Stepwise Tikhonov Regularisation: Application to the Prediction of HIV-1 Drug Resistance.

This paper focuses on constructing genotypic predictors for antiretroviral drug susceptibility of HIV. To this end, a method to recover the largest elements of an unknown vector in a least squares problem is developed. The proposed method introduces two novel ideas. The first idea is a novel forward stepwise selection procedure based on the magnitude of the estimates of the candidate variables. To implement this newly introduced procedure, we revise Tikhonov regularisation from a sparse representations' perspective. This analysis leads us to the second novel idea in the paper, which is the development of a new method to recover the largest elements of the unknown vector in the least squares problem. The method implements a sequence of Tikhonov regularisation problems which aim to recover the largest of the remaining elements of the unknown vector. Additionally, we derive sufficient conditions that ensure the recovery of the largest elements of the unknown vector. We perform numerical studies using simulated data and data from the Stanford HIV resistance database. The performance of the proposed method is compared against a state-of-the-art method.

A method for evaluating treatment quality using in vivo EPID dosimetry and statistical process control in radiation therapy.

Purpose Due to increasing complexity, modern radiotherapy techniques require comprehensive quality assurance (QA) programmes, that to date generally focus on the pre-treatment stage. The purpose of this paper is to provide a method for an individual patient treatment QA evaluation and identification of a "quality gap" for continuous quality improvement. Design/methodology/approach A statistical process control (SPC) was applied to evaluate treatment delivery using in vivo electronic portal imaging device (EPID) dosimetry. A moving range control chart was constructed to monitor the individual patient treatment performance based on a control limit generated from initial data of 90 intensity-modulated radiotherapy (IMRT) and ten volumetric-modulated arc therapy (VMAT) patient deliveries. A process capability index was used to evaluate the continuing treatment quality based on three quality classes: treatment type-specific, treatment linac-specific, and body site-specific. Findings The determined control limits were 62.5 and 70.0 per cent of the χ pass-rate for IMRT and VMAT deliveries, respectively. In total, 14 patients were selected for a pilot study the results of which showed that about 1 per cent of all treatments contained errors relating to unexpected anatomical changes between treatment fractions. Both rectum and pelvis cancer treatments demonstrated process capability indices were less than 1, indicating the potential for quality improvement and hence may benefit from further assessment. Research limitations/implications The study relied on the application of in vivo EPID dosimetry for patients treated at the specific centre. Sampling patients for generating the control limits were limited to 100 patients. Whilst the quantitative results are specific to the clinical techniques and equipment used, the described method is generally applicable to IMRT and VMAT treatment QA. Whilst more work is required to determine the level of clinical significance, the authors have demonstrated the capability of the method for both treatment specific QA and continuing quality improvement. Practical implications The proposed method is a valuable tool for assessing the accuracy of treatment delivery whilst also improving treatment quality and patient safety. Originality/value Assessing in vivo EPID dosimetry with SPC can be used to improve the quality of radiation treatment for cancer patients.

Investigation of a real-time EPID-based patient dose monitoring safety system using site-specific control limits.

PURPOSE: The aim of this study is to investigate the performance and limitations of a real-time transit electronic portal imaging device (EPID) dosimetry system for error detection during dynamic intensity modulated radiation therapy (IMRT) treatment delivery. Sites studied are prostate, head and neck (HN), and rectal cancer treatments. METHODS: The system compares measured cumulative transit EPID image frames with predicted cumulative image frames in real-time during treatment using a χ comparison with 4 %, 4 mm criteria. The treatment site-specific thresholds (prostate, HN and rectum IMRT) were determined using initial data collected from 137 patients (274 measured treatment fractions) and a statistical process control methodology. These thresholds were then applied to data from 15 selected patients including 5 prostate, 5 HN, and 5 rectum IMRT treatments for system evaluation and classification of error sources. RESULTS: Clinical demonstration of real-time transit EPID dosimetry in IMRT was presented. For error simulation, the system could detect gross errors (i.e. wrong patient, wrong plan, wrong gantry angle) immediately after EPID stabilisation; 2 seconds after the start of treatment. The average rate of error detection was 7.0 % (prostate = 5.6 %, HN= 8.7 % and rectum = 6.7 %). The detected errors were classified as either clinical in origin (e.g. patient anatomical changes), or non-clinical in origin (e.g. detection system errors). Classified errors were 3.2 % clinical and 3.9 % non-clinical. CONCLUSION: An EPID-based real-time error detection method for treatment verification during dynamic IMRT has been developed and tested for its performance and limitations. The system is able to detect gross errors in real-time, however improvement in system robustness is required to reduce the non-clinical sources of error detection.

A cine-EPID based method for jaw detection and quality assurance for tracking jaw in IMRT/VMAT treatments.

A new tool with the potential to verify and track jaw position during delivery has been developed. The method should be suitable for independent quality assurance for jaw position during jaw tracking dynamic IMRT and VMAT treatments. The jaw detection and tracking algorithm developed consists of five main steps. Firstly, the image is enhanced by removing a normalised predicted EPID image (that does not include the collimator transmission) from each cine EPID image. Then, using a histogram clustering technique a global intensity threshold level was determined. This threshold level was used to classify each pixel of the image as either under the jaws or under the MLC. Additionally, the collimator angle was automatically detected and used to rotate the image to vertical direction. Finally, this rotation allows the jaw positions to be determined using vertical and horizontal projection profiles. Nine IMRT fields (with static jaws) and a single VMAT clinical field (with dynamic jaws) were tested by determining the root mean square difference between planned and detected jaw positions. The test results give a detection accuracy of ±1 mm RMS error for static jaw IMRT treatments and ±1.5 mm RMS error for the dynamic jaw VMAT treatment. This method is designed for quality assurance and verification in modern radiation therapy; to detect the position of static jaws or verify the position of tracking jaws in more complex treatments. This method uses only information extracted from EPID images and it is therefore independent from the linear accelerator.

Equilibria and stability of a class of positive feedback loops.

Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops determines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain Theorem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biological meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability.

An independent system for real-time dynamic multileaf collimation trajectory verification using EPID.

A new tool has been developed to verify the trajectory of dynamic multileaf collimators (MLCs) used in advanced radiotherapy techniques using only the information provided by the electronic portal imaging devices (EPID) measured image frames. The prescribed leaf positions are resampled to a higher resolution in a pre-processing stage to improve the verification precision. Measured MLC positions are extracted from the EPID frames using a template matching method. A cosine similarity metric is then applied to synchronise measured and planned leaf positions for comparison. Three additional comparison functions were incorporated to ensure robust synchronisation. The MLC leaf trajectory error detection was simulated for both intensity modulated radiation therapy (IMRT) (prostate) and volumetric modulated arc therapy (VMAT) (head-and-neck) deliveries with anthropomorphic phantoms in the beam. The overall accuracy for MLC positions automatically extracted from EPID image frames was approximately 0.5 mm. The MLC leaf trajectory verification system can detect leaf position errors during IMRT and VMAT with a tolerance of 3.5 mm within 1 s.

Spatial Quantification of Cytosolic Ca²âº Accumulation in Nonexcitable Cells: An Analytical Study.

Calcium ions act as messengers in a broad range of processes such as learning, apoptosis, and muscular movement. The transient profile and the temporal accumulation of calcium signals have been suggested as the two main characteristics in which calcium cues encode messages to be forwarded to downstream pathways. We address the analytical quantification of calcium temporal-accumulation in a long, thin section of a nonexcitable cell by solving a boundary value problem. In these expressions we note that the cytosolic Ca(2+) accumulation is independent of every intracellular calcium flux and depends on the Ca(2+) exchange across the membrane, cytosolic calcium diffusion, geometry of the cell, extracellular calcium perturbation, and initial concentrations. In particular, we analyse the time-integrated response of cytosolic calcium due to i) a localised initial concentration of cytosolic calcium and ii) transient extracellular perturbation of calcium. In these scenarios, we conclude that i) the range of calcium progression is confined to the vicinity of the initial concentration, thereby creating calcium microdomains; and ii) we observe a low-pass filtering effect in the response driven by extracellular Ca(2+) perturbations. Additionally, we note that our methodology can be used to analyse a broader range of stimuli and scenarios.

A slow axon antidromic blockade hypothesis for tremor reduction via deep brain stimulation.

Parkinsonian and essential tremor can often be effectively treated by deep brain stimulation. We propose a novel explanation for the mechanism by which this technique ameliorates tremor: a reduction of the delay in the relevant motor control loops via preferential antidromic blockade of slow axons. The antidromic blockade is preferential because the pulses more rapidly clear fast axons, and the distribution of axonal diameters, and therefore velocities, in the involved tracts, is sufficiently long-tailed to make this effect quite significant. The preferential blockade of slow axons, combined with gain adaptation, results in a reduction of the mean delay in the motor control loop, which serves to stabilize the feedback system, thus ameliorating tremor. This theory, without any tuning, accounts for several previously perplexing phenomena, and makes a variety of novel predictions.

A single compartment model of pacemaking in dissasociated substantia nigra neurons: stability and energy analysis.

Spontaneous oscillations in the mid-brain dopaminergic neurons are an important feature of motor control. The degeneration of these neurons is involved in movement disorders, particularly Parkinson's Disease. Modelling of this activity is an important part of developing an understanding of the pathogenic process. We develop a mathematical paradigm to describe this activity with a single compartment approach and a CellML version is made publicly available. The model explicitly describes the dynamics of the transmembrane potential with changes in the levels of important cations and is consistent with two major observations in the literature regarding its behaviour in the presence of channel blockers. Stability of the model behaviour is determined from the properties of its Monodromy matrix. We also discuss from the perspective of energy, a pharmacological intervention suggested in the treatment of Parkinson's Disease.

Cumulative signal transmission in nonlinear reaction-diffusion networks.

Quantifying signal transmission in biochemical systems is key to uncover the mechanisms that cells use to control their responses to environmental stimuli. In this work we use the time-integral of chemical species as a measure of a network's ability to cumulatively transmit signals encoded in spatiotemporal concentrations. We identify a class of nonlinear reaction-diffusion networks in which the time-integrals of some species can be computed analytically. The derived time-integrals do not require knowledge of the solution of the reaction-diffusion equation, and we provide a simple graphical test to check if a given network belongs to the proposed class. The formulae for the time-integrals reveal how the kinetic parameters shape signal transmission in a network under spatiotemporal stimuli. We use these to show that a canonical complex-formation mechanism behaves as a spatial low-pass filter, the bandwidth of which is inversely proportional to the diffusion length of the ligand.

Modeling the three stages in HIV infection.

A typical HIV infection response consists of three stages: an initial acute infection, a long asymptomatic period and a final increase in viral load with simultaneous collapse in healthy CD4+T cell counts. The majority of existing mathematical models give a good representation of either the first two stages or the last stage of the infection. Using macrophages as a long-term active reservoir, a deterministic model is proposed to explain the three stages of the infection including the progression to AIDS. Simulation results illustrate how chronic infected macrophages can explain the progression to AIDS provoking viral explosion. Further simulation studies suggest that the proposed model retains its key properties even under moderately large parameter variations. This model provides important insights on how macrophages might play a crucial role in the long term behavior of HIV infection.

Positive feedback in the Akt/mTOR pathway and its implications for growth signal progression in skeletal muscle cells: an analytical study.

The IGF-1 mediated Akt/mTOR pathway has been recently proposed as mediator of skeletal muscle growth and a positive feedback between Akt and mTOR was suggested to induce homogeneous growth signals along the whole spatial extension of such long cells. Here we develop two biologically justified approximations which we study under the presence of four different initial conditions that describe different paradigms of IGF-1 receptor-induced Akt/mTOR activation. In first scenario the activation of the feedback cascade was assumed to be mild or protein turnover considered to be high. In turn, in the second scenario the transcriptional regulation was assumed to maintain defined levels of inactive pro-enzymes. For both scenarios, we were able to obtain closed-form formulas for growth signal progression in time and space and found that a localised initial signal maintains its Gaussian shape, but gets delocalised and exponentially degraded. Importantly, mathematical treatment of the reaction diffusion system revealed that diffusion filtered out high frequencies of spatially periodic initiator signals suggesting that the muscle cell is robust against fluctuations in spatial receptor expression or activation. However, neither scenario was consistent with the presence of stably travelling signal waves. Our study highlights the role of feedback loops in spatiotemporal signal progression and results can be applied to studies in cell proliferation, cell differentiation and cell death in other spatially extended cells.

Computer simulation of structured treatment interruption for HIV infection.

The use of highly active antiretroviral therapy (HAART) for suppression of measurable levels of virus in the body has greatly contributed to restoration and preservation of the immune system in HIV positive patients. However, short and long term problems associated with HAART have led to proposals for alternative treatment strategies for controlling HIV infection. In particular, structured treatment interruptions (STIs) that consist of therapy withdrawal and re-initiation according to specific criteria have been considered. The aim of these STIs was one or both of: (i) to stimulate the immune system to react to HIV, (ii) to allow re-emergence of wild-type virus and thereby reduce problems of drug resistance. However, a number of clinical trials of STIs have shown adverse outcomes for patients under discontinuous therapy, including serious health risks associated with treatment interruptions. In this paper we consider in some detail two of the larger clinical studies, namely, (a) strategies for management of anti-retroviral therapy (SMART); (b) Staccato study. For each of these studies we perform computer simulations of the treatment strategies. These simulations suggest several underlying reasons for the adverse outcomes during treatment interruption. In particular, HIV infection exhibits rapid dynamic load changes, and therefore measurement based treatment regimes need to be carefully designed to avoid large transients in healthy CD4+ T cell count. Furthermore, repeated treatment interruptions may accelerate the emergence of resistant mutant virus and may increase the infection of long term reservoirs such as macrophages which will accelerate progression to AIDS.

Quantifying uncertainty, variability and likelihood for ordinary differential equation models.

BACKGROUND: In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space. RESULTS: The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability. CONCLUSIONS: While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.

Sequential activation of metabolic pathways: a dynamic optimization approach.

The regulation of cellular metabolism facilitates robust cellular operation in the face of changing external conditions. The cellular response to this varying environment may include the activation or inactivation of appropriate metabolic pathways. Experimental and numerical observations of sequential timing in pathway activation have been reported in the literature. It has been argued that such patterns can be rationalized by means of an underlying optimal metabolic design. In this paper we pose a dynamic optimization problem that accounts for time-resource minimization in pathway activation under constrained total enzyme abundance. The optimized variables are time-dependent enzyme concentrations that drive the pathway to a steady state characterized by a prescribed metabolic flux. The problem formulation addresses unbranched pathways with irreversible kinetics. Neither specific reaction kinetics nor fixed pathway length are assumed.In the optimal solution, each enzyme follows a switching profile between zero and maximum concentration, following a temporal sequence that matches the pathway topology. This result provides an analytic justification of the sequential activation previously described in the literature. In contrast with the existent numerical approaches, the activation sequence is proven to be optimal for a generic class of monomolecular kinetics. This class includes, but is not limited to, Mass Action, Michaelis-Menten, Hill, and some Power-law models. This suggests that sequential enzyme expression may be a common feature of metabolic regulation, as it is a robust property of optimal pathway activation.