| Publications of Eduardo D. Sontag jointly with J.M. Greene |
| Articles in journal or book chapters |
| There is growing recognition that phenotypic plasticity enables cancer cells to adapt to various environmental conditions. An example of this adaptability is the persistence of an initially sensitive population of cancer cells in the presence of therapeutic agents. Understanding the implications of this drug-induced resistance is essential for predicting transient and long-term tumor tumor dynamics subject to treatment. This paper introduces a mathematical model of this phenomenon of drug-induced resistance which provides excellent fits to time-resolved in vitro experimental data. From observational data of total numbers of cells, the model unravels the relative proportions of sensitive and resistance subpopulations, and quantifies their dynamics as a function of drug dose. The predictions are then validated using data on drug doses which were not used when fitting parameters. The model is then used, in conjunction with optimal control techniques, in order to discover dosing strategies that might lead to better outcomes as quantified by lower total cell volume. |
| Different epidemiological models, from the classical SIR system to more sophisticated ones involving population compartments for socially distanced, quarantined, infection aware, asymptomatic infected, and other individuals, share some remarkable dynamic characteristics when contact rates are subject to periodic or one-shot changes. In simple pulsed isolation policies, a linear relationship is found among optimal start time and duration for reduction of the infected peak. If a single interval social distancing starts too early or too late it will be ineffective with respect to decreasing the peak of infection. On the other hand, the nonlinearity of epidemic models leads to non-monotone behavior of the peak of infected population under periodic relaxation policies. This observation led us to hypothesize that an additional single interval social distancing at a proper time can significantly decrease the infected peak of periodic policies, and we verified this improvement. |
| Motivated by the current COVID-19 epidemic, this work introduces an epidemiological model in which separate compartments are used for susceptible and asymptomatic "socially distant" populations. Distancing directives are represented by rates of flow into these compartments, as well as by a reduction in contacts that lessens disease transmission. The dynamical behavior of this system is analyzed, under various different rate control strategies, and the sensitivity of the basic reproduction number to various parameters is studied. One of the striking features of this model is the existence of a critical implementation delay in issuing separation mandates: while a delay of about four weeks does not have an appreciable effect, issuing mandates after this critical time results in a far greater incidence of infection. In other words, there is a nontrivial but tight "window of opportunity" for commencing social distancing. Different relaxation strategies are also simulated, with surprising results. Periodic relaxation policies suggest a schedule which may significantly inhibit peak infective load, but that this schedule is very sensitive to parameter values and the schedule's frequency. Further, we considered the impact of steadily reducing social distancing measures over time. We find that a too-sudden reopening of society may negate the progress achieved under initial distancing guidelines, if not carefully designed. |
| One of the most important factors 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. In this work, we expound on the details relating to an optimal control problem outlined in our previous paper (Greene et al., 2018). The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebraic 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. |
| Resistance to chemotherapy is a major impediment to the successful treatment of cancer. Classically, resistance has been thought to arise primarily through random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that the progression to resistance need not occur randomly, but instead may be induced by the therapeutic agent itself. This process of resistance induction can be a result of genetic changes, or can occur through epigenetic alterations that cause otherwise drug-sensitive cancer cells to undergo "phenotype switching". This relatively novel notion of resistance further complicates the already challenging task of designing treatment protocols that minimize the risk of evolving resistance. In an effort to better understand treatment resistance, we have developed a mathematical modeling framework that incorporates both random and drug-induced resistance. Our model demonstrates that the ability (or lack thereof) of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. The importance of induced resistance in treatment response led us to ask if, in our model, one can determine the resistance induction rate of a drug for a given treatment protocol. Not only could we prove that the induction parameter in our model is theoretically identifiable, we have also proposed a possible in vitro experiment which could practically be used to determine a treatment's propensity to induce resistance. |
| Utilizing the synthetic transcription-translation (TX-TL) system, this paper studies the impact of nucleotide triphosphates (NTPs) and magnesium (Mg2+), on gene expression, in the context of the counterintuitive phenomenon of suppression of gene expression at high NTP concentration. Measuring translation rates for different Mg2+ and NTP concentrations, we observe a complex resource dependence. We demonstrate that translation is the rate-limiting process that is directly inhibited by high NTP concentrations. Additional Mg2+ can partially reverse this inhibition. In several experiments, we observe two maxima of the translation rate viewed as a function of both Mg2+ and NTP concentration, which can be explained in terms of an NTP-independent effect on the ribosome complex and an NTP- Mg2+ titration effect. The non-trivial compensatory effects of abundance of different vital resources signals the presence of complex regulatory mechanisms to achieve optimal gene expression. |
| This paper describes a novel approach for characterization of chemosensitivity and prediction of clinical response in multiple myeloma. It relies upon a patient-specific computational model of clinical response, parameterized by a high-throughput ex vivo assay that quantifies sensitivity of primary MM cells to 31 agents or combinations, in a reconstruction of the tumor microenvironment. The mathematical model, which inherently accounts for intra-tumoral heterogeneity of drug sensitivity, combined with drug- and regimen-specific pharmacokinetics, produces patient-specific predictions of clinical response 5 days post-biopsy. |
| Conference articles |
| Due to the usage of social distancing as a means to control the spread of the novel coronavirus disease COVID-19, there has been a large amount of research into the dynamics of epidemiological models with time-varying transmission rates. Such studies attempt to capture population responses to differing levels of social distancing, and are used for designing policies which both inhibit disease spread but also allow for limited economic activity. One common criterion utilized for the recent pandemic is the peak of the infected population, a measure of the strain placed upon the health care system; protocols which reduce this peak are commonly said to "flatten the curve". In this work, we consider a very specialized distancing mandate, which consists of one period of fixed length of distancing, and addresses the question of optimal initiation time. We prove rigorously that this time is characterized by an equal peaks phenomenon: the optimal protocol will experience a rebound in the infected peak after distancing is relaxed, which is equal in size to the peak when distancing is commenced. In the case of a non-perfect lockdown (i.e. disease transmission is not completely suppressed), explicit formulas for the initiation time cannot be computed, but implicit relations are provided which can be pre-computed given the current state of the epidemic. Expected extensions to more general distancing policies are also hypothesized, which suggest designs for the optimal timing of non-overlapping lockdowns. |
| The primary factor limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. This work extends the work reported in "A mathematical approach to distinguish spontaneous from induced evolution of drug resistance during cancer treatment" by introducing a time-optimal control problem that is analyzed utilizing differential-geometric techniques: we seek a treatment protocol which maximizes the time of treatment until a critical tumor size is reached. The general optimal control structure is determined as a combination of both bang-bang and path-constrained arcs. Numerical results are presented which demonstrate decreasing treatment efficacy as a function of the ability of the drug to induce resistance. Thus, drug-induced resistance may dramatically effect the outcome of chemotherapy, implying that factors besides cytotoxicity should be considered when designing treatment regimens. |
| Internal reports |
| This paper continues the study of a model which was introduced in earlier work by the authors to study spontaneous and induced evolution to drug resistance under chemotherapy. The model is fit to existing experimental data, and is then validated on additional data that had not been used when fitting. In addition, an optimal control problem is studied numerically. |
| 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 . In this work, the control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie techniques. A structural identfiability analysis is also presented, demonstrating that patient-specfic parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. |
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