_{1}

This work proposed a coupled model of diffusion. It adopted two forms of coupled movement, the interacting and non-interacting driven forms of movement of a solution particle of efavirenz concentration measured in blood plasma. Data from projected pharmacokinetics in a patient on efavirenz were used. A relationship between interacting and non-interacting diffusion was suggested through a stochastic differential equation. The solution particle with a small value of relative acceleration drift to its active neighbourhood was projected to have a corresponding high transport/interacting diffusion.

The work attempted to combine two forms of diffusion starting with phenome- nological approach motivated by Fick’s laws and their mathematical solutions and the “random walk” of diffusing particles suggested by Robert Brown. This random walk of microscopic particles in suspension in a fluid has since been adopted as the “Brownian motion” in honour of Brown. The mathematical form of the Brownian motion was derived by Einstein [

The research encouraging modelling of random fluctuations found in PK/PD (Pharmacokinetic and Pharmacodynamic) relations has been suggested. It has been noted that there is an increasing need to extend the deterministic models which are currently favoured to models including a stochastic component in modelling pharmacological processes [

The transport diffusion is the concentration gradient dependent driven movement and describes the change in concentration of a solution particle. The transport diffusion can further be studied from Fick’s laws by use of partial differential equations, however reseachers have used different variable spaces to study this flow [

The work has allowed for a proposition on the possibility to study the two movements in the 24 h dosing period that aid the process of diffusion by considering stochastic differential equations [

Simulated projected data on secondary saturation movement, time and concentration was taken from pharmacokinetic projections made on Patient P on 600 mg dose considered in Nemaura [

Initially there was modelling of the concentration profile of a solution particle of patient P in time. Letting x ( t ) model concentration of solution particle at time t and was given by,

d x ( t ) d t = θ ( t ) , (1)

for some θ ( t ) . The process x ( t ) was sufficiently modelled by the following equation,

x ( t ) = q t e − r t . (2)

We modelled the gradient driven movement ( x G ) by initially considering, the derived secondary saturation movement ( F ( x , t ) ) , for some patient P, in Nemaura (2014) from relations found which were informed from Nemaura (2015). The diffusant was the drug efavirenz in blood plasma. We gave the estimation of the secondary saturation movement in patient P with respect to time and concentration,

F ( x , t ) = { λ 1 ( e − λ 2 t − e − λ 3 t ) , t ∈ [ 0 , 24 ] : = A , ( i ) u x v + x , x ∈ [ 0 , c max ] : = B . ( i i ) (3)

The following was noted,

d x G d t = d x d F d F d t , (4)

where x G : = x ( F ( t ) ) = x ( t ) . Thus,

d x G = d x d F d F d t d t = d x d F F ′ ( t ) d t , (5)

and,

d F d x = 1 / d x d F . (6)

The secondary saturation movement followed a one to one relation with concen- tration and was thus used as compared to other forms of secondary movement that is convection and advection [

d y ( t ) = d x G ︸ M D + μ ( t , x ( t ) ) d t ︸ M I + σ ( t , x ( t ) ) d b ( t ) ︷ E ϵ y ( 0 ) = 0 , (7)

where the process b ( t ) modelled the independent standard Brownian motion (Weiner process), μ ( t , x ( t ) ) ( h − 2 ) was the relative acceleration drift of a solution particle that allowed for the exchange of concentration material to its neighbourhood, M D -auxilliary concentration gradient dependant driven movement, M I -auxilliary concentration gradient independant driven movement and E ϵ -as the random term in M I . Additionally,

μ ( t , x ( t ) ) ≤ m , (8)

where at μ ( t , x ( t ) ) = m , ρ ( y ( t ) , x ( t ) ) = 0 for x ( t ) > 0 . The parameter, m is the least upper bound of drift for possible positive gradient driven process to occur. Thus the following:

d y ( t ) = ( v + x ( t ) ) 2 u v d F ( t ) + μ ( t , x ( t ) ) d t + σ ( t , x ( t ) ) d b ( t ) y ( 0 ) = 0 , (9)

d y ( t ) = ( ( v + x ( t ) ) 2 u v F ′ ( t ) + μ ( t , x ( t ) ) ) d t + σ ( t , x ( t ) ) d b ( t ) , (10)

d y ( t ) = φ ( t , x ( t ) ) d t + σ ( t , x ( t ) ) d b ( t ) , (11)

where,

φ ( t , x ( t ) ) = ( ( v + x ( t ) ) 2 u v F ′ ( t ) + μ ( t , x ( t ) ) ) . (12)

Furthermore, for φ ( t , x ) and σ ( t , x ) and ∀ t ∈ A ⊂ ℝ + , x , z ∈ B ⊂ ℝ + and some C < ∞ the following conditions are satistified,

| φ ( t , x ) − φ ( t , z ) | ≤ C | x − z | , | σ ( t , x ) − σ ( t , z ) | ≤ C | x − z | , (13)

| φ ( t , x ) | ≤ C ( 1 + | x | ) , | σ ( t , x ) | ≤ C ( 1 + | x | ) . (14)

Conditions 13 and 14 allowed for the existence and uniqueness of solutions for Equation (11) [

We estimated the parameters in Equation (1) for the concentration-time curve (

Furthermore, we estimated the parameter values for the derived secondary saturation F ( x , t ) with respect to time and concentration in Patient P (

Parameters | Estimate | Std Error | t value | |
---|---|---|---|---|

q | 2.7996445 | 0.0280670 | 99.75 | |

r | 0.1168398 | 0.0007906 | 147.79 |

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 | 0.005934 | 135.1 | |

v | 5.624198 | 0.126684 | 44.4 |

There was consideration of how the concentration x ( t ) and movement y ( t ) ( h − 1 ) processes informed potential relationships. The relationship of the concentration gradient driven movement and concentration rate of change was established by observing the relationship between equation 1 and 7. The following was initially considered where M I = 0 and with η ( t , x ( t ) ) = 0 and σ ( t , x ( t ) ) = 0. There were two types of movement considered that were postulated to contribute to diffusion. The part M D modelled the auxilliary transport constituent of diffusion and M I measures the auxilliary self aspect of diffusion. A high correlation value was obtained of ρ ( x ( t ) , y ( t ) ) = 0.9667 (estimated the extent of transport diffusion) in the 24 h dose interval, where x ( t ) , y ( t ) were solutions of Equations (1) and (7) (

Informed by results herewith there was an adoption that transport diffusion was driven by concentration. Thus concentration could be used as a substitute parameter for transport diffusion.

The following case was considered where μ ( t , x ( t ) ) ≥ 0 and 0 ≤ σ ( t , x ( t ) ) ≤ 1. However, it was projected that at μ = 3.2 h − 2 = m the solution particle movement ( y ( t ) ) had zero correlation with the concentration process x ( t ) . In the interval where μ ( t , x ( t ) ) ≥ 3.2 h − 2 , a solution particle movement was projected to have no transport diffusion. A solution particle with increasing relative acceleration drift to its neighbour would have less magnitude of extent with respect to transport diffusion. However, a solution particle with low relative drift to its neighbour had a high potential of transport diffusion. That is transport diffusion was more pronounced in neighbouring particles with relatively low acceleration drift. A relationship of the correlation between x ( t ) and y ( t ) , and the drift in self diffusion was proposed (

An equation for the logistic decay relationship [

ρ ( x ( t ) , y ( t ) ) = 1 1 + k e l μ ( t , x ( t ) ) , 0 ≤ μ ( t , x ( t ) ) ≤ 3.2 , (15)

where k and l (decay) are constants (

l m = 3.99269 × 3.2 = 12.776608 > 1.

where m generally defines the `carrying capacity’ however in this case it was the value of supremum of acceleration drift that enables transfer of concentration material [

Parameters | Estimate | Std Error | t value | |
---|---|---|---|---|

l | 3.99269 | 0.25592 | 15.601 | |

k | 0.10301 | 0.01616 | 6.374 |

It is important to note that the relationship between concentration change and solution particle movement was inferred to be predominantly affected by relative acceleration drift.

Other researchers have been able to correlate self and transport forms of diffusion from experimental observations [

Throughout this work, there was an inherent assumption that the particles were already in orientation that enabled transport diffusion. This was because for transport diffusion to occur the particles should be in such an orientation which enables progression of that process [

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. (2017) Stochastic Modelling of Solution Particle Movement: An Individual Case of Coupled Concentration Gradient Dependent and Independent Movements of Efavirenz. Journal of Applied Mathematics and Physics, 5, 1027-1034. https://doi.org/10.4236/jamp.2017.55090