liability performance of MC networks and enables high-accuracy disease detection and health monitoring.
Large-scale bacterial MC systems: In Chapter 4, we presented an analytically tractable model for characterizing i) QS signal propagation within randomly-distributed microorganisms and ii) the distribution of the number of responsive cooperative microorganisms. Different from prior studies, the motion of molecules undergoing independent diffusion and degradation is taken into consideration. Microorganisms are randomly distributed in a 2D environment where each one continuously releases molecules at random times. We derived the 2D channel response at an observer due to one bacterium or randomly-distributed bacteria. We then derived the expected probability of cooperation at the bacterium at a fixed location. We finally derived the approximate expressions for the MGF and different statistics of the number of cooperators.
The analytical results agree with simulation results where the Brownian motion of molecules is simulated by a particle-based method. Our results showed that the Poisson distribution provides the overall best approximation of the PDF and CDF of the number of cooperators, especially when the population density is low. Our model captures the basic features of QS and accounts for the diffusion of molecules.
The presented work serves as the first step to prevent undesirable bacterial infections and lead to new environmental remediation, since cooperative behaviors of microscopic popula-tions, e.g. the formation of biofilms and the production of antibiotics, play a crucial role in bacterial infections, environmental remediation, and wastewater treatment.
Realistic MC environment: In Chapter 5, we for the first time investigated communica-tion through realistic porous channels via statistical breakthrough curves. Assuming that the number of arrived molecules can be approximated as a Gaussian RV and using fully resolved computational fluid dynamics results for the breakthrough curves, we presented the numeri-cal results for the throughput, mutual information, error probability, and information diversity gain. Using these numerical results, we revealed the unique characteristics of the PM channel.
The presented work provides useful guidelines for designing the optimal MC system through porous media and predicting the system communication performance in a practical biological environment.
6.2 Future Work
The research of MC is still at a very early stage. In the future, the following work can be conducted based on the contributions presented in this thesis:
6.2.1 Theoretical Modeling
• Since the work in Chapters 2 and 3 adopted many unrealistic assumptions, interesting future work includes relaxing these assumptions:
–Modeling an imperfect TX: It is interesting to develop a mathematical model for an imperfect TX by considering the volume of the TX and the molecules that are randomly generated within the TX. This consideration is motivated by the fact that in real biologi-cal environments, the cells are spheribiologi-cal, rod-shaped or spiral-shaped and the molecules are generated in different sections of cells. The chemical reactions that occur in the real-istic generation and emission processes of molecules in cells can also be considered. As a starting point, the results in Chapter 4 can be used to establish the model for an imper-fect TX. For this model, stochastic processes such as the PPP will be used to model the random location of generated molecules.
–Modeling flow-aided propagation in a bounded environment: This modeling is in-spired by the fact that in blood vessels, the propagation of molecules is driven by the blood flow in a bounded channel. Considering laminar flows with non-uniform flow ve-locity, the mechanical dispersion coefficient of molecules caused by the interaction of diffusion and non-uniform flow can be derived. Using this coefficient, the PDF of the first arrival time of a molecule in the bounded environment can be derived by jointly solving (i) the Stokes equation for the flow velocity and (ii) the reaction-diffusion differ-ential equation for chemical reactions.
–Modeling channel response at multiple reactive RXs: Chapters 2 and 3 assumed transparent RXs for tractability due to the independence among observations at multiple transparent RXs. However, many practical RX surfaces may interact with the molecules of interest, e.g., by providing binding sites for absorption or other reactions [134]. In an environment where multiple non-transparent RXs co-exist, one non-transparent RX may impact the molecules received by other non-transparent RXs. Derivation of channel response at RXs when multiple reactive RXs co-exist, by taking into account the mutual influence between RXs, is interesting.
• Cooperative localization and channel estimation: Most of MC studies considered the estimation of location, distance, and channel response using the observation at one RX.
Our results in Chapters 2 and 3 focus on the estimation of transmitted symbols. Cooper-ative localization or channel estimation using multiple RXs has not been exploited in the MC area. Our work in cooperative detection provides useful guidelines for exploiting benefits of combining observations at multiple distributed RXs.
• Predicting dynamic behaviors of microorganisms: We can use game theory to un-derstand noisy real-time signaling and the resulting behavioral dynamics in microscopic
§6.2 Future Work 113
populations such as bacteria and other cells, as identified in [69]. In Chapter 4, we presented an analytically tractable model for predicting the distribution of the number of cooperative microorganisms, but did not consider evolutionary behavior coordination over time and did not apply game theory to our current model with elaborated payoffs and strategies. Based on Chapter 4, we can develop a model for predicting dynamic be-havior of a population of randomly-distributed microorganisms over time by considering real-time noisy signaling among microorganisms and applying game theory.
6.2.2 Validation of Theoretical Work
The theoretical results must be validated. There are three main methods, i.e., simulation, testbed, and experiments, to validate the theoretical results. To enable the future application of MC, the following advancements in simulation, testbed, and experiments should be done:
• Simulation validation: All of the theoretical results in Chapters 2–5 are validated by particle-based simulations where the location of each particle over time is tracked. Al-though simulation is cost-effective, but is still time-consuming, especially simulating multiple-TX, multiple-RX, and relay-aided MC systems. Developing more efficient sim-ulation methods to decrease the required time for simsim-ulation speeds up research progress of MC.
• Testbed development: York University, University of Warwick, Yonsei University, and Australian National University have developed macro-scale MC testbeds, e.g., multiple-input multiple-output macro-scale MC systems done by Yonsei University. Development of macro-scale testbeds for our cooperative MC systems considered in Chapters 2 and 3 and MC systems over porous media considered in Chapter 5 provides practical insights on reliability improvement brought by cooperative MC and unique characteristics of molecular information delivery over porous media.
• Biological experiments: Our work in 4 considered a 2D environment since a 2D envi-ronment facilitates experimental validation of our theoretical work. Biological experi-ments, especially with bacteria, are usually conducted in a 2D environment. Collabora-tion with biologists to conduct lab experiments to validate our theoretical results is an essential and promising direction of future work.
AppendixA
Appendix A
A.1 Proof of Theorem 2.1
The convexity of Pmd[j]K can be proven by showing that its second derivative with respect to ξRXis nonnegative [31]. We derive the second derivative of Pmd[j]Kas
∂2Pmd[j]K
∂ ξRX
2 = 1
2K
2(−1+K)K
π Ξ(−2+K, 2, 1) + r2
πK(0.5+U1[j]− ξRX) Ξ
−1+K, 1,3 2
, (A.1)
where
Ξ(α , β , γ) = (1+Λ(ξRX,U1[j]))αΘ(ξRX,U1[j])β
U1[j]γ (A.2)
andΘ(x, λ), exp −(0.5+λ − x)2/2λ. Due to the fact that the value ofΛ(x, λ)is be-tween −1 and 1 and the value ofΘ(x, λ)is always greater than zero, (A.1) is always nonneg-ative if we impose the constraint (2.22). Following a similar procedure, we prove that Pfa[j]K is also convex with respect to ξRX, if we impose the constraint (2.23).