The dependence on the overexpression of a single oncogene constitutes an exploitable weakness for molecular targeted therapy. These drugs can produce dramatic tumor regression by targeting the driving oncogene, but relapse often follows. Understanding the complex interactions of the tumor’s multifaceted response to oncogene inactivation is key to tumor regression. It has become clear that a collection of cellular responses lead to regression and that immunemediated steps are vital to preventing relapse. Our integrative mathematical model includes a variety of cellular response mechanisms of tumors to oncogene inactivation. It allows for correct predictions of the time course of events following oncogene inactivation and their impact on tumor burden. A number of aspects of our mathematical model have proven to be necessary for recapitulating our experimental results. These include a number of heterogeneous tumor cell states since cells following different cellular programs have vastly different fates. Stochastic transitions between these states are necessary to capture the effect of escape from oncogene addiction (i.e., resistance). Finally, delay differential equations were used to accurately model the tumor growth kinetics that we have observed. We use this to model oncogene addiction in MYCinduced lymphoma, osteosarcoma, and hepatocellular carcinoma.
Bernard Weinstein first proposed in 1997 that “oncogene addiction” is the phenomenon whereby the inactivation of a single oncogene, even if brief, may lead to sustained tumor regression, providing a weakness for a molecularly targeted therapy to exploit [
Inactivation of the oncogene by targeted therapy produces a complex array of responses at the cellular level including apoptosis, cell cycle arrest, differentiation, senescence, and inhibition of angiogenesis. In preclinical models, the oncogene may be inactivated using conditional expression in transgenic animals (e.g., Cre/LoxP, tamoxifen, or tetracycline systems). Some of these resultant cellular programs are cell intrinsic (i.e., not involving other cells) while others are cell extrinsic, involving complex host interactions with effector cells in the immune system. While these different response mechanisms have been studied and modeled individually, there has been far less investigation into integrating the overall sequence and interactions of tumor responses into a unified mathematical model that can inform the design and optimization of therapeutic strategies. Understanding how and why some tumors relapse while others do not, as well as how and why the specific cellular program responses depend on the tissuespecific and host immune background, is of crucial importance for designing the most effective therapies.
Previously, we have built and validated a model of tumor growth and regression kinetics in response to oncogene inactivation [
We will use the model to study the major cellular processes involved in MYCinduced lymphoma, osteosarcoma, and hepatocellular carcinoma, which involve difference combinations and sequences of these programs and to test different therapeutic strategies.
Much work has been done in characterizing tumor growth kinetics
We utilize the tetracycline (Tet) system to conditionally and reversibly control the expression of the MYC oncogene in mouse models [
In tumor dormancy, cells can restore their neoplastic properties upon MYC reactivation. In order to improve therapy, it is important to distinguish when MYC inactivation leads to complete tumor regression characterized by permanent loss of malignant phenotype and when it simply results in a reversible state of tumor dormancy [
MYC inactivation in MYCinduced lymphoma leads to differentiation, apoptosis, and complete tumor regression. Therefore, a permanent loss of a neoplastic phenotype occurs upon MYC inactivation. In osteogenic sarcoma, MYC inactivation induces differentiation and proliferative arrest but does not induce significant apoptosis. MYC reactivation in these apparently differentiated cells either has no consequences or leads to apoptosis. Only in very rare cells is there restoration of neoplastic properties. In hepatocellular carcinoma, MYC inactivation leads to differentiation and then eventually to gradual apoptosis of most of the tumor cells. Upon reactivation of MYC, these differentiated cells quickly become tumorigenic [
Senescence is the growtharrest process by which normal cells are restrained from malignant transformation. Oncogene inactivationinduced senescence (OIIS) is the irreversible cell cycle arrest of normal cells in response to inactivating an oncogene. We have recently shown cellular senescence resulting from MYC inactivation to depend on the host immune system [
The
Tumors undergo regression initially regardless of the status of the host immune system. But in hosts that are immune compromised, tumor elimination is incomplete and the tumors eventually relapse. An intact immune system is required for oncogene inactivationinduced senescence, inhibition of angiogenesis, and chemokine expression, which lead to sustained tumor regression. CD4+ (but not CD8+) Tcell deficiency was enough to impede sustained tumor regression. The secretion of TSP1 is markedly decreased in immune compromised versus wildtype hosts. TSP1 expression requires host immune cells particularly CD4+ T cells. Reconstitution of immune compromised mouse with CD8+ T cells still showed significant minimum residual disease, whereas reconstitution with CD4+ T cells showed no minimum residual disease, the same result found in wildtype hosts upon MYC inactivation. Hence, simply restoring CD4+ T cells was sufficient to eliminate minimum residual disease and to lead to sustained tumor regression. CD4+ T cells are the crucial host effector population necessary for tumor regression upon MYC inactivation. TSP1 is important in immune effectors for sustained regression upon MYC inactivation, and overexpression of TSP1 in immune compromised hosts is sufficient to increase the duration of sustained tumor regression upon MYC inactivation [
Our mathematical model uses various modeling techniques that were each shown to be necessary to accurately recapitulate the experimental observations.
We created a new mathematical model (Figure
Mathematical model of cellular states. Former model shown in gold with additions shown in blue. The arrows with slashes corresponding to ±dox indicate that this is an independent variable controlled experimentally. The arrows representing proliferative loops have an implicit state during proliferation representing the mitotic phase of the cell cycle.
The structure and topology of our mathematical model is based on
Tumor cells can exist in one of six different states; the number of cells in each state is represented as follows.
The experimental data shows the variability in relapse kinetics, which a deterministic model cannot capture. Hence, stochasticity was added to the model. We use random sampling from a multinomial distribution (well approximated by binomial due to very low pertime step probabilities) to represent a stochasticity in the number of cells transitioning from one state to another, enabling us to recapitulate the variability in tumor relapse (Figure
Tumor regression and relapse kinetics as measured by bioluminescence imaging. Wildtype (WT) are immunocompetent mice while SCID and RAG2^{−/−}c
Some parameters of the model are immune system dependent, some are MYC expression dependent, and others are tumor type dependent (Table
Model parameters and values governing the response to oncogene inactivation.
Parameter  Estimated value (day^{−1})  Description 


0.1  Transition coefficient from the “MYC off” state to proliferating 

0.02  Transition coefficient between “MYC off” state and differentiation state 

2  Transition coefficient between “MYC off” and apoptosis state 

0.001  Transition coefficient between “MYC off” state and Oncogene inactivation induced senescence state 

0/3 × 10^{−8}  Transition coefficient between “MYC off” state and escaped state (immunocompetent/immunodeficient) 

0.1  Transition coefficient between differentiated state and “MYC on” state 

0.5  Transition coefficient from the escaped state to proliferating 

0.05  Transition coefficient between “MYC on” and apoptosis state 

0.5  Transition coefficient from the “MYC on” state to proliferating 

0.01  Transition coefficient between escaped state and apoptosis state 

0.05/0.002/0.01  Transition coefficient between differentiated state and apoptosis state (tumor dependent: lymphoma, osteosarcoma, and hepatocellular carcinoma) 

0/2  Transition from “MYC on” to “MYC off” (depending on if MYC is on/off) 

2/0  Transition from “MYC off” to “MYC on” (depending on if MYC is on/off) 
Figure
Set of equations represented by the model shown in Figure
In biological processes, there are often “physical” delays making it vital to incorporate delays into the model in order to make the mathematical model closer to the real phenomenon. Examples of delay mathematical models in biology are population dynamics (e.g., Hutchinson’s equation), ecology (e.g., the LotkaVolterra predator prey), and immunology (e.g., delay in immune system response). We have implemented delay differential equations (on top of our stochastic framework) for the apoptosis state. There is a delay between when a cell commits to apoptosis and when cell death actually occurs. This is important since these cells that have committed to apoptosis (in state
Although our modeling philosophy has been to create the simplest possible model that could explain the salient features of our experimental data, we found multiple tumor states and stochastic transitions without explicit delays to be insufficient to model some of our observations from bioluminescent imaging and immunohistochemistry. In particular, we found that proliferative arrest occurs almost immediately after oncogene inactivation but apoptosis was delayed for approximately 45 days.
Figure
Simulated tumor growth kinetics with no delay and no escape of tumor cells from the conditional control of MYC due to an intact immune system. Note the lack of delay in the decrease of tumor cells as is observed in days 0–5.
By running simulations, our model recapitulates features such as the different rates and delays in the tumor kinetics measured from
Furthermore, the rate of mutations leading to tumor relapse (which is captured in the term
Simulated tumor growth kinetics with no escape of tumor cells from the conditional control of MYC due to an intact immune system. Note the correctly modeled delay in tumor cell death as is observed in Figure
Simulated tumor growth kinetics with a low (but nonzero) rate of tumor cells escaping from conditional control of MYC due to a compromised immune system. Note the early growth kinetics of the escaped tumor cell population demonstrates stochastic variability that affects the timing of the relapse.
Empirically, the regression kinetics had little variability (in Figure
Variability in relapse kinetics captured using 20 stochastic simulations.
While our model has yielded numerous hypotheses and insights about oncogene addiction, the model has various limitations. There are other important hostdependent factors that have yet to be added, namely, the inhibition of angiogenesis upon oncogene inactivation. Angiogenesis has a number of potential effects on portions of our model including growth rates, hypoxiarelated resistance, and lymphocyte access to the tumor. Another limitation is that we do not currently model tumor stem cell properties. Many models have been developed that model the spectrum of stemness in tumor cells but we do not know how much effect this will have on our results. Similarly, other effects such as DNA methylation and transcription factor networks have not been included [
The initial results from our new model are helping to quantitatively hypothesize about the sequence of cellular mechanisms involved in tumor response to oncogene inactivation such as might be encountered using targeted therapeutics, and the interactions among them. These hypotheses will be tested experimentally using the same conditional control mouse models and
This finding is captured in our model. If residual disease reaches a low enough level, relapse can be prevented. Large tumors have a great number of cells that can transition through an extremely lowlikelihood event to the “Escaped” state. If the tumor is small enough, not enough cells remain to make it likely that any one will achieve the very lowlikelihood “Escaped” state. Tumor burden is also an important factor in tumor regression mediated through the immune system. The immune system might not be able to attack and eliminate the tumor fast enough if the tumor burden is too high. Currently our model does not account for this but we are adding more sophisticated immune system components to the model in order to show this. Our model is simple in that all but one state (with explicit delay) have memoryless transitions and yet the model is able to recapitulate the complex response of tumors to oncogene inactivation. No age structuring of cells is required as with some other models [
Our model offers more fidelity than models in the literature that just capture tumor cells in a single variable [
The authors gratefully acknowledge support from NIH P50 CA114747 (ICMIC@Stanford) and R01 CA180387.