An imaging-based computational model for simulating angiogenesis and tumour oxygenation dynamics

Abstract

Tumour growth, angiogenesis and oxygenation vary substantially among tumours and significantly impact their treatment outcome. Imaging provides a unique means of investigating these tumour-specific characteristics. Here we propose a computational model to simulate tumour-specific oxygenation changes based on the molecular imaging data. Tumour oxygenation in the model is reflected by the perfused vessel density. Tumour growth depends on its doubling time (T_d) and the imaged proliferation. Perfused vessel density recruitment rate depends on the perfused vessel density around the tumour (sMVD_tissue) and the maximum VEGF concentration for complete vessel dysfunctionality (VEGF_max). The model parameters were benchmarked to reproduce the dynamics of tumour oxygenation over its entire lifecycle, which is the most challenging test. Tumour oxygenation dynamics were quantified using the peak pO₂ (pO_2peak) and the time to peak pO₂ (t_peak). Sensitivity of tumour oxygenation to model parameters was assessed by changing each parameter by 20%. t_peak was found to be more sensitive to tumour cell line related doubling time (~30%) as compared to tissue vasculature density (~10%). On the other hand, pO_2peak was found to be similarly influenced by the above tumour- and vasculature-associated parameters (~30–40%). Interestingly, both pO_2peak and t_peak were only marginally affected by VEGF_max (~5%). The development of a poorly oxygenated (hypoxic) core with tumour growth increased VEGF accumulation, thus disrupting the vessel perfusion as well as further increasing hypoxia with time. The model with its benchmarked parameters, is applied to hypoxia imaging data obtained using a [⁶⁴Cu]Cu-ATSM PET scan of a mouse tumour and the temporal development of the vasculature and hypoxia maps are shown. The work underscores the importance of using tumour-specific input for analysing tumour evolution. An extended model incorporating therapeutic effects can serve as a powerful tool for analysing tumour response to anti-angiogenic therapies.

1. Introduction

The network of blood vessels in the human body carries the important task of delivering oxygen and nutrient supply as well as removal of cell waste. Similarly, the growth and development of a tumour is also dependent on this vasculature (Folkman 1976). Angiogenesis, or the process of recruitment of new vessels from older ones, is critical to satiate the ever increasing demands of the tumour. Unlike vasculature in normal tissues, tumour vasculature is highly abnormal, immature, leaky and inefficient (Nagy et al 2009). This dysfunctional vasculature results in hypoxia, which can lead to both necrosis and an increase in tumour malignant potential (Vaupel and Harrison 2004). The abnormalities are a result of the haphazard regulation of numerous angiogenic processes which are tightly controlled in normal vessel growth (Carmeliet and Jain 2000). Understanding the biology and the timing behind these processes in a systematic manner is essential for designing treatment regimens for tumour eradication.

Tumour angiogenesis involves a cascade of complex signalling pathways. Tumour growth results in reduced vessel density and subsequently increased hypoxia. Hypoxic tumour cells secrete tumour angiogenic factors (TAFs) tilting the balance between the pro- and anti-angiogenic factors in the microenvironment towards increased pro-angiogenic factor concentration. The TAFs result in increased vessel permeability (Brown et al 1997) and angiogenesis. Vessel networking within the neovasculature establishes blood flow, decreases hypoxia and facilitates vessel maturation. A more detailed description and information on multiple modes of vasculature recruitment can be found in the literature (Paweletz and Knierim 1989, Carmeliet and Jain 2000, Bergers and Benjamin 2003, Karamysheva 2008). Typically the tumour is hypoxic as the perfused vasculature recruitment rate is unable to keep up with the tumour growth rate (Vaupel 1977). Thus the tumour has a high concentration of TAFs and its vasculature is leaky.

The biology of the angiogenic processes has been of interest to both experimentalists as well as computational researchers working on angiogenesis. Laboratory experiments and simulations complement each other to advance the science behind these processes (Sanga et al 2006). While laboratory experiments give researchers the much needed data and validations, computational models give researchers the fidelity to perform a variety of in silico experiments which would otherwise be highly expensive and time consuming to be carried out in a laboratory. In silico experiments also have the potential to give experimental researchers avenues to pursue in the laboratory (Jain et al 2007). To this end a number of models addressing different aspects of tumour growth and angiogenesis have been proposed.

Computational models of angiogenesis are typically categorized as discrete cellular level models, continuum models or hybrid models. Discrete models simulate cellular interactions at the microscopic scale demonstrating their capability to generate tree-like vascular structures. Examples of techniques used in discrete models for vasculature generation include cellular automation, fractals, cellular potts etc algorithms (Stokes et al 1991, Bauer et al 2007). Continuum models take a different approach by focussing on the macroscopic tumour and vasculature dynamics typically based on the foundation of reaction-diffusion equations (Orme and Chaplain 1996, Levine et al 2001, Hogea et al 2006) demonstrating their dynamics over a much bigger spatial scale. Hybrid models incorporate both of these characteristics by using continuum models to simulate the broad spatial dynamics and discrete models to explicitly simulate vasculature (Plank and Sleeman 2003, Sun et al 2005, Milde et al 2008). These models include phenomenon such as random motility, chemotaxis, haptotaxis, cellular adhesion, proliferation and branching. Several vasculature models focus on the incorporation of flow (Gödde and Kurz 2001, Alarcón et al 2005, Bartha and Rieger 2006, McDougall et al 2006, Welter et al 2008, Liu et al 2011) and vasculature maturity (Arakelyan et al 2005, Cai et al 2011, Gevertz 2011, Perfahl et al 2011). An in-depth review of vasculature models can be found in literature (Mantzaris et al 2004, Scianna et al 2013).

Although, tumour models such as the ones mentioned above can simulate hypothetical tumour and vasculature behaviour, it is not possible to predict the dynamics of individual tumours. This is due to the inherent complexity of these growth dynamics which depend on the microenvironment as well as the characteristics of the particular tumour in question. The heterogeneity of its growth and response to therapy also hampers the treatment of cancer (Michor and Polyak 2010). Imaging provides a unique opportunity to gain knowledge about the tumour condition. Based on imaging data short term predictions about its growth and development can also be made. Imaging techniques such as computed tomography (CT), positron emission tomography (PET), magnetic resonance imaging (MRI) and ultrasound are non-invasive and are routinely used in the clinics. PET and MRI are capable of providing functional as well as molecular information about the tumour microenvironment. Examples include perfusion, blood volume, permeability, angiogenesis, oxygenation levels, and cell proliferation (Matsumoto et al 2009, Michalski and Chen 2011, Stephen et al 2012, Buscombe and Wong 2013).

Acknowledging the power of imaging, modelling of tumour and vasculature based on imaging data has been steadily gaining importance. Swanson and colleagues developed a model describing the invasion patterns of gliomas using MRI images (Swanson et al 2000, 2003, Jbabdi et al 2005). This model has since been enhanced to investigate how the proliferation and dispersal of glioma cells can affect tumour malignancy (Swanson et al 2011). Recently, the model was applied incorporating the effects of hypoxia-mediated radiation resistance to predict therapeutic outcome (Rockne et al 2015). Yankeelov et al (2010) proposed a methodology to incorporate PET, MRI and SPECT imaging data into a reaction-diffusion based mathematical model though explicit imaging data was not used. In a different work (Yankeelov et al 2013) they used MRI and FDG imaging data to simulate tumour growth and the evolution of its glucose concentration with time. Titz and Jeraj (2008) proposed a PET imaging-based model to simulate the effect of radiotherapy on tumour growth and subsequently advanced the model to simulate response to anti-angiogenic therapy based on hypoxia and proliferation data (Titz et al 2012) although the explicit modelling of vasculature dynamics was not proposed. Explicit modelling of imaging-based vasculature growth is important to further advance our understanding of anti-angiogenic therapies and why and how they fail.

Here we propose an imaging-based model simulating the temporal changes in tumour vasculature and its related tumour oxygenation based on molecular imaging data. The model uses the hypoxia PET imaging data to characterize tumour oxygenation status and proliferation PET imaging data to characterize tumour growth. We build upon a previous work where a stochastic model for the simulation of the vasculature based on a hypoxia PET scan was proposed (Adhikarla and Jeraj 2012). The current work extends the model to incorporate tumour growth and angiogenesis based on the molecular imaging as well as literature-based experimental data. The model is benchmarked to reproduce the observed temporal dynamics of tumour oxygenation in a xenografted mice tumour. The effects of model parameters on tumour oxygenation were analysed and the benchmarked parameters were then applied to preclinical imaging data.

2. Description of the model

2.1. Model workflow

To simulate the growth of tumour vasculature over time, the model uses imaging inputs related to tumour oxygenation and tumour proliferation. Figure 1 demonstrates the model workflow.

Figure 1.

Flowchart demonstrating the dynamics between the different model blocks.

2.1.1. Calculation of perfused vessel density (Step I).

Tumour oxygenation map is derived from its hypoxia map using a functional relationship as described in earlier works (Lewis et al 1999, Titz and Jeraj 2008, Adhikarla and Jeraj 2012). A previous work (Adhikarla and Jeraj 2012) quantified the amount of vasculature present in the microenvironment based on its oxygen tension with higher vessel density reflecting higher oxygen tension (equation (1)).

$$p{\mathrm{O}}_2=60\frac{{\text{sMVD}}^{1.95}}{{\text{sMVD}}^{1.95}+{0.015}^{1.95}}\text{mmHg}$$ (1)

The simulated microvessel density (sMVD) used as a model parameter reflects the percentage of the tissue volume occupied by the capillaries. Thus given the hypoxia imaging data, the microvessel density of each voxel could be obtained.

2.1.2. Calculation of non-perfused vessel density.

At the start of the simulation all the vessels are assumed to be perfused. Alternately, estimates of the non-perfused vessel density could be obtained from additional imaging; such as measurement of tumour blood volume using dynamic PET or MR imaging data.

2.1.3. Tumour angiogenic factor concentration (Step II).

Hypoxic tumour cells secrete angiogenic factors which diffuse in the micro-environment to initiate the process of vascular recruitment. The concentration of the angiogenic factor in the tumour micro-environment can be calculated by numerically solving its diffusion equation (equation (2)). Data on vascular endothelial growth factor (VEGF) which is one of the prime angiogenic factors has been used.

$$\frac{\mathrm{d}{c}_{\text{VEGF}}}{\mathrm{d}t}=D_{\text{VEGF}}{\mathrm{\nabla }}^2{c}_{\text{VEGF}}-{\mathrm{\lambda }}_{\text{VEGF}}{c}_{\text{VEGF}}-Q_{\text{VEGF}}+S_{\text{VEGF}}$$ (2)

Where c_VEGF is the VEGF concentration, D_VEGF is the diffusion coefficient of VEGF in the tissue, λ_VEGF is the decay rate of VEGF in tissue, Q_VEGF is the VEGF consumption rate and S_VEGF is the VEGF secretion rate. Details regarding the implementation of the diffusion equation and the calculation of c_VEGF at the voxel level are presented in the supplementary material section (stacks.iop.org/PMB/61/3885/mmedia).

Increased hypoxia results in increased secretion of the angiogenic factors. Work done by Leith and Michelson (1995) on HCT-8 and Clone-A cells in vitro has been used to quantify this relationship (supplementary material) between the secretion rate of VEGF and the tumour oxygen tension. Unless explicitly stated, the secretion rate characteristics of VEGF by the HCT-8 cell line have been used in the presented work.

2.1.4. Change in total vessel density (Step III).

The growth rate of vasculature depends on the amount of angiogenic factor in the tumour microenvironment. Higher concentration of the angiogenic factor results in increased vessel growth. The amount of vessel growth depends specifically on the particular angiogenic factor. The total vessel density (sMVD_total) after time Δt_sim can be calculated as:

$${\text{sMVD}}_{\text{total}}(t+\mathrm{\Delta }{t}_{\text{sim}})=\text{VGR}(c_{\text{VEGF}},\mathrm{\Delta }{t}_{\text{sim}})\times {\text{sMVD}}_{\text{total}}(t)$$ (3)

Where VGR is the vasculature growth ratio reflecting the growth rate of the vasculature and is directly dependant on the concentration of VEGF and is calculated (supplementary material) based on the study by Ferrara and Henzel (1989).

2.1.5. Change in perfused vessel density (Step IV).

Vessel functionality depends on the growth of vessels, their subsequent networking with each other and the establishment of blood flow thereafter. Tumour blood vessels are however leaky, tortuous and of uneven diameter. Notably, VEGF was discovered initially as the vasculature permeability factor (VPF). This abnormal vasculature leads to erratic and reduced blood flow with tumour growth (Nagy et al 2009). Given these characteristics of tumour vessels, the following assumptions have been made to derive equation (4) for the generation of perfused vasculature:

1. Vessels with blood flow will originate from vessels that are already perfused.

2. Increased VEGF concentration will increase the permeability and hence reduce the flow downstream (Brown et al 1997).

$$\mathrm{\Delta }{\text{sMVD}}_{\text{perf}}=\text{PRC}\times \log \left(\frac{{\text{VEGF}}_{\max }}{c_{\text{VEGF}}}\right)\times {\text{sMVD}}_{\text{perf}}\times \mathrm{\Delta }{t}_{\text{sim}}$$ (4)

The amount of perfused vasculature recruited in a simulation time-step (Δt_sim) is limited by the total amount of vasculature recruited in that time-step. To evaluate vasculature perfusion, a vessel perfusion index (VPI) reflecting the percentage of perfused vasculature is defined.

$$\text{VPI}=\frac{{\text{sMVD}}_{\text{perf}}}{{\text{sMVD}}_{\text{total}}}$$ (5)

2.1.6. Tumour growth (Step V).

Changes in vessel density are governed by vascular as well as tumour growth rate. The growth rate of the tumour is determined by the doubling time of the particular tumour cell line in question. Titz and Jeraj (2008) simulated the growth in the number of tumour cells based on tumour proliferation map, which is the same as used in this work.

2.2. Model mechanics

2.2.1. Simulation of vasculature.

The simulation of vasculature structure based on tumour oxygenation has been previously described (Titz and Jeraj 2008, Adhikarla and Jeraj 2012). Briefly, all vessels are simulated as cylindrical elements of equal lengths and diameters and are assumed to have equal oxygen concentration within them. Further details regarding the simulation of vessels are provided in the supplementary materials and the previous publication (Adhikarla and Jeraj 2012).

2.2.2. Voxel coupling.

Tumours remain avascular till they attain a diameter of about 1–2 mm (Bergers and Benjamin 2003). After this stage they recruit vasculature from the feeding vessels which emerge from the existing vasculature and branch out to form a very dense network of vessels by the time they reach the tumour (Muthukkaruppan et al 1982). To simulate this phenomenon of vasculature recruitment from outside the tumour, the following method is employed. The tissue directly adjacent the tumour is kept at a constant vessel density of sMVD_tissue which is a model parameter. Folkman (1976) found the maximum velocity of vessel growth from the surrounding tissue to be 1 mm d⁻¹. Hence within a simulation time-step of Δt_sim days each voxel of the tumour is assumed capable of recruiting vasculature from a distance of d = Δt_sim * 1 mm. To incorporate this recruitment capability, the total and perfused vessel density recruited within a particular voxel is calculated as the average values of sMVD_total and sMVD_perf calculated within a cube of side 2d centred on that voxel.

2.2.3. Tumour growth.

The tumour is considered to have uniform cellular density. Since all the voxels are equi-volumetric, each voxel of the tumour is assumed to contain N tumour cells. The growth rate of tumour is determined by the doubling time of the particular cell line. Given this doubling time and the voxel growth fraction, the change in the number of tumour cells in a simulation time-step within the voxel can be calculated. A change in the number of tumour cells determines the change in volume of the voxel. Details on the algorithm employed to simulate changes in the simulation matrix are provided in the supplementary material. Volumetric increase of a particular voxel cuts down the corresponding vessel density of the voxel by the same factor.

2.2.4. Simulation procedure.

To simulate vasculature dynamics in silico, a spherical tumour is padded with tissue with a uniform vessel density of sMVD_tissue. The avascular tumour is then allowed to grow as previously explained. Once the tumour reaches the diameter of 2 mm, the angiogenic switch is switched ‘on’ and vessel recruitment begins. The VEGF concentration for each voxel is calculated as mentioned in section 2.1. The voxel coupled new total and perfused vessel densities are calculated using equations (3) and (4) while also taking the volumetric change of the tumour voxel into account. Voxel specific oxygen tensions of the expanded tumour are recalculated and the non-tumour tissue is reinitialized to sMVD_tissue again. For a given simulation, sMVD_tissue is assumed to be constant and the vessel perfusion outside the tumour is assumed to be 100%. Once the new vessel density map is calculated, the vasculature is again simulated as determined by the regional vessel density. Table 1 includes the parameters used for vasculature simulation.

Table 1.

Model parameters.

Model parameter/literature-based constant	Symbol	Value (range)	Reference/comments
Vasculature simulation parameters:
Vascular pO2	p0	60 mm Hg
Capillary radius	rc	10 μm (2–100 μm)	(Konerding et al 1999)
Average vessel length	SL	Capillaries: 120 μm (diameter:length = 1:6)	(Less et al 1991)
Oxygen consumption rate	Qmax	0.5ml ${\mathrm{O}}_2^{-1}$ 100g−1 min−1 (0.2–4ml ${\mathrm{O}}_2^{-1}$ 100g−1 min−1	(Vaupel et al 1989)
Minimum voxel resolution required for simulation (capillary diameter)		20 μm
Theoretical tumour simulation (base case):
Tumour doubling time	Td	3.4 d	(Nöth et al 2004)
Surrounding tissue vessel density	sMVDtissue	3.5%	Variable tuned
Perfusion rate constant	PRC	0.05 d−1	Variable tuned
VEGF concentration for complete vessel dysfunctionality	VEGFmax	1000 pg cm−3	Variable tuned
Cell density		1 × 106 cells per voxel	(Steel 2002)
VEGF diffusion coefficient	DVEGF	100 μm2 s−1	(Mac Gabhann et al 2007)
Decay rate of VEGF	λVEGF	0.65 h−1	(Serini et al 2003)
Voxel resolution for tumour matrix		112 μm
Voxel resolution for vasculature matrix		60 μm
Application to in vivo data (parameters that have been altered from the base case):
Tumour doubling time	Td	11.3 d	(Foster et al 1997)
Vessel density at the tumour edge	sMVDtissue	1.5%	Variable tuned
Perfusion rate constant	PRC	0.028 d−1	Variable tuned
VEGF concentration for complete vessel dysfunctionality	VEGFmax	1000 pg cm−3	From theoretical scenario
Voxel resolution for tumour matrix		250 μm
Voxel resolution for vasculature matrix		100 μm

3. Materials and methods

3.1. Benchmarking model parameters.

The dynamics of tumour oxygenation is the result of the interplay between the vasculature and tumour growth. Nöth et al (2004) evaluated the dynamics of tumour oxygenation in a xenografted mouse tumour over its entire lifecycle using ¹⁹F-MRI by studying the longitudinal relaxation rate (1/T₁) of perfluorocarbon that was injected along with tumour cells. Since the reproduction of vasculature dynamics over the entire tumour growth lifecycle is the most challenging test, this experiment was chosen as the benchmark experiment. The model was applied to reproduce these trends of a growing tumour by tuning the following parameters: sMVD_tissue, PRC and VEGF_max.

In order to reproduce experimental results, tumour was simulated as a volume with uniform vessel density (reflecting pO₂) on a Euclidian computational matrix. Initial tumour volume depended on the experimentally observed doubling time and the timing when the angiogenic switch is ‘on’ (diameter 2 mm). The timing of this switch (angiogenic switch ‘on’) was taken as the time when the pO₂ of the experimental tumour was at the minimum. The voxel resolution of the matrix (112 μm) was chosen such that a significant initial volume could be assigned to the tumour on the computational matrix, as well as to accommodate its volumetric growth for the required number of days.

3.2. Sensitivity study

To characterize the impact of the model parameters on the tumour oxygenation, a sensitivity study of the model parameters was performed by varying each parameter individually by ±20%. The changes in peak tumour pO₂ (pO_2peak) and the time to peak tumour pO₂ (t_peak) were quantitatively evaluated. These parameters include:

Variation in tumour-associated parameters: T_d and cell line.

Variation in vasculature-associated parameters: PRC, sMVD_tissue and VEGF_max.

3.3. Microenvironmental characteristics

Tumour-vasculature growth impacts a variety of microenvironmental characteristics that are important indicators of the tumour condition. Mean oxygenation and perfusion values mask the spatial variation of these characteristics. Extremely hypoxic and underperfused regions might increase tumour aggressiveness or might hamper drug delivery. Spatio-temporal quantification of the vessel perfusion index (VPI), VEGF concentration as well as tumour pO₂ has been done and shown.

3.4. Model application

The model was applied to imaging data obtained from imaging a TRAMP mouse tumour. Mice were injected with [⁶⁴Cu]Cu-ATSM radiotracer and a PET scan was acquired, the duration of which was 60 min. Average uptake over this scan period was used as a surrogate of hypoxia distribution. Further scan details have been described elsewhere (Adhikarla and Jeraj 2012). The original PET data was then resampled to the desired simulation resolution chosen based on the feasibility of the computational calculations involved. Using an empirical relationship between Cu-ATSM SUV and tumour pO₂ (Lewis et al 1999), the hypoxia scan was then converted to a map of tumour pO₂ as has been shown earlier (Titz and Jeraj 2008, Adhikarla and Jeraj 2012). The doubling time of the TRAMP cell line was assumed to be of the same for each voxel.

To simulate tumour oxygenation dynamics, the following two vasculature parameters were tuned from the theoretical simulation case: perfusion rate constant (PRC) and the sMVD_tissue. For this purpose, the pO₂ at the tumour edge was assumed constant and the rate of change in the mean tumour oxygenation compared to the theoretical case, was normalized by the doubling time ratio of the imaged TRAMP and the xenograft tumours.

4. Results

4.1. Benchmarking model parameters

Figure 2 shows the oxygenation trends observed in the xenograft tumours (Nöth et al 2004); with an initial decrease in tumour oxygenation followed by a sharp rise and eventually followed by a slow decrease in oxygen tension. The model parameters (PRC, sMVD_tissue, VEGF_max) were tuned to reproduce these trends. The initial decrease in tumour pO₂ reflects tumour growth without vasculature recruitment. The sharp rise and eventual decrease in tumour pO₂ results from vasculature recruitment from surrounding tissue and the tumour growth rate outpacing the vasculature recruitment rate respectively. Variations between the simulated and the observed curve can be explained on the basis of variation in the doubling time of the tumour from its average value.

Figure 2.

Reproducing trends in tumour oxygenation. Variations between the simulated and the observed data can be explained on the basis of variation in the tumour doubling time from its average value. Simulation parameters: PRC = 0.05 d⁻¹, sMVD_tissue = 3.5%, VEGF_max = 1000 pg cm⁻³.

4.2. Sensitivity study

4.2.1. Variation in tumour-associated characteristics.

Tumour doubling time demonstrated the most significant impact on the time to peak pO₂ (t_peak) and the peak value of pO₂ (pO_2peak). 20% reduction in tumour doubling time reduces t_peak and pO_2peak by 27% and 31% respectively. An increase in doubling time demonstrates exactly opposite characteristics with reduced slope after the peak demonstrating the reduced rate at which hypoxia develops in the tumour. The tumour is thus significantly oxygenated even by day 26 (figure S2(a)). An increase of 20% in the doubling time increases t_peak by 34% and pO_2peak by 37%. Slower growing tumour also results in a delayed timing for the angiogenic switch. Tumour cell line impacts its pO₂ via the secretion rate of the angiogenic factors. Clone A cells secrete more VEGF (figure S1(a)) than the HCT-8 cells. This significantly hampers the recruitment of perfused vessels leading to 16% reduction in the t_peak and 37% reduction in pO_2peak.

4.2.2. Variation in the vasculature-associated characteristics.

Among the vasculature-associated parameters, the tissue vessel density (sMVD_tissue) had the biggest effect while VEGF_max had the least effect on the tumour oxygenation. An increase in the pO₂ of the surrounding tissue leads to an increase in the tumour pO₂ due to the availability of increased number of perfused vessels from which the tumour can recruit its blood flow. While a 20% decrease in sMVD_tissue results in 10% decrease in t_peak and 32% decrease in the pO_2peak, a 20% increase results in 8% increase in t_peak and 36% increase in pO_2peak. The effect of VEGF_max on both t_peak and pO_2peak was about 5%. A 20% decrease in VEGF_max decreases t_peak by 1% and pO_2peak by 6%. Corresponding 20% increase in VEGF_max increases t_peak by 3% and pO_2peak by 4%.

The impact of PRC on tumour oxygenation was comparable to sMVD_tissue with a 20% decrease in PRC decreasing t_peak by 14% and pO_2peak by 31%. A 20% increase in PRC increased t_peak by 3% and pO_2peak by 30%.

Figure 3 summarizes the results of the sensitivity study of model parameters. In general the changes in the peak values of the pO₂ are much greater than the changes in the timing of the peak pO₂. The timing of the peak pO₂ (figure 3(a)) is influenced more by the tumour-associated parameters rather than vasculature-associated parameters. In contrast the value of peak tumour oxygenation (figure 3(b)) was similarly influenced by both tumour- and vasculature-associated parameters.

Figure 3.

Sensitivity study of model parameters. (a) Impact on time to peak tumour pO₂ (tpeak). (b) Impact on peak tumour pO₂ (pO_2peak). Both tumour- and vasculature-associated parameters have similar impact on pO_2peak. Tumour-associated parameters however, have more impact on t_peak as compared to vasculature-associated parameters. Note: since the VEGF secretion rates of only two kinds of cell lines have been used, the box for the cell line does not represent the ±20% change that is typical for the other parameters.

4.3. Microenvironmental characteristics

Figure 4(a) shows the temporal changes in the percentage of vessels perfused (VPI). At the time when the vessels are first recruited (day 3), the growth rate of vasculature (sMVD_total) is higher than the recruitment rate of perfused vasculature (sMVD_perf) due to higher VEGF concentration. This results in a sharp decrease in the vessel perfusion index corresponding to the sharp increase in pO₂. The decrease in VEGF due to this vasculature recruitment results in a lower vasculature growth rate (of sMVD_total) in comparison to the perfused vasculature (sMVD_perf) growth rate. Hence there is an increase in the VPI. However, as the tumour grows, the VEGF concentration inside the tumour starts accumulating due to sheer volume of the tumour (figure 4(b)). This results in increased total vasculature recruitment and a corresponding decrease in the recruitment of the perfused vessels.

Figure 4.

Temporal changes in perfusion and angiogenic factor characteristics. (a) Vessel perfusion index. (b) Tumour angiogenic factor (VEGF) concentration.

Line profiles across the tumour quantitatively evaluating the values of voxel pO₂ and vessel perfusion index (VPI) are shown in figures 5(a) and (b) respectively. The decrease of both pO₂ and VPI towards the centre of the tumour can be seen.

Figure 5.

Line profiles across the tumour. The curves demonstrate the changes in (a) tumour oxygenation and (b) vessel perfusion index at different time points. The edge of the tumour is reflected in the graphs when the pO₂ is roughly 50 mmHg (reflecting sMVD_tissue) and the VPI is 100%.

4.4. Application to imaging data

4.4.1. Visualization of the tumour and vasculature growth.

Figure 6 demonstrates the application of the model to [⁶⁴Cu]Cu-ATSM PET imaging data from which the oxygen map was generated. Increased hypoxia and tumour volume can be readily seen with tumour growth. Similar results can be seen from the cross-sectional views across the tumour. Decreased vessel density can be seen in areas of low oxygen tension although the total number of vessels increases with time due to vasculature growth. Comparison of the vasculature-simulated hypoxia maps (figures 6(d)–(f)) and these hypoxia maps at the PET resolution (figures 6(g) – (i)) shows how significant local hypoxia clusters needs to be present for it to be localized well on the PET resolution scale.

Figure 6.

Volumetric and cross-sectional visualization of tumour growth and change in tumour hypoxia obtained from simulations. (a)–(c) Volumetric visualization of [⁶⁴Cu] Cu-ATSM hypoxia PET maps along with the simulated vasculature on day 0, 7 and 14. (d)–(f) Cross-sections of the vasculature and the simulated hypoxia maps from the vasculature on day 0, 7 and 14. (g)–(i) Cross-sections of the vasculature and the simulated hypoxia maps (resampled to PET resolution) from the vasculature on day 0, 7 and 14. An increase in tumour volume and hypoxia along with reduced vessel density is immediately visible.

5. Discussion

The developed angiogenesis model successfully reproduces the temporal changes in tumour oxygenation observed in literature. Keeping personalized therapy in mind, the model is able to simulate the growth of tumour and its vasculature using patient-specific molecular imaging data. A key advantage of the model lies in the incorporation of microscopic vasculature along with the imaging data. This provides the model the capability to bridge the gap between the microscopic and the macroscopic (PET imaging) scales and study the oxygenation transients at both scales. Sensitivity studies of the model parameters show how drastically the tumour oxygenation can vary with the cell line specific parameters such as tumour doubling time and VEGF secretion characteristics, as well as the surrounding tissue characteristics. The impact of the surrounding tissue (sMVD_tissue) has a huge effect on the fast-growing xenograft tumour. However, human tumours typically grow slower and thus the vessel density in the immediate vicinity of an imaged voxel will have a significant impact on the vasculature growth rate. This vessel density reflected by tumour hypoxia makes the imaged hypoxia map a very strong candidate for the required imaging input for the model. While VEGF_max as a parameter had the least influence on evaluated peak tumour pO₂, its importance increases with tumour size as the concentration of the angiogenic factor rapidly increases within the tumour due to accumulation within it.

Different trends in tumour oxygenation (figure 2) can be attributed to different stages of tumour growth. The initial decrease is attributed to the tumour expansion and a corresponding decrease in the vessel density. At this point the tumour is still in its avascular stage. The sudden increase in the pO₂ after this minimum is related to vasculature recruitment from the surrounding tissue. At a later time point the tumour surface area exposed to perfused vessels relative to its volume decreases significantly. Meanwhile the vessels inside the tumour are significantly under-perfused due to increased VEGF concentration in the microenvironment which hampers the formation of a well networked and stable vasculature. This results in decrease in mean tumour pO₂ with time and the formation of a hypoxic core at a later stage.

Changes in tumour VEGF concentration are typically attributed to changes in its hypoxia status. In line with this assumption, figure 4(b) demonstrates VEGF concentration trends that are exactly opposite of the trends observed in tumour oxygenation (figure 2). The initial spike in VEGF concentration is small compared to the concentration when the tumour develops a hypoxic core. While the magnitude of this initial spike depends on the TAF secretion characteristics of the cell line, another factor that is of importance here is the tumour size. For a smaller tumour, with a large surface area to volume ratio, more TAF is able to diffuse into the surroundings than remain within the tumour leading to lower concentration of tumour TAF. The concentration of TAF might also be dependent on non-hypoxia related factors. Incipient angiogenesis independent of the tumour hypoxia status was observed by Hendriksen et al (2009). Thus owing to contributions from non-HIF-1α mediated pathways, the amount of angiogenic factor in the tumour microenvironment might be much higher than calculated by the model.

In the presented model, vasculature endothelial growth factor (VEGF) has been used as the sole angiogenic factor in question. Different tumour cell lines secrete both different concentrations as well as different types of angiogenic factors (Hasina et al 2008). The site of the tumour and the endothelial cells present are also potential variables affecting the response of vasculature to the angiogenic factor in question. Work by Ferrara and Henzel (1989) from which the vasculature growth ratio in response to VEGF (figure S1(b)) has been used, demonstrates how the growth rate of endothelial cells obtained from different tissue types can vary by a factor of four. Such changes in parameters influence the tumour oxygenation values significantly. Given how critical tumour oxygenation is for therapies (Nordsmark et al 2001) these factors show how important it is for models to incorporate tumour-specific data. These are however parameters that can be easily incorporated into the model by considering the tissue-specific vasculature recruitment characteristics of each type of angiogenic factor separately.

Simulations addressing large amounts of data often run into computational limitations and the current work is no exception. The tumour represented by a MATLAB matrix has to be small enough to ensure the computational efficiency of the tumour growth simulations. Thus, simulation of the tumour above a certain resolution is computationally challenging. Also, a sufficient number of tumour voxels need to be initialized such that the tumour growth algorithm registers a sufficient increase in the number of voxels. The simulation resolution of the tumour matrix (typically on the order of a few hundred micro-meters) is governed by these two conditions. The vasculature matrix could be simulated at a finer resolution than the tumour matrix and was done as such. However, it still fell short of the required minimum voxel resolution 20 μm (capillary diameter is assumed to be 20 μm), which would require a more powerful computational platform.

A potential problem that can arise while using voxel coupling to simulate vasculature recruitment from neighbouring tissue relates to the presence of hypoxia at the very edge of the tumour. This hypoxia at the tumour boundary decreased due to immediate access to well perfused vasculature from outside the tumour. This may or may not reflect the actual hypoxia progression. Such potential problems can arise either due to use of a very aggressive tumour segmentation technique or if the tumour is surrounded by some anatomical constraint like the bone. Additionally, the tumour segmentation based solely on hypoxia imaging might not reflect the actual extent of the tumour. In such cases either the segmentation approach should be rechecked and/or an imaging-based value of sMVD_tissue should be used.

In the presented model, the effect of tumour angiogenic factor on the vasculature is twofaced. On one hand, it increases the rate at which vasculature (sMVD_total) is recruited, and on the other it increases the permeability of the vessels leading to reduction vessel perfusion (sMVD_perf). A logarithmic dependence of the perfused vessel recruitment on the microenvironmental angiogenic factor concentration has been proposed. The advantage of this formulation is that it condenses the effect of vessel immaturity on the tumour oxygenation without incorporating additional parameters. Current imaging-based models do not incorporate vessel maturation in their framework. In the simulations performed, limited growth of vasculature (sMVD_total) has not had an impact on the oxygen dynamics since the total vasculature growth rate always being greater than the perfused vasculature recruitment rate. A more in depth modelling scheme explicitly modelling vessel permeability and maturation would be required to couple the total vessel density with the perfused vessel density.

While the parameters that impact the generation of vessel structure and its density are discussed in detail in a previous work (Adhikarla and Jeraj 2012), the presented model is calibrated (by tuning PRC, VEGF_max and sMVD_tissue) such that the rate of decrease of mean tumour pO₂ depends on the cell line doubling time and the rate of pO₂ decrease observed by Noth and colleagues in their experiment. However, further benchmarking the model against longitudinal imaging data will lend the model capability to successfully predict spatiotemporal changes in tumour hypoxia on a case to case basis. Apart from hypoxia and proliferation PET data that determine tumour oxygenation and growth rate, additional imaging data can be used to parameterize the model. Imaging α_vβ₃ integrins (Haubner et al 2005) using [¹⁸F] Galacto-RGD can provide direct input about the level of angiogenic stimulus. This can help quantify model vessel recruitment parameter PRC that might change in between tumours and cell lines. Further validation of vascular parameters PRC and VEGF_max and a comprehensive picture determining the amount of vasculature and its quality can be obtained by using other imaging-based metrics such as blood volume, perfusion and vessel permeability. These measures can be derived from kinetic analysis of dynamic PET imaging or MRI data.

Anti-angiogenic drugs targeting tumour angiogenesis have faced numerous challenges owing to the complexity of angiogenesis and very few anti-angiogenic drugs have received FDA approval (Samant and Shevde 2011). Normalization of vasculature leading to efficient vascular network and better blood flow has been postulated as a consequence of anti-angiogenic drug treatment (Jain 2005). However, evidence of normalization has been controversial, its efficacy moderate and our understanding shallow (Bruce et al 2014). The time scales of normalization are still under investigation. The proposed model can prove to be extremely beneficial in analyzing the maturation time frames affecting vascular perfusion. Evaluation of vessel maturation and perfusion changes to anti-angiogenic therapy and relating them to the doubling time of the tumour is a future direction of model application. Such modelling scheme might give possible hints on extrapolating the results to the timescales of human patients (order of months) and help in classifying patients who would benefit from the anti-angiogenic therapy. While modelling such therapies, it should not be forgotten that there are multiple angiogenic factors in play and the tumour might use other escape angiogenic pathways for vessel recruitment.

6. Conclusion

Incorporation of imaging data into tumour modelling is a powerful technique for studying the dynamics of tumour progression; eventually these models will find application in simulating tumour response to different therapies and for optimizing them. In the presented work, molecular imaging data related to tumour hypoxia has been used to simulate transients in tumour hypoxia and its growth. Quantification of the impact of tumour- and vasculature-associated parameters on tumour oxygenation demonstrates how cell line and imaging-based tumour-specific measures have a profound impact on tumour hypoxia. Further imaging data particularly related to angiogenesis will prove to be an important input to further the model. Simulating anti-angiogenic therapies in silico and analysing the reasons for their potential success or failure is a key application of the model.