A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations

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For each patient, the time from diagnosis to recurrence defined as second malignant event in any of the included head and neck sites , death, or censoring defined as the minimum of end-of-study, loss to follow-up, and 10 years was calculated. Kaplan-Meier curves were used to visualize the results. To characterize the field geometry at time of diagnosis with invasive cancer, we first derived the size distribution of the local field.

We refer to Supplemental Information S3 for details of the calculation. The corresponding density functions 4 for ages at diagnosis of 40, 60, and 80 years are shown in Figure 3A. Unfortunately, these results have an explicit dependence on the radial growth rate of the precancer field, c 2 s 1 , which cannot be directly estimated due to identifiability constraints we used literature-based estimates instead.

To overcome this issue, we focused on the relative size of a precancerous field compared to the field size at age More precisely, we introduced the relative mean field radius RMFR , defined as the mean field radius at a given age divided by the mean field radius at age 50 years. Based on the posterior distributions from the Bayesian inference, we computed the RMFR as a function of patient age at diagnosis in Figure 3B.

A The probability density function of the field radius at diagnosis is shown for patient ages 40, 60 and 80 years, respectively. B Based on the posterior distributions, the statistics of the relative mean field radius RMFR for different ages at diagnosis were computed. C The probability distribution of the number of distant fields is shown for patient ages 40, 60 and 80 years, respectively. Parameter values as in A.

D The probability of multiple unrelated fields at time of diagnosis was computed based on the posterior distributions. In addition to the local field size at diagnosis, we estimated the probability of harboring distant fields in addition to the local field. The exact distribution of the total number M t of clonally independent fields, including the local and all distant fields, is found in Supplemental Information S3. Using this distribution, we then computed the probability of harboring at least two clonally unrelated fields in the head and neck region at time of diagnosis,.

Accounting for the posterior distributions of the parameter groups, the probability of harboring at least one distant field is shown as a function of age at diagnosis in Figure 3D. To test the validity of our model, we made model-based predictions and tested them against outcome data from SEER. Considering the predicted increase of local field size with age at diagnosis Figure 3A , an age-independent margin width implies that, for the same tumor size, the area of precancerous tissue left behind after resection of a primary tumor is bigger in older patients with larger precancer fields compared to younger patients with smaller precancer fields , see Figure 4.

This increase in recurrence risk in older patients is further increased by the elevated probability of harboring distant fields Figure 3B that may not be affected by local excision of the primary tumor. The corresponding Kaplan-Meier curve for recurrence-free survival is shown in Figure 5A. A Illustration of age-related differences in local field size and number of unrelated fields. Before surgery, only one local field may be present in a younger patient left , whereas a larger local field and additional distant fields may be present in an older patient right.

B During surgery the local field is removed in the younger patient left , but only partially resected in the older patient right , where the residual field portions elevate the risk of recurrence. Because adjuvant radiation therapy after surgery targets the margins of the resected tissue portion, the combination of surgery and radiation is more likely to eradicate the entire local field and hence diminish the recurrence risk from a residual field portion. Therefore, we hypothesized a reduction in the age-related difference in recurrence risk among patients who received radiation in addition to surgery.

This prediction was corroborated by the analysis of outcome data. Finally, we note that for the patient group with radiation but without surgery Figure 5C , no significant difference in recurrence risk between the two age groups was observed. However, we were not able to draw any definite conclusions because of a small sample size and patient attrition in the younger age group.

Effect of Dedifferentiation on Time to Mutation Acquisition in Stem Cell-Driven Cancers

Leftover fields of precancerous tissue that are not removed during resection of HPV-negative, tobacco-related HNSCC lead to an increased risk of local recurrence. Due to their visually normal appearance, epithelial precancer fields are generally not detectable at the time of surgery without harmful biopsies of the surrounding tissue portions. Thus there is a need for non-invasive methods to predict the extent of the field and to make patient-tailored decisions with respect to optimal treatment modality, size of excision margins, and frequency of postoperative surveillance.

In this work, we developed and calibrated a mechanistic model of spatial tumorigenesis in tobacco-related HNSCC. Following the lines of previous multi-scale modeling work 40 — 43 , we linked biological mechanism at the tissue scale to data at the population scale and used a Bayesian framework to ensure compatibility of the model dynamics across scales. Using the model we made predictions about the geometry of precancerous fields at the time of diagnosis with invasive cancer and tested them against outcome data.

The models predicted a strong dependence of the local field size on age at diagnosis. More precisely, we found the expected field size of the local precancerous field surrounding the primary tumor to double between the ages of 50 and 90 years. Together, these findings indicate that the current one-size-fits-all approach to the width of surgical excision margins may need to be critically reevaluated. Based on the model predictions, we hypothesized that larger local fields and a higher risk of harboring multiple fields in older patients would translate into a higher recurrence risk in older compared to younger patients when treated by surgery only.

As predicted, we found a higher risk of local recurrence in older patients compared to younger patients. In contrast, we did not find a statistically significant age effect in patients who received adjuvant radiation therapy. This is consistent with the observation that radiation is less focalized than surgical excision and hence is more likely to remove the surrounding field portions, even in older patients with larger fields.

In addition, radiation may have an age-specific effect on local recurrence that is independent of the field size.

A generalized theory of age-dependent carcinogenesis

For example, studies on the effects of radiotherapy in patients with ductal carcinoma in situ reported a higher likelihood of recurrence in patients who received adjuvant radiation therapy compared to those who did not 44 , It is important to note that the observed recurrence patterns may be due to a combination of several biological and clinical factors, not just the age-related size of the precancer field. Our study has several limitations.

First, inherent limitations of the SEER database such as ascertainment biases and incomplete recurrence records may impact the validity of our results Third, although the incidence data used for model calibration was restricted to a relatively short period — , it is likely subject to secular trends that are at least partially due to changing smoking patterns in the population 4 , In addition, it has been shown that smoking cessation leads to a slow decrease in head and neck cancer risk 48 , which may result in differential field sizes between former and current tobacco users.

In future work, the use of more granular smoking prevalence and cancer incidence data, adjusted for secular trends, is expected to address these issues and improve the model predictions. Fourth, a limitation shared with most multistage modeling analyses is the assumption of identical parameters for all individuals.

A stochastic model for immunological feedback in carcinogenesis : analysis and approximations

This issue is partially mitigated by the Bayesian approach, which provides posterior distribution of parameters rather than point estimates. However, incorporating patient-level heterogeneity into the modeling framework constitutes a critical next step toward the long-term goal of developing personalized approaches to head and neck cancer care.

Finally, we did not explicitly account for the role of the immune system as a first line of defense against neoplastic progression. Although immune effects could be incorporated into the model, current knowledge about the exact mechanisms of immune response to neoplastic transformation seems insufficient to develop meaningful models.

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Historically, different types of multistage models have been used to infer the nature of cancer-causing mechanisms based on incidence and mortality data. The first such models 49 were based on the assumption that cancer arises as the product of an organ-specific number of rare mutations. These models assumed a well-mixed population of cells and neglected cellular dynamics and spatial tissue structure.

Later, these models were extended to account for clonal expansions of precancerous cells 40 , 41 , 50 , and used to analyze the number and size of premalignant clones in non-spatial populations, both for exponential mean growth 51 and more general growth dynamics 52 , In parallel to multistage models, population dynamic models such as the Wright-Fisher and Moran processes have also been used to model cancer initiation under well-mixed assumptions 54 , In these models, the expansion of premalignant clones is constrained by competition with healthy cells or premalignant cells at other stages.

Our current work constitutes an extension of both multistage and population dynamic models. In particular, we accounted for the spatial structure of the epithelial lining of head and neck sites, and we developed a mechanistic model based on the current understanding of tissue homeostasis and molecular biology of HPV-negative HNSCC.

For the sake of comparison, we also fitted the classical multistage model 49 to the incidence data, see Supplemental Information S4.