Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology—defined here simply as the use of mathematics in cancer research—complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

Objective IBD confers an increased lifetime risk of developing colorectal cancer (CRC), and colitis-associated CRC (CA-CRC) is molecularly distinct from sporadic CRC (S-CRC). Here we have dissected the evolutionary history of CA-CRC using multiregion sequencing. Design Exome sequencing was performed on fresh-frozen multiple regions of carcinoma, adjacent non-cancerous mucosa and blood from 12 patients with CA-CRC (n=55 exomes), and key variants were validated with orthogonal methods. Genome-wide copy number profiling was performed using single nucleotide polymorphism arrays and low-pass whole genome sequencing on archival non-dysplastic mucosa (n=9), low-grade dysplasia (LGD; n=30), high-grade dysplasia (HGD; n=13), mixed LGD/HGD (n=7) and CA-CRC (n=19). Phylogenetic trees were reconstructed, and evolutionary analysis used to reveal the temporal sequence of events leading to CA-CRC. Results 10/12 tumours were microsatellite stable with a median mutation burden of 3.0 single nucleotide alterations (SNA) per Mb, ~20% higher than S-CRC (2.5 SNAs/Mb), and consistent with elevated ageing-associated mutational processes. Non-dysplastic mucosa had considerable mutation burden (median 47 SNAs), including mutations shared with the neighbouring CA-CRC, indicating a precancer mutational field. CA-CRCs were often near triploid (40%) or near tetraploid (20%) and phylogenetic analysis revealed that copy number alterations (CNAs) began to accrue in non-dysplastic bowel, but the LGD/HGD transition often involved a punctuated ‘catastrophic’ CNA increase. Conclusions Evolutionary genomic analysis revealed precancer clones bearing extensive SNAs and CNAs, with progression to cancer involving a dramatic accrual of CNAs at HGD. Detection of the cancerised field is an encouraging prospect for surveillance, but punctuated evolution may limit the window for early detection.

Cancer arises through a multistage process, but it is not fully clear how this process influences the age-specific incidence curve. Studies of colorectal and pancreatic cancer using the multistage clonal expansion (MSCE) model have identified two phases of the incidence curves. One phase is linear, beginning about age of 60 years, suggesting that at least two rare rate-limiting mutations occur before clonal expansion of premalignant cells. A second phase is exponential, seen in early-onset cancers occurring before the age of 60 years that are associated with premalignant clonal expansion. Here, we extend the MSCE model to include clonal expansion of malignant cells, an advance that permits study of the effects of tumor growth and extinction on the incidence of colorectal, gastric, pancreatic, and esophageal adenocarcinomas in the digestive tract. After adjusting the age-specific incidence for birth-cohort and calendar-year trends, we found that initiating mutations and premalignant cell kinetics can explain the primary features of the incidence curve. However, we also found that the incidence data of these cancers harbored information on the kinetics of malignant clonal expansion before clinical detection, including tumor growth rates and extinction probabilities on three characteristic time scales for tumor progression. In addition, the data harbored information on the mean sojourn times for premalignant clones until occurrence of either the first malignant cell or the first persistent (surviving) malignant clone. Finally, the data also harbored information on the mean sojourn time of persistent malignant clones to the time of diagnosis. In conclusion, cancer incidence curves can harbor significant information about hidden processes of tumor initiation, premalignant clonal expansion, and malignant transformation, and even some limited information on tumor growth before clinical detection. Cancer Res; 73(3); 1086–96. ©2012 AACR.

Background: The incidence of esophageal adenocarcinoma (EAC) has increased five-fold in the United States since 1975. The aim of our study was to estimate future U.S. EAC incidence and mortality and to shed light on the potential drivers in the disease process that are conduits for the dramatic increase in EAC incidence. Methods: A consortium of three research groups calibrated independent mathematical models to clinical and epidemiologic data including EAC incidence from the Surveillance, Epidemiology, and End Results (SEER 9) registry from 1975 to 2010. We then used a comparative modeling approach to project EAC incidence and mortality to year 2030. Results: Importantly, all three models identified birth cohort trends affecting cancer progression as a major driver of the observed increases in EAC incidence and mortality. All models predict that incidence and mortality rates will continue to increase until 2030 but with a plateauing trend for recent male cohorts. The predicted ranges of incidence and mortality rates (cases per 100,000 person years) in 2030 are 8.4 to 10.1 and 5.4 to 7.4, respectively, for males, and 1.3 to 1.8 and 0.9 to 1.2 for females. Estimates of cumulative cause-specific EAC deaths between both sexes for years 2011 to 2030 range between 142,300 and 186,298, almost double the number of deaths in the past 20 years. Conclusions: Through comparative modeling, the projected increases in EAC cases and deaths represent a critical public health concern that warrants attention from cancer control planners to prepare potential interventions. Impact: Quantifying this burden of disease will aid health policy makers to plan appropriate cancer control measures. Cancer Epidemiol Biomarkers Prev; 23(6); 997–1006. ©2014 AACR.

Objective Barrett’s oesophagus (BE) is a known precursor to oesophageal adenocarcinoma (OAC) but current clinical data have not been consolidated to address whether BE is the origin of all incident OAC, which would reinforce evidence for BE screening efforts. We aimed to answer whether all expected prevalent BE, diagnosed and undiagnosed, could account for all incident OACs in the US cancer registry data. Design We used a multiscale computational model of OAC that includes the evolutionary process from normal oesophagus through BE in individuals from the US population. The model was previously calibrated to fit Surveillance, Epidemiology and End Results cancer incidence curves. Here, we also utilised age-specific and sex-specific US census data for numbers at-risk. The primary outcome for model validation was the expected number of OAC cases for a given calendar year. Secondary outcomes included the comparisons of resulting model-predicted prevalence of BE and BE-to-OAC progression to the observed prevalence and progression rates. Results The model estimated the total number of OAC cases from BE in 2010 was 9970 (95% CI: 9140 to 11 980), which recapitulates nearly all OAC cases from population data. The model simultaneously predicted 8%–9% BE prevalence in high-risk males age 45–55, and 0.1%–0.2% non-dysplastic BE-to-OAC annual progression in males, consistent with clinical studies. Conclusion There are likely few additional OAC cases arising in the US population outside those expected from individuals with BE. Effective screening of high-risk patients could capture the majority of population destined for OAC progression and potentially decrease mortality through early detection and curative removal of small (pre)cancers during surveillance.