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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

Typical phase planes. (A) Typical phase plane with (non-zero) clines for case (a), here for δ/η0 < 1, showing a steady state which is a stable spiral. (B) Typical phase plane with (non-zero) clines for case (b), δ/η0 > 1. The steady state is unstable and the shown trajectory evolves into a stable limit cycle. We remark that compared to (A) we have varied δ, ds and σp (ds and σp were chosen so that the co-existence steady state would be unstable). Key: green dashed-dotted lines: non-zero p-clines; solid red lines: non-zero v-clines; open circle: linearly unstable steady state; closed circle: linearly stable steady state; blue bold lines with solid arrow: trajectories for the initial conditions (p(0), v(0)) = (0.01, 1) (A) and (p(0), v(0)) = (0.5, 0.6) (B). Parameter values: η0 = 0.02, dp = 0.8, ds = 0.9, δ = 0.01, sβ = 0.1 and σp = 0.08 (A) and η0 = 0.02, dp = 0.8, ds = 0.4, δ = 0.03, sβ = 0.1 and σp = 0.01 (B).
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Figure 1: Typical phase planes. (A) Typical phase plane with (non-zero) clines for case (a), here for δ/η0 < 1, showing a steady state which is a stable spiral. (B) Typical phase plane with (non-zero) clines for case (b), δ/η0 > 1. The steady state is unstable and the shown trajectory evolves into a stable limit cycle. We remark that compared to (A) we have varied δ, ds and σp (ds and σp were chosen so that the co-existence steady state would be unstable). Key: green dashed-dotted lines: non-zero p-clines; solid red lines: non-zero v-clines; open circle: linearly unstable steady state; closed circle: linearly stable steady state; blue bold lines with solid arrow: trajectories for the initial conditions (p(0), v(0)) = (0.01, 1) (A) and (p(0), v(0)) = (0.5, 0.6) (B). Parameter values: η0 = 0.02, dp = 0.8, ds = 0.9, δ = 0.01, sβ = 0.1 and σp = 0.08 (A) and η0 = 0.02, dp = 0.8, ds = 0.4, δ = 0.03, sβ = 0.1 and σp = 0.01 (B).

Mentions: It is straightforward to show that g(v) is positive only if 0 ≤ v ≤ 1 (see Figure 1 where the v-clines are indicated by solid red lines). Therefore, for the p- and v-clines to intersect and a co-existence steady state to exist, the zero of f (v), fzero, must fulfill(24)


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

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

Typical phase planes. (A) Typical phase plane with (non-zero) clines for case (a), here for δ/η0 < 1, showing a steady state which is a stable spiral. (B) Typical phase plane with (non-zero) clines for case (b), δ/η0 > 1. The steady state is unstable and the shown trajectory evolves into a stable limit cycle. We remark that compared to (A) we have varied δ, ds and σp (ds and σp were chosen so that the co-existence steady state would be unstable). Key: green dashed-dotted lines: non-zero p-clines; solid red lines: non-zero v-clines; open circle: linearly unstable steady state; closed circle: linearly stable steady state; blue bold lines with solid arrow: trajectories for the initial conditions (p(0), v(0)) = (0.01, 1) (A) and (p(0), v(0)) = (0.5, 0.6) (B). Parameter values: η0 = 0.02, dp = 0.8, ds = 0.9, δ = 0.01, sβ = 0.1 and σp = 0.08 (A) and η0 = 0.02, dp = 0.8, ds = 0.4, δ = 0.03, sβ = 0.1 and σp = 0.01 (B).
© Copyright Policy - open-access
Related In: Results  -  Collection

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Figure 1: Typical phase planes. (A) Typical phase plane with (non-zero) clines for case (a), here for δ/η0 < 1, showing a steady state which is a stable spiral. (B) Typical phase plane with (non-zero) clines for case (b), δ/η0 > 1. The steady state is unstable and the shown trajectory evolves into a stable limit cycle. We remark that compared to (A) we have varied δ, ds and σp (ds and σp were chosen so that the co-existence steady state would be unstable). Key: green dashed-dotted lines: non-zero p-clines; solid red lines: non-zero v-clines; open circle: linearly unstable steady state; closed circle: linearly stable steady state; blue bold lines with solid arrow: trajectories for the initial conditions (p(0), v(0)) = (0.01, 1) (A) and (p(0), v(0)) = (0.5, 0.6) (B). Parameter values: η0 = 0.02, dp = 0.8, ds = 0.9, δ = 0.01, sβ = 0.1 and σp = 0.08 (A) and η0 = 0.02, dp = 0.8, ds = 0.4, δ = 0.03, sβ = 0.1 and σp = 0.01 (B).
Mentions: It is straightforward to show that g(v) is positive only if 0 ≤ v ≤ 1 (see Figure 1 where the v-clines are indicated by solid red lines). Therefore, for the p- and v-clines to intersect and a co-existence steady state to exist, the zero of f (v), fzero, must fulfill(24)

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