Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics

Abstract

Background: One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity.

Results: Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus. We show that several phenomena that are absent in homogeneous models, such as cancer recurrence, tumor dormancy, and others, appear in heterogeneous setting. We also demonstrate that, within the applied modeling framework, to overcome the adverse effect of tumor cell heterogeneity on the outcome of cancer treatment, a heterogeneous population of an oncolytic virus must be used. Heterogeneity in parameters of the model, such as tumor cell susceptibility to virus infection and the ability of an oncolytic virus to infect tumor cells, can lead to complex, irregular evolution of the tumor. Thus, quasi-chaotic behavior of the tumor-virus system can be caused not only by random perturbations but also by the heterogeneity of the tumor and the virus.

Conclusion: The modeling approach described here reveals the importance of tumor cell and virus heterogeneity for the outcome of cancer therapy. It should be straightforward to apply these techniques to mathematical modeling of other types of anticancer therapy.

Reviewers: Leonid Hanin (nominated by Arcady Mushegian), Natalia Komarova (nominated by Orly Alter), and David Krakauer.

Figure 1

Phase-parameter portrait of system (2) given as a cut of the positive parameter space (γ, β, δ) for an arbitrary fixed value of 0 <γ < 1 (a) and 1 <γ (b). The boundaries of domains are α₁ = {(δ, β, γ): δ - β = 0},α₂ = {(δ, β, γ): γ -δ = 0}, α₃ = {(δ, β, γ): γβ -δ = 0}, and α₄ = {(δ, β, γ): γ - δ - 1 + β = 0}.

Figure 2

Solutions of system (5)–(6) with gamma distributed parameter β on [1.5, ∞). Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green and black, respectively. The initial conditions X(0) = 0.5, Y(0) = 0.1, parameter values γ = 1, δ = 2. The initial mean of distribution E_β (0) = 2.5, the initial variances 0.06 (a), 0.1(b), 0.3(c), 0.4(d).

Figure 3

E_β (t) versus time for the cases presented in Fig. 2. The curves from top to bottom correspond to panels (a)–(d) in Fig. 2.

Figure 4

Solutions of system (5)–(6) with gamma distributed parameter β on [0.1, ∞). Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green, and black, respectively. The initial conditions are X(0) = 0.5, Y(0) = 0.1, parameter values γ = 0.5, δ = 0.3. The initial mean of distribution E_β (0) = 3.5, the initial variances 1.5 (a) and 0.5(b).

Figure 5

(a) Solutions of system (9)–(10) with gamma distributed parameters β on [1, ∞) and δ on [1, ∞). Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green and black, respectively. The initial means of the distributions are E_β (0) = 11 and E_δ (0) = 10, initial variances are σβ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8NSdigabaGaeGOmaidaaaaa@3131@ (0) = 8 and σδ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8hTdqgabaGaeGOmaidaaaaa@3135@ (0) = 0.5. The initial conditions are X(0) = 0.5, Y(0) = 0.1, and γ = 1. (b) The parametric curve (E_β (t), E_δ (t)) in the parameter space (compare with Fig. 1b)

Figure 6

(a) Solutions of system (9)–(10) with gamma-distributed parameters β on [2, ∞) and δ on [0.9, ∞). Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green and black, respectively. The initial means of distributions are E_β (0) = 11 and E_δ (0) = 10, initial variances are σβ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8NSdigabaGaeGOmaidaaaaa@3131@ (0) = 8 and σδ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8hTdqgabaGaeGOmaidaaaaa@3135@ (0) = 0.5. The initial conditions X(0) = 0.5, Y(0) = 0.1 and γ = 0.7. (b) The parametric curve (E_β (t), E_δ (t)) in the parameter space

Figure 7

Solutions of system (13)–(14) with both uninfected cell specific and infected cell specific distributions of transmission coefficient β. β₁ is gamma-distributed on [0.6, ∞) and β₂ is beta-distributed on [0,2.5]. Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green and black, respectively. The initial means of distributions are Eβ1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWgaaWcbaacciGae8NSdi2aaSbaaWqaaiabigdaXaqabaaaleqaaaaa@30BB@ (0) = 2.9 and Eβ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWgaaWcbaacciGae8NSdi2aaSbaaWqaaiabikdaYaqabaaaleqaaaaa@30BD@ (0) = 0.6, initial variances are σβ12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8NSdi2aaSbaaWqaaiabigdaXaqabaaaleaacqaIYaGmaaaaaa@3259@ (0) = 1.9 and σβ22 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8NSdi2aaSbaaWqaaiabikdaYaqabaaaleaacqaIYaGmaaaaaa@325B@ (0) = 0.12 (a) σβ22 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGae8NSdi2aaSbaaWqaaiabikdaYaqabaaaleaacqaIYaGmaaaaaa@325B@ (0) = 0.11 (b). In panels (c) and (d), the mean parameter values Eβ1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWgaaWcbaacciGae8NSdi2aaSbaaWqaaiabigdaXaqabaaaleqaaaaa@30BB@ (t), Eβ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWgaaWcbaacciGae8NSdi2aaSbaaWqaaiabikdaYaqabaaaleqaaaaa@30BD@ (t) and E(t) = Eβ1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWgaaWcbaacciGae8NSdi2aaSbaaWqaaiabigdaXaqabaaaleqaaaaa@30BB@ (t) Eβ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWgaaWcbaacciGae8NSdi2aaSbaaWqaaiabikdaYaqabaaaleqaaaaa@30BD@ (t) are shown for cases (a) and (b), respectively