Contributed reagents/materials/analysis tools: KE JR BF. Wrote the paper: KE JR. Designed the software: KE JR BF. Contributed to concept and design of the model: KE JR BF PJ RH.
The authors have declared that no competing interests exist.
We propose a computational simulation framework for describing cancertherapeutic transport in the lung. A discrete vascular graph model (VGM) is coupled to a doublecontinuum 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 showoff 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
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
The vascular graph model describes the processes occurring in the arteries, arterioles, venules and veins. The alveolus model, a doublecontinuum approach, represents the processes occurring in the capillary bed and the surrounding tissue (right image according to Terese Winslow).
The vascular graph model developed by Reichold and coworkers
A collection of nodes
The nodes are the locations at which the vessels bifurcate or end. The edges represent the blood vessels themselves. The diameters of blood vessels vary along their length; typically they are widest at the points of bifurcation. The VGM assigns a mean diameter to each vessel and computes its conductance based on this value. If two adjacent nodes (vertices that are connected by an edge, e.g. node 1 and 2 in
The mass flow rate between two nodes
By inserting (2) into the continuity equation (1) for all vertices, one obtains a linear system of equations whose solution yields the vertex pressures. The flow in the pulmonary vasculature can then be computed from the pressure field using (2). The system of equations is linear due to the fact that the coupling variable
The distribution of blood in the lung is a function of the cardiac output, gravity, and pulmonary vascular resistance. An average human lung is about 30 cm long from the base (bottom of the lung) to the apex (top of the lung). The pulmonary artery enters each lung about midway between base and apex. Due to the influence of gravity, most of the blood flows through the lower half of the lung
The resistance
The volume
The transport of the dissolved drug molecules with the blood stream is modeled by the subsequent equation:
The events occurring in the arteries, arterioles, veins and venules are described by the vascular graph model. The flow, transport, and reaction processes within the capillaries around a single alveolus and the surrounding tissue are modeled using a doublecontinuum approach (see
There are two main possibilities to describe the flow and transport processes in a biological system: the molecular approach and the continuum approach. The molecular approach considers the movement of single molecules or particles and their interactions under external influences. As the inner diameter of an alveolus is in the order of 140
The structure of the biological system can also be considered either in a discrete or continuous fashion. To make the transition from a discrete to a continuum description, the concept of a representative elementary volume (REV) is used
As shown on the left image in
In summary, the doublecontinuum approach treats pulmonary tissue and capillary bed as two separate porous media continua. The interactions between them are taken into account by transfer functions.
The phase moving within the tissue continuum consists of two components, namely the interstitial fluid and the therapeutic agent. It is assumed that the fluid phase is incompressible. Thus, the movement of the dissolved drug molecules in the interstitial tissue of the lung is modeled using a singlephase twocomponent approach in a rigid, porous medium. The influence of the respiratory movement on the pulmonary tissue is not considered. The drug molecules are completely miscible with the interstitial fluid. The interstitial fluid is treated as a Newtonian fluid because it consists mainly of water. It has a composition similar to blood plasma, which consists of 90 percent of water, nine percent organic, and one percent inorganic substances that are dissolved in water
Due to the assumption of an incompressible fluid phase and a constant tissue porosity, the temporal variation of the product of porosity
Here,
The exchange of fluid and dissolved components between the tissue and capillary continuum is a surface related process. Therefore, the intercompartmental exchange rate (and thus, the coupling variable
At the arterial side of the capillary bed, about 0.5 percent of the plasma that flows through the capillaries is filtered out into the surrounding tissue. 90 percent of this extravasated fluid is reabsorbed at the venous side of the capillary bed. The remaining 10 percent of the extravasated fluid is removed by the lymphatic system from the interstitial space
The transport of the dissolved therapeutic agent in the pulmonary tissue is described by the following equation:
The first term of (11) is the socalled storage term. It describes the temporal variation of the product of tissue volume fraction
This sink term is defined in a similar way as the term for the flow reduction by the lymphatic system (see (10)), except that the mole fraction
The sink term
Here,
The sink term
The capillary continuum represents the pulmonary capillary bed around one alveolus as an averaged quantity. The movement of the dissolved drug molecules within the pulmonary capillaries is described with a singlephase twocomponent approach. The incompressible fluid phase consists of the two, completely miscible, components: blood and therapeutic agent. According to
As the capillary bed is treated as a porous media continuum, Darcy's law may be applied to determine the blood flow velocity. This has been demonstrated by
Fixed pressures
Hexagonal mesh of pulmonary capillaries embedded in a cuboid.
With the assumptions made at the beginning of Section 1.2.3 and given the intrinsic permeability tensor of the capillary continuum, the flow of blood and dissolved therapeutic agent can be described with the following continuity equation:
The transport of the dissolved therapeutic agent is represented by the subsequent equation:
The flow and transport processes between tissue and capillary continuum are described by the coupling functions
The transport across the capillary wall mainly depends on relative pressure and concentration gradients (see
The outflow of fluid from the capillaries into the interstitium across the microvascular wall is called filtration or extravasation. The inflow of fluid is termed reabsorption. The extravasated fluid can either be reabsorbed by the same or a different capillary, or it can leave the tissue via the lymphatic system
The StavermannKedemKatchalsky equation describes the advective and diffusive transport of the therapeutic agent across the microvascular wall
The vascular graph model and the alveolus model have been described in Sections 1.1 and 1.2 respectively. This section gives an overview of the approach used to couple the two models. The coupling of VGM and DCM has the advantage that one obtains a discrete representation of the vasculature where it is computationally affordable (i.e. at the noncapillary level) and a continuum representation where a fullyresolved approach would be too expensive (i.e. at the level of the capillary bed and its surrounding tissue). The vascular graph model describes the flow and transport of the therapeutic agent within the noncapillary pulmonary vasculature. Each precapillary arteriole and each postcapillary venule are connected via an upscaled edge to a socalled upscaled node, representing the capillary bed of a single alveolus and its associated tissue. These upscaled nodes are in turn described by the doublecontinuum model. It is at these sites that the administered drugs and blood plasma can leave the blood compartment and enter the surrounding tissue. The doublecontinuum model is used to compute the amount of therapeutic agent and fluid leaving the blood stream and provides this information to the VGM, where it is incorporated as additional sink terms
This section demonstrates how the coupled model described in Section 1.3 can be applied to simulate therapeutic agent kinetics in the lung. The computational grid is based on the literature values given in
The black lines represent the blood vessels, which form the unstructured grid of the VGM. The red nodes and the blue rectangles symbolize the capillary bed and pulmonary tissue around an alveolus, which are simulated using the DCM. The red nodes are the healthy upscaled nodes and the blue rectangles are the tumorous ones. The blood vessels above/below the alveoli are arteries/veins of the order one to four, with a morphology according to
order  number of branches  diameter in mm  length in mm 
17  1.000  30.000  90.50 
16  3.000  14.830  32.00 
15  8.000  8.060  10.90 
14 

5.820  20.70 
13 

3.650  17.90 
12 

2.090  10.50 
11 

1.330  6.60 
10 

0.850  4.69 
9 

0.525  3.16 
8 

0.351  2.10 
7 

0.224  1.38 
6 

0.138  0.91 
5 

0.086  0.65 
4 

0.054  0.44 
3 

0.034  0.29 
2 

0.021  0.20 
1 

0.013  0.13 
order  number of branches  diameter in mm  length in mm 
15  4.000  13.88  36.7 
14 

5.23  39.0 
13 

2.90  25.4 
12 

1.90  18.5 
11 

1.21  11.0 
10 

0.61  3.20 
9 

0.39  2,54 
8 

0.22  1.98 
7 

0.14  1.34 
6 

0.096  0.910 
5 

0.064  0.617 
4 

0.043  0.418 
3 

0.029  0.283 
2 

0.019  0.192 
1 

0.013  0.130 
A subset of the full pulmonary vasculature is chosen as the domain for the simulation. Starting from Strahler order 4, a dichotomous branching tree of arterioles is generated, leading to 21 precapillary arteries of order 1. These connect to as many upscaled nodes, which in turn are connected to 21 postcapillary venules of order 1. The veins dichotomously reunite until order 4 is reached (see
The diameters and lengths of the individual vessels are assigned according to the measurement results of
This setup is meant to demonstrate the functionality of the coupled model rather than being a realistic description of the spatial and temporal distribution of a therapeutic agent within the human lung for cancer therapy. Two simplifications are made. First, the crosssectional compliance, the pressure dependent change of the crosssection of a vessel segment, is not considered in this example. Second, the influence of the gravity on the blood flow through the vessel segments is neglected. The data of
The system of equations of the vascular graph model and the doublecontinuum model are solved for two primary variables: the pressure
The results, shown in Section 2.4, are based on the parameter values presented here. The vascular graph model requires the diameters and lengths of the vessel segments that make up the vascular graph. The numerical values of these vessel properties are taken from
parameter  symbol  value  parameter range 
tissue continuum  
diffusion coefficient 
  
dynamic viscosity 

hydraulic conductivity of lymphatic vessel wall 
n: 
  
initial receptor concentration  t: 
  
kinetic constant: forward reaction  t: 
  
kinetic constant: backward reaction  t: 
  
interstitial fluid pressure  n: −1064 
  
t: 133  133–3591 

lymphatic pressure 
n: −1200[see text]    
mass density 
1030 
  
molar density 
303.5 
  
permeability  n: 

t: 

porosity  n: 0.13  0.13–0.3 

t: 0.27  0.21–0.37 

surface area of lymph vessels per unit volume of tissue 
n: 3.0 [see text]    
tortuosity  n: 0.28 
  
t: 0.71  0.60–0.84 

volume fraction of tissue  n: 0.9[see text]    
t: 0.8  0.80–0.99 

capillary continuum  
capillary volume fraction  n: 0.1[see text]    
t: 0.2  0.01–0.20 

diffusion coefficient 
  
dynamic viscosity 
0.0021 
  
halflife of therapeutic agent  21600[see text]    
mass density 
1050  1040–1060 

molar density 
284 
  
permeability  n: see 
  
t: 
  
porosity 
1[see text]    
tortuosity 
1[see text]    
transfer equations  
capillary oncotic pressure  n: 3724 
  
t: 2660 
  
diffusive permeability  n: 

t: 

hydraulic conductivity  n: 

t: 

interstitial oncotic pressure  n: 1862 
  
t: 1995 
  
molar density 
293.75[see text]    
osmotic reflection coefficient 
0.8 
  
solventdrag reflection coefficient  n: 0.91 
  
t: 0.82 
  
surface area of capillaries per unit volume of tissue  n: 
  
t: 
 
Fluid properties do not change in a tumor.
There is no lymphatic system in a tumor.
The healthy parameter value is also taken for the tumor area.
As it has already been discussed in Section 1.2.3, the processes in the alveolar capillary bed are described with a porous media approach. The results of the permeability field calculation for a healthy upscaled node are shown in
cuboid with hexagonal network  0.15  0.22  0.00 
The spatiotemporal distribution of a therapeutic agent in the lung is studied numerically using a subset of the full pulmonary vasculature (see
The nonlinear system of equations of the DCM is numerically solved using a fully upwind vertex centered finite volume method, also called fully upwind box method (see
The simulation is performed as detailed in the previous sections. Initially, the pressure and flow fields of the vascular graph are computed. Then, a therapeutic agent is introduced at the arterial root vertex. The dissolved drug molecules are advected through the vasculature. At the alveoli, a fraction of the blood plasma and therapeutic agent migrate into the tissue. The exchange rates and phamacokinetics of this process differ between healthy and tumorous alveoli. Due to the intercompartmental exchange, the pressure and flow field of the vascular graph have to be recomputed at each time step.
(a) Pressure distribution [Pa]. (b) A therapeutic agent is introduced at the arterial root vertex. Drug distribution [
(a) Pressure distribution within the pulmonary capillary bed continuum [Pa]. (b) Drug distribution [
(a) Pressure distribution within the pulmonary capillary bed continuum [Pa]. (b) Drug distribution [
The model presented in this work describes the flow, transport and reaction processes of a therapeutic agent in the pulmonary circulation, and in healthy, as well as tumorous pulmonary tissue. It accounts for the influence of micturition and metabolic transformation reactions on the agent concentration. Moreover, the role of the lymphatic system as well as the binding of the drug molecules to tumor cells are captured. As such, the model can predict the distribution of a drug administered by continuous bolus injection for the therapy of alveolar cell carcinoma.
In order to guide cancertherapeutic strategies, however, several important extensions need to be made. The reaction of cancer cells to therapeutic agent binding, the proapoptotic signaling cascade, and the interactions between the individual tumor cells have to be modeled in addition. The model consists of two interconnected submodels, namely the vascular graph model that describes the processes occurring at the noncapillary level, and the alveolus model that simulates the processes within the alveolar capillary bed and tissue (both in the healthy and disease state). The focus of the model is on predicting the spatiotemporal distribution of therapeutic agents. Angiogenesis and tumor growth are currently not considered. However, these effects occur at timescales that are much larger than those of drug transport and adsorption
As blood is a heterogeneous, nonNewtonian fluid that exhibits pseudoplastic behavior, the VGM determines the blood viscosity within the vessel segments of the considered vascular graph from the hematocrit value using the relation derived by
Until now, the simplifying assumption of a nonpulsating flow through the vessel segments of the vascular graph and the capillary bed of the alveolus model is made. However, the arterial blood pressure rises and falls due to the phases of the cardiac cycle
It is our goal to further improve the accuracy of the model by the extensions outlined above such that the model can be of high clinical value. However, a model which strives to be of clinical value requires a thorough sensitivity analysis. In our particular case, the sensitivity analysis needs to determine how the uncertainty in the model parameters affects the primary variables of VGM and DCM, namely the pressure and the mole fraction of dissolved therapeutic agent. In other words, the sensitivity analysis needs to identify the most important influences on the coupled discrete/continuum model for describing cancer therapeutic transport in the lung.
Special thanks go to Holger Class and Jan Hasenauer for the discussions and helpful suggestions during the critical phases of the model development and the numerical implementation of the model.