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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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Solution profiles in the case of unstable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when irregular 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 ((p, v) = (0.16, 0.3)), irregular spatio-temporal oscillations develop. Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2,  = 0.2761 (using (A-11) in the Appendix (additional file 1)), δ = 1.0, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are unstable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3500, 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 5: Solution profiles in the case of unstable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when irregular 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 ((p, v) = (0.16, 0.3)), irregular spatio-temporal oscillations develop. Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2, = 0.2761 (using (A-11) in the Appendix (additional file 1)), δ = 1.0, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are unstable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3500, 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 5 the waves that develop in the wake of the invading front are unstable, and irregular dynamics are observed. Since the system is close to the Hopf bifurcation, i.e. it is weakly unstable, the oscillations that develop behind the invading front are initially regular and only become irregular at later times (similar behaviour was observed in [17]). Once the cancer cells have colonised the entire tissue, irregular spatio-temporal oscillations persist.


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 unstable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when irregular 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 ((p, v) = (0.16, 0.3)), irregular spatio-temporal oscillations develop. Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2,  = 0.2761 (using (A-11) in the Appendix (additional file 1)), δ = 1.0, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are unstable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3500, 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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Figure 5: Solution profiles in the case of unstable wave trains. Series of profiles (solutions of equations (25)-(26)) showing how the tumour cell density evolves when irregular 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 ((p, v) = (0.16, 0.3)), irregular spatio-temporal oscillations develop. Parameter values: η0 = 0.5, dp = 0.6, ds = 0.2, = 0.2761 (using (A-11) in the Appendix (additional file 1)), δ = 1.0, Dv = 1, sβ = 0.4 and σp = 0. For these parameter values the waves are unstable (see Figure 3). For the numerical simulations we fix Δx = 1/3 and L = 3500, 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 5 the waves that develop in the wake of the invading front are unstable, and irregular dynamics are observed. Since the system is close to the Hopf bifurcation, i.e. it is weakly unstable, the oscillations that develop behind the invading front are initially regular and only become irregular at later times (similar behaviour was observed in [17]). Once the cancer cells have colonised the entire tissue, irregular spatio-temporal oscillations persist.

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