Calculation of the uncertainty in complication probability for various dose-response models, applied to the parotid gland.

PURPOSE: Usually, models that predict normal tissue complication probability (NTCP) are fitted to clinical data with the maximum likelihood (ML) method. This method inevitably causes a loss of information contained in the data. In this study, an alternative method is investigated that calculates the parameter probability distribution (PD), and, thus, conserves all information. The PD method also allows the calculation of the uncertainty in the NTCP, which is an (often-neglected) prerequisite for the intercomparison of both treatment plans and NTCP models. The PD and ML methods are applied to parotid gland data, and the results are compared. METHODS AND MATERIALS: The drop in salivary flow due to radiotherapy was measured in 25 parotid glands of 15 patients. Together with the parotid gland dose-volume histograms (DVH), this enabled the calculation of the parameter PDs for three different NTCP models (Lyman, relative seriality, and critical volume). From these PDs, the NTCP and its uncertainty could be calculated for arbitrary parotid gland DVHs. ML parameters and resulting NTCP values were calculated also. RESULTS: All models fitted equally well. The parameter PDs turned out to have nonnormal shapes and long tails. The NTCP predictions of the ML and PD method usually differed considerably, depending on the NTCP model and the nature of irradiation. NTCP curves and ML parameters suggested a highly parallel organization of the parotid gland. CONCLUSIONS: Considering the substantial differences between the NTCP predictions of the ML and PD method, the use of the PD method is preferred, because this is the only method that takes all information contained in the clinical data into account. Furthermore, PD method gives a true measure of the uncertainty in the NTCP.

Using miniature sensor coils for simultaneous measurement of orientation and position of small, fast-moving animals.

A system is described that measures, with a sampling frequency of 1 kHz, the orientation and position of a blowfly (Calliphora vicina) flying in a volume of 0.4 x 0.4 x 0.4 m3. Orientation is measured with a typical accuracy of 0.5 degrees, and position with a typical accuracy of 1 mm. This is accomplished by producing a time-varying magnetic field with three orthogonal pairs of field coils, driven sinusoidally at frequencies of 50, 68, and 86 kHz, respectively. Each pair induces a voltage at the corresponding frequency in each of three miniature orthogonal sensor coils mounted on the animal. The sensor coils are connected via thin (12-microm) wires to a set of nine lock-in amplifiers, each locking to one of the three field frequencies. Two of the pairs of field coils produce approximately homogeneous magnetic fields, which are necessary for reconstructing the orientation of the animal. The third pair produces a gradient field, which is necessary for reconstructing the position of the animal. Both sensor coils and leads are light enough (0.8-1.6 mg for three sensor coils of 40-80 windings, and 6.7 mg/m for the leads, causing a maximal load of approximately 5.7 mg) not to hinder normal flight of the animal (typical weight 80 mg). In general, the system can be used for high-speed recordings of head, eye or limb movements, where a wire connection is possible, but the mechanical load on the moving parts needs to be very small.

Blowfly flight and optic flow. I. Thorax kinematics and flight dynamics

The motion of the thorax of the blowfly Calliphora vicina was measured during cruising flight inside a cage measuring 40 cmx40 cmx40 cm. Sensor coils mounted on the thorax picked up externally generated magnetic fields and yielded measurements of the position and orientation of the thorax with a resolution of 1 ms, 0.3 degrees and 1 mm. Flight velocities inside the cage were up to 1.2 m s-1, and accelerations were up to 1 g ( approximately 10 m s-2) vertically and 2 g horizontally. During flight, blowflies performed a series of short (approximately 20-30 ms) saccade-like turns at a rate of approximately 10 s-1. The saccades consisted of a succession of rotations around all axes, occurring in a fixed order. First, a roll was started. Second, the rolled thorax pitched (pulling the nose up) and yawed, resulting in a turn relative to the outside world. Finally, the thorax rolled back to a level position. Saccades had yaw amplitudes of up to 90 degrees, but 90 % were smaller than 50 degrees. Maximum angular velocities were 2000 degrees s-1, and maximum accelerations were 10(5 ) degrees s-2. The latter correspond to torques consistent with the maximal force (2x10(-3 )N) that can be generated by the flight motor as inferred from the maximal linear acceleration. Furthermore, the sequence of energy investment in consecutive rotations around different axes appears to be optimized during a saccade.

Blowfly flight and optic flow. II. Head movements during flight

The position and orientation of the thorax and head of flying blowflies (Calliphora vicina) were measured using small sensor coils mounted on the thorax and head. During flight, roll movements of the thorax are compensated by counter rolls of the head relative to the thorax. The yaw turns of the thorax (thorax saccades) are accompanied by faster saccades of the head, starting later and finishing earlier than the thorax saccades. Blowfly flight can be divided into two sets of episodes: 'during saccades', when high angular velocities of up to a few thousand degrees per second are reached by both the thorax and head, and 'between saccades', when the orientation of the thorax and, in particular, the head is well stabilized. Between saccades, the angular velocities of the head are approximately half those of the thorax and lie mostly in the range 0-100 degrees s-1 for any rotation (yaw, pitch and roll). These velocities are low enough to limit the visual blur attributable to rotation. It is argued that the split into periods during which either rotational optic flow ('during saccades') or translatory optic flow ('between saccades') dominates is helpful for processing optic flow when signals and neurons are noisy.

Estimation of parameters of dose-volume models and their confidence limits.

Predictions of the normal-tissue complication probability (NTCP) for the ranking of treatment plans are based on fits of dose-volume models to clinical and/or experimental data. In the literature several different fit methods are used. In this work frequently used methods and techniques to fit NTCP models to dose response data for establishing dose-volume effects, are discussed. The techniques are tested for their usability with dose-volume data and NTCP models. Different methods to estimate the confidence intervals of the model parameters are part of this study. From a critical-volume (CV) model with biologically realistic parameters a primary dataset was generated, serving as the reference for this study and describable by the NTCP model. The CV model was fitted to this dataset. From the resulting parameters and the CV model, 1000 secondary datasets were generated by Monte Carlo simulation. All secondary datasets were fitted to obtain 1000 parameter sets of the CV model. Thus the 'real' spread in fit results due to statistical spreading in the data is obtained and has been compared with estimates of the confidence intervals obtained by different methods applied to the primary dataset. The confidence limits of the parameters of one dataset were estimated using the methods, employing the covariance matrix, the jackknife method and directly from the likelihood landscape. These results were compared with the spread of the parameters, obtained from the secondary parameter sets. For the estimation of confidence intervals on NTCP predictions, three methods were tested. Firstly, propagation of errors using the covariance matrix was used. Secondly, the meaning of the width of a bundle of curves that resulted from parameters that were within the one standard deviation region in the likelihood space was investigated. Thirdly, many parameter sets and their likelihood were used to create a likelihood-weighted probability distribution of the NTCP. It is concluded that for the type of dose response data used here, only a full likelihood analysis will produce reliable results. The often-used approximations, such as the usage of the covariance matrix, produce inconsistent confidence limits on both the parameter sets and the resulting NTCP values.