Limits...
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.

Show MeSH

Related in: MedlinePlus

Partition of (δ, dp)-space based on stability of wave trains. Figure showing where in (δ, dp)-space (as determined by (A-7) in the Appendix (additional file 1)) the wake behind the invading wave of tumour cells is stable (shaded region) and where it is unstable (white region). The stability analysis requires ds ≈ , where  denotes the value of ds at which the Hopf bifurcation occurs; thus in our (δ, dp)-space we mark by solid lines contours where  (as calculated from (A-11) in the Appendix (additional file 1)) is constant (corresponding values are labelled on the contours). For a case of stability ((δ, dp) = (0.65, 0.6) and ds ≈ 0.2) tumour cell invasion into the spatially homogeneous vessel-only steady state results in regular spatio-temporal oscillations (see Figure 4), while in the case of instability ((δ, dp) = (1.0, 0.6) and ds ≈ 0.2), irregular spatio-temporal oscillations develop (see Figure 5). Key: shaded region: wave stability; white region: wave instability; solid lines: lines where  is constant. Parameter values: η0 = 0.5, sβ = 0.4 and σp = 0.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC2877015&req=5

Figure 3: Partition of (δ, dp)-space based on stability of wave trains. Figure showing where in (δ, dp)-space (as determined by (A-7) in the Appendix (additional file 1)) the wake behind the invading wave of tumour cells is stable (shaded region) and where it is unstable (white region). The stability analysis requires ds ≈ , where denotes the value of ds at which the Hopf bifurcation occurs; thus in our (δ, dp)-space we mark by solid lines contours where (as calculated from (A-11) in the Appendix (additional file 1)) is constant (corresponding values are labelled on the contours). For a case of stability ((δ, dp) = (0.65, 0.6) and ds ≈ 0.2) tumour cell invasion into the spatially homogeneous vessel-only steady state results in regular spatio-temporal oscillations (see Figure 4), while in the case of instability ((δ, dp) = (1.0, 0.6) and ds ≈ 0.2), irregular spatio-temporal oscillations develop (see Figure 5). Key: shaded region: wave stability; white region: wave instability; solid lines: lines where is constant. Parameter values: η0 = 0.5, sβ = 0.4 and σp = 0.

Mentions: Following [17,27] (see also [28]) we analyse equations (25)-(26) (here with σp = 0 and Dv = 1) by using normal form analysis to approximate it by one of λ-ω type near the Hopf bifurcation. The result of our analysis (for details of the analysis see section B in the Appendix (additional file 1)) is a prediction about the stability of the periodic waves behind the invading front. By keeping all other parameters fixed, and also ensuring that the system is close to Hopf bifurcation, it is possible to show that the stability of a wave train depends solely on the maximum rate of vessel occlusion, δ, and the tumour apoptosis rate, dp. In Figure 3, we have partitioned (δ, dp)-space into different regions according to whether the wave behind the invading front is stable (in the shaded region) or unstable (in the white region). We remark that once the other parameters have been fixed, the value of ds, the oxygen consumption rate of other cellular components, determines whether the system is near Hopf bifurcation, and thus whether the stability prediction is reliable.


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

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

Partition of (δ, dp)-space based on stability of wave trains. Figure showing where in (δ, dp)-space (as determined by (A-7) in the Appendix (additional file 1)) the wake behind the invading wave of tumour cells is stable (shaded region) and where it is unstable (white region). The stability analysis requires ds ≈ , where  denotes the value of ds at which the Hopf bifurcation occurs; thus in our (δ, dp)-space we mark by solid lines contours where  (as calculated from (A-11) in the Appendix (additional file 1)) is constant (corresponding values are labelled on the contours). For a case of stability ((δ, dp) = (0.65, 0.6) and ds ≈ 0.2) tumour cell invasion into the spatially homogeneous vessel-only steady state results in regular spatio-temporal oscillations (see Figure 4), while in the case of instability ((δ, dp) = (1.0, 0.6) and ds ≈ 0.2), irregular spatio-temporal oscillations develop (see Figure 5). Key: shaded region: wave stability; white region: wave instability; solid lines: lines where  is constant. Parameter values: η0 = 0.5, sβ = 0.4 and σp = 0.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 3: Partition of (δ, dp)-space based on stability of wave trains. Figure showing where in (δ, dp)-space (as determined by (A-7) in the Appendix (additional file 1)) the wake behind the invading wave of tumour cells is stable (shaded region) and where it is unstable (white region). The stability analysis requires ds ≈ , where denotes the value of ds at which the Hopf bifurcation occurs; thus in our (δ, dp)-space we mark by solid lines contours where (as calculated from (A-11) in the Appendix (additional file 1)) is constant (corresponding values are labelled on the contours). For a case of stability ((δ, dp) = (0.65, 0.6) and ds ≈ 0.2) tumour cell invasion into the spatially homogeneous vessel-only steady state results in regular spatio-temporal oscillations (see Figure 4), while in the case of instability ((δ, dp) = (1.0, 0.6) and ds ≈ 0.2), irregular spatio-temporal oscillations develop (see Figure 5). Key: shaded region: wave stability; white region: wave instability; solid lines: lines where is constant. Parameter values: η0 = 0.5, sβ = 0.4 and σp = 0.
Mentions: Following [17,27] (see also [28]) we analyse equations (25)-(26) (here with σp = 0 and Dv = 1) by using normal form analysis to approximate it by one of λ-ω type near the Hopf bifurcation. The result of our analysis (for details of the analysis see section B in the Appendix (additional file 1)) is a prediction about the stability of the periodic waves behind the invading front. By keeping all other parameters fixed, and also ensuring that the system is close to Hopf bifurcation, it is possible to show that the stability of a wave train depends solely on the maximum rate of vessel occlusion, δ, and the tumour apoptosis rate, dp. In Figure 3, we have partitioned (δ, dp)-space into different regions according to whether the wave behind the invading front is stable (in the shaded region) or unstable (in the white region). We remark that once the other parameters have been fixed, the value of ds, the oxygen consumption rate of other cellular components, determines whether the system is near Hopf bifurcation, and thus whether the stability prediction is reliable.

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