Lumped Parameter Modelling with Ordinary Differential Equations
This chapter presents an overview of ordinary differential equations and their use in lumped parameter modelling of physical systems and physiological processes. Methods are described on how to solve some ODEs analytically, followed by a brief overview of
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Lumped Parameter Modelling with Ordinary Differential Equations
2.1 Overview of Ordinary Differential Equations An ordinary differential equation (ODE) is used to express a relationship between a function of one independent variable (typically time) and its derivatives. If no derivatives are present, the relationship is characterised by an algebraic equation (AE). ODEs are often used in lumped parameter modelling to approximate the behaviour a physical system by separating it into discrete parts, each characterised by one or more dependent variables. An example of a simple ODE is: r N (K − N ) dN = , dt K
N (0) = N0
(2.1)
where N represents the population of, say, bacteria in a Petri-dish, r is the growth rate when N = 0, and K is the maximum population capacity of the system. For this simple example, it is possible to obtain an exact closed-form solution for N as a function of t using the method of separation of variables, in which the variables are grouped on each side of the equality. Thus, we can rewrite Eq. 2.1 in the form: dt dN = r N (K − N ) K Integrating both sides, we obtain
dN = r N (K − N )
dt t = + C0 K K
(2.2)
where C0 is a constant of integration. To integrate the left-hand side, we rewrite the integrand using the partial fraction expansion: A1 A2 1 = + r N (K − N ) rN K−N © Springer-Verlag Berlin Heidelberg 2017 S. Dokos, Modelling Organs, Tissues, Cells and Devices, Lecture Notes in Bioengineering, DOI 10.1007/978-3-642-54801-7_2
(2.3) 29
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2 Lumped Parameter Modelling with Ordinary Differential Equations
where A1 , A2 are constants to be determined. Multiplying both sides of Eq. 2.3 by r N , then setting N = 0, yields A1 = 1/K . Similarly, multiplying both sides of Eq. 2.3 by K − N , then setting N = K , yields A2 = 1/r K . Hence, the left-hand side of Eq. 2.2 can be written as:
dN 1 1 dN dN = + r N (K − N ) K rN rK (K − N ) ln N ln(K − N ) = − rK rK N 1 = ln rK K−N
Substituting this into the left-hand side of Eq. 2.2 and multiplying both sides by r K yields: N = r t + Co r K ln K−N and since N = N0 when t = 0, we have C0 =
N ln K−N
1 rK
ln
N0 K −N0
N0 = r t + ln K − N0
. Thus,
and taking the exponential of both sides: N = K−N
N0 er t K − N0
Finally, after a little algebraic manipulation, we obtain the closed-form solution for N as: K er t N= K −N0 + er t N0 which is known as the logistic equation. In general, however, when modelling with ODEs we must numerically-integrate to obtain an approximate solution. When more than one ODE is involved, the set of equations is known as a system of ordinary differential equations. If the multiple set of equations includes a combination of ODEs and AEs, it is termed a differential-algebraic equation (DAE) system. If any of the differential equations involves multiple independent variables (such as time and space), then it is referred to a partial differential equation (or PDE). PDEs are discussed further in the next chapter. Example 2.1 Consider a mass m connected to a spring, moving in the pre
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