The PI3K/Akt pathway upregulates Id1 and integrin α4 to enhance recruitment of human ovarian cancer endothelial progenitor cells

Yajuan Su
Lei Zheng
Qian Wang
Jie Bao
Zhen Cai
Ailan Liu
The PI3K/Akt pathway upregulates Id1 and integrin α4 to enhance recruitment of human ovarian cancer endothelial progenitor cells
R E S E A R C H A R T I C L E Open Access
Background: Endothelial progenitor cells (EPCs) contribute to tumor angiogenesis and growth. We aimed to determine whether inhibitors of differentiation 1 (Id1) were expressed in circulating EPCs of patients with ovarian cancer, whether Id1 could mediate EPCs mobilization and recruitment, and, if so, what underlying signaling pathway it used. Methods: Circulating EPCs cultures were from 25 patients with ovarian cancer and 20 healthy control subjects. Id1 and integrin α4 expression were analyzed by real-time reverse transcription-polymerase chain reaction and western blot. EPCs proliferation, migration, and adhesion were detected by MTT, transwell chamber, and EPCs-matrigel adhesion assays. Double-stranded DNA containing the interference sequences were synthesized according to the structure of a pGCSIL-GFP viral vector and then inserted into a linearized vector. Positive clones were identified as lentiviral vectors that expressed human Id1 short hairpin RNA (shRNA). Results: Id1 and integrin α4 expression were increased in EPCs freshly isolated from ovarian cancer patients compared to those obtained from healthy subjects. siRNA-mediated Id1 downregulation substantially reduced EPCs function and integrin α4 expression. Importantly, Inhibition of PI3K/Akt inhibited Id1 and integrin α4 expression, resulting in the decreasing biological function of EPCs.Conclusions: Id1 induced EPCs mobilization and recruitment is mediated chiefly by the PI3K/Akt signaling pathway and is associated with activation of integrin α4.

Background

Numerous studies have indicated that angiogenesis, a process mediated by endothelial progenitor cells (EPCs) derived from the bone marrow, is increased in many tumors due to elevated levels of angiogenic factors in the peripheral blood. An increase in EPCs supply and mobilization from the bone marrow can accelerate tumor angiogenesis [1][2][3]. A number of reports have described the incorporation of EPCs into tumor vessels in both tumor models and human patients. However the mechanisms that govern the behavior of EPCs, from their origin in the BM to their release into the circulation in response to pro-angiogenic stimuli, are still poorly understood [4,5].

Id1 is a member of a family of 4 proteins (Id1-4) known to inhibit the activity of basic helix loop helix transcription factors by blocking their ability to bind DNA [6]. Loss of Id1 in the BM leads to a complete loss of EPCs in peripheral blood, which has been correlated with a block in tumor neovascularization and delayed tumor growth [7]. However, the actual role of Id1 in regulating EPCs mobilization or recruitment remains unknown. Given the key roles that EPCs migration and adhesion may play in tumor metastasis, we tried to investigate the effect of Id1 on circulating EPCs mobilization and recruitment and the possible signal transduction pathways involved in the process.

We knocked down the expression of Id1 by an siRNAmediated Id1 lentiviral construct to determine the functional importance of Id1 in EPCs of patients with ovarian cancer,. Our results indicate that Id1 contributes to the migration and adhesion of EPCs in ovarian cancer patients and that Id1 may be important in the pathogenesis of ovarian cancer. Next, we evaluated the effects of inhibiting the phosphatidylinositol 3-kinase (PI3K)/Akt signaling pathway on Id1 and integrin α4 in EPCs of patients with ovarian cancer.

The identification of Id1 as a common target gene in EPCs migration and adhesion suggested that Id1 might serve as a novel therapeutic target in ovarian cancer. Id1 is expressed in bone marrow-derived EPCs [8] and is highly expressed in ovarian cancer cells [9,10]. Inhibiting Id1 can therefore both disrupt ovarian cancer cells growth and prevent blood vessels from feeding the ovarian cancer cells.

Methods

Patients

This study was approved by the local ethics committee in China and informed consent was obtained from all study participants. Twenty-five patients (median age, 41 years old; age range, 21-59 years old) with histologically proven ovarian cancer, including serous cancer (n = 14), mucinous cancer (n = 7), and endometrioid cancer (n = 4), were studied along with a control group of healthy women (n = 20, age range, 18-35 years old). These diagnosed ovarian cancer patients had no additional malignant, inflammatory, or ischemic disease; wounds; or ulcers that could influence the number of EPCs.

EPCs isolation and characterization

Total MNCs were isolated from 20 ml human peripheral blood samples from ovarian cancer patients and healthy women by density gradient centrifugation with Histopaque-1077 (density 1.077 g/ml; Sigma). MNCs were plated in 1 ml endothelial growth medium (EGM-2; Lonza) on fibronectin-coated (Sigma) twenty-four-well plates. After 24 h of culturing, unattached cells were discarded and attached cells were cultured as before. Medium was replaced every 2 days thereafter, and each colony/cluster was followed up. After 7 days in culture, colony forming cells were recognized as attached spindle-shaped cells. The adherent cells were incubated with DiI-acLDL (Molecule Probes) and then fixed in 2% paraformaldehyde and counterstained with fluorescein isothiocyanate (FITC)-labeled lectin from ulex europaeus agglutinin (UEA-1) (Sigma). The fluorescent images were recorded under a fluorescent microscope. Cells were also characterized by immunofluorescence staining for von Willebrand factor (vWF) and expression of CD31 and VEGFR2 (Becton Dickinson).

Quantitative real-time RT-PCR

Total RNA isolation and cDNA synthesis from cultured EPCs were performed using Trizol and the SuperScript II Reverse Transcriptase kit (Invitrogen, USA) according to the manufacturer's instructions.

Real-time PCR was performed with the Mx3000p Real Time PCR System (Stratagene, USA) using the following thermal cycling conditions: 10 sec at 95°C followed by 40 cycles of 15 sec at 95°C, 20 sec at 60°C, and 7 sec at 72°C. SYBR® GreenER qPCR SuperMix Universal S (Invitrogen, USA) (25 μl) were performed in triplicate. A no-template control (replacing RNA with water) was used as a negative control. Id1 and integrin α4 mRNA in the EPCs was determined by relative quantitation, interpolating from a standard curve of template DNA of known concentration and then normalized using β-actin as an internal control. Data were analyzed by 2 -ΔΔCt .

The primer sequences used for real-time PCR were as follows: Id1, 5-GTAAACGTGCTGCTCTACGA-CATGA-3 and 5-AGCTCCAACTGAAGGTCCCTGA-3; integrin α4, 5-TTGACAACAACGGTACTGCTAC-3 and 5-TGGTGAACACTGTGCTGATTAC-3; and β-actin, 5-TGGCACCCAGCACAATGAA-3 and 5-CTAAGTCATAGTCCGCCTAGAAGCA-3.

Western blots

The EPCs were collected with sample buffer. Cell lysates were centrifuged at 10000 g for 10 min at 4°C and the supernatant was stored at -70°C. Protein concentrations were determined with a Bio-Rad kit. 50-μg aliquots of protein were subjected to 12% and 6% SDS-PAGE gels. Then the protein was blotted onto a PVDF membrane. Primary antibodies against Id1(1:1000, Becton Dickinson), integrin α4 (1:1000, Becton Dickinson), Phospho-Akt (ser473) (1:1000, Cell Signaling), Total-Akt (1:2000, Cell Signaling), and β-actin (1:2000, Becton Dickinson) were used according to the manufacturer's recommendations. After washing the membrane, a second antibody (HRP-conjugated anti-mouse IgG) (1:2000, Becton Dickinson) was used to detect Id1, integrin α4, Phospho-Akt, Total-Akt, and β-actin. The bands were visualized using the ECL detection system (Pierce) with 5 to 30 min exposure after washing the membrane. β-actin were used as protein loading controls.

Construction of lentiviral vector expressing Id1-specific shRNA

Three different Id1-specific target sequences were chosen according to online siRNA tools provided by Invitrogen (http://www.invitrogen.com/rnai) using the Id1 reference sequence (Gene Bank Accession No NM_002165). Double-stranded DNA containing the interference sequences were synthesized according to the structure of a pGCSIL-GFP viral vector (Gikai gene company, Shanghai, China) and then inserted into a linearized vector. All the constructs were cloned and sequenced to confirm their structure. The positive clones were identified as lentiviral vectors that expressed human Id1 short hairpin RNA (shRNA), hereafter designated pGCSIL/Id1-1, pGCSIL/Id1-2, and pGCSIL/Id1-3. The three lentiviral vectors were transfected into HEK 293 cells separately to evaluate their RNA interference effects and pGCSIL/Id1-3 (sequence: 5'-gaCAT-GAACGGCTGTTACTCA-3') induced the highest levels of downregulation. So pGCSIL/Id1-3 vector and viral packaging system (Gikai gene company, Shanghai, China, containing an optimized mixture of two packaging plasmids: pHelper 1.0 vector and pHelper 2.0 vector) were cotransfected into 293 cells to replicate competent lentivirus. Viral supernatant was harvested 48 h after transfection, filtered through a 0.45-mm cellulose acetate filter and frozen at -70°C. The lentivirus (LV) containing the human Id1 shRNA-expressing cassette (pGCSIL/Id1-3) was used as a positive control for lentivirus production and denoted as Id1-RNAi-LV in the next experiments. The pGCSIL/U6 mock vector was also packaged and used as a negative control, denoted as NC-GFP-LV. Viral concentrations were determined by serial dilutions of the concentrated vector stocks in 293 cells in 96-well plates. The number of green fluorescent protein (GFP)-positive cells was measured 4 d post-transduction under microscopy. The titers averaged 2 × 10 9 TU/mL.

In vitro transduction of EPCs

For lentiviral transduction, the primary EPCs were passaged into 6-well plates at a density of 1 × 10 5 cells/ well. When cells reached 30% confluence (typically on the third day after subculturing) the medium was replaced with 1 mL of fresh medium containing lentivirus at an MOI of 150 and 6 μg/mL polybrene (Gikai gene company, Shanghai, China). The medium was replaced with fresh medium on the following day. Five days after transduction cells were analyzed by flow cytometry using a BD FACSCalibur cell analyzer (BD Biosciences). The percentage of GFP-positive cells and mean fluorescence intensity (MFI) of GFP-positive cells were determined with WinMDI 2.8 software (J. Trotter, Flow Cytometry Core Facility, Scripps Research Institute, La Jolla, CA). Means and standard deviations from experiments performed in triplicate are given.

Proliferation assay

The 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay was used to determine EPCs proliferation. EPCs were incubated with 50-400 μg/ml AGE-albumin or 400 μg/ml unmodified albumin for 24 h, then supplemented with MTT (0.5 mg/ml, Sigma) and incubated for a further 4 h. The blue formazan thus produced was solubilized with dimethyl sulfoxide and absorbance was measured at 550-650 nm.

Transwell Migration Assay

After 7 days of incubation, culture medium was removed and replaced with EBM-2 without any supplement 12 hours before the migration assay. EPCs migration was evaluated using a transwell migration assay. Briefly, 5 × 10 4 cells were suspended in 100 μL of EBM-2 supplemented with 0.1% BSA and placed in the upper chamber of an 8.0-mm pore size transwell (Costar, Cambridge, MA). 600 μL of the final dilution was placed in the lower chamber. After incubation for 6 hours at 37°C in 5% CO 2 , the cells that had not migrated were removed from the upper surface of the filters using cotton swabs and those that migrated to the lower surface of the filters were fixed in methanol and stained with Giemsa's Stain Solution. Migration was determined by counting the cell number with a microscope. Five visual fields were randomly chosen for each assay. The average number of the migrating cells in these 5 fields was taken as the cell migration number of the group.

Adhesion assay

EPCs were labeled with 4, 6-diamidino-2-phenylindole (Roche Applied Science, Indianapolis, IN). After detachment and centrifugation, EPCs were resuspended in adhesion buffer (0.5% bovine serum albumin in EBM-2) and identical numbers of cells were replated in fibronectin-coated 24-well culture plates, incubated for 60 min at 37°C, and then washed 3 times gently with adhesion buffer to remove the nonadherent cells. The adherent cells were counted by fluorescence microscopy at × 200 magnification. Five independent fields were assessed for each well, and the average number of adherent cells per × 200 field was determined.

To evaluate the ability of EPCs to adhere to confluent vascular endothelial cells, human vascular endothelial cells were cultured in 1640 medium containing 10% fetal bovine serum (Gibco) and served as a matrix for the adhesion assay. Briefly, a monolayer of vascular endothelial cells was prepared in 24-well culture plates 48 h before the assay. EPCs were labeled with 5, 6-carboxyfluorescein diacetate, succinimidyl ester (CFSE). Finally, EPCs were added to each well and incubated for 3 h at 37°C. Nonattached cells were gently removed with PBS and adherent cells were fixed with 4% paraformaldehyde and counted in five random fields under 200 × magnification. The results obtained revealed essentially unchanged results during a 4-week observational period with a variation coefficient less than 6%.

Statistical analysis

Statistical analyses were performed with Statistical Package for Social Sciences 13.0 software program (SPSS, USA). The Mann-Whitney U test and Student's t-test were used to compare variables between the two groups. Multiple comparisons were analyzed by Anova followed by post-hoc analysis to adjust the significance level. Data are shown as means ± S.E. Statistical significance was considered as P < 0.05.

Results

Characterization of EPCs

After 7 days of culture, ex vivo expanded EPCs derived from peripheral blood of healthy human volunteers and ovarian cancer patients exhibited spindle-shaped morphology. EPCs were characterized as adherent and double positive for Dil-Ac-LDL uptake and lectin binding based on their appearance in a fluorescent microscope. A total of 93.8 ± 4.5% of adherent cells showed uptake of Dil-Ac-LDL and lectin binding after 7 days of culture in our study. The endothelial phenotype of these expanded EPCs was further characterized by the expression of endothelial markers such as von willebrand factor, CD31, and VEGFR2. Immunofluorescence showed that the cells were positive for vWF, CD31, and VEGFR2 ( Figure. 1).

Id1-RNAi-LV transfection inhibited the expression of Id1 mRNA and protein in EPCs

We used real-time RT-PCR to examine mRNA expression of Id1 in EPCs of 25 patients with ovarian cancer, and Western blot analysis revealed a higher Id1 protein expression in human ovarian cancer EPCs than in 20 healthy controls. To confirm the role of Id1 in EPCs of patients with ovarian cancer we performed gene-silencing experiments. EPCs were transfected with lentiviral-Id1-siRNA vector. The negative control containing pGCSIL/U6 mock vector only was denoted NC-GFP-LV. After the Id1-RNAi-LV construct was transfected into EPCs of 6 patients with ovarian cancer, Id1 mRNA expression levels in transfected cells were compared to those in nontransfected and control-transfected (NC-GFP-LV) EPCs of 6 patients with ovarian cancer by quantitative RT-PCR. Cells with Id1-RNAi-LV transfection showed a 72% reduction in the level of Id1mRNA expression (Figure 2A). To further confirm the specificity of Id1-RNAi-LV-mediated Id1 silencing, Id1 protein expression was determined by Western blot. As shown in Figure 2B-C, Id1 protein expression in EPCs of 6 patients with ovarian cancer transfected with Id1-RNA-LV was significantly decreased compared to that in control EPCs. These results indicate that lentivirusmediated RNAi was an effective way of modulating Id1 expression in cultured EPCs.

Effects of Id1-RNAi-LV on EPCs proliferation, migration and adhesion

To examine EPCs functions in ovarian cancer we examined the proliferation, migration, and adhesion of EPCs. Proliferation, migration, and adhesion of EPCs are important for mobilization and recruitment. The EPCs of ovarian cancer patients showed increased migration and adhesion to fibronectin and endothelial cells. Statistical analyses revealed that ovarian cancer enhanced EPCs proliferation, migration, and adhesion.

After the Id1-RNAi-LV construct was transfected into EPCs, the cells were cultured for 7 days and we then performed EPCs proliferation, migration, and adhesion analysis. Id1-RNAi-LV markedly reduced EPCs functions. Cells transfected with Id1-RNAi-LV displayed less proliferation, migration, and adhesion abilities compared to non-transfected control cells, as shown in Figure 3A-D. Cells transfected with NC-GFP-LV, on the other hand, exhibited no change in cell proliferation, migration, and adhesion abilities, compared to non-transfected control EPCs. These data indicate that Id1 is crucial for the mobilization and recruitment of EPCs in ovarian cancer.

Id1-RNAi-LV transfection inhibited the expression of integrin a4 mRNA and protein in EPCs

To explain the effect of Id1 on migration toward peripheral blood and recruitment to tumor tissues we explored the expression of integrin α4 on the EPCs surface of 25 patients with ovarian cancer. In keeping with the mRNA results, Western blots showed increased integrin α4 protein expression in EPCs.

After the Id1-RNAi-LV construct was transfected into EPCs of 6 patients with ovarian cancer, integrin α4 mRNA expression levels in transfected cells were compared to those in nontransfected and controltransfected (NC-GFP-LV) EPCs of 6 patients with ovarian cancer by quantitative RT-PCR. Cells with Id1-RNAi-LV transfection showed a 65% reduction in integrin α4 mRNA expression (Figure 2A). Integrin α4 protein expression was then determined by Western blot. As shown in Figure 2B, integrin α4 protein expression in EPCs transfected with Id1-RNA-LV was significantly decreased compared to that in control EPCs. These results indicate that the effect of Id1 on adhesion and angiogenesis is associated with activation of integrin α4. Effects of PI3K/Akt on Id1, integrin a4, and EPCs functions EPCs use a broad spectrum of mobilization and recruitment mechanisms to achieve with enhanced tumor metastasis . To begin to determine which signaling transduction pathways might participate in Id1-mediated cell mobilization and recruitment in EPCs, we investigated PI3K/AKT pathway using pharmacological inhibitors. Elevated AKT-Ser473 phosphorylation was observed in EPCs, and it was completely abolished by LY294002 (1 μmol/L). As Figure 4A-B shows, Id1 and integrin α4 expression were also strongly decreased by LY294002. Figure 3A-D show that EPCs of patients with ovarian cancer had increased levels of cell proliferation, migration and adhesion, which were strongly decreased by incubation with LY294002. These results indicate that Id1-induced EPCs mobilization and recruitment is mediated by the PI3K/AKT pathway. Thus Id1 induced migration and adhesion of EPCs is mediated by an increase in integrin α4 expression and is regulated by the PI3K/Akt pathway.

Discussion

Ovarian cancer is one of the most aggressive gynecological malignancies and its high mortality is most often a direct result of delays in diagnosis. Only 25% of ovarian cancers are diagnosed while the malignancy is still confined to the ovary, and the cure rate in these patients can reach 90%. The remaining 75% of ovarian tumors have spread beyond the ovary by the time of diagnosis and the cure rate for these patients is less than 20% [11]. With the advent of molecular-targeted therapies, treatment for ovarian cancer is now moving beyond conventional chemotherapy. Inhibition of the specific cytokines essential for tumor vascularization is such a therapy [12]. Thus anti-angiogenesis therapy has become a new strategy for ovarian cancer treatment [13].

Evidence continues to accumulate confirming the importance of EPCs in the neovascularization of tumor tissues. Many researchers have focused on EPCs biology and potential clinical applications [14][15][16]. Previous studies have suggested that bone marrow-derived EPCs can migrate to tumor foci and promote tumor growth and metastasis. However, the signals that mediate mobilization and recruitment of these cells to tumors are not well understood. We showed previously that EPCs numbers are increased in ovarian cancer patients and EPCs numbers were related to tumor progression and angiogenesis. Here, using blood samples from 25 ovarian cancer patients, we demonstrated that Id1 is involved in enhancing EPCs migration and adhesion. Id1 may mediate EPCs mobilization and recruitment to ovarian cancer tissues.

We further explored the effect of Id1 and related signaling pathways on EPCs of patients with ovarian cancer.

Id1 has been implicated in a variety of cellular processes including cell growth, differentiation, angiogenesis, and neoplastic transformation. Id1 is expressed in various tumor tissues and cells [17]. Id1 knockout mice were critical in demonstrating that BM-derived progenitors are the source of tumor endothelium in some tumor types and grades, as Id1 knockout mice failed to mobilize these progenitors and transplantation of Id1 knockout mice with wild type BM was shown to rescue the observed vascular defects [8,18]. The importance of Id1+ progenitor cells in vascular rebound was confirmed recently by results from vascular disrupting therapies [19]. The mechanism by which Id1 controls the generation of EPCs is also beginning to be explored. In recent studies, Id1 was shown to be expressed in long term repopulating hematopoietic stem cells (lin-Sca+ kit+ CD34-) in the BM and Id1 loss was shown to lead to an upregulation of the expression of the cyclin-dependent kinase inhibitor p21. The expression of p21 in turn drives the stem cells towards a more committed myeloid state, as assessed by gene expression profiling; this myeloid differentiation is associated with the depletion of cells capable of endothelial cell fate commitment [20,21]. These results suggest that Id1 is required in early hematopoietic stem cells to restrain the commitment to the myeloid lineage and preserve a pool of cells that give rise to endothelial progenitors in response to vasculogenic growth signals. Although Id1 and Id3 are thought to be functionally redundant in many cell types [1,22]. It is not yet known if this is the case in EPCs.

In this study we show an essential role for Id1 in EPCs mobilization and recruitment to the tumor. Our real-time RT-PCR analysis indicates that Id1 expression in ovarian cancer is significantly increased compared to that in healthy subjects. Importantly, EPCs isolated from ovarian cancer patients display increased integrin α4 expression and baseline migration and adhesion. This is interesting in light of a recent report showing that integrin α4 promotes the homing of circulating progenitor cells and other bone marrow-derived mononuclear circulating cells not only to tumor tissues but also to inflamed and ischemic tissues [23]. To determine the role of Id1 in EPCs, we repressed its expression. As expected, transfection of EPCs with Id1-special siRNA silenced Id1 expression and abolished upregulation of integrin α4. Furthermore, migration and adhesion were both inhibited in EPCs when they were transduced with Id1-RNAi-LV. This indicates that Id1 is important for EPCs migration and adhesion. Our data demonstrate that down-regulation of Id1 may reduce the expression of integrin α4, thereby contributing to the functional inhibition we observed in our study. Little is known about the molecular signaling pathways underlying EPCs mobilization and recruitment, particularly in ovarian cancer patients. It has been shown that Id1 is an important regulator for the fate of hematopoietic stem cells. In this study, we showed that high Id1 in EPCs is mediated by the PI3K/Akt pathway, a key pathway in mediating vasculogenesis. We also demonstrated that ovarian cancer activated the PI3K/ Akt signaling pathway in EPCs. This is a novel finding. Therefore, we concluded that signaling through the P13K/Akt signaling pathway upregulates Id1, which enhances mobilization and recruitment via integrin α4. We showed that Akt is phosphorylated in EPCs of ovarian cancer patients, and inhibition of PI3K/Akt downregulated the expression level of Id1 and integrin α4 and reduced EPCs functions. Taken together, our data support the notion that ovarian cancer EPCs migration and recruitment via the PI3K/Akt-Id1-integrin α4 signaling pathway may be responsible for increased EPCs levels in ovarian cancer.

Conclusions

We showed that Id1 induced proliferation, migration and adhesion of human EPCs via the PI3K and Akt signaling pathway. Our data also provide further insights into the understanding of neovascularization. These findings raise the possibility that therapeutic strategies aiming to reduce EPCs angiogenesis have a two pronged effect: they might enhance the efficacy of certain cytotoxic anti-angiogenesis ovarian cancer therapies while at the same time reducing the risk of ovarian cancer metastases.

Figure 1
1Phenotypic characterization of EPCs from human peripheral blood. After a week in culture, EPCs were stained with DIL-labeled ac-LDL, FITC-conjugated Ulex europaeus lectin and double stained with DIL-labeled ac-LDL and FITC-conjugated Ulex europaeus lectin (overlay). von Willebrand factor (vWF), VEGFR2, CD31 and double stained with VEGFR2 and CD31 (overlay) analysis were assessed by immunofluorescence (× 20).

Figure 2
2Id1-RNAi-LV transfection silenced the mRNA and protein expression of Id1 and integrin a4 in EPCs. (A) Id1 and integrin α4 mRNA expression by real-time RT-PCR. Data are expressed as means ± S.E. **p < 0.01 vs control, # p < 0.05 vs ovarian cancer. (B) Typical Western blot images showing protein expression of Id1 and integrin α4 (β-actin is shown as a housekeeping control). (C) The graph showing the relative Id1 and integrin α4 protein levels normalized to β-actin. The results were expressed as the mean ± S.E. **p < 0.01 vs. control, # p < 0.05 vs. ovarian cancer.

Figure 3
3Effects of Id1-RNAi-LV and LY294002 on EPCs proliferation, migration, and adhesion functions. (A) Typical images of migration and adhesion as measured by transwell chamber and ECMatrix gel assay, respectively. (B) Effects of Id1-RNAi-LV and LY294002 on EPCs proliferation. **p<0.01 vs. control, #,$p<0.05 vs. ovarian cancer. (C-D) Accumulated data showing EPCs migration and adhesion functions. *p<0.05, **p<0.01 vs. control, #,$ p<0.05 vs. ovarian cancer.

Figure 4
4Effects of PI3K/Akt on the protein expression of Id1, integrin a4, p-Akt in EPCs. (A) Typical Western blot images showing protein expression of Id1, integrin α4, p-Akt and t-Akt (βactin is shown as a housekeeping control). (B) The graph showing the relative Id1, integrin α4, and p-Akt protein levels normalized to actin. The results are expressed as the mean ± S.E. *p < 0.05 vs. control, #,$ p < 0.05 vs. ovarian cancer.
Su et al. BMC Cancer 2010, 10:459
AcknowledgementsThe study was supported in part by the Ministry of Health research funds of China (No. WKJ2007-3-001) and the Provincial Natural Science Foundation (No. 07300312).Authors' contributions YS participated in study design, carried out most of the experiments, and drafted the manuscript. LZ participated in collecting samples and manuscript preparation. QW conceived of the study, and participated in its design and coordination. JB assisted with cell culture. ZC participated in study design and statistical analysis. AL assisted with the critical revision of the manuscript. All authors read and approved the final manuscript.Competing interestsThe authors declare that they have no competing interests.
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A Novel Method for Simulating the Extracellular Matrix in Models of Tumour Growth
Hindawi Publishing CorporationCopyright Hindawi Publishing Corporation

Alina Toma
Institute of Medical Engineering
University of Lübeck
23562LübeckGermany

Centre of Excellence for Technology and Engineering in Medicine (TANDEM)
University of Lübeck
23562LübeckGermany

Andreas Mang
Institute of Medical Engineering
University of Lübeck
23562LübeckGermany

Tina A Schuetz
Institute of Medical Engineering
University of Lübeck
23562LübeckGermany

Graduate School for Computing in Medicine and Life Sciences
University of Lübeck
23562LübeckGermany

Stefan Becker
Institute of Medical Engineering
University of Lübeck
23562LübeckGermany

Centre of Excellence for Technology and Engineering in Medicine (TANDEM)
University of Lübeck
23562LübeckGermany

Thorsten M Buzug firstname.lastname@example.com
Institute of Medical Engineering
University of Lübeck
23562LübeckGermany

Thorsten M Buzug
A Novel Method for Simulating the Extracellular Matrix in Models of Tumour Growth

Computational and Mathematical Methods in Medicine
Hindawi Publishing Corporation20121110.1155/2012/109019Received 17 April 2012; Revised 1 June 2012; Accepted 11 June 2012Research Article Correspondence should be addressed to Alina Toma, email@example.com Academic Editor: Quan Long
A novel hybrid continuum-discrete model to simulate tumour growth on a cellular scale is proposed. The lattice-based spatiotemporal model consists of reaction-diffusion equations that describe interactions between cancer cells and their microenvironment. The fundamental ingredients that are typically considered are the nutrient concentration, the extracellular matrix (ECM), and matrix degrading enzymes (MDEs). The in vivo processes are very complex and occur on different levels. This in turn leads to huge computational costs. The main contribution of the present work is therefore to describe the processes on the basis of simplified mathematical approaches, which, at the same time, depict realistic results to understand the biological processes. In this work, we discuss if we have to simulate the MDE or if the degraded matrix can be estimated directly with respect to the cancer cell distribution. Additionally, we compare the results for modelling tumour growth using the common and our simplified approach, thereby demonstrating the advantages of the proposed method. Therefore, we introduce variations of the positioning of the nutrient delivering blood vessels and use different initializations of the ECM. We conclude that the novel method, which does not explicitly model the matrix degrading enzymes, provides means for a straightforward and fast implementation for modelling tumour growth.

Introduction

Cancer is one of the most common diseases in adulthood and has gained more and more attention in research of different scientific disciplines in recent years. We focus on malignant brain tumours such as gliomas, but the models presented in this paper are generic and can also be applied on different tumour entities and other solid tumours through variation of the corresponding parameters.

Generally, the models of tumour growth are described either on a macroscopic [1,2], microscopic [3][4][5], or molecular level [6,7]. Multiscale approaches, that is a combination of two [8,9] or all three levels [10], are rarely used. The macroscopic models are generally based on continuous deterministic reaction-diffusion formalisms [1] and lead to a global description of the tumour. Thus, they depict a clinically significant size as it can typically be observed in magnetic resonance imaging or computed tomography. At this level the associating processes of tumour growth such as pressure and the therewith induced deformation of the environment tissue can be well observed and modelled. Such methods, though allowing for visual comparisons with medical, noninvasive in vivo imaging data, neglect the complex processes on the microscopic and molecular level. Mathematical approaches for simulating tumour growth on the cellular level are typically formulated in terms of discrete methods [11] like cellular automata [12,13] or agent-based models [14,15]. Nowadays hybrid modelling [4,[16][17][18] has become very prominent due to an improved description of the complicated processes by combining the continuous and discrete methods.

For understanding the in vivo behaviour of cancer, the fundamental challenge for mathematical simulation is the simplification of the underlying complex processes while 2 Computational and Mathematical Methods in Medicine maintaining realistic findings. Typically, a system of partial differential equations is used to model interactions between cancerous cells, the extracellular matrix (ECM), and matrix degrading enzymes (MDEs) that describe the haptotactic and/or chemotactic movement of the cells [16,[18][19][20][21][22][23][24]. For the ECM degradation and remodelling is considered. Degradation occurs where the proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs) are located. It has been proven that the degradation of the ECM occurs because MDEs are secreted by tumour cells (cf. [25]). To be able to model these complex interactions on a standard computer for a large number of cells it is inevitably necessary to develop a simplified description, that is, to identify processes that are of less significance for the overall spatiotemporal dynamics of cancerous cells. To this end, we introduce a novel simplified model describing the same processes as in the literature [16,[19][20][21][22][23][24], without explicitly modelling the MDE. We assume in our implementation that the cancer cells themselves are able to degrade the matrix. This assumption is motivated by the fact that the tumour cells are overexpressing the responsible enzymes. To get similar results, we determine the level of the degradation on the basis of a degrading parameter α f and thus also the remodelling parameter β f . To validate the findings, we introduce different arrangements of the nutrient delivering blood vessels. Further, we compare the novel proposed model with common ones by simulating different initial conditions for the ECM. In this work, we do not consider cell-cell adhesion of cancerous cells not only due to the decreased expression of neural cell adhesion molecules (NCAM) in aggressive gliomas, such as the glioblastoma (GBM) [26] but also due to the focus on the haptotactic migration of cells through the extracellular matrix.

In this paper, we aim at simplifying the mathematical modelling of the complex biological processes and speeding computational time. In Section 2, we study the common method for cellular interactions and the simplest possible case. Finally, in Section 3, we discuss our results for different positions of capillaries and for a randomly or constantly distributed extracellular matrix, highlighting the successes of our approach. In Section 4 we conclude with a short outlook.

Materials and Methods

For simulating brain tumour growth, we consider a twodimensional area of brain tissue Ω = [0, 1] × [0, 1] which corresponds to 4 mm × 4 mm with boundary Γ := ∂Ω. A 400 × 400 grid over Ω with a space step of h = 0.01 mm forms the basis for the discrete method since each square of the grid corresponds approximately to the area of a tumour cell, that is 6.25 × 10 −6 cm 2 (cf. [16]). The grid is introduced for computed tumour or necrotic cells but not for the host tissue, because of the clumps-like growth of the tumour. For interactions of the tumour with the host tissue, we recommend to have a look at the macroscopic models [27,28]. At this level one achieves a better representation of processes like deformation of the environmental host tissue.

Hybrid

Model. The hybrid model (cf. [4]) is composed of a continuous and a discrete part. For modelling tumour growth, a discrete description is used to account for the motion of individual cells. The remaining factors are based on partial-differential equations. For the extracellular matrix it has been assumed to be directly affected by the cancerous tissue. To this end, the complete system of equations consisting of the distribution of cancer cells c, nutrient concentration u and extracellular matrix (ECM) density f is given by:
∂c ∂t = D c ∇ 2 c − χ∇ · (c∇u) − ρ∇ · c∇ f ,(1a)∂u ∂t = D u ∇ 2 u − α u uc, (1b) ∂ f ∂t = −α f f c + β f f ,(1c)
in Ω × [0, T], where T defines the end of a given time interval, D c and D u denotes the diffusion coefficients of the tumour cell and nutrients, respectively. Furthermore, χ is the chemotaxis coefficient and ρ the haptotaxis coefficient. Uptake and decay of particular substances due to tumour growth are α u and α f (uptake rate for nutrients and ECM), β f represents the remodelling parameter for the ECM.

To compare our results with common ones, we outline also the approach for modelling the extracellular matrix effected by enzymes. The complete system including the matrix degrading enzymes' (MDEs) concentration m is defined in Ω × [0, T] as follows:
∂c ∂t = D c ∇ 2 c − χ∇ · (c∇u) − ρ∇ · c∇ f ,(2a)∂u ∂t = D u ∇ 2 u − α u uc,(2b)∂ f ∂t = − α f f m + β f f ,(2c)∂m ∂t = D m ∇ 2 m + β m c − α m m,(2d)
where D m denote the diffusion coefficients of the enzymes, α m the decay coefficient, and β m represents the production constant for MDEs. Due to the diffusive behaviour of the MDEs, the values are different for the uptake and the remodelling of the extracellular matrix comparing to (1c). Hence, we label them for the system ((2a), (2b), (2c), (2d)) α f and β f , respectively.

For the initialisation of both models 441 tumour cells are placed in the middle of domain Ω. As for the different initial conditions for the ECM we refer to Section 2.3. The amount Computational and Mathematical Methods in Medicine 3 of initially available nutrients is estimated from the solution of the equation:
−D u ∇ 2 u + α u uc ini = 1,(3)
where c ini represents the initial tumour with 441 cells in the middle of the domain. This equation denotes the availability of nutrients in the cerebrospinal fluid (CSF), which is a supplier of essential nutrients to the brain [29]. The initial nutrient concentration for two parallel blood vessels is shown in Figure 1. Depending on the location of the blood vessels, we have Dirichlet and Neumann boundary conditions (see Section 2.3). The initial concentration for MDE is set to zero throughout the domain. For the concentration of the MDE and ECM we assume zero flux boundary conditions.

Nondimensionalisation.

To obtain a similar magnitude in the range [0, 1] for all computed quantities, we rescale and nondimensionalise the variables and parameters of the proposed method ((1a), (1b), (1c)) and the common approach ((2a), (2b), (2c), (2d)). The dimensionless variables are defined as:
x = x L , t = t τ , c = c c 0 ,(4)u = u u 0 , f = f f 0 , m = m m 0 .(5)
For a more detailed description we refer to Table 1 For the appropriate length scale L we use 0.1 cm (taken from [19,23]), for the time τ = L 2 /D, where D = 10 −6 cm 2 /s is a representative diffusion coefficient [23]. For the tumour cell density c 0 , the nutrient concentration u 0 and ECM density f 0 following [16], the matrix degrading enzyme density m 0 = 0.1 nM is taken from [19]. For the dimensionless cell diffusion coefficient of the tumour cells we get D c = τD c /L 2 = 10 −5 and for the dimensionless haptotaxis parameter ρ = τρ f 0 /L 2 = 0.26 [16,23]. We assume the chemotactic parameter χ to be equal to the haptotactic parameter, so that the cells are equally attracted to nutrients and to interact with the ECM. The parameter D u = 10 −5 cm 2 /s is taken from [16], so we get D u = τD u /L 2 = 10. The uptake rate is assumed to be α u = τc 0 α 0 /u 0 = 6.25 · 10 −5 . The parameters of the MDE, β m = 1, and α m = 0 are taken from [16]. The dimensionless diffusion coefficient of the MDE is assumed to be 0.08. For the MDE dependent model ((2a), (2b), (2c), (2d)), we follow [30] for the ECM uptake α f and the remodelling part β f . Apart from that (system (1a), (1b), (1c)) we make a parameter estimation for α f and β f (see Section 2.4), since we are not aware of any values in the literature. For notational convenience, we drop the hats in the following.

Numerical Implementation.

For discretizing the systems of partial differential equations in ((1a), (1b), (1c)) and ((2a), (2b), (2c), (2d)) we use standard finite-difference and finite element method. For computing the nutrients (1b), (2b), and the MDE (2d), we use the method of finite elements. Because of the discrete-continuum interaction in every time step, we have to solve the equations in the steady state. The boundary and initial conditions for nutrients depend on their position relative to the oxygen and glucose (nutrients) delivering blood vessels or capillaries. This can be modelled by placing them at all four surrounding boundaries, at two of them, or only on a single side. For the sites occupied by blood vessels we apply a Dirichlet boundary condition with a constant function u d . For the remaining boundaries we use zero flux boundary conditions (Neumann). Thus at any time t ∈ [0, T]:
∂u ∂n = 0 on Γ N , u = u d on Γ D ,(6)
where Γ N and Γ D are the Neumann boundary and the Dirichlet boundary, respectively, with Γ N ∪ Γ D = Γ. We set u d = 1 since the concentration of nutrients, such as glucose and oxygen is highest in the capillaries. For the tumour cell equations (1a) and (2a), respectively, we use the resulting coefficients of the five-and ninepoint finite-difference stencil to generate the probabilities of the movement of an individual cell in response to its local milieu. The 5-point stencil is equivalent to the von Neumann neighbourhood and the 9-point stencil to the Moore neighbourhood. We implement both and use a switching mechanism to select one of them for each iteration and each cell (see Section 2.6). With t = mk, x = ih, and y = jh (m, k, i, j, h > 0), we use forward differences at time point t m and second order central differences for the spatial derivative at point x i, j . The resulting equation for the 5-point stencil governing the chemotactic-haptotactic migration of a tumour cell in (1a) and (2a) has thus the form Cancer cell [16,19,23] where P 0 is proportional to the quiescent cells. P 1 , P 2 , P 3 , P 4 are probabilities that are proportional to a movement of the cell to the right, left, up, or down, respectively. For the 9point stencil, the resulting equation is straight forward:
c m+1 i, j = P 0 · c m i, j + P 1 · c m i+1, j + P 2 · c m i−1, j + P 3 · c m i, j+1 + P 4 · c m i, j−1 ,(7)Cells cm −3 c/c 0 [16, 23] u(x, t) Nutrient concentration M O 2 cm −3 u/u 0 [16, 23] f (x, t) E C M M f / f 0 [16, 23] m(x, t) M D E M m / m 0c m+1 i, j = P 0 · c m i, j + P 1 · c m i+1, j + P 2 · c m i−1, j + P 3 · c m i, j+1 + P 4 · c m i, j−1 + P 5 · c m i−1, j+1 + P 6 · c m i+1, j−1 + P 7 · c m i+1, j+1 + P 8 · c m i−1, j−1 .(8)
Here, we have additionally the probabilities P 5 , P 6 , P 7 , P 8 , which are proportional to a movement of the cell to the top left, bottom right, bottom left, and top right. For the extracellular matrix we model different initial conditions. On one side we take random values between 0 and 1, which corresponds to the density of the ECM [16]. On the other side, for a homogeneous behaviour a constant value f (x, t) = 0.8 is taken to allow for the assumption that the density of the ECM is high at the beginning, but smaller than 1, which would be equivalent to the maximal density and the value chosen in [16]. The choice of the value in this work is taken in reference to [31], where the extracellular space of gliomas has been described. For solving the ECM equations (1c), (2c) we use Euler finite difference approximations. The density of the matrix is continuous, therefore a continuous description of the cells is needed. We use the tumour cell density c = 1 when a tumour cell is occupying the current location and take c = 0 otherwise. Intuitively, we use the highest value for the density, since the grid point is occupied by the cell. In this way we have a binary description of the tumour cells.

Parameter Estimation.

In this section we estimate the parameter α f and β f used in the newly proposed model, while the remaining parameters are taken from literature (c.f. Section 2.2). In order to estimate the parameters and for the sensitivity analysis, we examine the behaviour of ECM f by varying α f and β f within a certain range.

The MDEs are diffusing, so that the concentration at positions where tumour cells are located is not so high as for the tumour cells, where we have the highest value c = 1. For the degradation parameter α f of the extracellular matrix it is hence important to choose a value that is smaller than the value in (2c) which is 1. For the lower bound of the range, if the value 0 is included, in this case we will not have any degradations effects, which is not what we actually want. To this end, the parameter α f is varying in the range (0, 1).

The parameter β f is 0.015. Since the degradation parameter α f is smaller than α f and the equilibrium between degradation and remodelling of the ECM should remain valid, the upper bound of the remodelling parameter β f will be 0.015. On that account and because 0 would mean no remodelling effects, the parameter is varying in the range (0, 0.015].

When combining these two parameters, the stability has still to be ensured. Because of the explicit Euler method, which we use to solve the equation of the ECM and because the maximum of the cancer density c is 1, the stability requirement is
1 − kα f + kβ f ≤ 1,(9)
for t = mk and c → 1. We start by fixing the parameter β f and vary α f , since the range of the remodelling parameter is much smaller. First β f has to be chosen as high as possible, that is, β f = 0.015. Now, we vary α f just in a range of [0.02, 1) for not breaking the stability requirement (9). The resulting ECM in the middle of the domain is shown in Figure 2. The density of the ECM is greater than 1, that is the production parameter is too big. For a value of β f = 0.0075, the ECM density is approximately 1, and for β f = 0.005 the ECM is 0.92 (see Figure 2). At last the production parameter was chosen to be β f = 0.001 and β f = 0.0005. For each value for β f mentioned, we tested different values for α f , whereas α f was selected w.r.t. β f with the aim to archive a realistic, not too big production of the ECM and a degradation of the matrix, so that the cells can become invasive. A selection of different combinations of α f and β f as well as the resulting ECM f are given in Figure 2. The values for which the best equilibrium is archived are β f = 0.001 and α f = 0.01 (see Figure 2). These values reproduce the tumour growth extremely well (see Figure 8) throughout this work.

Cell

Actions. The tumour equations (1a) and (2a) describe solely the motility of the cells. For modelling the cell actions proliferation, death and quiescence we consider the following criteria. In each time step and for each tumour cell we account for the local nutrient concentration and decide based on this how the cell will react. In case that the nutrient value for the respective cell is under a critical threshold u crit = 0.4, we assume that the cell will die due to insufficient nutrients. Consequently, the cell is marked as necrotic tissue. In contrast to [16] the necrotic cells are not considered for the next step, since this material is not degraded by the macrophages of the brain (microglia) like it is the case for the apoptotic material due to phagocytosis [32]. Having checked the necrosis criterion, each cell moves according to the scheme described in Section 2.3. In case the nutrient concentration is high enough, that is, u crit ≥ 0.4, the cell is selected to divide. The duration of the cell cycle is in general one day [6], in our case we assume that the proliferation takes eight hours, which is realistic for malignant tumours [16]. The grid location onto which the daughter cells are placed depends upon the cells occupying the neighbourhood of the mother cell. One daughter cell will always replace the mother cell, which is common to do so [16,33]. Instead to place the second daughter cell randomly as usual [16], we placed it chemotacticially, that is in the nontumorous and nonnecrotic location with the highest nutrient concentration (if there is more than one free) in the neighbourhood of the mother cell. The choice of the neighbourhood is described in Section 2.6. This chemotacticially placement is also valid for the movement of a tumour cell.

In case there is no free space in the neighbourhood, the tumour cell becomes quiescent until free space is available or the cell becomes necrotic due to insufficient nutrient [16,18,34,35]. Of course one cell can also get into a quiescent state because of the migration scheme.

Invasive glioma cells are resistant against apoptosis [36,37] in favour of the cell survival. Furthermore, [38] proved by means of in vitro experiments that apoptosis is suppressed in GBM, hence we do not consider apoptosis in the proposed model in contrast to [16].

Neighbourhood and Update.

The choice of the local neighbourhood of an individual cell is crucial for the update of its state for lattice-based models [39][40][41]. To avoid unrealistic pattern formation, we use a novel effective and simply method [11], which introduces a fifty-fifty chance to decide if we take the eight neighbours into account or the four orthogonally surrounding cells. For each cell, at each discrete time point we generate a random number to decide whether the Moore or the von Neumann neighbourhood is favoured.

Another problem can arise from the strategy for updating the state of a tumour cell inside the lattice. If we would run sequentially to look at every tumour cell one by one, the first cell has often more possibilities for migration (in case of division: to place the daughter cells) than the one being located very next to it. Consequently, cells are not updated in a left-to-right or top-to-bottom manner but randomly [16].

Results and Discussion

The distribution of the extracellular matrix, the progression of nutrients, and the tumour cell arrangement over time are shown in Figures 3 and 4. There, the obtained tumour distributions are illustrated for up to 700 iteration steps, which is equivalent to a period of time of 350 h. At this point in time the tumour is hypoxic, that is, the mean value of nutrients and especially of the oxygen concentration is under a nondimensional value of u = 0.5. With hypoxia the tumour growth will transfer into the vascular stage through the process of angiogenesis [42].

In a first step, a homogeneous ECM (see also Section 2.3) and two neighbored blood vessels at the left side and at the bottom of the domain has been assumed. The results are illustrated in Figure 3. An increasing size of the tumour can be observed over the time. The steep rise can also be observed in Figure 8, bottom right. Likewise the distinct behaviour of the invasive cells migrating in the direction of the higher nutrient concentration is clearly visible and a large necrotic region. The nutrients are shown in the bottom row in Figure 3. The big consumption rate of the tumour cells is notable visible especially in the middle of the domain where the nutrient concentrations becomes smaller over time. The density of the ECM is shown in the middle row of Figure 3. The shape of the degradation is similar to the shape of the tumour cells. The boundary of the degraded matrix is less apparent, this is because individual tumour cells can not degrade the matrix just by moving through it just one time. In its centre, the ECM has been degraded more. There, the tumour bulk is located.

Computer simulations using two parallel blood vessels and a constantly initialized extracellular matrix is shown in Figure 4. The distribution of the tumour is invasive and a necrotic region due to the lack of nutrients (lower row) can be observed. Once more, the close interaction of tumour cells (upper row) and ECM (middle row) is clearly visible by means of the similar shape of both.

The simulated tumour depicts a distribution that can typically be observed in vivo A large necrotic core surrounded by a rim of quiescent cells and an outer rim of strongly diffusive glioma cells. As a further step and to show the importance of the novel method, we compare results of the proposed approach with results of common ones. To this end, computer simulations using system ((1a), (1b), (1c)) and ((2a), (2b), (2c), (2d)) have been run. In a first step, one nutrient delivering blood vessel at the right side  of the domain has been assumed. Given a homogeneous extracellular matrix distribution the results are shown in Figure 5. Comparing these data, differences for the ECM in Figures 5(c) and 5(f) can be observed, since the assumption for degradation of the matrix is different. The extracellular matrix modelled on the basis of system ((2a), (2b), (2c), (2d)) has been reduced through the diffusive MDEs shown in Figure 5(g) and the matrix simulated with the new introduced system ((1a), (1b), (1c)) is degraded by the tumour cells shown in Figure 5(a).

For the tumour cell distribution a quite similar size of the bulk can be observed, however, there are differences concerning the invasive properties of the cells. For the tumour progression modelled with the proposed approach ((1a), (1b), (1c)) the invasive properties of the cells are more distinct. This is indeed characteristic for malignant tumours and consistent with in vivo and in vitro glioma experiments (cf. [37,43] and references therein). However, this feature cannot be seen in Figure 5(d). The novel introduced method seems to capture this behaviour better. As expected, the nutrient progression is similar in all figures, since these are modelled in the same way in both models.

At last, we compare the tumour distribution assuming the typical Dirichlet boundary conditions for the nutrients, that is, we assume blood vessels located at all boundaries of the domain. Furthermore, a randomly initialised ECM ( Figures 6(c), and 6(f)) is assumed. The tumour cell arrangement is shown together with the nutrients, ECM, and MDE in Figure 6. We notice a quite similar structure of the tumour cells ( Figure 6(a)) simulated with the proposed method ((1a), (1b), (1c)) and the tumour cell distribution shown in Figure 6(d) modelled using ((2a), (2b), (2c), (2d)). There are only subtle differences, that have to be attributed to the probabilistically controlled model. For a better comparison of the differences of the tumour cell distributions obtained using ((1a), (1b), (1c)) and ((2a), (2b), (2c), (2d)), the difference image of Figures 6(a) and 6(d) is given in Figure 7.

Apart of a small rim, only zero entries occur. The white points, representing a value 1, and black points (with values −1) show the missing tumour cells in either of the two images, the entries coloured with grey denote different states of the cancerous cell (proliferating, quiescent, necrotic). As mentioned before, this must not be misunderstood as a quantitative error difference or estimation because of the probabilities of cell movement and the randomly-guided model (see Section 2).

However, we have simulated the lastly described model (cf. Figure 6) one hundred times with different random number seeds and show the average behaviour of the different cells with the respective standard error for both methods (the proposed method ((1a), (1b), (1c)) and the common method ((2a), (2b), (2c), (2d))) in Figure 8.     ). The same behaviour can be seen in the lower two plots, where the distribution of the migrating/proliferating and all tumour cells, including quiescent cells are displayed. These highly similar results indicate the potential for the simplified method without explicitly simulating the matrix degrading enzymes. At this, another important advantage of the introduced model is the computation time ( Table 2).

The proposed model ((1a), (1b), (1c)) takes about 2.5 less time compared to model ((2a), (2b), (2c), (2d)) in case the ECM is initialized with a random distribution ( Table 2). This reduction in run-time does even increase up to a factor of 6.3 in case the ECM is initialized to a constant value and just one blood vessel is assumed. Simulations have been carried out using a single-threaded MATLAB implementation and have been run on a Pentium i7920 with 2.67 GHz and 12 GB of RAM.

Conclusions and Outlook

A hybrid approach has been developed that uses a coupled continuum-discrete model to simulate tumour growth. This method is devoted to the modelling of cellular processes of tumour cells, which is itself part of a complex system. The availability of efficient approaches therefore is an essential prerequisite for modelling tumour growth. To this end, a novel lattice-based approach has been developed that does not only provide a significant simplification compared to previous models, but also is computationally efficient ( Table 2) and, moreover, depicts a more invasive behaviour of tumour cells, which is an important character of gliomas. The degradation of the ECM in the area covered by the tumour can clearly be seen in all simulation results. This behaviour confirms our hypothesis of the direct interaction between tumour cells and the extracellular matrix. As an important feature the novel method provides the best results using a constant matrix, where the invasive behaviour of tumour cells can clearly be observed, in accordance to in vivo and in vitro behaviours [37,43].

Furthermore, the proposed model requires about 2.5 less computation time compared to the common model in case  the ECM is heterogeneous (Table 2). This reduction in runtime does even increase up to a factor of 6.3 in case the ECM is chosen to be homogeneous (assuming one blood vessel).

Comparing the results for modelling haptotacticchemotactic cancer growth using the common choice of the environment and the novel introduced approach, a rather similar size of the tumour can be observed, in case a heterogeneous ECM is used. Faced with the requirement of modelling much more complex processes than the degradation of the ECM through the MDE (in order to adequately model tumour growth) the proposed model provides a reasonable tradeoff between complexity and accuracy.

Prospectively, it will also be essential to extend the devised model by cell-cell interactions such as pressure on neighbouring cells caused by mitosis or cell migration. We expect, that this pressure will make the thickness of the proliferating rim more distinct [44]. For a more realistic description of the in vivo processes, incorporating the immune system [45] and effects of therapies [46,47] forms the central focus of our current work. In this case, the established efficiency gain might pay off even more through alternative numerical solvers than the implemented finite element method. Further, we aim at devising multiscale tumour growth models that not only account for cell-cell interaction but also for molecular events as well as for information available from the macroscopic embedding. The important effects of, for example, epidermal growth factor receptor (EGFR) which are overexpressed in gliomas [48] on the molecular level have to be included.

Figure 1 :
1Initial distribution of the nutrients for blood vessels placed at two parallel boundaries.

01 Figure 2 :
012= 0.015, α f = 0.5 β f = 0.015, α f = 0.1 β f = 0.0075, α f = 0.2 β f = 0.0075, α f = 0.1 β f = 0.005, α f = 0.1 β f = 0.001, α f = 0.1 β f = 0.001, α f = 0.05 β f = 0.001, α f = 0.01 β f = 0.0005, α f = 0.05 β f = 0.0005, α f = 0.ECM profiles at t = 325 h for different α f and β f .

Figure 3 :
3Simulation results for the proposed model ((1a), (1b), (1c)) at points in time t = 50 h, 150 h, 250 h, and 350 h. The initial condition of the nutrients is given by the blood vessel placed at the left side and at the bottom of the domain and are shown in the third row. The extracellular matrix is initialised with a constant value and is illustrated in the second row. For the cancerous cells in the first row light grey represents necrotic tissue, dark grey quiescent cells and black proliferating and migrating cells. The colouration for the remaining data is given in the colour bar.

Figure 4 :
4Simulation results for the proposed model at points in time t = 50 h, 150 h, 250 h, and 350 h. The initial condition of the nutrients is given by the blood vessel placed at the left and right sides of the domain and are shown in the third row. The extracellular matrix is initialised with a constant value and is illustrated in the second row. The first row is showing the distribution of the cancerous cells. Colouration as given inFigure 3.

Figure 5 :
5Simulation results (a-c) of the proposed method ((1a), (1b), (1c)) and (d-g) of the common method ((2a), (2b), (2c), and (2d)), respectively, at time t = 375 h. The initial condition of the nutrients is given by the blood vessel placed at the right side of the domain. The extracellular matrix is initialised with a constant value. Colouration as given inFigure 3.

Figure 6 :
6Simulation results (a-c) of the proposed method ((1a), (1b), (1c)) and (d-g) of the common method ((2a), (2b), (2c), (2d)), respectively, at time t = 375 h. The initial condition of the nutrients is given by blood vessel placed at all sides surrounding the domain. The extracellular matrix is initialised with random values (heterogeneous). Colouration as given inFigure 3.

Figure 7 :
7Difference image of the tumour cell distributions (Figures 6(a) and 6(d)) computed with the two described methods at time t = 375 h. For the distribution of the necrotic cells shown in the upper left figure the differences are hardly visible. Only after approximately 110 h small differences can be observed for the progression of the quiescent tumour cells (upper right figure

Figure 8 :
8Average number of necrotic cells (upper row, left), quiescent cells (upper row, right), proliferating and migrating tumour cells (lower row, left), and total number of nonnecrotic tumour cells (lower row, right) over time with the respective standard error.

Table 1 :
1Variables used in the hybrid model: The variable of the dimensional system ((1a), (1b), (1c)) and ((2a), (2b), (2c), (2d)), a description, their units, the computation for the corresponding nondimensional parameters and references.Variable y
Description
Unit
Formula of y
Reference
c(x, t)

Table 2 :
2Computation time in minutes for the common model ((2a), (2b), (2c), (2d)) and the proposed approach (without explicitly modeling MDE, ((1a), (1b), (1c)) at different configurations.Initial ECM
Capillaries
Standard model
New model
All sides
85
32
Random
Two sides
125
48
One side
138
61
All sides
92
28
Constant
Two sides
133
31
One sides
151
24

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