Back-Translation to a Core-Box Model

The determination of t has many advantages. First, it is a way to compare in-vivo and in-vitro parameters for the same system [11]. Second, it gives an interpretation of the results obtained in the system identification step of the core model. Nevertheless, the final step in the core-box modeling framework is not achieved until the gray-box model has obtained all the estimated features of the estimated core model, 

There is a rich literature for the determination of these symmetry families [31], and this might be of use in the back-translation problem. In any case, for structural unidentifiability caused by conserved moieties the back-translation may be formulated by simple linear combinations [3, 6]. Another special case where improved solutions of the back-translation problem are possible is term elimination and lumping [6] examples of both these cases in the model for insulin signaling are provided in the next section. First, however, we must consider the back-translation problem in the example introduced above. 

Example 3. In Example 1, the core model was obtained using model reduction, and the mapping t is thus already given 

Since p2 is not part of the image in J , the p2 value in the back-translation will not depend on the value of p. We therefore choose p2 always to be mapped to its previous in-vitro estimate, denoted by X. The back-translation of p and p2 may be obtained by choosing the value of one of them (here p ) as the other design variable, X2, and the other as p — X2. This gives the following analytical translation formulas  

The back-translation of the parameter values are given directly by The back-translation of the parameter uncertainties are also given by F using some simple calculations. Simply vary p within its in-vivo uncertainties, and for each p value vary X within the regions allowed by the in-vitro uncertainties. The back-translated values of p are the back-translated uncertainties. Note that these uncertainties are a combination of the in-vivo uncertainties with the in-vitro uncertainties, but that the uncertainties in p are more purely based in the in-vivo data alone. 

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