A Framework for Performing Data-Driven Modeling of Tumor Growth with Radiotherapy Treatment

Abstract

Recent technological advances make it possible to collect detailed information about tumors, and yet clinical assessments about treatment responses are typically based on sparse datasets. In this work, we propose a workflow for choosing an appropriate model, verifying parameter identifiability, and assessing the amount of data necessary to accurately calibrate model parameters. As a proof-of-concept, we compare tumor growth models of varying complexity in an effort to determine the level of model complexity needed to accurately predict tumor growth dynamics and response to radiotherapy. We consider a simple, one-compartment ordinary differential equation model which tracks tumor volume and a two-compartment model that accounts for tumor volume and the fraction of necrotic cells contained within the tumor. We investigate the structural and practical identifiability of these models, and the impact of noise on identifiability. We also generate synthetic data from a more complex, spatially-resolved, cellular automaton model (CA) that simulates tumor growth and response to radiotherapy. We investigate the fit of the ODE models to tumor volume data generated by the CA in various parameter regimes, and we use sequential model calibration to determine how many data points are required to accurately infer model parameters. Our results suggest that if data on tumor volumes alone is provided, then a tumor with a large necrotic volume is the most challenging case to fit. However, supplementing data on total tumor volume with additional information on the necrotic volume enables the two compartment ODE model to perform significantly better than the one compartment model in terms of parameter convergence and predictive power.

Authors Heyrim Cho, Allison L. Lewis, and Kathleen M. Storey have equally contributed to this chapter.

Appendix: Structural Identifiability for the Two-Compartment Model

Below, we investigate the structural identifiability of the two-compartment model with radiotherapy, given in Equations (19) and (20), using the same techniques as presented in Sect. 3.1.

1.1 Case 1: No Radiation

In this case the model reads

$$\displaystyle \begin{aligned} \frac{dY}{dt}&=\lambda(1-\varPhi)Y(1-(1-\varPhi)\frac{Y}{K})-\xi\varPhi Y\\ \frac{d\varPhi}{dt}&=(1-\varPhi)\left[ \eta-\lambda\varPhi(1-(1-\varPhi)\frac{Y}{K}-\xi\varPhi) \right] \end{aligned} $$

with unknown parameters p = {λ, K, ξ, η}, observable quantities y(p;t) = {Y, Φ}, and known initial conditions. We repeat the analysis as before, using the Taylor coefficients. We define the following known quantities:

$$\displaystyle \begin{aligned} a_{0}{\kern-1pt}={\kern-1pt}Y(0^{+}) \enspace b_0{\kern-1pt}={\kern-1pt}\phi(0^{+}) \enspace a_1{\kern-1pt}={\kern-1pt}Y'(0^{+}) \enspace b_1{\kern-1pt}={\kern-1pt}\phi'(0^{+})\quad a_2{\kern-1pt}={\kern-1pt}Y''(0^{+}) \enspace b_2=\phi''(0^{+}). \end{aligned}$$

We substitute these quantities into the model system to obtain:

$$\displaystyle \begin{aligned} a_1&=\lambda(1-b_0)a_0[1-(1-b_0)\frac{a_0}{K}]-\xi b_0a_0\\ &=[-(1 - b_0)^{2}a_{0}^2]\frac{\lambda}{K} + [(1 - b_0)a_0]\lambda - [b_0a_0]\xi \\ b_1&=(1-b_0)\left[ \eta-\lambda b_0(1-(1-b_0)\frac{a_0}{K}-\xi b_0) \right]\\ &+[-(1 - b_0)b_0]\lambda + [(1 - b_0)^2b_0a_0]\frac{\lambda}{K} - [(1 - b_0)b_0]\xi + [(1-b_0)]\eta. \end{aligned} $$

We differentiate the model equations once more to obtain:

$$\displaystyle \begin{aligned} a_2&=\frac{(-2\lambda(-1 + b_0)^2a_0 - ((\xi + \lambda)b_0 - \lambda)K)a_1 - a_0b_1(2(-1 + b_0)\lambda a_0 + (\xi + \lambda)K)}{K}\\ b_2&=\frac{(3a_0b_0^{2}\lambda + (-4\lambda a_0 + 2(\xi + \lambda)K)b_0 + \lambda a_0 - K(\xi + \eta + \lambda))b_1 + b_0a_1\lambda(-1 + b0)^2)}{K}. \end{aligned} $$

The above four equations can now be used to solve for each parameter as follows:

$$\displaystyle \begin{aligned} K&=\frac{1}{[((b_0-1)b_2 - b_{1}^2 )a_0^2 + a_2(b_0-1)^2a_0 - a_1^2(b_0-1)^2]b_1} \\ &\quad \times [(b_1(b_0 {\kern-1pt}+{\kern-1pt} 1)(b_0b_2 {\kern-1pt}-{\kern-1pt} b_1^2 {\kern-1pt}-{\kern-1pt} b_2)a_0^3 {\kern-1pt}+{\kern-1pt} ((a_1b_2 {\kern-1pt}+{\kern-1pt} a_2b_1)b_0^2 {\kern-1pt}+{\kern-1pt} ((-b_1^2 {\kern-1pt}-{\kern-1pt} b_2)a_1 {\kern-1pt}+{\kern-1pt} a_2b_1)b_0 \\ \\ &- a_1b_1^2 - a_2b_1)(-1 + b0)a_0^2- (-a_2b_0^3 + (a_1b_1 + a_2)b_0^2 + 2a_1b_1b_0 - 2a_1b_1) \\ &\quad \times a_1(-1 + b_0)a_0 - a_1^3b_0^2(-1 + b_0)^2)(-1 + b_0)],\\ \lambda&=\frac{-1}{(a_0b_0b_1 + a_1b_0^2 - a_0b_1 - 2a_1b_0 + a_1)a_0^2b_1}\\ &\quad \times [a_0^3b_0^2b_1b_2 - a_0^3b_0b_1^3 + a_0^2a_1b_0^3b_2 - a_0^2a_1b_0^2b_1^2 \\ &\quad + a_0^2a_2b_0^3b_1 - a_0a_1^2b_0^3b_1 + a_0a_1a_2b_0^4 - a_1^3b_0^4 - a_0^3b_1^3\\ &\quad - 2a_0^2a_1b_0^2b_2 - a_0a_1^2b_0^2b_1 - 2a_0a_1a_2b_0^3\\ &\quad + 2a_1^3b_0^3 - a_0^3b_1b_2 + a_0^2a_1b_0b_2 + a_0^2a_1b_1^2 - 2a_0^2a_2b_0b_1 + 4a_0a_1^2b_0b_1 \\ &\quad + a_0a_1a_2b_0^2- a_1^3b_0^2+ a_0^2a_2b_1 - 2a_0a_1^2b_1]\\ \eta&=-\frac{a_0^2b_0b_2 - a_0^2b_1^2 + a_0a_2b_0^2 - a_1^2b_0^2}{b_1a_0^2}\\ \xi &=\frac{a_0^2b_0b_2 - a_0^2b_1^2 + a_0a_2b_0^2 - a_1^2b_0^2 - a_0^2b_2 - a_0a_1b_1 - a_0a_2b_0 + a_1^2b_0}{b_1a_0^2} \end{aligned} $$

Since we are able to obtain unique solutions for each of the four parameters, we declare them to be structurally identifiable. For GenSSI, only two Lie derivatives are needed which yield rank 4, and thus results show all four parameters are structurally identifiable, in agreement with our calculations above.

In addition to the above analysis, we also repeated the analysis in the case in which only the tumor volume could be observed, (i.e., y(t;p) = Y (t)), but with known initial conditions in tumor volume and necrotic fraction. In this case, we took higher order Taylor series coefficients (up to order 4) and obtained that p = {λ, K, ξ, η} were structurally identifiable. Similarly, GenSSI took Lie derivatives up to order 4 and confirmed that all parameters were structurally identifiable.

1.2 Case 2: With Radiation Treatment

Similar to the single compartment model, here we examine the effect of a point treatment. The model equations read:

$$\displaystyle \begin{aligned} \frac{dY}{dt}&=\lambda(1-\varPhi)Y(1-(1-\varPhi)\frac{Y}{K})-\xi\varPhi Y\\ \frac{d\varPhi}{dt}&=(1-\varPhi)\left[ \eta-\lambda\varPhi(1-(1-\varPhi)\frac{Y}{K})-\xi\varPhi\right],\quad \mbox{for} \quad \; t_i^+ < t < t_{i+1}^-\\ \varPhi(t_{i}^{+})&=\varPhi(t_{i}^{-})+(1-\varPhi(t_{i}^{-}))(1-\varGamma), \end{aligned} $$

where $\varGamma =\exp (-\alpha d-\beta d^2)$. Since the other parameters are known and measured prior to treatment, as in the previous section, we want to solve for p = {α, β} assuming y(p;t) = {Y, Φ} as observable quantities. We let

$$\displaystyle \begin{aligned} A_0=Y(t_{i}^{+}) \quad B_0=\varPhi(t_{i}^{-}) \quad A_1=Y'(t_{i}^{+}) \quad B_1=\varPhi'(t_{i}^{+}), \end{aligned} $$

and substitute these quantities into the model equations:

$$\displaystyle \begin{aligned} A_1&=\frac{-A_0^2\lambda(B_0 - 1)^2}{K}\varGamma^2 - A_0(\xi + \lambda)(B_0 - 1)\varGamma - A_0\xi\\ B_1&=\frac{[A_0\lambda(B_0 {\kern-1pt}-{\kern-1pt} 1)^2\varGamma^2 {\kern-1pt}+{\kern-1pt} (B_0 {\kern-1pt}-{\kern-1pt} 1)((K {\kern-1pt}+{\kern-1pt} A_0)\lambda {\kern-1pt}+{\kern-1pt} \xi K)\varGamma {\kern-1pt}+{\kern-1pt} K(\xi {\kern-1pt}-{\kern-1pt} \eta + \lambda)]\varGamma(B_0 - 1)}{K} \end{aligned} $$

As with the one-compartment model, we find that the equations are not informative for α and β simultaneously, thus, we again declare the pair (α, β) to be non-identifiable in this setting. As before, we choose to fix α for all subsequent model calibrations and measure the ratio α∕β to use as a measure of radiosensitivity.