Parameter estimation and model reduction for retinal laser treatment

Abstract

Laser photocoagulation is one of the most frequently used treatment approaches for retinal diseases such as diabetic retinopathy and macular edema. The use of model-based control, such as Model Predictive Control (MPC), enhances a safe and effective treatment by guaranteeing temperature bounds. In general, real-time requirements for model-based control designs are not met since the temperature distribution in the eye fundus is governed by a heat equation with a nonlinear parameter dependency. This issue is circumvented by representing the model by a lower-dimensional system which well-approximates the original model, including the parametric dependency. We combine a global-basis approach with the discrete empirical interpolation method, tailor its hyperparameters to laser photocoagulation, and show its superiority in comparison to a recently proposed method based on Taylor-series approximation. Its effectiveness is measured in computation time for MPC. We further present a case study to estimate the range of absorption parameters in porcine eyes, and by means of a theoretical and numerical sensitivity analysis we show that the sensitivity of the temperature increase is higher with respect to the absorption coefficient of the retinal pigment epithelium (RPE) than of the choroid's.

State and parameter estimation for retinal laser treatment

“…In our previous work [10], we performed a parameter study with 250 treatment spots and obtained the empirical means ᾱrpe = 0.7636 and ᾱch = 0.0986 and the empirical standard deviations σ rpe = 0.1907 and σ rpe = 0.0281, respectively. Thus, the parameter domain that was used for parametric model reduction and that will be used in the MHE in Subsec.…”

“…We consider two different scenarios: first, a simpler one where the absorption coefficient in the choroid is assumed to be constant and only the absorption coefficient in the RPE is estimated. This is motivated by our case study [10], which showed that the inputoutput sensitivity is higher with respect to the RPE absorption coefficient than to the choroid absorption coefficient. In a second scenario, both absorption coefficients of the RPE and the choroid are estimated independently.…”

“…In our previous work [8], we used a parametric model order reduction technique from [9] and extended it to polynomial parameter dependencies to obtain a parametric reduced order model that allows for estimation of the absorption coefficients after model reduction. In [10], we performed a thorough case study using 250 measurement spots from 25 porcine eyes in order to determine the range for the absorption coefficients located between the photoreceptors of the neural retina and the choroid (cf. Fig.…”

“…2). Moreover, in [10] we examined an alternative model reduction technique applying a global basis approach [11] in combination with a discrete empirical interpolation method [12], which was shown to reduce the model order reduction error compared to the method previously proposed in [8].…”

“…A preliminary version of parts of this paper has appeared in the conference proceedings [17], where a first simple EKF and MHE design was proposed for the case of one parameter (RPE absorption coefficient to be estimated) based on the reduced order model from [8]. Compared to [17], in this work we (i) use the better suited MOR technique proposed in [10], (ii) provide a much more comprehensive simulation case study, (iii) evaluate the performance of both estimators using real measurement data from porcine eyes in detail, and (iv) develop estimation schemes in case that both the RPE and choroid absorption coefficients are estimated independently.…”