My current research interests fall into two main areas:

Previously I was interested in using machine learning to predict T-cell epitopes and I also studied zero-divisor graphs of matrices.

Mathematical models of cancer evolution

In healthy tissues, cell division and cell death are tightly controlled processes, which enable a precise balance assuring that the number of cells in the body remains approximately constant. However, during each cell division mistakes in DNA replication can occur, leading to accumulation of mutations in individual cells. The majority of such mutations are effectively neutral (passengers), but some of them (drivers) can provide selective advantage to the cell, by tipping the balance of cell division and death in favor of increased proliferation. This unwanted evolution of somatic cells can lead to a clonal expansion of cells with driver mutations, which can ultimately result in the formation of tumors and seeding of new lesions in distant tissues. I am interested in studying how mutations accumulate both in healthy tissue and in cancer. My approach combines developing and studying mathematical models of genetic evolution together with genomic data to obtain insights into these important, but hidden evolutionary processes.

1. Stochastic evolution of driver and passenger mutations

We developfig2_clonal_v2ed a stochastic model for accumulation of driver and passenger mutations during cancer evolution and studied the relationship between the number of drivers and passengers in individual cancers. After a cell with the first driver initiates clonal expansion, at every cell division its progeny can collect new driver mutations with some small probability, forming a new cell type. Each subsequent driver provides additional selective advantage to cells, leading to an even faster clonal expansion. In this multi-type branching process model, cells can also collect passenger mutations, which have no effect on fitness. We fit our theory to sequencing data from glioblastoma and pancreatic cancer and show that the selective advantage of driver mutations in human cancers in vivo is surprisingly small, and amounts to less than 1% reduction in the death rate. Studying the effect of a single additional driver on tumor evolution using a two-type branching process, we observe that early, small lesions need additional drivers, while late stage metastases are only marginally affected by them.


I Bozic et al. (2010) Accumulation of driver and passenger mutations during tumor progression. PNAS 107, 18545-18550.
JG Reiter*, I Bozic* et al. (2013) The effect of one additional driver mutation on tumor progression. Evolutionary Applications 6, 34-45.

2. Passenger mutations as a molecular clock in healthy stem cells and in cancer

Using a novel methodolofig1_v2gy that utilizes cancer sequencing data, we are able to estimate the rate of accumulation of mutations in healthy stem cells of the colon, blood and head and neck tissues and refute the immortal strand hypothesis. This hypothesis has been proposed as a mechanism used by stem cells in order to minimize the accumulation of mutations in their genomes. The number of mutations in cancer tissues correlates significantly with the age of the patient, independent of the cancer stage. Difference in the numbers of somatic mutations detected in the cancers of patients of different ages can be explained by the difference in the length of the self-renewal phase in the lineage that lead to cancer, which allows us to infer the normal rate at which mutations accumulate in healthy stem cells.

We also use passenger mutations as a molecular clock to study the length of pancreatic cancer evolution. Most pancreatic cancer patients are diagnosed with metastatic disease and have median survival of less than a year. Whether the dismal prognosis of patients with pancreatic cancer compared to patients with other types of cancer is a result of late diagnosis or early dissemination of disease to distant organs is not known. Relying on data generated by sequencing the genomes of multiple metastases and primary tumors from seven pancreatic cancer patients, we performed a phylogenetic analysis to evaluate the clonal relationships among primary and metastatic cancers in individual patients. Using the fact that passenger mutations are collected with some probability at each cell division, we then performed a quantitative analysis of the timing of the genetic evolution of pancreatic cancer. Our results indicate at least a decade between the occurrence of the tumor initiating mutation and the acquisition of metastatic ability and define a broad time window of opportunity for early detection to prevent deaths from metastatic disease.


C Tomasetti and I Bozic (2015) The (not so) immortal strand hypothesis. Stem Cell Research 14, 238-241.
S Yachida, S Jones, I Bozic et al. (2010) Distant metastasis occurs late during the genetic evolution of pancreatic cancer. Nature 467, 1114-1117.

3. Spatial model of cancer evolution

Wespatial study the genetic evolution of cancer in three dimensional space. We are interested in how drivers and passengers accumulate in the presence of spatial constraints due to crowding and nutrient deprivation, which play a significant role in non-microscopic solid tumors. In this model, cells occupy sites of a 3D lattice and their division rate depends on the number of empty neighboring sites, hence replication is faster at the edge of the tumor. At each division, daughter cell occupies a site adjacent to the cell that divided, and both cells can gain new driver and passenger mutations with some probabilities. We also study the effects of migration and cell death on genetic heterogeneity of cancer.  


B Waclaw, I Bozic et al. (2015) A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity. Nature 525, 261-264.


Mathematical models of resistance to cancer therapy

Acquired resistance to treatment is a major impediment to successful eradication of cancer. Patients presenting with early-stage cancers can often be cured surgically, but patients with metastatic disease must be treated with systemic therapies. Targeted therapies, a new class of drugs, inhibit specific molecules implicated in tumor development and are typically less harmful to normal cells compared with chemotherapy and radiation. In the case of many targeted treatments, patients initially have a dramatic response, only to be followed by a regrowth of most of their lesions several months later. I study mathematical evolutionary models of resistance to cancer therapy in order to quantify the dynamics of treatment and resistance and predict optimal treatment strategies.

1. Evolution of resistance to single targeted therapies

We TMM_2012_Bozic1studied the fully stochastic version of the Luria-Delbrück model, which was originally used to study the evolution of resistance in bacteria, to show that acquired resistance to EGFR-inhibitor panitumumab in colorectal cancer is a fait accompli, and due to a small preexisting resistant subpopulation. We also studied the effects of density limitations in tumor growth on the evolution of resistance to therapy, using a density-dependent generalization of a branching process. In our model, tumor growth is initially exponential, but slows as the tumor size increases, and the tumor eventually reaches a steady state. In this steady state, density constraints prevent further growth, unless new mutations arise that allow the tumor to overcome these constraints. After the tumor has been at steady state for time T, treatment is initiated, and sensitive cells decline exponentially. Resistance mutations can appear in any of the phases of tumor evolution (expansion, steady state or treatment) and can be selectively neutral or deleterious before treatment. We calculate the overall probability of tumor eradication and time until there is probability p that all sensitive cells have been eradicated.


I Bozic, B Allen and MA Nowak (2012) Dynamics of targeted cancer therapy. Trends in Molecular Medicine 18, 311-316.
LA Diaz Jr, R Williams, J Wu, J R Hecht, J Berlin, B Allen, I Bozic et al. (2012) The molecular evolution of acquired resistance to targeted EGFR blockade in colorectal cancers. Nature 486, 537-540.

2. Evolution of resistance to combination therapy

Combination targeted therapies are widely believed to be the one of the best hopes for achieving long-term remissions in cancer patients. We study a mathematical model for the evolution of resistance to combination targeted therapy in order to critically evaluate the potential and limitations of such therapies in actual disease settings. Our model is based on a multitype branching process and includes cells sensitive and resistant to each of the drugs in the combination. We derive formulas for the probability that cells resistant to all drugs exist at the start of therapy, expected number of such cells, and the overall probability of treatment success. To obtain key parameters for the model, we studied the dynamics of 68 index lesions in 20 melanoma patients receiving the BRAF inhibitor vemurafenib. To determine the total extent of disease in typical patients who enroll for clinical trials, we quantified all radiographically detectable metastases in 22 independent patients, seven with pancreatic ductal adenocarcinomas, eleven with colorectal carcinomas, and six with melanomas.

We fiOLYMPUS DIGITAL CAMERAnd that combination therapy will not lead to long-term disease control in most patients if there is even one possible mutation that has the potential to cause cross-resistance to the administered drugs. In other words, if any of the 6.6 billion base pairs in a typical diploid cell can be mutated to a form that causes cross-resistance, the effects of combination therapy will not be dramatically better than monotherapy. Moreover, even if there is no possibility of a mutation that could cause resistance to two drugs, dual therapy will fail in patients with large disease burdens. In such cases, triple therapy will be needed – and this will only be successful if there is no possibility that a mutation could cause cross-resistance to all three drugs. Finally, we demonstrate that the current practice of administering targeted agents sequentially is a recipe for certain treatment failure and that simultaneous administration of targeted agents is essential for long-term disease control.


I Bozic*, JG Reiter*, B Allen* et al. (2013) Evolutionary dynamics of cancer in response to targeted combination therapy. eLife 2, e00747.
S Misale*, I Bozic* et al. (2015) Vertical suppression of the EGFR pathway prevents onset of resistance in colorectal cancers. Nature Communications 6, 8305.

3. Heterogeneity of resistance mutations

Failure of systemic cancer therapies is often due to a pre-existing resistant subpopulation of tumor cells, but little is known about genetic heterogeneity of this resistant subpopulation. We use mathematical modeling to study the evolution of resistance to anti-cancer treatment and describe for the first time the entire ensemble of resistant subclones in cancer therapy. We model the growth of a metastatic lesion as a branching process that starts from a single cell (the founder cell of the metastasis) that is sensitive to treatment. Sensitive cells divide with rate b and die with rate d. During cell division, one of the daughter cells receives a resistance mutation with probability u. Resistant mutations can be neutral in the absence of treatment, which means they have the same birth and death rates as sensitive cells, or they can be deleterious or advantageous, and we study all these cases. We label the resistance mutations that survive stochastic drift and establish a resistant subclone in their order of appearance and show that radiographically detectable lesions of solid malignancies contain multiple resistant subclones of comparable size. Surprisingly, we find that the ratio of the median numbers of cells in successive resistant subclones is independent of any model parameters: for example, the median of the second clone divided by the median of the first is √2 – 1. Finally, we show that our model is in excellent agreement with liquid biopsy data on resistance-associated mutations in colorectal cancer patients.


I Bozic and MA Nowak (2014) Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers. PNAS 111, 15964-15968.