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The primary objective of this article is to present a general framework for users and applications of the master equation approach in extended mean field type control, for- mulated with a McKean-Vlasov stochastic differential equation that depends on the law of both the control and state variables. This control problem has recently gained significant attention and has been extensively studied at the level of the Bellman equa- tion. Here, we extend the analysis to the master equation and derive the corresponding Hamilton-Jacobi-Bellman equation. A key novelty of our approach is that we do not directly rely on the Fokker-Planck equation, which surprisingly leads to a significant simplification. We provide a concise theoretical presentation with proofs, as the stan- dard theory of stochastic control is not directly applicable. In the current work, the solution is constructed using an ansatz-based approach to dynamic programming via the master equation.We illustrate this method with a practical example. All proofs are presented in a self-contained manner. This paper offers a structured presentation of the extended mean field type control problem, serving as a valuable toolbox for users who are less focused on mathematical intricacies but seek a general framework for application. 

We developed a computational model of sodium fluorescein (SF) biliary excretion in ex vivo machine perfusion and used this model to assess changes in model parameters associated with the activity of MRP2, a hepatocyte membrane transporter, in response to increasing warm ischemia time. We found a significant decrease in the parameter value describing MRP2 activity, consistent with a role of decreased MRP2 function in ischemia-reperfusion injury leading to decreased secretion of SF into bile. 

Abstract— During liver transplantation, ischemia-reperfusion injury (IRI) is inevitable and decreases the overall success of the surgery. While guidelines exist, there is no reliable way to quantitatively assess the degree of IRI present in the liver. Our recent study has shown a correlation between the bile-to-plasma ratio of FDA-approved sodium fluorescein (SF) and the degree of hepatic IRI, presumably due to IRI-induced decrease in the activity of the hepatic multidrug resistance-associated protein 2 (MRP2); however, the contribution of SF blood clearance via the bile is still convoluted with other factors, such as renal clearance. In this work, we sought to computationally model SF blood clearance via the bile. First, we converted extant SF fluorescence data from rat whole blood, plasma, and bile to concentrations using calibration curves. Next, based on these SF concentration data, we generated a “liver-centric”, physiologically-based pharmacokinetic (PBPK) model of SF liver uptake and clearance via the bile. Model simulations show that SF bile concentration is highly sensitive to a change in the activity of hepatic MPR2. These simulations suggest that SF bile clearance along with the PBPK model can be used to quantify the effect of IRI on the activity of MRP2. Clinical Relevance— This study establishes the theory necessary to generate a model for predicting the degree of IRI during liver transplantation. 

The objective of this paper is to study the optimal consumption and portfolio choice problem of risk-controlled investors who strive to maximize total expected discounted utility of both consumption and terminal wealth. Risk is measured by the variance of terminal wealth, which introduces a nonlinear function of the expected value into the control problem. The control problem presented is no longer a standard stochastic control problem but rather, a mean field-type control problem. The optimal portfolio and consumption rules are obtained explicitly. Numerical results shed light on the importance of controlling variance risk. The optimal investment policy is nonmyopic, and consumption is not sacrificed.