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ABSTRACTWe expand the standard FRAP model introduced by Axelrod et al. in 1976. Our goal is to capture some common artifacts observed in the fluorescence measurements obtained with a confocal laser scanning microscope (CLSM) in biofilms: 1) linear drift, 2) exponential decrease (due to bleaching during the measurements), 3) stochastic Gaussian noise, and 4) uncertainty in the exact time point of the onset of fluorescence recovery. In order to fit the resulting stochastic model to data from FRAP measurements and to estimate all unknown model parameters, we apply a suitably adapted Metropolis-Hastings algorithm. In this way, a more accurate estimation of the diffusion coefficient of the fluorophore is achieved. The method was tested on data obtained from FRAP measurements on a cultivated biofilm.STATEMENT OF SIGNIFICANCEDiffusion and mass transport in biofilms is presumed to play an important role in the resistance against antimicrobial agents and the secretion thereof. FRAP measurements give insight into these transport processes. In this article, the authors expand the standard FRAP model by a blackbox part which addresses some artifacts commonly observed in the measurements. This is done in order to improve the estimations of diffusion coefficients considerably, aiming at a more comprehensive description of diffusion processes inside the biofilm. We expect that the methods are transferable to FRAP measurements on materials other than biofilms.

The stochastic H ∞ -norm is defined as the L 2induced norm of the input-output operator of a stochastic linear system. Like the deterministic H ∞ -norm it is characterised by a version of the bounded real lemma, but without a frequency domain description or a Hamiltonian condition. Therefore, we base its computation on a parametrised algebraic Riccati-type matrix equation and a Newton iteration. For large dimensions, our algorithm outperforms LMI-methods.

We expand the standard FRAP model introduced by Axelrod et al. in 1976. Our goal is to capture some common artifacts observed in the fluorescence measurements obtained with a confocal laser scanning microscope (CLSM) in biofilms: 1) linear drift, 2) exponential decrease (due to bleaching during the measurements), 3) stochastic Gaussian noise, and 4) uncertainty in the exact time point of the onset of fluorescence recovery. In order to fit the resulting stochastic model to data from FRAP measurements and to estimate all unknown model parameters, we apply a suitably adapted Metropolis-Hastings algorithm. In this way, a more accurate estimation of the diffusion coefficient of the fluorophore is achieved. The method was tested on data obtained from FRAP measurements on a cultivated biofilm. STATEMENT OF SIGNIFICANCEDiffusion and mass transport in biofilms is presumed to play an important role in the resistance against antimicrobial agents and the secretion thereof. FRAP measurements give insight into these transport processes. In this article, the authors expand the standard FRAP model by a blackbox part which addresses some artifacts commonly observed in the measurements. This is done in order to improve the estimations of diffusion coefficients considerably, aiming at a more comprehensive description of diffusion processes inside the biofilm. We expect that the methods are transferable to FRAP measurements on materials other than biofilms.

BackgroundWhen mathematical modelling is applied to many different application areas, a common task is the estimation of states and parameters based on measurements. With this kind of inference making, uncertainties in the time when the measurements have been taken are often neglected, but especially in applications taken from the life sciences, this kind of errors can considerably influence the estimation results. As an example in the context of personalized medicine, the model-based assessment of the effectiveness of drugs is becoming to play an important role. Systems biology may help here by providing good pharmacokinetic and pharmacodynamic (PK/PD) models. Inference on these systems based on data gained from clinical studies with several patient groups becomes a major challenge. Particle filters are a promising approach to tackle these difficulties but are by itself not ready to handle uncertainties in measurement times.ResultsIn this article, we describe a variant of the standard particle filter (PF) algorithm which allows state and parameter estimation with the inclusion of measurement time uncertainties (MTU). The modified particle filter, which we call MTU-PF, also allows the application of an adaptive stepsize choice in the time-continuous case to avoid degeneracy problems. The modification is based on the model assumption of uncertain measurement times. While the assumption of randomness in the measurements themselves is common, the corresponding measurement times are generally taken as deterministic and exactly known. Especially in cases where the data are gained from measurements on blood or tissue samples, a relatively high uncertainty in the true measurement time seems to be a natural assumption. Our method is appropriate in cases where relatively few data are used from a relatively large number of groups or individuals, which introduce mixed effects in the model. This is a typical setting of clinical studies. We demonstrate the method on a small artificial example and apply it to a mixed effects model of plasma-leucine kinetics with data from a clinical study which included 34 patients.ConclusionsComparisons of our MTU-PF with the standard PF and with an alternative Maximum Likelihood estimation method on the small artificial example clearly show that the MTU-PF obtains better estimations. Considering the application to the data from the clinical study, the MTU-PF shows a similar performance with respect to the quality of estimated parameters compared with the standard particle filter, but besides that, the MTU algorithm shows to be less prone to degeneration than the standard particle filter.