A concept of a prognostic system for personalized anti ...

Available online at www.sciencedirect.com

ScienceDirect Procedia Computer Science 108C (2017) 1832–1841

International Conference on Computational Science, ICCS 2017, 12-14 June 2017, Zurich, Switzerland

A concept of a prognostic system for A concept of a prognostic system for personalized anti-tumor therapy based on personalized anti-tumor therapy based on supermodeling supermodeling

Witold Dzwinel, Adrian Kłusek, Maciej Paszyński Witold Adrian Kłusek, Paszyński AGH University of Science and Dzwinel, Technology, Department of Computer Science, Krakow, Poland Witold Dzwinel, Adrian Kłusek, Maciej Maciej Paszyński AGH University of Science and Technology, Department of Computer {dzwinel,klusek,paszynsk}@agh.edu.pl AGH University of Science and Technology, Department of Computer Science, Science, Krakow, Krakow, Poland Poland {dzwinel,klusek,paszynsk}@agh.edu.pl {dzwinel,klusek,paszynsk}@agh.edu.pl Abstract Abstract Abstract Application of computer simulation for predicting cancer progression/remission/ Application of computer simulation cancer progression/remission/ recurrence is still underestimated by clinicians. This is mainly to the lack of tumor Application of computer simulation for for predicting predicting cancerdue progression/remission/ recurrence is still underestimated by clinicians. This is mainly due tumor modeling approaches, which are both reliable and realistic computationally. Weof recurrence is still underestimated by clinicians. This is mainly due to to the the lack lack ofpresent tumor modeling approaches, which are both reliable and realistic computationally. We present here the concept of a viable prediction/correction system for predicting tumor modeling approaches, which are both reliable and realistic computationally. We present here aa viable prediction/correction predicting tumor dynamics. It is veryof in spirit, to that used in system weatherfor forecast and climate here the the concept concept of similar, viable prediction/correction system for predicting tumor dynamics. It is very similar, in spirit, to that used in weather forecast and climate modeling. The system is based on the supermodeling technique where the supermodel dynamics. It is very similar, in spirit, to that used in weather forecast and climate modeling. system is on supermodeling the consists ofThe a few coupled instances of technique a generic where coarse-grained tumor modeling. The system is based based on the the(sub-models) supermodeling technique where the supermodel supermodel consists of a few coupled instances (sub-models) of a generic coarse-grained model. Consequently, the latent and fine-grained not included tumor in the consists of a few coupled instances (sub-models)cancer of a properties, generic coarse-grained tumor model. the and cancer properties, not in generic model, e.g., reflecting phenomena other unpredictable model. Consequently, Consequently, the latent latentmicroscopic and fine-grained fine-grained cancerand properties, not included included events in the the generic model, e.g., reflecting microscopic phenomena and other unpredictable events influencing tumor dynamics, are hidden in sub-models coupling parameters, which can generic model, e.g., reflecting microscopic phenomena and other unpredictable events influencing tumor dynamics, are hidden in sub-models coupling parameters, which be learned from real Thus of matching of parameters influencing tumorincoming dynamics, aredata. hidden in instead sub-models couplinghundreds parameters, which can can be learned from incoming real data. Thus instead of matching hundreds of parameters for multi-scale tumor models, we need to fit only several values of coupling coefficients be learned from incoming real data. Thus instead of matching hundreds of parameters for tumor we need to only between sub-models simulate current status. values for multi-scale multi-scale tumortomodels, models, wethe need to fit fittumor only several several values of of coupling coupling coefficients coefficients between sub-models to simulate the current tumor status. between sub-models to simulate the current tumor status.

© 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Computational Science

1 Introduction 1 Introduction 1 Even Introduction though a tremendous effort in terms of intellectual and financial resources was spent on

Even aa tremendous effort in intellectual and resources was on understanding cancerogenesis developing anticancer therapies, in general, remains Even though though tremendousand effort in terms termsanof ofeffective intellectual and financial financial resources wasitspent spent on understanding cancerogenesis and developing an effective anticancer therapies, in general, it remains extremely difficult to control by using up-to-date anti-cancer therapies. However, the remission rate understanding cancerogenesis and developing an effective anticancer therapies, in general, it remains extremely difficult to by using anti-cancer However, the remission considerably increases when the is personalized and istherapies. focused on a particular (Ungerrate et extremely difficult to control control by therapy using up-to-date up-to-date anti-cancer therapies. However, the case remission rate considerably increases when the therapy is personalized and is focused on a particular case (Unger al., 2015). Especially, in the context of continual monitoring of a patient allowing for modifications of considerably increases when the therapy is personalized and is focused on a particular case (Unger et et al., Especially, continual monitoring of allowing modifications of the anticancer therapy in onthe thecontext basis ofof prognosis of tumor dynamics. Withfor this in mind, great al., 2015). 2015). Especially, in the context ofupdated continual monitoring of aa patient patient allowing for modifications of the anticancer therapy on of prognosis of dynamics. With in great hopes are pinned on predictive power of mathematical computer simulation cancer evolution the anticancer therapy on the the basis basis of updated updated prognosisand of tumor tumor dynamics. Withofthis this in mind, mind, great hopes are pinned on predictive power of mathematical and computer simulation of cancer evolution (see, e.g., Wodarz and Komarova, 2014; Agur et al., 2016). However, despite 40-year history of hopes are pinned on predictive power of mathematical and computer simulation of cancer evolution (see, e.g., Wodarz and Komarova, 2014; Agur et al., 2016). However, despite 40-year history cancer modeling, the clinical usage of computer simulation for predicting tumor dynamics in (see, e.g., Wodarz and Komarova, 2014; Agur et al., 2016). However, despite 40-year history of of cancer modeling, the clinical usage of computer simulation for predicting tumor dynamics personalized anti-cancer therapies is still unrealistic. In our opinion, mainly because of inappropriate cancer modeling, the clinical usage of computer simulation for predicting tumor dynamics in in personalized personalized anti-cancer anti-cancer therapies therapies is is still still unrealistic. unrealistic. In In our our opinion, opinion, mainly mainly because because of of inappropriate inappropriate

1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Computational Science 10.1016/j.procs.2017.05.013

Witold Dzwinel et al. / Procedia Computer Science 108C (2017) 1832–1841

conceptual approaches represented by the state-of-art research in tumor modeling. So called, “multiscale modeling” (still very fashionable term and often overused), which tries to follow intrinsic complexity of cancer phenomenon across many spatio-temporal scales, can be a way to nowhere having in mind its direct application in clinical medicine. Consequently, by bridging many nonlinearly coupled scales and scores of accompanying biological phenomena, the multi-scale cancer models become simultaneously: ill conditioned, overfitted and irreducible computationally. Moreover, due to the continuous and partly chaotic evolution of cancer, we will never know, which tumor growth/regression factors (microscopic, macroscopic, internal and external ones) trigger its dynamics, and which are omitted or yet unknown. For these reasons, multi-scale models can have only a cognitive value. In fact, cancer dynamics scenario is very individual and should be considered on a case by case basis, considering real initial and current tumor proliferation conditions, which can be changed anytime in an unpredictable way. The same problems are encountered in many complex phenomena including weather forecast and climate prediction. Therefore, to employ computer simulation for elaborating prognoses of cancer dynamics, one can apply a very intuitive prediction/correction scheme, such as that exploited in weather forecast and climate modeling, but built around a computer model of tumor. This way numerical simulations could be continually verified by incoming data and reinforced by data models. The principal question arises: What kind of tumor model should be used as the simulation engine? Seemingly, such the model should be as accurate as possible, following all known micro and macroscopic processes influencing tumor dynamics. However, as we have argued above, such the multi-scale tumor models can not be useful in a clinical practice due to their intrinsic complexity. On the other hand, all the reported single-scale models have pros and cons and their quality highly depends on the particular situation they simulate. One of the intuitive issue is to develop a framework generating a set of tumor models, which could be then selected and matched to the current stage of the real tumor evolution. As shown in (Lima et al. 2016), there exist a general continuous approach that exploits mixturetheory representations of tissue behavior (Rajagopal and Tao, 1995) while accounting for a range of relevant biological factors. It yields many potentially predictive cancer models. Lima et al. show that, a proper tumor model, approximating observed growth pattern, could be selected on the basis of the Occam Plausibility Algorithm (OPAL) platform. OPAL is able to assist in this selection from a defined set of models of various complexity. The models can be fitted to current data by using approximate Bayesian computation (ABC) scheme (Toni et al., 2009). However, the problems with real data adaptation and matching scores of parameters for all inspected tumor models, simultaneously taking into account their high spatio-temporal variability in 3-D, make this approach very demanding computationally. Much simpler approach is in a great demand. To face with this challenge, we propose in (Dzwinel et al. 2016; Kłusek et al., 2016) the supermodeling technique for computer simulation of melanoma dynamics. The supermodel represents the coupled dynamical system (Hiemstra et al., 2012). It consists of a few numerically linked instances of the same or various computer models (sub-models). The sub-models synchronize their spatio-temporal behaviors, which allow for penetrating phase space domains unreachable for a single sub-model. In (Dzwinel et al., 2016) we present the supermodel of melanoma cancer. We demonstrated that by coupling three instances of the same melanoma model, one can control the scenario of tumor growth by selecting a small number of coefficients representing coupling strength between the sub-models. In this position paper we summarize our previous findings and we formulate a concept of prediction/correction scheme based on supermodeling technique, which could be used in the future as a prognostic tool in the personalized anti-cancer therapy. We describe its main components and we discuss the problems of its realization. The success of the proposed approach is seriously constrained by the high computational complexity of cancer models. It is determined by the realistic size of tumor and the high resolution of computational grid, which is necessary to follow fine-grained spatio-temporal dynamics of cancer. That is why we have to consider the computational aspects of this system such as the

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computational complexity of tumor model, its numerical realization and computer implementation. Therefore, we will mention and comment here also the problems of:  defining a generic coarse-grained computer model of cancer, which can be used as a computational framework for developing high quality supermodels;  developing a numerical engine, which will be fast enough to follow reverse optimization computations, which has to be performed during each learning phase;  implementation issues.

2 Supermodel Similarly to weather and climate supermodels (van den Berge et al.,2006 ; Mirchev et al., 2012), we define the cancer supermodel as an ensemble of a few (μ=1,...,M) instances (the sub-models) of a generic tumor model. This generic model can be described by a set of parabolic partial differential equations: 𝑖𝑖 𝜕𝜕𝜕𝜕𝜇𝜇

𝜕𝜕𝜕𝜕

= 𝑓𝑓𝜇𝜇𝑖𝑖 (∆𝑥𝑥𝜇𝜇𝑖𝑖 , 𝒙𝒙𝜇𝜇 , 𝑡𝑡𝑡

(1)

where xμ =(xμ1,..., xμm) is the state vector with m dynamical system variables, Δ is a spatial operator and μ is the number of sub-models. We define the coupling tensor C={Ciμv}where Ciμv are the coupling coefficients between sub-models μ and v for a dynamical variable xi such that: 𝑖𝑖 𝜕𝜕𝜕𝜕𝜇𝜇

𝜕𝜕𝜕𝜕

= 𝑓𝑓𝜇𝜇𝑖𝑖 (∆𝑥𝑥𝜇𝜇𝑖𝑖 , 𝒙𝒙𝜇𝜇 , 𝑡𝑡𝑡 𝑡 ∑𝑣𝑣 𝐶𝐶𝜇𝜇𝜇𝜇 ∙ (𝑥𝑥𝜇𝜇𝑖𝑖 − 𝑥𝑥𝑣𝑣𝑖𝑖 ).

(2)

The supermodel evolution xsumo with time t is described by the ensemble average, i.e.: �

𝒙𝒙sumo �C, 𝑡𝑡� ≡

𝑚𝑚

∑𝜇𝜇 𝒙𝒙𝜇𝜇 �C, 𝑡𝑡�.

(3)

The quality of synchronization between μ and v sub-models can be defined as the average of squared distances between corresponding points of μ and v trajectories (see (Duane, 2009; Duane et al., 2006; Mirchev et al., 2012)). The C tensor coefficients can be learned from the “ground truth” vector xgt (repesented by the real or artificial data) by minimizing a squared error E(C): 𝐸𝐸�𝐂𝐂� =

�

𝑘𝑘𝑘𝑘𝑘𝑘

𝑡𝑡 +∆𝑡𝑡

𝑖𝑖 ∑𝐾𝐾 𝑖𝑖�� ∫𝑡𝑡 𝑖𝑖

|𝒙𝒙sumo �C, 𝑡𝑡� − 𝒙𝒙gt �𝑡𝑡�|

�

(4)

in K subsequent timesteps ∆t, where γt (γ(0,1)) is a discount factor responsible for decreasing the contribution of internal error increase. To illustrate the supermodeling concept, we present in Fig.1 a toy supermodel, which consists of three coupled various “incomplete Lorentz systems”. As shown in Fig.1a, it can reproduce the original Lorenz strange attractor for selected coupling coefficients. On the basis of this supermodeling definition, the results of SUMO FET project (http://projects.knmi.nl/sumo/) clearly show that by coupling a few climate models one can obtain more accurate results than those they produce separately. What is even more impressing, Yang, et al., (2006) and van den Berge, et al., (2011) demonstrated that the supermodel was able to follow global warming effect not present in the sub-models but reflected by data used for learning the coupling coefficients. Nevertheless, as shown in (Yang, et al., 2006; van den Berge, et al., 2011), many fundamental problems are still opened, e.g., the lack of both a proper definition of the reliable supermodel and the formal methods of its creation. As shown in (Duane et al., 2006; Yang, et al., 2006; van den Berge, et al., 2011), not all dynamical system variables have to be coupled, what is particularly important in the context of a prognostic system implementability, mainly, in terms of its computational complexity. Therefore, we postulate that the number of coupling coefficients has to be substantially smaller than the number of model parameters.


Simultaneously, we assume that the choice of the model parameter set Sμ can be done only once, on the basis of very general premises (e.g. S1-6 in Fig.1b). These inaccurate and rule-of-thumb based values of parameters can be treated as “imperfections” in sub-models. The “exact” parameters adaptation is unrealistic anyway, due to confounding uncertainties: in observational data ... in model selection, and in the features targeted in the prediction (Lima et al. 2016). Also other model inaccuracies resulting from: the lack of some components of unknown fine-grained phenomena, the model initial conditions different than the real ones as well as the truncation and round-off errors, are “corrected” by the procedure of the supermodel data adaptation. Such the “corrections” can be valid on a longer time-scale giving acceptable predictions (Yang, et al., 2006; van den Berge, et al., 2011). This suggests that the finegrained features and other unpredictable events accompanying cancer dynamics and not included in the tumor model can be hidden in data. From the point of view of machine learning, the formal mathematical framework plays the role of an additional knowledge, which defines more precisely the feature space topology. The synchronization of the supermodel with real data can be achieved by employing a prediction/correction learning scheme focused on fitting several values of coupling coefficients C between sub-models, instead of matching scores of tumor model parameters S0, as it is in the classical data adaptation techniques used for multi-scale models. This advantage of supermodeling over multi-scale approach in terms of data adaptation is depicted in Fig.1b.

a

b

Figure 1: a) The attractors obtained for three “imperfect” Lorenz systems (Model 1, Model 2, Model 3) and the supermodel for two different coupling tensors (Supermodel 1, Supermodel 2) compared to original strange Lorenz attractor. b) The schematic comparing multi-scale and supermodel data adaptation.

3 Melanoma supermodel and prognostic system In (Dzwinel et al., 2016 and Kłusek et al. 2016) we adopt supercomputing idea in simulation of melanoma skin cancer, which belongs to the most aggressive and malignant tumors (Manning, 2013). Unlike “a general” tumor model, frequently referenced as a metaphor of tumor (Chaplain et al., 2006), melanoma proliferates in a computationally complex environment. It grows on the skin surface and invades its subsequent layers of various mechanical and physical properties (see Fig.3a). Moreover, as shown in Fig.2, the vascular network in skin differs considerably from that in a homogeneous tissue. This structured vascular network requires developing heterogeneous continuum/discrete numerical models. Apart from continuous concentration fields we have to simulate remodeling process of discrete blood vasculature. For simulation of realistic situations, the computational layout should be reconstructed from, e.g., CT images (such as in (Lima et al. 2016)).

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Figure 2: The initial multi-layer layout (a) and vascular network (b) used in modeling of melanoma dynamics. The vascular network generated for a homogeneous tissue (c) is depicted for comparison (Łazarz, 2017).

In (Dzwinel et al., 2016) we describe in details our “generic” melanoma model. It represents a singlephase continuum tumor model (Vittorio and Lowengrub, 2010) and is based on models presented earlier in (Chaplain et al., 2006; Ramis-Conde et al., 2008; Welter and Rieger, 2010). It is described by means of mainly diffusion-reaction parabolic partial differential equations (PDEs) coupled by algebraic constitutive relations (Dzwinel et al., 2016). The numerical integration of PDEs simulates spatiotemporal evolution of concentration fields xμ=(xμ1,..., xμm). We use m=7 dynamical variables representing concentrations of: tumor cells, extracellular matrix (ECM), endothelial cells, tumor angiogenic factors (TAF), oxygen, fibronectin, and enzymes influencing ECM degradation. The additional discrete model simulates the process of vascular remodeling (Łoś et al., 2016; Łazarz, 2016). Our model neglects many microscopic and discrete cancer growth factors. Particularly, in the context of melanoma growth, the mechanical properties of skin layers and biological processes resulting in the sudden deep vertical growth of tumor (such as Langerhans mechanism (Clausen and Grabbe, 2015)) are neglected. Various skin layers differ only in values of diffusion coefficients and the vascularization displayed in Fig.2. The model uses 30 free parameters, which approximate values were taken from (Chaplain et al., 2006; Ramis-Conde et al., 2008; Welter and Rieger, 2010) and are collected in our recent paper (Kłusek et al. 2016). The supermodel of melanoma progression (Dzwinel et al., 2016) was made of three sub-models. They differ in one or a few parameters which control the cancer growth speed such as: interaction between degraded extracellular matrix and cancer cells, pressure and diffusion coefficient of tumor cells. The spatio-temporal cancer cells concentration field xμtc(r,t) (rposition, t - time, μ- the sub-model index) is the most important variable in modeling of tumor dynamics. Therefore, we have decided to couple the sub-models only via this variable. Thus according to Eqs. (2,3), we have: 𝜕𝜕𝜕𝜕�𝑡𝑡𝑡𝑡

𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕�𝑡𝑡𝑡𝑡

𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕�𝑡𝑡𝑡𝑡 𝜕𝜕𝜕𝜕

= −∇ ∙ 𝑱𝑱� + 𝑏𝑏�+ + 𝑏𝑏�− + 𝐶𝐶�� 𝑥𝑥�𝑡𝑡𝑡𝑡 − 𝑥𝑥�𝑡𝑡𝑡𝑡 � + 𝐶𝐶�� 𝑥𝑥�𝑡𝑡𝑡𝑡 − 𝑥𝑥�𝑡𝑡𝑡𝑡 �

(5)

= −∇ ∙ 𝑱𝑱� + 𝑏𝑏�+ + 𝑏𝑏�− + 𝐶𝐶�� 𝑥𝑥�𝑡𝑡𝑡𝑡 − 𝑥𝑥�𝑡𝑡𝑡𝑡 � + 𝐶𝐶�� 𝑥𝑥�𝑡𝑡𝑡𝑡 − 𝑥𝑥�𝑡𝑡𝑡𝑡 �

(7)

= −∇ ∙ 𝑱𝑱� +

𝑏𝑏�+

+

𝑏𝑏�−

+

𝐶𝐶�� 𝑥𝑥�𝑡𝑡𝑡𝑡

−

𝑥𝑥�𝑡𝑡𝑡𝑡 �

+

𝐶𝐶�� 𝑥𝑥�𝑡𝑡𝑡𝑡

−

𝑥𝑥�𝑡𝑡𝑡𝑡 �

(6)

where J1,2,3 are the tumor cell fluxes in corresponding sub-models (1,2,3), b+1,2,3 and b-1,2,3 are the sources (birth) and sinks (death) terms, respectively. The full set of equations one can find in (Dzwinel et al., 2016). The rest of the model equations are identical in the three sub-models. We demonstrated in (Dzwinel et al., 2016; Kłusek et. al., 2016) that depending on values of the coupling coefficients we can obtain all recognized melanoma growth scenarios. In (Kłusek et al., 2016) we show that for thicker melanoma we are able to simulate the development of massive necrotic core in


the neoplasm center. This result correlates well with the real observation that ulceration occurs more frequently in thick tumors. The exemplary results from our simulations are displayed in Fig.3.

Figure 3: The snapshots from simulations of melanoma growth by using the supermodel with various coupling strengths C={Ciμv} between sub-models a) no couplings (C=0) b) very high c) medium. The side and top views are displayed. Dark gray color corresponds to the highest tumor density while the black one shows necrotic spots. The timings for 2∙103 timesteps for AWESUMM and our explicit solver are compared.

Our melanoma model is consistent with single-phase models, which can be generated within the OPAL framework (Lima et al. 2016). More simple models can also be constructed and then matched to the current tumor growth scenario, automatically calibrated and validated by incoming data (Lima et al. 2016). However, the maximum number of dynamic variables in OPAL is m=14 what allows for developing also more complex two-phase models. Because the OPAL framework allows for constructing many classes of tumor models, we find it extremely useful in selecting a candidate set of the most parsimonious “generic” tumor models. They could be further used as the sub-models for constructing the most reliable and computationally efficient supermodel for considered tumor type and its growth stage. The ABC Bayesian approach, used in OPAL for model calibration, could be directly exploited in learning coupling coefficients in supermodels. The melanoma model was implemented by using standard FDM (finite difference method) and the Euler numerical scheme. The computational box is discretized on 250x250x150 grid, which represents 5mmx5mmx4mm fragment of skin. The numerical model was implemented in C++ in CUDA environment on the ZEUS GPGPU cluster (ACK CYFRONET, Kraków) equipped with 20 computational nodes. Each of node consists of two Intel Xeon X5645 (6 cores), 96 GB RAM and 8 Nvidia®TeslaTM M2090 (512 cores, 6GB GDDR5) boards. For a single and longer simulation we have used only one CPU with 4 GPUs attached. For the short tests we have employed a stand-alone server with a single CPU boosted by 4 GPU processors. In both cases, three GPUs were used for calculations of concentration fields while the fourth one for blood flow calculation. The simulation snapshots were visualized by using ParaViewTM visualization interface. The computational time for a single simulation, allowing for the growth of tumor of the size of the computational box, is about 1 hour (2∙105 timesteps) (see Fig.3 for timings). This is much better than other more sophisticated solvers (Fig.3) but still

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unsatisfactory, having in mind that the coupling coefficients have to be learned from data. The learning phase will be, in fact, the reverse optimization task, minimizing the weighted squared error E(C) from Eq.(4). Therefore, we expect that the model has to be run many times. Moreover, to employ the supermodeling for predicting realistic tumor sizes (a few centimeters in the case of melanoma) and taking into account its internal dynamics in an adequate precision, the computational time should be decreased at least one-two orders of magnitude. This can be achieved by using novel multi-scale numerical engines, allowing for dynamic mesh adaptation.

Figure 4: Initialization of the prediction/correction scheme for melanoma prognostic system.

Figure 5: Prediction/correction loop step for melanoma prognostic system.

To this end, we have tested adaptive numerical framework AWESUMM (Adaptive Wavelet Environment for in Silico Universal Multiscale Modeling) based on second-generation wavelet collocation method invented by Vasilyev et al. (2005). We have employed its recent implementation which is used for complex multi-scale CFD (computational fluid dynamics) simulations such as combustion phenomenon (Regele et al., 2016). However, despite this code is parallel, uses MPI interface and can be run on multiprocessor servers, it cannot beat very efficient GPU implementation of our direct FDM solver. Moreover, it appeared that the collocation method cannot tackle the problem of discontinuity of density fields (of oxygen and TAF - tumor angiogenic factors), generated by the complex topology of the vascular network. We believe that the application of a new isogeometric method (rIGA – refined Isogeometric Analysis) implemented in GPU environment, which is now tested on 2D tumor model, will considerably improve the efficiency of the melanoma model. On the base of


preliminary results (Łoś et al., 2016) we estimate that the computational time of tumor simulation can be reduced at least one order of magnitude in comparison with our FDM model. In Fig.4 and Fig.5 we present the block schemes demonstrating a general idea of the melanoma prognostic system based on the supermodeling approach. It consists of two phases: initialization and control. During the initialization phase, the 3-D image of melanoma, obtained by using advanced imaging techniques such as Optical Projection Tomography, is pre-processed to reconstruct early pretumor tissue (see Fig.4). This reconstruction would be further used as the layout for supermodeling. The coupling coefficients C are learned in the prediction/correction loop by minimizing the E(C) error (see Eq.(4)) between cancer cell concentration fields from the real image and produced by the model. It can be done in a similar way as it is in the OPAL framework (Lima et. al 2016) by using ABC scheme (Toni et al., 2009). If the error appears satisfactory, i.e., less than a given tolerance T, one can simulate and observe cancer proliferation scenario and estimate its growth rate. On the basis of these results, an oncologist can plan a proper drug treatment by manipulating the model parameters responsible for drug delivery, function and activity. This way, various treatment scenarios can be checked and the best one – e.g., resulting in cancer remission – can be finally chosen. Whereas, in the control phases shown in Fig.5, the simulations go with diagnostic checkpoints from one checkpoint to another, following the prediction/correction loop briefly described above. This prediction/correction system is very similar to that used in numerical weather forecast systems (see e.g. (Fischer et al., 2005)). However, the most important difference is that the pressure and temperature fields can be directly measured, while the methods of medical imaging, which would be able to display 3D concentration fields of tumor cells, oxygen, TAF and the structure of vascular network, are still very much in its infancy.

4 Conclusions and discussion In this position paper we postulate that it is possible to obtain reliable prognoses about cancer dynamics by creating the supermodel of cancer, which consists of a few coupled instances (the submodels) of a generic cancer model. We present a concept of supermodel integration with real data by employing a prediction/correction learning scheme. It is focused on fitting several values of coupling coefficients between sub-models, instead of matching scores of tumor model parameters as it is in the classical data adaptation techniques. We postulate also that there exist such the generic coarse-grained computer model of cancer, which can be used as a computational framework for developing high quality supermodels. The latent fine-grained tumor features, e.g., microscopic processes and other unpredictable events accompanying its proliferation not included in the model, are hidden in incoming real data. This concept can be merged with OPAL approach, which rather cannot be considered in the present form as a viable cancer prognosis method. Apart from high computational complexity, the OPAL models represent fully continuous approach. Such the models cannot account of the discrete vessels remodeling process, which is crucial for tumor growth/recurrence evolution (Rieger et al., 2016). In Fig.6 we compare the snapshots from two simulations of similar tumor growth scenarios displaying oxygen distribution for continuous/discrete (with discrete vascularization) and purely continuous (AWESUMM) model. The first figure (Fig.6a) clearly displays some remaining functional perfusive blood capillaries inside the tumor while for the continuous model in the same one can observe place a necrotic spot (Fig.6b). This, seemingly small, difference can produce completely different results of recurrence process after drag treatment. In the nearest future we plan the following research steps: 1. We will select the candidate set of simplified tumor models by using both the OPAL framework (Lima et al. 2016) extended by other advanced continuous/discrete cancer multi-scale model, from which the candidate set will be generated by neglecting or approximating its parts.

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2. We will develop a general procedure for creating the supermodel of cancer by coupling several sub-models represented by either various instances of a single cancer model or different models from the candidate set. 3. The method of learning the coupling coefficients will be developed in the prediction/correction process to match the results produced by the supermodel with real data. 4. The generic (the most parsimonious) cancer models will be selected from the candidate set, i.e., the model (or models) which produces the best predictions. 5. Simultaneously we will develop the fast numerical solver based on refined Isogeometric Analysis concept (Łoś et al., 2016) to speed up the process of learning coupling coefficients. For elaborating a prognosis for the real cancer evolution, the 3D spatio-temporal concentrations of tumor cells (oxygen, nutrients, TAF etc.) and the structure of vascular networks should be available as the input data. Till now this type of diagnostics remains a dream of the future. Therefore, as a proofof-concept, we plan to use the advanced (multi-scale) model of tumor, as a source of artificial data. The model will mimic the “ground truth”, i.e., the real tumor dynamics. We also expect to couple our model with real data coming from in vitro images of tumor tomography. The “ground truth” data will be supplied continually every given time interval. In our opinion, the coupling of the formal submodels with data-based models will be, in the future, the key modeling components in predicting behavior of biological systems. The formal mathematical models could play the role of an additional knowledge, which defines more precisely the feature space topology for machine learning tools.

Figure 6: The oxygen distribution (brown and red means the highest concentration) in melanoma environment simulated by (a) continuous/discrete and (b) purely continuous AWESUMM (Regele et al.201) tumor models.

Acknowledgements: The work has been supported by the Polish National Science Center (NCN) projects 2013/10/M/ST6/00531 (WD and AK) and 2016/21/B/ST6/01539 (MP). Thanks are due to professor dr A. Dudek (University of Illinois Cancer Center) for discussions and comments.

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