Reliability for drug targeting in cancer treatment through nanotechnology (A Stochastic differential equation-based flexible model)

The lack of a unified definition of reliability function in system design phase may contribute to the difficulty in improving the reliability of nanotechnology drug delivery systems. The reliability function is frequently used in reliability engineering and gives the probability of an item operating for a certain amount of time without failure. As such, the reliability function is a function of time. In this paper we define the notion of reliability function in the context of drug targeting in cancer treatment. This function gives the probability of all cancerous cells being destroyed in a certain amount of time. Having defined our notion of the reliability function, we propose a model for this function based on Ito type stochastic differential equations. We also describe an algorithm for computing the minimum dose level to achieve a desired value for expected time needed to destroy all cancerous cells. Notation C the number of nanoconjugates that are carrying the drug and they are injected into the blood stream S (t) the number of nanoconjugates that are still at the site of administration at time t S1 (t) the number of nanoconjugates that are in non-targeted tissues at time t I1 (t) the number of nanoconjugates that are in cancerous cells population at time t I2 (t) the number of nanoconjugates that are in healthy cells population at time t R (t) the number of nanoconjugates that are eliminated at time t d1 the average number of nanoconjugates that penetrate cancerous cells population in the targeted tissue per unit time d2 the average number of nanoconjugates that penetrate healthy cells population in the targeted tissue per unit time d3 the average number of nanoconjugates that penetrate the non-targeted tissues per unit time d4 the average number of nanoconjugates that are eliminated from non-targeted tissues per unit time d* =(d1+d2+d3) γ1 the average number of nanoconjugates that are eliminated from cancerous cells population in the targeted tissue per unit time γ2 the average number of nanoconjugates that are eliminated from healthy cells population in the targeted tissue per unit time λ the average number of cancerous cells that are killed per unit time λ1 the average number of cancerous cells population is expanded per unit time d1* the average number of cancerous cells being eliminated per unit time= λ1 k Number of nanocojugates, carrying the drug, needed to destroy a cancerous cell in the tissue N(t) the number of cancerous cells in the tissue at time t n the number of cancerous cells in the tissue at time 0 T the first time all cancerous cells will be destroyed P(t) the reliability of a nanodrug delivery system = P(T≤ t) E(T) expected time required to kill all cancerous cells Bj (t) standard Brownian motion, j=1,2,3,4,5 σj (t) diffusion coefficient, j=1,2,3,4,5. Introduction The focus of the nanotechnology therapeutic approaches to treat cancer has been on early disease detection, drug discovery and monitoring, controlled release of therapeutic agents, and targeted drug delivery. See Amiji [1] for more details. Targeted drug delivery requires statistical modeling to establish the reliability in success of drug targeting. Unreliable nanotechnology drug delivery systems are a Correspondence to: Ali Mansoori G, Departments of Bioengineering, Chemical Engineering and Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA; E-mail: mansoori@uic.edu