The Microsoft Research  University of Trento Centre for Computational and Systems Biology, Rovereto, Italy
Abstract
Background
Reactiondiffusion based models have been widely used in the literature for modeling the growth of solid tumors. Many of the current models treat both diffusion/consumption of nutrients and cell proliferation. The majority of these models use classical transport/mass conservation equations for describing the distribution of molecular species in tumor spheroids, and the Fick's law for describing the flux of uncharged molecules (i.e oxygen, glucose). Commonly, the equations for the cell movement and proliferation are first order differential equations describing the rate of change of the velocity of the cells with respect to the spatial coordinates as a function of the nutrient's gradient. Several modifications of these equations have been developed in the last decade to explicitly indicate that the tumor includes cells, interstitial fluids and extracellular matrix: these variants provided a model of tumor as a multiphase material with these as the different phases. Most of the current reactiondiffusion tumor models are deterministic and do not model the diffusion as a local statedependent process in a nonhomogeneous medium at the micro and mesoscale of the intra and intercellular processes, respectively. Furthermore, a stochastic reactiondiffusion model in which diffusive transport of the molecular species of nutrients and chemotherapy drugs as well as the interactions of the tumor cells with these species is a novel approach. The application of this approach to he scase of nonsmall cell lung cancer treated with gemcitabine is also novel.
Methods
We present a stochastic reactiondiffusion model of nonsmall cell lung cancer growth in the specification formalism of the tool Redi, we recently developed for simulating reactiondiffusion systems. We also describe how a spatial gradient of nutrients and oncological drugs affects the tumor progression. Our model is based on a generalization of the Fick's first diffusion law that allows to model diffusive transport in nonhomogeneous media. The diffusion coefficient is explicitly expressed as a function depending on the local conditions of the medium, such as the concentration of molecular species, the viscosity of the medium and the temperature. We incorporated this generalized law in a reactionbased stochastic simulation framework implementing an efficient version of Gillespie algorithm for modeling the dynamics of the interactions between tumor cell, nutrients and gemcitabine in a spatial domain expressing a nutrient and drug concentration gradient.
Results
Using the mathematical framework of model we simulated the spatial growth of a 2D spheroidal tumor model in response to a treatment with gemcitabine and a dynamic gradient of oxygen and glucose. The parameters of the model have been taken from recet literature and also inferred from real tumor shrinkage curves measured in patients suffering from nonsmall cell lung cancer. The simulations qualitatively reproduce the time evolution of the morphologies of these tumors as well as the morphological patterns follow the growth curves observed in patients.
Conclusions
s This model is able to reproduce the observed increment/decrement of tumor size in response to the pharmacological treatment with gemcitabine. The formal specification of the model in Redi can be easily extended in an incremental way to include other relevant biophysical processes, such as local extracellular matrix remodelling, active cell migration and traction, and reshaping of host tissue vasculature, in order to be even more relevant to support the experimental investigation of cancer.
Background
As the name indicates, reactiondiffusion models consist of two components. For systems of molecules and atoms, the first component is a set of biochemical reactions which produce, transform or remove chemical species. The second component is a mathematical description of the diffusion process. At molecular level, diffusion is due to the motion of the molecules in a medium. If solutions of different concentrations are brought into contact with each other, the solute molecules tend to flow from regions of higher concentration to regions of lower concentration, and there is ultimately an equalization of concentration. The conceptual framework of a microscale reactiondiffusion system can be also adopted to describe the phenomenology of cellular proliferation in tumor growth. Indeed, reactiondiffusion equations based models have been widely used in the literature for modeling the tumor growth. A comprehensive review of reactiondiffusion models and spatial dynamics of tumor growth can be found in
In this application domain, the reactiondiffusion models describe the evolution of the tumors via proliferation of malignant cells and their infiltration into the surrounding healthy tissue (see
In this study we present a multiscale reactiondiffusion model linking the proliferation of malignant cells to (i) the upshot of the interactions between the oncological drug and the tumor cell, (ii) the availability and the rate of uptake of nutrient by the tumor cells, (iii) and, finally the availability and the rate of consumption of oxygen. Moreover, unlike the majority of the existing models of tumor growth, our model is stochastic, i.e. the interactions between tumor cells and drugs, as well as the events of uptake and consumption of nutrients and oxygen are stochastic Markov events. All the reactiondiffusion events are parallel and concurrent. The probability of a given event to be executed is proportional to the number of substrate molecules (for biochemical reactions) or to the number of cells (for interactions like cell proliferation and tumor growth). Recent studies
We developed a generalization of the Fick's laws to model diffusion of drugs, nutrients and oxygen in the tissue, whereas we use the standard Fick's law to model the tumor cell proliferation and invasion following the gradient of nutrients and oxygen. Namely, the number of tumor cells and their spatial proliferation depend on the diffusion of nutrients, oxygen and drugs through the space and on the results of the interaction of these cell with the anticancer drug.
Before proceeding with a detailed explanation of our model of reactiondiffusion system, we briefly introduce the motivations and the guidelines of our work.
The tumor size provides a measure that is useful for describing the time course of tumor response to the chemotherapic treatment. However, tumor growth changes can be observed only through repeat followingup visits and may require sophisticated and expensive hardware and software imaging techniques especially for monitoring the size of in deepseated tumors. Due to this reason, the measurements of tumor progression in time and space have yet to gain wide application as an end point for drug effects modelling in clinical trials. The measurements of tumor size are still principally used for tumor stage categorization, whereas in the earlyphase clinical trials the measurements of changes in hematologic variables have been used as pharmacodynamic targets
In order to build an accurate and predictive model of tumor growth, the physical and biochemical nonhomogeneous environments in which the tumor arises and progresses, a generalization of the current mathematical formalization of reactiondiffusion systems is needed both at the microscale of the intracellular phenomena and at the mesoscale of intercellular and tissue processes. A preponderance of reactiondiffusion models of intra and inter cellular kinetics is usually performed on the premise that diffusion is so fast that all concentrations are maintained homogeneous in space. However, recent experimental data on intra and intercellular diffusion constants, indicate that this supposition is not necessarily valid even for small prokaryotic cells
In order to tackle this problem, this paper presents a new model of diffusion coefficient for a nonhomogeneous nonwellstirred reactiondiffusion system. In this model the diffusion coefficient explicitly depends on the local concentration, frictional coefficient of the particles of the systems, and of the temperature of the reaction environment. In turn, the rates of diffusion of the biochemical species are expressed in terms of these concentrationdependent diffusion coefficients. In this study the purely diffusive transport phenomena of noncharged particles, and, in particular, the case in which diffusion is driven by a chemical potential gradient in the
The diffusion events are modeled as reaction events and the spatial domain of the reaction chamber is divided into
The paper outlines as follows. First we describe our generalization of the Fick's law, then we briefly describe how it can be incorporated in a stochastic simulation framework. Finally, we present a model of tumor growth and the simulation results.
Methods
We summarize here the main passages of the generalization of the Fick's first law. We refer the reader to Lecca
A generalization of Fick's law for modeling diffusion
In a chemical system the driving force for diffusion of each species is the gradient of chemical potential
where
The quantity
where
where
where
Moreover, if the solvent is not turbulent, the
i.e. the number of molecules per unit volume multiplied by the linear distance traveled per unit time.
Since the virtual force on the solute is balanced by the drag force (i. e.
so that Eq. (5) becomes
where
are the diffusion coefficients. The Eq. (7) states that, in general, the flux of one species depends on the gradients of all the others, and not only on its own gradient. However, here it is supposed that the chemical activity
Substituting Eq. (7) into Eq. (6), and then substituting the obtained expression for
so that
Let
By using Eq. (9) into Eq. (6) the diffusive flux of species
where
The rate of diffusion of substance
and thence
To determine completely the righthand side of Eq. (11) is now necessary to find an expression for the activity coefficient,
where the frictional coefficient is assumed to be linearly dependent on the concentration of the solute like in sedimentation processes, i.e. in a mesh
where
Although Eq. (13) is a simplified linear model of the frictional forces, it works quite well in many case studies and can be easily extended to treat more complex frictional effects (see
Let us focus now on calculation of the activity coefficients: a way to estimate the frictional coefficients will be presented in the next subsection. By using the subscript '1' to denote the solvent and '2' to denote the solute, it can be written that
where
The chemical potential of the solvent is related to the osmotic pressure (II) by
where
Now, from the GibbsDuhem relation
where
where
Introducing Eq. (20) into Eq. (18) gives
From Eq. (15) and Eq. (21) it can be obtained that
so that
On the grounds that
The molecular mass
Substituting the expression in Eq. (22) gives, for the activity coefficient of the solute of species
Therefore, substituting Eq. (13) and Eq. (24) into Eq. (12), we obtain the following expression for a time and spacedependent diffusion coefficient
We finally estimated in the following way the second virial coefficient
where
By expanding the term exp
where
An estimate of
Modelling the stochasticity
Both diffusion and reactions are modelled as reaction events whose dynamics is driven by the First Reaction Method of the Gillespie algorithm.
In particular, the diffusion events are modeled as firstorder reactions. namely, the movement of a molecule
The space domain of the system is divided into a number
The time at which each event is expected to occur is a random variable extracted by an exponential distribution
where
where
where
From Eq. (11), the rate coefficient of the first order reaction representing a diffusion event is recognized to be as follows
Results
Here we describe the model of tumor growth implemented with the toll Redi. Redi is a software prototype
Model of tumor growth
The reactiondiffusion system modelling tumor growth involves four components: (i) the drug, gemcitabine, (ii) the tumor cell, (iii) oxygen, and (iv) glucose. The reaction events we modeled are the following:
R1. gemcitabine injection;
R2. gemcitabine diffusion;
R3. gemcitabine degradation (rate parameter
R4. effective interaction of gemcitabine and death of tumor cell (rate parameter
R5. ineffective interaction of gemcitabine: the tumor cell survives to the drug (rate parameter
R6. tumor growth (rate parameter
R7. glucose uptake (rate parameter
R8. oxygen uptake (rate parameter
R9. glucose diffusion;
R10. oxygen diffusion;
R11. tumor turnover (rate parameter
With regard to the dosage schedule of gemcitabine (event R1), we simulated the administration regime proposed by Tham et al.
values of parameters and variables in the three models.
Variable
Model 1
Model 2
Model 3
Nr. of tumor cell per mesh
2547
100
2457
Amount of gemcitabine per mesh
1
1
10
Amount of glucose per mesh
792


Amount of oxygen per mesh
3


Parameter
Model 1
Model 2
Model 3
Gemcitabine infusion rate (
0.56 pg/sec


Gemcitabine degradation (
2.78 × 10^{5} sec^{1}


Gemcitabine efficacy (
8.33 × 10^{7} (mm · sec) ^{1}


Rate constant of resistance appearance (
2.78 × 10^{8}


Tumor growth rate (
5.56 × 10^{5} mm/sec


Glucose uptake rate constant(
10.4 pg/sec
0.0104 pg/sec
10.4 pg/sec
Oxygen uptake rate constant (
0.16 pg/sec


Tumor turnover (
218 mm · week


Molecular weight of gemcitabine
0.29966 kD


Molecular weight of gslucose
0.18016 kD


Molecular weight of oxygen
0.01801528 kD


The estimate of
^{*} This dose correspond to 10,600 mg body concentration of drug (optimal dose estimated in
categorization of patients and average values of gemcitabine efficacy.
Category of patient
Median value of efficacy
Male
0.03817219 (cm · hours)^{}1
Female
0.03815441 (cm · hours)^{}1
Smoker
0.02937583 (cm · hours)^{}1
Exsmoker
0.07753538 (cm · hours)^{}1
NonSmoker
0.03815441 (cm · hours)^{}1
Finally, the parameters of reactions R7 (
According to reactions R9 and R10, tissues receive glucose and oxygen perfusing through the vessel wall and diffusing in the extracellular space.
Finally, the event R11 (tumor turnover) refers to the replacement of old tumor cells with newly generated ones from the existing ones. Tumor turnover is measured in units of sec · mm, and its value (
We simulated the morphological changes of an irregular 2D spheroidal tumor having an initial diameter of 3 mm. The size of the computational space is 40 × 40 squared meshes each of which represents a squared portion of tissue having a side of 1 mm. If we assume that the cells have a diameter of 50
We assumed that the initial spatial distribution of gemcitabine exhibits a gradient pointing outside the tumor. Furthermore we assumed that the tumor as well as the surrounding healthy tissue are crossed by a vascular network of capillaries separated by a distance of 80
A simple model of the vascular network innervating the tumor
A simple model of the vascular network innervating the tumor. The distance between capillaries is 80
All the events R1R11 are modelled as reactionevents as in the following
R1. gemcitabine injection: zerothorder reaction ∅ → gemcitabine
R2. gemcitabine diffusion: first order reaction modelling the movement of gemcitabine molecules from mesh
R3. gemcitabine degradation (rate parameter
R4. effective interaction of gemcitabine and death of tumor cell (rate parameter
R5. ineffective interaction of gemcitabine: the tumor cell survives to the drug (rate parameter
R6. tumor growth (rate parameter
R7. glucose uptake (rate parameter
R8. oxygen uptake (rate parameter
R9. glucose diffusion: first order reaction modelling the movement of gemcitabine molecules from mesh
R10. oxygen diffusion: first order reaction modelling the movement of gemcitabine molecules from mesh
R11. tumor turnover (rate parameter
We developed three models, by changing the values of the glucose uptake, the dose of gemcitabine and the number of tumor cell per mesh. In vivo and in vitro experiments carried on in the last decade highlight the crucial role of these variables in governing the dynamics of tumor growth. Some reference experimental studies in this regards are reported in
The average of 100 simulations for each model (Model 1, 2 and 3) is showed in Figures
Simulation of Model 1
Simulation of Model 1. The time unit is the week. The time separating a screenshot from the previous one is 10 weeks. The parameters of the model are listed in Table 1. The longitudinal initial size of the tumor spheroid is 3 mm. Screenshot number "0" is the state of the tumor after 10 weeks of treatment. In the spatial domain of tumor lesion each mesh hosts only tumor cells2. Blue regions are those occupied by more that 2000 tumor cells, yellow regions corresponds to areas of tissue with a number of tumor cells between 100 and 2000, and orange regions are those occupied by less that 100 tumor cells. The extension of the tumor increases linearly in time.
Simulation of Model 2
Simulation of Model 2. The time unit is the week. The time separating a screenshot from the previous one is 10 weeks. The parameters of the model are listed in Table 1. The initial diameter of the tumor ellipsoid is 3 mm. Screenshot number "0" is the state of the tumor after 10 weeks of treatment. In this model, in the spatial domain of tumor lesion, a mesh hosts both healthy and tumor cells. The number of tumor cells is 100 per mesh and the rate of glucose uptake is two order of magnitude smaller than in rate of glucose uptake in Model 1. As in Figure 1, blue regions are those occupied by more that 2000 tumor cells, yellow regions corresponds to areas of tissue with a number of tumor cells between 100 and 2000, and orange regions are those occupied by less that 100 tumor cells. The size of the tumor is approximately constant, but filaments of tumor cells propagate from the border of the tumor.
Simulation of Model 3
Simulation of Model 3. The time unit is the week. The parameters of the model are listed in Table 1. The initial diameter of the tumor ellipsoid is 3 mm. Screenshot number "0" is the state of the tumor after 10 weeks of treatment. In this model, in the spatial domain of tumor lesion, a mesh hosts both healthy and tumor cells. The number of tumor cells per mesh is 2457 as in Model 1 and the rate of glucose uptake is two order of magnitude smaller than the rate of glucose uptake in Model 1. As in Figure 1 and Figure 2, blue regions are those occupied by more that 2000 tumor cells, yellow regions corresponds to areas of tissue with a number of tumor cells between 100 and 2000, and orange regions are those occupied by less that 100 tumor cells. The size of the tumor is approximately constant, but filaments of tumor cells propagate from the border of the tumor, but are disrupted by the action of gemcitabine.
Figure
As expected, from these simulations we deduced that the effect of gemcitabine is stronger (i) at the early stage of the tumor (i.e. when the number of tumor cells is still low) and the rate of glucose uptake is also low (Model 2), or (ii) if the dose if greater them 1,000 mg.
At the best of our knowledge our study is the first attempt to model and simulate the tumor growth of nonsmall cell lung cancer in space and time. We validated our models by comparing the time behavior of the longitudinal size of the tumor ellipsoid with the theoretical and experimental results of Tham et al.
Conclusions
We have presented a computational framework for modeling and simulating the spatial dynamics of the diffusion of biological entities at micro and mesoscale in a nonhomogeneous medium. We use these mathematical and computational structure to model and simulate a nonsmall cell lung cancer treated with gemcitabine. The drug efficacy and the rate constant of resistance appearance have been estimated from real tumor growth curves recorded in 56 patients. The other parameters have been obtained from the literature reporting the in vitro experiments of the last decade. We explored the behavior of the model under different conditions concerning the rate of glucose uptake, the number of tumor cells and the dose of gemcitabine. The proposed models reproduce the expected tumor growth rate at the optimal body concentration of gemcitabine and confirm the correlation between glucose uptake and the response to the chemotherapy. At the best of our knowledge, this study is the first attempt to build a reactiondiffusion model of nonsmall cell lung cancer by integrating data from in vivo experiments and by inferring kinetic parameters from the tumor shrinkage curves of patients with the purpose to provide in silicogenerated dynamical images of the morphology of this kind of tumor.
Nonlinear models of cancer growth are needed to understand the phenomenon of realistic cancer growth. Simulations of such models conducted to determine the patterns of cancer growth and cancer response to drug and nutrient supply could support the design of the administration schedule and the duration of the therapy. Moreover, a computational model of a reactiondiffusion system taking into account the stochasticity of the interaction between drugs and tumor cells as well as the nonhomogeneity of the intra and intercellular medium may be a contribution toward this direction. Further extensions of this study are in progress and consider the opportunity to include immunological and angiogenic factors and interactions to make the current models more accurate, realistic and of greater medical interest.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Each author contributed to this work in compliance with his/her expertise field. Paola Lecca developed the mathematical model of the stochastic dynamics of a nonhomogeneous reactiondiffusion systems. Paola Lecca also designed the in silico experiments and wrote the scripts for the simulations of the tumor growth with Redi. Daniele Morpurgo contributed to the literature referencing of the study, to the calibration of the models, and to the analysis and validation of the results.
Acknowledgements
The authors would like to thank Corrado Priami of the Microsoft Research  University of Trento Centre for Computational and Systems Biology of Trento (Italy), and University of Trento, for his valuable suggestions, R. A. Soo of the Department of HematologyOncology, National University Hospital of Singapore, and Lai San Tham of the LillyNUS Centre for Clinical Pharmacology for their indications.
This article has been published as part of