Reduced Parameter Space Kinetic Modeling for Robust Nonlinear Fitting
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Solution Overview
Problem
Current kinetic modeling techniques face challenges in fitting nonlinear equations due to complex multi-dimensional spaces with local minima, ridges, and valleys, leading to dependency on initial conditions and high computational demands, especially when dealing with noisy measurements.
Innovation Solution
The method reformulates kinetic modeling equations to be linear in one parameter and nonlinear in another, reducing the parameter space to simplify the fitting problem, allowing for faster and more robust fits by constraining the solution space to minimize the objective function in a linear sense.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If iterative nonlinear fitting algorithms are used to minimize least squares objective function, then robust kinetic model fits are obtained, but the solution becomes dependent on initial conditions and requires careful management of iterations and stopping criteria
Solution Approach 1:
The patent segments the parameter space by identifying and exploiting the linear subspace within the nonlinear fitting problem. By separating linear parameters from nonlinear parameters, the method creates a simplified fitting landscape that eliminates dependence on initial conditions for the linear parameters, while reducing the complexity of the nonlinear parameter search.
Solution Approach 2:
The patent transforms the fitting problem by changing the parameter representation - specifically by reformulating the model equations to identify which parameters appear linearly and which appear nonlinearly. This parameter transformation allows the linear parameters to be solved analytically rather than iteratively, eliminating the initial condition dependency for those parameters.
2Reliability
If robust versions of NLLS fitting algorithms are used with restarting fits over and over again with changing conditions, then dependency on initial conditions is reduced, but computational demand increases significantly
Solution Approach 1:
By segmenting the parameters into linear and nonlinear components, the patent eliminates the need for repeated restarts. The linear parameters can be solved directly without iteration, and the reduced nonlinear parameter space can be searched more efficiently, dramatically reducing computational demand while maintaining robustness.
Solution Approach 2:
The patent extracts the linear parameters from the iterative fitting process entirely, solving them analytically outside the iteration loop. This removes the source of computational inefficiency (repeated evaluation of linear terms) while preserving the ability to find global minima through the reduced nonlinear parameter search.
3Reliability
If alternative algorithms such as simulated annealing are used to increase chances of obtaining true global minimum fit, then robustness is improved, but extensive computer power and time are required for each fit
Solution Approach 1:
The patent changes the dimensionality of the fitting problem by reducing it from a multi-dimensional nonlinear search to a lower-dimensional nonlinear search combined with analytical linear parameter calculation. This dimensional reduction makes exhaustive or near-exhaustive search feasible, guaranteeing global minimum identification without requiring simulated annealing's extensive computational resources.
Solution Approach 2:
The patent replaces the mechanical iterative optimization system (which requires repeated function evaluations and gradient calculations) with a hybrid analytical-numerical system. The linear parameters are solved analytically, and only the essential nonlinear parameters require numerical optimization, dramatically reducing computational time while maintaining accuracy.
4Productivity
If linearized fitting approaches are used, then computational speed is improved, but certain approximations are made that fall short of the ideal fitting solution
Solution Approach 1:
Rather than approximating the nonlinear model with a linearized version, the patent changes the parameter representation to exactly identify which parameters are linear and which are nonlinear. This allows the linear parameters to be solved exactly (not approximately) while the nonlinear parameters are handled with appropriate numerical methods, achieving both speed and accuracy.
Data Source
AI summary
Systems, methods and devices are provided for fitting kinetic models to measurements of dynamic curves where the kinetic models give rise to nonlinear fitting equations in two or more unknowns that can be formulated so that they are linear in one or more of the unknown parameters and nonlinear in one or more of the unknown parameters. Such systems, methods and devices may be utilized to monitor and characterize the attributes of a given tracer such as a radioactive substance within a body, a drug within the body, a concentration of a substance within a particular medium, and the like.