Multi-scale mathematical modelling of tumour growth and microenvironments in anti-angiogenic therapy
- Yan Cai^{1, 2}Email author,
- Jie Zhang^{3} and
- Zhiyong Li^{1, 2}
https://doi.org/10.1186/s12938-016-0275-x
© The Author(s) 2016
Published: 28 December 2016
Abstract
Background
Angiogenesis, a process of generation of new blood vessels from the pre-existing vasculature, has been demonstrated to be a basic prerequisite for sustainable growth and proliferation of tumour. Anti-angiogenic treatments show normalization of tumour vasculature and microenvironment at least transiently in both preclinical and clinical settings.
Methods
In this study, we proposed a multi-scale mathematical model to simulate the dynamic changes of tumour microvasculature and microenvironment in response to anti-angiogenic drug endostatin (ES). We incorporated tumour growth, angiogenesis and vessel remodelling at tissue level, by coupling tumour cell phenotypes and endothelial cell behaviour in response to local chemical and haemodynamical microenvironment.
Results
Computational simulation results showed the tumour morphology and growth curves in general tumour progression and following different anti-angiogenic drug strategies. Furthermore, different anti-angiogenic drug strategies were designed to test the influence of ES on tumour growth and morphology. The largest reduction of tumour size was found when ES is injected at simulation time 100, which was concomitant with the emergence of angiogenesis phase.
Conclusion
The proposed model not only can predict detailed information of chemicals distribution and vessel remodelling, but also has the potential to specific anti-angiogenic drugs by modifying certain functional modules.
Keywords
Multi-scale mathematical model Numerical simulation Solid tumour growth and angiogenesis Tumour microenvironment Anti-angiogenic therapyBackground
Angiogenesis is a physiological process of generation of new blood vessels required for normal body functioning, such as wound healing, endometrial growth during the menstrual cycle, tissue grafting, inflammation, and hypoxia. However, clinically and pathologically it has been demonstrated as a basic prerequisite for sustainable growth, proliferation and metastatic spread of solid tumours [1]. Cascade of events activated by several pro-angiogenic factors produced by hypoxic tumour cells happens during tumour angiogenesis including dissolution of vascular basal membrane, increased vascular permeability and degradation of extracellular matrix (ECM) resulting in endothelial cell (EC) migration, invasion, proliferation and tube formation [2]. The growth of new blood vessels requires the concerted action of activators and inhibitors of angiogenesis. Chemicals that can activate angiogenesis include vascular endothelial growth factor (VEGF), matrix metalloproteinases (MMPs), placenta growth factor (PlGF), fibroblast growth factor (FGF) and hepatocyte growth factor (HGF) [3]. Endogenous inhibitors of angiogenesis include angiostatin (AS) and endostatin (ES) [4]. The imbalance between angiogenic growth factors and inhibitors is believed to be significantly important in the development of the tumour vasculature [5].
Given the prominent role of angiogenesis in cancer, many cancer therapies aim to block angiogenesis and thereby inhibit tumour growth. These approaches undertaken towards that goal include inhibition of growth factors required for ECs proliferation and survival, such as VEGF [6]; inhibition of MMPs [7]; or direct inhibition of EC migration [8]. In addition, vascular disrupting agents (VDA) have been shown to inhibit tumour growth through induction of vascular collapse [9]. This research was transformed into therapeutic modality which led to the development of bevacizumab (Avastin), the first VEGF targeted anti-angiogenesis cancer therapy, and approved by the FDA in 2004 [10].
While the anti-angiogenic therapy have been approved for cancer treatment, it appears that the clinical application of anti-angiogenic therapy is more complex than originally thought. Since the anti-angiogenic drugs target tumour vasculatures without tumour cells, resistance to anti-angiogenic therapy caused by the pathological microenvironment in tumour is a prominent issue which may explains the variable results observed in the experiment and clinic using this approach [11]. Unlike normal blood vessels, tumour vasculature has abnormal structure and function [12]. The microvessels inside the tumour are leaky, which causes the high interstitial fluid pressure (IFP) in tumours. The blood perfusion in the abnormal microvasculature is heterogeneous and often compromised [13]. In addition, host-tumour interactions regulate the dynamic balance of angiogenic growth factors and inhibitors and result in the pathophysiological characteristics of the tumour [14]. Therefore, it is important to understand the mechanisms of interactions between tumour cells and microenvironment in response to anti-angiogenic therapy.
Mathematical modelling and numerical simulation offer powerful tools with which to study complex biological processes, such as tumour growth and angiogenesis. Billy et al. [15] developed a model of tumour growth that includes inhibitors (ES and Ang2) and promoters (VEGF and Ang1) of angiogenesis. Sleeman and coworkers’ [16] recent study proposed a model that combined angiogenesis and haemodynamic simulations in metastatic tumours. Their model predicted that treatment with angiostatin affected tumour vessels through the process called ‘vessel normalization’. Jain and coworkers developed a model to predict changes in tumour microenvironment including interstitial fluid pressure after vessel normalization treatment [17, 18]. Alarcon et al. [19] proposed a model of VEGFR association with VEGF and internalization to investigate how these processes influence the response to anti-angiogenic therapy. Using numerical algorithms, such as partial differential equations, agent-based modelling, and continuous–discrete hybrid modelling, computational modelling and numerical simulation provide platforms to study the complexity of tumour growth and angiogenesis process and the therapeutic strategies of anti-angiogenesis in cancer.
Based on the continuous–discrete hybrid model proposed by Chaplain’s group for modelling angiogenesis [20], we started our work by investigating the dynamic interactions between local haemodynamics and angiogenesis in a 2D model [21] and then developed it to a 3D model [22]. The inclusion of different tumour cell phenotypes into the angiogenesis simulation forms our second generation of model in which the tumour growth, vessel remodelling and blood perfusion were incorporated [23]. The present study is a further development in which a multi-scale 3D model was established aiming to simulate the dynamic changes of tumour microvasculature and microenvironment in response to anti-angiogenic therapy. The model focused on the anti-angiogenic drug endostatin (ES), which can inhibit endothelial cell (EC) proliferation, migration, invasion and tube formation. We envision that the proposed model will serve as a simulation framework for studying tumour growth and the changes of tumour microenvironment in response to anti-angiogenic therapy.
Mathematical model and methods
Tumour growth
Parameters of different phenotypes of tumour cells
Phenotypes | MDE production | VEGF production | Oxygen consumption |
---|---|---|---|
Migrating cells (M) | 2μ _{T} | χ × 4 | 2γ |
Proliferating cells (P) | μ _{T} | χ | γ |
Quiescent cells (Q) | μ _{T}/5 | χ × 2 | γ/2 |
Necrotic cells (N) | μ _{T}/10 | χ × 4 | γ/4 |
Proliferative cells (PC)
If there is enough oxygen (\(C_{o}^{ex} \ge \uptheta_{\text{prol}}\)) and space is available, a tumour cell will proliferate into two daughter cells with a probability, defined as T_{age}/T_{TC}. T_{age} is the tumour cell age, ranging from 1 to T_{TC} and with an incremental 1 in each simulation time step. T_{TC} is the tumour cell proliferation time (set to be 9 h, equal to six time steps). One of the two daughter cells will replace the parent cell and the other cell will move to a neighbouring element.
Quiescent cells (QC)
When a tumour cell satisfies the survival condition but there is no neighbouring space for it to proliferate, it will go quiescent. When the neighbouring space of one quiescent cell has been released, the quiescent cell will turn back into a proliferating cell if the local oxygen supply is sufficient.
Necrotic cells (NC)
When the local oxygen concentration at a tumour site is less than the cell survival threshold θ_{surv}, the tumour cell is marked as a necrotic cell and will not be revisited at the next time step. A necrotic cell has a probability of 20% to disappear and release the space for a tumour cell or an endothelial cell if it stays necrotic for more than 45 h (30 time steps).
Migrating cells (MC)
When local oxygen level is higher than θ_{surv} but lower than θ_{prol} and a space is available, a proliferating cell has a probability (50%) to become a migrating cell (MC), and will migrate to a neighbouring space which has the highest oxygen concentration in the neighbouring elements. It was also assumed that the migrating cells adjacent to the pre-existing vessel wall have higher probability of moving in the longitudinal direction (vessel axial direction) than the radial direction. The migration speeds of the two directions are the same. After a migrating cell completes its movement, the space it originally occupied will be released for other cells.
Angiogenesis and vessel remodelling
Based on the above equations, when the vessel segment becomes immature, L_{p} will increase which causes higher P_{i}, and consequently vessel will be compressed. A compressed vessel, on the other hand will induce a higher flow resistance, lower flow which will then decrease the wall shear stress (WSS) level for the vessel. Vessel collapse will occur by either a significant reduced R or WSS criteria [24].
Chemical microenvironment
To obtain a more realistic oxygen concentration field, the advection and diffusion of oxygen in the vessel network are introduced [27]. The computational space is separated into three domains to characterize three distinct physiological processes, which are (a) the oxygen advection equation inside the vessel, (b) the oxygen flux across the vessel wall and (c) the free oxygen diffusion in the tissue. The detailed equations and methodology of oxygen delivery can be found in Cai et al. [28].
Haemodynamical microenvironment
The haemodynamic model in this study is based on our previous work on the coupled modelling of intravascular blood flow with interstitial fluid flow [21, 22]. Briefly, the basic equation for the intravascular blood flow is the flux concentration and incompressible flow at each node. Flow resistance is assumed to follow Poiseuille’s law in each vessel segment. The interstitial fluid flow is controlled by Darcy’s law. The intravascular and interstitial flow is coupled by the transvascular flow, which is described by Starling’s law. Blood viscosity is a function of vessel diameter, local haematocrit, and plasma viscosity [29]. In addition, vessel compliance and wall shear stress are correlated to vessel remodelling and vessel collapse (see “Angiogenesis and vessel remodelling” section).
Anti-angiogenic therapy
D_{ES} is the diffusion coefficient of ES. R_{ES} is the ES elimination rate in the plasma. U_{I,ex} is the ES injection rate. V_{p} is the volume of the plasma. λ_{ES} is the positive coefficient for ES decay of itself.
Simulation setup
Dirichlet boundary conditions of the chemicals are used in the simulation field. The initial condition of ECM density is set to be 1and other chemicals’ concentration (oxygen, VEGF and MDEs) are 0. The chemicals’ concentrations are solved to steady state at each time step of the simulation with an inner iteration step of 5 s. The oxygen concentration has been normalized to be 0–1 in the "Results" section.
The total difference of P_{v} through the two main parallel arterioles from plane x = 100 to x = 0 is set to be 3.5 mmHg as the driving force of blood in the network (or the boundary condition). The distribution of red blood cells (RBCs) at a microvascular bifurcation is calculated based on the blood rheology simulation (see Wu et al. [21]).
Parameter values used in the simulation
Parameter | Value | Description | Reference |
---|---|---|---|
Δl | 10 μm | Lattice constant | |
R_{0} | 4 μm | Origin radius of the capillary | |
D_{e} | 10^{−9} cm^{−3} s^{−1} | EC diffusion coefficient | [20] |
\(\upphi_{\text{c}}\) | 2.6 × 10^{3} cm^{−3} M^{−1} s^{−1} | EC chemotaxis coefficient | [20] |
\(\upphi_{\text{h}}\) | 10^{3} cm^{−3} M^{−1} s^{−1} | EC haptotaxis coefficient | [20] |
\({\text{L}}_{\text{p}}^{\text{T}}\) | 2.8 × 10^{−7} cm (mmHg s)^{−1} | Vessel permeability in tumour tissue | [31] |
\({\text{L}}_{\text{p}}^{\text{N}}\) | 0.36 × 10^{−7} cm (mmHg s)^{−1} | Vessel permeability in normal tissue | [31] |
P_{c} | 3 mmHg | Vessel collapse pressure | [26] |
E | 6.5 mmHg | Vessel compliance coefficient | [26] |
b | 0.1 | Vessel compliance index | [26] |
D_{m} | 10^{−9} cm^{−3} s^{−1} | MDE diffusion coefficient | [24] |
δ | 1.3 × 10^{2} cm^{−3} M^{−1} s^{−1} | ECM degradation coefficient | [23] |
μ_{T} | 1.7 × 10^{−18} Mcells^{−1} s^{−1} | MDE production by TC | [23] |
μ_{E} | 0.3 × 10^{−18} Mcells^{−1} s^{−1} | MDE production by EC | [23] |
λ | 1.7 × 10^{−8} s^{−1} | MDE decay coefficient | [24] |
D_{v} | 2.9 × 10^{−7} cm^{−3} s^{−1} | VEGF diffusion coefficient | [20] |
χ | 10^{−17} Mcells^{−1} s^{−1} | VEGF production by TC | [32] |
ξ | 10^{−3} cm^{−3} s^{−1} | VEGF production in ECM | [23] |
ε | 10^{−20} Mcells^{−1} s^{−1} | VEGF consumption by EC | [32] |
θ | 10^{−8} s^{−1} | VEGF decay coefficient | [32] |
e_{0} | 2.0 × 10^{−9} mol L^{−1} | Initial EC density | [33] |
ɛ_{max} | 1 | Max inhibiting effect of ES on ECs | [33] |
C_{ES50} | 2.288 × 10^{−8} mol L^{−1} | ES concentration that induces 50% of the maximum inhibiting effect | [33] |
D_{ES} | 2.9 × 10^{−7} cm^{−3} s^{−1} | Diffusion coefficient of ES | [33] |
R_{ES} | 5.54 × 10^{−5} L s^{−1} | ES elimination rate in the plasma | [33] |
U_{I,ex} | 20 mg (kg × day)^{−1} | ES injection rate | [33] |
V_{p} | 10^{−3} L | Volume of the plasma | [33] |
λ_{ES} | 10^{−8}/s | ES decay coefficient | Estimated |
Results
Tumour growth and angiogenesis without anti-angiogenic drug
Tumour growth and angiogenesis with anti-angiogenic drug
Metabolical and haemodynamical microenvironment
Parameter sensitivity
Discussion
In this paper, we proposed a multi-scale coupled modelling system to investigate the dynamic process of tumour growth and angiogenesis in response to the changes in chemical and haemodynamical microenvironment caused by anti-angiogenic therapy. At tissue level, we considered tumour growth, tumour angiogenesis and vessel remodelling. Tumour growth and neo-vessel growth are coupled by the chemical microenvironment including oxygen, VEGF, ECM and MDE, which are described by continuous partial differential equations. At the same time, haemodynamic calculation is carried out by coupling intravascular blood flow and interstitial fluid flow. According to the microenvironmental information, detailed and comprehensive mechanisms of tumour cell phenotype and endothelial cell proliferation and migration are generated at cellular level, which in turn influence the modelling at intratumoral and tissue levels. In addition, we extended the model to study the dynamic changes of tumour growth in response to anti-angiogenic therapy. Anti-angiogenic drug ES is assumed to target immature vessels and may cause endothelial cell apoptosis.
Simulation results revealed multiple characteristics of tumour progression, including tumour expansion, morphology changes and angiogenesis. The growth curves of cell population with different cell phenotypes are consistent with the pathophysiological knowledge. Furthermore, different anti-angiogenic drug strategies are designed to test the influence of ES on tumour growth and morphology. The largest reduction of tumour size is found when ES is injected at simulation time 100, which is concomitant with the emergence of angiogenesis phase. With pharmacokinetics information of specific anti-angiogenic drugs, the proposed model can be adopted for cancer drug discovery and testing.
There are a few major limitations of the present work. On calculating the tumour mechanical environment, only interstitial fluid static pressure was included. Mechanical stress caused by the rapid proliferation of tumour cells was not simulated in the model. This solid mechanical stress may influence tumour cell behaviour. In addition, although most of the simulation parameters were set based on published experimental data, some of them cannot be found, such as tumour migration speed.
Conclusions
A multi-scale coupled modelling system to investigate the dynamic process of tumour growth and angiogenesis in response to the changes in chemical and haemodynamical microenvironment during the anti-angiogenic therapy, is established. The simulation results demonstrate the inhibition of tumour growth and progression due to the reduction of neo-vasculature in response to anti-angiogenic therapy. Different anti-angiogenic drug strategies can be designed based on the proposed mathematical model, to test the influence of anti-angiogenic drug on tumour growth and morphology, with pharmacokinetics information of specific drugs
Declarations
Declarations
Authors’ contributions
YC and ZL were responsible for the design and computational modeling. YC and JZ were responsible for data analysis part. All authors read and approved the final manuscript.
Acknowledgements
This research is supported by the National Basic Research Program of China (973 Program) (No. 2013CB733800), the National Nature Science Foundation of China (No. 11302050, No. 11272091), the Nature Science Foundation of Jiangsu Province (No. BK20130593).
Competing interests
The authors declare that they have no competing interests.
About this supplement
This article has been published as part of BioMedical Engineering OnLine Volume 15 Supplement 2, 2016. Computational and experimental methods for biological research: cardiovascular diseases and beyond. The full contents of the supplement are available online http://biomedical-engineering-online.biomedcentral.com/articles/supplements/volume-15-supplement-2.
Availability of data and materials
All relevant data are within the paper.
Funding
Publication charges for this article have been funded by the National Nature Science Foundation of China (No. 11302050).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
- Folkman J, Bach M, Rowe JW, et al. Tumor angiogenesis-therapeutic implications. N Engl J Med. 1971;285:1182–6.View ArticleGoogle Scholar
- Ferrara N, Gerber HP, LeCouter J. The biology of VEGF and its receptors. Nat Med. 2003;9(6):669–76.View ArticleGoogle Scholar
- Potente M, Gerhardt H, Carmeliet P. Basic and therapeutic aspects of angiogenesis. Cell. 2011;146:873–87.View ArticleGoogle Scholar
- Vasudev NS, Reynolds AR. Anti-angiogenic therapy for cancer: current progress, unresolved questions and future directions. Angiogenesis. 2014;17:471–94.View ArticleGoogle Scholar
- Fukumura D, Jain RK. Tumor microvasculature and microenvironment: targets for anti-angiogenesis and normalization. Microvasc Res. 2007;74:72–84.View ArticleGoogle Scholar
- Ellis LM, Hicklin DJ. VEGF-targeted therapy: mechanisms of anti-tumour activity. Nat Rev Cancer. 2008;8(8):579–91.View ArticleGoogle Scholar
- Gialeli C, et al. Roles of matrix metalloproteinases in cancer progression and their pharmacological targeting. FEBS J. 2011;278:16–27.View ArticleGoogle Scholar
- Ribatii D. Endogenous inhibitors of angiogenesis: a historical review. Leuk Res. 2009;33:638–44.View ArticleGoogle Scholar
- Zulato E, Curtarello M, Nardo G, et al. Metabolic effects of anti-angiogenic therapy in tumors. Biochimie. 2012;94:925–31.View ArticleGoogle Scholar
- Lambrechts D, et al. Markers of response for the antiangiogenic agent bevacizumab. J Clin Oncol. 2013;31:1219–30.View ArticleGoogle Scholar
- Roodink I, Leenders WPJ. Targeted therapies of cancer: angiogenesis inhibition seems not enough. Cancer Lett. 2010;299:1–10.View ArticleGoogle Scholar
- Jain RK. Normalizing tumor microenvironment to treat cancer: bench to bedside to biomarkers. J Clin Oncol. 2013;31:2205–18.View ArticleGoogle Scholar
- Harris AL. Hypoxia: a key regulatory factor in tumour growth. Nat Rev Cancer. 2002;2:38–47.View ArticleGoogle Scholar
- Jain RK. Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy. Science. 2005;307:58–62.View ArticleGoogle Scholar
- Billy F, et al. A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J Theor Biol. 2009;260:545–62.MathSciNetView ArticleGoogle Scholar
- Plank MJ, Sleeman BD. A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies. Math Med Biol. 2003;20:135–81.View ArticleMATHGoogle Scholar
- Jain RK, et al. Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis: insights from a mathematical model. Cancer Res. 2007;67:2729–35.View ArticleGoogle Scholar
- Phipps C, Kohandel M. Mathematical model of the effect of interstitial fluid pressure on angiogenic behavior in solid tumors. Comput Math Methods Med. 2011;2011:843765.MathSciNetView ArticleMATHGoogle Scholar
- Alarcón T, Page KM. Mathematical models of the VEGF receptor and its role in cancer therapy. J R Soc Interface. 2007;4:283–304.View ArticleGoogle Scholar
- Anderson ARA, Chaplain MAJ. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol. 1998;60:857–900.View ArticleMATHGoogle Scholar
- Wu J, Xu SX, Long Q, et al. Coupled modeling of blood perfusion in intravascular, interstitial spaced in tumor microvasculature. J Biomech. 2008;41:996–1004.View ArticleGoogle Scholar
- Wu J, Long Q, Xu SX, et al. Study of tumor blood perfusion and its variation due to vascular normalization by anti-angiogenic therapy based on 3D angiogenic microvasculature. J Biomech. 2009;42:712–21.View ArticleGoogle Scholar
- Cai Y, Xu SX, Wu J, et al. Coupled modelling of tumour angiogenesis, tumour growth, and blood perfusion. J Theor Biol. 2011;279:90–101.View ArticleGoogle Scholar
- Cai Y, Wu J, Li ZY, et al. Mathematical modelling of a brain tumour initiation and early development: a coupled model of glioblastoma growth, pre-existing vessel co-option, angiogenesis and blood perfusion. PLoS ONE. 2016;11(3):e0150296.View ArticleGoogle Scholar
- Welter M, Bartha K, Rieger H. Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth. J Theor Biol. 2009;259:405–22.MathSciNetView ArticleGoogle Scholar
- Netti PA, Roberge S, Boucher Y, et al. Effect of transvascular fluid exchange on pressure–flow relationship in tumors: a proposed mechanism for tumor blood flow heterogeneity. Microvasc Res. 1996;52:27–46.View ArticleGoogle Scholar
- Fang Q, Sakadžić S, Ruvinskaya L, et al. Oxygen advection and diffusion in a three dimensional vascular anatomical network. Opt Express. 2008;16:17530–41.View ArticleGoogle Scholar
- Cai Y, Zhang J, Wu J, et al. Oxygen transport in a three-dimensional microvascular network incorporated with early tumour growth and preexisting vessel cooption: numerical simulation study. Biomed Res Int. 2015;2015:476964.Google Scholar
- Pries AR, Secomb TW. Microvascular blood viscosity in vivo and the endothelial surface layer. Am J Physiol Heart Circ Physiol. 2005;289:H2657–64.View ArticleGoogle Scholar
- O’Reilly MS, et al. Endostatin: an endogenous inhibitor of angiogenesis and tumor growth. Cell. 1997;88(2):277–85.View ArticleGoogle Scholar
- Baxter LT, Jain RK. Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. Microvasc Res. 1989;37:77–104.View ArticleGoogle Scholar
- Alarcón T, Owen MR, Byrne HM, et al. Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy. Comput Math Methods Med. 2006;7(2–3):85–119.MathSciNetView ArticleMATHGoogle Scholar
- Cai Y, Wu J, Xu SX, et al. Numerical simulation of inhibiting effects on solid tumour cells in anti-angiogenic therapy: application of coupled mathematical model of angiogenesis with tumour growth. Appl Math Mech. 2011;32(10):1287–96.View ArticleMATHGoogle Scholar