We propose a dynamic mathematical model of tissue oxygen transport by a preexisting three-dimensional microvascular network which provides nutrients for an in situ cancer at the very early stage of primary microtumour growth. The expanding tumour consumes oxygen during its invasion to the surrounding tissues and cooption of host vessels. The preexisting vessel cooption, remodelling and collapse are modelled by the changes of haemodynamic conditions due to the growing tumour. A detailed computational model of oxygen transport in tumour tissue is developed by considering (a) the time-varying oxygen advection diffusion equation within the microvessel segments, (b) the oxygen flux across the vessel walls, and (c) the oxygen diffusion and consumption within the tumour and surrounding healthy tissue. The results show the oxygen concentration distribution at different time points of early tumour growth. In addition, the influence of preexisting vessel density on the oxygen transport has been discussed. The proposed model not only provides a quantitative approach for investigating the interactions between tumour growth and oxygen delivery, but also is extendable to model other molecules or chemotherapeutic drug transport in the future study.
There are a lot of controversies about exactly how a tumour is initiated, but it is generally known that the interactions between tumour cells with the host microenvironment have played a predominant role in the pathophysiological mechanisms of tumourigenesis and progression. Hypoxia is believed to be one of the important hallmarks of the abnormal metabolic microenvironments in malignant tumours [
Hypoxic tumour cells can express high levels of angiogenic regulators including vascular endothelial growth factor (VEGF). The new vasculature in response to the upregulated angiogenic factors provides essential nutrients for rapid neoplastic expansion of tumours [
Mathematical models for oxygen transport in capillary-perfused tissue have been studied extensively from last century. As a famous pioneering study, Krogh [
For the modelling of oxygen delivery in tumours, oxygen concentration field is generally introduced to simulate the dynamic interactions of tumour cells with the surrounding tissues, by considering one or more pathophysiological characteristics during the process of malignant tumours growth. The reaction-diffusion equation is often used to describe the distribution of oxygen concentration. However, the source term of the equation is difficult to handle without the inclusion of preexisting vessels. Anderson [
The main aim of this study is to establish a dynamic mathematical model of tissue oxygen transport by a preexisting three-dimensional microvascular network which provides nutrients for an in situ cancer at the very early stage of primary microtumour growth. At the same time, the expanding tumour will consume oxygen during its invasion to the surrounding tissues and cooption of host vessels. In addition, the preexisting vessel cooption, remodelling, and collapse are modelled by the changes of haemodynamic conditions due to the growing tumour. Based on our previous haemodynamical calculation in solid tumour [
In this section, we first describe the generation of initial preexisting vessel network in the model (Section
For the morphological analysis we consider vessel segments within a cube simulation domain Ω of 1 mm3. A basic grid of 100 × 100 × 100 is generated uniformly in the cube with a distance of 10
Vessel diameter and proportion of three orders of arterioles and capillaries in the model.
Vessel diameter ( |
Proportion | |
---|---|---|
Arterioles | ||
Order 1 | 50 | 10% |
Order 2 | 30 | 20% |
Order 3 | 10 | 70% |
Capillaries | 8 | NaN |
(a) 3D preexisting microvascular network with typical pattern of a normal arteriolar network, including parallel distributed vessels with varying vessel diameter and capillaries for cross link. (b) The enlarged view of local microvascular network. Different colours represent different orders of arterioles and capillaries (arterioles order 1: black; arterioles order 2: blue; arterioles order 3: green; capillaries: red).
The haemodynamic model in this study is based on our previous work on the coupled modelling of intravascular blood flow with interstitial fluid flow [
The main equations for blood flow calculation are as follows:
The velocity of intravascular and interstitial flow satisfies
The distribution of red blood cells (RBCs) at microvascular bifurcation is calculated based on the approach proposed by Pries and Secomb [
From the haemodynamic simulation, we are able to obtain the intravascular flow velocity
The glioma cell and endothelial cell behaviours are coupled by the changes of the chemicals in the extracellular matrix (ECM), such as oxygen and matrix degradation enzymes (MDEs). The transport of these chemicals (oxygen and MDEs) is modelled by quasisteady reaction-diffusion equations. The ECM is treated as a continuum substance and can be degraded by MDEs, while the MDEs are governed by diffusion, produced by TCs and ECs, and decay of itself. The equations describing the interactions of TCs and ECs with ECM and MDE are
To obtain more realistic oxygen concentration field, the advection and diffusion of oxygen in the vessel network are introduced [
Specifically, the oxygen transport inside the vessel is represented by the advection equation
The equation between the hemoglobin bound oxygen and free oxygen is
The free oxygen flux across the vessel wall satisfies Fick’s law:
The interstitial fluid velocity is very slow due to the low interstitial pressure gradient in the tumour region. In fact,
Consider
The initial condition of ECM density is set to be 1 and other chemicals’ concentrations (oxygen and MDEs) are 0. No-flux boundary conditions are used in the simulation field. Since chemicals are transported much faster than the characteristic time for cell proliferation and migration, the chemicals’ concentrations are solved to steady state at each time step of the simulation with an inner iteration step of 5 s.
The probabilistic hybrid model for tumour cell growth is based on the previous work [
We assumed three different phenotypes of glioma cells: the proliferating cells (
Each phenotype of tumour cell has different coefficients of the consumption rate of oxygen and the production rate of MDEs [
Parameters of different phenotypes of glioma cells.
Phenotypes | MDE production | Oxygen consumption |
---|---|---|
Proliferating cells (P) |
|
|
Quiescent cells (Q) |
|
|
Necrotic cells (N) |
|
|
Experimental and clinical studies both revealed that microvessel diameter increases in response to growth factors. Döme et al. [
For a preexist vessel, once vessel dilation occurs, the vessel segment was treated as an immature vessel and was allowed to vary in radius in response to the difference between intravascular pressure and interstitial pressure. The capillary compliance satisfies the empirical equation of Netti et al. [
Based on the above equation, when the immaturation level of the vessel segment become more serious, the
Each time step increment (
Parameter values used in the simulation.
Parameter | Value | Description | Reference |
---|---|---|---|
|
10 |
Lattice constant | |
|
0.82 | Average osmotic reflection coefficient for plasma proteins | Baxter and Jain (1989) [ |
|
20 mmHg | Colloid osmotic pressure of plasma | Baxter and Jain (1989) [ |
|
15 mmHg | Colloid osmotic pressure of interstitial fluid | Baxter and Jain (1989) [ |
|
4.13 × 10−8 cm2/mmHg s | Hydraulic conductivity coefficient of the interstitium | Baxter and Jain (1989) [ |
|
200 cm−1 | Surface area per unit volume for transport in the interstitium | Baxter and Jain (1989) [ |
|
10−9 cm2s−1 | MDE diffusion coefficient | Anderson (2005) [ |
|
1.3 × 102 cm3M−1s−1 | ECM degradation coefficient | Cai et al. (2011) [ |
|
1.7 × 10−18 Mcells−1s−1 | MDE production by TC | Cai et al. (2011) [ |
|
0.3 × 10−18 Mcells−1s−1 | MDE production by EC | Cai et al. (2011) [ |
|
1.7 × 10−8 s−1 | MDE decay coefficient | Anderson (2005) [ |
|
1.27 × 10−15 |
Bunsen solubility coefficient | Fang et al. (2008) [ |
|
10−5 cm2s−1 | Oxygen diffusion coefficient | Anderson (2005) [ |
|
6.25 × 10−17 Mcells−1s−1 | Oxygen consumption coefficient | Anderson (2005) [ |
|
2.8 × 10−7 cm/mmHg s | Vessel permeability in tumour tissue | Baxter and Jain (1989) [ |
|
6.5 mmHg | Vessel compliance coefficient | Netti et al. (1996) [ |
|
0.1 | Vessel compliance index | Netti et al. (1996) [ |
Figure
The distributions of oxygen concentration (left column) at plane
To access the sensitivities to the preexisting vascular network, we changed the microvessel density (MVD) to values of 0.5 time (Case
Distribution of integrated oxygen concentration
The oxygen concentration at the central region around
In the present study, flow-dependent oxygen transport was used instead of the simple treatment of oxygen in our previous studies [
The proportion of oxygen supply to the tumour tissue by every arterioles order and capillaries in the basic case and the control case at
In this work, we have proposed a dynamic mathematical modelling system to investigate the oxygen transport in a three-dimensional preexisting vessel network during the early tumour growth process. A 3D tree-like architecture network with different arterioles orders for vessel diameter is generated as preexisting vasculature in host tissue. To obtain more realistic oxygen concentration field, the advection and diffusion of oxygen in the vessel network are introduced. The computational space is separated into three domains to characterize three distinct physiological processes, which are the oxygen advection equation inside the vessel, the oxygen flux across the vessel wall, and the free oxygen diffusion in the tissue. The oxygen advection inside the vessel and the oxygen flux across the vessel wall are calculated based on the coupling haemodynamic environment including the intravascular blood flow and the interstitial fluid flow. In addition, the dynamic changes of vessel diameter and vessel wall permeability are introduced to reflect a series of pathological characteristics of abnormal blood perfusion in tumours such as vessel dilation, leakage, cooption, remodelling, and collapse.
The simulation focuses on the avascular phase of tumour development and stops before the emergence of angiogenesis phase. The results show the oxygen concentration distribution at different time points of tumour growth. In addition, the influence of preexisting vessel density on the oxygen transport has been discussed. In a case of blood-supply-deficient microenvironment, that is, preexisting vessel network with low MVD, the significant oxygen consumption will lead to the uneven distribution of oxygen concentration through the tumour tissue and eventually upregulate the hypoxia-induced growth factors such as VEGF to activate angiogenesis. Compared with the simple treatment of oxygen transport in which the vessel was treated as point source of oxygen, the modelling of flow-dependent oxygen transport can offer more realistic oxygen concentration field since the blood perfusion is known to be heterogeneous in the tumour tissue. The proposed model not only provides a quantitative approach for investigating the interactions between tumour growth and oxygen delivery, but also is extendable to model other molecules or chemotherapeutic drug transport in the future study.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Yan Cai and Jie Zhang contributed equally to this work.
This research is supported by the National Basic Research Program of China (973 Program) (no. 2013CB733800), the National Natural Science Foundation of China (no. 11302050 and no. 11272091), the Nature Science Foundation of Jiangsu Province (no. BK20130593), the Fundamental Research Funds for the Central Universities (Yan Cai, SEU), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (Yan Cai, SEU).