Fig. 1

Example of a general multi-infusion set-up. In this example, the number of pumps is three. The method, however, allows for any number of pumps. a The catheter \(\mathcal {C}\) contains a mixture of the fluids “R”, “G”, and “B”, in which “R”, “G”, and “B” are the contents of syringe 1, syringe 2, and syringe 3, respectively. We consider this mixture to be completely homogeneous, despite the “stratified” rendering of the three colours in catheter \(\mathcal {C}\) in this figure. b Situation at \(t=t_{delay}\); since \(t_{delay}\) is by definition the time needed to travel through the catheter \(\mathcal {C}\), the contents \(\xi \) that was inside the voxel \(k=N\) (near the mixing point \(\mathcal {M}\)) at time \(t = 0\) in a, has now reached the very tip \(\mathcal {P}\) of the catheter at time \(t=t_{delay}\), and is entering the blood stream of the patient