Mechanistic Mathematical Model Research Articles

Experimentally-driven mathematical model to understand the effects of matrix deprivation in breast cancer metastasis.

Normal epithelial cells receive proper signals for growth and survival from attachment to the underlying extracellular matrix (ECM). They perceive detachment from the ECM as a stress and die - a phenomenon termed as 'anoikis'. However, metastatic cancer cells acquire anoikis-resistance and circulate through the blood and lymphatics to seed metastasis. Under normal (adherent) growth conditions, the serine-threonine protein kinase Akt stimulates protein synthesis and cell growth, maintaining an anabolic state in the cancer cell. In contrast, previously we showed that the stress due to matrix deprivation is sensed by yet another serine-threonine kinase, AMP-activated protein kinase (AMPK), that inhibits anabolic pathways while promoting catabolic processes. We illustrated a switch from Akthigh/AMPKlow in adherent condition to AMPKhigh/Aktlow in matrix-detached condition, with consequent metabolic switching from an anabolic to a catabolic state, which aids cancer cell stress-survival. In this study, we utilized these experimental data and developed a deterministic ordinary differential equation (ODE)-based mechanistic mathematical model to mimic attachment-detachment signaling network. To do so, we used the framework of insulin-glucagon signaling with consequent metabolic shifts to capture the pathophysiology of matrix-deprived state in breast cancer cells. Using the developed metastatic breast cancer signaling (MBCS) model, we identified perturbation of several signaling proteins such as IRS, PI3K, PKC, GLUT1, IP3, DAG, PKA, cAMP, and PDE3 upon matrix deprivation. Further, in silico molecular perturbations revealed that several feedback/crosstalks like DAG to PKC, PKC to IRS, S6K1 to IRS, cAMP to PKA, and AMPK to Akt are essential for the metabolic switching in matrix-deprived cancer cells. AMPK knockdown simulations identified a crucial role for AMPK in maintaining these adaptive changes. Thus, this mathematical framework provides insights on attachment-detachment signaling with metabolic adaptations that promote cancer metastasis.

Multi-strain modeling of influenza vaccine effectiveness in older adults and its dependence on antigenic distance

Influenza vaccine effectiveness (VE) varies seasonally due to host, virus and vaccine characteristics. To investigate how antigenic matching and dosage impact VE, we developed a mechanistic knowledge-based mathematical model. Immunization with a split vaccine is modeled for exposure to A/H1N1 or A/H3N2 virus strains. The model accounts for cross-reactivity of immune cells elicited during previous immunizations with new antigens. We simulated vaccine effectiveness (sVE) of high dose (HD) versus standard dose (SD) vaccines in the older population, from 2011 to 2022. We find that sVE is highly dependent on antigenic matching and that higher dosage improves immunogenicity, activation and memory formation of immune cells. In alignment with clinical observations, the HD vaccine performs better than the SD vaccine in all simulations, supporting the use of the HD vaccine in the older population. This model could be adapted to predict the impact of alternative virus strain selection on clinical outcomes in future influenza seasons.

Modelling post-chemotherapy stem cell dynamics in the bone marrow niche of AML patients

Acute myeloid leukemia (AML) is a stem cell-driven malignancy of the blood forming (hematopoietic) system. Despite of high dose chemotherapy with toxic side effects, many patients eventually relapse. The “7+3 regimen”, which consists of 7 days of cytarabine in combination with daunorubicin during the first 3 days, is a widely used therapy protocol. Since peripheral blood cells are easily accessible to longitudinal sampling, significant research efforts have been undertaken to characterize and reduce adverse effects on circulating blood cells. However, much less is known about the impact of the 7+3 regimen on human hematopoietic stem cells and their physiological micro-environments, the so-called stem cell niches. One reason for this is the technical inability to observe human stem cells in vivo and the discomfort related to bone marrow biopsies. To better understand the treatment effects on human stem cells, we consider a mechanistic mathematical model of the stem cell niche before, during and after chemotherapy. The model accounts for different maturation stages of leukemic and hematopoietic cells and considers key processes such as cell proliferation, self-renewal, differentiation and therapy-induced cell death. In the model, hematopoietic (HSCs) and leukemic stem cells (LSCs) compete for a joint niche and respond to both systemic and niche-derived signals. We relate the model to clinical trial data from literature which longitudinally quantifies the counts of hematopoietic stem like (CD34+CD38-ALDH+) cells at diagnosis and after therapy. The proposed model can capture the clinically observed interindividual heterogeneity and reproduce the non-monotonous dynamics of the hematopoietic stem like cells observed in relapsing patients. Our model allows to simulate different scenarios proposed in literature such as therapy-related impairment of the stem cell niche or niche-mediated resistance. Model simulations suggest that during the post-therapy phase a more than 10-fold increase of hematopoietic stem-like cell proliferation rates is required to recapitulate the measured cell dynamics in patients achieving complete remission. We fit the model to data of 7 individual patients and simulate variations of the treatment protocol. These simulations are in line with the clinical finding that G-CSF priming can improve the treatment outcome. Furthermore, our model suggests that a decline of HSC counts during remission might serve as an indication for salvage therapy in patients lacking MRD (minimal residual disease) markers.

Emergence of cyclic hypoxia and the impact of PARP inhibitors on tumor progression.

Tumor hypoxia is a dynamic phenomenon marked by fluctuations in oxygen levels across both rapid (seconds to minutes) and slow (hours to days) time scales. While short hypoxia cycles are relatively well understood, the mechanisms behind longer cycles remain largely unclear. In this paper, we present a novel mechanistic mathematical model that explains slow hypoxia cycles through feedback loops involving vascular expansion and regression, oxygen-regulated tumor growth, and toxic cytokine production. Our study reveals that, for the emergence of slow hypoxia cycles, endothelial cells must adapt by decreasing receptor activation as ligand concentration increases. Additionally, the interaction between tumor cells and toxic cytokines influences frequency and intensity of these cycles. By examining the effects of pharmacological interventions, specifically poly (ADP-ribose) polymerase inhibitors, we also demonstrate how targeting cell proliferation can help regulate oxygen levels. Our findings enhance the understanding of hypoxia regulation and suggest PARP proteins as promising therapeutic targets.

Utilising an in silico model to predict outcomes in senescence-driven acute liver injury

Currently liver transplantation is the only treatment option for liver disease, but organ availability cannot meet patient demand. Alternative regenerative therapies, including cell transplantation, aim to modulate the injured microenvironment from inflammation and scarring towards regeneration. The complexity of the liver injury response makes it challenging to identify suitable therapeutic targets when relying on experimental approaches alone. Therefore, we adopted a combined in vivo-in silico approach and developed an ordinary differential equation model of acute liver disease able to predict the host response to injury and potential interventions. The Mdm2fl/fl mouse model of senescence-driven liver injury was used to generate a quantitative dynamic characterisation of the key cellular players (macrophages, endothelial cells, myofibroblasts) and extra cellular matrix involved in liver injury. This was qualitatively captured by the mathematical model. The mathematical model was then used to predict injury outcomes in response to milder and more severe levels of senescence-induced liver injury and validated with experimental in vivo data. In silico experiments using the validated model were then performed to interrogate potential approaches to enhance regeneration. These predicted that increasing the rate of macrophage phenotypic switch or increasing the number of pro-regenerative macrophages in the system will accelerate the rate of senescent cell clearance and resolution. These results showcase the potential benefits of mechanistic mathematical modelling for capturing the dynamics of complex biological systems and identifying therapeutic interventions that may enhance our understanding of injury-repair mechanisms and reduce translational bottlenecks.

Practical parameter identifiability and handling of censored data with Bayesian inference in mathematical tumour models

Mechanistic mathematical models (MMs) are a powerful tool to help us understand and predict the dynamics of tumour growth under various conditions. In this work, we use 5 MMs with an increasing number of parameters to explore how certain (often overlooked) decisions in estimating parameters from data of experimental tumour growth affect the outcome of the analysis. In particular, we propose a framework for including tumour volume measurements that fall outside the upper and lower limits of detection, which are normally discarded. We demonstrate how excluding censored data results in an overestimation of the initial tumour volume and the MM-predicted tumour volumes prior to the first measurements, and an underestimation of the carrying capacity and the MM-predicted tumour volumes beyond the latest measurable time points. We show in which way the choice of prior for the MM parameters can impact the posterior distributions, and illustrate that reporting the most likely parameters and their 95% credible interval can lead to confusing or misleading interpretations. We hope this work will encourage others to carefully consider choices made in parameter estimation and to adopt the approaches we put forward herein.

GIS-ODE: linking dynamic population models with GIS to predict pathogen vector abundance across a country under climate change scenarios.

Mechanistic mathematical models such as ordinary differential equations (ODEs) have a long history for their use in describing population dynamics and determining estimates of key parameters that summarize the potential growth or decline of a population over time. More recently, geographic information systems (GIS) have become important tools to provide a visual representation of statistically determined parameters and environmental features over space. Here, we combine these tools to form a 'GIS-ODE' approach to generate spatiotemporal maps predicting how projected changes in thermal climate may affect population densities and, uniquely, population dynamics of Ixodes ricinus, an important tick vector of several human pathogens. Assuming habitat and host densities are not greatly affected by climate warming, the GIS-ODE model predicted that, even under the lowest projected temperature increase, I. ricinus nymph densities could increase by 26-99% in Scotland, depending on the habitat and climate of the location. Our GIS-ODE model provides the vector-borne disease research community with a framework option to produce predictive, spatially explicit risk maps based on a mechanistic understanding of vector and vector-borne disease transmission dynamics.

Normoglycemia and physiological cortisone level maintain glucose homeostasis in a pancreas-liver microphysiological system

Current research on metabolic disorders and diabetes relies on animal models because multi-organ diseases cannot be well studied with standard in vitro assays. Here, we have connected cell models of key metabolic organs, the pancreas and liver, on a microfluidic chip to enable diabetes research in a human-based in vitro system. Aided by mechanistic mathematical modeling, we demonstrate that hyperglycemia and high cortisone concentration induce glucose dysregulation in the pancreas-liver microphysiological system (MPS), mimicking a diabetic phenotype seen in patients with glucocorticoid-induced diabetes. In this diseased condition, the pancreas-liver MPS displays beta-cell dysfunction, steatosis, elevated ketone-body secretion, increased glycogen storage, and upregulated gluconeogenic gene expression. Conversely, a physiological culture condition maintains glucose tolerance and beta-cell function. This method was reproducible in two laboratories and was effective in multiple pancreatic islet donors. The model also provides a platform to identify new therapeutic proteins, as demonstrated with a combined transcriptome and proteome analysis.

Systems modeling of oncogenic G-protein and GPCR signaling reveals unexpected differences in downstream pathway activation

Mathematical models of biochemical reaction networks are an important and emerging tool for the study of cell signaling networks involved in disease processes. One promising potential application of such mathematical models is the study of how disease-causing mutations promote the signaling phenotype that contributes to the disease. It is commonly assumed that one must have a thorough characterization of the network readily available for mathematical modeling to be useful, but we hypothesized that mathematical modeling could be useful when there is incomplete knowledge and that it could be a tool for discovery that opens new areas for further exploration. In the present study, we first develop a mechanistic mathematical model of a G-protein coupled receptor signaling network that is mutated in almost all cases of uveal melanoma and use model-driven explorations to uncover and explore multiple new areas for investigating this disease. Modeling the two major, mutually-exclusive, oncogenic mutations (Gαq/11 and CysLT2R) revealed the potential for previously unknown qualitative differences between seemingly interchangeable disease-promoting mutations, and our experiments confirmed oncogenic CysLT2R was impaired at activating the FAK/YAP/TAZ pathway relative to Gαq/11. This led us to hypothesize that CYSLTR2 mutations in UM must co-occur with other mutations to activate FAK/YAP/TAZ signaling, and our bioinformatic analysis uncovers a role for co-occurring mutations involving the plexin/semaphorin pathway, which has been shown capable of activating this pathway. Overall, this work highlights the power of mechanism-based computational systems biology as a discovery tool that can leverage available information to open new research areas.

Computational Model Predicts Patient Outcomes in Luminal B Breast Cancer Treated with Endocrine Therapy and CDK4/6 Inhibition.

Development of a computational biomarker to predict, prior to treatment, the response to CDK4/6 inhibition (CDK4/6i) in combination with endocrine therapy in patients with breast cancer. A mechanistic mathematical model that accounts for protein signaling and drug mechanisms of action was developed and trained on extensive, publicly available data from breast cancer cell lines. The model was built to provide a patient-specific response score based on the expression of six genes (CCND1, CCNE1, ESR1, RB1, MYC, and CDKN1A). The model was validated in five independent cohorts of 148 patients in total with early-stage or advanced breast cancer treated with endocrine therapy and CDK4/6i. Response was measured either by evaluating Ki67 levels and PAM50 risk of relapse (ROR) after neoadjuvant treatment or by evaluating progression-free survival (PFS). The model showed significant association with patient's outcomes in all five cohorts. The model predicted high Ki67 [area under the curve; AUC (95% confidence interval, CI) of 0.80 (0.64-0.92), 0.81 (0.60-1.00) and 0.80 (0.65-0.93)] and high PAM50 ROR [AUC of 0.78 (0.64-0.89)]. This observation was not obtained in patients treated with chemotherapy. In the other cohorts, patient stratification based on the model prediction was significantly associated with PFS [hazard ratio (HR) = 2.92 (95% CI, 1.08-7.86), P = 0.034 and HR = 2.16 (1.02 4.55), P = 0.043]. A mathematical modeling approach accurately predicts patient outcome following CDK4/6i plus endocrine therapy that marks a step toward more personalized treatments in patients with Luminal B breast cancer.

Numerical exploration of a fully mechanistic mathematical model of aqueous CO2 corrosion in steel pipelines

A numerical exploration of a comprehensive mechanistic aqueous carbon dioxide (CO2) corrosion model is conducted across a range of temperatures (273–313 K), CO2 partial pressures (0.1–1 bar), and bulk pHs (5–6.5). Contour plots are produced to examine the impact on corrosion rate, surface pH, and surface saturation index of iron carbonate (FeCO3). Two response patterns are identified depending upon the limiting behaviour of the system, with a transition from charge-transfer control to mass-transport control as temperature is increased and partial pressure is reduced. FeCO3 surface saturation shows a strong correlation with the release of Fe2+ and increase in bulk pH.

Advancing cancer drug development with mechanistic mathematical modeling: bridging the gap between theory and practice.

Quantitative predictive modeling of cancer growth, progression, and individual response to therapy is a rapidly growing field. Researchers from mathematical modeling, systems biology, pharmaceutical industry, and regulatory bodies, are collaboratively working on predictive models that could be applied for drug development and, ultimately, the clinical management of cancer patients. A plethora of modeling paradigms and approaches have emerged, making it challenging to compile a comprehensive review across all subdisciplines. It is therefore critical to gauge fundamental design aspects against requirements, and weigh opportunities and limitations of the different model types. In this review,we discuss three fundamental types of cancer models: space-structured models, ecological models, and immune system focused models. For each type, it is our goal to illustrate which mechanisms contribute to variability and heterogeneity in cancer growth and response, so that the appropriate architecture and complexity of a new model becomes clearer. We present the main features addressed by each of the three exemplary modeling types through a subjective collection of literature and illustrative exercises to facilitate inspiration and exchange, with a focus on providing a didactic rather than exhaustive overview. We close by imagining a future multi-scale model design to impact critical decisions in oncology drug development.

Examining the Effects of Temperature on the Evolution of Bacterial tRNA Pools.

The genetic code consists of 61 codons coding for 20 amino acids. These codons are recognized by transfer RNAs (tRNAs) that bind to specific codons during protein synthesis. All organisms utilize less than all 61 possible anticodons due to base pair wobble: the ability to have a mismatch with a codon at its third nucleotide. Previous studies observed a correlation between the tRNA pool of bacteria and the temperature of their respective environments. However, it is unclear if these patterns represent biological adaptations to maintain the efficiency and accuracy of protein synthesis in different environments. A mechanistic mathematical model of mRNA translation is used to quantify the expected elongation rates and error rate for each codon based on an organism's tRNA pool. A comparative analysis across a range of bacteria that accounts for covariance due to shared ancestry is performed to quantify the impact of environmental temperature on the evolution of the tRNA pool. We find that thermophiles generally have more anticodons represented in their tRNA pool than mesophiles or psychrophiles. Based on our model, this increased diversity is expected to lead to increased missense errors. The implications of this for protein evolution in thermophiles are discussed.

Understanding the impact of HIV on mpox transmission in the MSM population: A mathematical modeling study

The recent mpox outbreak (in 2022–2023) has different clinical and epidemiological features compared with previous outbreaks of the disease. During this outbreak, sexual contact was believed to be the primary transmission route of the disease. In addition, the community of men having sex with men (MSM) was disproportionately affected by the outbreak. This population is also disproportionately affected by HIV infection. Given that both diseases can be transmitted sexually, the endemicity of HIV, and the high sexual behavior associated with the MSM community, it is essential to understand the effect of the two diseases spreading simultaneously in an MSM population. Particularly, we aim to understand the potential effects of HIV on an mpox outbreak in the MSM population. We develop a mechanistic mathematical model of HIV and mpox co-infection. Our model incorporates the dynamics of both diseases and considers HIV treatment with anti-retroviral therapy (ART). In addition, we consider a potential scenario where HIV infection increases susceptibility to mpox, and investigate the potential impact of this mechanism on mpox dynamics. Our analysis shows that HIV can facilitate the spread of mpox in an MSM population, and that HIV treatment with ART may not be sufficient to control the spread of mpox in the population. However, we showed that a moderate use of condoms or reduction in sexual contact in the population combined with ART is beneficial in controlling mpox transmission. Based on our analysis, it is evident that effective control of HIV, specifically through substantial ART use, moderate condom compliance, and reduction in sexual contact, is imperative for curtailing the transmission of mpox in an MSM population and mitigating the compounding impact of these intertwined epidemics.

A mechanistic model to predict headspace concentration profile of carvacrol for controlled release packaging

The efficacy of an antimicrobial agent in food packaging depends largely on its concentration profile in the package headspace. In this study, a mechanistic mathematical model was developed to predict the headspace concentration profile of carvacrol, a prominent antimicrobial agent, using the physical principles that govern the movement of carvacrol molecules in a controlled release packaging system. The model and its estimated parameter values were validated using three different sets of gas chromatography measurements—this rigorous process confirms the accuracy and reliability of the model. Advancing beyond existing models, our model enhances the understanding of the dynamics between the concentration profile and its interacting processes including evaporation, adsorption, and gas transmission. It provides packaging developers with a superior alternative to the trial-and-error approach to tailor concentration profiles for optimizing antimicrobial efficacy. It leads to more efficient and better design of controlled release packaging.

Kinetic modelling of glycolytic oscillations.

Glycolytic oscillations have been studied for well over 60 years, but aspects of their function, and mechanisms of regulation and synchronisation remain unclear. Glycolysis is amenable to mechanistic mathematical modelling, as its components have been well characterised, and the system can be studied at many organisational levels: in vitro reconstituted enzymes, cell free extracts, individual cells, and cell populations. In recent years, the emergence of individual cell analysis has opened new ways of studying this intriguing system.

Putting theory to the test: An integrated computational/experimental chemostat model of the tragedy of the commons.

Cooperation via shared public goods is ubiquitous in nature, however, noncontributing social cheaters can exploit the public goods provided by cooperating individuals to gain a fitness advantage. Theory predicts that this dynamic can cause a Tragedy of the Commons, and in particular, a 'Collapsing' Tragedy defined as the extinction of the entire population if the public good is essential. However, there is little empirical evidence of the Collapsing Tragedy in evolutionary biology. Here, we experimentally demonstrate this outcome in a microbial model system, the public good-producing bacterium Pseudomonas aeruginosa grown in a continuous-culture chemostat. In a growth medium that requires extracellular protein digestion, we find that P. aeruginosa populations maintain a high density when entirely composed of cooperating, protease-producing cells but completely collapse when non-producing cheater cells are introduced. We formulate a mechanistic mathematical model that recapitulates experimental observations and suggests key parameters, such as the dilution rate and the cost of public good production, that define the stability of cooperative behavior. We combine model prediction with experimental validation to explain striking differences in the long-term cheater trajectories of replicate cocultures through mutational events that increase cheater fitness. Taken together, our integrated empirical and theoretical approach validates and parametrizes the Collapsing Tragedy in a microbial population, and provides a quantitative, mechanistic framework for generating testable predictions of social behavior.

Quantitative evaluation of trastuzumab deruxtecan pharmacokinetics and pharmacodynamics in mouse models of varying degrees of HER2 expression.

Trastuzumab deruxtecan (T-DXd; DS-8201; ENHERTU®) is a human epithelial growth factor receptor 2 (HER2)-directed antibody drug conjugate (ADC) with demonstrated antitumor activity against a range of tumor types. Aiming to understand the relationship between antigen expression and downstream efficacy outcomes, T-DXd was administered in tumor-bearing mice carrying NCI-N87, Capan-1, JIMT-1, and MDA-MB-468 xenografts, characterized by varying HER2 levels. Plasma pharmacokinetics (PK) of total antibody, T-DXd, and released DXd and tumor concentrations of released DXd were evaluated, in addition to monitoring γΗ2AX and pRAD50 pharmacodynamic (PD) response. A positive relationship was observed between released DXd concentrations in tumor and HER2 expression, with NCI-N87 xenografts characterized by the highest exposures compared to the remaining cell lines. γΗ2AX and pRAD50 demonstrated a sustained increase over several days occurring with a time delay relative to tumoral-released DXd concentrations. Invitro investigations of cell-based DXd disposition facilitated the characterization of DXd kinetics across tumor cells. These outputs were incorporated into a mechanistic mathematical model, utilized to describe PK/PD trends. The model captured plasma PK across dosing arms as well as tumor PK in NCI-N87, Capan-1, and MDA-MB-468 models; tumor concentrations in JIMT-1 xenografts required additional parameter adjustments reflective of complex receptor dynamics. γΗ2AX longitudinal trends were well characterized via a unified PD model implemented across xenografts demonstrating the robustness of measured PD trends. This work supports the application of a mechanistic model as a quantitative tool, reliably projecting tumor payload concentrations upon T-DXd administration, as the first step towards preclinical-to-clinical translation.

Abstract 3872: Multi-ancestral origins of colorectal lesionsMulti-ancestral origins of colorectal lesions

Abstract The widely accepted paradigm that cancer initiation is a monoclonal event has not been definitively resolved. In sporadic cancers this is due to the requirement for longitudinal sampling to study this occult process, which occurs over years to decades. Individuals with familial adenomatous polyposis (FAP) possess a germline APC mutation, leading to hundreds of colonic polyps in adolescence and a 100% lifetime risk of colorectal cancer. Resected colons from FAP patients contain normal mucosa, benign and dysplastic polyps that embed the natural history of colorectal tumorigenesis at different points of malignant progression. Through genome sequencing of 127 independent polyps or mucosa samples from six FAP patients, we characterized the genomic landscape, relative timing of somatic alterations, and clonal dynamics in normal, dysplastic, and invasive colon tissue. Comparison with multi-region sequencing data from FAP patients and sporadic colorectal adenomas and carcinomas indicates similar molecular profiles, with canonical cancer driver genes present across benign and dysplastic premalignant lesions and shared mutational orders. Pairwise sample comparisons and phylogenetic reconstructions revealed extensive heterogeneity and subclonal sharing within a patient’s benign and dysplastic lesions, irrespective of location in the colon. These data imply that polyps were initiated from multiple cells from distinct lineages whose most recent common ancestor arose during development. We corroborate these findings using a mechanistic mathematical model, which infers polyclonality in 100% of benign and 83% of dysplastic samples. Based on these findings, we propose a conceptual model of the multi-ancestral origin of intestinal malignancies. Citation Format: Ryan O. Schenck, Aziz Khan, Aaron Horning, Edward D. Esplin, Debra Van Egeren, HTAN–FAP Consortium, William Greenleaf, James M. Ford, Michael P. Snyder, Christina Curtis. Multi-ancestral origins of colorectal lesionsMulti-ancestral origins of colorectal lesions [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 3872.

A review of mechanistic learning in mathematical oncology.

Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework.