Tumor Biochemical Heterogeneity and Cancer Radiochemotherapy: Network Breakdown Zone-Model.

Breakdowns of two-zone random networks of the Erdos-Rényi type are investigated. They are used as mathematical models for understanding the incompleteness of the tumor network breakdown under radiochemotherapy, an incompleteness that may result from a tumor's physical and/or chemical heterogeneity. Mathematically, having a reduced node removal probability in the network's inner zone hampers the network's breakdown. The latter is described quantitatively as a function of reduction in the inner zone's removal probability, where the network breakdown is described in terms of the largest remaining clusters and their size distributions. The effects on the efficacy of radiochemotherapy due to the tumor micro-environment (TME)'s chemical make-up, and its heterogeneity, are discussed, with the goal of using such TME chemical heterogeneity imaging to inform precision oncology.

Tumor Hypoxia Heterogeneity Affects Radiotherapy: Inverse-Percolation Shell-Model Monte Carlo Simulations.

Tumor hypoxia was discovered a century ago, and the interference of hypoxia with all radiotherapies is well known. Here, we demonstrate the potentially extreme effects of hypoxia heterogeneity on radiotherapy and combination radiochemotherapy. We observe that there is a decrease in hypoxia from tumor periphery to tumor center, due to oxygen diffusion, resulting in a gradient of radiative cell-kill probability, mathematically expressed as a probability gradient of occupied space removal. The radiotherapy-induced break-up of the tumor/TME network is modeled by the physics model of inverse percolation in a shell-like medium, using Monte Carlo simulations. The different shells now have different probabilities of space removal, spanning from higher probability in the periphery to lower probability in the center of the tumor. Mathematical results regarding the variability of the critical percolation concentration show an increase in the critical threshold with the applied increase in the probability of space removal. Such an observation will have an important medical implication: a much larger than expected radiation dose is needed for a tumor breakup enabling successful follow-up chemotherapy. Information on the TME's hypoxia heterogeneity, as shown here with the numerical percolation model, may enable personalized precision radiation oncology therapy.

A network SIRX model for the spreading of COVID-19.

Infectious diseases, such as the current COVID-19, have a huge economic and societal impact. The ability to model its transmission characteristics is critical to minimize its impact. In fact, predicting how fast an infection is spreading could be a major factor in deciding on the severity, extent and strictness of the applied mitigation measures, such as the recent lockdowns. Even though modelling epidemics is a well studied subject, usually models do not include quarantine or other social measures, such as those imposed in the recent pandemic. The current work builds upon a recent paper by Maier and Brockmann (2020), where a compartmental SIRX model was implemented. That model included social or individual behavioural changes during quarantine, by introducing state X , in which symptomatic quarantined individuals are not transmitting the infection anymore, and described well the transmission in the initial stages of the infection. The results of the model were applied to real data from several provinces in China, quite successfully. In our approach we use a Monte-Carlo simulation model on networks. Individuals are network nodes and the links are their contacts. We use a spreading mechanism from the initially infected nodes to their nearest neighbours, as has been done previously. Initially, we find the values of the rate constants (parameters) the same way as in Maier and Brockmann (2020) for the confirmed cases of a country, on a daily basis, as given by the Johns Hopkins University. We then use different types of networks (random Erdos-Rényi, Small World, and Barabási-Albert Scale-Free) with various characteristics in an effort to find the best fit with the real data for the same geographical regions as reported in Maier and Brockmann (2020). Our simulations show that the best fit comes with the Erdos-Rényi random networks. We then apply this method to several other countries, both for large-size countries, and small size ones. In all cases investigated we find the same result, i.e. best agreement for the evolution of the pandemic with time is for the Erdos-Rényi networks. Furthermore, our results indicate that the best fit occurs for a random network with an average degree of the order of 〈 k 〉 ≈ 10-25, for all countries tested. Scale Free and Small World networks fail to fit the real data convincingly.

An electrostatics method for converting a time-series into a weighted complex network.

This paper proposes a new method for converting a time-series into a weighted graph (complex network), which builds on electrostatics in physics. The proposed method conceptualizes a time-series as a series of stationary, electrically charged particles, on which Coulomb-like forces can be computed. This allows generating electrostatic-like graphs associated with time-series that, additionally to the existing transformations, can be also weighted and sometimes disconnected. Within this context, this paper examines the structural similarity between five different types of time-series and their associated graphs that are generated by the proposed algorithm and the visibility graph, which is currently the most popular algorithm in the literature. The analysis compares the source (original) time-series with the node-series generated by network measures (that are arranged into the node-ordering of the source time-series), in terms of a linear trend, chaotic behaviour, stationarity, periodicity, and cyclical structure. It is shown that the proposed electrostatic graph algorithm generates graphs with node-measures that are more representative of the structure of the source time-series than the visibility graph. This makes the proposed algorithm more natural rather than algebraic, in comparison with existing physics-defined methods. The overall approach also suggests a methodological framework for evaluating the structural relevance between the source time-series and their associated graphs produced by any possible transformation.

Spontaneous repulsion in the A+B→0 reaction on coupled networks.

We study the transient dynamics of an A+B→0 process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions q of cross couplings, the concentration of A (or B) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time t_{x}. By numerical and analytical arguments, we show that for symmetric and homogeneous structures t_{x}∝(〈k〉/q)log(〈k〉/q) where 〈k〉 is the mean degree of both networks. Being this behavior is in marked contrast with a purely diffusive process, where the mixing time would go simply like 〈k〉/q, we identify the logarithmic slowing down in t_{x} to be the result of a spontaneous mechanism of repulsion between the reactants A and B due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.

Matrix Density Engineering of Hydrogel Nanoparticles with Simulation-Guided Synthesis for Tuning Drug Release and Cellular Uptake.

The use of a nanoparticle (NP)-based antitumor drug carrier has been an emerging strategy for selectively delivering the drugs to the tumor area and, thus, reducing the side effects that are associated with a high systemic dose of antitumor drugs. Precise control of drug loading and release is critical so as to maximize the therapeutic index of the NPs. Here, we propose a simple method of synthesizing NPs with tunable drug release while maintaining their loading ability, by varying the polymer matrix density of amine- or carboxyl-functionalized hydrogel NPs. We find that the NPs with a loose matrix released more cisplatin, with up to a 33 times faster rate. Also, carboxyl-functionalized NPs loaded more cisplatin and released it at a faster rate than amine-functionalized NPs. We performed detailed Monte Carlo computer simulations that elucidate the relation between the matrix density and drug release kinetics. We found good agreement between the simulation model and the experimental results for drug release as a function of time. Also, we compared the cellular uptake between amine-functionalized NPs and carboxyl-functionalized NPs, as a higher cellular uptake of NPs leads to improved cisplatin delivery. The amine-functionalized NPs can deliver 3.5 times more cisplatin into cells than the carboxyl-functionalized NPs. The cytotoxic efficacy of both the amine-functionalized NPs and the carboxyl-functionalized NPs showed a strong correlation with the cisplatin release profile, and the latter showed a strong correlation with the NP matrix density.

Efficiency of message transmission using biased random walks in complex networks in the presence of traps.

We study the problem of a particle or message that travels as a biased random walk towards a target node in a network in the presence of traps. The bias is represented as the probability p of the particle to travel along the shortest path to the target node. The efficiency of the transmission process is expressed through the fraction f(g) of particles that succeed to reach the target without being trapped. By relating f(g) with the number S of nodes visited before reaching the target, we first show that, for the unbiased random walk, f(g) is inversely proportional to both the concentration c of traps and the size N of the network. For the case of biased walks, a simple approximation of S provides an analytical solution that describes well the behavior of f(g), especially for p>0.5. Also, it is shown that for a given value of the bias p, when the concentration of traps is less than a threshold value equal to the inverse of the mean first passage time (MFPT) between two randomly chosen nodes of the network, the efficiency of transmission is unaffected by the presence of traps and almost all the particles arrive at the target. As a consequence, for a given concentration of traps, we can estimate the minimum bias that is needed to have unaffected transmission, especially in the case of random regular (RR), Erdos-Rényi (ER) and scale-free (SF) networks, where an exact expression (RR and ER) or an upper bound (SF) of the MFPT is known analytically. We also study analytically and numerically, the fraction f(g) of particles that reach the target on SF networks, where a single trap is placed on the highest degree node. For the unbiased random walk, we find that f(g)∼N(-1/(γ-1)), where γ is the power law exponent of the SF network.

Self-intermediate scattering function of strongly interacting three-dimensional lattice gases: time- and wave-vector-dependent tracer diffusion coefficient.

We investigate the self-intermediate scattering function (SISF) in a three-dimensional (3D) cubic lattice fluid (interacting lattice gas) with attractive nearest-neighbor interparticle interactions at a temperature slightly above the critical one by means of Monte Carlo simulations. A special representation of SISF as an exponent of the mean tracer diffusion coefficient multiplied by the geometrical factor and time is considered to highlight memory effects that are included in time and wave-vector dependence of the diffusion coefficient. An analytical expression for the diffusion coefficient is suggested to reproduce the simulation data. It is shown that the particles' mean-square displacement is equal to the time integral of the diffusion coefficient. We make a comparison with the previously considered 2D system on a square lattice. The main difference with the two-dimensional case is that the time dependence of particular characteristics of the tracer diffusion coefficient in the 3D case cannot be described by exponentially decreasing functions, but requires using stretched exponentials with rather small values of exponents, of the order of 0.2. The hydrodynamic values of the tracer diffusion coefficient (in the limit of large times and small wave vectors) defined through SIFS simulation results agree well with the results of its direct determination by the mean-square displacement of the particles in the entire range of concentrations and temperatures.

Reaction efficiency effects on binary chemical reactions.

We study the effect of the variation of reaction efficiency in binary reactions. We use the well-known A + B → 0 model, which has been extensively studied in the past. We perform simulations on this model where we vary the efficiency of reaction, i.e., when two particles meet they do not instantly react, as has been assumed in previous studies, but they react with a probability γ, where γ is in the range 0 < γ < 1. Our results show that at small γ values the system is reaction limited, but as γ increases it crosses over to a diffusion limited behavior. At early times, for small γ values, the particle density falls slower than for larger γ values. This fall-off goes over a crossover point, around the value of γ = 0.50 for high initial densities. Under a variety of conditions simulated, we find that the crossover point was dependent on the initial concentration but not on the lattice size. For intermediate and long times simulations, all γ values (in the depleted reciprocal density versus time plot) converge to the same behavior. These theoretical results are useful in models of epidemic reactions and epidemic spreading, where a contagion from one neighbor to the next is not always successful but proceeds with a certain probability, an analogous effect with the reaction probability examined in the current work.

Fractal dimension of microbead assemblies used for protein detection.

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 µm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles.

Random walk with priorities in communicationlike networks.

We study a model for a random walk of two classes of particles (A and B). Where both species are present in the same site, the motion of A's takes precedence over that of B's. The model was originally proposed and analyzed in Maragakis et al. [Phys. Rev. E 77, 020103(R) (2008)]; here we provide additional results. We solve analytically the diffusion coefficients of the two species in lattices for a number of protocols. In networks, we find that the probability of a B particle to be free decreases exponentially with the node degree. In scale-free networks, this leads to localization of the B's at the hubs and arrest of their motion. To remedy this, we investigate several strategies to avoid trapping of the B's, including moving an A instead of the hindered B, allowing a trapped B to hop with a small probability, biased walk toward non-hub nodes, and limiting the capacity of nodes. We obtain analytic results for lattices and networks, and we discuss the advantages and shortcomings of the possible strategies.

Generalized Achlioptas process for the delay of criticality in the percolation process.

We extend the Achlioptas model for the delay of criticality in the percolation problem. Instead of having a completely random connectivity pattern, we generalize the idea of the two-site probe in the Achlioptas model for connecting smaller clusters, by introducing two models: the first one by allowing any number k of probe sites to be investigated, k being a parameter, and the second one independent of any specific number of probe sites, but with a probabilistic character which depends on the size of the resulting clusters. We find numerically the complete spectrum of critical points and our results indicate that the value of the critical point behaves linearly with k after the value of k = 3. The range k = 2-3 is not linear but parabolic. The more general model of generating clusters with probability inversely proportional to the size of the resulting cluster produces a critical point which is equivalent to the value of k being in the range k = 5-7.

Anomalous biased diffusion in networks.

We study diffusion with a bias toward a target node in networks. This problem is relevant to efficient routing strategies in emerging communication networks like optical networks. Bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node. We investigate the scaling of the mean first passage time (MFPT) with the size of the network. We find by using theoretical analysis and computer simulations that for random regular (RR) and Erdos-Rényi networks, there exists a threshold probability, p(th), such that for pp(th), the MFPT scales logarithmically with N. The threshold value p(th) of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes. An exact solution for every value of p is given for the scaling of the MFPT in RR networks. The regime transition is also observed for the second moment of the probability distribution function, the standard deviation. For the case of scale-free (SF) networks, we present analytical bounds and simulations results showing that the MFPT scales at most as lnN to a positive power for any finite bias, which means that in SF networks even a very small bias is considerably more efficient in comparison to unbiased walk.

Experimental system for one-dimensional rotational brownian motion.

We present here an experimental, strictly one-dimensional rotational system, made by using single magnetic Janus particles in a static magnetic field. These particles were half-coated with a thin metallic film, and by turning on a properly oriented external static magnetic field, we monitor the rotational brownian motion of single particles, in solution, around the desired axis. Bright-field microscopy imaging provides information on the particle orientation as a function of time. Rotational diffusion coefficients are derived for one-dimensional rotational diffusion, both for a single rotating particle and for a cluster of four such particles. Over the studied time domain, up to 10 s, the variation of the angle of rotation is strictly brownian; its probability distribution function is gaussian, and the mean squared angular displacement is linear in time, as expected for free diffusion. Values for the rotational diffusion coefficients were also determined. Monte Carlo and hydrodynamic simulations agree well with the experimental results.

Explosive site percolation and finite-size hysteresis.

We report the critical point for site percolation for the "explosive" type for two-dimensional square lattices using Monte Carlo simulations and compare it to the classical well-known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the explosive site percolation threshold as p(c) = 0.695 and we find evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms. Finally, we study and compare the direct and reverse processes, showing that while the reverse process is different from the direct process for finite size systems, the two cases become equivalent in the thermodynamic limit of large L.

Spreading of infection in a two species reaction-diffusion process in networks.

We study the dynamics of the infection of a two mobile species reaction from a single infected agent in a population of healthy agents. Historically, the main focus for infection propagation has been through spreading phenomena, where a random location of the system is initially infected and then propagates by successfully infecting its neighbor sites. Here both the infected and healthy agents are mobile, performing classical random walks. This may be a more realistic picture to such epidemiological models, such as the spread of a virus in communication networks of routers, where data travel in packets, the communication time of stations in ad hoc mobile networks, information spreading (such as rumor spreading) in social networks, etc. We monitor the density of healthy particles ρ(t), which we find in all cases to be an exponential function in the long-time limit in two-dimensional and three-dimensional lattices and Erdos-Rényi (ER) and scale-free (SF) networks. We also investigate the scaling of the crossover time t(c) from short- to long-time exponential behavior, which we find to be a power law in lattices and ER networks. This crossover is shown to be absent in SF networks, where we reveal the role of the connectivity of the network in the infection process. We compare this behavior to ER networks and lattices and highlight the significance of various connectivity patterns, as well as the important differences of this process in the various underlying geometries, revealing a more complex behavior of ρ(t).

Absence of depletion zone effects for the trapping reaction in complex networks.

In the present work we examine in detail the formation of a depletion zone in the trapping reaction in networks, with a single perfect trap. We monitor the particle density rho(r) with respect to the distance r from the trap. We show using Monte Carlo simulations that the depletion zone is absent in regular, Erdos-Renyi (ER), and scale-free (SF) networks. The density profiles show significant differences for these cases. The particles are homogeneously distributed in regular and ER networks with the depletion effect appearing in very sparse ER networks. In SF networks we reveal the important role of the hubs, which due to their high random walk centrality are critical in the trapping reaction. In addition, the degree distribution plays a significant role in the distribution of the particles recovering the depletion zone formation for high gamma values. The mean connectivity of the network is found to play a significant role in both ER and SF networks.

Priority diffusion model in lattices and complex networks.

We introduce a model for diffusion of two classes of particles (A and B ) with priority: where both species are present in the same site the motion of A's takes precedence over that of B's. This describes realistic situations in wireless and communication networks. In regular lattices the diffusion of the two species is normal, but the B particles are significantly slower due to the presence of the A particles. From the fraction of sites where the B particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where B's are free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes, leads to trapping of the low priority particles in scale-free networks.

Improving immunization strategies.

We introduce an immunization method where the percentage of required vaccinations for immunity are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our strategy retains the advantage of being purely local, without the need for knowledge on the global network structure or identification of the highest degree nodes. The method consists of selecting a random node and asking for a neighbor that has more links than himself or more than a given threshold and immunizing him. We compare this method to other efficient strategies on three real social networks and on a scale-free network model and find it to be significantly more effective.

Effect of a slit-shaped trap on depletion kinetics within a microchannel.

The diffusion-limited trapping reaction kinetics of the growth of the depletion zone within and around a "slit-shaped" trap in a flat microchannel was studied experimentally and numerically. In the experiment, an ellipse-shaped laser beam acted as a slit trap in a long, flat capillary, and the trapping reaction is photobleaching of fluorescein dye. The parameter studied was the theta distance, i.e., the distance from the trap to the point where the reactant concentration has been locally depleted to the specific survival fraction [theta] of its initial bulk value. When the trap is perfect, then, due to the geometry of the trap and the reactor, as many as three time regimes can be found, with up to two crossover transitions. The number of crossovers is determined by the relative sizes of the trap and the microreactor. In the case of two crossovers, we show that the first crossover relates to the length of the trap, while the second crossover relates to the width of the reactor. When the slit trap is imperfect and its width cannot be neglected, as is the case in the experiments, a nontrivial early behavior is observed, followed by two regions in time, separated by a single crossover only.