Two compartmental fractional derivative model with general fractional derivative.

This study presents a new two compartmental model with, recently defined General fractional derivative. We review that concept of General fractional derivative and use the kernel function that generalizes the classical Caputo derivative in a mathematically consistent way. Next we use this model to study the release of antibiotic gentamicin in poly (vinyl alcohol)/gentamicin(PVA/Gent) hydrogel aimed for wound dressing in medical treatment of deep chronical wounds. The PVA/Gent hydrogel was prepared by physical cross linking of poly (vinyl alcohol) dispersion using freezing-thawing method, and then was swollen in gentamicin solution at 37 °C during 48 h. The concentration of released gentamicin was determined using a high-performance liquid chromatography coupled with mass spectrometer. The advantage of this model is the existence of new parameters in the definition of fractional derivative, as compared with classical fractional compartmental models. The model proposed here in the special case reduces to the classical (integer order) linear two compartmental model as well as classical fractional order two compartmental model since it has more parameters that are determined from the experimental results.

On the thermodynamical restrictions in isothermal deformations of fractional Burgers model.

We investigate, in the distributional setting, the restrictions on the constitutive equation for a fractional Burgers model of viscoelastic fluid that follow from the weak form of the entropy inequality under isothermal conditions. The results are generalized, from the Burgers model, to an arbitrary class of linear constitutive equations with fractional derivatives. Our results show that the restrictions obtained here on the coefficients of constitutive equations are weaker when compared with the restrictions obtained by Bagley-Torvik method. We show the precise relation between restrictions derived here and those derived by Bagley-Torvik. We deal with the creep test, for the case when Bagley-Torvik conditions are violated, and new conditions obtained in this work are satisfied. The results show a qualitative difference in the form of creep function. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

A Method for Prediction of Femoral Component of Hip Prosthesis Durability due to Aseptic Loosening by Using Coffin/Manson Fatigue Model.

The purpose of this work is to develop a new model estimate of the fatigue life of a hip prosthesis due to aseptic loosening as a multifactorial phenomenon. The formula developed here is a three-parameter model based on Basquin's law for fatigue, eccentric compression formula for the compressive stress and torsion in the prosthesis due to the horizontal components of the contact force. With our model, we can accurately predict the durability of a hip prosthesis due to the following four parameters: body weight, femoral offset, duration, and intensity of daily physical activities of a patient. The agreement of the prediction with the real life of the prosthesis, observed on 15 patients, is found to be adequate. Based on the formula derived for a particular implant, there was a high degree of concurrence between the model-predicted and actual values of aseptic loosening (durability) proved by the Mann-Whitney U test. By virtue of the validated model, it is possible to predict, quantitatively, the influence of various factors on the hip life. For example, we can conclude that a 10% decrease of a patient's body mass, with all other conditions being the same, causes 5% increase of the hip fatigue life.

Viscoelastic properties of uncured resin composites: Dynamic oscillatory shear test and fractional derivative model.

OBJECTIVE: In this study we analyze viscoelastic properties of three flowable (Wave, Wave MV, Wave HV) and one universal hybrid resin (Ice) composites, prior to setting. We developed a mathematical model containing fractional derivatives in order to describe their properties. METHODS: Isothermal experimental study was conducted on a rheometer with parallel plates. In dynamic oscillatory shear test, storage and loss modulus, as well as the complex viscosity where determined. We assumed four different fractional viscoelastic models, each belonging to one particular class, derivable from distributed-order fractional constitutive equation. The restrictions following from the Second law of thermodynamics are imposed on each model. The optimal parameters corresponding to each model are obtained by minimizing the error function that takes into account storage and loss modulus, thus obtaining the best fit to the experimental data. RESULTS: In the frequency range considered, we obtained that for Wave HV and Wave MV there exist a critical frequency for which loss and storage modulus curves intersect, defining a boundary between two different types of behavior: one in which storage modulus is larger than loss modulus and the other in which the situation is opposite. Loss and storage modulus curves for Ice and Wave do not show this type of behavior, having either elastic, or viscous effects dominating in entire frequency range considered. SIGNIFICANCE: The developed models may be used to predict behavior of four tested composites in different flow conditions (different deformation speed), thus helping to estimate optimal handling characteristics for specific clinical applications.

A model of the viscoelastic behavior of flowable resin composites prior to setting.

OBJECTIVE: The aim of this study is to develop fractional derivative models for the assessment of viscoelastic properties related to handling characteristics of dental resin composites belonging to two classes: flowable (Revolution Formula 2 and Filtek Ultimate) and posterior "bulk-fill" flowable base (Smart Dentin Replace). METHODS: Rheological measurements on all materials tested in this study were performed using dynamic oscillatory rheometer at temperature of 23°C. A parallel plates module with a diameter of 20mm was used to measure the properties of the resin composites tested. We developed two models to describe the obtained data: the generalized Newton model and the generalized Zener model (the so-called three parameter model). Both models contain fractional derivatives of Riemann-Liouville type. By determining the parameters of the model we were able to fit experimental data with high accuracy. RESULTS: Our results show that flowable "bulk-fill" resin-composite material (SDR) has distinct properties as compared to other two flowable resin composite materials (Revolution Formula 2 and Filtek Ultimate). Thus, previously found SDR properties as "bulk-fill" flowable base results in the fact that it is described by generalized Zener model (i.e., it has properties of solid like viscoelastic material). SIGNIFICANCE: Our model may be used to predict behavior of tested composites in different flow conditions. The SDR has initially small almost constant complex viscosity showing that it has good self-leveling property.

A biomechanical model for a new incremental technique for tooth restoration.

The objective of this study is to introduce a modified incremental technique that leads to improved marginal adaptation and to develop a mathematical model that explains the results obtained. The technique proposed is a two-step incremental technique that reduces volume of a resin that is polymerized at each step and eliminates the central point in resin, so that the stresses are additionally reduced. In the first step, the resin is placed in the cylindrical cavity with a conical dental instrument embedded in the middle of restoration. After polymerization, the conical dental instrument is removed and the conical hole is filled with the second layer of composite and polymerized. This technique is a variant of a method where singular stress point is eliminated. We modified the previous technique by introducing a conical dental instrument into the centre of the cavity. The procedure proposed was compared with the bulk and horizontal layer incremental technique. This study confirmed that the incremental type placement technique used here has better marginal adaptation than bulk technique and horizontal two-layer incremental technique although it has larger C-factor in the first step than the two-layer incremental technique. Thus, the elimination of the central point of restoration leads to better marginal adaptation. Conical shape of the cavity that is filled in the second step makes this technique easy to apply in clinical conditions. A mathematical model describing stresses in the restoration shows stress reduction as a consequence of applying the procedure proposed.

Predicting in vivo failure of rotary nickel-titanium endodontic instruments under cyclic fatigue.

The aim of this study was to examine the lifespan or number of cycles to failure of tapered rotary nickel-titanium (Ni-Ti) endodontic instruments. Simulated root canals with different curvatures were used to determine a relation between canal curvature and instrument lifespan. Using a novel mathematical model for the deformation of pseudoelastic Ni-Ti alloy, it was shown that maximum stress need not necessarily occur at the outer layer. On the basis of this observation, the Coffin-Manson relation was modified with parameters determined from this experiment. Results showed that the number of cycles to failure was influenced by the angle and radius of canal curvature and the size of instrument at the beginning of canal curvature. The resulting quantitative mathematical relation could be used to predict the lifespan of rotary Ni-Ti endodontic instruments under clinical conditions and thereby reduce the incidence of instrument failure in vivo.

A nonlinear two compartmental fractional derivative model.

This study presents a new nonlinear two compartmental model and its application to the evaluation of valproic acid (VPA) pharmacokinetics in human volunteers after oral administration. We have used literature VPA concentrations. In the model, the integer order derivatives are replaced by derivatives of real order often called fractional order derivatives. Physically that means that the history (memory) of a biological process, realized as a transfer from one compartment to another, is taken into account with the mass balance conservation observed. Our contribution is the analysis of a specific nonlinear two compartmental model with the application in evaluation of VPA pharmacokinetics. The agreement of the values predicted by the proposed model with the values obtained through experiments is shown to be good. Thus, pharmacokinetics of VPA after oral application can be described well by a nonlinear two compartmental model with fractional derivatives of the same order proposed here. Parameters in the model are determined by the least-squares method and the particle swarm optimization (PSO) numerical procedure is used. The results show that the nonlinear fractional order two compartmental model for VPA pharmacokinetics is superior in comparison to the classical (integer order) linear two compartmental model and to the linear fractional order two compartmental model.

A method of improving marginal adaptation by elimination of singular stress point in composite restorations during resin photo-polymerization.

OBJECTIVES: To reduce the effect of stresses due to volumetric shrinkage the authors propose an incremental technique for placing composite restorations. METHODS: The goal of the method is to reduce the volume of the resin that is polymerized and eliminate a stress singular point in the resin that is positioned at the geometric center of the cavity. This is achieved by a two step type incremental technique. In the first step the resin is placed in the cylindrical cavity with a metal pin embedded in the middle of the composite restoration. After polymerization, the metal pin is removed and the cylindrical hole is filled with the second layer of composite. Finally, the second layer in the center of the composite restoration is polymerized. RESULTS: This study confirmed that the proposed incremental type placement technique reduces marginal debonding. SIGNIFICANCE: The main hypothesis is that the elimination of a stress singular point at the center of the restoration results in the reduction of stresses at tooth-composite interface and therefore improve the marginal adaptation (reduces length of the contraction gap at tooth-composite interface).

A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac.

This study presents a new two compartmental model and its application to the evaluation of diclofenac pharmacokinetics in a small number of healthy adults, during a bioequivalence trial. In the model the integer order derivatives are replaced by derivatives of real order often called fractional order derivatives. Physically that means that a history (memory) of a biological process, realized as a transfer from one compartment to another one with the mass balance conservation, is taken into account. This kind of investigations in pharmacokinetics is founded by Dokoumetzidis and Macheras through the one compartmental models while our contribution is the analysis of multi-dimensional compartmental models with the applications of the two compartmental model in evaluation of diclofenac pharmacokinetics. Two experiments were preformed with 12 healthy volunteers with two slow release 100 mg diclofenac tablet formulations. The agreement of the values predicted by the proposed model with the values obtained through experiments is shown to be good. Thus, pharmacokinetics of slow release diclofenac can be described well by a specific two compartmental model with fractional derivatives of the same order. Parameters in the model are determined by the least-squares method and the Particle Swarm Optimization (PSO) numerical procedure is used. The results show that the fractional order two compartmental model for diclofenac is superior in comparison to the classical two compartmental model. Actually this is true in general case since the classical one is a special case of the fractional one.

A model for shrinkage strain in photo polymerization of dental composites.

OBJECTIVES: We formulate a new model for the shrinkage strain developed during photo polymerization in dental composites. The model is based on the diffusion type fractional order equation, since it has been proved that polymerization reaction is diffusion controlled (Atai M, Watts DC. A new kinetic model for the photo polymerization shrinkage-strain of dental composites and resin-monomers. Dent Mater 2006;22:785-91). Our model strongly confirms the observation by Atai and Watts (see reference details above) and their experimental results. The shrinkage strain is modeled by a nonlinear differential equation in (see reference details above) and that equation must be solved numerically. In our approach, we use the linear fractional order differential equation to describe the strain rate due to photo polymerization. This equation is solved exactly. RESULTS: As shrinkage is a consequence of the polymerization reaction and polymerization reaction is diffusion controlled, we postulate that shrinkage strain rate is described by a diffusion type equation. We find explicit form of solution to this equation and determine the strain in the resin monomers. Also by using equations of linear viscoelasticity, we determine stresses in the polymer due to the shrinkage. The time evolution of stresses implies that the maximal stresses are developed at the very beginning of the polymerization process. SIGNIFICANCE: The stress in a dental composite that is light treated has the largest value short time after the treatment starts. The strain settles at the constant value in the time of about 100s (for the cases treated in Atai and Watts). From the model developed here, the shrinkage strain of dental composites and resin monomers is analytically determined. The maximal value of stresses is important, since this value must be smaller than the adhesive bond strength at cavo-restoration interface. The maximum stress determined here depends on the diffusivity coefficient. Since diffusivity coefficient increases as polymerization proceeds, it follows that the periods of light treatments should be shorter at the beginning of the treatment and longer at the end of the treatment, with dark interval between the initial low intensity and following high intensity curing. This is because at the end of polymerization the stress relaxation cannot take place.

On a mathematical model of a human root dentin.

OBJECTIVE: On the basis of recent experimental data, a new mathematical model that predicts creep in human root dentin has been developed. METHOD: The chosen constitutive model comprises fractional derivatives of stress and strain and the restrictions on the coefficients that follow from the Clausius-Duhem inequality. RESULTS: The four constants describing mechanical properties of the human dentin at constant temperature are calculated from a highly non-linear system involving Mittag-Leffler-type functions. Special attention is paid to thermodynamical restrictions that should be observed in determining parameters of the model from experimental results. SIGNIFICANCE: The proposed model allows us to predict behavior of a human dentin in different load situations. Also it could be used for describing mechanical properties of dentin that are important in the development of 'dentin-like' restorative materials.