**Purpose**:
Manual, individual adjustment of the laser power in retinal laser therapies is time-consuming, is inaccurate with respect to uniform effects, and can only prevent over- or undertreatment to a limited extent. Automatic closed-loop temperature control allows for similar temperatures at each irradiated spot despite varying absorption. This is of crucial importance for subdamaging hyperthermal treatments with no visible effects and the safety of photocoagulation with short irradiation times. The aim of this work is to perform extensive experiments on porcine eye explants to demonstrate the benefits of automatic control in retinal laser treatments.

**Methods**:
To ensure a safe and reliable temperature rise, we utilize a model predictive controller. For model predictive control, the current state and the spot-dependent absorption coefficients are estimated by an extended Kalman filter (EKF). Therein, optoacoustic measurements are used to determine the temperature rise at the irradiated areas in real time. We use fluorescence vitality stains to measure the lesion size and validate the proposed control strategy.

**Results**:
By comparing the lesion size with temperature values for cell death, we found that the EKF accurately estimates the peak temperature. Furthermore, the proposed closed-loop control scheme works reliably with regard to similar lesion sizes despite varying absorption with a smaller spread in lesion size compared to open-loop control.

**Conclusions**:
Our closed-loop control approach enables a safe subdamaging treatment and lowers the risk for over- and undertreatment for mild coagulations in retinal laser therapies.

**Translational Relevance**:
We demonstrate that modern control strategies have the potential to improve retinal laser treatments for several diseases.

^{1}Depending on the disease, a different intensity of the coagulation is desired: strong coagulations to induce scars in order to prevent retinal detachment

^{2}or mild coagulations to treat diseases such as diabetic retinopathy

^{3}or diabetic macular edema.

^{4}Besides coagulation, subdamaging treatments are under investigation that stimulate retinal metabolism by hyperthemia.

^{5}

^{–}

^{8}Irradiation times are usually 30 to 200 ms, which is below the physician’s reaction time. However, as the light-scattering and retinal absorption vary within the eye,

^{9}manual adjustment of the laser power at each irradiation spot would be necessary for the same effect. This titration procedure is time-consuming, and prevention from overtreatment or undertreatment is only possible to a certain extent.

^{10}For reliable subdamaging treatments, it is inevitable to have additional information (e.g., about the temperature at the irradiated spot). Otherwise, due to a lack of visibility, the ophthalmologist cannot determine whether the treatment was successful. The measurement of an average depth-weighted temperature at the irradiated spot

^{11}allows for temperature feedback and, hence, control strategies toward safer and faster treatment for a variety of diseases.

^{12}

^{–}

^{15}In Baade et al.

^{14}and Schlott et al.,

^{15}different open-loop strategies are presented, whereas Abbas et al.

^{12}and Herzog et al.

^{13}proposed closed-loop control approaches. All methods are based on controlling the hottest temperature at the irradiated spot, that is, the peak temperature in the center of the RPE. To this end, the measured depth-weighted temperature is translated to the peak temperature by means of an offline identified conversion function.

^{16}This function is only valid under some assumptions (e.g., constant laser power or constant relation of the absorption coefficients between different tissue layers), and might lead to errors in the approximation of the peak temperature (if the assumptions are not met). The proportional-integral-derivative (PID) controller design

^{12}

^{,}

^{13}is based on a linear first-order model. The (unknown) parameters (i.e., the static gain and the time constant) were identified offline. Due to patient and spot-dependent tissue parameters, such as the absorption coefficients, a wide parameter range needs to be taken into account to design the controller. This may lead to a lack of performance because the controller gains need to be robust against the whole parameter range. Furthermore, bounds on the peak temperature, which are crucial for safety reasons, cannot be incorporated in the proposed schemes.

^{12}

^{,}

^{13}Nevertheless, first preclinical experiments based on the PID controller with anesthetized rabbits show that closed-loop control can improve retinal laser treatments.

^{17}

^{18}This proposed method includes four major differences to the aforementioned approaches: (1) modeling of the heat diffusion during laser irradiation by a partial differential equation,

^{19}(2) conducting a parametric model order reduction to obtain a (parameter-dependent) low-dimensional model,

^{19}

^{,}

^{20}(3) designing a suitable observer for joint (online) state and parameter estimation,

^{21}

^{,}

^{22}and, finally, (4) designing a model predictive controller that is suitable for retinal laser treatments.

^{18}In doing so, we can overcome the drawbacks of previous controllers in the following way. First, by modeling the heat diffusion, we can consider the peak temperature as one output of the system and the volume temperature as another one without approximating any conversion function. Second, estimating the absorption coefficient allows for an adaptive (and hence in general better performing) controller. Third, limits on the laser power and peak temperature can be implemented in the optimal control problem, which improves the safety of closed-loop control. We found that an extended Kalman filter (EKF) is a suitable estimator for our purposes.

^{22}Therefore, we proposed an EKF-MPC approach in Schaller et al.

^{18}for temperature-controlled retinal laser treatments, where also some first proof-of-concept closed-loop experimental results were reported.

^{1}We show that the proposed technique is especially suited for nondamaging, invisible treatments as the peak temperature is estimated reliably, and consequently, undesired coagulation or insufficient heating can be avoided. Furthermore, a major objective of the proposed closed-loop control is to reduce the variation in lesion size for mild coagulation compared to the case where a constant open-loop laser power is applied. This is achieved by aiming for the same peak temperature independent of the absorption.

*D*= 50 µm and numerical aperture of

*NA*= 0.1. After leaving the slit lamp, the laser beam is imaged onto the tissue sample by means of an ophthalmic contact lens (Mainster Focal Grid; Ocular Instruments, Bellevue, Washington, USA) with a spot diameter

*D*= 200 µm on the sample’s surface. The contact lens was customized with a ring-shaped piezo-ceramic transducer (Medical Laser Center Lübeck, Lübeck, Germany) as a pressure sensor and is attached to a sample cuvette filled with sodium chloride solution of \(0.9\% \). The pressure transient depends on the laser pulse energy and the tissue’s temperature. A depth-weighted average temperature can be calculated as in [Ref. 11, Eq. 9] if the pressure transient and the pulse energy are known. A detailed description on the temperature determination can be found in Brinkmann et al.

^{11}and Mordmüller et al.

^{23}Both pressure transient and laser pulse signals are recorded by a fast data acquisition board (DAQ) (ADQ14; Teledyne SP Devices, Linköping, Sweden) and processed with C/C++ software.

^{24}and depends nonlinearly on the absorption at each spot. The absorption of laser light with a wavelength of 527 nm mainly takes place in the RPE and choroid

^{25}; hence, we consider absorption in these two layers with µ = [µ

_{rpe}, µ

_{ch}]

^{⊤}for the RPE and the choroid. As the absorption is spot-dependent, we introduce an absorption parameter α = [α

_{rpe}, α

_{ch}]

^{⊤}to scale the absorption, that is, µ = αµ

_{0}with µ

_{0}= [120,400, 27,000] m

^{−1}from [Ref. 20, Table 1]. In this way, only the unitless absorption parameter needs to be estimated at each spot. To explain the fundamentals of the model used for control, we denote by

*x*(

*t*, ω) the temperature difference with respect to the ambient temperature at time

*t*and spatial coordinate ω and the laser power at time

*t*by

*u*(

*t*). Then, by means of the abovementioned Beer’s law, we get the heat equation

_{I}(ω) ∈ {0, 1} is the characteristic function of the domain that is heated by the laser, that is, χ

_{I}= 1 inside the irradiated region. We choose the outer cylinder large enough such that the temperature increase at the boundary is negligible during typical treatment duration. Hence, we consider homogeneous Dirichlet boundary conditions and a homogeneous initial condition due to the fact that the eye explants are at room temperature. For the values of the positive parameters,

*C*

_{p},

*k*,

*R*

_{I}, we refer to [Ref. 20, Table 1]. Whereas the left-hand side of Equation (1) models the heat diffusion in the tissue, the right-hand side accounts for the amount of heat absorbed within the interval [0, ω

_{3}] by means of the exponential function. Leveraging rotational symmetry [Ref. 26, Section 2.2], the output corresponding to the volume temperature is defined by

*x*

_{mean}(ω

_{3}) is an average of the (two-dimensional) heat distribution at depth ω

_{3}and

*z*

_{b},

*z*

_{e}denote the beginning and the end of the considered region. Correspondingly, the peak temperature is defined via

*T*

_{peak}lies in the center of the RPE layer, except for a very short initial phase. The coagulation and hence the visible damage (lesion size) is expected to start at the hottest point inside the tissue. Thus, the peak temperature can be considered as the threshold temperature for thermal denaturation. As mentioned before, the peak temperature

*T*

_{peak}(

*t*) cannot be measured. However, we consider

*T*

_{peak}(

*t*) as an additional output to allow for peak temperature control.

^{26}and spatial discretization of the heat diffusion Equation (1), a high-dimensional state-space model is obtained that is too large for real-time control in 1 kHz. Therefore, a low-dimensional surrogate model of order

*n*

_{x}= 6 is obtained by a parametric model order reduction (pMOR) that preserves the parameter-dependency as described in Schaller et al.

^{20}Temporal discretization yields the discrete-time surrogate model in state-space

*y*(

*k*) = (

*T*

_{vol}(

*k*),

*T*

_{peak}(

*k*))

^{⊤}, time instant

*k*, and input

*u*(

*k*) that is the laser power.

^{27}or the Gaussian heat kernel.

^{28}However, to evaluate these formulas, truncation of the Fourier series or an approximation of the convolution integrals is necessary. Thus, in this application, we utilize the sophisticated framework of finite differences methods combined with model order reduction to obtain a discrete-time model that can be efficiently evaluated and allows for optimal control in real time.

^{18}

*x*and the absorption parameter α in a joint fashion, we extend the state-space model Equation (2) by adding α as an additional state with constant dynamics, that is, α(

*k*+ 1) = α(

*k*) to the state-space model. As this yields a nonlinear model, we employ an EKF for estimation. The EKF is a well-known state estimator for nonlinear systems that is based on successive linearization at each time step (see, e.g., Chui and Chen

^{29}). A thorough evaluation of the EKF (in comparison with moving horizon estimation) in the context of retinal laser treatment can be found in Kleyman et al.

^{22}The estimation of both absorption parameters α

_{rpe}and α

_{ch}is difficult for various reasons. First, the input needs to be exciting enough to allow for estimation of both parameters. This is especially difficult to achieve in closed-loop identification. Second, convergence can, in general, not be guaranteed in a joint state and parameter estimation resulting in biased estimates.

^{30}However, for the estimation of only one absorption parameter, a constant input is exciting enough and convergence was reached in all cases in Kleyman et al.

^{22}As the influence of the RPE absorption on the peak and volume temperature is higher than the influence of the choroidal absorption,

^{20}only α

_{rpe}is estimated. The absorption parameter in the choroid layer α

_{ch}is chosen as the (constant) mean value that was found in [Ref. 20, Table 2].

^{31}

^{–}

^{33}MPC predicts the future behavior of the system over a finite horizon \(N \in \mathbb {N}\) to obtain the optimal input trajectory \({\bf u}=\left(u_{k|k},\ldots , u_{k+N-2|k}\right)\in \mathbb {R}^{N-1}\). Here,

*u*

_{i|k}denotes the optimal input for time

*i*(with

*k*⩽

*i*⩽

*k*+

*N*− 2) predicted at current time

*k*. The optimal control problem (OCP)

*k*and the first (optimal) input

*u*

_{k|k}is applied to the system until the next time step. Then, the horizon is shifted and the OCP is solved again to obtain

*u*

_{k + 1|k + 1}. This procedure is repeated until the end of the treatment. To avoid undesired side effects caused by (peak) temperature overshoots during control, the maximum allowed peak temperature

*T*

_{max}is constrained to 2°C above the aim temperature

*T*

_{aim}, that is,

*T*

_{max}=

*T*

_{aim}+ 2°C. As there are physical limits on the laser power, hard constraints on the maximum laser power

*P*

_{L, max}are included. At each time step, the initial condition for the state predictions,

*x*

_{k|k}, and the parameter vector are set to the current EKF estimates \(\tilde{x}\) and \(\tilde{\alpha }\), respectively. A short horizon

*N*= 5 is used so that the OCP can be solved in less then 1 ms. The optimization problem Equation (3) is solved by means of a C++-implementation with the solver OSQP.

^{34}Further details and a in-depth comparison concerning different cost functions, horizon lengths, and sampling rates can be found in Schaller et al.

^{18}

*x*in Equation (2) do not have a physical interpretation, we concentrate on the estimated peak temperature (i.e., the second output of Equation (2)) as a measure to assess the functionality of the observer. Moreover, the peak temperature has a major influence on the lesion size that is used to further verify the EKF.

*y*

_{meas}and estimated peak temperature

*T*

_{peak}are shown for one spot where a laser power of 15 mW for 100 ms was applied. We observe that α converges after approx. 20 ms to a constant value, as shown in blue in Figure 4b as a representative example of the convergence speed. However, for other spots, the parameter estimation shows a positive drift (after some settling time), as depicted in Figure 5 in purple or a negative drift as depicted in green. Note that these are different spots, and hence it is not expected that one obtains convergence to the same value.

*T*

_{peak, max}at the end of the treatment and the corresponding measured lesion diameter

*d*

_{lesion}caused by laser powers of 10 mW (blue), 15 mW (orange), 20 mW (yellow), 25 mW (purple), and 30 mW (green). For peak temperatures below 48°C, there is no visible lesion except for some outliers. For peak temperatures above 48°C, the lesion size increases with increasing peak temperature. For a constant laser power

*P*

_{L}= {25, 30} mW over 100 ms, the mean of the lesion diameter \(\bar{d}_\mathrm{lesion}\), the standard deviation σ

_{d}, and the coefficients of variation \(c = \frac{\sigma _\mathrm{d}}{\bar{d}_\mathrm{lesion}}\) are shown in Table 1. The values for

*P*

_{L}⩽ 20 mW are not presented as there was no visible lesion for most of the spots (cf. Fig. 6).

^{35}However, we consider a 10-times smaller value in the choroid layer compared to Hammer et al.,

^{35}according to the value that we identified in Schaller et al.

^{20}A recently published article

^{36}that identified the absorption for the RPE and choroid together (as one part of the whole eye) suggests that the absorption is in the same order of magnitude as our (offline identified) values.

^{20}The parameters that are estimated online via the EKF are in a similar range as in Regal et al.

^{36}and Schaller et al.,

^{20}which indicates that the online parameter identification works reliably. This was also shown in Kleyman et al.

^{22}for a slightly different setup.

^{1}where the threshold temperature in laser-induced damage was investigated. The authors found out that cell death starts at 53°C ±2°C for an irradiation duration between 100 ms and 1 s. These values are close to the estimated peak temperature at which visible damage starts in Figure 6, which leads to the conclusion that the employed EKF estimates reasonable peak temperatures and absorption parameters, which is of crucial importance for a reliable close-loop control. A possible reason why in Figure 6, in some (few) cases, lesions are already visible at temperatures below 50°C is the fact that the same (constant) choroidal absorption coefficient is used for all spots, which can lead to a small inaccuracy in the peak temperature estimation.

^{13}

^{,}

^{35}In vivo experiments with chinchilla gray rabbits

^{37}have shown a larger standard deviation for open-loop measurements as found here. This can be explained by additional influencing parameters like the light scattering within the eye, as in the case of explants consisting of RPE, choroid, and sclera only. An open-loop control cannot react to varying parameters as there is no feedback of the current system’s state by means of measurements, and hence, no knowledge about spot-dependent parameters can be incorporated. Spot-dependent tissue parameters lead to different peak temperatures and, thus, to different lesion sizes. Therefore, the lesion size can only be influenced very roughly by open-loop control. However, the lesion size also varies for similar peak temperatures. One reason could be that convergence is not always reached, as shown in Figure 5. As we would expect convergence much faster, we will discuss the influence of a drifting parameter in the following.

*C*

_{p},

*k*, ρ) and/or different choroidal absorption parameters α

_{ch}. When using different tissue parameters, we observed some offsets for the identified parameters, but there was no drift (except for a very slight one in case of a different thermal conductivity

*k*). On the other hand, a different choroidal absorption parameter α

_{ch}in the reduced model (compared to the “real” one, i.e., the one used in the full-order model) indeed can result in a parameter drift and very slow convergence after several hundred milliseconds to a false parameter value. In particular, we considered in the full-order plant model a lower and a higher choroidal absorption parameter α

_{ch, low}and α

_{ch, high}, respectively. We choose α

_{ch, low}and α

_{ch, high}such that 95% of the identified absorption parameters α

_{ch}of the case study in Schaller et al.

^{20}are within these values, that is, α

_{ch, low}= 0.04 and α

_{ch, high}= 0.15. If α

_{ch}is higher, the tissue will heat up more intensely, which has also an influence on the volume temperature. If the volume temperature is higher than expected (by the EKF model), the estimated absorption parameter α

_{rpe}will compensate for this. Figure 7 shows the estimated absorption parameters α

_{rpe}for different choices of α

_{ch}in the full-order plant model. In the EKF, α

_{ch}is set constant to the mean value α

_{ch, mean}from Schaller et al.,

^{20}which means that there is a mismatch between the plant and the EKF model. If α

_{ch}is higher than assumed by the EKF, there is a negative drift of α

_{rpe}as depicted in blue and for the opposite case (α

_{ch}is smaller than assumed) in purple. Based on these simulations, we conclude that the drift is mainly caused by a different absorption in the choroid. This conclusion is strengthened by the fact that the drift is spot-dependent as well as the absorption in the choroid. Independent of the cause, the drift has an influence on the estimated peak temperature and therefore on the closed-loop performance, as we will discuss in “Discussion of Closed-Loop Experiments.”

*T*

_{peak}and α

_{rpe}properly. The evaluation of the open-loop data showed that our approach enables the estimation of both in a reliable fashion. However, the open-loop data also showed that there is room for improvement by means of additional estimation of α

_{ch}. Future work could, for example, focus on experiment design techniques

^{38}

^{,}

^{39}to design suitable (exciting enough) input signals (for open-loop experiments) and/or on using a laser with two different wavelengths to be able to reliably estimate both absorption parameters. In the following, we show and discuss the closed-loop results. We will also determine errors arising from the simplification of constant choroidal absorption in closed loop.

*T*

_{aim}is held until the end of the measurement.

*T*

_{aim}is shown in Table 2. We tested the closed-loop performance using two different laser power limits

*P*

_{L, max}= {30, 40} mW in the control design (compare Equation (3)). The average time

*t*

_{{30, 40}}until the (estimated) peak temperature reaches the aim temperature for the first time increases with increasing

*T*

_{aim}. The rise times are different for the two laser power limits with longer rise times for

*P*

_{L, max}= 30 mW. The amount of spots where

*T*

_{aim}is not reached increases for higher

*T*

_{aim}. In the case of

*P*

_{L, max}= 40 mW, the desired temperature is reached at all spots. Due to measurement noise and input noise (laser energy fluctuations), we count estimated peak temperatures that are at least 0.3°C above

*T*

_{aim}as an overshoot and define the overshoot as the difference between the estimated (maximum) peak temperature during the irradiation interval and the aim temperature (i.e., the desired peak temperature)

*T*

_{os}=

*T*

_{peak}−

*T*

_{aim}. The duration of the overshoot

*t*

_{os}is the time until the peak temperature is (again) inside

*T*

_{aim}± 0.3°C. We found that the duration of the overshoot is similar for all

*T*

_{aim}. The intensity of the overshoot decreases with increasing

*T*

_{aim}, whereas an overshoot occurs less frequently for higher

*T*

_{aim}. We did not observe any constraint violation for the maximum allowed peak temperature (i.e.,

*T*

_{peak}>

*T*

_{max}=

*T*

_{aim}+ 2°C) or for the maximum laser power

*P*

_{L, max}.

*T*

_{aim}). The higher

*T*

_{aim}, the higher is the average power for similar absorption coefficients α.

*d*

_{lesion}for the corresponding

*T*

_{aim}for all measurements where the aim temperature was reached is shown in Figure 10. The means over all lesions \(\bar{d}_\mathrm{lesion}\) for a certain

*T*

_{aim}are depicted as a black line. The standard deviation for each

*T*

_{aim}is depicted as an error bar. The lesion diameter \(\bar{d}_\mathrm{lesion}\) increases with increasing

*T*

_{aim}. However, there are some outliers. A statistical evaluation of the lesion sizes is given in Table 3. With increasing

*T*

_{aim}, the coefficient of variation \(c = \sigma _{d}/\bar{d}_\mathrm{lesion}\) of the lesion diameter decreases.

*T*

_{aim}, the longer it takes (in average) to reach

*T*

_{aim}. This is to be expected due to active input (i.e., laser power) constraints that limit the speed of the temperature increase. Due to varying absorption, it is not possible to reach

*T*

_{aim}in all cases if the laser power is limited to 30 mW. From

*T*

_{aim}= 55°C on, for more than half of the measurements, the peak temperature is not reached. For

*P*

_{L, max}= 40 mW, the aim temperature is reached in all cases. For future experiments, the laser power limit can either be increased or the duration of the treatment can be extended. As a longer treatment is not in favor from an application point of view,

*P*

_{L, max}should be higher to guarantee that the aim temperature can be reached at each spot independent of the absorption. The average deviation σ

_{aim}of the rise time

*t*

_{{30, 40}}over all spots (with the same

*T*

_{aim}) increases also due to different absorption that determines

*t*

_{{30, 40}}, that is, the time to reach

*T*

_{aim}(at bounded laser power). To lower the influence of the absorption on the time to reach

*T*

_{aim}, a tracking control with a desired temperature rise profile

*T*

_{aim}(

*t*) could be used instead of the current setpoint control (i.e.,

*T*

_{aim}(

*t*) = const∀

*t*⩾ 0). This might also have a positive effect on the deviation of the lesion size if the temperature rise is similar in all cases.

*T*

_{aim}) with increasing frequency for smaller

*T*

_{aim}. The duration of the overshoot is on average three time steps (i.e., 3 ms) independent of the aim temperature. The overshoot results from a time delay of two time steps that is present in the current setup and not from the control algorithm itself, that is, the solution of the optimal control problem Equation (3). The overshoot is in all cases below 2°C (no constraint violation), and the duration is short in comparison to the treatment time. Therefore, we do not expect a significant effect on the lesion size. Nevertheless, delay compensation MPC schemes

^{40}

^{,}

^{41}could prevent an overshoot.

*P*

_{L}= 30 mW (compare the second row of Table 1). However, when comparing the standard deviation (and hence the coefficient of variation) in these cases, one finds that the values are smaller for closed-loop compared to open-loop control. This means that as desired, the proposed closed-loop control scheme decreases the spread in the lesion size, that is, the lesion size depends less on the absorption than in open loop, which is of crucial importance for a reliable therapy, and underlines ones more a benefit of closed-loop control. We observe that there is still a significant spread in lesion size for a given aim temperature

*T*

_{aim}. In the following, we discuss several possible explanations for this observation and ways to reduce the spread, if desired from a therapeutic perspective. Before that, we would like to point out that controlling the temperature in biological tissue is demanding due to its strong individual properties and that a certain spread in the lesion size will be inevitable.

_{ch, high}, α

_{ch, mean}, and α

_{ch, low}(cf. “Discussion of Open-Loop Experiments” and Fig. 5). The (reduced) EKF model is the same in all three cases, with α

_{ch}= α

_{ch, mean}. The EKF estimates the peak temperature to be the same as

*T*

_{aim}(after

*T*

_{aim}was reached for the first time). Due to the model mismatch caused by model reduction and falsely assumed choroidal absorption, the peak temperature of the (simulated) plant

*T*

_{peak, plant}is different. Figure 11 shows the relative error between

*T*

_{aim}and

*T*

_{peak, plant}over time, that is, \(e_\mathrm{rel}(t) = \frac{|T_\mathrm{aim}-T_\mathrm{peak, plant}(t)|}{T_\mathrm{aim}}\) for

*T*

_{aim}= 55°C. The error is the largest at the beginning, since the sample is not yet heated. For a correctly assumed choroidal absorption, the error is only caused by a model reduction error and a state estimation error. Hence, it is the smallest with

*e*

_{rel}= 0.0008 after

*t*

_{e}= 100 ms, that is,

*e*

_{mean}(

*t*

_{e}) =

*T*

_{aim}−

*T*

_{peak, plant}(

*t*

_{e}) = −0.02°C. For a higher choroidal absorption, α

_{rpe}is overestimated and

*T*

_{aim}is not reached at all. Because of the parameter drift, the estimation error of the peak temperature increases also over time, as shown in Figure 11. This leads to an increasing offset (after the initial temperature rise) between the estimated and the real (simulated) peak temperature with

*e*

_{rel}= 0.03 after 100 ms, that is,

*e*

_{high}(

*t*

_{e}) = 0.9°C. In the case of a smaller choroidal absorption, the peak temperature is underestimated, which leads to a peak temperature above

*T*

_{aim}. After 100 ms, the absolute error is slightly above the maximum allowed temperature

*T*

_{max}in Equation (3), that is,

*T*

_{peak, plant}>

*T*

_{max}with

*e*

_{low}(

*t*

_{e}) = −2.3°C. Keep in mind that these (rather large) errors are the worst-case scenarios in terms of changing choroidal absorption, meaning in 95% of the case study in Schaller et al.,

^{20}the error is smaller (for the same aim temperature). Nevertheless, this could be one reason why the deviation in the lesion diameter in Table 3 for the same

*T*

_{aim}is higher than we expected by closed-loop control.

*tophat*beam with a constant and uniform intensity but rather as a Gaussian beam. Further intensity modulations due to coherent mode matching in the multimode fiber are not considered, which can lead to strong intensity peaks at the irradiated spot. The influence of the beam profile might be stronger for temperatures close to the coagulation threshold, where the profile is decisive for (visible) damage.

^{11}During coagulation, the expansion properties of the tissue (especially the Grüneisen parameter) change. In addition, light scattering takes place in coagulated tissue, which has an influence on the light distribution and thus absorption profile at the irradiated site. Both changing tissue parameters and the onset of light scattering override the temperature calibration and lead to a falsified volume temperature.

^{42}Although the influence on the volume temperature is small for mild coagulations,

^{15}it can still have an effect on the peak temperature estimation. Therefore, our method is especially suitable for subdamaging hyperthermia treatments and mild coagulations to a certain extent. In the case of strong coagulation, no reliable determination of the temperature can be guaranteed, and therefore no reliable feedback for the estimator/controller can be expected. However, the associated sudden change in the pressure signal at the onset of coagulation could also be used to stop the treatment at an early stage to prevent a larger coagulation or even tissue bleeding and ruptures.

^{43}and help to verify/support our conjectures made in this section. Also, as already discussed in “Discussion of Open-Loop Experiments”, a second wavelength could be used to gain more information about a depth-dependent absorption and to facilitate the estimation of the choroidal absorption. Closed-loop estimation for more than one parameter can be implemented by means of sensitivity updates.

^{44}

^{45}Therefore, we expect similar results with human eyes. However, by irradiating RPE/choroid/sclera explants instead of living eyes, some limitations need to be considered. First of all, heat convection due to blood flow is not included in the modeling. This can be done by an additional heat sink; see Sandeau et al.,

^{46}where a perfusion term is added to the heat diffusion equation. However, it was found that heat loss by blood perfusion can be neglected for such a short irradiation time of 100 ms.

^{47}If the tissue is irradiated for several seconds, perfusion cannot be neglected, as shown for living and dead rabbits in Hermann et al.

^{48}Hence, the model needs to be adapted when it comes to such long irradiation times. Independent of in vivo or ex vivo irradiation, the absence of cornea, lens, and vitreous body simplifies the parameter identification. In particular, scattering in the lens affects the amount of light that reaches the eye fundus. As scattering varies individually and depends, among others things, on the age of the patient,

^{49}it may be necessary to identify an additional transmission factor that scales the laser power reaching the eye fundus. Since this parameter can be assumed to remain constant throughout the eye, it can be identified initially and then kept constant during treatment at different spots. Concerning the explant model in this work, we have assumed that the density, thermal conductivity, and heat capacity are equal to those of water as it is the main component of the tissue. However, the model could be refined by including layer-dependent parameters in following versions.

^{50}during temperature-controlled patient treatment by means of a PID controller. This change can be detected by sudden shifts in the parameter estimation. Thus, the EKF can be utilized as a fault detector. The treatment could then simply be stopped or paused until the remaining heat has been dissipated and the spot will then be reirradiated.

**V. Kleyman**, None;

**S. Eggert**, None;

**C. Schmidt**, None;

**M. Schaller**, None;

**K. Worthmann**, None;

**R. Brinkmann**, Medical Laser Center Lübeck (P);

**M.A. Müller**, None

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