Understanding glioblastoma invasion using physically-guided neural networks with internal variables

Microfluidic capacities for both recreating and monitoring cell cultures have opened the door to the use of Data Science and Machine Learning tools for understanding and simulating tumor evolution under controlled conditions. In this work, we show how these techniques could be applied to study Glioblastoma, the deadliest and most frequent primary brain tumor. In particular, we study Glioblastoma invasion using the recent concept of Physically-Guided Neural Networks with Internal Variables (PGNNIV), able to combine data obtained from microfluidic devices and some physical knowledge governing the tumor evolution. The physics is introduced in the network structure by means of a nonlinear advection-diffusion-reaction partial differential equation that models the Glioblastoma evolution. On the other hand, multilayer perceptrons combined with a nodal deconvolution technique are used for learning the go or grow metabolic behavior which characterises the Glioblastoma invasion. The PGNNIV is here trained using synthetic data obtained from in silico tests created under different oxygenation conditions, using a previously validated model. The unravelling capacity of PGNNIV enables discovering complex metabolic processes in a non-parametric way, thus giving explanatory capacity to the networks, and, as a consequence, surpassing the predictive power of any parametric approach and for any kind of stimulus. Besides, the possibility of working, for a particular tumor, with different boundary and initial conditions, permits the use of PGNNIV for defining virtual therapies and for drug design, thus making the first steps towards in silico personalised medicine.

The authors claim that the presented method is more flexible than previous frameworks (PINNs and BINNs), as well as generally performing better than standard parametric learning frameworks both in terms of predicting future cell culture evolution as well as learning the unknown functions governing the go-or-grow mechanism. The former could benefit from a more thorough discussion/motivation, and the latter claim is indeed demonstrated for a number of different in-silico experimental conditions. From the discussion and general theme of the paper, focus is on understanding glioblastoma, however due to the lack of experimental data specific conclusions cannot be drawn.
Following is a list of comments and questions, ranging from major to minor, followed by a list of typos.

MAJOR
1. The work is interesting and the method appears promising, however the results would be greatly improved and method's utility more believable if it was applied to experimental data. The title suggests that the understanding of glioblastoma invasion is furthered, and mechanisms are unraveled, although no data is present, and no conclusions regarding glioblastoma were drawn.
2. The go or grow hypothesis is often postulated as migration and proliferation being spatiotemporally exclusive processes. There are many instances where this has been modelled using two subpopulations, one representing cells which only migrate and one representing cells which only proliferate (e.g. Fedotov and Iomin PRL 2007, Gerlee and Nelander PLOS Computational Biology 2012, Stepien et al. Siam J. Appl. Math. 2018). In such models the switching terms between either state has an easily interpretable meaning, namely the rate at which cells change phenotype. Is there any deeper interpretation of the two functions Π go and Π gr , in particular in the case that their sum is not 1?
As a follow-up question, what is the motivation for assuming that the go-term represented by Π go only influences the advection term and not the random motility term? Since therapeutic evaluations are discussed, the random motility would presumably be of equal importance, and in the current framework it is specified beforehand and not inferred from the data. Would not this be a limitation when it comes to generality of the model?
3. It is mentioned that the proposed framework is more flexible compared to PINNs and BINNs. Could you please elaborate on this and explain the differences in greater detail?
4. It would be interesting to know more about the practical aspects, such as the computational demands and the hardware/software used. Is it possible to train your network on the dataset used on a standard PC within reasonable times, or would it require special hardware such as GPUs? It would also be interesting to know if the proposed method is faster or slower than the parametric approach used for comparison.

MINOR
1. On lines 74-82: In my opinion it should be emphasized that the method is used on synthetic data and not experimental data. First time reading the paper I was under the impression at this point that experimental data would be used.

Line 275:
Here R is used to represent residual, whereas on line 237 and Eq. 8 it is used to describe the physical constraints. This is confusing.
3. Figure 3 is almost unreadable when zooming in. This may be due to formatting of images for review, in that case disregard the comment. If not, it needs to be made larger or increased resolution. Moreover, there is a tiny red underscoring of D x in Fig 3a that should be removed.
4. The notation ofû shows up in Fig 3 and Eq 19 but it is not described in the text.
5. Should the arguments of R in Eq. 20 be u n andû n+1 (u n )?
6. Line 261-262: What is this function F used in F (u 2j )? Is this at all related to the bold F described previously, or merely used to symbolize an arbitrary relationship? 7. Line 333: A single sentence about the numerical method referenced would be useful to the reader.
8. The description of batch in the section "Feeding the network": Is your use of the word "batch" the traditional one? i.e. the network parameters are updated after each such batch? 9. Regarding the section "Training process". Is the data shuffled at any point (before initiation, between epochs, the order of batches, etc.)?