Mathematical Details on a Cancer Resistance Model

Abstract

The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy [11,10]. In this work, we expound on the details relating to an optimal control problem outlined in [10]. The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebra techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.

Integrating multimodal data sets into a mathematical framework to describe and predict therapeutic resistance in cancer

“…To compare the effects of a constant versus pulsed dose, we simulated the effect of a constant dose (black line Fig 1F) equal to the mean value over the time interval simulated of the pulsed dose (blue line Fig 1F) in an attempt to reflect realistic toxicity constraints that would be present in a clinical setting when developing treatment regimens. This analysis, as well as further work to utilize this modeling framework to develop optimal treatment protocols (16) indicates that identifying these model parameters is essential to implementing more sophisticated treatment strategies in a practical clinical setting. While (16) show that the critical model parameters are theoretically structurally identifiable from population size data alone, we seek to demonstrate how this model can be practically identified from in vitro data using both longitudinal population size data (N(t)) and snapshot outputs of the phenotypic composition (f(t)) at a few time points, enabled by recent advances in lineage tracing (11,25) and scRNA-seq technologies.…”

“…This analysis, as well as further work to utilize this modeling framework to develop optimal treatment protocols (16) indicates that identifying these model parameters is essential to implementing more sophisticated treatment strategies in a practical clinical setting. While (16) show that the critical model parameters are theoretically structurally identifiable from population size data alone, we seek to demonstrate how this model can be practically identified from in vitro data using both longitudinal population size data (N(t)) and snapshot outputs of the phenotypic composition (f(t)) at a few time points, enabled by recent advances in lineage tracing (11,25) and scRNA-seq technologies. We present this project workflow in the experimental setting as proof-of-concept of the ability to properly identify key model parameters from multimodal data sets, with the hopes that the approach of integrating snapshot with longitudinal data sets will eventually be brought to the clinic to develop optimized treatment regimens for existing therapeutic agents.…”

“…= (1 − ( + )) − ( ) − E ( ) = 4 (1 − ( + )) + ( ) − 4 ( ) ( ) = ( ) + ( ) ( ) = ( ) ( ) + ( ) We assume all parameters are non-negative and 0 < 9 < 1 represents the relative growth rate of the resistant population with respect to the sensitive population scaled by the carrying capacity, and 9 < 1 assumes that resistant cells grow more slowly than sensitive cells. In work by Greene et al (16), they demonstrate that, if they assume dr=0, i.e. resistant cells are not killed by drug, and that the initial state of the population is completely comprised of sensitive cells (i.e.…”

“…We follow the same arguments outlined in (16), along with the complete explanation of the approach with illustrative examples, for the case of multiple outputs from (62). We start by formulating the dynamical system relevant to our in vitro experimental system.…”

“…These models describe cancer cells dynamically growing and responding to drug with differential growth rates and drug sensitivities. Knowledge of these model parameters have enabled the theoretical optimization of treatment protocols (16)(17)(18), and have been applied successfully to prolong tumor control in both mice (12) and patients (14,19).…”