Patientspecific simulation of tumor growth, response to the treatment, and relapse of a lung metastasis: a clinical case
 Thierry Colin^{1, 3, 4}Email author,
 François Cornelis^{1, 4, 5},
 Julien Jouganous^{1, 4},
 Jean Palussière^{6} and
 Olivier Saut^{1, 2, 4}
https://doi.org/10.1186/s4024401400141
© Jouganous et al.; licensee Springer. 2015
Received: 7 May 2014
Accepted: 13 October 2014
Published: 4 February 2015
Abstract
In this paper, a parametrization strategy based on reduced order methods is presented for tumor growth PDE models. This is applied to a new simple spatial model for lung metastasis including angiogenesis. The goal is to help clinicians monitoring tumors and eventually predicting their evolution or response to a particular kind of treatment. To illustrate the whole approach, a clinical case including the natural history of the lesion, the response to a chemotherapy, and the relapse before a radiofrequency ablation is presented.
Keywords
Clinical data assimilation Medical imaging Partial differential equations Tumor growth modelingBackground
Introduction
The metastatic disease to the lung is frequently encountered in patients with cancer whatever the primary location, and it has been associated with poor prognosis. The incidence of such disease in patients who have died of an extrathoracic malignancy is reported to be of 20% to 54% [1]. Nevertheless, limited pulmonary metastatic disease can now be successfully treated not only for palliative reasons. By controlling the primary tumor and in the absence of widely disseminated disease in many organs, the resection or the ablation of pulmonary metastases may prolong survival, improve the quality of life, and, in some cases, ensure cure [2]. During the last decade, a better management of the metastatic disease to the lung has been achieved by the evolution of imaging, medical oncology, and surgical techniques. There has been improvements in CT imaging quality and scan time [3], as well as advances in the field of nuclear medicine and MRI [4] which can give more precise information on the location and extent of the disease. In particular, there has been widespread use of PET/CT for evaluating patients with metastatic pulmonary disease, which can early detect metabolically active metastatic disease [5]. The targeted oncology treatments can achieve improved responsiveness, and the resection of the lesion turns to be possible now with minimally invasive surgical techniques as well as percutaneous thermal ablations (see [68]). In order to continue this trend and to improve the knowledge of the pulmonary metastatic disease, some authors advocated the development of new tools able to explore the first steps of metastatic implantation [9]. The majority of pulmonary metastases are derived from cancer cells that enter the lungs through the pulmonary arteries and disperse in alveolar capillaries. More rarely, metastatic disease is the consequence of lymphatic spreading or development directly in the bronchial tree (see [10]). Most of these cancer cells are able to adhere to the endothelium of the capillaries. However, the cell survival is ultimately determined by local and oncology factors. Although most of these malignant cells do not survive [2], those that survive penetrate the endothelium of the capillaries and install in the pulmonary parenchyma, where they grow. Therefore, a better understanding of this tumor growth could widely have direct clinical applications. Cancer growth modeling aims at describing, understanding, and predicting the evolution of tumors using numerical models. This science is constantly evolving, from the first cellular automata adapted to the microscopic scale to ordinary differential equations describing the global dynamics of the tumor or partial differential equations taking the spatial distribution of the cancer cells into account. If modeling can be used to help biologists understand the complex mechanisms of tumor growth, in this work, the motivation is to develop a tool for clinical oncologists to improve patient monitoring and eventually to predict the response to a treatment of some metastases to the lung. As we deal with medical images, the spatial information is important and partial differential equations seem to be the most adapted method to model tumor growth in our context. A true difficulty is to include in a model the biological interactions, as complex as they are, responsible for the tumor evolution without neglecting that the model must be relevant for in vivo applications, and its parameters need to be recovered from clinical images. Among the diversity of PDE models existing on this topic, we chose, as a starting point, the one developed in [11] for the promising results obtained. The diffusion term for nutrient is replaced by a micro model of angiogenesis (see [12]) which seems more relevant at the medical imaging scale. The goal is to simulate the evolution of a tumor from a given patient so we had to define a calibration method to personalize the model. This method must be a good tradeoff between accuracy and computation time. For that purpose, the approach chosen is based on a reduced order method named proper orthogonal decomposition (POD) (see [13] or [14]). The way we address this problem is similar to that of [11]: we first design a PDE model of tumor growth dealing with the spatial distribution of cancer cells with respect to time T(t,x,y). This PDE model contains some parameters that are patient specific. The problem is therefore to determine for which parameters the numerical solution fits the data. Once these parameters have been estimated (we call this step the calibration of the model), one can perform a prediction for a longer time scale. The outline of this paper is the following: in the following paragraph, we present a clinical case of a single metastasis including natural growth, response to a treatment, and relapse. The ‘Methods’ section is devoted to the presentation of the PDE model of tumor growth that we use and the data assimilation technics. Then, the results are discussed in a dedicated part.
A typical clinical case
Methods
A mathematical model for lung metastasis

One single type of nutrient is considered. Its concentration controls cellular proliferation and death.

The amount of nutrients available is proportional to the quantity of blood vessels in the tissue. The nutrient diffusion is not taken into account as it is not relevant at the scale we consider.

The only cellular motion we take into account is the passive transport due to volume variations caused by mitosis or cellular death.

Cancer cells are continuously switching from the proliferative to the quiescent phenotype.
Cell behavior
This formulation amounts to saying that tumor cells are pushed out if they are proliferating or pulled in if they are dying. Here, we could also use a Stokes equation to describe the velocity (see [15]) but it complicates the model without improving significantly its accuracy or biological relevance.
Angiogenesis
The model takes into account some important mechanisms involved in tumor growth such as proliferation, death, or angiogenesis. Moreover, it is simpler than the previous one [11] as we just have one sort of cancer cells and there is no transport or diffusion of the vasculature. This tradeoff is made to keep the model biologically relevant yet simple enough to be parametrized.
Taking therapeutical effects into account
We postpone the modeling of this effect of antiangiogenic drugs to a future work.
Data assimilation technique
The idea is to use the model that has been described in the previous section to obtain a forecast of the evolution of the lung tumor presented in the ‘Background’. This model contains numerous parameters that have to be recovered using the two images in order to perform the prediction. Some appear explicitly in the equations, such as the scalars α, β, γ _{0}, γ _{1}, η, λ, and M _{th}. Others are implicit and imposed as initial conditions, such as the scalar field M(t=0) and the scalar ξ(t=0), and in the case where the patient is under treatment, at least one additional parameter has to be determined. The goal of this section is to build a calibration method fast and accurate enough to recover an adequate set of parameters describing the tumor evolution.
Simplifying assumptions
As mentioned before, we have eight scalars and one spatial field to identify. These quantities cannot be estimated by in vitro or in vivo experiments. Furthermore, the parameterization problem is ill posed. We can fix ξ(t=0) arbitrarily without loss of generality since the variations of ξ(t=0) can be taken into account with parameters α, β, and λ. For convenience, concerning the order of magnitude in our numerical code, we take ξ(t=0)=0.1. For M, the situation is more complex and we have no universal solution to propose. We choose to take M(t=0)=0.8×T+S. This means that the available quantity of nutrients is lower inside the lesion than outside and that initially these quantities are constant. Note that this property is not satisfied for t>0 because of Equation 9. This is one of the strengths of the model: it is possible to obtain an heterogeneous distribution of nutrients within the tumor during the evolution that accounts for complex evolution and that makes the difference with scalar models dealing only with the volume of the tumor. Again, the ratio 0.8 in the initialization of M is arbitrary and has to be related to M _{th} and K.
Sensitivity analysis
Parameter space used for the sensitivity analysis
Parameter  Variation range 

α  [ 5;25] 
β  [ 5;25] 
η  [ 0.1;1] 
γ _{0}  [ 0.2;1.4] 
γ _{1}  [ 0;0.3] 
λ  [ 0.2;1.2] 
M _{th}  [ 0.7;1.1] 
The influence of the parameter can be established by the distance to the origin of the corresponding point (m _{ i },S _{ i }). In our case, we can see that the most influent parameter is γ _{0} which is not surprising as it rules the exponential growth of the tumor. Conversely, the model is almost not sensitive to variations of α and β in the ranges we consider so a part of the inverse problem can be simplified by fixing them thus reducing the degrees of freedom.
Formulation of the inverse problem
where (p _{1},…,p _{ k }) is the parameter set used in the model simulation, w _{1} and w _{2} are weights to balance the influence of each term, and n _{ s } is the number of snapshots we have. The first term in the objective function accounts for the shape while the second one only accounts for its volume. Of course, as initial data for our simulation, we use the ‘exact’ value given by the data at time t=t _{1}, therefore the sum starts at i=2. As a consequence, we need at least two images of the metastasis at different times to personalize the model. The tumor cell density T _{data}(t _{ i }) is extracted from the i ^{th} snapshot (corresponding to time t _{ i }). The quantity T _{model}(t _{ i },p _{1},…,p _{ k }) is the tumor cell density given by the simulation at time t _{ i } and for the parameters (p _{1},…,p _{ k }). To compute the L ^{2} distance, a data registration is necessary. We simply translate the images in order that their centers of mass match with the center of mass of the first image T _{0}. The masses are computed by integrating the density on the domain Ω. From a computational point of view, whatever the minimization method is, one has to evaluate many times the objective function. It implies to simulate the model for lots of parameter sets which could be quite expensive. For instance, if we use a gradient algorithm to estimate seven parameters, at each iteration, the model is simulated eight times. To make the calibration faster, we have developed a strategy based on a reduced order method called proper orthogonal decomposition (POD) (see [21]).
Building a reduced order model to speed up computations
where \({\lambda _{i}^{T}}\) denotes the i ^{th} eigenvalue of M ^{ T } and \({v_{i}^{T}}[\!k]\) the k ^{th} component of the i ^{th} eigenvector of M ^{ T }.Once we have obtained these POD modes, we compute the projection of the PDE system on the approximation space E _{ h } using the eigenmodes \({\Phi _{i}^{T}} (X)\). We use the POD approach on both the tumor cell density T and the pressure field π which are the two fields driven by PDEs in our system. Thus, we obtain the approximations \(T(X,t)=\sum _{i=1}^{n_{T}} {a_{i}^{T}}(t) {\Phi _{i}^{T}}(X)\) and \(\pi (X,t)=\sum _{i=1}^{n_{\pi }} a_{i}^{\pi }(t) \Phi _{i}^{\pi }(X)\), n _{ T } and n _{ π } being the numbers of modes we use to represent T and π.
Reduced order model on T:
Therefore, we avoid the time interpolation phase that was essential in [11] to approximate the time derivative. Consequently, the hypothesis that two consecutive medical data are close in time is not necessary in our case and the accuracy of the scheme is improved.
Reduced order model on π:
Initialization:
For the simulations, we also need to set initial values for the coefficients \(\left ({a_{i}^{T}}\right)_{i}\) and \(\left (a_{i}^{\pi }\right)_{i})\). These values are obtained by projecting the initial configurations of the fields associated on the POD basis: \(\forall i \in \{1,...n_{T}\}, {a_{i}^{T}}(t=0)=\left (T(t=0)  {\Phi _{i}^{T}}\right)\) and \(\forall j \in \{1,...n_{\textit {pi}}\}, a_{i}^{\pi }(t=0)=\left (\pi (t=0)  \Phi _{i}^{\pi }\right)\).
Resolution of the inverse problem
The aim of this section is to sum up the algorithm we use in order to solve the inverse problem associated to the cost function (Equation 12), that is, find (p _{1},…,p _{7})=argmin(f _{obj}(p _{1},…,p _{7})). We proceed in two steps.
Step 1:
We perform a Monte Carlo method in order to determine a first approximation of the parameters and to avoid a local minimizer. This consists in comparing simulations performed with parameter sets randomly chosen in an empirical parameter space. We keep the parameters set corresponding to the lower value of the cost function f _{obj} (see Equation 12).
Step 2:
Once we have this first approximation, we start a gradient descent method using the result of the first step as initialization. In this second step, the value of parameters α and β are fixed to those obtained in step 1, since the sensitivity analysis performed in paragraph 4.2 shows a low variability of the results with respect to their values.
Note that the use of the POD method implies that we deal with the evaluation of solutions of ODEs and not of PDEs during this process, and this fact saves a lot of CPU time.
Results and discussion
Tumor growth
Parameter set obtained by the inverse problem
α  Angiogenic agent synthesis factor  8.109 
β  Angiogenic growth factor  7.241 
η  Vasculature destruction factor  0.673 
γ _{0}  Proliferation rate  1.116 
γ _{1}  Death by hypoxia rate  0 
λ  Angiogenic agent destruction rate  0.865 
M _{th}  Hypoxia threshold  1.045 
These parameters are used to make a simulation on the direct model and try to predict the evolution of the tumor. A third CT scan on 2008/12/10 is available to compare our prediction and the real evolution.
Scalar indicators for the tumor growth of the first clinical case: DICE, volume concordance, and delays
2008/09/22  2008/12/10  

DICE  90.96%  87.21% 
Volume concordance  82.54%  77.76% 
Delay (days)  0  −6.7 
Normalized delay  0%  −3.6% 
Chemotherapy
Given the fast growth of this metastasis, clinicians decided to treat the patient with a chemotherapy. The treatment starts just after the last scan on 2008/12/10 and ends on 2009/06/29. To monitor the efficiency of the treatment, three control scans were planned: two during the treatment (on 2009/03/21 and 2009/05/27) and a last one, 1 month after the end of the therapy (on 2009/07/27). This last one shows a relapse as the tumor starts growing again from the end of the treatment. It is interesting for us to calibrate the model on this decreasing phase to try to predict the response to the treatment and possibly the relapse. As the model has already been parametrized on the growth phase, we keep the same parameters from this calibration and use the cytotoxic drug modeling given in Equations 10 and 11. This second inverse problem is easier as the only parameter to estimate is δ. This time again, we need two images. We use the last one before treatment (on 2008/12/10) as the initial condition and the first control (on 2009/03/21) to parametrize. We find δ=1.05 and then the direct model is simulated up to after the date of the last scan (2009/07/27).
Scalar indicators for the tumor under chemotherapy and rebound of the first clinical case: DICE, volume concordance, and delays
2009/03/21  2009/05/27  2009/07/27  

DICE  92.26%  87.44%  84.79 
Volume concordance  84.41%  74.56%  69.9% 
Delay (days)  0.5  −0.84  −6.4 
Normalized delay  0.5%  −0.5%  −2.8% 
The tumor shape is quite well reproduced by the model for the first control scan which we use to find the treatment parameter δ. However, on the last control scan made during the cure (on 2009/05/27), the tumor is quite hard to delineate due to various physiological phenomena that are not taken into account in the model (such as edema or fibrosis). It results from the cytotoxic effects of the treatment on the tumor cells. Finally, the model thus calibrated gives a good prediction of the relapse after the end of the chemotherapy. For this last time point, shape comparison provides a good result.
Another clinical test case
Scalar indicators for the tumor growth of the second test case: DICE, volume concordance, and delays
2010/03/11  2010/07/16  

DICE  85.41%  88.69% 
Volume concordance  70.59%  76.45% 
Delay (days)  0  5.6 
Normalized delay  0%  2.3% 
Conclusion
To try to answer the questions raised by our clinical case, a new approach was developed. First, a model was written that takes important mechanisms of tumor growth like hypoxia or angiogenesis into account. Yet, the model is simple enough to be calibrated for a specific patient. We have described a new calibration method based on proper orthogonal decomposition to fit the patient data and perform a prediction of the tumor evolution; prediction which is confirmed qualitatively and quantitatively by the medical imaging. As the model contains various kinds of treatments, we could, on the same clinical case, establish a precise quantitative prediction of the response to the treatment at the end of the protocol. As shown in Figure 3, the evolution is really quick between 2008/09/22 and 2008/12/10 and is difficult to predict by the clinicians only using the scans. In that kind of circumstances, a mathematical approach like ours could have helped the oncologist in his diagnosis. On this particular case, the interest of the simulation is clear. For the time being, we do not use any data on the real vasculature. With the constant progress of medical imaging, one can imagine that such data will soon be available. This advancement will validate our modeling of the vasculature M. We mainly used here two dimensional data extracted from scans (that naturally give a 3D view of the tumor). The whole method was coded for threedimensional data, from the direct model simulation to the POD projection and the calibration algorithm. Currently, it is computationally too expensive to build a database in 3D so we cannot use the whole 3D process on the clinical case. However, by improving and making the database generation faster, it could be interesting to use a 3D method as the reduced model simulation calculation time is almost the same as in 2D. Moreover it would allow us to free ourselves from the choice of a particular slice.
Declarations
Acknowledgements
This study has been carried out within the frame of the LABEX TRAIL, ANR10LABX0057 with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the ‘Investments for the future’ Programme IdEx (ANR10IDEX0302). Experiments presented in this paper were carried out using the PLAFRIM experimental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil Régional d’Aquitaine, FeDER, Université de Bordeaux, and CNRS (see https://plafrim.bordeaux.inria.fr/).
Authors’ Affiliations
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