Advances in Bioinformatics

Volume 2016, Article ID 8054219, 12 pages

http://dx.doi.org/10.1155/2016/8054219

## Agent-Based Deterministic Modeling of the Bone Marrow Homeostasis

Department of Computer Science and Engineering, VNIT Nagpur, Nagpur 440010, India

Received 20 November 2015; Revised 1 April 2016; Accepted 5 May 2016

Academic Editor: Nurit Haspel

Copyright © 2016 Manish Kurhekar and Umesh Deshpande. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Modeling of stem cells not only describes but also predicts how a stem cell’s environment can control its fate. The first stem cell populations discovered were hematopoietic stem cells (HSCs). In this paper, we present a deterministic model of bone marrow (that hosts HSCs) that is consistent with several of the qualitative biological observations. This model incorporates stem cell death (apoptosis) after a certain number of cell divisions and also demonstrates that a single HSC can potentially populate the entire bone marrow. It also demonstrates that there is a production of sufficient number of differentiated cells (RBCs, WBCs, etc.). We prove that our model of bone marrow is biologically consistent and it overcomes the biological feasibility limitations of previously reported models. The major contribution of our model is the flexibility it allows in choosing model parameters which permits several different simulations to be carried out in silico without affecting the homeostatic properties of the model. We have also performed agent-based simulation of the model of bone marrow system proposed in this paper. We have also included parameter details and the results obtained from the simulation. The program of the agent-based simulation of the proposed model is made available on a publicly accessible website.

#### 1. Introduction

Stem cells and their descendants are the building blocks of life. How stem cell populations guarantee their maintenance and self-renewal and how individual stem cells decide to transit from one cell state to another to generate mature differentiated cells are long standing and fascinating questions [1]. There is a significant interest in studying stem cells, both to elucidate their basic biological functions and to learn how to utilize them as new sources of specialized cells for tissue repair [2].

Blood is the life preserving fluid, whose major functions are supply of nutrients and oxygen to the tissues, self-immunity, and defense against pathogens. In order to carry out these tasks, human blood contains a variety of cells, each precisely adapted to its specific objective. All the different blood cells develop from a kind of master cell, called the hematopoietic (blood forming) stem cell (HSC). Incidentally, the first stem cell populations discovered were HSCs. HSCs are primarily present in the bone marrow. HSCs are stem cells that give rise to all the differentiated blood cell types including white blood cells (WBC), red blood cells (RBCs), and platelets. Fully mature differentiated cells migrate into the blood stream. The transition of HSCs from quiescence (not undergoing any cell cycle) into proliferation, or differentiation, is governed by their internal state and by chemicals secreted by neighboring cells in their immediate microenvironment.

It is believed that a single HSC is sufficient to reconstitute the entire blood system [3, 4]. This extraordinary regenerative ability of the bone marrow is not surprising, considering that it has a vital role that must remain unaffected by stem cells depletion that might occur, for example, as a result of chemotherapy, radiation, or disease. It should be emphasized that though the supply of blood cells in the periphery is steady, the bone marrow is not static. It is dynamic in the sense that it constantly changes in its constitution and arrangement, and these changes occur at varying rates. The bone marrow is in a state of homeostasis that can be considered as a dynamic equilibrium between its constituents.

Theise and Harris [5] describe how stem cells and their lineages are examples of complex adaptive systems. Profound understanding of a complex adaptive system can be gathered by generating computer models using computational techniques. Agent-based modeling is a way to represent such complex adaptive systems in software. An agent is a high-level software abstraction that provides a convenient and powerful way to describe a complex software entity in terms of its behavior within a contextual computational environment. Agents are flexible problem-solving computational entities that are reactive (respond to the environment) and autonomous (not externally controlled) and interact with other such entities.

One of the significant contributions to stem cell modeling was by Agur et al. [6]. The main aim of their paper was to provide a mathematical basis for the bone marrow homeostasis. More precisely, they wanted to define the properties that enabled the bone marrow to rapidly return to a steady supply of blood cells after relatively large perturbations in stem cell numbers. Their model is represented as a family of cellular automata on a connected, locally finite undirected graph. Their model can be briefly described as follows: It has three types of cells, stem cells, differentiated cells, and null cells. Each cell has an internal counter. Stem cells differentiate when their immediate neighborhood is saturated with stem cells and their internal counter reaches a certain threshold. A differentiated cell converts to a null cell after its internal counter crosses the required threshold, a process that denotes the passing of a differentiated cell to blood stream leaving the place it had earlier occupied in the bone marrow empty. A null cell, with a stem cell neighbor, is converted to a stem cell when its internal counter reaches a particular threshold.

D’Inverno and Saunders [7] have listed the following drawbacks of Agur et al.’s [6] model:(1)The specification of Agur et al.’s model reveals that the null cells must have counters. In a sense, an empty space has to do some computational work. This lacks biological feasibility and is against what the paper states about modeling cells having counters, rather than empty locations.(2) Stem cell division is not explicitly represented; instead, stem cells are brought into existence in empty spaces.(3) A stem cell appears when a null cell has been surrounded by at least one stem cell for a particular period. However, the location of the neighboring stem cell can vary at each step. As an effect of the drawback mentioned above, a stem cell can potentially differentiate more than once in the same time instant since it might be surrounded by more than one null cell. Hence, potentially more than one neighboring null cells can get converted to stem cells.(4) The state of a stem cell after division is not defined. Nothing is said about what happens to a stem cell after a new stem cell appears in the null cell space. There is no provision of any preconditions on the stem cell division.(5) There is no provision for stem cell apoptosis.

D’Inverno and Saunders [7] provided an agent-based simulation for the model described by Agur et al. They needed to overcome the third limitation mentioned above for creating a deterministic agent-based simulation. In order to overcome the limitation, they introduced the concept of a controlling microenvironment that links a null cell, which has reached a threshold, with a stem cell that can differentiate. All the cells send and receive signals from the microenvironment and act on its suggestions. They performed agent-based implementation with the incorporation of Agur et al.’s model in two dimensions. However, the improvement suggested by them, of a controlling microenvironment, does not have any biological basis.

Moreover, there is an additional limitation of the model described by Agur et al. The limitation is that there are no intermediate cells, also called transitive cells, in the model proposed by them. Transitive cells have limited stem cell-like properties that decreases with each subsequent generation and they are eventually converted to differentiated cells. For hematopoietic system, common lymphoid progenitor (CLP) and common myeloid progenitor (CMP) are examples of transitive cells [8]. As there are no transitive cells, there cannot be any conversion of a transitive cell to a stem cell, which can help bone marrow system to recover in case of severe perturbations.

Some other novel models of HSCs are proposed. Roeder and Loeffler [9] propose a stochastic model with two growth environments where a stem cell remains quiescent for longer periods of time when it is in first environment and proliferates when it transitions to the other environment. The proliferation and transition depend only upon two stochastic parameters. Glauche et al. [10] provide two independent compartments for fast proliferating HSCs and slow proliferating HSCs to explain simultaneous occurrence of self-renewal and differentiation. Glauche et al. [11] further improved their model by considering the effects of aging on stem cell population. These models show within-tissue plasticity and proliferation and self-renewal potential of stem cells. Stem cells moving between two different compartments or environments with a fixed probability are an artifact that is not biologically consistent. The probability might change since it is dependent on the local environment of the stem cell. Another limitation is that these models show homeostasis for only a limited range of parameter values.

The model proposed in this paper is an enhancement over our earlier model [12], with incorporation of stem cell death (apoptosis) after certain number of cell divisions. In our model, we have addressed all the limitations listed above by extending and augmenting the model originally proposed by Agur et al., thereby making the model close to biological observations. The model we present is aimed at simulating a situation in which a cell’s behavior is determined only by a combination of the types and states of cells in its proximity and its own cell cycle represented by its internal counters. The main assumptions of our model are as follows:(i)Cell behavior is determined by the number and type of its neighbors. This assumption is aimed at describing the fact that cytokines, secreted by cells into the microenvironment, are capable of activating cells into changing their types [1, 3].(ii) Each cell has an internal counter that determines the time required for it to mature. The duration for maturity is fixed for each type of cell. After maturity, the cell changes its type or its generation.(iii) Stem cell apoptosis occurs after certain number of renewals.(iv) Every cell possesses a directional component and it proliferates in the direction of that component. The directional component is updated after each proliferation. Although the directional component has no biological significance, it allows the model to be fully deterministic.(v) The model captures emergent behavior of the bone marrow that is consistent with several biological observations:(a) The model has high resilience for the bone marrow homeostasis as shown in [13].(b) The model incorporates intermediate transitive cells and quiescent stem cells as described in [8].(c) In [14], the authors mention that the transitive cells can become stem cells in exceptional circumstances. The model supports this observation.(d) The model also incorporates stem cell apoptosis as given in [15].(vi) The model does not account for leukemia causing abnormal stem cell behaviors as is done in [16].

We have performed agent-based simulation of the model of bone marrow stem cell system proposed in this paper. The details and the results of this simulation are provided in the Appendix.

The paper is organized as follows: In the next section, we describe our model and the rules that govern it. In Section 3, we show how a single stem cell can populate the entire bone marrow and also prove the homeostatic properties of the proposed model. In Section 4, we show that the model provides a steady supply of differentiated cells to the blood stream and we also show that several stem cells remain in quiescent state. In Section 5, we describe the steady states and death states of our proposed model. We discuss the theoretical results in Section 6. The results of the agent-based simulation are included in the Appendix.

#### 2. Description of the Model

Our model contains three basic types of cells and a notation for empty space:(i)*Stem cell*, denoted by , either can proliferate generating new stem cells or can convert to a transitive cell. They can become quiescent. In the event of the death of a stem cell, it can be considered to be converting to an empty space.(ii)*Transitive cell*, denoted by , either can convert to a differentiated cell or can convert back to a stem cell when there are no stem cells in its near neighborhood.(iii)*Differentiated cell*, denoted by , is the final product of a stem cell. After maturation, these cells leave the bone marrow.(iv)*Empty space*, represented by , denotes space in the bone marrow that can be occupied by either a stem cell or a transitive cell or a differentiated cell.

In our model, the bone marrow is represented as a connected, locally finite undirected graph. This describes the neighborhood of bone marrow cells.

Let be a connected, locally finite undirected graph that denotes the bone marrow. Its vertex set denotes the cells and the set of edges describes the neighboring cells to which a cell is connected in the bone marrow (Figure 1).