Stefano Casagranda^{1}, Marco Pizzolato^{2}, Francisco Torrealdea^{3}, Xavier Golay^{4}, and TimothÃ© Boutelier^{1}

^{1}Olea Medical, La Ciotat, France, ^{2}EPFL, Lausanne, Switzerland, ^{3}Centre for Medical Imaging, UCL, London, United Kingdom, ^{4}Institute of Neurology, UCL, London, United Kingdom

### Synopsis

GlucoCEST is an MRI contrast enhancement technique
sensitive to the concentration of sugar in the tissue. Because of a difference
in metabolism, it is thought that tumors consume more sugar than normal tissue.
However, glucose metabolism is complex and depends on many processes, which are
all important to understand the origin of the measured signal. To
achieve this goal we apply here a process analysis method to a deterministic
system describing the metabolism of glucose in the tissue.

### Introduction

Chemical Exchange Saturation Transfer (CEST) is an MRI contrast enhancement technique that enables the indirect detection of molecules with exchangeable protons^{1}. GlucoCEST is a CEST technique that measures a signal related to the concentration of injected glucose and its derivates^{1}. It is expected that metabolic anomalies due to the presence of cancerous tissue could be measured or characterized by means of glucoCEST, and in particular to its dynamic characteristics. There the signal is analyzed as a function of time following glucose metabolism, which is paramount to pathological tissue assessment. Different subvoxel compartments contribute to the signal. Particularly, a simplified model made as a system composed by 19 Ordinary Differential Equations (ODEs) can be used in a first step, where different parameters can be set to simulate healthy or tumor-like conditions^{2}. The system accounts for the presence of vascular glucose, its equilibrium within the interstitial environment, and the transport into the intracellular space. The purpose of this work is to identify the processes that contribute more to the signal. This analysis allows designing a reduced model, dedicated to the fitting of experimental data, and quantification of physiological parameters. To do so we apply Principal Process Analysis (PPA)^{3,4}, a numerical method for the analysis and reduction of biological systems designed with ODE formalism. PPA allows associating a dynamic weight to all involved processes and inferring their importance during the acquisition time. ### Methods

We
consider each ODE of the system as a sum of biological processes. Let $$\dot{x_i}=\sum_{j} f_{i,j}(x,p)$$ where $$$x=(x_1,x_2,\ldots,x_n)~\epsilon~\Re
^n$$$ is the vector of concentrations of glucose in different compartments
and $$$p~\epsilon~\Re ^b$$$ the vector of parameters. $$$f_{i,j}(x,p)$$$
is the $$$j^{th}$$$ process involved in the dynamical evolution of the
$$$i^{th}$$$ variable of the system over the acquisition time $$$[0,T]$$$. A
relative weight $$$W_{i,j}(t,p)$$$ is associated to each process
$$$f_{i,j}(x(t),p)$$$, to study its influence in the evolution of the variable
$$$x_i$$$ over time $$$t$$$:
$$W_{i,j}(t,p)=\frac{|f_{i,j}(x(t),p)|}{\sum_{j}
|f_{i,j}(x(t),p)|}$$
where
$$$0\leq W_{i,j} (t,p)\leq 1$$$ and $$$\sum_{j} W_{i,j}(t,p)=1$$$. We use the
thresholds $$$\delta$$$ and $$$\nu$$$ to detect not only the *inactive* processes
but also to make a further distinction between processes with a *moderate
activity* and a *fully activity*. We call
a process $$$f_{i,j}(x(t),p)$$$* inactive* at time
$$$t$$$ when $$$W_{i,j}(t,p)\leq \delta$$$, *moderately active* at
time $$$t$$$ when $$$\delta \leq W_{i,j}(t,p) \leq \nu$$$, *fully
active* at time $$$t$$$ when $$$W_{i,j}(t,p)\geq \delta$$$.
Furthermore, because the glucoCEST signal $$$y$$$ is seen as a sum of the
glucose concentration in different compartments (described by six variables of
the model in^{2}) we also compute the
contribution of each variable $$$x_i(t)$$$ for the system output $$$y(t)=\sum_{i}
(x_{i}(t))$$$:
$$M_{i}(t,p)=\frac{(x_{i}(t))}{\sum_{i}
(x_{i}(t))}$$
where
$$$0\leq M_{i} (t)\leq 1$$$ and $$$\sum_{i} M_{i}(t)=1$$$. Their *activity *is
dictated in the same manner by the thresholds $$$\delta$$$ and $$$\nu$$$, set at $$$0.1$$$
and $$$0.4$$$. The outcome of the analysis is summarized with an extended *Boolean
Process Map*^{3 } called *3-Level
Process Map* with three possible *activity* outcomes.### Results

Fig. 1 shows the two different CEST signals simulated in
healthy and cancerous tissue after an IV bolus, using the parameters described
in^{2}. The two* 3-Level Process Maps* in Fig. 2
reports the *activity* of the system variables that contribute to the CEST signal
and of their relative processes, in both healthy and cancerous tissue.### Discussion

The map in Fig. 2A shows that in a healthy tissue the
highest contribution in the generation of CEST signal is given by the glucose
in the interstitial compartment as seen during an IP
bolus situation^{5}. Fig. 2B shows that in the tumor case the
interstitial contribution becomes* inactive* because the glucose carriers bring the
glucose faster into the cell. For the first minutes after the injection the
vascular compartment is *fully active* and then it becomes moderately active because most of the glucose bolus
has passed through the tissue. From our theoretical developments, the intracellular compartment seems to be dominated by glucose
and fructose-1,6-biphosphate within healthy tissue (Fig. 2A), while
glucose-6-phosphate and fructose-6-phosphate dominates in cancerous tissue
(Fig. 2B).### Conclusion

We have presented the PPA of a glucose metabolism model,
in the context of dynamic glucoCEST imaging. It shows that dominant processes are
different depending on the properties of the tissue. It would be interesting to
apply the technique for different kind of tumors and glucose administration.
Based on this analysis we can build different reduced models where the processes that remain *inactive* have been removed.### Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 667510 and the Department of Health’s NIHR-funded Biomedical Research Centre at University College London. ### References

1. Kogan, F., Hariharan, H., & Reddy, R. (2013). Chemical exchange saturation transfer (CEST) imaging: description of technique and potential clinical applications. Current radiology reports, 1(2), 102-114.

2. Torrealdea, F. (2016). Investigation of brain tumour metabolism using naturally occurring chemical exchange saturation transfer agents with magnetic resonance imaging (Doctoral dissertation, UCL (University College London)).

3. Casagranda, S., Ropers, D., & Gouzé, J. L. (2015, June). Model reduction and process analysis of biological models. In Control and Automation (MED), 2015 23th Mediterranean Conference on(pp. 1132-1139). IEEE.

4. Casagranda, S., & Gouzé, J. L. (2017, July). Principal Process Analysis and reduction of biological models with order of magnitude. In IFAC World Congress.

5. Walker-Samuel, S., Ramasawmy, R., Torrealdea, F., Rega, M., Rajkumar, V., Johnson, S. P., Richardson, S., Goncalves, M., Parkes, H. G., Arstad, E., Thomas, D. L., Pedley, B., Lynthgoe, M.F., & Golay X. (2013). In vivo imaging of glucose uptake and metabolism in tumors. Nature medicine, 19(8), 1067-1072.