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In this paper, we propose the conditional optimal spacing defined as the optimal spacing after specifying a predetermined order statistic. If we specify a censoring time, then the optimal inspection times for grouped inspection can be determined from this conditional optimal spacing. We take an example of exponential distribution, and provide a simple method of finding the conditional optimal spacing.
A simple example of intravenous theophylline therapy is used to present and compare various drug administration policies based on stochastic control theory. The simplest approach (Heuristic-Certainty-Equivalence (HCE) control) assumes that the model parameters are known. Prior uncertainty on these parameters can be taken into account by using average optimal (AO) control. The available knowledge about the system can be improved by measuring the drug concentration some time after the beginning of the treatment. This corresponds to the notion of feedback and leads to the HCE feedback (HCEF) and AO feedback (AOF) policies. A further step towards optimality consists in choosing the optimal measurement time given that the final purpose is the control of the system and not the estimation of its parameters. Finally, closed-loop optimal (CLO) control optimally chooses both the dosage regimen and measurement time.
This paper deals with accurate estimation of the degrees of freedom (DOF) of a body segment, starting from the trajectories of clusters of markers acquired with a suitable measurement system. The most commonly employed estimation procedures involve two sequential steps, a trajectory smoothing algorithm and a DOF reconstruction routine. Three optimized smoothing and reconstruction schemes are described, analyzed and tested and their performances are compared with each other and with those of a more traditional technique which contains no optimization criteria. All three schemes include an iterative, weighted-least-squares DOF reconstruction routine and a self-tuning, zero-phase-shift, 4th-order Butterworth filter. Both routines are extensively described and validated on the basis of numerically-simulated marker trajectories. Test results, analyzed on a statistical basis, show that the use of an optimization routine provides a visible improvement in DOF reconstruction. This performance has also been confirmed using stereophotogrammetric data collected on a subject wearing an external fracture fixation device which provides reference values for the bone DOF. Angular DOF estimated applying the optimized method to skin technical clusters are much closer to the reference values than the non-optimized values. Smoothing of data further improves the reconstruction accuracy while a far less crucial role is played by the order in which smoothing and reconstruction routines are applied.
This paper focuses on the solution of two problems related to cycling. One is to determine the velocity as a function of distance which minimizes the cyclist's energy expenditure in covering a given distance in a set time. The other is to determine the velocity as a function of the distance which minimizes time for fixed energy expenditure. To solve these problems, an equation of motion for the cyclist riding over arbitrary terrain is written using Newton's second law. This equation is used to evaluate either energy expenditure or time, and the minimization problems are solved using an optimal control formulation in conjunction with the method of Miele [Optimization Techniques with Applications to Aerospace Systems, pp. 69-98 (1962) Academic Press, New York]. Solutions to both optimal control problems are the same. The solutions are illustrated through two examples. In one example where the relative wind velocity is zero, the optimal cruising velocity is constant regardless of terrain. In the second, where the relative wind velocity fluctuates, the optimal cruising velocity varies.
The goals of the work reported by this article are two-fold. The first is to develop a dynamic optimization framework for analysis of equipment setup problems in endurance cycling. The second is to illustrate the application of the approach by determining an optimal chainring shape. To achieve these goals, a mathematical model of the pedaling motion for given trajectories of the net joint moments and the rate of change of the chainring radius was derived, and chainring optimization was posed as an optimal control problem. The cost functional produced a chainring shape that reduced the cost of endurance cycling at 250 W and 90 rpm, apparently by taking advantage of mechanical interactions that arise as a natural consequence of the movement. However, the predicted joint moments required larger peak values during phases of significantly increased joint velocity. Thus, the 'optimal' performance predicted by the cost functional appears opposed to expectations based on muscle mechanics and illustrates the need for further analysis of endurance cycling with a physiologically based cost functional.
Our previous research and clinical tests of a self-contained powered above-knee prosthesis (AKP) showed that a knee joint with one degree of freedom (DOF) increases the energy cost of walking with respect to able-bodied subjects. Better symmetry of the gait can improve performance, so we suggest here the integration of a second powered DOF into the knee joint mechanism to control the internal-external rotation of the shank-foot complex. The control for the AKP with two DOFs is based on a method of optimal tracking. The data used for analysis were collected in able-bodied subjects braced with an ankle splint to experimentally duplicate a gait like that of amputees using a two-DOF prosthesis. The simulation showed the following: (1) the technique of optimal programming can be used for simulation of the artificial leg during locomotion; (2) the optimal tracking method is an efficient tool for selection of actuators for the above-knee prosthesis, ensuring that the tracking remains within limits. Limitation of joint torque is desirable in order to reduce the size of the motor, but beyond a certain point limiting maximal torques lead to tracking errors that are associated with higher energy costs and hence the need for a larger power source. The errors are also associated with higher forces at the interface between the socket and the prosthesis. The optimal tracking method allows the optimization of tracking with constraints on the size of the motor used and its energy cost.
This paper presents an exact analytical solution to the problem of locating the junction point between three branches so that the sum of the total costs of the branches is minimized. When the cost per unit length of each branch is known the angles between each pair of branches can be deduced following reasoning first introduced to biology by Murray. Assuming the outer ends of each branch are fixed, the location of the junction and the length of each branch are then deduced using plane geometry and trigonometry. The model has applications in determining the optimal cost of a branch or branches at a junction. Comparing the optimal to the actual cost of a junction is a new way to compare cost models for goodness of fit to actual junction geometry. It is an unambiguous measure and is superior to comparing observed and optimal angles between each daughter and the parent branch. We present data for 199 junctions in the pulmonary arteries of two human lungs. For the branches at each junction we calculated the best fitting value of x from the relationship that flow alpha (radius)x. We found that the value of x determined whether a junction was best fitted by a surface, volume, drag or power minimization model. While economy of explanation casts doubt that four models operate simultaneously, we found that optimality may still operate, since the angle to the major daughter is less than the angle to the minor daughter. Perhaps optimality combined with a space filling branching pattern governs the branching geometry of the pulmonary artery.
The D-optimal design, a minimal sample design that minimizes the volume of the joint confidence region for the parameters, was used to evaluate binding parameters in a saturation curve with a view to reducing the number of experimental points without loosing accuracy in binding parameter estimates. Binding saturation experiments were performed in rat brain crude membrane preparations with the opioid mu-selective ligand [3H]-[D-Ala2,MePhe4,Gly-ol5]enkephalin (DAGO), using a sequential procedure. The first experiment consisted of a wide-range saturation curve, which confirmed that [3H]-DAGO binds only one class of specific sites and non-specific sites, and gave information on the experimental range and a first estimate of binding affinity (Ka), capacity (Bmax) and non-specific constant (k). On this basis the D-optimal design was computed and sequential experiments were performed each covering a wide-range traditional saturation curve, the D-optimal design and a splitting of the D-optimal design with the addition of 2 points (+/- 15% of the central point). No appreciable differences were obtained with these designs in parameter estimates and their accuracy. Thus sequential experiments based on D-optimal design seem a valid method for accurate determination of binding parameters, using far fewer points with no loss in parameter estimation accuracy.
This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and stance model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses.
Optimization theory is used more often than any other method to predict individual muscle forces in human movement. One of the limitations frequently associated with optimization algorithms based on efficiency criteria is that they are thought to not provide solutions containing antagonistic muscular forces; however, it is well known that such forces exist. Since analytical solutions of nonlinear optimization algorithms involving multi-degree-of-freedom models containing multijoint muscles are not available, antagonistic behavior in such models is not well understood. The purpose of this investigation was to study antagonistic behavior of muscles analytically, using a three-degree-of-freedom model containing six one-joint and four two-joint muscles. We found that there is a set of general solutions for a nonlinear optimal design based on a minimal cost stress function that requires antagonistic muscular force to reach the optimal solution. This result depends on a system description involving multijoint muscles and contradicts earlier claims made in the biomechanics, physiology, and motor learning literature that consider antagonistic muscular activities inefficient.
In plate assays in the presence of S. typhimurium TA100 and various amounts of liver 9000 X g supernatant (S9) from either untreated, phenobarbitone- (PB) or Aroclor-treated rats, the S9 concentration required for optimal mutagenicity of aflatoxin B1 (AFB) depended both on the source of S9 and on the concentration of the test compound. In these assays, the water-soluble procarcinogen, dimethylnitrosamine (DMN) was mutagenic in S. typhimurium TA1530 only in the presence of a 35-fold higher concentration of liver S9 from PB-treated rats than that required for AFB, a lipophilic compound. In liquid assays, a biphasic relationship was observed in the mutagenicities in S. typhimurium TA100 of benzo[a]pyrene (BP) and AFB and the concentration of liver S9. For optimal mutagenesis of BP, the concentration of liver S9 from rats treated with methylcholanthrene (MC) was 4.4% (v/v); for AFB it was 2.2% (v/v) liver S9 from either Aroclor-treated or untreated rats. At higher concentrations of S9 the mutagenicity of BP and of AFB was related inversely to the amount of S9 per assay. The effect of Aroclor treatment on the microsomemediated mutagenicity of AFB was assay-dependent: in the liquid assay, AFB mutagenicity was decreased, whereas in the plate assay it did not change or was increased. As virtually no bacteria-bound microsomes were detected by electron microscopy, after the bacteria had been incubated in a medium containing 1-34% (v/v) MC-treated rat-liver S9, it is concluded that, in mutagenicity assays, mutagenic metabolites generated by microsomal enzymes from certain pro-carcinogens have to diffuse through the assay medium before reaching the bacteria. Thus the mutagenicity of BP was dependent on both the concentration of rat-liver microsomes and that of total cytosolic proteins and other soluble nucleophiles such as glutathione. At a concentration of 4.4% (v/v) liver S9, the mutagenicity of BP was about 3.6 times higher than in assays containing a 4-fold higher concentration of cytosolic fraction. Studies on the glutathione-dependent reduction of BP mutagenicity in plate assays has shown that, in the presence of liver S9 concentrations greater than that required for optimal mutagenicity, the reduction in mutagenicity was related directly to the concentration of liver S9. Thus, in the Salmonella/microsome assay, when the concentration of rat-liver S9 was increased over and above the amount required for the optimal mutagenicity of BP, the mutagenic metabolites of BP were inactivated (by being trapped with cytosolic nucleophiles and/or by enzymic conjugation with glutathione); this effect increased more rapidly than their rate of formation. The concentration of liver S9 for optimal mutagenicity of test compounds requiring activation catalyzed by mono-oxygenases seems, therefore, to be related to the departure from linearity of the relationship between the rate of formation of mutagenic metabolites and the concentration of liver S9.
With inverse radiation therapy planning methods, both biological and physical objective functions can be used to perform the optimization. A biological objective function, namely the probability of achieving tumor control without causing severe complications in normal tissues, P+, has been used to evaluate six different optimization methods. All six methods have been tested on two different clinically relevant treatment geometries. The results show that optimization with a physical objective function which gives the best possible agreement with the desired dose distribution in the least squares sense, may result in severe loss of complication-free tumor control due to insufficient consideration of the organs at risk. It is generally better to use a physical objective function which minimizes the over-dosage when the desired dose distribution can not be exactly reproduced. In all cases the use of physical objective functions results in a lower probability of controlling the tumor without causing severe normal tissue reactions than if the biological objective function, P+, is used. However, the results also show the importance of accurately accounting for beam divergence, dose build-up, beam attenuation, and lateral scatter during the optimization procedure, particularly when the biological objective function is used. The loss in P+ by assuming that all energy deposition kernels are identical and that all the constituent beams have fixed relative weights can be 15% or more. When lateral scatter is not accounted for during the optimization, serious injury to organs at risk may result. This problem is specially severe for organs that are partly or totally encapsulated by the target volume. For superficial target volumes accurate consideration of the dose build-up of the incident pencil beams is fundamental.
To overcome the limitations of the intensity modulation optimization techniques based on dose criteria, we introduce a method for optimizing intensity distributions in which we employ an objective function based on biological indices. The objective function also includes constraints on dose and dose-volume combinations to ensure that the results are consistent with the physician's judgement. We apply a variant of the steepest-descent method to optimize the objective function. The method is three-dimensional and incorporates scattered radiation in the optimization process using an iterative scheme employing the pencil beam convolution method. Previously we had shown that the inverse technique of obtaining optimum intensity distributions, for which the objectives are defined in terms of a desired uniform dose to the target volume and desired upper limits of dose to normal organs, produces satisfactory approximations of the desired dose distributions for prostate plans. However, for lung, the performance of this technique was considerably inferior. Our conclusion was that, in general, it is not sufficient to specify the objectives of optimization purely in terms of a desired pattern of dose and that the objectives should also incorporate biology, perhaps in the form of biological indices. We demonstrate that the biology-based approach produces lung plans that are superior to those produced when only dose-based objectives are used. For the treatment of prostate, the two methods produce comparable dose distributions.
Models for material disposition and dosimetry involve predictions of the biokinetics of the material among compartments representing organs and tissues in the body. Because of a lack of human data for most toxicants, many of the basic data are derived by modeling the results obtained from studies using laboratory animals. Such a biomathematical model is usually developed by adjusting the model parameters to make the model predictions match the measured retention and excretion data visually. The fitting process can be very time-consuming for a complicated model, and visual model selections may be subjective and easily biased by the scale or the data used. Due to the development of computerized optimization methods, manual fitting could benefit from an automated process. However, for a complicated model, an automated process without an optimization strategy will not be efficient, and may not produce fruitful results. In this paper, procedures for, and implementation of, an optimization strategy for a complicated mathematical model is demonstrated by optimizing a biokinetic model for 144Ce in fused aluminosilicate particles inhaled by beagle dogs. The optimized results using SimuSolv were compared to manual fitting results obtained previously using the model simulation software GASP. Also, statistical criteria provided by SimuSolv, such as likelihood function values, were used to help or verify visual model selections.
1. There is an exponential relationship between blood viscosity (cP) and hematocrit (%) for the bullfrog; eta = 1.81 e0.033Hct. The in vitro optimal hematocrit calculated for blood flow through tubes, from this relationship for bullfrog blood, is 30%. 2. Amphibian blood is a non-Newtonian fluid with viscosity dependent on shear rate. It has a finite yield shear stress of about 1.5 dynes cm-2. 3. Hematocrit of bullfrogs was increased from 27% (control) to 57% by isovolemic erythrocythemia (constant volume blood-doping). There was a slight increase in systolic, diastolic and venous blood pressure with elevated hematocrit. 4. Systemic arch blood flow rate was inversely related to blood viscosity for erythrocythemic bullfrogs. The decrease in systemic arch blood flow at high hematocrits was due primarily to reduced pulse volume rather than reduced heart rate. 5. Systemic arch blood flow, when standardised between individuals, was inversely related to blood viscosity; Qbl = 0.185 + 3.73 eta -1. This relationship was significantly different from that predicted by the Poiseuille-Hagen flow formula. The in vivo optimal hematocrit calculated from this relationship was 41%. 6. Optimal hematocrit theory appears to be generally applicable for Rana catesbeiana in vitro and in vivo. Most individuals had an in vivo optimal hematocrit, but the absence of a clear optimal hematocrit for some individuals could reflect methodological variability, or in vivo physiological compensation for the increased blood viscosity at high hematocrit.
One of the systems under investigation for producing hyperthermia noninvasively for treating deep-seated tumors is the annular phased array. This device consists of two rings of eight electromagnetic apertures that are placed concentrically about the long axis of the patient and radiate energy toward the center. Previous theoretical and clinical studies have concentrated primarily on systems where the amplitude and phase of the signal applied to each aperture were the same, and these studies have shown that the system is capable of depositing power deep within the patient. Nevertheless, in many situations the system was not capable of producing desirable temperature distributions in the tumor and normal tissue. In this paper we report on a 2-dimensional theoretical investigation where an optimization routine was used to select the amplitude and phases of each of eight apertures. The optimization procedure and resulting calculations were based on CT scans of patients with tumors. The electrical and thermal properties of the different organs and tissues were taken into account. The optimization routine tried to achieve uniform absorbed power in the tumor region with zero absorbed power outside. Using the optimized amplitudes and phases, the SAR (specific absorption rate, W/kg) was calculated for the array. The results show that in general the optimization procedure was successful in that the power deposited within the tumor volume was increased with less power deposited into normal tissue when compared to the equal amplitude and phase case. This SAR data was then used as the input to a program based on the bioheat transfer equation, which calculated the temperature distribution in the patient model for an assumed set of blood perfusion rates. Depending on the location, size of the tumor, and blood perfusion rates, the improvement in the percentage of the tumor brought to therapeutic temperature varied from 0% to as much as 80%.
An interactive system has been developed to aid in determining optimal photon and electron beams and beam weights for radiotherapy treatment planning. Dose constraints at various points are selected and an algorithm searches for a set of beams and weighting factors that satisfy these constraints. In the event that no combination of beam weights satisfies the choice of treatment modalities and dose constraints, the treatment modalities and dose constraints can be modified interactively. The goal of this procedure is different from that of more conventional optimization schemes in which optimal dose values are specified and the optimization algorithm determines the set of beam weights that yields a dose distribution closest to optimal.
Computer controlled high dose-rate (HDR) brachytherapy afterloading machines are equipped with a single, miniaturized, high activity Ir-192 source that can be rapidly moved in fine increments among several channels. Consequently, by appropriate programming of source dwell positions and times, the dose distribution can be optimized as desired. We have explored the optimization potential of this new technology for two applications: (a) cervix brachytherapy, and (b) transvaginal irradiation. Cervix brachytherapy with a gynecologic ring applicator was simulated by 48 sources of relative activities ranging from 0.17 to 1.00 that were equally distributed between the tandem and the ring. The results confirmed that the optimized distribution of physical doses are superior to those achievable with standard brachytherapy sources and applicators. For example, with five-point optimization, the relative dose-rate in the rectum was only 47% of that in point A; for standard application the dose rate was 47% higher. For transvaginal application 27 sources of relative activities between 0.07-0.79 were placed in the ring and a single source of unit strength in the tandem. Using dose distribution homogeneity as an optimization criterion, the results (+/- 2.5%) were again superior to those obtained for commonly used double ovoid (+/- 15%), linear cylinder (+/- 27%), or a "T" source (31%).