A Coupled Discrete/Continuum Model for Describing Cancer-Therapeutic Transport in the Lung

Helmig R (2012) A Coupled Discrete/Continuum Model for Describing Cancer-Therapeutic Transport in the Lung. PLoS ONE 7(3): e31966. doi:10.1371/journal.pone.0031966 A Coupled Discrete/Continuum Model for Describing Cancer-Therapeutic Transport in the Lung Karin Erbertseder 0 Johannes Reichold 0 Bernd Flemisch 0 Patrick Jenny 0 Rainer Helmig 0 Rongling Wu, Pennsylvania State University, United States of America 0 1 Department of Hydromechanics and Modeling of Hydrosystems, Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart , Stuttgart, Germany , 2 Department of Mechanical and Process Engineering, Institute of Fluid Dynamics, ETH Zurich , Zurich , Switzerland We propose a computational simulation framework for describing cancer-therapeutic transport in the lung. A discrete vascular graph model (VGM) is coupled to a double-continuum model (DCM) to determine the amount of administered therapeutic agent that will reach the cancer cells. An alveolar cell carcinoma is considered. The processes in the bigger blood vessels (arteries, arterioles, venules and veins) are described by the VGM. The processes in the alveolar capillaries and the surrounding tissue are represented by a continuum approach for porous media. The system of equations of the coupled discrete/continuum model contains terms that account for degradation processes of the therapeutic agent, the reduction of the number of drug molecules by the lymphatic system and the interaction of the drug with the tissue cells. The functionality of the coupled discrete/continuum model is demonstrated in example simulations using simplified pulmonary vascular networks, which are designed to show-off the capabilities of the model rather than being physiologically accurate. - According to the World Health Organization, lung cancer kills more people than any other type of cancer and is responsible for 1.4 million deaths worldwide yearly [1]. Often, drug treatments employ a trial and error procedure to determine the most effective dosage. A predictive mathematical model suitable to guide cancertherapeutic strategies is still lacking. There exist plenty of publications about the modeling of fluid flow and delivery of macromolecules in solid tumors, for example: [2], [3], [4], [5], [6] and [7]. Further, there are several publications about blood flow simulations in vascular networks, for example: [8], [9], [10], [11] and [12]. While the application of these models is restricted to tumor tissue or to vascular networks, the modeling concept presented here is designed for the simulation of the fluid and drug transport in the entire organ affected by the cancer: the macrocirculation, the microcirculation, the tissue and the tumor. A mathematical and a numerical model are developed that describe the distribution of a targeted protein therapeutic within the human lung for cancer therapy. The developed model concept is based on these former publications about the flow and transport processes in the macrocirculation, in the microcirculation and in tumors. However, the coupling of a model for the macrocirculation to a second model for the microcirculation and the surrounding tissue and the representation of a whole organ affected by a tumor are new. To model the delivery of the therapeutic agent to the tumor cells, the transport of the dissolved drug molecules within the blood vessels, the flow across the vasculature walls into the surrounding tissue, and the transport through the interstitial space towards the tumor have to be described. If the tumor exceeds a diameter of about three millimeters, tumor induced angiogenesis will occur [13]. In this case, a direct transport of the therapeutic agent via the blood vessels to the targeted cells is possible. The model has to account for all aforementioned modes of transport. The development of a mathematical and a numerical model that are suitable to guide lung cancer therapeutic strategies is an ambitious aim. This work does not claim to fully achieve this ultimate goal. However, it is a first step towards it. This paper focuses on the model development taking into account a number of simplifying assumptions. Figure 1 depicts the general concept of the model. It includes the transport of the injected therapeutic agent through the pulmonary circulation, the transition of the dissolved drug molecules from the blood vessels into the tissue and the processes occurring within the pulmonary tissue. The advection and reaction of the blood-dissolved drug within the non-capillary part of the vasculature is simulated using the previously presented vascular graph model (VGM, see Section 1.1 and [9]). The abundance of pulmonary capillaries (about 1800 capillary segments per alveolus [14]) prevents the application of this discrete approach to the capillary bed due to the high computational cost incurred. Therefore, the flow, transport and reaction processes within the capillary bed and the surrounding tissue are described by the alveolus model instead, which is a double-continuum approach (see Section 1.2). This approach is based on two separated continua: the pulmonary tissue, and the pulmonary capillaries that are coupled by transfer functions (see Section 1.2.4). Thus, so-called upscaled nodes are inserted into the computational lattice of the VGM, which represent the capillary Figure 1. General model concept. The vascular graph model describes the processes occurring in the arteries, arterioles, venules and veins. The alveolus model, a double-continuum approach, represents the processes occurring in the capillary bed and the surrounding tissue (right image according to Terese Winslow). doi:10.1371/journal.pone.0031966.g001 bed described by the alveolus model. In this way, the VGM blood flow simulations are corrected for the loss of therapeutic agent by the transfer of the dissolved drug molecules through the capillary walls into the tissue. The coupling of the alveolus model and the vascular graph model is described in more detail in Section 1.3. An alveolar cell carcinoma (cancer cells located in the alveolar tissue) is modeled by introducing two kinds of upscaled nodes, representing healthy and tumor tissue respectively. The concentration distribution of a therapeutic agent administered via a bolus injection is determined within the blood vessel network and the surrounding tissue. Due to the different physiological properties in a tumor, the drug concentration in the cancer region differs from the one in the healthy pulmonary tissue. The simulation results, which demonstrate the functionality of the coupled discrete/ continuum model, are presented in Section 2.4. 1.1 Vascular Graph Model (VGM) The vascular graph model developed by Reichold and coworkers [9] describes flow and transport processes in vascular networks. Here it is used to compute the spatial and temporal distribution of a therapeutic agent in the pulmonary arteries, arterioles, venules and veins: (...truncated)