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Without loss of generality, we assume that the first edge with velocity [MATH] secretes cancer biomarkers at location [MATH] into the medium, and is joined with the [MATH] other edges at [MATH] to yield a larger edge with flow velocity [MATH] . We are particularly interested in finding the concentration of biomarkers i... |
[EQUATION] where steady-state operation was assumed, i.e., [MATH] Since [MATH] [MATH] , and [MATH] are all positive, the solution of ( 15 ) will be a combination of exponential functions of the form [MATH] where the coefficients [MATH] must be roots of following algebraic equation |
[EQUATION] which has two possible solutions. Furthermore, based on the fundamental laws of diffusion and conservation of mass, the following two conditions have to be satisfied: i) the concentration of the molecules must be continuous, ii) the flux must be continuous and can be obtained as |
[EQUATION] In the following, we analytically derive the concentration of the biomarkers and provide a suitable approximation for simpler evaluation. |
Exact Solution: Based on the above discussion, the steady-state concentration of the biomarkers can be shown to have the following general form |
[EQUATION] where [EQUATION] Here, [MATH] [MATH] [MATH] , and [MATH] are unknown parameters that have to be determined. Combining ( 15 )–( 22 ), we arrive at the linear equation [MATH] with |
[EQUATION] The unique solution for [MATH] can be obtained as [MATH] , where [MATH] is the inverse of matrix [MATH] Approximate Solution: |
Due to the possibly large values for [MATH] and [MATH] , matrix [MATH] may be ill-conditioned, which makes the determination of unknown parameter vector [MATH] numerically challenging. Hence, in the following, we propose a simple approximate solution to ( 15 ). In particular, we consider the following approximations: |
I: Suppose that the flow in edge [MATH] with velocity [MATH] is not influenced by the other connected blood vessels at [MATH] . Then, we use the solution for a local straight edge in ( 11 ) for [MATH] |
II: Consider the concentration at [MATH] as a virtual source for the blood vessel with outgoing flow with velocity [MATH] . Then, we employ ( 11 ) for [MATH] and flow velocity [MATH] |
Considering these approximations, the approximated solution can be obtained as follows [EQUATION] Remark 3 Similar to Remark , for typical parameter values for biomarkers and blood vessels, we can assume that [MATH] and [MATH] . Hence, ( 27 ) can be simplified as |
[EQUATION] III-C Bifurcation Node For a bifurcation node, we assume that one edge is divided into [MATH] edges. Furthermore, let us assume that the incoming flow velocity is [MATH] and the outgoing flow velocity for the [MATH] -th edge ( [MATH] ) is [MATH] , see Fig. b). The derivation of an expression for the velocity... |
. Therefore, similar to Section III-C , we assume that the flow velocities of the edges are given. Then, using the principle of mass conservation |
, we can derive the probability that a biomarker secreted at [MATH] in the incoming edge, enters the [MATH] -th outgoing edge at [MATH] as follows |
[EQUATION] where [MATH] is the cross-sectional area of the [MATH] -th outgoing edge. By using a similar approach as in Section III-C , we arrive at the following analytical expression for the concentration of the biomarkers at the bifurcation node |
[EQUATION] where [MATH] and [MATH] are the positive and negative roots of [MATH] , respectively, and [MATH] is the negative root of [MATH] Similar to the approach proposed in Section III-C , by considering the initial value and boundary conditions at [MATH] and [MATH] , we can construct a linear system of equations to ... |
Based on the results obtained in this section, for any given network, inside each blood vessel of the network, we can derive the concentrations of the biomarkers, [MATH] , secreted by cancer cells. That is, by plugging the 1-D expressions into ( 10 ) we obtain the 2-D concentration of the biomarkers. |
III-D PMF of the Number of Biomarkers In the following, we derive the spatial PMF of the number of biomarkers inside a small virtual observation area [MATH] representing one MNS. In the next section, we use this PMF to derive the activation level of the MNSs. With [MATH] derived in Sections III-B and III-C and assuming... |
Let us denote the PMF for having at time [MATH] [MATH] biomarkers out of the [MATH] biomarkers released in total, which undergo advection and diffusion and may degrade until time [MATH] , inside the observation area centered at [MATH] by [MATH] To derive [MATH] , suppose that the release time for the [MATH] -th biomark... |
[EQUATION] where [MATH] is the set of all subsets of [MATH] integers [MATH] , with [MATH] , where [MATH] denotes the cardinality of a set . |
The Poisson Binomial distribution is cumbersome to work with, but can often be approximated by a Poisson distribution when the number of trials is high and the success probability is small, cf. |
To this end and for simplicity, we approximate ( 37 ) by a Poisson PMF with mean [MATH] For our system model, this approximation is accurate when the number of released biomarkers is high; and for each biomarker, the probability of being observed inside the MNS is small. In Section VI , we validate the accuracy of this... |
[EQUATION] where [MATH] denotes a Poisson distribution with mean [MATH] . Here, we have denoted the number of biomarkers, secreted by healthy cells, inside the observation area by a Poisson RV with mean [MATH] |
IV MNS Activation Level Statistic In the previous section, we derived the concentration of the biomarkers inside a network of blood vessels. Using this result, we then derived the PMF of the number of biomarkers inside a small area in the network. In this section, we use the results of Section III , to determine the PM... |
To derive the PMF of the activation level of the MNSs, we have to consider all possible routes inside the network between the injection site and the FC as well as the distribution of the biomarkers inside each edge of the network. To formulate this rigorously, we assume that the network between the injection site and t... |
Now, based on the definition of the activation level [MATH] given in Section II-C , we can express the PMF for the activation level of the [MATH] -th MNS as |
[EQUATION] To evaluate ( IV ), first we need to determine the PMF of [MATH] . However, since the position of the MNS in [MATH] -direction is generally not deterministic due to diffusion, to derive the PMF of [MATH] , we employ a simplification. In particular, due to the small value of the diffusion coefficient, [MATH] ... |
, advection is dominant over diffusion regarding the movement of the MNSs along the [MATH] -axis. For instance, for capillaries, normally [MATH] cm [MATH] -1 [MATH] cm, and [MATH] cm [MATH] -1 holds |
In this case, the distance that an MNS can travel by advection during a test time interval of [MATH] s, is [MATH] cm which is much larger than the standard deviation of the distance that the MNS typically moves by diffusion during the same time interval, [MATH] |
[MATH] cm Hence, the locations where measurement are taken along the [MATH] -axis are practically deterministic. Thus, we define the set of deterministic measurement positions for an MNS during its journey inside the [MATH] -th edge via the [MATH] -th route by [MATH] , where [MATH] is the flow velocity in the [MATH] -t... |
[EQUATION] Note that in ( 40 ) the contributions of both cancerous and healthy cells are included. For ease of presentation, we define [MATH] and [MATH] as the mean values of the conditional RV [MATH] under hypotheses [MATH] and [MATH] , respectively, which can be written as |
[EQUATION] Therefore, the activation level given in ( IV ) can be represented for hypotheses [MATH] and [MATH] as follows [EQUATION] |
Detector Design at FC Based on the results derived in Section IV , in this section, we derive the optimal decision rule at the FC with respect to the Neyman-Pearson criterion |
, which is referred to as the likelihood ratio test (LRT) and relies on the knowledge of the network topology between the injection site and the FC. However, in practice, we do not know the biomarker secretion rate and the location of the cancer cell. Therefore, we also propose a simple suboptimal detector that does no... |
Let [MATH] denote an RV vector modeling the vector containing the activation levels of all [MATH] MNSs observed during the observation window, i.e., [MATH] , and let [MATH] be a realization of [MATH] According to the Neyman-Pearson criterion, the goal is to design the optimal detector that minimizes the missed detectio... |
[EQUATION] where [MATH] and [MATH] are the false alarm and missed detection probabilities, respectively. The solution to ( 47 ) is the well-known LRT |
that compares the log-likelihood ratio (LLR) with a threshold denoted by [MATH] , where [MATH] is chosen such that [MATH] In particular, the decision rule for the LRT can be characterized as |
[EQUATION] where [MATH] means that the detector selects hypothesis [MATH] . In ( 50 ), [MATH] is given by [EQUATION] In the following, we derive [MATH] for the problem at hand. Since the movements of different MNSs are independent, the activation levels of different MNSs are statistically independent, which results in |
[EQUATION] Therefore, based on ( 51 ) and ( 52 ), we can express [MATH] as [EQUATION] where using ( 46 ) we obtain [EQUATION] for the LLR of the observation of the [MATH] -th MNS. |
Since in reality the network topology, and hence the values of [MATH] and [MATH] are not known at the FC, we propose a simple alternative but suboptimal detector that does not need knowledge of the network topology. The decision rule of this detector, which we refer to as sum detector , is as follows |
[EQUATION] where [MATH] is chosen such that the decision rule in ( 57 ) yields [MATH] In Section VI-B , we evaluate the performance of the proposed detectors in terms of [MATH] and [MATH] |
VI Simulation Results In this section, we first validate the system model and the assumptions made for the analysis presented in this paper via particle-based simulation. Then, we provide Monte Carlo simulation results to assess the performance of the proposed detection schemes. |
VI-A Model and Assumption Verification We first consider a single blood vessel which is sufficient to verify most of our assumptions. Subsequently, we introduce an example network for a more thorough analysis. |
VI-A Single Blood Vessel To verify our analysis, we perform a 2-D particle-based simulation of both the MNSs and the biomarkers secreted by the cancer cells. Table summarizes the system parameters that are used for all simulations, unless stated otherwise. |
Simulation results are averaged over [MATH] independent realizations of the biomarker release, where for each realization, we assume that the biomarkers are released into the blood vessel with rate [MATH] -1 . This rate is calculated via ( ) for [MATH] months, i.e., the early stage of a cancer. |
Figs. a) and b) depict the average total number of biomarkers inside the blood vessel over time, along with their distribution with respect to [MATH] , where the biomarkers are released into the medium at time [MATH] and position [MATH] . From Fig. a) we observe that about [MATH] hours (i.e., [MATH] s) after the start ... |
Fig. shows the histograms for the numbers of biomarkers inside rectangles of area [MATH] cm [MATH] cm [MATH] cm centered at four different sample positions inside the blood vessel, along with their Poisson PMF approximations according to ( 38 ). As can be seen, the histogram for the number of biomarkers inside the rect... |
To investigate the effect of the blood vessel height on the distribution of the biomarkers, in Figs. a), b), c), and d), we plot the biomarker concentration versus [MATH] for different values of [MATH] and [MATH] . As can be observed, the distribution of the mean number of biomarkers inside the blood vessel in the vici... |
Fig. depicts the trajectories of four independent MNSs entering a blood vessel of length [MATH] m at random [MATH] positions. In this figure, we investigate the effect of MNS diffusion on the mean value of the activation levels of the MNSs during their passage of the blood vessel. We adopt time intervals of length [MAT... |
VI-A Network of Blood Vessels In the following, we study the methods proposed in Section III-C for evaluating the biomarker concentration in a network of blood vessels. In Fig. , we show a sample network of blood vessels which is a simplified version of the network given in Fig. , where all blood vessels have identical... |
. Hence, the distance between the injection site and the FC could be from tens of centimeters to a few meters. In addition, the number of blood vessels inside the network between an injection point and an FC could be on the order of thousands. However, to gain insight into the impact of the different parameters for sys... |
. We also assume equal heights of [MATH] m for all edges. The biomarkers are released at edge E [MATH] , at a distance of [MATH] cm from node N [MATH] , as depicted in Fig. In addition, we consider typical flow velocities for all edges. In particular, we assume that the flow velocity in edge E [MATH] is [MATH] cm [MATH... |
In Fig. , the concentration of the biomarkers, secreted by cancer cells, inside the sample network, [MATH] , is plotted versus the local [MATH] -axis of the edges, where we have averaged over [MATH] independent realizations of the particle-based simulations. We have plotted the concentrations of the biomarkers in edges... |
In addition, we observe that at junction nodes at which the velocity of the outgoing flow is higher than that of the incoming flow, for instance at node N [MATH] , two phenomena occur. First, at node N [MATH] , the concentration at the beginning of the outgoing edge E [MATH] is lower than that at the end of edge E [MAT... |
VI-B Performance Evaluation In the following, we evaluate the performances of the proposed LRT and sum detectors and compare them with a benchmark scheme where the nanosensors are not mobile and are fixed at the FC. In particular, the benchmark scheme, which is similar to a conventional blood test, measures the biomark... |
Furthermore, we assume a large enough time interval of [MATH] s between consecutive measurements of the MNSs to have independent measurements on each edge, cf. |
. Hence, the number of measurements taken on the edges having flow velocities of [MATH] cm [MATH] -1 [MATH] cm [MATH] -1 [MATH] cm [MATH] -1 , and [MATH] cm [MATH] -1 is [MATH] [MATH] [MATH] , and [MATH] , respectively. Besides, to have a fair comparison between the proposed scheme and the benchmark scheme, we employ t... |
In Fig. , we compare the performance of the proposed MNS-based approach with that of the benchmark scheme. In particular, in this figure, we plot the probability of missed detection versus the biomarker secretion rate for a given probability of false alarm of [MATH] [MATH] , and [MATH] by employing Monte Carlo simulati... |
, we expect that using the proposed MNS-based approach significantly improves the probability of detecting cancer compared to the blood sample tests which are widely used now and are similar to our benchmark scheme. |
Finally, in Fig. 10 , we investigate the impact of the number of the MNSs observed at the FC on the performance of the proposed sum detector. We show the probability of missed detection versus the probability of false alarm based on Monte Carlo simulation of [MATH] realizations. The number of used MNSs is an important ... |
VII Conclusions In this paper, we studied anomaly detection inside blood vessels where multiple MNSs are injected and pass through the blood vessels. The MNSs sense the presence of an anomaly by detecting the biomarkers secreted by cancer cells. The final decision regarding the presence of anomaly is made at an FC base... |
# Source: arxiv 1805.09554 # Title: Hierarchical burst model for complex bursty dynamics # Sections: all # Downloaded: 2026-03-03T05:15:52.283579+00:00 |
\useunder \ul Hierarchical burst model for complex bursty dynamics Abstract Temporal inhomogeneities observed in various natural and social phenomena have often been characterized in terms of scaling behaviors in the autocorrelation function with a decaying exponent [MATH] , the interevent time distribution with a powe... |
Introduction Events in temporal patterns of natural and social phenomena have often been found to be inhomogeneously distributed in time. Examples include solar flares Wheatland et al. 1998 , earthquakes Corral ( 2004 ); de Arcangelis et al. 2006 , neuronal firing Kemuriyama et al. 2010 , and human social activities Ba... |
In order to characterize the temporal inhomogeneities in event sequences, we first denote the event sequence by [MATH] that has a value of [MATH] at the moment of event occurred, [MATH] otherwise. Then one can measure an autocorrelation function with delay time [MATH] as |
[EQUATION] where [MATH] is a time average. For event sequences with long-range memory effects, the autocorrelation function often shows a power-law decaying behavior as |
[EQUATION] with a decaying exponent [MATH] . In general, temporal correlations characterized by [MATH] can be understood in terms of (i) interevent times and (ii) correlations between interevent times Jo ( 2017 . Here the interevent time is defined as a time interval between two consecutive events, denoted by [MATH] . ... |
In many empirical datasets showing temporal inhomogeneities, the interevent time distribution [MATH] has been characterized by a power-law function as |
[EQUATION] with [MATH] denoting the power-law exponent Karsai et al. 2018 . It has been proved that when interevent times are fully uncorrelated with each other, the power-law exponent [MATH] of the interevent time distribution is related to the decaying exponent [MATH] of the autocorrelation function such that [MATH] ... |
Lowen and Teich ( 1993 ); Vajna et al. 2013 . On the other hand, the correlations between interevent times have been studied in terms of bursty trains Karsai et al. 2012a . A bursty train or burst is defined as a set of events such that interevent times between any two consecutive events in the same burst are less than... |
[EQUATION] with [MATH] denoting the power-law exponent. Here one can ask a question about how strong correlations between interevent times should be present to violate the scaling relation [MATH] derived for the uncorrelated case. Our understanding on this issue is far from complete, except for few recent works Rybski ... |
Along with various characterization methods for the bursty temporal patterns, a number of modeling approaches have been suggested to understand the underlying mechanisms behind such temporal inhomogeneities Karsai et al. 2018 . In the case with human dynamics, we find several modeling approaches, such as priority queui... |
Our paper is organized as follows: In Sec. II , after introducing the hierarchical burst model, we study our model analytically and numerically in terms of the scaling behaviors of the fractal temporal structure, the autocorrelation function, and the interevent time distribution. Then we numerically obtain the stretche... |
II Model and results II.1 Model definition We introduce the hierarchical burst model by assuming that each observed event in an event sequence might be a consequence of the multi-level causal or decision-making process: A seed event at the zeroth level induces other events at the first level, each of which in turn lead... |
We focus on the case with one seed event, enabling us to set [MATH] without loss of generality. Then the case with multiple seed events will be briefly discussed in Subsec. II.5 . Since [MATH] and [MATH] , the induction interval decreases exponentially as a function of the level index [MATH] , while the number of event... |
II.2 Temporal scaling behaviors Once an event sequence of [MATH] events is For calculating the fractal dimension [MATH] of the event sequence, the box-counting method is used: We count the number of boxes of size [MATH] needed to cover all events, which is denoted by [MATH] . If [MATH] decays as a power law according t... |
[EQUATION] When the box size is given as [MATH] for [MATH] , we get [MATH] , leading to [EQUATION] Here we have assumed that the boxes covering events or induction intervals at the same level do not necessarily overlap, or that even when they overlap, its effect would be negligible in estimating the fractal dimension. ... |
As evident in Eq. ( ), the autocorrelation function [MATH] with delay time [MATH] essentially measures the possibility of finding two events observed in [MATH] and [MATH] , no matter how many events occur between them. The number of events within the range of [MATH] from any event is of the order of [MATH] using Eq. ( ... |
[EQUATION] Next, we derive the interevent time distribution [MATH] . Let us consider [MATH] events at the [MATH] th level. Among them, events induced by the same event in [MATH] at the [MATH] th level will be found in the range of [MATH] . Thus, the interevent times between events induced by the same [MATH] th-level ev... |
[EQUATION] By using [MATH] and the relation [MATH] in Eq. ( ), one gets [MATH] , leading to [EQUATION] Since [MATH] in our model, the value of [MATH] is limited to the range of [MATH] |
Finally, combining the results in Eq. ( ) and Eq. ( ), we obtain the scaling relation between [MATH] and [MATH] as follows: [EQUATION] |
which turns out to hold irrespective of [MATH] , i.e., irrespective of [MATH] and [MATH] . This scaling relation has been derived for the case that interevent times are fully uncorrelated with each other Lowen and Teich ( 1993 ); Vajna et al. 2013 . Hence, this result in Eq. ( 10 ) may indicate that the correlations be... |
For the numerical simulations of our model, we begin with one seed event in [MATH] at the zeroth level. Then [MATH] events are uniformly distributed in the range of [MATH] at the first level. Each of these induced events in turn induces [MATH] events at the second level in the range of [MATH] , with [MATH] denoting the... |
We analyze the generated event sequences for various values of [MATH] . For example, Fig. (a–c) shows the numerical results of [MATH] [MATH] , and [MATH] , all averaged over [MATH] event sequences using [MATH] . We find that the estimated values of corresponding power-law exponents, i.e., [MATH] [MATH] , and [MATH] , a... |
II.3 Burst size distributions In order to scrutinize the existence of correlations between interevent times, we measure the burst size distributions [MATH] for various values of the time window [MATH] . For example, the numerical results for [MATH] are shown in Fig. (a). The curves of [MATH] for a wide range of [MATH] ... |
[EQUATION] with [MATH] and [MATH] denoting a proper coefficient depending on [MATH] . The fact that [MATH] deviates from the exponential function indicates the existence of correlations between interevent times. At the same time, such correlations depicted in terms of stretched exponential functions might not be strong... |
In order to understand why burst size distributions observed in our model are better described by a stretched exponential function rather than a power-law function, we study how likely it is to cluster events induced by the different events to the same burst for a given time window. The more likely such case happens, t... |
For the analysis we consider the minimal case with [MATH] . When the time window is given as [MATH] , we only need to consider the events at the [MATH] th, [MATH] th, and [MATH] th levels, as depicted in Fig. (b). The timescales at other levels are either too large or too small to be relevant to the analysis. We denote... |
[EQUATION] These two events at the [MATH] th level induce four events at the [MATH] th level, whose timings are respectively [MATH] [MATH] [MATH] , and [MATH] , satisfying |
[EQUATION] That is, the events in [MATH] and [MATH] are induced by the event in [MATH] , while the events in [MATH] and [MATH] are induced by the event in [MATH] . By assuming that [MATH] , we have three interevent times between events at the [MATH] th level: |
[EQUATION] see Fig. (b). Using the order statistics David and Nagaraja ( 2003 ); Kivelä et al. 2012 ); Kim and Jo ( 2016 , we get the distribution of [MATH] as |
[EQUATION] for [MATH] , which is the same as [MATH] . Then the probability of clustering two events induced by the same [MATH] th-level event for a given [MATH] is calculated as |
[EQUATION] Next, in order to derive the distribution of [MATH] , we rewrite [MATH] in Eq. ( 16 ) as [EQUATION] where [EQUATION] see Fig. (b). Here [MATH] is indeed the interevent time between events at the [MATH] th level, leading to its distribution as |
[EQUATION] for [MATH] . We also get the distribution of [MATH] as [EQUATION] for [MATH] , which is the same as [MATH] . We now calculate [MATH] , i.e., |
[EQUATION] As [MATH] s for [MATH] are statistically independent of each other, using [MATH] we rewrite the above equation as [EQUATION] |
where [MATH] is the Heaviside step function. Taking the Laplace transform, one gets [EQUATION] By plugging the Laplace transforms of [MATH] in Eqs. ( 24 ) and ( 25 ) into the above equation, and then taking the inverse Laplace transform of [MATH] , one can get [MATH] . We finally obtain |
[EQUATION] for the entire range of [MATH] . By comparing Eq. ( 19 ) with Eq. ( II.3 ), we conclude that for [MATH] [EQUATION] as numerically shown in Fig. (c). This inequality holds for any level index [MATH] , implying that the chance of clustering events induced by the different events at the previous level must be l... |
II.4 Effect of non-overlapping induction intervals Our model allows induction intervals at the same level to overlap with each other, although its effects turn out to be irrelevant to the scaling relations between [MATH] [MATH] , and [MATH] , as discussed in Subsec. II.2 . Let us consider two events at the [MATH] th le... |
[EQUATION] By the first condition in Eq. ( 31 ) any descendent events of the event in [MATH] are forced to remain in the range of [MATH] . The second condition in Eq. ( 32 ) prohibits the induction intervals at the same level from overlapping with each other. |
By performing numerical simulations, we find that the fractal dimension, the interevent time distribution, and the burst size distributions show overall the same behaviors as in the original version of our model (not shown). However, the autocorrelation function shows a qualitatively different behavior, as shown in Fig... |
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