Differential Equations in MATLAB
Differential equations play an important role in describing time-dependent quantities such as drug concentrations in the human body following drug treatment. In this chapter, we introduce the main types of differential equations used in the pharmacologic
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Differential Equations in MATLAB
Differential equations play an important role in describing time-dependent quantities such as drug concentrations in the human body following drug treatment. In this chapter we introduce the main types of differential equations used in the pharmacologic modeling of drug concentrations and effects. We show how they are treated in MATLAB, from both the symbolic and numeric viewpoints. Several interrelated quantities are often considered together, giving rise to differential equation systems also called dynamic systems [1]. In pharmacologic modeling, dynamic systems representing the PK of a drug receive external input due to drug treatment. Forcing (driving, or control) functions are the mathematical construct for modeling this phenomenon. Dynamic systems may be steered or controlled in two ways: either by scheduling times for input (‘time scheduling’) or by specifying events that should impact on input (‘event scheduling’). We will show how both types of scheduling can be used with MATLAB. Empirical models described by differential equations usually contain model parameters to be determined from data. This regression problem can be solved in MATLAB, as shown at the end of this chapter.
3.1 Concepts Differential equations started their triumphant march with the work of Isaac Newton (1642–1727) who established the field of classical mechanics by inventing differential and integral calculus. Classical mechanics deals with the motion, i.e., the change in position over time, of bodies under the influence of external forces. Differential equations became the predominant mathematical construct to describe any quantities that change over time. If the quantities are interrelated, we refer to them as a dynamic system. In ecology, the interactions between two species, predator and prey, were independently investigated about 100 years ago by Alfred Lotka (1880–1949) and Vito Volterra (1860–1940) by means of nonlinear dynamic systems.
R. Gieschke and D. Serafin, Development of Innovative Drugs via Modeling with MATLAB, DOI: 10.1007/978-3-642-39765-3_3, Springer-Verlag Berlin Heidelberg 2014
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Differential Equations in MATLAB
Differential equations are equations that contain derivatives of an unknown function y, as shown below: 9 dy ¼ fðt; yÞ ; t 2 ½t0 ; tf = dt ð3:1Þ ; yðt0 Þ ¼ y0 Here, y refers to a vector-valued function of time t, y = y(t). The right-hand side of (3.1) specifies a vector-valued function f depending on y and t. y is a solution of this equation if its derivative with respect to time matches f for the interval [t0; tf]. 9 dyðtÞ fðt; yðtÞÞ ; t 2 ½t0 ; tf = ð3:2Þ dt ; yðt0 Þ ¼ y0 Without further constraints, differential equations have no unique solution. The second condition in (3.1) poses a constraint on y by specifying a value, y0, at initial time t0. For this initial value problem (IVP) uniqueness and existence of a solution are guaranteed for a wide class of functions f. A differential equation is said to be autonomous if it does not explicitly depend on
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