The role of viral dynamics and infectivity in models of oncolytic virotherapy for tumours with different motility

TL;DR

The results show that the ability of viruses to infect tumours seems, in certain cases, more important to a final positive outcome than tumours'motility or even reproducibility.

Abstract

The use of ad-hoc engineered viruses in the fight against tumours is one of the greatest ideas in cancer therapeutics within the last three decades. Together with other strategies such as immunotherapies, nanoparticles and adjunct therapies, the use of viral vectors in clinical trials and in the clinics has been and is still widely studied and pursued. The ability of those vectors to infiltrate and infect tumours represents one of the key attributes that regulates the success of such a strategy. Although some remarkable successes have been obtained, it is still not entirely clear how to achieve reliable protocols that can be routinely employed with confidence on a significant range of tumours. In this work, we thus concentrate on the study of different mathematical descriptions of virotherapy with the aim of better understanding the role of viral infectivity and viral dynamics in positive therapeutic outcomes. In particular, we compare probabilistic, individual approaches with continuous, spatially inhomogeneous models and investigate the importance of different tumour motility and different mathematical representations of viral infectivity. These formulations also allow us to arrive at better analytical characterisation of how waves of viral infections arise and propagate in tumours, providing interesting insights into therapy dynamics. Similarly to previous studies, oscillatory behaviours, stochasticity and cancers' diffusivities are all central to the eradication or the escape of tumours under virotherapy. Here, though, our results also show that the ability of viruses to infect tumours seems, in certain cases, more important to a final positive outcome than tumours' motility or even reproducibility. This could hopefully represent a first step into better insights into viral dynamics that may help clinicians to achieve consistently better outcomes.

Figures (13)

• Figure 1: (a) Theoretical speed of the travelling wave of viral infection invading an uninfected tumour at carrying capacity, computed following the procedure of Ref. baabdulla23 (as explained in the Appendix \ref{['app:wave']}). The parameters employed are the ones given in Table \ref{['tab:parametersB1']}, with three values of $D_v$; $q_v$ and $\alpha$ are varied so that the value of $\tilde{\beta}$ is constantly equal to $1.02\times 10^{-1}\;$h$^{-1}$. The dashed cyan line shows the speed of the infection wave obtained for Eq. \ref{['eq:l']} with the same parameters. (b) Numerical solutions of Eq. \ref{['eq:lv']} compared with the position of the theoretical front (shown as a dashed red line). The parameters employed are the ones given in Table \ref{['tab:parametersB1']}, with $\alpha=580\;$viruses/cells and $q_v=1.67\times 10^{-1}\;$h$^{-1}$; we also set $u(0,\cdot)=K$ to focus on the infection inside an established tumour.
• Figure 2: Comparison in one spatial dimension between numerical simulations of the discrete model with undirected cell movement (solid lines), the numerical solution of Eq. \ref{['eq:l']} (dashed lines) and the numerical solution of Eq. \ref{['eq:lv']} (dotted black lines) at three different times, with the parameters given in Table \ref{['tab:parametersB1']}, $\alpha=580\;$viruses/cells and $q_v=1.67\times 10^{-1}\;$h$^{-1}$. The viral density of the agent-based model is multiplied by $q_v/(\alpha q)$ to allow the comparison with cell numbers; note that the viral density for Eq. \ref{['eq:lv']} is not shown, as it would superimpose with infected cells. For the agent-based model, the densities of uninfected cells, infected cells and virus are represented respectively in blue, red and purple; the numerical solutions of Eq. \ref{['eq:l']} are represented using the same colours. The vertical black lines represent the expected positions of the invasion fronts: the dash-dotted line in panel (a) refers to the infected front, whose speed is depicted in Fig. \ref{['fig:speedqv']}, and the dashed line in panels (b) and (c) refers to the uninfected front, travelling at speed $2\sqrt{D_Up}$. The horizontal solid black lines show the equilibrium of the ODE given by Eq. \ref{['eq:eq_lv']}. The horizontal dashed yellow line represents the expected uninfected density at the front for Eq. \ref{['eq:l']} given by Eq. \ref{['eq:ubar']} and the horizontal dash-dotted green line shows the analogue quantity in the case of Eq. \ref{['eq:lv']} (both are only relevant in panel (c)). The results of the agent-based model are averaged over five simulations. The maximum of the cell density axes in panels (b) and (c) corresponds to the maximum over time of this average (which is larger than the carrying capacity); on the other hand, in panel (a), it has been reduced to allow the comparison between infected cells and virus (and, as a consequence, uninfected cells are not shown).
• Figure 3: Comparison in one spatial dimension between numerical simulations of the discrete model with pressure-driven cell movement (solid lines), the numerical solution of Eq. \ref{['eq:l']} (dashed lines) and the numerical solution of Eq. \ref{['eq:pv']} (dotted black lines) at three different times, with the parameters given in Table \ref{['tab:parametersB1']} and both combinations of $\alpha$ and $q_v$. We remark that the diffusion coefficient in Eq. \ref{['eq:l']} was modified to keep the propagation speed constant (see Appendix \ref{['app:wave_u']}). All the graphical elements have the same meaning as in Fig. \ref{['fig:lv_low']}. Observe that, this time, the maximum value reached by the average uninfected cell density is lower than the carrying capacity.
• Figure 4: Comparison in one spatial dimension between the various models with a low value of viral diffusion $D_v=10^{-5}\;$mm$^2$/h at time $t=1500\;$h. Panels (a-b) show numerical simulations of the discrete model with undirected cell movement (solid lines), the numerical solution of Eq. \ref{['eq:l']} (dashed lines) and the numerical solution of Eq. \ref{['eq:lv']} (dotted black lines) with the parameters given in Table \ref{['tab:parametersB1']}, low viral diffusion and both combinations of $\alpha$ and $q_v$. Panel (c) shows numerical simulations of the discrete model with pressure-driven cell movement (solid lines), the numerical solution of Eq. \ref{['eq:p']} (dashed lines) and the numerical solution of Eq. \ref{['eq:pv']} (dotted black lines) with the parameters given in Table \ref{['tab:parametersB1']}, low viral diffusion, $\alpha=3500\;$viruses/cells and $q_v=1\;$h$^{-1}$; we also consider $R_v=R_u$ in the initial conditions to model a wide initial infection. All the graphical elements have the same meaning as in Fig. \ref{['fig:lv_low']}.
• Figure 5: Numerical simulation of the discrete model with pressure-driven movement in two spatial dimensions at three different times. The parameters employed are the ones given in Table \ref{['tab:parametersB1']}, with the exception of the diffusion coefficient of viral particles $D_v$ (which is set to $10^{-5}\;$mm$^2$/h), the death rate of infected cells $q$ (which is set to $8.33\times 10^{-3}\;$h$^{-1}$, i.e. one-fifth of the reference values) and the initial radius of viral infection $R_v$ (which is set equal to $R_u$); furthermore, we use the values $\alpha=3500\;$viruses/cells and $q_v=1\;$h$^{-1}$. The dashed cyan circles in the top panels represent the expected positions of the uninfected invasion fronts in the absence of treatment, travelling at speed $\sqrt{D_P p/2}$. The dashed red circles in the bottom panels represent the front of the infected cells given by the numerical solution of Eq. \ref{['eq:pv']}.