Mathematical study of controlled release drug delivery to biological tissues

Motivation

These limitations lead to the development of various drug delivery devices with sustained release and controlled release modes. Furthermore, it supports to help the researchers experimenting in pharmaceutical industries in the formulation and design of drug delivery systems to have an effective drug release and drug bioavailability.

Background concepts

Polymeric drug delivery systems
Particulate drug delivery systems
Drug release mechanisms
Drug transport mechanisms

The main application of such a drug delivery system lies in transdermal drug delivery systems in the form of transdermal patches and others. The controlled release drug delivery systems are of three types depending on the drug release mechanisms, namely diffusion controlled, chemically controlled and swelling controlled drug delivery systems.

Current status and objectives of drug delivery

Research objectives

Model development requires an understanding of drug release kinetics and the underlying physiological phenomena governing drug transport. Here, a mathematical model of drug release from the polymer matrix and the resulting intracellular drug transport is proposed to be analyzed.

Thesis structure

Polymeric matrix and its degradation
Release mechanism from matrices
Mathematical modelling efforts formulating drug release
Drug delivery devices

Drug release has been the most important process in the field of drug delivery for decades. A drug release study performed on a polyanhydride (poly (fatty acid dimer-sebacic acid), p (FAD-SA), 50:50 w/w) based implant for local drug delivery claimed that the release kinetics depended on the drug solubility and the intrinsic dissolution rate [104].

Drug transport mechanisms

Drug transport in solid tumour

A major advantage of controlled drug delivery is that it not only prolongs the action of the drug, but also maintains drug levels within the therapeutic range (range of drug dosages that can effectively treat diseases while remaining within the safety range) of the drug. The drug release process is described by considering the solubility dynamics of drug crystallites and the diffusion of the solubilized drug through the microparticle.

Formulation of the problem

Model solutions

Numerical results and discussion

The influence of binding rate constant (δ0) on the free drug concentration in the microparticle phase is shown in Fig. The influence of unbinding rate constant (kd) on the free drug concentration C1 in the tissue is shown in Fig.

Figure 3.4: Time variant concentration profile of C 1 .

Conclusions

It is the formulation of the drug into a dosage form or drug delivery system that, in reality, transforms research involving drug discovery and other aspects of pharmacology into clinical practice. Although drug delivery can, in principle, be monitored, the most important risk is the design of the drug delivery system. In the present study, a generalized mathematical model of drug release from a local drug delivery device and its transport into biological tissue is proposed.

Formulation of the problem

Some preliminary concepts

The Vrevis length scale is assumed to be much larger than the pore size and smaller than the model length scale of the phenomenon. Porosity ε0 is generally described as the ratio between the volume of voids (Vrevf) and the total volume (Vrev=Vrevf +Vrevs), where f and s mean the liquid and solid phases respectively [11]. Since part of the void is accessible to the drug particles, a new parameter, the partition coefficient (k), is introduced so that kε0 becomes the available void volume.

Drug release mechanism and its transport phenomena

In the continuum approach, the porous medium is generally treated as a homogeneous material by defining the average variables in a sufficiently large volume, the representative base volume Vrev. The valid equations that describe the dynamics of the release of the active ingredient in the polymer matrix phase are. DRUG TRANSPORT THROUGH ENDOCYTOSIS 43 where Φ0(= 1−εkε0 . 0) is the ratio between the accessible void volume and the solid volume, CL indicates the available molar concentration of the solid drug, C0 is the available molar concentration of the free drug, k means the partition coefficient, ε0 indicates the porosity, l0 is the length of the polymer matrix, kmis the mass transfer coefficient, Climstands for the solubilization limit of the drug, β0 is the dissociation rate constant, δ0 is the association rate constant, α0 denotes the solid-liquid mass transfer coefficient and D0 is the diffusion coefficient of the free drug into the matrix.

Figure 4.1: Schematic diagram of drug transport

Dimensionless equations

Method of solution

Numerical simulation and discussion

Model validation

The time-variant concentration profiles for fixed (charged) and free drug particles in the polymeric matrix phase at four different axial locations spanning the entire domain are shown in Fig. In the case of decreasing porosity (ε0), the free drug concentration becomes higher due to lower diffusion rate. The time-varying drug concentration profiles in the tissue are largely perturbed by the dissociation rate parameter (kd) as shown in Fig.

Figure 4.2: Agreement of the present results with experimental data in the absence of recrystallization [115].

Conclusions

Physiologically based pharmacokinetic modeling has been developed to overcome the limitations of compartmental modeling. These polymers are suitable for the effective release of the drug as a result of the combined action of degradation and erosion. The objective of this study is to solve the governing system of differential equations analytically and have a thorough sensitivity analysis of the key parameters that produce phenomenal brushstrokes on the canvas of drug release kinetics.

Formulation of the problem

Degradation and erosion work together to synchronize the rate of drug release in the polymeric coatings [40]. The tissue section highlights the reversible dissolution/binding and internalization processes of drugs. The governing equations representing the dynamics of drug release in the polymeric matrix are given by.

Figure 5.1: Schematic diagram of drug transport along with polymer degradation Monomers:

Model solutions

Dimensionless equations

Experimental validation

In contrast to the release of PTX from the PLGA matrix, the addition of 20% PEG stimulates the initial burst release of PTX in the first 8 h and then a fairly stable pattern is followed (Fig. 5A of [72]). The experimental setting [72] of Paclitaxel distribution in poly(ethylene glycol)/poly(lactide-co-glycolic acid) blend films has been investigated and correlated with the time-percentage profile of drug release (non-dimensionalized parameters are used here) , which is procured by the proposed mathematical model (Fig. 5.2). It can be noted that the agreement of the experimental data [72] with the theoretical profile of the percentage of drug released is quite worthy of mention.

Sensitivity analysis

Mathematically, the sensitivity of the drug concentration function with respect to model parameters is characterized as the partial derivative of the concentration function with respect to those parameters. The local method involves the partial derivatives of the drug concentration functionCk(Xp,t) with respect to model input parametersXp, defined by. The partial derivatives of the drug concentrations corresponding to both the polymeric matrix and biological tissue phases with respect to the perturbed parameters provide a measure of model sensitivity to each key parameter.

Figure 5.3: Time variant concentration profile of C 0 at different k.

Numerical simulation and discussion

The significant variation of the results increases the credibility of the model regarding the significance of diffusivity in the system. Although significant variations of the output profiles with respect to the increasing parameter δ0 of the model as shown in Fig. Simulated results of the partial derivative of the free drug concentration profile for perturbations in the binding rate coefficient (ka) are given in Fig.

Figure 5.6: Space-time variant concentration profile of C b .

Conclusions

Furthermore, the advocated mathematical model explains the complex process of drug release from the polymeric matrix as an outcome of the coupled action of polymer degradation and erosion. Moreover, the porous structure of the biological tissue controls the ability of free drug to diffuse into the tissue. Consequently, an effective diffusivity within the porous biological tissue is responsible for free drug concentration in the tissue phase, which can be highly influenced by the compression of the porous wall.

Model development

In our study, the effective diffusivity of the polymer matrix is used to explain the kinetics of drug release from the polymer matrix into the biological tissue due to the synchronized action of polymer degradation and erosion. Eventually, water penetrates the polymer matrix and the solid drug embedded in it becomes wet, leading to solubilization of the loaded drug particles into free drug (C0). The free drug (C0) self-diffuses from the matrix according to the effective diffusivity of the polymer component of the matrix and enters the biological tissue as C1.

Figure 6.1: Schematic diagram of drug transport along with polymer degradation

Mathematical formulation

Nondimensionalization

All variables and parameters are non-dimensional so that we can proceed to convert the equations into a simple form to perform computations in a systematic approach. In the process of eliminating dimensionality in the system of reaction-diffusion-advection equations, corresponding numbers such as Peclet (Pe) and Damkohler (Da) numbers are obtained. In the following, prime numbers (0) are omitted from all dimensionless variables and parameters for convenience.

Numerical solution

Model validation

The primary objective of this study is to validate the applicability of the model through a direct comparison of experimental results based on transdermal drug delivery. To show the strength of the recommended mathematical model, validation of the theoretical results with the results of the experimental study by Argemí et al. It can be observed that the experimental data agreed well with the theoretical drug-released percentage profile, which speaks volumes for the authenticity of the proposed model.

Figure 6.2: Comparison of the present results with experimental data (cf. [3]).

Results and discussion

Therefore, it is worth noting a sharp decrease in the CL concentration profile accompanied by a strong increase in C0 in case of an increase in the dissociation rate constant. 6.16, it is observed that with the increase of kr1 the concentration profile of the free drug in the biological tissue increases. The lysosomal degradation rate constant describes the rate of degradation of drug particles in the lysosomes of the cell.

Figure 6.4: Time variant spatially averaged C 0 for biodegradable and biodurable polymer coating cases.

Conclusions

Moreover, the influence of model parameters is also well perceived through graphical portrayals. This chapter is completely defined by the model framework of liposomal drug delivery in solid tumor by meeting the objectives described in chapter 1. Temperature-sensitive drug-loaded liposomes after administration pass through the walls of blood vessels due to their size. their small and accumulate in extracellular spaces in the tumor.

Formulation of the problem

Physical background
Mathematical modelling of drug release
Mathematical modelling of drug transport
Nondimensionalization

LIPOSOMAL DRUG DELIVERY 111 The drug released from the liposomes in the interstitial fluid is depicted as free drug (CFT IF). Free drug (CFT IF) binds to the proteins (CPT IF) in the interstitial fluid to form a free drug-protein complex, ie. the free drug particles in the tumor interstitial fluid (CT IFF), are qualified to enter intracellular space interact with the cell surface receptors (RTCS) to form free drug-receptor complex, i.e. cell surface bound drug (CBSTC).

Figure 7.1: Schematic diagram of liposomal drug release and drug transport to tumour compartment In order to elucidate the drug release and drug transport mechanism to the targeted tumour cells, the physical, biological and biochemical phenomena are discus

Numerical solution

Model validation

The experimental set-up of thermosensitive liposomes is examined and matched with the profile of percentage release of the drug as a function of time (non-dimensional parameters are used here) obtained from the presented mathematical model (Figure 7.2). It is worth noting that the experimental data were in good agreement with the present theoretical results, which speaks to the strength of the advocated model. It can be observed in Figure 7.2 that there is a slight disagreement between the theoretical result of this study and the experimental result [88].

Results and discussion

7.4(a) in this panel, illustrates the change of bound drug (CBT P) with time in tumor plasma. Over time, CBT IFcontour also rapidly declines due to the separation of the free drug-protein complex in the interstitial fluid of the tumor. Figure 7.11 shows the change in the time-varying concentration profile for free drug in tumor plasma (CFT P) with changing plasma clearance rate (ke1).

Figure 7.3: Time variant concentration profiles of C F S and C L S .

Conclusions

This chapter concludes with the stability analysis of the drug dynamics model to get an overview of the use of pharmacokinetic mathematical models in the present thesis. The concerning work presents a coupled mathematical model for drug release from polymer matrix and subsequent drug transport into the intracellular domain. Stability analysis as carried out in the present study thus contributes to a better understanding of the underlying biochemical processes that control drug transport to the biological cells.

Model development

STABILITY ANALYSIS OF DRUG DYNAMICS 132 study is to perform a local stability analysis, which is probably the first time to our knowledge, taking into account the dynamics of the proposed mathematical model of drug release from a local drug delivery device and subsequent drug transport. Additionally, the stability analysis study describes the power of the mathematical model to be realistic enough to model drug kinetics. ANALYSIS OF DRUG DYNAMICS 133 shown and visualized through the given schematic representation in the form of fig.

Mathematical formulation

Dimensionless equations

Boundedness of the system

Stability analysis

Reduced order model

Boundedness of the system

Stability analysis

Numerical simulation and discussion

Due to the scale of the present original dynamical system, attention is focused on the stability of several subsystems in the reduced model with the inclusion of phase por-. Figures 8.8, 8.9 & 8.10 show the influence of the recycling rate constant (kx) on cell surface bound drug (x5), internalized bound drug (x6) and internalized free drug (x7) respectively over a set period of time. Similar framework consisting of the dynamics of protein-free drug complex, internalized bound drug and internalized free drug is presented in Fig.

Figure 8.2: Time variant x 2 profile for different α 0

Conclusions

Both analytical and numerical studies are performed for local stability analysis of both the full and reduced model systems. Fifth, the complex features of various subsystems of the reduced model are also explored through graphical representations of three-dimensional subsystem projection and phase portraits. Finally, the sensitivity of drug dynamics to the model parameters is demonstrated through graphical representations, which determine the applicability of drug delivery in the treatment of patients in general.

Appendix

Detailed impact for each of the mathematical models developed in the dissertation on pharmacokinetics can be categorically stated as follows. Its impact is also significant in the aspect of studying dynamic stability of the models representing drug release phenomena in pharmacokinetics. Cn: concentration of polymers, depicts the length of the polymer. D0: diffusion coefficient of free drug in the polymeric matrix D1: diffusion coefficient of free drug in the biological tissue.