Deterministic Compartmental Models

This is Polyfemos the copper Cyclops whose body is full of water and someone has given him one eye, one mouth and one hand to each of which a tube is attached. Water appears to drip from his body and to gush from his mouth, all the tubes have regular flow. When the tube connected to his hand is opened his body will empty within 3 days, while the one from his eye will empty in one day and the one from his mouth in 2/5 of a day. Who can tell me how much time is needed to empty him when all three are opened together  

We assume that compartment i is occupied at time 0 by qto amount of material and we denote by q, (t) the amount in the compartment i at time t. We also assume that no material enters in the compartments from the outside of the compartmental system and we denote by Rj,0 (t) the rate of elimination from compartment i to the exterior of the system. Let also Rji (t) be the transfer rate of material from the jth to ith compartment. Because the material 

Mathematics is now called upon to describe the compartmental configurations and then to simulate their dynamic behavior. To build up mathematical equations expressing compartmental systems, one has to express the mass balance equations for each compartment i  

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From the above it can be concluded that in many instances the introduction of an artificial radionuclide into the environment provides us with a natural tracer experiment. Indeed, this is the basis for the application of deterministic compartmental models, based on tracer kinetics, to radioecology (Whicker and Schultz, 1982). This approach is largely based on the assumption that radionuclide movements will exhibit first order kinetics although the existence of naturally-occurring tracees (stable isotopes) at relatively high abundance may result in more complex concentration-dependent kinetics. Furthermore, nutrient analogues may exert even more complex effects on processes such as radioion absorption across root plasma membranes this will become evident later in the chapter. 

The real world of compartmental systems has a strong stochastic component, so we will present a stochastic approach to compartmental modeling. In deterministic theory developed in Chapter 8, each compartment is treated as being both homogeneous and a continuum. But ... 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

As previously, initial conditions for the compartmental model and the enzymatic reaction were set to tiq = [100 50], and so = 100, eo = 50, and cq = 0, respectively. Figures 9.31 and 9.32 show the deterministic prediction, a typical run, and the average and confidence corridor for 100 runs from the stochastic simulation algorithm for the compartmental system and the enzyme reaction, respectively. Figures 9.33 and 9.34 show the coefficient of variation for the number of particles in compartment 1 and for the substrate particles, respectively. 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

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