## Mechanical Engineering ETDs

## Publication Date

Spring 8-1-2019

## Abstract

To better understand and to improve therapies for complex diseases such as cancer or diabetes, it is not sufficient to identify and characterize the interactions between molecules and pathways in complex biological systems, such as cells, tissues, and the human body. It also is necessary to characterize the response of a biological system to externally supplied agents (e.g., drugs, insulin), including a proper scheduling of these drugs, and drug combinations in multi drugs therapies. This obviously becomes important in applications which involve control of physiological processes, such as controlling the number of autophagosome vesicles in a cell, or regulating the blood glucose level in patients affected by diabetes. A critical consideration when controlling physiological processes in biological systems is to reduce the amount of drugs used, as in some therapies drugs may become toxic when they are overused. All of the above aspects can be addressed by using tools provided by the theory of optimal control, where the externally supplied drugs or hormones are the inputs to the system. Another important aspect of using optimal control theory in biological systems is to identify the drug or the combination of drugs that are effective in regulating a given therapeutic target, i.e., a biological target of the externally supplied stimuli.

The dynamics of the key features of a biological system can be modeled and described as a set of nonlinear differential equations. For the implementation of optimal control theory in complex biological systems, in what follows we extract \textit{a network} from the dynamics. Namely, to each state variable $x_i$ we will assign a network node $v_i$ ($i=1,...,N$) and a network directed edge from node $v_i$ to another node $v_j$ will be assigned every time $x_j$ is present in the time derivative of $x_i$. The node which directly receives an external stimulus is called a \emph{driver nodes} in a network. The node which directly connected to an output sensor is called a \emph{target node}. %, and it has a prescribed final state that we wish to achieve in finite time.

From the control point of view, the idea of controllability of a system describes the ability to steer the system in a certain time interval towards thea desired state with a suitable choice of control inputs. However, defining controllability of large complex networks is quite challenging, primarily because of the large size of the network, its complex structure, and poor knowledge of the precise network dynamics. A network can be controllable in theory but not in practice when a very large control effort is required to steer the system in the desired direction. This thesis considers several approaches to address some of these challenges. Our first approach is to reduce the control effort is to reduce the number of target nodes. We see that by controlling the states of a subset of the network nodes, rather than the state of every node, while holding the number of control signals constant, the required energy to control a portion of the network can be reduced substantially. The energy requirements exponentially decay with the number of target nodes, suggesting that large networks can be controlled by a relatively small number of inputs as long as the target set is appropriately sized. We call this strategy \emph{target control}.

As our second approach is based on reducing the control efforts by allowing the prescribed final states are satisfied approximately rather than strictly. We introduce a new control strategy called \textit{balanced control} for which we set our objective function as a convex combination of two competitive terms: (i) the distance between the output final states at a given final time and given prescribed states and (ii) the total control efforts expenditure over the given time period. Based on the above two approaches, we propose an algorithm which provides a locally optimal control technique for a network with nonlinear dynamics. We also apply pseudo-spectral optimal control, together with the target and balance control strategies previously described, to complex networks with nonlinear dynamics. These optimal control techniques empower us to implement the theoretical control techniques to biological systems evolving with very large, complex and nonlinear dynamics. We use these techniques to derive the optimal amounts of several drugs in a combination and their optimal dosages. First, we provide a prediction of optimal drug schedules and combined drug therapies for controlling the cell signaling network that regulates autophagy in a cell. Second, we compute an optimal dual drug therapy based on administration of both insulin and glucagon to control the blood glucose level in type I diabetes. Finally, we also implement the combined control strategies to investigate the emergence of cascading failures in the power grid networks.

## Keywords

Complex Networks, Optimal Control, Nonlinear Dynamics, Biological Systems, Drugs Scheduling, Combinatorial Therapy

## Degree Name

Mechanical Engineering

## Level of Degree

Doctoral

## Department Name

Mechanical Engineering

## First Committee Member (Chair)

Francesco Sorrentino

## Second Committee Member

Jens Lorenz

## Third Committee Member

Chaouki T. Abdallah

## Fourth Committee Member

Asal Naseri

## Fifth Committee Member

William S. Hlavacek

## Document Type

Dissertation

## Language

English

## Recommended Citation

Shirin, Afroza. "Optimal Control Strategies for Complex Biological Systems." (2019). https://digitalrepository.unm.edu/me_etds/171