Differences in predictions of ODE models of tumor growth: a cautionary example

The initial engraftment of tumor cells is critical for the future growth pattern: a mathematical study based on simulations and animal experiments

Background

Xenograft mouse tumor models are used to study mechanisms of tumor growth and metastasis formation and to investigate the efficacy of different therapeutic interventions. After injection the engrafted cells form a local tumor nodule. Following an initial lag period of several days, the size of the tumor is measured periodically throughout the experiment using calipers. This method of determining tumor size is error prone because the measurement is two-dimensional (calipers do not measure tumor depth). Primary tumor growth can be described mathematically by suitable growth functions, the choice of which is not always obvious. Growth parameters provide information on tumor growth and are determined by applying nonlinear curve fitting.

Methods

We used self-generated synthetic data including random measurement errors to research the accuracy of parameter estimation based on caliper measured tumor data. Fit metrics were investigated to identify the most appropriate growth function for a given synthetic dataset. We studied the effects of measuring tumor size at different frequencies on the accuracy and precision of the estimated parameters. For curve fitting with fixed initial tumor volume, we varied this fixed initial volume during the fitting process to investigate the effect on the resulting estimated parameters. We determined the number of surviving engrafted tumor cells after injection using ex vivo bioluminescence imaging, to demonstrate the effect on experiments of incorrect assumptions about the initial tumor volume.

Results

To select a suitable growth function, measurement data from at least 15 animals should be considered. Tumor volume should be measured at least every three days to estimate accurate growth parameters. Daily measurement of the tumor volume is the most accurate way to improve long-term predictability of tumor growth. The initial tumor volume needs to have a fixed value in order to achieve meaningful results. An incorrect value for the initial tumor volume leads to large deviations in the resulting growth parameters.

Conclusions

The actual number of cancer cells engrafting directly after subcutaneous injection is critical for future tumor growth and distinctly influences the parameters for tumor growth determined by curve fitting.

Understanding the effect of measurement time on drug characterization

In order to determine correct dosage of chemotherapy drugs, the effect of the drug must be properly quantified. There are two important values that characterize the effect of the drug: εmax is the maximum possible effect of a drug, and IC50 is the drug concentration where the effect diminishes by half. There is currently a problem with the way these values are measured because they are time-dependent measurements. We use mathematical models to determine how the εmax and IC50 values depend on measurement time and model choice. Seven ordinary differential equation models (ODE) are used for the mathematical analysis; the exponential, Mendelsohn, logistic, linear, surface, Bertalanffy, and Gompertz models. We use the models to simulate tumor growth in the presence and absence of treatment with a known IC50 and εmax. Using traditional methods, we then calculate the IC50 and εmax values over fifty days to show the time-dependence of these values for all seven mathematical models. The general trend found is that the measured IC50 value decreases and the measured εmax increases with increasing measurement day for most mathematical models. Unfortunately, the measured values of IC50 and εmax rarely matched the values used to generate the data. Our results show that there is no optimal measurement time since models predict that IC50 estimates become more accurate at later measurement times while εmax is more accurate at early measurement times.

Mathematical oncology and it's application in non melanoma skin cancer - A primer for radiation oncology professionals

Cancers of the skin (the majority of which are basal and squamous cell skin carcinomas, but also include the rarer Merkel cell carcinoma) are overwhelmingly the most common of all types of cancer. Most of these are treated surgically, with radiation reserved for those patients with high risk features or anatomical locations less suitable for surgery. Given the high incidence of both basal and squamous cell carcinomas, as well as the relatively poor outcome for Merkel cell carcinoma, it is useful to investigate the role of other disciplines regarding their diagnosis, staging and treatment. Mathematical modelling is one such area of investigation. The use of mathematical modelling is a relatively recent addition to the armamentarium of cancer treatment. It has long been recognised that tumour growth and treatment response is a complex, non-linear biological phenomenon with many mechanisms yet to be understood. Despite decades of research, including clinical, population and basic science approaches, we continue to be challenged by the complexity, heterogeneity and adaptability of tumours, both in individual patients in the oncology clinic and across wider patient populations. Prospective clinical trials predominantly focus on average outcome, with little understanding as to why individual patients may or may not respond. The use of mathematical models may lead to a greater understanding of tumour initiation, growth dynamics and treatment response.

Recent advances in physiologically based pharmacokinetic and pharmacodynamic models for anticancer nanomedicines

Nanoparticles (NPs) have distinct pharmacokinetic (PK) properties and can potentially improve the absorption, distribution, metabolism, and elimination (ADME) of small-molecule drugs loaded therein. Owing to the unwanted toxicities of anticancer agents in healthy organs and tissues, their precise delivery to the tumor is an essential requirement. There have been numerous advancements in the development of nanomedicines for cancer therapy. Physiologically based PK (PBPK) models serve as excellent tools for describing and predicting the ADME properties and the efficacy and toxicity of drugs, in combination with pharmacodynamic (PD) models. The recent preliminary application of these modeling approaches to NPs demonstrated their potential benefits in research and development processes relevant to the ADME and pharmacodynamics of NPs and nanomedicines. Here, we comprehensively review the pharmacokinetics of NPs, the developed PBPK models for anticancer NPs, and the developed PD model for anticancer agents.

The Application of the Logistic Equation Model to Predict the Remineralization Characteristics of Desensitizing Paste

Objectives

A mathematical model making using of the Verhulst logistic equation was developed to predict the remineralization behaviors of desensitizing paste.

Methods

The input parameter used for the model was obtained experimentally by brushing twenty-one simulated dentin specimens for seven days with three sample groups, namely, EB@TiO₂, Colgate Pro-relief, and Sensodyne repair (n = 7). A field emission scanning electron microscope (FESEM) and ImageJ software were used to observe and measure the % occluded ratio of the dentin surface. The model fittings for the three sample groups were carried out using the built-in MATLAB least-squares fitting routine fmincon in the optimization toolbox.

Results

The results suggest that the experimental parameter were in agreement with the model. It was found that the logistic equation model can make a future prediction of the remineralization pattern for EB@TiO₂ and Colgate Pro-relief. It was, however, found that the trajectory for the Sensodyne repair was a bit complex, thus making the prediction difficult.

Conclusions

Overall, the salient feature of this study suggests that the logistic equation could be used to predict the remineralization behavior of desensitizing paste in the management of sensitive tooth.

Towards quantitative imaging biomarkers of tumor dissemination: A multi-scale parametric modeling of multiple myeloma

The advent of medical imaging and automatic image analysis is bringing the full quantitative assessment of lesions and tumor burden at every clinical examination within reach. This opens avenues for the development and testing of functional disease models, as well as their use in the clinical practice for personalized medicine. In this paper, we introduce a Bayesian statistical framework, based on mixed-effects models, to quantitatively test and learn functional disease models at different scales, on population longitudinal data. We also derive an effective mathematical model for the crossover between initially detected lesions and tumor dissemination, based on the Iwata-Kawasaki-Shigesada model. We finally propose to leverage this descriptive disease progression model into model-aware biomarkers for personalized risk-assessment, taking all available examinations and relevant covariates into account. As a use case, we study Multiple Myeloma, a disseminated plasma cell cancer, in which proper diagnostics is essential, to differentiate frequent precursor state without end-organ damage from the rapidly developing disease requiring therapy. After learning the best biological models for local lesion growth and global tumor burden evolution on clinical data, and computing corresponding population priors, we use individual model parameters as biomarkers, and can study them systematically for correlation with external covariates, such as sex or location of the lesion. On our cohort of 63 patients with smoldering Multiple Myeloma, we show that they perform substantially better than other radiological criteria, to predict progression into symptomatic Multiple Myeloma. Our study paves the way for modeling disease progression patterns for Multiple Myeloma, but also for other metastatic and disseminated tumor growth processes, and for analyzing large longitudinal image data sets acquired in oncological imaging. It shows the unprecedented potential of model-based biomarkers for better and more personalized treatment decisions and deserves being validated on larger cohorts to establish its role in clinical decision making.

Mathematical Models of Cancer: When to Predict Novel Therapies, and When Not to

The number of publications on mathematical modeling of cancer is growing at an exponential rate, according to PubMed records, provided by the US National Library of Medicine and the National Institutes of Health. Seminal papers have initiated and promoted mathematical modeling of cancer and have helped define the field of mathematical oncology (Norton and Simon in J Natl Cancer Inst 58:1735-1741, 1977; Norton in Can Res 48:7067-7071, 1988; Hahnfeldt et al. in Can Res 59:4770-4775, 1999; Anderson et al. in Comput Math Methods Med 2:129-154, 2000. https://doi.org/10.1080/10273660008833042 ; Michor et al. in Nature 435:1267-1270, 2005. https://doi.org/10.1038/nature03669 ; Anderson et al. in Cell 127:905-915, 2006. https://doi.org/10.1016/j.cell.2006.09.042 ; Benzekry et al. in PLoS Comput Biol 10:e1003800, 2014. https://doi.org/10.1371/journal.pcbi.1003800 ). Following the introduction of undergraduate and graduate programs in mathematical biology, we have begun to see curricula developing with specific and exclusive focus on mathematical oncology. In 2018, 218 articles on mathematical modeling of cancer were published in various journals, including not only traditional modeling journals like the Bulletin of Mathematical Biology and the Journal of Theoretical Biology, but also publications in renowned science, biology, and cancer journals with tremendous impact in the cancer field (Cell, Cancer Research, Clinical Cancer Research, Cancer Discovery, Scientific Reports, PNAS, PLoS Biology, Nature Communications, eLife, etc). This shows the breadth of cancer models that are being developed for multiple purposes. While some models are phenomenological in nature following a bottom-up approach, other models are more top-down data-driven. Here, we discuss the emerging trend in mathematical oncology publications to predict novel, optimal, sometimes even patient-specific treatments, and propose a convention when to use a model to predict novel treatments and, probably more importantly, when not to.

Best fitting tumor growth models of the von Bertalanffy-PütterType

Background

Longitudinal studies of tumor volume have used certain named mathematical growth models. The Bertalanffy-Pütter differential equation unifies them: It uses five parameters, amongst them two exponents related to tumor metabolism and morphology. Each exponent-pair defines a unique three-parameter model of the Bertalanffy-Pütter type, and the above-mentioned named models correspond to specific exponent-pairs. Amongst these models we seek the best fitting one.

Method

The best fitting model curve within the Bertalanffy-Pütter class minimizes the sum of squared errors (SSE). We investigate also near-optimal model curves; their SSE is at most a certain percentage (e.g. 1%) larger than the minimal SSE. Models with near-optimal curves are visualized by the region of their near-optimal exponent pairs. While there is barely a visible difference concerning the goodness of fit between the best fitting and the near-optimal model curves, there are differences in the prognosis, whence the near-optimal models are used to assess the uncertainty of extrapolation.

Results

For data about the growth of an untreated tumor we found the best fitting growth model which reduced SSE by about 30% compared to the hitherto best fit. In order to analyze the uncertainty of prognosis, we repeated the search for the optimal and near-optimal exponent-pairs for the initial segments of the data (meaning the subset of the data for the first n days) and compared the prognosis based on these models with the actual data (i.e. the data for the remaining days). The optimal exponent-pairs and the regions of near-optimal exponent-pairs depended on how many data-points were used. Further, the regions of near-optimal exponent-pairs were larger for the first initial segments, where fewer data were used.

Conclusion

While for each near optimal exponent-pair its best fitting model curve remained close to the fitted data points, the prognosis using these model curves differed widely for the remaining data, whence e.g. the best fitting model for the first 65 days of growth was not capable to inform about tumor size for the remaining 49 days. For the present data, prognosis appeared to be feasible for a time span of ten days, at most.

Computational modelling of resistance and associated treatment response heterogeneity in metastatic cancers

Metastatic cancer patients invariably develop treatment resistance. Different levels of resistance lead to observed heterogeneity in treatment response. The main goal was to evaluate treatment response heterogeneity with a computation model simulating the dynamics of drug-sensitive and drug-resistant cells. Model parameters included proliferation, drug-induced death, transition and proportion of intrinsically resistant cells. The model was benchmarked with imaging metrics extracted from 39 metastatic prostate cancer patients who had ¹⁸F-NaF-PET/CT scans performed at baseline and at three cycles into chemotherapy or hormonal therapy. Two initial model assumptions were evaluated: considering only inter-patient heterogeneity and both inter-patient and intra-patient heterogeneity in the proportion of intrinsically resistant cells. The correlation between the median proportion of intrinsically resistant cells and baseline patient-level imaging metrics was assessed with Spearman's rank correlation coefficient. The impact of model parameters on simulated treatment response was evaluated with a sensitivity study. Treatment response after periods of six, nine, and 12 months was predicted with the model. The median predicted range of response for patients treated with both therapies was compared with a Wilcoxon rank sum test. For each patient, the time was calculated when the proportion of disease with a non-favourable response outperformed a favourable response. By taking into account inter-patient and intra-patient heterogeneity in the proportion of intrinsically resistant cells, the model performed significantly better ([Formula: see text]) than by taking into account only inter-patient heterogeneity ([Formula: see text]). The median proportion of intrinsically resistant cells showed a moderate correlation (ρ  =  0.55) with mean patient-level uptake, and a low correlation (ρ  =  0.36) with the dispersion of mean metastasis-level uptake in a patient. The sensitivity study showed a strong impact of the proportion of intrinsically resistant cells on model behaviour after three cycles of therapy. The difference in the median range of response (MRR) was not significant between cohorts at any time point (p   >  0.15). The median time when the proportion of disease with a non-favourable response outperformed the favourable response was eight months, for both cohorts. The model provides an insight into inter-patient and intra-patient heterogeneity in the evolution of treatment resistance.

Single-cell imaging of CAR T cell activity in vivo reveals extensive functional and anatomical heterogeneity

Cazaux et al. use intravital imaging to dissect anti-CD19 CAR T cell activity. This study uncovers both anatomical and functional diversity in the outcome of anti-CD19 CAR T cell interactions with tumor cells impacting engraftment, killing dynamics, and tumor immunoediting.

CAR T cells represent a potentially curative strategy for B cell malignancies. However, the outcome and dynamics of CAR T cell interactions in distinct anatomical sites are poorly understood. Using intravital imaging, we tracked interactions established by anti-CD19 CAR T cells in B cell lymphoma–bearing mice. Circulating targets trapped CAR T cells in the lungs, reducing their access to lymphoid organs. In the bone marrow, tumor apoptosis was largely due to CAR T cells that engaged, killed, and detached from their targets within 25 min. Notably, not all CAR T cell contacts elicited calcium signaling or killing while interacting with tumors, uncovering extensive functional heterogeneity. Mathematical modeling revealed that direct killing was sufficient for tumor regression. Finally, antigen-loss variants emerged in the bone marrow, but not in lymph nodes, where CAR T cell cytotoxic activity was reduced. Our results identify a previously unappreciated level of diversity in the outcomes of CAR T cell interactions in vivo, with important clinical implications.

Impact of primary cancer features on behaviour of colorectal liver metastases and survival after hepatectomy

Background

Markers of tumour biology may be valuable prognostic indicators after hepatic resection of colorectal cancer liver metastases (CRLMs). Identification of the aggressiveness of these metastases might inform the appropriateness of hepatic surgery.

Methods

Patients undergoing liver resection for CRLMs between January 2001 and July 2013 in four tertiary hospitals were reviewed. A mathematical model to estimate CRLM doubling times was constructed for patients with metachronous metastases. Tumour doubling time was investigated in relation to the features of colorectal cancer, including KRAS status. The hazard rate for recurrence and death following hepatectomy was explored through the Kernel‐smoothed estimator.

Results

Of 1063 patients undergoing liver resection for CRLMs, 361 with metachronous metastases undergoing single‐stage hepatectomy were analysed. The mean doubling time in patients not receiving chemotherapy between surgery for colorectal cancer and CRLM was 71·4 days. Tumour doubling time was shorter in patients with more advanced primary tumour stages, with mutant KRAS and in those who did not receive chemotherapy. For fast‐growing CRLMs (doubling time less than 48 days), the risk of recurrence was highest within the first postoperative year, and was about 7 per cent per month.

Conclusion

Primary features of colorectal cancer were linked to aggressiveness of CRLMs as measured by doubling time.

Volumetric tumor growth rates of meningiomas involving the intracranial venous sinuses

OBJECT

There is currently no consensus as to whether meningiomas located inside the venous sinuses should be aggressively or conservatively treated. The goals of this study were to identify how sinus-invading meningiomas grow, report and compare growth rates of tumor components inside and outside the different venous sinuses, identify risk factors associated with increased tumor growth, and determine the effects of the extent of tumor resection on recurrence for meningiomas that invade the dural venous sinuses.

METHODS

Adult patients who underwent primary, non-biopsy resection of a WHO grade 1 meningioma invading the dural venous sinuses at a tertiary care institution between 2007 and 2015 were retrospectively reviewed. Rates of tumor growth were fit to several growth models to evaluate the most accurate model. Cohen's d analysis was used to identify associations with increased growth of tumor in the venous sinuses. Logistic regression was used to compare extent of resection with recurrence.

RESULTS

Of the 68 patients included in the study, 34 patients had postoperative residual tumors in the venous sinuses that were measured over time. The growth model that best fit the growth of intrasinus meningiomas was the Gompertzian growth model (r² = 0.93). The annual growth rate of meningiomas inside the sinuses was 7.3%, compared to extrasinus tumors with 13.6% growth per year. The only factor significantly associated with increased tumor growth in sinuses was preoperative embolization (effect sizes (ES) [95% CI], 1.874 [7.633-46.735], p = 0.008).

CONCLUSIONS

This study shows that meningiomas involving the venous sinuses have a Gompertzian-type growth with early exponential growth followed by a slower growth rate that plateaus when they reach a certain size. Overall, the growth rate of the intrasinus portion is low (7.3%), which is half of the reported growth rates for other studies involving primarily extrasinus tumors.

Metabolism is the tie: The Bertalanffy-type cancer growth model as common denominator of various modelling approaches

Cancer or tumour growth has been addressed from a variety of mathematical modelling perspectives in the past. Examples are single variable growth models, reaction diffusion models, compartment models, individual cell-based models, clonal competition models, to name only a few. In this paper, we show that the so called Bertalanffy-type growth model is a macroscopic model variant that can be conceived as an optimal condensed modelling approach that to a high degree preserves complexity with respect to the aforementioned more complex modelling variants. The derivation of the Bertalanffy-type model is crucially based on features of metabolism. Therefore, this model contains a shape parameter that can be interpreted as a resource utilisation efficiency. This shape parameter reflects features that are usually captured in much more complex models. To be specific, the shape parameter is related to morphological structures of tumours, which in turn depend on metabolic conditions. We, furthermore, show that a single variable variant of the Bertalanffy-type model can straightforwardly be extended to a multiclonal competition model. Since competition is crucially based on available shared or clone-specific resources, the metabolism-based approach is an obvious candidate to capture clonal competition. Depending on the specific context, metabolic reprogramming or other oncogene driven changes either lead to a suppression of cancer cells or to an improved competition resulting in outgrowth of tumours. The parametrisation of the Bertalanffy-type growth model allows to account for this observed variety of cancer characteristics. The shape parameter, conceived as a classifier for healthy and oncogenic phenotypes, supplies a link to survival and evolutionary stability concepts discussed in demographic studies, such as opportunistic versus equilibrium strategies.

Rate-based structural health monitoring using permanently installed sensors

Permanently installed sensors are becoming increasingly ubiquitous, facilitating very frequent in situ measurements and consequently improved monitoring of 'trends' in the observed system behaviour. It is proposed that this newly available data may be used to provide prior warning and forecasting of critical events, particularly system failure. Numerous damage mechanisms are examples of positive feedback; they are 'self-accelerating' with an increasing rate of damage towards failure. The positive feedback leads to a common time-response behaviour which may be described by an empirical relation allowing prediction of the time to criticality. This study focuses on Structural Health Monitoring of engineering components; failure times are projected well in advance of failure for fatigue, creep crack growth and volumetric creep damage experiments. The proposed methodology provides a widely applicable framework for using newly available near-continuous data from permanently installed sensors to predict time until failure in a range of application areas including engineering, geophysics and medicine.

Mechanistic Model for Cancer Growth and Response to Chemotherapy

Cancer treatment has developed over the years; however not all patients respond to this treatment, and therefore further research is needed. In this paper, we employ mathematical modeling to understand the behavior of cancer and its interaction with therapy. We study a drug delivery and drug-cell interaction model along with cell proliferation. Due to the fact that cancer cells grow when there are enough nutrients and oxygen, proliferation can be a barrier against a response to therapy. To understand the effect of this factor, we perform numerical simulations of the model for different values of the parameters with a continuous delivery of the drug. The numerical results showed that cancer dies after short apoptotic cycles if the cancer is highly vascularized or if the penetration of the drug is high. This suggests promoting angiogenesis or perfusion of the drug. This result is similar to the situation where proliferation is not considered since the constant release of drug overcomes the growth of the cells and thus the effect of proliferation can be neglected.

Fighting Cancer with Mathematics and Viruses

After decades of research, oncolytic virotherapy has recently advanced to clinical application, and currently a multitude of novel agents and combination treatments are being evaluated for cancer therapy. Oncolytic agents preferentially replicate in tumor cells, inducing tumor cell lysis and complex antitumor effects, such as innate and adaptive immune responses and the destruction of tumor vasculature. With the availability of different vector platforms and the potential of both genetic engineering and combination regimens to enhance particular aspects of safety and efficacy, the identification of optimal treatments for patient subpopulations or even individual patients becomes a top priority. Mathematical modeling can provide support in this arena by making use of experimental and clinical data to generate hypotheses about the mechanisms underlying complex biology and, ultimately, predict optimal treatment protocols. Increasingly complex models can be applied to account for therapeutically relevant parameters such as components of the immune system. In this review, we describe current developments in oncolytic virotherapy and mathematical modeling to discuss the benefit of integrating different modeling approaches into biological and clinical experimentation. Conclusively, we propose a mutual combination of these research fields to increase the value of the preclinical development and the therapeutic efficacy of the resulting treatments.

Predicting Patient-Specific Radiotherapy Protocols Based on Mathematical Model Choice for Proliferation Saturation Index

Radiation is commonly used in cancer treatment. Over 50% of all cancer patients will undergo radiotherapy (RT) as part of cancer care. Scientific advances in RT have primarily focused on the physical characteristics of treatment including beam quality and delivery. Only recently have inroads been made into utilizing tumor biology and radiobiology to design more appropriate RT protocols. Tumors are composites of proliferating and growth-arrested cells, and overall response depends on their respective proportions at irradiation. Prokopiou et al. (Radiat Oncol 10:159, 2015) developed the concept of the proliferation saturation index (PSI) to augment the clinical decision process associated with RT. This framework is based on the application of the logistic equation to pre-treatment imaging data in order to estimate a patient-specific tumor carrying capacity, which is then used to recommend a specific RT protocol. It is unclear, however, how dependent clinical recommendations are on the underlying tumor growth law. We discuss a PSI framework with a generalized logistic equation that can capture kinetics of different well-known growth laws including logistic and Gompertzian growth. Estimation of model parameters on the basis of clinical data revealed that the generalized logistic model can describe data equally well for a wide range of the generalized logistic exponent value. Clinical recommendations based on the calculated PSI, however, are strongly dependent on the specific growth law assumed. Our analysis suggests that the PSI framework may best be utilized in clinical practice when the underlying tumor growth law is known, or when sufficiently many tumor growth models suggest similar fractionation protocols.

Cadherin- and Rigidity-Dependent Growth of Lung Cancer Cells in a Partially Confined Microenvironment

During tumor development, cancer cells constantly confront different types of extracellular barriers. However, fundamental questions like whether tumor cells will continue to grow against confinement or away from it and what key factors govern this process remain poorly understood. To address these issues, here we examined the growth dynamics of human lung epithelial carcinoma A549 cells partially confined in micrometer-sized cylindrical pores with precisely controlled wall stiffness. It was found that, after reaching confluency, the cell monolayer enclosed by a compliant wall was able to keep growing and pushing the boundary, eventually leading to a markedly enlarged pore. In contrast, a much reduced in-plane growth and elevated strain level among cells were observed when the confining wall becomes stiff. Furthermore, under such circumstance, cells switched their growth from within the monolayer to along the out-of-plane direction, resulting in cell stacking. We showed that these observations can be well explained by a simple model taking into account the deformability of the wall and the threshold stress for inhibiting cell growth. Interestingly, cadherins were found to play an important role in the proliferation and stress buildup within the cell monolayer by aggregating at cell-cell junctions. The stiff confinement led to an elevated expression level of cadherins. Furthermore, inhibition of N-cadherin resulted in a significantly suppressed cell growth under the same confining conditions.