We study a finite{dimensional approach to the Heath Jarrow Morton model
for interest rate and introduce a notion of approximate consistency for a family of functions
in a deterministic and stochastic framework. This amounts to asking the decrease
of the minimum distance in least squares sense. We start from a general linearly parameterized
set of functions and extend the theory to a nonlinear Nelson Siegel family.
Necessary and sufficient condition to have approximately consistency are given as well
as a criterion of stability for the approximation.
Cancer immunotherapy aims at eliciting an immune system response against the tumor. However, it is often characterized by toxic side-effects. Limiting the tumor growth and, concurrently, avoiding the toxicity of a drug, is the problem of protocol design. We formulate this question as an optimization problem and derive an algorithm for its solution.
Unlike the standard optimal control approach, the algorithm simulates impulse-like drug administrations. It relies on an exact computation of the gradient of the cost function with respect to any protocol by means of the variational equations, that can be solved in parallel with the system. In comparison with previous versions of this method [F. Castiglione, B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy, Bull. Math. Biol. 68 (2006) 255274; B. Piccoli, F. Castiglione, Optimal vaccine scheduling in cancer immunotherapy, Physica A. 370 (2) (2007) 672680], we optimize both the timing and the dosage of each administration and introduce a penalty term to avoid clustering of subsequent injections, a requirement consistent with the clinical practice. In addition, we implement the optimization scheme to simulate the case of multitherapies.
The procedure works for any ODE system describing the pharmacokinetics and pharmacodynamics of an arbitrary number of therapeutic agents. In this work, it was tested for a well known model of the tumorimmune system interaction [D. Kirschner, J.C. Panetta, Modeling immunotherapy of tumorimmune interaction, J. Math. Biol. 37 (1998) 235252]. Exploring three immunotherapeutic scenarios (CTL therapy, IL-2 therapy and combined therapy), we display the stability and efficacy of the optimization method, obtaining protocols that are successful compromises between various clinical requirements.
Optimization
Protocol design
mmunotherapy
Cancer
Drug holidays
