Streptavidin Recombinant br Fig Partition of the model
Fig. 9. Partition of the model data set according to the four methods.
the number of clusters to 12, the complete Streptavidin Recombinant remains as the best method, but with smaller differences.
At the first glance, the complete linkage seems to be the method with better performance, having substantial differences compared with the k-means. However, k-means agglomerates the models in a more uniformed way, yielding a better sample representation. This is mainly because the complete linkage tends to make partitions in regions where the models are further apart, creating also clusters with a large number of elements in areas where they are closer to each other. In this situation the ultimate check is only made by testing the controller selected on the model.
3. Results and discussion
Several simulations were performed for a model data set that is distinct from the test set used in Section 2.4. Comparison met-rics were used in order to evaluate the results, which include the number of true and false switches, the time delay between refer-ence and true switch, and also the Mean Absolute Percentage Error 100
, where R is the reference
and V the tumor volume.
This test simulates a scenario of a patient in which his dynam-ics, described by the equations summarized in Section 2.1, is time varying. That is, the parameters and ˇ vary in a non linear way over time, as illustrated in Fig. 9 by a solid black curve for the two different clustering algorithms with 6 and 12 clusters.
The results of the MMAC algorithm simulations performed for the four methods are described in Table 1. All the four simulations present satisfactory results, with levels of MAPE around 3% and low toxicity levels. What directly stands out is the large number of false switches for the 12 clusters algorithms with 5 and 8 false switches for k-means and complete linkage, respectively. This case
Evaluation metrics for the simulations performed.
Algorithm Test MAPE [%] T.S. F.S. mean [days] max [days]
happens because the computed centroids are closer to each other, which means that a small variation in the plant dynamics can result in a big oscillation in the selected model, until the stabilization of the control system. Aside from that, most of the false switches are delayed true switches, i.e., the right switches are performed but not in the correct time. This situation indicates that the variations occur too fast, and since the system has to dwell 30 days after a switch, the performance can be deteriorated for methods with more clusters.
It was also expected that the MAPE would decrease with the increase of clusters quantity, since each controller was designed for a smaller neighborhood. However, the MAPE is slightly greater for simulations with 12 clusters, possibly due to the oscillations described.
The algorithms with 6 clusters showed better results in terms of performance, particularly, the k-means, whose simulations are illustrated in Fig. 10, having 100% accuracy of true switches, false switches and the lowest MAPE. Thus, the k-means with 6 clusters is used in the next tests.
Fig. 10. Simulation of the MMAC algorithm using the k-means with 6 clusters.
Fig. 11. Selected patient dynamics depending on the dwell time D .
3.1. Dwell time variation
In this test – Fig. 11 and Table 1 – the effect of the dwell time can be observed. All three different curves are identical except at the beginning of therapy. For D = 30 and 75 days, there is a switch from the model M1c to M2c , which is kept during their respective dwell times. However, for the case where no dwell time condition is involved, the system cannot prevent the bump at t = 0.4 days.
Therefore, in this case, dwelling in the model M2c for D con-tributes positively to the performance. This fact can be evidenced if the MAPE values for the three dwell time tests are compared. Without the dwell time condition, i.e. for D = days, the system presents the largest MAPE, since model M2c was not kept during the first instants. The smallest MAPE was obtained for D = 30 days, having thus the best performance.
3.2. Pharmacodynamical simulation error variation
In order to study the uncertainty influence on the PS parameters, several simulations were performed for different values of PS – Fig. 12 and Table 1.
A higher value for PS basically represents an increase in parameters of the block DR’, belonging to the PS, namely in the parameters Lr , Kr , and C50base – Section 2.2.2. Increasing Lr means that the drug resistance threshold is higher, but the same variation in the other two parameters is a warning for the control system that the mutant cells are more drug resistant. In this way, the control system is going to react by administrating a higher drug concen-tration in order to achieve the same desired effect, as illustrated in Fig. 12 where the drug concentration is higher for larger values of