Lecture notes in biomathematics: Compartmental modeling and tracer kinetics

Lecture 8 Biomathematics ( FMAN 01 )

So what happens when b is small? The graph below illustrates (for a = 1, d = 0.1) what happens when we increase b from 0.1 to 0.5. For b ≥ 0.26 we have a stable equilibrium, whereas when b < 0.26 the orbits approch a so-called limit cycle. In contrast to the Lotka-Volterra model this cycle is independent of starting conditions (but depends on parameters). Why there is a limit cycle and why it i...

Modeling Tracer Kinetics in Dynamic GdlDTPA

Three major models (from Tofts, Larsson, and Brix) for collecting and analyzing dynamic MRI gadolinium-diethylenetriamine penta-acetic acid (GI-DTPA) data are examined. All models use compartments representing the blood plasma and the abnormal extravascular extracellular space (EES), and they are intercompatible. All measure combinations of three parameters: (1) kPSp is the influx volume transf...

Tracer Kinetics Modeling Basics: Model Formulation

Introduction With the advances in fast MR imaging techniques, it is possible to monitor the traversal of a MR contrast agent within tissues at short time intervals (of the order of a few seconds or less), thereby allowing the possibility to study blood flow at the tissue level. Dynamic contrastenhanced MRI (DCE MRI) is emerging as a promising approach for in vivo assessment of tissue microcircu...

Lecture 6 Biomathematics ( FMAN 01 ) Anders Källén

of which the second requires α > 1 to exist biologically. When α < 1 the first is stable; it is unstable when the second exist, which in such a case is stable. To get further insight we look at the isoclines, i.e., the curves where the trajectories are parallel to one of the axes. In this case the equation s′ = 0 corresponds to the curve b = (s0 − s) 1+s s and the equation b′ = 0 to straignt li...

Lecture Notes #13: Reliability and Structural Equation Modeling 13-1