) alone (WVK) or in association with amlodipine (15 mg/kg per day) for 4 weeks or during the last week or last 4 weeks of an 8-week WVK treatment (two regression groups). Results Inactivation of matrix Gla protein by WVK for 4 or 8 weeks increased the calcium content 10-fold in the aorta, inducing a significant elevation of pulse wave velocity and pulse pressure by selective augmentation of systolic blood pressure. Amlodipine prevented aortic MEC, pulse wave velocity and pulse pressure elevation, but reversed only MEC and pulse pressure when administered for 4 weeks. One week of amlodipine administered after 7 weeks of WVK partially decreased pulse pressure without modifying aortic MEC. Amlodipine did not reduce the fibrosis associated with calcified areas in the WVK model during the regression protocols. Conclusion The clinical efficacy of amlodipine in improving hemodynamic variables and reducing cardiovascular events in isolated systolic hypertension could be explained by its beneficial effect on vascular calcification. Amlodipines lack of effect on pulse wave velocity and collagen deposition, however, suggests that it may reduce pulse pressure by means other than improving arterial stiffness.

Intersection of triadic Cantor sets with their translates— I. Fundamental properties

Abstract Motivated by Mandelbrots [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.

Peripheral link model as an alternative for pharmacokinetic-pharmacodynamic modeling of drugs having a very short elimination half-life

Attempts to obtain estimates of pharmacokinetic–pharmacodynamic (PK–PD) parameters for mivacurium with traditional central link models were unsuccessful in many patients. We hypothesized that a link model with the peripheral compartment would be more appropriate for mivacurium in view of its extremely rapid plasma clearance and its potential elimination by tissue pseudocholinesterases. For validation purposes, the peripheral link model was applied to other neuromuscular blocking agents (NMBA), i.e., atracurium and doxacurium which have respectively an intermediate and a long elimination half-life. Assuming peripheral elimination in PK–PD modeling was investigated but found to have no impact on the estimation of PK–PD parameters. Our results indicate that, for drugs having intermediate and long elimination half-lives, EC50 values are similar with either the central or peripheral link model. For mivacurium, a peripheral link model enables PK–PD modeling in all subjects, with more precision in the PK–PD parameter estimates and a better fitting of the effect data when compared to the central link model. For these reasons, a peripheral link model should be preferred for mivacurium.

A Pharmacokinetic Formalism Explicitly Integrating the Patient Drug Compliance

The adherence phenomenon is now well recognized to seriously compromise drug efficacy. In this paper, we analyze the role of compliance through drug intake history as an integral part of the pharmacokinetic process. Being concerned with what is accessible in medical practice, we develop a stochastic approach to model the drug intake behavior that we combine with a conventional pharmacokinetic model in order to investigate the effect of drug intake history on the pharmacokinetic time-course. For this purpose, we explicitly formalize the plasma concentration variations for the most common administration routes. This analytical approach allows to characterize drug concentration variations directly inherited from patient compliance.

Intersection of triadic Cantor sets with their translates. II. Hausdorff measure spectrum function and its introduction for the classification of Cantor sets

Abstract Initiated by the purpose of classification of sets having the same fractal dimension, we continue, in this second paper of a series of two, our investigation of intersection of triadic Cantor sets and their use in the classification of fractal sets. We exploit the infinite tree structure of translation elements to give the exact expressions of these elements. We generalize this result to a family of uniform Cantor sets for which we also give the Hausdorff measure spectrum function (HMSF). We develop three algorithms for the construction of HMSF of triadic Cantor sets. Then, we introduce a new method based on HMSF as a way for tracing the geometrical organization of a fractal set. The HMSF does carry a huge amount of information about the set to likely be explored in a chosen way. To extract this information, we develop a one by one step method and apply it to typical fractal sets. This results in a complete identification of fractals.

Neutrophil dynamics during concurrent chemotherapy and G-CSF administration: Mathematical modelling guides dose optimisation to minimise neutropenia.

The choice of chemotherapy regimens is often constrained by the patients tolerance to the side effects of chemotherapeutic agents. This dose-limiting issue is a major concern in dose regimen design, which is typically focused on maximising drug benefits. Chemotherapy-induced neutropenia is one of the most prevalent toxic effects patients experience and frequently threatens the efficient use of chemotherapy. In response, granulocyte colony-stimulating factor (G-CSF) is co-administered during chemotherapy to stimulate neutrophil production, increase neutrophil counts, and hopefully avoid neutropenia. Its clinical use is, however, largely dictated by trial and error processes. Based on up-to-date knowledge and rational considerations, we develop a physiologically realistic model to mathematically characterise the neutrophil production in the bone marrow which we then integrate with pharmacokinetic and pharmacodynamic (PKPD) models of a chemotherapeutic agent and an exogenous form of G-CSF (recombinant human G-CSF, or rhG-CSF). In this work, model parameters represent the average values for a general patient and are extracted from the literature or estimated from available data. The dose effect predicted by the model is confirmed through previously published data. Using our model, we were able to determine clinically relevant dosing regimens that advantageously reduce the number of rhG-CSF administrations compared to original studies while significantly improving the neutropenia status. More particularly, we determine that it could be beneficial to delay the first administration of rhG-CSF to day seven post-chemotherapy and reduce the number of administrations from ten to three or four for a patient undergoing 14-day periodic chemotherapy.

A MODIFIED BOX-COUNTING METHOD

Fractal geometry has been widely used to characterize irregular structures. Our interest in applying this concept in biomedical research leads us to the conclusion that there are no standard methods. In order to objectively set parameters involved in the estimation of fractal dimension, a significantly more accurate and efficient box-counting method based on a new algorithm was developed. Measurements of mathematical objects with known fractal dimension was performed using the traditional method and the proposed modification. The latter always yields results with less than 1% difference from the theoretical value, which represents a significant improvement.