On the control of transport of a medicinal product in an organism

1. Introduction

In the modern pharmaceutical market, there are a huge number of drugs used in the treatment of various diseases. At the same time, there are various factors that can affect pharmacokinetics and pharmacodynamics drugs: age, diseases, bad habits, lifestyle of patients, gender, and mutations in genes. To take into account a maximal number of the above factors one can find a fast increase of the quantity of new medicinal products as well as intensive development of old medicinal products with questionable efficacy and safety.^1,2,3,4,5 Usually, the influence of medicinal products on organisms could be done experimentally. Sometimes, the required dose of the considered products with influence on the organism could be estimated. At present several models to analyze the transport of medicinal products through organisms have been elaborated.^6,7 In the framework of these models one can analyze mass transport of medicinal products in an organism: input, absorption and removal of medicinal products (with constant values or variable parameters). In this paper, we consider a model for estimation of spatio-temporal distribution of concentration of a medicinal product. Based on the model, we analyzed the above concentration taking into account the possible changing properties of organisms.

2. Method of Solution

In this section, we introduce a model for estimation and analysis of spatio-temporal distribution of concentration of a medicinal product in an organism taking into account the possible changing properties. In comparison with models in Refs. 6 and 7, the introduced model gives a possibility to analyze the distribution of concentration of a medicinal product in space and time with spatial and temporal variations of parameters of the model. We calculate the required distribution as the solution of the second Fick’s law in the following form :

where $C(x,t)$ is the spatio-temporal distribution of concentration of the considered medicinal product; $D(x,t)$ is the diffusion coefficient of the product, which depends on tissue conditions in the organism; $K(x,t)$ is the parameter of interaction of the considered product with other substances in the organism; $N(x,t)$ is the concentration of other substances in the organism, which interactes with the infused product. Initial distribution of concentration of the considered medicinal product depends on the type of infusion (single or continuous infusion) and could be written in the following form:

$$\mathrm{\text{For single infusion of product}}:C(x,0)=f_C(x);$$(2a)

$$\mathrm{\text{For continuous infusion of product}}:C(0,0)=C_0,C(x>0,0)=0.$$(2b)

Boundary values of concentration of the considered medicinal product also depend on the type of infusion

$$\mathrm{\text{For single infusion of product}}:{\left.\frac{\partial C(x,t)}{\partial x}\right|}_{x=0}=0,C(L,t)=0.$$(3a)

$$\mathrm{\text{For continuous infusion of product}}:C(0,t)=C_0,C(L,t)=0,$$(3b)

where $C_0$ is the concentration of product in place of infusion. In this situation, we assume that the initial distribution of medicinal product presents near the left boundary. The right boundary presents very far from the initial distribution of the medicinal product. In this situation, the product cannot achieve the right boundary. Next, let us solve Eq. (1) with conditions (2) and (3) by the method of averaging of function corrections.^8,9,10 First of all, we transform the differential equation (1) to the following integral form (for single or continuous infusion, respectively) by integrating left and right sides of Eq. (1) on x and t and using boundary and initial conditions.

$$\begin{array}{l}C(x,t)=C(x,t)+\frac{1}{L^2}\left\{{\int }_0^t{\int }_0^{x}D(v,\tau )C(v,\tau )dvd\tau \right.\\ -{\int }_0^t{\int }_0^x(x-v)C(v,\tau )\frac{\partial D(v,\tau )}{\partial v}dvd\tau \\ -{\int }_0^t{\int }_0^x(x-v)K(v,\tau )C(v,\tau )N(v,\tau )dvd\tau \\ +{\int }_0^L(L-v)C(v,t)dv\\ -{\int }_0^t{\int }_0^{L}D(v,\tau )C(v,\tau )dvd\tau \\ +{\int }_0^x(x-v)f(v)dv+{\int }_0^t{\int }_0^L(L-v)C(v,\tau )\\ \left.\times \frac{\partial D(v,\tau )}{\partial v}dvd\tau -{\int }_0^x(x-v)C(v,t)dv\right\},\end{array}$$(4a)

$$\begin{array}{l}C(x,t)=C(x,t)+\frac{1}{L^2}\left\{{\int }_0^t{\int }_0^{x}C(v,\tau )\frac{\partial D(v,\tau )}{\partial v}dvd\tau \right.\\ -{\int }_0^t{\int }_0^x(x-v)K(v,\tau )C(v,\tau )N(v,\tau )dvd\tau \\ +{\int }_0^x(x-v)f(v)dv-{\int }_0^{t}D(x,\tau )C(x,\tau )d\tau \\ +C_0{\int }_0^{t}D(x,\tau )d\tau \\ -\frac{x}{L}\left[{\int }_0^t{\int }_0^{L}C(v,\tau )\frac{\partial D(v,\tau )}{\partial v}dvd\tau \right.\\ -{\int }_0^t{\int }_0^L(L-v)K(v,\tau )C(v,\tau )N(v,\tau )dvd\tau \\ +\left.{\int }_0^L(L-v)f(v)dv-{\int }_0^L(L-v)C(v,t)dv\right]\\ -\left.{\int }_0^x(x-v)C(v,t)dv\right\}.\end{array}$$(4b)

It should be noted that we add functions $C(x,t)$ to both sides of the obtained integral equation: the transformation gives a possibility to use the considered method of averaging of function corrections. To use the above method we substitute the not-yet-known average value of the required concentration ${\alpha }_1$ instead of the above concentration on the right sides of Eq. (4). The substitution gives a possibility to obtain equations to calculate the first-order approximations of concentration of the considered product in the following form (for single or continuous infusion, respectively):

$$\begin{array}{l}{C}_1(x,t)={\alpha }_1+\frac{1}{L^2}\left[{\int }_0^x(x-v)f(v)dv-{\alpha }_1{\int }_0^t{\int }_0^x(x-v)\right.\\ \left.\times K(v,\tau )N(v,\tau )dvd\tau +{\alpha }_1\frac{L^2-x^2}{2}\right],\end{array}$$(5a)

$$\begin{array}{l}{C}_1(x,t)={\alpha }_1+\frac{1}{L^2}\left\{{\alpha }_1{\int }_0^t[D(x,\tau )-D(0,\tau )]d\tau \right.\\ -{\alpha }_1{\int }_0^t{\int }_0^x(x-v)K(v,\tau )N(v,\tau )dvd\tau \\ +{\int }_0^x(x-v)f(v)dv-{\alpha }_1{\int }_0^{t}D(x,\tau )d\tau \\ +C_0{\int }_0^{t}D(x,\tau )d\tau \\ -\left[{\alpha }_1{\int }_0^t[D(L,\tau )-D(0,\tau )]d\tau \right.\\ -{\alpha }_1{\int }_0^t{\int }_0^L(L-v)K(v,\tau )N(v,\tau )dvd\tau \\ \left.\left.+{\int }_0^L(L-v)f(v)dv\right]\frac{x}{L}-{\alpha }_1(L+x)\frac{x}{2}\right\}.\end{array}$$(5b)

Average value of concentration of medical product ${\alpha }_1$ could be obtained by the following standard relation^8,9,10 :

Substitution of relations (5) into relation (6) gives a possibility to obtain relations to determine the average value ${\alpha }_1$ in the final form (for single or continuous infusion, respectively)

$$\begin{array}{l}{\alpha }_1=\left[2\mathrm{\Theta }{\int }_0^L{\int }_0^x(x-v)f(v)dvdx\right.\\ +\mathrm{\Theta }L{\int }_0^L(L-x)f(x)dx\\ +\left.2C_0{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}D(x,t)dxdt\right]\\ \times \left[{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L(L^2-x^2)K(x,t)N(x,t)dxdt\right.\\ +{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}D(x,t)dxdt+L^3\frac{5\mathrm{\Theta }}{12}\\ -{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}x(L-x)K(x,t)N(x,t)dxdt\\ +\frac{L}{2}{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L(L-x)K(x,t)N(x,t)dxdt\\ -\left.{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}D(x,t)dxdt\right].\end{array}$$(7b)

The second-order approximation of the considered concentration in the framework of the method of averaging of function corrections could be determined by using the following standard procedure: replacement of the considered concentration on the right sides of Eq. (4) on the sum $C(x,t)\to {\alpha }_2+C_1(x,t)$.^8,9,10 The replacement gives a possibility to obtain the following equations to determine the concentration of the considered medicinal product (for single or continuous infusion, respectively) :

$$\begin{array}{l}{C}_2(x,t)={\alpha }_2+C_1(x,t)+\frac{1}{L^2}\left\{{\int }_0^t{\int }_0^x[{\alpha }_2+C_1(v,\tau )]\right.\\ \times \frac{\partial D(v,\tau )}{\partial v}dvd\tau +{\int }_0^x(x-v)f(v)dv\\ -{\int }_0^t{\int }_0^x(x-v)K(v,\tau )[{\alpha }_2+C_1(v,\tau )]\\ \times N(v,\tau )dvd\tau -{\int }_0^{t}D(x,\tau )[{\alpha }_2+C_1(x,\tau )]d\tau \\ +C_0{\int }_0^{t}D(x,\tau )d\tau -{\int }_0^x(x-v)[{\alpha }_2+C_1(v,t)]dv\\ -\frac{x}{L}\left[{\int }_0^t{\int }_0^L[{\alpha }_2+C_1(v,\tau )]\frac{\partial D(v,\tau )}{\partial v}dvd\tau \right.\\ -{\int }_0^t{\int }_0^L(L-v)K(v,\tau )[{\alpha }_2+C_1(v,\tau )]\\ \times N(v,\tau )dvd\tau +{\int }_0^L(L-v)f(v)dv\\ \left.\left.-{\int }_0^L(L-v)\times [{\alpha }_2+C_1(v,t)]dv\right]\right\}.\end{array}$$(8b)

Average value of the second-order approximation of the above concentration ${\alpha }_2$ could be calculated by using the following standard relation^8,9,10 :

Substitution of relations (8) into relation (9) gives a possibility to obtain the following relations for the required average value ${\alpha }_2$ (for single or continuous infusion, respectively):

$$\begin{array}{l}{\alpha }_2=\left[\frac{1}{2}{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L{(L-x)}^{2}K(x,t)C_1(x,t)N(x,t)dxdt\right.\\ +{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L{x}^{2}K(x,t)C_1(x,t)\times N(x,t)dxdt\\ -{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L(L-x)C_1(x,t)\frac{\partial D(x,t)}{\partial x}dxdt\\ -\frac{\mathrm{\Theta }}{2}{\int }_0^L{(L-x)}^{2}f(x)dx-\mathrm{\Theta }{\int }_0^L{x}^{2}f(x)dx\\ +C_0{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}D(x,t)dxdt+{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t)\\ \times {\int }_0^{L}D(x,t)C_1(x,t)dxdt\\ +2{\int }_0^{\mathrm{\Theta }}{\int }_0^L{x}^2{C}_1(x,t)dxdt-\frac{L}{2}{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t)\\ \times {\int }_0^L(L-x)K(x,t)C_1(x,t)N(x,t)dxdt\\ +\frac{1}{2}{\int }_0^{\mathrm{\Theta }}{\int }_0^L(L^2-x^2)\times C_1(x,t)dxdt\\ +\mathrm{\Theta }{\int }_0^L(L-v)f(v)dv\\ -{\int }_0^{\mathrm{\Theta }}{\int }_0^{L}x(L-x)C_1(x,t)dxdt\\ +\left.\frac{L}{2}{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L{C}_1(x,t)\times \frac{\partial D(x,t)}{\partial x}dxdt\right]\\ \times \left[{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}D(x,t)dxdt-\frac{1}{2}{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t)\right.\\ \times {\int }_0^L{(L-x)}^{2}K(x,t)N(x,t)dxdt-\mathrm{\Theta }\frac{L^3}{3}\\ -{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L{x}^{2}K(x,t)N(x,t)dxdt\\ +\frac{L}{2}{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^L(L-x)K(x,t)N(x,t)dxdt\\ {\left.-{\int }_0^{\mathrm{\Theta }}(\mathrm{\Theta }-t){\int }_0^{L}D(x,t)dxdt\right]}^{-1}.\end{array}$$(10b)

Spatio-temporal distribution of concentration of medicinal products was analyzed analytically by using the second-order approximation in the framework of the method of averaging of function corrections. The approximation is usually a good approximation to make qualitative analysis and to obtain some quantitative results. All the obtained results have been checked by comparison with the results of numerical simulations in the framework of explicit different scheme.

3. Discussion

In this section, we present an analysis of the obtained second-order approximation of the spatio-temporal distribution of concentration of medicinal products in organisms. Figures 1 and 2 show typical dependences of the considered approximation of concentration (8a) and (8b) on time with several examples of parameters for testing of the introduced model. Figures 3 and 4 show typical dependences of the considered approximation of concentration (8a) and (8b) on coordinate with several examples of parameters for testing of the introduced model. The obtained dependences qualitatively coincide with analogous experimental distributions. Increasing temperature of the organism leads to acceleration (in the considered interval of temperature) of interaction of the considered medicinal product with other substances of the organism.

4. Conclusion

In this paper, we consider the analysis of transport of a medicinal product in an organism. The analysis is based on estimation of spatio-temporal distribution of concentration of the above product. We introduce an analytical approach for analysis of the above transport taking into account its changing conditions. We consider the possibility to accelerate and decelerate the transport of medicinal product in organisms.