_{1}

This work describes the deterministic interaction of a diffusing particle of efavirenz through concentration gradient. Simulated pharmacokinetic data from patients on efavirenz are used. The Fourier’s Equation is used to infer on transfer of movement between solution particles. The work investigates diffusion using Fick’s analogy, but in a different variable space. Two important movement fluxes of a solution particle are derived an absorbing one identified as conductivity and a dispersing one identified as diffusivity. The Fourier’s Equation can be used to describe the process of gain/loss of movement in formation of a solution particle in an individual.

The derivation of an important equation by Fourier brought so many insights in the study of dynamics of flow through his law of thermal conduction [

Further, the work notes that “diffusivity’’ is not independent of concentration as proposed by Fick’s laws and proposes composition-dependent diffusion coefficients as suggested by Boltzmann, but differs in form [

Conductivity flux in this work defines the zero sum inward acquiring “concentration” movement of an interacting solution particle in a volume space from a neighbouring solution particle or central volume space in a solution particle bridge. A solution particle bridge is a state of exchange of concentration material between two interacting solution particles. Diffusivity flux is the zero sum outward dispersing concentration movement generated from an interacting solution particle to a neighbouring solution particle or peripheral volume space in a solution particle bridge. A total movement flux is postulated to be comprised of four main entities. These components have been identified as advective, saturation, passive and convective. Advective diffusivity is the deceleration of the advective diffusivity flux. Advective conductivity is the acceleration of advective conductivity flux.

The work considers definitions of important terms of advective movement flux. They are saturation kinetic advective conductivity and saturation kinetic advective diffusivity. Saturation kinetic advective conductivity flux is defined as the solution particle’s inward formulation velocity rate constant of the secondary saturation movement with respect to advection. Additionally, saturation kinetic advective diffusivity flux as the interacting solution particle’s outward formulation velocity of the secondary saturation movement of an interacting solution particle with respect to advection (rate of change of velocity rate constant of the volume (ml) that dissolves a mass of 1 μg of efavirenz with respect to advection). These two important parameters vary with the state and properties of the volumetric spaces (Central and Peripheral), the concentration of the solution particle and possibly the factors that affect absorption and elimination of solution particle.

Diffusivity and conductivity fluxes can be tracked using the saturation relation that governs kinetic solubility movement in individuals to infer on the probable resultant effects. The solution particle characterisation is tracked in an individual through time and concentration. The work characterizes conductivity and diffusivity fluxes by studying the secondary saturation movement dynamics in solution particle’s bridge. The calculations are done in the concentration-time space.

This work can further be developed to construct similar arguments for the primary saturation [

Simulated projected data on secondary saturation movement, time and concentration was obtained from pharmacokinetic projections made on patients on 600 mg dose of efavirenz considered in Nemaura (2014 & 2015). Partial and Ordinary Differential equations are used in the development of models. A statistical Package R is used to develop nonlinear regression models.

Derivation of Advection Kinetic Flow for the Secondary Saturation MovementLet

・ The amount of movement required to raise the saturation in a solution particle by

・ The rate at which saturation movement crosses a solution particle is proportional to concentration, time and secondary saturation gradient of solution particle. The following proposition is made that the amount of advective conductivity flux of solu-

tion particle

conductivity flux which is generated by movement density

Considering an abitrary solution particle bridge (A) (see

In a homogenous mix,

tive conductivity flux in the solvent of a solute of x, in the absence of the solute x (before mixing)] is constant and Equation (1) reduces to,

Assume that secondary saturation is increasing from left to right. The saturation gra-

dient at the right end is

to regions of lower concentration, movement thus should be entering the infinitesmal bridge through the right end and exiting the infinitesimal bridge through the left end. So in an infinitesimal time interval

Considering a small time interval

changes by

which increase by

particle bridge is not generating or destroying movement itself. The following conclusion is made that it is equal to the amount of movement that entered the solution particle bridge in the interval time

Dividing both sides by

Further, division of Equation (5) both sides by,

The following result is immediate,

where,

is the specific secondary saturation advective conductivity flux (/h) associated with a unit amount of movement in a solution bridge. This reduces to,

where,

The terms,

are specific secondary saturation base-advection diffusivity flux (/h), secondary satura-

tion advection diffusivity

advection conductivity flux (/h) respectively.

The following relation from Equations (6) and (7) is established between advection conductivity flux and advection diffusivity flux,

where

It is important to note that

There is use of the relationship developed from [

These two relations (

where, x is the concentration and t is time. These two variables track solution particle dynamics. Furthermore,

Following from Equation (9), Equation (6) assumes the form of,

and Equation (8) becomes,

The following conditions holds for secondary saturation advective diffusivity

It is noted that for

The part CC shows the advective diffusivity solution particle space (Central volume space) and CE shows the diffusivity in the complement space of the solution particle (neighbourhood volume space-Peripheral volume space) (

Parameters | Estimate | Std Error | t Value | |
---|---|---|---|---|

0.623106 | 0.012214 | 51.02 | ||

0.022465 | 0.001203 | 18.68 | ||

0.439258 | 0.017354 | 25.31 |

Parameters | Estimate | Std Error | t Value | |
---|---|---|---|---|

u | 0.801936 | 0.05934 | 135.1 | |

v | 5.624198 | 0.126684 | 44.4 |

is affected by two neighbourhoods, one that is in the central volume space (CC) and the other is related to the complement (CE) immediate peripheral volume space. Movement in these two spaces varies according to the characterisation of these two spaces. The advective diffusivity flux’s deceleration within the central compartment (that volume space which the solution particle is taken from) is given by

A positive value for advective diffusivity in the volume space (CC) implies that the movement is “concentrating” within CC space. However, a negative value shows movement is “concentrating” in the complement space (CE).

The diffusivity flux is an important parameter which is further investigated to see its dynamics in terms of the four entities that formed the basis of a solution particle. The existence of negative values for advection diffusivity shows presence of negative advective diffusivity flux (

where,

The characterisation of the four main diffusivity flux entities are shown and magnitudes of effects (

From Equation (8), we can infer that auxilliary conductivity flux consists of four main components and affects the movement into-solution particle and diffusivity flux affects

Parameters | Estimate | Std Error | t Value | |
---|---|---|---|---|

w | −37.240855 | 1.320478 | −28.203 | |

n | 0.416443 | 0.007034 | 59.207 | |

b | 87.672322 | 12.141991 | 7.221 | |

d | 0.157279 | 0.002074 | 75.816 | |

f | 99.406672 | 13.520844 | 7.352 | |

g | 1.979299 | 0.179995 | 10.996 |

w―residence rate of the convective diffusivity flux, n―declining rate of the convective diffusivity flux, b―maximum passive diffusivity flux rate constant, d―declining rate of the passive diffusivity flux, f―maximum saturation dif- fusivity flux rate constant and, g―time at which the saturation diffusivity flux was half of f.

the movement out-of-solution particle. Diffusivity flux is symmetrical to conductivity flux in solution particle formation,

where,

One notices a range of interpretations that can be deduced from the numerics done for the saturation dynamics of a solution particle (

This work proposes possible markers in the transportation of a drug which are conductivity and diffusivity. The partial differential equation derived has been shown to model the advection component of diffusivity and conductivity.

A new development is inferred where the movement diffusivity flux is shown to be a function of four primary components. The primary components have been identified as

advection, saturation, convection and passive. The total movement diffusivity flux in the system is zero for any given concentration at a given time. The two movement diffusivity fluxes convection and passive are inferred to exert their influence in the peripheral volume system. However, saturation and advection predominantly exerts influence in the central volume space. The increase in advective diffusivity flux shows a corresponding reduction of advective conductivity, this is postulated from the negative correlation that these two parameters share in this system. It has been shown that, not only concentration affect diffusivity and conductivity, but also the state of the neighbourhood. Additionally, there are potential factors that could be involved in the processes of transportation, absorption and elimination of the drug.

The advective diffusivity of the drug has partly negative values. Furthermore, it is not a constant and the regions of negative advective diffusivity are relatively very high as compared to positive diffusivity in the volume spaces. Initially, the negative values of advective diffusivity indicate that the peripheral volume space has relatively higher levels of flow. The drug is flowing more in the peripheral space. Thus the drug is quickly available to the peripheral space initially and furthermore is “concentrating” in this space [

Inference on in-vivo release can be made using models developed. The interpretation of kinetic conductivity developed here allowed study of diffusivity. The work made use of data informed directly from the developed PK/PD models [

Diffusivity and conductivity of drugs in patients finds its use in studying drug release kinetics. This is important to the successful design of polymeric delivery systems bearing in mind that in-vivo release models are laborious and costly [

Diffusivity is symmetrical to conductivity in solution particle formation. These two processes occur concurrently. The pro-gradient-driven-movement entities of a solution particle with respect to conductivity and diffusivity are convective and passive. While, anti-gradient-driven-movement entities are saturation and advection. This analysis is inferred from the characterisation of a solution particle in Nemaura (2015) [

With the aid of the characterisation of a saturation kinetic flow (strictly positively correlated relation to concentration), one can use this to infer the behaviour of the solution particle in relation to its neghbourhood through advective diffusivity. This work shows the existence of a model which is deterministic and describes the behaviour of how a solution particle gain/lose movement relative to its local neighbourhood. The gain of movement is synonymous with conductivity and loss of movement is synonymous with diffusivity. The two advection movement fluxes are symmetric about half the initial value of advection conductivity (M/2) and also all the other subsequent corresponding components in the 24 h.

The author would like to thank the following; C. Nhachi, C. Masimirembwa, and G. Kadzirange, AIBST and The College of Health Sciences, University of Zimbabwe.

Nemaura, T. (2016) The Advection Diffusion-in-Secondary Satu- ration Movement Equation and Its Application to Concentration Gradient-Driven Satu- ration Kinetic Flow. Journal of Applied Mathe- matics and Physics, 4, 1998-2010. http://dx.doi.org/10.4236/jamp.2016.411200