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Oscillatory dynamics in a model of vascular tumour growth--implications for chemotherapy.

Stamper IJ, Owen MR, Maini PK, Byrne HM - Biol. Direct (2010)

Bottom Line: In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth.Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics.This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel.

View Article: PubMed Central - HTML - PubMed

Affiliation: Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK.

ABSTRACT

Background: Investigations of solid tumours suggest that vessel occlusion may occur when increased pressure from the tumour mass is exerted on the vessel walls. Since immature vessels are frequently found in tumours and may be particularly sensitive, such occlusion may impair tumour blood flow and have a negative impact on therapeutic outcome. In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth.

Results: By analysing a spatially uniform submodel, we identify regions of parameter space in which the combination of tumour cell proliferation and vessel occlusion give rise to sustained temporal oscillations in the tumour cell population and in the vessel density. Alternatively, if the vessels are assumed to be less prone to collapse, stable steady state solutions are observed. When spatial effects are considered, the pattern of tumour invasion depends on the dynamics of the spatially uniform submodel. If the submodel predicts a stable steady state, then steady travelling waves are observed in the full model, and the system evolves to the same stable steady state behind the invading front. When the submodel yields oscillatory behaviour, the full model produces periodic travelling waves. The stability of the waves (which can be predicted by approximating the system as one of lambda-omega type) dictates whether the waves develop into regular or irregular spatio-temporal oscillations. Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics. In particular, if the dynamics are oscillatory, then therapeutic efficacy is difficult to assess since the fluctuations in the size of the tumour cell population are enhanced, compared to untreated controls.

Conclusions: We have developed a mathematical model of vascular tumour growth formulated as a system of partial differential equations (PDEs). Employing a combination of numerical and analytical techniques, we demonstrate how the spatio-temporal dynamics of the untreated tumour may influence its response to chemotherapy.

Reviewers: This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel.

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Related in: MedlinePlus

Solution profiles in the case of stable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when regular waves develop behind the invading tumour front (similar profiles of the vessel density not presented). Panel A (B) depicts the behaviour at t = 2000 (t = 4000). Behind the invading front, which travels with constant shape and connects the tumour-free steady state with the unstable co-existence steady state, regular spatio-temporal oscillations develop. Since the invading front moves faster than the evolving regular wave train, a large portion of the domain is at the unstable steady state ((p, v) = (0.35, 0.3)). Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2,  = 0.2163 (using (A-11) in the Appendix (additional file 1)), δ = 0.65, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are stable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3000 with (p(0, 0), v(0, 0)) = (0.01, 1) and (p(x, 0), v(x, 0)) = (0, 1) for x: 0 <x ≤ 1.
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Figure 4: Solution profiles in the case of stable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when regular waves develop behind the invading tumour front (similar profiles of the vessel density not presented). Panel A (B) depicts the behaviour at t = 2000 (t = 4000). Behind the invading front, which travels with constant shape and connects the tumour-free steady state with the unstable co-existence steady state, regular spatio-temporal oscillations develop. Since the invading front moves faster than the evolving regular wave train, a large portion of the domain is at the unstable steady state ((p, v) = (0.35, 0.3)). Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2, = 0.2163 (using (A-11) in the Appendix (additional file 1)), δ = 0.65, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are stable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3000 with (p(0, 0), v(0, 0)) = (0.01, 1) and (p(x, 0), v(x, 0)) = (0, 1) for x: 0 <x ≤ 1.

Mentions: In Figure 4, we present a simulation for which the λ-ω analysis, as presented in Figure 3, predicts stable periodic travelling waves behind the invading front. We note that damped oscillations connect the tumour-free steady state ahead of the invading front with the co-existence steady state, a solution which is qualitatively similar to those presented in [17]. Since the co-existence steady state is unstable, oscillations develop; in this case these are regular spatio-temporal waves because the periodic wave is stable. By comparing panels A and B in Figure 4, we see that the portion of the spatial domain in which the system is at the co-existence steady state increases over time; thus the spatial domain appears to become transiently spatially uniform. As established for a predator-prey system in [29], this occurs because the speed of the interface between the "plateau" (where the system is at steady state) and the region of regular oscillations is less than the speed of the invading front. We remark that for large times, after the invading front has reached the right-hand boundary, regular temporal oscillations persist at fixed spatial positions. After the invading front has reached the right-hand boundary it is reflected (due to the no-flux boundary conditions), and regular waves, travelling in the opposite direction to that of the original front, develop. The no-flux conditions imply that the boundaries are impermeable to the tumour cells, an assumption whose validity depends on the tissue of interest. For example, while cancer cells may be able to colonise a softer tissue, a more rigid material, such as bone, may halt cancer invasion.


Oscillatory dynamics in a model of vascular tumour growth--implications for chemotherapy.

Stamper IJ, Owen MR, Maini PK, Byrne HM - Biol. Direct (2010)

Solution profiles in the case of stable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when regular waves develop behind the invading tumour front (similar profiles of the vessel density not presented). Panel A (B) depicts the behaviour at t = 2000 (t = 4000). Behind the invading front, which travels with constant shape and connects the tumour-free steady state with the unstable co-existence steady state, regular spatio-temporal oscillations develop. Since the invading front moves faster than the evolving regular wave train, a large portion of the domain is at the unstable steady state ((p, v) = (0.35, 0.3)). Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2,  = 0.2163 (using (A-11) in the Appendix (additional file 1)), δ = 0.65, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are stable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3000 with (p(0, 0), v(0, 0)) = (0.01, 1) and (p(x, 0), v(x, 0)) = (0, 1) for x: 0 <x ≤ 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC2877015&req=5

Figure 4: Solution profiles in the case of stable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when regular waves develop behind the invading tumour front (similar profiles of the vessel density not presented). Panel A (B) depicts the behaviour at t = 2000 (t = 4000). Behind the invading front, which travels with constant shape and connects the tumour-free steady state with the unstable co-existence steady state, regular spatio-temporal oscillations develop. Since the invading front moves faster than the evolving regular wave train, a large portion of the domain is at the unstable steady state ((p, v) = (0.35, 0.3)). Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2, = 0.2163 (using (A-11) in the Appendix (additional file 1)), δ = 0.65, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are stable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3000 with (p(0, 0), v(0, 0)) = (0.01, 1) and (p(x, 0), v(x, 0)) = (0, 1) for x: 0 <x ≤ 1.
Mentions: In Figure 4, we present a simulation for which the λ-ω analysis, as presented in Figure 3, predicts stable periodic travelling waves behind the invading front. We note that damped oscillations connect the tumour-free steady state ahead of the invading front with the co-existence steady state, a solution which is qualitatively similar to those presented in [17]. Since the co-existence steady state is unstable, oscillations develop; in this case these are regular spatio-temporal waves because the periodic wave is stable. By comparing panels A and B in Figure 4, we see that the portion of the spatial domain in which the system is at the co-existence steady state increases over time; thus the spatial domain appears to become transiently spatially uniform. As established for a predator-prey system in [29], this occurs because the speed of the interface between the "plateau" (where the system is at steady state) and the region of regular oscillations is less than the speed of the invading front. We remark that for large times, after the invading front has reached the right-hand boundary, regular temporal oscillations persist at fixed spatial positions. After the invading front has reached the right-hand boundary it is reflected (due to the no-flux boundary conditions), and regular waves, travelling in the opposite direction to that of the original front, develop. The no-flux conditions imply that the boundaries are impermeable to the tumour cells, an assumption whose validity depends on the tissue of interest. For example, while cancer cells may be able to colonise a softer tissue, a more rigid material, such as bone, may halt cancer invasion.

Bottom Line: In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth.Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics.This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel.

View Article: PubMed Central - HTML - PubMed

Affiliation: Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK.

ABSTRACT

Background: Investigations of solid tumours suggest that vessel occlusion may occur when increased pressure from the tumour mass is exerted on the vessel walls. Since immature vessels are frequently found in tumours and may be particularly sensitive, such occlusion may impair tumour blood flow and have a negative impact on therapeutic outcome. In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth.

Results: By analysing a spatially uniform submodel, we identify regions of parameter space in which the combination of tumour cell proliferation and vessel occlusion give rise to sustained temporal oscillations in the tumour cell population and in the vessel density. Alternatively, if the vessels are assumed to be less prone to collapse, stable steady state solutions are observed. When spatial effects are considered, the pattern of tumour invasion depends on the dynamics of the spatially uniform submodel. If the submodel predicts a stable steady state, then steady travelling waves are observed in the full model, and the system evolves to the same stable steady state behind the invading front. When the submodel yields oscillatory behaviour, the full model produces periodic travelling waves. The stability of the waves (which can be predicted by approximating the system as one of lambda-omega type) dictates whether the waves develop into regular or irregular spatio-temporal oscillations. Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics. In particular, if the dynamics are oscillatory, then therapeutic efficacy is difficult to assess since the fluctuations in the size of the tumour cell population are enhanced, compared to untreated controls.

Conclusions: We have developed a mathematical model of vascular tumour growth formulated as a system of partial differential equations (PDEs). Employing a combination of numerical and analytical techniques, we demonstrate how the spatio-temporal dynamics of the untreated tumour may influence its response to chemotherapy.

Reviewers: This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel.

Show MeSH
Related in: MedlinePlus