Optimal controlling of anti-TGF-[Formula: see text] and anti-PDGF medicines for preventing pulmonary fibrosis.

In the repair of injury, some transforming growth factor-[Formula: see text]s (TGF-[Formula: see text]s) and platelet-derived growth factors (PDGFs) bind to fibroblast receptors as ligands and cause the differentiation of fibroblasts into myofibroblasts. When the injury repair is repeated, the myofibroblasts proliferate excessively, forming fibrotic tissue. We goal to control myofibroblasts proliferation and apoptosis with anti-transforming growth factor-[Formula: see text] (anti-TGF-[Formula: see text]) and anti-platelet-derived growth factor (anti-PDGF) medicines. The novel optimal regulator control problem with two controls (medicines) is proposed to simulate how to the preventing pulmonary fibrosis. Idiopathic pulmonary fibrosis (IPF) consists of restoring a system of cells, protein, and tissue networks with injury and scar. Myofibroblasts proliferation back to its equilibrium position after it has been disturbed by abnormal repair. Thus, the optimal regulator control problem with a parabolic partial differential equation as a constraint, zero flux boundary, and given specific initial conditions, is considered. The myofibroblast diffusion equation stands as a governing dynamic system while the objective function is the summation of myofibroblast, anti-TGF-[Formula: see text] and anti-PDGF medicines for the fixed final time. Here, myofibroblast is a nonlinear state of time while anti-TGF-[Formula: see text] and anti-PDGF are two unknown control functions. In order to solve the corresponding problem a weighted Galerkin method is used. Firstly, we convert the myofibroblast diffusion equation to a system of ordinary differential equations using the Lagrangian interpolation polynomials defined at Gauss-Lobatto integration points. Secondly, by the calculus of variations, the optimal control problem is solved successfully using canonical Hamiltonian and extended Riccati equations. Numerical results are given, and the plots are depicted. Moreover, solutions to the problem in which there is no control are compared. Numerical results show that, over time, the myofibroblast increases and then remains constant when there is no control. In contrast, the current solution decreases and vanishes after 300 days by prescribing controller medicines for anti-TGF-[Formula: see text] and anti-PDGF. The optimal strategy proposed in this paper helps practitioners to reduce myofibroblasts by controlling both anti-TGF-[Formula: see text] and anti-PDGF medicines.

Optimal control of TGF-ß to prevent formation of pulmonary fibrosis.

In this paper, three optimal control problems are proposed to prevent forming lung fibrosis while control is transforming growth factor-ß (TGF-ß) in the myofibroblast diffusion process. Two diffusion equations for fibroblast and myofibroblast are mathematically formulated as the system's dynamic, while different optimal control model problems are proposed to find the optimal TGF-ß. During solving the first optimal control problem with the regulator objection function, it is understood that the control function gets unexpected negative values. Thus, in the second optimal control problem, for the control function, the non-negative constraint is imposed. This problem is solved successfully using the extended canonical Hamiltonian equations with no flux boundary conditions. Pontryagin's minimum principle is used to solve the related optimal control problems successfully. In the third optimal control problem, the fibroblast equation is added to a dynamic system consisting of the partial differential equation. The two-dimensional diffusion equations for fibroblast and myofibroblast are transferred to a system of ordinary differential equations using the central finite differences explicit method. Three theorems and two propositions are proved using extended Pontryagin's minimum principle and the extended Hamiltonian equations. Numerical results are given. We believe that this optimal strategy can help practitioners apply some medication to reduce the TGF-ß in preventing the formation of pulmonary fibrosis.

Optimal control and differential game solutions for social distancing in response to epidemics of infectious diseases on networks.

In this paper, the problem of social distancing in the spread of infectious diseases in the human network is extended by optimal control and differential game approaches. Hear, SEAIR model on simulation network is used. Total costs for both approaches are formulated as objective functions. SEAIR dynamics for group k that contacts with k individuals including susceptible, exposed, asymptomatically infected, symptomatically infected and improved or safe individuals is modeled. A novel random model including the concept of social distancing and relative risk of infection using Markov process is proposed. For each group, an aggregate investment is derived and computed using adjoint equations and maximum principle. Results show that for each group, investments in the differential game are less than investments in an optimal control approach. Although individuals' participation in investment for social distancing causes to reduce the epidemic cost, the epidemic cost according to the second approach is too much less than the first approach.

Solving Multiextremal Problems by Using Recurrent Neural Networks.

In this paper, a neural network model for solving a class of multiextremal smooth nonconvex constrained optimization problems is proposed. Neural network is designed in such a way that its equilibrium points coincide with the local and global optimal solutions of the corresponding optimization problem. Based on the suitable underestimators for the Lagrangian of the problem, one give geometric criteria for an equilibrium point to be a global minimizer of multiextremal constrained optimization problem with or without bounds on the variables. Both necessary and sufficient global optimality conditions for a class of multiextremal constrained optimization problems are presented to determine a global optimal solution. By study of the resulting dynamic system, it is shown that under given assumptions, steady states of the dynamic system are stable and trajectories of the proposed model converge to the local and global optimal solutions of the problem. Numerical results are given and related graphs are depicted to illustrate the global convergence and performance of the solver for multiextremal constrained optimization problems.

Heat treatment modelling using strongly continuous semigroups.

In this paper, mathematical simulation of bioheat transfer phenomenon within the living tissue is studied using the thermal wave model. Three different sources that have therapeutic applications in laser surgery, cornea laser heating and cancer hyperthermia are used. Spatial and transient heating source, on the skin surface and inside biological body, are considered by using step heating, sinusoidal and constant heating. Mathematical simulations describe a non-Fourier process. Exact solution for the corresponding non-Fourier bioheat transfer model that has time lag in its heat flux is proposed using strongly continuous semigroup theory in conjunction with variational methods. The abstract differential equation, infinitesimal generator and corresponding strongly continuous semigroup are proposed. It is proved that related semigroup is a contraction semigroup and is exponentially stable. Mathematical simulations are done for skin burning and thermal therapy in 10 different models and the related solutions are depicted. Unlike numerical solutions, which suffer from uncertain physical results, proposed analytical solutions do not have unwanted numerical oscillations.