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Modeling of 2D diffusion processes based on microscopy data: parameter estimation and practical identifiability analysis.

Hock S, Hasenauer J, Theis FJ - BMC Bioinformatics (2013)

Bottom Line: However, such a model-based analysis is still challenging due to measurement noise and sparse observations, which result in uncertainties of the model parameters.Our novel approach for the estimation of model parameters from image data as well as the proposed identifiability analysis approach is widely applicable to diffusion processes.The profile likelihood based method provides more rigorous uncertainty bounds in contrast to local approximation methods.

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ABSTRACT

Background: Diffusion is a key component of many biological processes such as chemotaxis, developmental differentiation and tissue morphogenesis. Since recently, the spatial gradients caused by diffusion can be assessed in-vitro and in-vivo using microscopy based imaging techniques. The resulting time-series of two dimensional, high-resolutions images in combination with mechanistic models enable the quantitative analysis of the underlying mechanisms. However, such a model-based analysis is still challenging due to measurement noise and sparse observations, which result in uncertainties of the model parameters.

Methods: We introduce a likelihood function for image-based measurements with log-normal distributed noise. Based upon this likelihood function we formulate the maximum likelihood estimation problem, which is solved using PDE-constrained optimization methods. To assess the uncertainty and practical identifiability of the parameters we introduce profile likelihoods for diffusion processes.

Results and conclusion: As proof of concept, we model certain aspects of the guidance of dendritic cells towards lymphatic vessels, an example for haptotaxis. Using a realistic set of artificial measurement data, we estimate the five kinetic parameters of this model and compute profile likelihoods. Our novel approach for the estimation of model parameters from image data as well as the proposed identifiability analysis approach is widely applicable to diffusion processes. The profile likelihood based method provides more rigorous uncertainty bounds in contrast to local approximation methods.

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Parameters estimation for the diffusion model. (A) Shows the source term Q for an early time point. (B) Shows  for the last time point t5. (C) Likelihood ratio calculated for the five dynamic parameters D, α, k1, k−1 and γ are shown in red. The second-order local approximation used for asymptotic confidence intervals is given in blue. The x-axis is given as the logarithm of the parameters, which was also used for the estimation process.
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Figure 2: Parameters estimation for the diffusion model. (A) Shows the source term Q for an early time point. (B) Shows for the last time point t5. (C) Likelihood ratio calculated for the five dynamic parameters D, α, k1, k−1 and γ are shown in red. The second-order local approximation used for asymptotic confidence intervals is given in blue. The x-axis is given as the logarithm of the parameters, which was also used for the estimation process.

Mentions: To calculate the output function we discretized (7)-(9) by finite differences and numerically integrated the discretized state variable c. For the estimation process, data are generated via model simulation (for parameters see Table 1). Forthese simulations we chose a y-shape source term imitating a lymphoid vessel branch (see Figure 2A). We consider images taken at five time points tk ∈ (0, 1], k = 1, . . . , 5 with 50 pixels each , j = 1, . . . , 50. To account for measurement noise, log-normal noise was added (according to (3)) with σ = 10−2 and b = 10−4 (see Figure 2B for one representative image).


Modeling of 2D diffusion processes based on microscopy data: parameter estimation and practical identifiability analysis.

Hock S, Hasenauer J, Theis FJ - BMC Bioinformatics (2013)

Parameters estimation for the diffusion model. (A) Shows the source term Q for an early time point. (B) Shows  for the last time point t5. (C) Likelihood ratio calculated for the five dynamic parameters D, α, k1, k−1 and γ are shown in red. The second-order local approximation used for asymptotic confidence intervals is given in blue. The x-axis is given as the logarithm of the parameters, which was also used for the estimation process.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3750519&req=5

Figure 2: Parameters estimation for the diffusion model. (A) Shows the source term Q for an early time point. (B) Shows for the last time point t5. (C) Likelihood ratio calculated for the five dynamic parameters D, α, k1, k−1 and γ are shown in red. The second-order local approximation used for asymptotic confidence intervals is given in blue. The x-axis is given as the logarithm of the parameters, which was also used for the estimation process.
Mentions: To calculate the output function we discretized (7)-(9) by finite differences and numerically integrated the discretized state variable c. For the estimation process, data are generated via model simulation (for parameters see Table 1). Forthese simulations we chose a y-shape source term imitating a lymphoid vessel branch (see Figure 2A). We consider images taken at five time points tk ∈ (0, 1], k = 1, . . . , 5 with 50 pixels each , j = 1, . . . , 50. To account for measurement noise, log-normal noise was added (according to (3)) with σ = 10−2 and b = 10−4 (see Figure 2B for one representative image).

Bottom Line: However, such a model-based analysis is still challenging due to measurement noise and sparse observations, which result in uncertainties of the model parameters.Our novel approach for the estimation of model parameters from image data as well as the proposed identifiability analysis approach is widely applicable to diffusion processes.The profile likelihood based method provides more rigorous uncertainty bounds in contrast to local approximation methods.

View Article: PubMed Central - HTML - PubMed

ABSTRACT

Background: Diffusion is a key component of many biological processes such as chemotaxis, developmental differentiation and tissue morphogenesis. Since recently, the spatial gradients caused by diffusion can be assessed in-vitro and in-vivo using microscopy based imaging techniques. The resulting time-series of two dimensional, high-resolutions images in combination with mechanistic models enable the quantitative analysis of the underlying mechanisms. However, such a model-based analysis is still challenging due to measurement noise and sparse observations, which result in uncertainties of the model parameters.

Methods: We introduce a likelihood function for image-based measurements with log-normal distributed noise. Based upon this likelihood function we formulate the maximum likelihood estimation problem, which is solved using PDE-constrained optimization methods. To assess the uncertainty and practical identifiability of the parameters we introduce profile likelihoods for diffusion processes.

Results and conclusion: As proof of concept, we model certain aspects of the guidance of dendritic cells towards lymphatic vessels, an example for haptotaxis. Using a realistic set of artificial measurement data, we estimate the five kinetic parameters of this model and compute profile likelihoods. Our novel approach for the estimation of model parameters from image data as well as the proposed identifiability analysis approach is widely applicable to diffusion processes. The profile likelihood based method provides more rigorous uncertainty bounds in contrast to local approximation methods.

Show MeSH