A mathematical model of the chemotherapeutic treatment of acute myeloblastic leukemia.

Based on our previous mathematical model of the acute myeloblastic leukemic (AML) state in man, we superimpose a chemotherapeutic drug treatment regimen. Our calculations suggest that small changes in the protocol can have significant effects on the result of treatment. Thus, the optimal period between drug doses is the S-phase interval of the leukemic cells--about 20h--and the greater the number of doses administered in a given course treatment, the longer the rest interval should be before the next course is administered. For a patient with a "slow" growing AML cell population, remission can be achieved with one or two courses of treatment, and further suppression of the leukemic population can be achieved with continued courses of treatment. However, for patients with a "fast" growing AML cell population, a similar aggressive treatment regimen succeeds in achieving remission status only at the cost of very great toxic effects on the normal neutrophil population and its precursors.     

A pharmacogenomic method for individualized prediction of drug sensitivity

Identifying the best drug for each cancer patient requires an efficient individualized strategy. We present MATCH (M erging genomic and pharmacologic A nalyses for T herapy CH oice), an approach using public genomic resources and drug testing of fresh tumor samples to link drugs to patients. Valproic acid (VPA) is highlighted as a proof‐of‐principle. In order to predict specific tumor types with high probability of drug sensitivity, we create drug response signatures using publically available gene expression data and assess sensitivity in a data set of >40 cancer types. Next, we evaluate drug sensitivity in matched tumor and normal tissue and exclude cancer types that are no more sensitive than normal tissue. From these analyses, breast tumors are predicted to be sensitive to VPA. A meta‐analysis across breast cancer data sets shows that aggressive subtypes are most likely to be sensitive to VPA, but all subtypes have sensitive tumors. MATCH predictions correlate significantly with growth inhibition in cancer cell lines and three‐dimensional cultures of fresh tumor samples. MATCH accurately predicts reduction in tumor growth rate following VPA treatment in patient tumor xenografts. MATCH uses genomic analysis with in vitro testing of patient tumors to select optimal drug regimens before clinical trial initiation.     

Prediction of weed density: the increase of error with prediction interval, and the use of long-term prediction for weed management

1. This paper addresses the errors that are associated with the long-term prediction of weed densities, and the effect of these errors on the performance of weed management decisions based on those long-term predictions.
2. A model of weed population dynamics was constructed and its parameters were estimated from experimental observations of population dynamics of the weed species Stellaria media in a crop rotation.
3. The observations showed that estimates of weed population growth rate differed between two locations.
4. The model was used to analyse error propagation for predicted weed densities in an enlarged prediction interval. It is concluded that errors due to an uncertain population growth rate increase linearly with the length of the prediction interval, and thus pose an upper limit to the horizon for long-term predictions.
5. It is shown that a limited ability to predict weed densities does not necessarily impair the practical use of weed population dynamic models in planning for long-term weed control programmes.     

Filtration and digestion responses of an elementally homeostatic consumer to changes in food quality: a predictive model

In the study of the stoichiometric relationship between autotrophs and herbivores, attention has been largely focused on effects of the encountered mismatch between needs and supplies of an element on herbivore growth and ecosystem processes. Herbivore adaptation to poor food quality has rarely been investigated. This study presents a predictive model of feeding, assimilation, digestion and excretion of Daphnia facing a dietary deficiency in phosphorus. Biochemical compounds in the food were divided into phosphorous and non-phosphorus compounds. It was assumed that Daphnia is able to differently assimilate both types of compounds by regulation of target specific digestive enzymes. Feeding rate was regulated by optimal gut residence time of food particles, and assimilation efficiency by gut residence time and optimal secretion of both classes of gut enzymes. The model predicted the optimal strategy for a consumer facing an elementally imbalanced diet: (1) increase the ingestion rate, and (2) increase the secretion rate of both classes of gut enzymes. It resulted in decreased C and nutrient assimilation efficiencies, increased C feeding costs, and reduced growth rate. Sensitivity analysis showed that these predictions were qualitatively not influenced by parameter values. An alternative model was tested that includes an additive term allowing the direct excretion of C assimilated in excess. Results showed that this strategy is not optimal for the consumer growth rate. In conclusion, the model supports the hypothesis that carbon ingested in excess may generate energy that can be used to obtain more nutrients by increased feeding rate.     

Modeling multi-drug chemotherapy: tailoring treatment to individuals

BACKGROUND: Predicting and tailoring optimal cancer treatments presents a major challenge. METHODS: A computational model (kinetically tailored treatment, or KITT model) is developed to predict drug combinations, doses, and schedules likely to be effective in reducing tumor size and prolonging patient life. Treatment strategies may be tailored to individuals based on tumor cell kinetics. The model incorporates intra-tumor heterogeneity and evolution of drug resistance, apoptotic rates, and cell division rates. Tumor growth may follow an exponential or a Gompertzian trajectory. Drug pharmacodynamic and pharmacokinetic models are used. Toxicity is modeled in several ways. RESULTS: A key prediction of KITT is that including cytostatic drugs like tamoxifen and herceptin during treatment with cytotoxic drugs substantially increases the probability of cure and prolongs patient life. Results also suggest that altering drug scheduling may be more effective but not more toxic than dose escalation. CAF chemotherapy (cyclophosphamide, adriamycin, and 5-fluorouracil) is predicted to be more effective than CMF (cyclophosphamide, methotrexate, and 5-fluorouracil). KITT also suggests that tumors with a high proliferative index (PI) may respond better to drug combinations incorporating two cell-cycle phase-specific drugs than do tumors with a low PI. Tumors with a low PI, in contrast, are predicted to respond better to regimens involving two cell-cycle phase-non-specific drugs than do tumors with a high PI. These predictions are borne out by clinical trial results published in the literature, which are discussed.Simulated predictions of the model match well with results from a clinical trial by Silvestrini et al. (2000. Int. J. Cancer 87, 405). The results of simulating the growth of 26896 tumors are used to construct a decision tree for prognosis to identify the key tumor and treatment variables. CONCLUSION: Additional tests of the model are needed in which physicians collect information on apoptotic and proliferative indices, cell-cycle times, and drug resistance from biopsies of each individual's tumor. Computational models may become important tools to help optimize and tailor cancer treatments.     

Alternative Predictions for Optimal Dispersal in Response to Local Catastrophic Mortality

What dispersal strategy should be employed by an organism in response to local catastrophic mortality? Here we contrast predictions from an analytical solution derived from an ESS model which optimizes competitive ability (Comins et al., 1980) with those from a stochastic, branching process model (Karlson and Taylor, 1992) which maximizes survivorship of a clonal lineage. The optimal dispersal fraction varies directly with the probability of local extinction in the ESS model, yet varies inversely with this probability over much of the parameter space in the latter model. In order to conform more closely with the assumptions of the ESS model, we have modified the branching process model to have a random, Poisson-distributed number of offspring and compared the predictions of these models. Both models invoke dispersal as an escape from local extinction and predict mixed dispersal strategies over a wide range of conditions. However, increasing local catastrophic mortality favors more dispersal in the ESS model, but it can be so severe in the branching process model that no dispersal strategy is adaptive. In this model, the predicted optimal proportion of dispersed offspring is highest at low to intermediate levels of catastrophic mortality depending on the total number of offspring produced. We suggest that this observed discrepancy is sufficiently large to warrant empirical tests of these qualitatively different predictions.     

Fluctuating selection on reproductive timing in Digitalis purpurea

Despite recent, strong interest in the modelling of monocarpic perennial flowering strategies, little is known about how variation in demographic rates affects selection on optimal timing of flowering. Temporal variation may yield fluctuating selective pressures, or, if individuals experience time trends, selection for phenotypic plasticity. Here we report the results of a 3-year study in a large field population of the facultative biennial herb Digitalis purpurea , where we use field data on size-dependent growth, survival and fecundity to parameterize an existing optimisation model. We compare results from models using either deterministic or individually varying demographic rates to address the degree of fluctuating selection on the flowering strategy. In addition, we explore whether recent growing conditions influence the size-specific liability to flower. Model results differed widely between years; immediate onset of reproduction was predicted in 1999, strongly delayed reproduction in 2000. This reflected large differences in both growth and survival rates between years. Observed flowering sizes also varied between years, but were larger in 1999 than in 2000, contrary to model predictions. Incorporating individual variation in growth increased predicted optimal flowering sizes compared to models using deterministic growth, whereas the inclusion of individual survival variation had little effect. There was no significant effect of recent growth rate on flowering probability. Taken together, these results indicate highly fluctuating selection on the flowering strategy in D. purpurea , but no evidence of adaptive plasticity in response to current growing conditions. Fluctuating selection may contribute to maintain genetic variation for threshold size for flowering, and may partly explain the large within-season size-variation in flowering individuals found in natural populations of D. purpurea .     

Using a kinetic model that considers cell segregation to optimize hEGF expression in fed-batch cultures of recombinant Escherichia coli

Growth inhibition of recombinant Escherichia coli during the expression of human epidermal growth factor was observed. The recombinant cells could be segregated into three populations based on their cell division and plasmid maintenance abilities: dividing and plasmid-bearing cells, dividing and plasmid-free cells, and viable-but-non-culturable (VBNC) cells. Fed-batch fermentations were performed to investigate the effect of cell segregation on the kinetics of growth and foreign protein production. The results showed that a low concentration of inducer caused weak induction, whereas high levels cause strong induction, resulting in cells segregating into VBNC bacteria and producing a low foreign protein yield. A kinetic model for cell segregation was proposed and its predictions correlated well with experimental data for cell growth and protein expression. The optimal induction strategy could then be predicted by the model, and this prediction was then verified by experimentally deriving the conditions necessary for maximum expression of recombinant protein.     

Foraging for intermittently refuged prey: theory and field observations of a parasitoid

Many insect herbivores feed in concealed locations but become accessible intermittently, creating windows of greater vulnerability to attack, and generating a proportion of the prey population that is readily accessible to foraging natural enemies. We incorporated accessible prey into an extant optimal foraging model, and found that this addition allowed opportunistic exploitation of prey that have already emerged from refugia (the leaving strategy) as a viable strategy, in addition to waiting at refugia for prey to emerge (the waiting strategy). We parameterized the model empirically for the parasitoid Macrocentrus grandii and its host, Ostrinia nubilalis, under field conditions. The model predicted that M. grandii should adopt a leaving strategy when host patch density is high (travel time between patches is short), but a waiting strategy when host patch density is low (travel time between patches is long). Field observations of M. grandii patch tenure were consistent with model predictions, indicating that M. grandii exhibited flexible behaviour based on experience within a foraging bout, and that these behavioural shifts improved foraging efficiency. Behaviour of M. grandii was responsive to heterogeneity in host emergence rates, and appeared to be driven by the relatively small proportion of the host population that became accessible at a fast rate. Therefore understanding forager responses to intermittently refuged prey may require characterization of the behaviour of a subset of the prey population, rather than the average prey individual. The model can potentially be used as a framework for comparative studies across forager taxa, to understand when foragers on intermittently accessible prey should adopt fixed waiting or leaving strategies vs. a flexible strategy that is responsive to the current environment.     

Mixed immunotherapy and chemotherapy of tumors: feedback design and model updating schemes

In this paper, a recently developed model governing the cancer growth on a cell population level with combination of immune and chemotherapy is used to develop a reactive (feedback) mixed treatment strategy. The feedback design proposed here is based on nonlinear constrained model predictive control together with an adaptation scheme that enables the effects of unavoidable modeling uncertainties to be compensated. The effectiveness of the proposed strategy is shown under realistic human data showing the advantage of treatment in feedback form as well as the relevance of the adaptation strategy in handling uncertainties and modeling errors. A new treatment strategy defined by an original optimal control problem formulation is also proposed. This new formulation shows particularly interesting possibilities since it may lead to tumor regression under better health indicator profile.     

Quantification of long-term doxorubicin response dynamics in breast cancer cell lines to direct treatment schedules

While acquired chemoresistance is recognized as a key challenge to treating many types of cancer, the dynamics with which drug sensitivity changes after exposure are poorly characterized. Most chemotherapeutic regimens call for repeated dosing at regular intervals, and if drug sensitivity changes on a similar time scale then the treatment interval could be optimized to improve treatment performance. Theoretical work suggests that such optimal schedules exist, but experimental confirmation has been obstructed by the difficulty of deconvolving the simultaneous processes of death, adaptation, and regrowth taking place in cancer cell populations. Here we present a method of optimizing drug schedules in vitro through iterative application of experimentally calibrated models, and demonstrate its ability to characterize dynamic changes in sensitivity to the chemotherapeutic doxorubicin in three breast cancer cell lines subjected to treatment schedules varying in concentration, interval between pulse treatments, and number of sequential pulse treatments. Cell populations are monitored longitudinally through automated imaging for 600–800 hours, and this data is used to calibrate a family of cancer growth models, each consisting of a system of ordinary differential equations, derived from the bi-exponential model which characterizes resistant and sensitive subpopulations. We identify a model incorporating both a period of growth arrest in surviving cells and a delay in the death of chemosensitive cells which outperforms the original bi-exponential growth model in Akaike Information Criterion based model selection, and use the calibrated model to quantify the performance of each drug schedule. We find that the inter-treatment interval is a key variable in determining the performance of sequential dosing schedules and identify an optimal retreatment time for each cell line which extends regrowth time by 40%-239%, demonstrating that the time scale of changes in chemosensitivity following doxorubicin exposure allows optimization of drug scheduling by varying this inter-treatment interval.     

Statistical optimization of single-cell production from Taxus cuspidata plant cell aggregates

Flow-cytometric characterization of plant cell culture growth and metabolism at the single-cell level is a method superior to traditional culture average measurements for collecting population information. Investigation of culture heterogeneity and production variability by obtaining information about different culture subpopulations is crucial for optimizing bio-processes for enhanced productivity. Obtaining high yields of intact and viable single cells from aggregated plant cell cultures is an enabling criterion for their analysis and isolation using high-throughput flow cytometric methods. The critical parameters affecting the enzymatic isolation of single cells from aggregated Taxus cuspidata plant cell suspensions were optimized using response-surface methodology and factorial central composite design. Using a design of experiments approach, the output response single-cell yield (SCY, percentage of cell clusters containing only a single cell) was optimized. Optimal conditions were defined for the independent parameters cellulase concentration, pectolyase Y-23 concentration, and centrifugation speed to be 0.045% (w/v), 0.7% (w/v), and 1200?×?g, respectively. At these optimal conditions, the model predicted a maximum SCY of 48%. The experimental data exhibited a 72% increase over previously attained values and additionally validated the model predictions. More than 99% of the isolated cells were viable and suitable for rapid analysis through flow cytometry, thus enabling the collection of population information from cells that accurately represent aggregated suspensions. These isolated cells can be further studied to gain insight into both growth and secondary metabolite production, which can be used for bio-process optimization.     

PERIODIC THERMAL STRESS: OPTIMAL SPACING PREDICTED BY A SIMPLE SET-BACK MODEL OF CELL DIVISION SYNCHRONIZATION

Digital computer simulations have been used to make quantitative predictions based on a simple set-back model of cell division synchronization. According to the model appropriate thermal stress reverses progress within a segment of the division cycle called the set-back interval. In the simulations normally distributed cell-to-cell variations in division cycling rate between periodic thermal shocks were produced with the Monte Carlo method.
The simulations have shown that reasonably good synchronization with the single shock per division strategy requires a relatively long set-back interval and small cell-to-cell variations in rate of progress through the division cycle. The simulations have shown that the degree of synchrony produced by such periodic shocks is highly dependent on the time interval between shocks—with a series of as many as seven shocks inappropriately spaced producing less synchrony than a single shock! The optimal time interval between successive thermal shocks was found approximately equal to the mode division cycle time at synchrony equilibrium multiplied by 1 plus half the fraction of the division cycle occupied by the set-back interval. Position of the set-back interval within the division cycle had little effect on synchrony at the end of the final shock.     

Optimal resource allocation to seeds and vegetative propagules under density-dependent regulation in Syneilesis palmata (Compositae)

Syneilesis palmata reproduces by both seeds and vegetative propagules (short rhizomes). The latter result in the production of new plants that are larger in size and hence have a higher survival probability and a higher growth rate than seeds. A previous study predicted that the optimal reproductive strategy, in terms of maximizing population growth rate (a fitness measure under no density regulations), was pure vegetative reproduction. However, high resource investment to vegetative propagules can cause local crowding resulting in reduced demographic performances of the plants, because the vegetative propagules of Syneilesis are produced close to one another. We examined, in this situation, the impact of allocating a certain proportion of reproductive resource to seeds with relatively greater capacity for dispersal. We simulated dynamics of hypothetical Syneilesis populations with various reproductive resource allocation balances (from pure seed to pure vegetative reproduction), using a density-dependent matrix model. In the model, it was assumed that plants from vegetative propagules experienced density-dependent reduction in their survival probabilities, but this was not the case for plants originating from seeds. Each allocation strategy was evaluated based on an equilibrium population density, a fitness measure under density-dependent regulations. The optimal reproductive strategy predicted was pure vegetative reproduction. Unrealistic conditions were required for seed reproduction to be favoured, such as the production of seeds one hundred times the normal number per unit resource investment. However, the conditions were fairly relaxed compared with those required in the model where no density effects were incorporated. This indicates that escape from local crowding is likely to be one of the roles of seed production in Syneilesis.     

Adjusted Prediction of Plant Height with Application to Greenhouse Grown Poinsettias

The objective is to predict future plant growth using data from greenhouse grown poinsettias. A population‐mean growth trajectory can be predicted from growth conditions using previously published results. Four different predictors are used to identify the local group‐effects that come in addition to the population‐mean response to growth conditions, in order to improve the predictions of the final plant height for specific plant groups. A search for optimal model complexity and use of growth history during calibration is conducted using cross‐validation.     

Optimal metabolic regulation along resource stoichiometry gradients

Most heterotrophic organisms feed on substrates that are poor in nutrients compared to their demand, leading to elemental imbalances that may constrain their growth and function. Flexible carbon (C)‐use efficiency (CUE, C used for growth over C taken up) can represent a strategy to reduce elemental imbalances. Here, we argue that metabolic regulation has evolved to maximise the organism growth rate along gradients of nutrient availability and translated this assumption into an optimality model that links CUE to substrate and organism stoichiometry. The optimal CUE is predicted to decrease with increasing substrate C‐to‐nutrient ratio, and increase with nutrient amendment. These predictions are generally confirmed by empirical evidence from a new database of c. 2200 CUE estimates, lending support to the hypothesis that CUE is optimised across levels of organisation (microorganisms and animals), in aquatic and terrestrial systems, and when considering nitrogen or phosphorus as limiting nutrients.     

Design of optimal cancer chemotherapy using a continuous-time state model of cell kinetics

A continuous bilinear model in state space is used to describe the cell kinetics of a tumor-cell population under the effects of chemotherapy. Firstly, the time-course behavior of a Chinese-hamster-ovary (CHO) cell population is simulated to demonstrate the utility of the model. Then, an optimal strategy for cancer treatment is derived, based on the need to balance the effects on both cancerous and normal tissues. The performance index minimized is the sum of the weighted tumor population and the weighted total drug dosage. The optimization problem has resulted in a two-point boundary-value problem (TPBVP) with a bang-bang control policy, which is solved by the switching-time variation method (STVM). Computer simulation of CHO cells is also carried out as a numerical example of determining optimal cancer chemotherapy.     

Regulatory control and the costs and benefits of biochemical noise

Experiments in recent years have vividly demonstrated that gene expression can be highly stochastic. How protein concentration fluctuations affect the growth rate of a population of cells is, however, a wide-open question. We present a mathematical model that makes it possible to quantify the effect of protein concentration fluctuations on the growth rate of a population of genetically identical cells. The model predicts that the population's growth rate depends on how the growth rate of a single cell varies with protein concentration, the variance of the protein concentration fluctuations, and the correlation time of these fluctuations. The model also predicts that when the average concentration of a protein is close to the value that maximizes the growth rate, fluctuations in its concentration always reduce the growth rate. However, when the average protein concentration deviates sufficiently from the optimal level, fluctuations can enhance the growth rate of the population, even when the growth rate of a cell depends linearly on the protein concentration. The model also shows that the ensemble or population average of a quantity, such as the average protein expression level or its variance, is in general not equal to its time average as obtained from tracing a single cell and its descendants. We apply our model to perform a cost-benefit analysis of gene regulatory control. Our analysis predicts that the optimal expression level of a gene regulatory protein is determined by the trade-off between the cost of synthesizing the regulatory protein and the benefit of minimizing the fluctuations in the expression of its target gene. We discuss possible experiments that could test our predictions.     

Modelling chemotherapy resistance in palliation and failed cure

The goal of palliative cancer chemotherapy treatment is to prolong survival and improve quality of life when tumour eradication is not feasible. Chemotherapy protocol design is considered in this context using a simple, robust, model of advanced tumour growth with Gompertzian dynamics, taking into account the effects of drug resistance. It is predicted that reduced chemotherapy protocols can readily lead to improved survival times due to the effects of competition between resistant and sensitive tumour cells. Very early palliation is also predicted to quickly yield near total tumour resistance and thus decrease survival duration. Finally, our simulations indicate that failed curative attempts using dose densification, a common protocol escalation strategy, can reduce survival times.     

Optimal control for a cancer chemotherapy problem with general growth and loss functions

We investigate a model of a cancer chemotherapy problem where the aim is to minimize the tumor burden at the end of the treatment period while maintaining a normal cell population above a lower level as a limit of toxicity. The analysis is performed for general classes of growth and loss functions. The optimal drug dose is maximum initially so that the normal cell population is driven down to its lower level, and then the drug level is chosen to maintain the normal cell population there until the end of treatment. During treatment the number of tumor cells is always decreasing.