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1.  Modeling the Natural History and Detection of Lung Cancer Based on Smoking Behavior 
PLoS ONE  2014;9(4):e93430.
In this study, we developed a method for modeling the progression and detection of lung cancer based on the smoking behavior at an individual level. The model allows obtaining the characteristics of lung cancer in a population at the time of diagnosis. Lung cancer data from Surveillance, Epidemiology and End Results (SEER) database collected between 2004 and 2008 were used to fit the lung cancer progression and detection model. The fitted model combined with a smoking based carcinogenesis model was used to predict the distribution of age, gender, tumor size, disease stage and smoking status at diagnosis and the results were validated against independent data from the SEER database collected from 1988 to 1999. The model accurately predicted the gender distribution and median age of LC patients of diagnosis, and reasonably predicted the joint tumor size and disease stage distribution.
PMCID: PMC3976286  PMID: 24705368
2.  Relative contribution of free-virus and synaptic transmission to the spread of HIV-1 through target cell populations 
Biology Letters  2013;9(1):20121049.
Human immunodeficiency virus can spread through target cells by transmission of cell-free virus or directly from cell-to-cell via formation of virological synapses. Although cell-to-cell transmission has been described as much more efficient than cell-free infection, the relative contribution of the two transmission pathways to virus growth during multiple rounds of replication remains poorly defined. Here, we fit a mathematical model to previously published and newly generated in vitro data, and determine that free-virus and synaptic transmission contribute approximately equally to the growth of the virus population.
PMCID: PMC3565528  PMID: 23269844
human immunodeficiency virus; mathematical model; virological synapse
3.  Dependence of the Firearm-Related Homicide Rate on Gun Availability: A Mathematical Analysis 
PLoS ONE  2013;8(7):e71606.
In the USA, the relationship between the legal availability of guns and the firearm-related homicide rate has been debated. It has been argued that unrestricted gun availability promotes the occurrence of firearm-induced homicides. It has also been pointed out that gun possession can protect potential victims when attacked. This paper provides a first mathematical analysis of this tradeoff, with the goal to steer the debate towards arguing about assumptions, statistics, and scientific methods. The model is based on a set of clearly defined assumptions, which are supported by available statistical data, and is formulated axiomatically such that results do not depend on arbitrary mathematical expressions. According to this framework, two alternative scenarios can minimize the gun-related homicide rate: a ban of private firearms possession, or a policy allowing the general population to carry guns. Importantly, the model identifies the crucial parameters that determine which policy minimizes the death rate, and thus serves as a guide for the design of future epidemiological studies. The parameters that need to be measured include the fraction of offenders that illegally possess a gun, the degree of protection provided by gun ownership, and the fraction of the population who take up their right to own a gun and carry it when attacked. Limited data available in the literature were used to demonstrate how the model can be parameterized, and this preliminary analysis suggests that a ban of private firearm possession, or possibly a partial reduction in gun availability, might lower the rate of firearm-induced homicides. This, however, should not be seen as a policy recommendation, due to the limited data available to inform and parameterize the model. However, the model clearly defines what needs to be measured, and provides a basis for a scientific discussion about assumptions and data.
PMCID: PMC3724776  PMID: 23923062
4.  Synaptic transmission and the susceptibility of HIV infection to anti-viral drugs 
Scientific Reports  2013;3:2103.
Cell-to-cell viral transmission via virological synapses has been argued to reduce susceptibility of the virus population to anti-viral drugs through multiple infection of cells, contributing to low-level viral persistence during therapy. Using a mathematical framework, we examine the role of synaptic transmission in treatment susceptibility. A key factor is the relative probability of individual virions to infect a cell during free-virus and synaptic transmission, a currently unknown quantity. If this infection probability is higher for free-virus transmission, then treatment susceptibility is lowest if one virus is transferred per synapse, and multiple infection of cells increases susceptibility. In the opposite case, treatment susceptibility is minimized for an intermediate number of virions transferred per synapse. Hence, multiple infection via synapses does not simply lower treatment susceptibility. Without further experimental investigations, one cannot conclude that synaptic transmission provides an additional mechanism for the virus to persist at low levels during anti-viral therapy.
PMCID: PMC3696900  PMID: 23811684
5.  Evolutionary dynamics of giant viruses and their virophages 
Ecology and Evolution  2013;3(7):2103-2115.
Giant viruses contain large genomes, encode many proteins atypical for viruses, replicate in large viral factories, and tend to infect protists. The giant virus replication factories can in turn be infected by so called virophages, which are smaller viruses that negatively impact giant virus replication. An example is Mimiviruses that infect the protist Acanthamoeba and that are themselves infected by the virophage Sputnik. This study examines the evolutionary dynamics of this system, using mathematical models. While the models suggest that the virophage population will evolve to increasing degrees of giant virus inhibition, it further suggests that this renders the virophage population prone to extinction due to dynamic instabilities over wide parameter ranges. Implications and conditions required to avoid extinction are discussed. Another interesting result is that virophage presence can fundamentally alter the evolutionary course of the giant virus. While the giant virus is predicted to evolve toward increasing its basic reproductive ratio in the absence of the virophage, the opposite is true in its presence. Therefore, virophages can not only benefit the host population directly by inhibiting the giant viruses but also indirectly by causing giant viruses to evolve toward weaker phenotypes. Experimental tests for this model are suggested.
PMCID: PMC3728950  PMID: 23919155
Evolutionary dynamics; giant viruses; mathematical models; virophages
6.  Methylation kinetics and CpG-island methylator phenotyope status in colorectal cancer cell lines 
Biology Direct  2013;8:14.
Hypermethylation of CpG islands is thought to contribute to carcinogenesis through the inactivation of tumor suppressor genes. Tumor cells with relatively high levels of CpG island methylation are considered CpG island methylator phenotypes (CIMP). The mechanisms that are responsible for regulating the activity of de novo methylation are not well understood.
We quantify and compare de novo methylation kinetics in CIMP and non-CIMP colon cancer cell lines in the context of different loci, following 5-aza-2’deoxycytidine (5-AZA)-mediated de-methylation of cells. In non-CIMP cells, a relatively fast rate of re-methylation is observed that starts with a certain time delay after cessation of 5-AZA treatment. CIMP cells, on the other hand, start re-methylation without a time delay but at a significantly slower rate. A mathematical model can account for these counter-intuitive results by assuming negative feedback regulation of de novo methylation activity and by further assuming that this regulation is corrupted in CIMP cells. This model further suggests that when methylation levels have grown back to physiological levels, de novo methylation activity ceases in non-CIMP cells, while it continues at a constant low level in CIMP cells.
We propose that the faster rate of re-methylation observed in non-CIMP compared to CIMP cells in our study could be a consequence of feedback-mediated regulation of DNA methyl transferase activity. Testing this hypothesis will involve the search for specific feedback regulatory mechanisms involved in the activation of de novo methylation.
Reviewers’ report
This article was reviewed by Georg Luebeck, Tomasz Lipniacki, and Anna Marciniak-Czochra
PMCID: PMC3691599  PMID: 23758948
Methylation kinetics; Methylator phenotype; Methylation rates; Mathematical modeling
7.  Stem Cell Control, Oscillations, and Tissue Regeneration in Spatial and Non-Spatial Models 
Normal human tissue is organized into cell lineages, in which the highly differentiated mature cells that perform tissue functions are the end product of an orderly tissue-specific sequence of divisions that start with stem cells or progenitor cells. Tissue homeostasis and effective regeneration after injuries requires tight regulation of these cell lineages and feedback loops play a fundamental role in this regard. In particular, signals secreted from differentiated cells that inhibit stem cell division and stem cell self-renewal are important in establishing control. In this article we study in detail the cell dynamics that arise from this control mechanism. These dynamics are fundamental to our understanding of cancer, given that tumor initiation requires an escape from tissue regulation. Knowledge on the processes of cellular control can provide insights into the pathways that lead to deregulation and consequently cancer development.
PMCID: PMC3625858  PMID: 23596567
tissue regeneration; cell linage control; tissue stability; mathematical models; cancer
8.  Accelerated crossing of fitness valleys through division of labor and cheating in asexual populations 
Scientific Reports  2012;2:917.
Complex traits can require the accumulation of multiple mutations that are individually deleterious. Their evolution requires a fitness valley to be crossed, which can take relatively long time spans. A new evolutionary mechanism is described that accelerates the emergence of complex phenotypes, based on a “division of labor” game and the occurrence of cheaters. If each intermediate mutation leads to a product that can be shared with others, the complex type can arise relatively quickly as an emergent property among cooperating individuals, without any given individual having to accumulate all mutations. Moreover, the emergence of cheaters that destroy cooperative interactions can lead to the emergence of individuals that have accumulated all necessary mutations on a time scale that is significantly faster than observed in the absence of cooperation and cheating. Application of this mechanism to somatic and microbial evolution is discussed, including evolutionary processes in tumors, biofilms, and viral infections.
PMCID: PMC3512085  PMID: 23209877
9.  Effect of Synaptic Transmission on Viral Fitness in HIV Infection 
PLoS ONE  2012;7(11):e48361.
HIV can spread through its target cell population either via cell-free transmission, or by cell-to-cell transmission, presumably through virological synapses. Synaptic transmission entails the transfer of tens to hundreds of viruses per synapse, a fraction of which successfully integrate into the target cell genome. It is currently not understood how synaptic transmission affects viral fitness. Using a mathematical model, we investigate how different synaptic transmission strategies, defined by the number of viruses passed per synapse, influence the basic reproductive ratio of the virus, R0, and virus load. In the most basic scenario, the model suggests that R0 is maximized if a single virus particle is transferred per synapse. R0 decreases and the infection eventually cannot be maintained for larger numbers of transferred viruses, because multiple infection of the same cell wastes viruses that could otherwise enter uninfected cells. To explain the relatively large number of HIV copies transferred per synapse, we consider additional biological assumptions under which an intermediate number of viruses transferred per synapse could maximize R0. These include an increased burst size in multiply infected cells, the saturation of anti-viral factors upon infection of cells, and rate limiting steps during the process of synapse formation.
PMCID: PMC3499495  PMID: 23166585
10.  Complex Spatial Dynamics of Oncolytic Viruses In Vitro: Mathematical and Experimental Approaches 
PLoS Computational Biology  2012;8(6):e1002547.
Oncolytic viruses replicate selectively in tumor cells and can serve as targeted treatment agents. While promising results have been observed in clinical trials, consistent success of therapy remains elusive. The dynamics of virus spread through tumor cell populations has been studied both experimentally and computationally. However, a basic understanding of the principles underlying virus spread in spatially structured target cell populations has yet to be obtained. This paper studies such dynamics, using a newly constructed recombinant adenovirus type-5 (Ad5) that expresses enhanced jellyfish green fluorescent protein (EGFP), AdEGFPuci, and grows on human 293 embryonic kidney epithelial cells, allowing us to track cell numbers and spatial patterns over time. The cells are arranged in a two-dimensional setting and allow virus spread to occur only to target cells within the local neighborhood. Despite the simplicity of the setup, complex dynamics are observed. Experiments gave rise to three spatial patterns that we call “hollow ring structure”, “filled ring structure”, and “disperse pattern”. An agent-based, stochastic computational model is used to simulate and interpret the experiments. The model can reproduce the experimentally observed patterns, and identifies key parameters that determine which pattern of virus growth arises. The model is further used to study the long-term outcome of the dynamics for the different growth patterns, and to investigate conditions under which the virus population eliminates the target cells. We find that both the filled ring structure and disperse pattern of initial expansion are indicative of treatment failure, where target cells persist in the long run. The hollow ring structure is associated with either target cell extinction or low-level persistence, both of which can be viewed as treatment success. Interestingly, it is found that equilibrium properties of ordinary differential equations describing the dynamics in local neighborhoods in the agent-based model can predict the outcome of the spatial virus-cell dynamics, which has important practical implications. This analysis provides a first step towards understanding spatial oncolytic virus dynamics, upon which more detailed investigations and further complexity can be built.
Author Summary
Traditional chemotherapy of cancers is characterized by strong side effects, while showing a low success rate in the long term control of tumors. Besides small molecule inhibitors, which have shown great promise, oncolytic viruses present an emerging specific treatment approach. They are engineered viruses that spread from tumor cell to tumor cell, killing them in the process. Non-tumor cells are generally not infected. While clinical trials have given rise to promising results, reliable success remains elusive. Besides experiments, computational approaches provide a valuable tool to better understand the dynamics of virus spread through a growing population or tumor cells. Combining in vitro experimental approaches with computational models, we study the principles of virus spread through a spatially structured population of cells, which is of fundamental importance to understanding virus treatment of solid tumors. We describe different growth patterns that can occur, interpret them, and explore how they relate to the ability of the virus to induce tumor regression. We further define how these spatial dynamics relate to settings where cells and viruses mix more readily, such as in many cell culture experiments that are used to evaluate candidate viruses.
PMCID: PMC3375216  PMID: 22719239
11.  Increased burst size in multiply infected cells can alter basic virus dynamics 
Biology Direct  2012;7:16.
The dynamics of viral infections have been studied extensively in a variety of settings, both experimentally and with mathematical models. The majority of mathematical models assumes that only one virus can infect a given cell at a time. It is, however, clear that especially in the context of high viral load, cells can become infected with multiple copies of a virus, a process called coinfection. This has been best demonstrated experimentally for human immunodeficiency virus (HIV), although it is thought to be equally relevant for a number of other viral infections. In a previously explored mathematical model, the viral output from an infected cell does not depend on the number of viruses that reside in the cell, i.e. viral replication is limited by cellular rather than viral factors. In this case, basic virus dynamics properties are not altered by coinfection.
Here, we explore the alternative assumption that multiply infected cells are characterized by an increased burst size and find that this can fundamentally alter model predictions. Under this scenario, establishment of infection may not be solely determined by the basic reproductive ratio of the virus, but can depend on the initial virus load. Upon infection, the virus population need not follow straight exponential growth. Instead, the exponential rate of growth can increase over time as virus load becomes larger. Moreover, the model suggests that the ability of anti-viral drugs to suppress the virus population can depend on the virus load upon initiation of therapy. This is because more coinfected cells, which produce more virus, are present at higher virus loads. Hence, the degree of drug resistance is not only determined by the viral genotype, but also by the prevalence of coinfected cells.
Our work shows how an increased burst size in multiply infected cells can alter basic infection dynamics. This forms the basis for future experimental testing of model assumptions and predictions that can distinguish between the different scenarios.
This article was reviewed by RJdeB, RMR and MK.
PMCID: PMC3482397  PMID: 22569346
Multiple infection of cells; Increased burst size; HIV; Mathematical models; Virus dynamics
12.  Multiploid Inheritance of HIV-1 during Cell-to-Cell Infection ▿ †  
Journal of Virology  2011;85(14):7169-7176.
During cell-to-cell transmission of human immunodeficiency virus type 1 (HIV-1), many viral particles can be simultaneously transferred from infected to uninfected CD4 T cells through structures called virological synapses (VS). Here we directly examine how cell-free and cell-to-cell infections differ from infections initiated with cell-free virus in the number of genetic copies that are transmitted from one generation to the next, i.e., the genetic inheritance. Following exposure to HIV-1-expressing cells, we show that target cells with high viral uptake are much more likely to become infected. Using T cells that coexpress distinct fluorescent HIV-1 variants, we show that multiple copies of HIV-1 can be cotransmitted across a single VS. In contrast to cell-free HIV-1 infection, which titrates with Poisson statistics, the titration of cell-associated HIV-1 to low rates of overall infection generates a constant fraction of the newly infected cells that are cofluorescent. Triple infection was also readily detected when cells expressing three fluorescent viruses were used as donor cells. A computational model and a statistical model are presented to estimate the degree to which cofluorescence underestimates coinfection frequency. Lastly, direct detection of HIV-1 proviruses using fluorescence in situ hybridization confirmed that significantly more HIV-1 DNA copies are found in primary T cells infected with cell-associated virus than in those infected with cell-free virus. Together, the data suggest that multiploid inheritance is common during cell-to-cell HIV-1 infection. From this study, we suggest that cell-to-cell infection may explain the high copy numbers of proviruses found in infected cells in vivo and may provide a mechanism through which HIV preserves sequence heterogeneity in viral quasispecies through genetic complementation.
PMCID: PMC3126592  PMID: 21543479
13.  Heterogeneity in chronic myeloid leukaemia dynamics during imatinib treatment: role of immune responses 
Previous studies have shown that during imatinib therapy, the decline of chronic myeloid leukaemia BCR-ABL transcript numbers involves a fast phase followed by a slow phase in averaged datasets. Drug resistance leads to regrowth. In this paper, variation of treatment responses between patients is examined. A significant positive correlation is found between slopes of the fast and the slow phase of decline. A significant negative correlation is found between slopes of the slow phase of decline and the regrowth phase. No correlation is found between slopes of the fast phase of decline and the regrowth phase. A mathematical model that is successfully fitted to diverse clinical profiles explains these correlations by invoking the immune response as a key determinant of tumour decline during treatment. Boosting immunity during drug therapy could enhance the response to treatment in patients.
PMCID: PMC2871874  PMID: 20147328
mathematical models; population dynamics; immune system; dynamics; cancer treatment dynamics
14.  ODE models for oncolytic virus dynamics 
Journal of theoretical biology  2010;263(4):530-543.
Replicating oncolytic viruses are able to infect and lyse cancer cells and spread through the tumor, while leaving normal cells largely unharmed. This makes them potentially useful in cancer therapy, and a variety of viruses have shown promising results in clinical trials. Nevertheless, consistent success remains elusive and the correlates of success have been the subject of investigation, both from an experimental and a mathematical point of view. Mathematical modeling of oncolytic virus therapy is often limited by the fact that the predicted dynamics depend strongly on particular mathematical terms in the model, the nature of which remain uncertain. We aim to address this issue in the context of ODE modeling, by formulating a general computational framework that is independent of particular mathematical expressions. By analyzing this framework, we find some new insights into the conditions for successful virus therapy. We find that depending on our assumptions about the virus spread, there can be two distinct types of dynamics. In models of the first type (the “fast spread” models), we predict that the viruses can eliminate the tumor if the viral replication rate is sufficiently high. The second type of models is characterized by a suboptimal spread (the “slow spread” models). For such models, the simulated treatment may fail, even for very high viral replication rates. Our methodology can be used to study the dynamics of many biological systems, and thus has implications beyond the study of virus therapy of cancers.
PMCID: PMC2839021  PMID: 20085772
15.  Effect of different modes of viral spread on the dynamics of multiply infected cells in human immunodeficiency virus infection 
Infection of individual cells with more than one HIV particle is an important feature of HIV replication, which may contribute to HIV pathogenesis via the occurrence of recombination, viral complementation and other outcomes that influence HIV replication and evolutionary dynamics. A previous mathematical model of co-infection has shown that the number of cells infected with i viruses correlates with the ith power of the singly infected cell population, and this has partly been observed in experiments. This model, however, assumed that virus spread from cell to cell occurs only via free virus particles, and that viruses and cells mix perfectly. Here, we introduce a cellular automaton model that takes into account different modes of virus spread among cells, including cell to cell transmission via the virological synapse, and spatially constrained virus spread. In these scenarios, it is found that the number of multiply infected cells correlates linearly with the number of singly infected cells, meaning that co-infection plays a greater role at lower virus loads. The model further indicates that current experimental systems that are used to study co-infection dynamics fail to reflect the true dynamics of multiply infected cells under these specific assumptions, and that new experimental techniques need to be designed to distinguish between the different assumptions.
PMCID: PMC3033025  PMID: 20659927
HIV; multiple infection; mathematical model; spatial; virus spread
16.  Use of oncolytic viruses for the eradication of drug-resistant cancer cells 
Targeted therapy using small-molecule inhibitors is a promising new therapy approach against cancer, but drug-resistant mutants present an obstacle to success. Oncolytic virus therapy, where viruses replicate specifically in cancer cells and kill them, is another promising therapy approach against cancer. While encouraging results have been observed in clinical trials, consistent success has not been possible so far. Based on a computational framework, I report that even if oncolytic virus therapy fails to eradicate a cancer, it can have the potential to eradicate the sub-population of drug-resistant cancer cells. Once this has occurred, targeted drug therapy can be used to induce cancer remission. For this to work, a drug resistance mutation must confer a certain fitness cost to the cell, as has been documented in the literature. The reason for this finding is that in the presence of a shared virus, the faster growing (drug-sensitive) cell population produces an amount of virus that is too much for the slower growing (drug-resistant) cell population to survive. This is derived from a population dynamic principle known as apparent competition. Therefore, a sequential combination of oncolytic virus and targeted therapies can overcome major weaknesses of either approach alone.
PMCID: PMC2658788  PMID: 18664430
mathematical models; oncolytic virus therapy; apparent competition
17.  Immunity and protection by live attenuated HIV/SIV vaccines 
Virology  2008;378(2):299-305.
Live attenuated virus vaccines have shown the greatest potential to protect against simian immunodeficiency virus (SIV) infection, a model for human immunodeficiency virus (HIV). Immunity against the vaccine virus is thought to mediate protection. However, it is shown computationally that the opposite might be true. According to the model, the initial growth of the challenge strain, its peak load, and its potential to be pathogenic is higher if immunity against the vaccine virus is stronger. This is because the initial growth of the challenge strain is mainly determined by virus competition rather than immune suppression. The stronger the immunity against the vaccine strain, the weaker its competitive ability relative to the challenge strain, and the lower the level of protection. If the vaccine virus does protect the host against a challenge, it is because the competitive interactions between the viruses inhibit the initial growth of the challenge strain. According to these arguments, an inverse correlation between the level of attenuation and the level of protection is expected, and this has indeed been observed in experimental data.
PMCID: PMC2613491  PMID: 18586297
HIV; vaccine; live attenuated; protection; theoretical biology; dynamics; mathematical model
18.  Model-driven Approaches for in Vitro Combination Therapy using ONYX-015 Replicating Oncolytic Adenovirus 1 
Replicating genetically modified adenoviruses have shown promise as a new treatment approach against cancer. Recombinant adenoviruses replicate only in cancer cells which contain certain mutations, such as the loss of functional p53, as is the case in the virus ONYX-015. The successful entry of the viral particle into target cells is strongly dependent on the presence of the main receptor for adenovirus, the coxsackie- and adeno-virus receptor (CAR). This receptor is frequently down-regulated in highly malignant cells, rendering this population less vulnerable to viral attack. It has been shown that use of MEK inhibitors can up-regulate CAR expression, resulting in enhanced adenovirus entry into the cells. However, inhibition of MEK results in G1 cell cycle arrest, rendering infected cells temporarily unable to produce virus. This forces a tradeo1. While drug mediated up-regulation of CAR enhances virus entry into cancer cells, the consequent cell cycle arrest inhibits production of new virus particles and the replication of the virus. Optimal control-based schedules of MEK inhibitor application should increase the efficacy of this treatment, maximizing the overall tumor toxicity by exploiting the dynamics of CAR expression and viral production. We introduce a mathematical model of these dynamics and show simple optimal control based strategies which motivate this approach.
PMCID: PMC2712658  PMID: 17095020
Oncolytic Viruses; ONYX-015; Optimal Control; Mathematical Modeling
19.  Combination of Two but Not Three Current Targeted Drugs Can Improve Therapy of Chronic Myeloid Leukemia 
PLoS ONE  2009;4(2):e4423.
Chronic myeloid leukemia (CML) is a cancer of the hematopoietic system and has been treated with the drug Imatinib relatively successfully. Drug resistance, acquired by mutations, is an obstacle to success. Two additional drugs are now considered and could be combined with Imatinib to prevent resistance, Dasatinib and Nilotinib. While most mutations conferring resistance to one drug do not confer resistance to the other drugs, there is one mutation (T315I) that induces resistance against all three drugs. Using computational methods, the combination of two drugs is found to increase the probability of treatment success despite this cross-resistance. Combining more than two drugs, however, does not provide further advantages. We also explore possible combination therapies using drugs currently under development. We conclude that among the targeted drugs currently available for the treamtent of CML, only the two most effective ones should be used in combination for the prevention of drug resistance.
PMCID: PMC2635955  PMID: 19204794
20.  Towards Predictive Computational Models of Oncolytic Virus Therapy: Basis for Experimental Validation and Model Selection 
PLoS ONE  2009;4(1):e4271.
Oncolytic viruses are viruses that specifically infect cancer cells and kill them, while leaving healthy cells largely intact. Their ability to spread through the tumor makes them an attractive therapy approach. While promising results have been observed in clinical trials, solid success remains elusive since we lack understanding of the basic principles that govern the dynamical interactions between the virus and the cancer. In this respect, computational models can help experimental research at optimizing treatment regimes. Although preliminary mathematical work has been performed, this suffers from the fact that individual models are largely arbitrary and based on biologically uncertain assumptions. Here, we present a general framework to study the dynamics of oncolytic viruses that is independent of uncertain and arbitrary mathematical formulations. We find two categories of dynamics, depending on the assumptions about spatial constraints that govern that spread of the virus from cell to cell. If infected cells are mixed among uninfected cells, there exists a viral replication rate threshold beyond which tumor control is the only outcome. On the other hand, if infected cells are clustered together (e.g. in a solid tumor), then we observe more complicated dynamics in which the outcome of therapy might go either way, depending on the initial number of cells and viruses. We fit our models to previously published experimental data and discuss aspects of model validation, selection, and experimental design. This framework can be used as a basis for model selection and validation in the context of future, more detailed experimental studies. It can further serve as the basis for future, more complex models that take into account other clinically relevant factors such as immune responses.
PMCID: PMC2629569  PMID: 19180240
21.  Human immunodeficiency virus evolution towards reduced replicative fitness in vivo and the development of AIDS 
Human immunodeficiency virus (HIV) infection progresses to AIDS following an asymptomatic period during which the virus is thought to evolve towards increased fitness and pathogenicity. We show mathematically that progression to the strongest HIV-induced pathology requires evolution of the virus towards reduced replicative fitness in vivo. This counter-intuitive outcome can happen if multiple viruses co-infect the same cell frequently, which has been shown to occur in recent experiments. According to our model, in the absence of frequent co-infection, the less fit AIDS-inducing strains might never emerge. The frequency of co-infection can correlate with virus load, which in turn is determined by immune responses. Thus, at the beginning of infection when immunity is strong and virus load is low, co-infection is rare and pathogenic virus variants with reduced replicative fitness go extinct. At later stages of infection when immunity is less efficient and virus load is higher, co-infection occurs more frequently and pathogenic virus variants with reduced replicative fitness can emerge, resulting in T-cell depletion. In support of these notions, recent data indicate that pathogenic simian immunodeficiency virus (SIV) strains occurring late in the infection are less fit in specific in vitro experiments than those isolated at earlier stages. If co-infection is blocked, the model predicts the absence of any disease even if virus loads are high. We hypothesize that non-pathogenic SIV infection within its natural hosts, which is characterized by the absence of disease even in the presence of high virus loads, could be explained by a reduced occurrence of co-infection in this system.
PMCID: PMC2274968  PMID: 17666377
HIV dynamics; mathematical models; evolution; disease progression; coinfection; fitness
22.  Dynamics of killer T cell inflation in viral infections 
Upon acute viral infection, a typical cytotoxic T lymphocyte (CTL) response is characterized by a phase of expansion and contraction after which it settles at a relatively stable memory level. Recently, experimental data from mice infected with murine cytomegalovirus (MCMV) showed different and unusual dynamics. After acute infection had resolved, some antigen specific CTL started to expand over time despite the fact that no replicative virus was detectable. This phenomenon has been termed as ‘CTL memory inflation’. In order to examine the dynamics of this system further, we developed a mathematical model analysing the impact of innate and adaptive immune responses. According to this model, a potentially important contributor to CTL inflation is competition between the specific CTL response and an innate natural killer (NK) cell response. Inflation occurs most readily if the NK cell response is more efficient than the CTL at reducing virus load during acute infection, but thereafter maintains a chronic virus load which is sufficient to induce CTL proliferation. The model further suggests that weaker NK cell mediated protection can correlate with more pronounced CTL inflation dynamics over time. We present experimental data from mice infected with MCMV which are consistent with the theoretical predictions. This model provides valuable information and may help to explain the inflation of CMV specific CD8+T cells seen in humans as they age.
PMCID: PMC2373406  PMID: 17251133
mathematical models; virus dynamics; CMV; CTL memory inflation; competition
23.  Effect of Cellular Quiescence on the Success of Targeted CML Therapy 
PLoS ONE  2007;2(10):e990.
Similar to tissue stem cells, primitive tumor cells in chronic myelogenous leukemia have been observed to undergo quiescence; that is, the cells can temporarily stop dividing. Using mathematical models, we investigate the effect of cellular quiescence on the outcome of therapy with targeted small molecule inhibitors.
Methods and Results
According to the models, the initiation of treatment can result in different patterns of tumor cell decline: a biphasic decline, a one-phase decline, and a reverse biphasic decline. A biphasic decline involves a fast initial phase (which roughly corresponds to the eradication of cycling cells by the drug), followed by a second and slower phase of exponential decline (corresponding to awakening and death of quiescent cells), which helps explain clinical data. We define the time when the switch to the second phase occurs, and identify parameters that determine whether therapy can drive the tumor extinct in a reasonable period of time or not. We further ask how cellular quiescence affects the evolution of drug resistance. We find that it has no effect on the probability that resistant mutants exist before therapy if treatment occurs with a single drug, but that quiescence increases the probability of having resistant mutants if patients are treated with a combination of two or more drugs with different targets. Interestingly, while quiescence prolongs the time until therapy reduces the number of cells to low levels or extinction, the therapy phase is irrelevant for the evolution of drug resistant mutants. If treatment fails as a result of resistance, the mutants will have evolved during the tumor growth phase, before the start of therapy. Thus, prevention of resistance is not promoted by reducing the quiescent cell population during therapy (e.g., by a combination of cell activation and drug-mediated killing).
The mathematical models provide insights into the effect of quiescence on the basic kinetics of the response to targeted treatment of CML. They identify determinants of success in the absence of drug resistant mutants, and elucidate how quiescence influences the emergence of drug resistant mutants.
PMCID: PMC1991595  PMID: 17912367
24.  Differences in Reactivation of Tuberculosis Induced from Anti-TNF Treatments Are Based on Bioavailability in Granulomatous Tissue 
PLoS Computational Biology  2007;3(10):e194.
The immune response to Mycobacterium tuberculosis (Mtb) infection is complex. Experimental evidence has revealed that tumor necrosis factor (TNF) plays a major role in host defense against Mtb in both active and latent phases of infection. TNF-neutralizing drugs used to treat inflammatory disorders have been reported to increase the risk of tuberculosis (TB), in accordance with animal studies. The present study takes a computational approach toward characterizing the role of TNF in protection against the tubercle bacillus in both active and latent infection. We extend our previous mathematical models to investigate the roles and production of soluble (sTNF) and transmembrane TNF (tmTNF). We analyze effects of anti-TNF therapy in virtual clinical trials (VCTs) by simulating two of the most commonly used therapies, anti-TNF antibody and TNF receptor fusion, predicting mechanisms that explain observed differences in TB reactivation rates. The major findings from this study are that bioavailability of TNF following anti-TNF therapy is the primary factor for causing reactivation of latent infection and that sTNF—even at very low levels—is essential for control of infection. Using a mathematical model, it is possible to distinguish mechanisms of action of the anti-TNF treatments and gain insights into the role of TNF in TB control and pathology. Our study suggests that a TNF-modulating agent could be developed that could balance the requirement for reduction of inflammation with the necessity to maintain resistance to infection and microbial diseases. Alternatively, the dose and timing of anti-TNF therapy could be modified. Anti-TNF therapy will likely lead to numerous incidents of primary TB if used in areas where exposure is likely.
Author Summary
Tuberculosis (TB) is the leading cause of death due to infectious disease in the world today. It is estimated that 2 billion people are currently infected, and although most people have latent infection, reactivation occurs due to factors such as HIV-1 and aging. Antibiotic treatments exist; however, there is still no cure and the current vaccine has proven to be unreliable. Experimental science has uncovered a plethora of immune factors that help the host control infection and maintain latency. One such factor, tumor necrosis factor alpha (TNF), is a protein that facilitates cell–cell communication during an inflammatory immune response. Animal models have shown that TNF is necessary for control of TB infection. Different types of anti-TNF drugs were developed for patients with non-TB related inflammatory diseases such as rheumatoid arthritis and Crohn's disease. Some of these patients who had latent TB suffered reactivation, especially with one drug type. Because these studies cannot be performed in the mouse, and nonhuman primates are expensive, we developed a computational model to perform virtual clinical trials (VCTs) that predicted why reactivation occurs and why it happens differentially between the two classes of drugs tested. We make recommendations on how this issue can be combated.
PMCID: PMC2041971  PMID: 17953477
25.  Immune responses and the emergence of drug-resistant virus strains in vivo. 
The treatment of viral infections using antiviral drugs has had a significant public health benefit in the setting of human immunodeficiency virus (HIV) infection, and newly developed drugs offer potential benefits in the management of other viral infections, including acute self-limiting infections such as influenza and picornaviruses (including the rhinoviruses that are responsible for a large proportion of 'common colds'). A serious concern with such treatments is that they may lead to the selection of drug-resistant strains. This has been a significant problem in the case of HIV infection. Existing mathematical-modelling studies of drug resistance have focused on the interactions between virus, target cells and infected cells, ignoring the impact of immune responses. Here, we present a model that explores the role of immune responses in the rise of drug-resistant mutants in vivo. We find that drug resistance is unlikely to be a problem if immune responses are maintained above a threshold level during therapy. Alternatively, if immune responses decline at a fast rate and fall below a threshold level during treatment (indicating impaired immunity), the rise of drug-resistant mutants is more likely. This indicates an important difference between HIV, which impairs immunity and for which immune responses have been observed to vanish during treatment, and viral infections such as influenza and rhinoviruses, for which such immune impairment is not present. Drug resistance is much more likely to be a problem in HIV than in acute and self-limiting infections.
PMCID: PMC1691709  PMID: 15306358

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