Unlocking the Genetic Enigma: Estimating the Hidden Horde of Barcoded Cancer Cells using a Molecular German Tank Problem Approach

doi:10.21203/rs.3.rs-3212854/v1

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Unlocking the Genetic Enigma: Estimating the Hidden Horde of Barcoded Cancer Cells using a Molecular German Tank Problem Approach

https://doi.org/10.21203/rs.3.rs-3212854/v1

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Accurate estimation of the number of undetected cancer cells after treatment is of paramount importance in oncology to gauge treatment efficacy and potential disease recurrence. Drawing inspiration from the German tank problem, we present a novel sampling-based molecular approach to estimate the population of elusive cancer cells that remain undetected by current molecular techniques. This study proposes an innovative approach to be undertaken in the future for estimating the number of undetected cancer cells post-treatment. Inspired by the renowned German tank problem, our sampling-based molecular methodology involves creating a unique molecular barcode representing the increasing mutations in cancerous cells over time. By collecting serial samples from cancer patients after treatment, we aim to statistically analyze and develop a robust estimation model to infer the total number of undetected cancer cells. The validation of this approach using an independent patient cohort would help ensure its reliability. Successful implementation of this methodology has the potential to revolutionize cancer research, providing crucial insights into residual cancer cell populations and facilitating the development of personalized treatment strategies. Ultimately, this future research endeavor could significantly impact cancer care, leading to improved patient outcomes and advancing our understanding of post-treatment cancer dynamics. This innovative approach may transform cancer research and improve patient outcomes by offering insights into the post-treatment cancer landscape that was hitherto concealed.

Health sciences/Oncology/Cancer/Cancer screening

Biological sciences/Cancer/Cancer screening

residual cancer cells

molecular barcoding

german tank problem

estimation methodology

Cancer remains one of the most challenging and complex diseases confronting modern medicine. While significant progress has been made in cancer treatment, the accurate assessment of treatment outcomes and the presence of residual cancer cells continue to be critical factors influencing patient prognosis and therapeutic strategies. Undetected cancer cells, despite conventional treatment, can serve as a source of disease relapse and progression, necessitating innovative approaches to estimate their prevalence.

In recent years, advances in molecular techniques have revolutionized cancer research and diagnostics, enabling a deeper understanding of the genomic landscape of cancer cells (Vogelstein et al., 2013). Molecular barcoding techniques, which involve the characterization of unique molecular signatures of cancer cells, have emerged as powerful tools for monitoring tumor evolution and treatment response (Kim et al., 2018). Leveraging such techniques, our proposed study aims to estimate the number of undetected cancer cells post-treatment through a sampling-based approach. Inspired by the renowned German tank problem (Simon et al. 2023), a method used during World War II to estimate enemy tank production based on captured serial numbers, our approach relies on the concept of statistical sampling to infer the total number of undetected cancer cells. By collecting serial samples from cancer patients at different time points following treatment, we can construct a unique molecular barcode that reflects the increasing number of mutations in cancerous cells over time. The statistical analysis of these barcodes will allow us to estimate the elusive population of undetected cancer cells, thereby providing valuable insights into post-treatment residual disease burden. The importance of accurately estimating undetected cancer cells cannot be overstated. Identifying residual disease at an early stage can guide the implementation of appropriate treatment interventions, potentially improving patient outcomes and reducing the risk of disease relapse (Altman et al., 2012). Moreover, this estimation approach holds promise for the development of personalized treatment regimens that consider the unique dynamics of cancer evolution in individual patients (Gerlinger et al. 2012). To ensure the reliability and applicability of our proposed sampling-based molecular approach, we plan to validate our findings using an independent cohort of cancer patients. The results of this study could significantly contribute to the field of oncology, offering valuable information about the hidden cancer cell population that has remained undetectable with current molecular techniques.

In summary, our project represents a cutting-edge initiative to estimate undetected cancer cells post-treatment through the innovative fusion of statistical sampling and molecular barcoding techniques. By shedding light on the dynamics of residual cancer cells, this study has the potential to drive advancements in cancer research, treatment, and personalized medicine, ultimately improving patient care and outcomes.

To estimate the number of barcoded cancer cells using the German tank problem approach (Simon et al., 2023), we can adapt the mathematical model used in the original problem. In the German tank problem, the goal is to estimate the total number of items (tanks) based on observing a NGS sequence of serial numbers from a sample (Vogelstein et al., 2013). In our case, we want to estimate the total number of barcoded cancer cells based on the molecular barcodes observed in serial samples.

Patient Selection: Identify a cohort of cancer patients who have undergone treatment and are in remission or stable disease status.
Sample Collection: Obtain serial samples (biopsies or blood samples) from each patient at multiple time points following treatment, such as 3 months, 6 months, 1 year, etc.
Molecular Barcode Creation: Perform next-generation sequencing (NGS) on the samples to identify and characterize the mutations present in cancer cells (Kamps et al., 2017). Use this information to create a unique molecular barcode for each sample, representing the accumulated mutations in cancer cells over time.
Counting Reference: Establish a reference count of the number of cancer cells present in a subset of patients using a highly sensitive and quantitative detection technique, such as digital PCR or single-cell RNA sequencing (Sefrioui et al., 2017; Kuksin et al., 1990).
Estimation Model: Develop a statistical model based on the German tank problem to estimate the total number of undetected cancer cells in each patient's samples (Simon et al., 2023). This model should consider the molecular barcode, time since treatment, and the known reference count.
Validation Cohort: Utilize an independent cohort of cancer patients to validate the accuracy and reliability of the estimation model.
Descriptive Statistics: Calculate summary statistics (mean, median, standard deviation, etc.) for the number of mutations in the molecular barcode at each time point.
Correlation Analysis: Assess the relationship between the number of mutations and the time since treatment using correlation coefficients.
Estimation Model: Apply the sampling-based estimation model to the patient samples to estimate the total number of undetected cancer cells at each time point.
Confidence Intervals: Calculate confidence intervals around the estimated values to quantify the uncertainty in the estimates.
Validation: Evaluate the performance of the estimation model in the validation cohort using measures such as root mean square error (RMSE) and coefficient of determination (R-squared).
Sensitivity Analysis: Perform sensitivity analysis to investigate the impact of different assumptions and parameters on the estimated values.

Let's define the following variables:

N: Total number of barcoded cancer cells in the patient's body (unknown parameter we want to estimate).

n: Number of serial samples collected from the patient after treatment.

M_i: The number of unique molecular barcodes observed in the i-th sample (i = 1 to n) (Kim et al., 2018).

Now, we can derive the mathematical model for estimation:

Sample Range (R):

The range R is defined as the difference between the maximum and minimum observed molecular barcode counts in the samples.

R = max(M_i) - min(M_i)

Estimated Total (E[N]): The estimated total (E[N]) is an estimator for the total number of barcoded cancer cells. In the German tank problem, this estimator is calculated as follows:

E[N] = max(M_i) + R − 1

Confidence Intervals: To estimate the uncertainty in our estimation, we can calculate the confidence intervals (Altman et al., 2012). The lower and upper bounds of the confidence interval are given by:

Lower Bound = E[N] - Z * (R / √(n)) Upper Bound = E[N] + Z * (R / √(n))

Where Z is the critical value of the standard normal distribution corresponding to the desired confidence level (e.g., 95% confidence level corresponds to Z ≈ 1.96).

The proposed sampling-based molecular approach, inspired by the German tank problem, presents a novel and promising method to estimate the number of undetected barcoded cancer cells after treatment. By leveraging the unique molecular barcodes and analyzing serial samples, we aimed to shed light on the elusive population of residual cancer cells that may evade current molecular detection methods.

One of the significant advantages of our methodology lies in its potential to offer personalized treatment insights. Estimating the number of undetected cancer cells post-treatment could provide critical information for clinicians, enabling them to tailor therapies based on the individualized cancer dynamics of each patient (Shemesh et al., 2021). By gaining a better understanding of the residual cancer cell burden, treatment plans can be optimized to address the unique characteristics of each patient's cancer, potentially leading to improved treatment responses and outcomes. Furthermore, accurate estimation of residual cancer cells could prove instrumental in predicting early disease relapse. Identifying even a small number of undetected cancer cells can be indicative of potential disease progression or recurrence (Gatenby et al., 2018). Early prediction of relapse can prompt timely interventions, such as additional treatments or close monitoring, which may prevent disease escalation and improve long-term survival rates.

Our approach also has the potential to reduce overtreatment, a significant concern in cancer management. With precise estimation of residual cancer cells, unnecessary aggressive treatments can be avoided, sparing patients from potential side effects and enhancing their quality of life (Marzo-Castillejo et al., 2018). Tailoring treatments based on accurate estimates may lead to more personalized and less toxic therapeutic strategies, enhancing patient well-being during their cancer journey. Moreover, the dynamic monitoring capability of our methodology represents a major advantage. The serial sampling approach allows for continuous tracking of cancer cells over time, providing insights into the evolution of cancer populations during and after treatment (Burrell et al., 2014). This longitudinal view of cancer dynamics can help identify patterns of resistance or response to therapy, facilitating the development of adaptive treatment plans as cancer cells evolve.

This mathematical model assumes that the barcodes are randomly distributed among the cancer cells and that the samples are collected independently and randomly from the patient's body. It's important to validate and fine-tune this mathematical model using simulated data or data from actual patient samples to assess its performance and accuracy in estimating the total number of undetected barcoded cancer cells after treatment. Additionally, as the field of oncology is highly complex and dynamic, the actual mathematical model might require additional refinements and adjustments based on the specific characteristics of the cancer type, treatment response, and evolution of cancer cells over time.

While our methodology shows promising advantages, it is essential to acknowledge potential disadvantages. Implementing molecular barcoding and next-generation sequencing techniques for creating unique molecular barcodes may present technical challenges and require advanced expertise (Schmitt et al., 2012). Ensuring the accuracy and reproducibility of these techniques is critical to obtain reliable estimates of undetected cancer cells. Another potential limitation of our methodology is the variability in the composition of serial samples. Factors such as spatial heterogeneity within the tumor and sampling different tumor regions may impact the accuracy of estimation (Swanton et al., 2012). Careful consideration and standardization of sampling protocols are necessary to minimize biases and improve the precision of the results. Furthermore, the applicability of our methodology may vary across different cancer types and treatment modalities. The genetic and molecular characteristics of various cancers can differ significantly, influencing the accuracy and generalizability of our estimates (Dienstmann et al., 2017). Therefore, validation studies across diverse cancer types are essential to assess the robustness and reliability of our approach.

In conclusion, our sampling-based molecular approach, akin to the German tank problem, holds significant promise for estimating undetected barcoded cancer cells post-treatment. By providing personalized treatment insights, early relapse prediction, and reduced overtreatment, our methodology could revolutionize cancer care and management. The ability to dynamically monitor cancer cell populations represents a unique advantage, fostering adaptive treatment strategies. Nevertheless, technical challenges and potential limitations call for careful validation and refinement. As we embark on future research endeavors, continuous improvement and validation of this methodology are vital to unlock the full potential of unraveling the enigma of residual cancer cells.

Funding statement: Funded by Uskudar University.

Conflict of interest: No authors have any conflict of interest.

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Unlocking the Genetic Enigma: Estimating the Hidden Horde of Barcoded Cancer Cells using a Molecular German Tank Problem Approach

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