In-Silico Modeling of Tumor Spheroid Formation and Growth.

Mathematical modeling has significant potential for understanding of biological models of cancer and to accelerate the progress in cross-disciplinary approaches of cancer treatment. In mathematical biology, solid tumor spheroids are often studied as preliminary in vitro models of avascular tumors. The size of spheroids and their cell number are easy to track, making them a simple in vitro model to investigate tumor behavior, quantitatively. The growth of solid tumors is comprised of three main stages: transient formation, monotonic growth and a plateau phase. The last two stages are extensively studied. However, the initial transient formation phase is typically missing from the literature. This stage is important in the early dynamics of growth, formation of clonal sub-populations, etc. In the current work, this transient formation is modeled by a reaction-diffusion partial differential equation (PDE) for cell concentration, coupled with an ordinary differential equation (ODE) for the spheroid radius. Analytical and numerical solutions of the coupled equations were obtained for the change in the radius of tumor spheroids over time. Human glioblastoma (hGB) cancer cells (U251 and U87) were spheroid cultured to validate the model prediction. Results of this study provide insight into the mechanism of development of solid tumors at their early stage of formation.

Hybrid models of genetic networks: Mathematical challenges and biological relevance.

We review results concerning dynamics in a class of hybrid ordinary differential equations which incorporates logical control to yield piecewise linear equations. These equations relate qualitative features of the structure of networks to qualitative properties of the dynamics. Because of their simple structure, they have been studied using techniques from discrete mathematics and nonlinear dynamics. Initially developed as a qualitataive description of gene regulatory networks, many generalizations of the basic approach have been developed. In particular, we show how this qualitative approach may be adapted to switching biochemical systems without degradation, illustrated by an example of a motif in which two branches of a pathway may be regulated differently when the thresholds for the two pathways are separated.

Modeling the behavior of human induced pluripotent stem cells seeded on melt electrospun scaffolds.

BACKGROUND: Human induced pluripotent stem cells (hiPSCs) can form any tissue found in the body, making them attractive for regenerative medicine applications. Seeding hiPSC aggregates into biomaterial scaffolds can control their differentiation into specific tissue types. Here we develop and analyze a mathematical model of hiPSC aggregate behavior when seeded on melt electrospun scaffolds with defined topography. RESULTS: We used ordinary differential equations to model the different cellular populations (stem, progenitor, differentiated) present in our scaffolds based on experimental results and published literature. Our model successfully captures qualitative features of the cellular dynamics observed experimentally. We determined the optimal parameter sets to maximize specific cellular populations experimentally, showing that a physiologic oxygen level (∼ 5%) increases the number of neural progenitors and differentiated neurons compared to atmospheric oxygen levels (∼ 21%) and a scaffold porosity of ∼ 63% maximizes aggregate size. CONCLUSIONS: Our mathematical model determined the key factors controlling hiPSC behavior on melt electrospun scaffolds, enabling optimization of experimental parameters.

A scaling law for random walks on networks.

The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.

Using ecological whole body kinematics to evaluate effects of medication adjustment in Parkinson disease.

BACKGROUND: Functional motor impairments including mobility are major reasons for clinical intervention and medication adjustment in symptomatic therapy for Parkinson's disease (PD). Outcome measures used to assess the impact of medication are mostly based on patients' memory or diaries which, considering the gaps between visits, are neither objective nor very reliable. OBJECTIVE: Investigating the feasibility of using movement features extracted from ecological whole-body kinematics recordings to measure the quantitative and qualitative changes in multiple aspects of mobility after medication changes in PD. METHODS: Eleven patients with PD (PwPD) performed mobility tasks in their own home, wearing a full body wireless inertial sensing based motion capture system. Three scripted walking tasks (walking, fast walking, and walk turns) were examined at baseline and two weeks after medication changes. Clinical scales, including investigator-rated clinical global impression of improvement (CGI-I), were collected at both visits. RESULTS: Out of 59 recorded body joint variables, five were identified as pertinent. Changes were represented in vector space as a plot of mean versus peak amplitude. Regression analysis was used to predict clinical improvement or worsening based on these vector features. The predictors were able to explain (>98.5% of variance) patients' clinical global impression of improvement, thus correctly predicting 5 cases of improvement and 2 cases of worsening. CONCLUSIONS: This study provided a method of extracting clinically meaningful reports from ecological kinematic data showing changes after drug adjustments. The results are presented using a novel concept called change space that may be more understandable for clinical staff.

Comparing movement patterns associated with Huntington's chorea and Parkinson's dyskinesia.

Involuntary movements such as levodopa-induced dyskinesia in Parkinson's disease (PD) and chorea in Huntington's disease (HD) are the consequence of two distinct basal ganglia dysfunctions. Yet, their clinical manifestations seem to resemble each other. We seek to determine how to detect PD dyskinesia and HD chorea during quiet stance using healthy control subjects' postural sway as a base measure and identify means to distinguish mathematically HD chorea from PD dyskinesia. Movements were recorded using a magnetic tracker system with fifteen sensors placed strategically to capture whole-body displacement. Choreic and dyskinetic patients as well as healthy controls were asked to stand with arms stretched horizontally in front of them for 60 s. We examined amplitude, frequency dispersion, proportional energy, sample entropy, kurtosis, skewness, amplitude fluctuation, maximum coherency between 44 pairs of body segments. The choreic and dyskinetic movements revealed similar patterns of sample entropy, amplitude fluctuation, and coherencies between body segments. However, skewness and kurtosis for velocity of movements were found to be higher in HD chorea than in PD dyskinesia, reflecting rapid movements in HD patients. There was also a tendency for the frequency composition of PD dyskinesia to be more concentrated in the 1.0-1.5 Hz range. Our results show that despite their similarities in apparent randomness and lack of coordination, dyskinesia associated with treatment of PD and chorea in HD each have their own distinctive characteristics which may be related to their specific pathophysiology.

Control design for sustained oscillation in a two-gene regulatory network.

Control strategies for gene regulatory networks have begun to be explored, both experimentally and theoretically, with implications for control of disease as well as for synthetic biology. Recent work has focussed on controls designed to achieve desired stationary states. Another useful objective, however, is the initiation of sustained oscillations in systems where oscillations are normally damped, or even not present. Alternatively, it may be desired to suppress (by damping) oscillations that naturally occur in an uncontrolled network. Here we address these questions in the context of piecewise-affine models of gene regulatory networks with affine controls that match the qualitative nature of the model. In the case of two genes with a single relevant protein concentration threshold per gene, we find that control of production terms (constant control) is effective in generating or suppressing sustained oscillations, while control of decay terms (linear control) is not effective. We derive an easily calculated condition to determine an effective constant control. As an example, we apply our analysis to a model of the carbon response network in Escherichia coli, reduced to the two genes that are essential in understanding its behavior.

Capturing whole-body mobility of patients with Parkinson disease using inertial motion sensors: expected challenges and rewards.

While many studies have reported on the use of kinematic analysis on well-controlled, in-laboratory mobility tasks, few studies have examined the challenges of recording dynamic mobility in home environments. This preliminary study evaluated whole body mobility in eleven patients with Parkinson disease (H&Y 2-4). Patients were recorded in their home environment during scripted and non-scripted mobility tasks while wearing a full-body kinematic recording system using 11 inertial motion sensors (IMU). Data were analyzed with principal component analysis (PCA) in order to identify kinematic variables which best represent mobility tasks. Results indicate that there was a large degree of variability within subjects for each task, across tasks for individual subjects, and between scripted and non-scripted tasks. This study underscores the potential benefit of whole body multi-sensor kinematic recordings in understanding the variability in task performance across patients during daily activity which may have a significant impact on rehabilitation assessment and intervention.

Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease.

We propose a new method for selective modulation of cortical rhythms based on neural field theory, in which the activity of a cortical area is extensively monitored using a two-dimensional microelectrode array. The example of Parkinson's disease illustrates the proposed method, in which a neural field model is assumed to accurately describe experimentally recorded activity. In addition, we propose a new closed-loop stimulation signal that is both space- and time- dependent. This method is especially designed to specifically modulate a targeted brain rhythm, without interfering with other rhythms. A new class of neuroprosthetic devices is also proposed, in which the multielectrode array is seen as an artificial neural network interacting with biological tissue. Such a bio-inspired approach may provide a solution to optimize interactions between the stimulation device and the cortex aiming to attenuate or augment specific cortical rhythms. The next step will be to validate this new approach experimentally in patients with Parkinson's disease.

Intra- and inter-limb coherency during stance in non-dyskinetic and dyskinetic patients with Parkinson's disease.

OBJECTIVE: Examine the level of intra- and inter-limb coherency in non-dyskinetic and dyskinetic patients with Parkinson's disease (PD). PATIENTS & METHODS: Using a magnetic tracking system, whole-body 3D movements were assessed in 10 dyskinetic patients with clear monophasic peak-dose levodopa-induced dyskinesia (LID), in 10 non-dyskinetic patients and in 10 control subjects, standing with their arms out. Patients were tested during their best ON period. Coherency in the kinematics of pairs of body segments was assessed by spectral analysis. For each pair examined, we calculated the highest coherency between 0.5 and 3.0Hz and the frequency at which this maximum coherency occurred. RESULTS: Analysis of variance showed that for 34 out of the 44 (77.3%) comparisons we studied, there were significant differences between the means of coherencies of the groups. Typically, the control group had the highest coherency and the patients with LID had the lowest. Patients with LID also tended to have their maximum coherency at higher frequencies than the control and non-dyskinetic patient groups (30 out of 44 comparisons were significant). These trends appeared in all types of inter-segment comparisons, including bilaterally symmetric segments, biomechanically linked segments (in which coherencies were higher overall in all groups, but still different between groups) and in other comparisons, but the trends were not so clear for comparisons involving the feet. CONCLUSION: LID is indeed incoherent in the frequency domain, suggesting that body segments may be driven by different neural outputs. The challenges of dealing with these incoherent involuntary movements when planning and executing voluntary movements must certainly play a role in motor difficulties observed in patients with LID. The fact that both dyskinetic and non-dyskinetic patients showed less coherency than controls suggests that levodopa may alter postural control by decreasing stiffness and increasing limb independence.

Structural principles for periodic orbits in glass networks.

A piecewise-linear differential equation model framework for gene regulatory interactions (Glass networks) has allowed considerable analysis of qualitative dynamics in such systems, including periodicity, an important class of regulatory behaviors. Here, we present new results relating the structure of the network to its dynamics (structural principles). The structure we refer to is the state space of the network, which is a digraph on an n-cube in the case of a single threshold per gene. In particular, we show that for a wide class of cycles in the state space there exist parameter values, consistent with the graph structure, for which a periodic orbit exists in the network. For some classes, we show in addition that stable periodic orbits exist. These results extend greatly earlier work by Glass and Pasternack (J Math Biol 6:207-223, 1978).

Movement patterns of peak-dose levodopa-induced dyskinesias in patients with Parkinson's disease.

The present study characterized involuntary movements associated with levodopa-induced dyskinesias (LID) in patients with Parkinson's disease. We used amplitude, proportional energy, frequency dispersion and sample entropy to determine whether LID movement patterns are truly random, as clinical description seems to suggest, or possess some underlying pattern that is not visible to the naked eye. LID was captured using a magnetic tracker system, which provided 3D rendering of whole-body LID. Patients were instructed to maintain a standing position, with arms extended in front of them. We compared the measurements of the dyskinetic PD group (DPD) with 10 patients without dyskinesias (NDPD) and 10 control subjects. In comparison to the other two groups, movement patterns from the DPD group had significantly higher amplitude, confirming the presence of dyskinesias. In addition, higher frequency components in the power spectrum of velocity were detected, suggestive of higher velocity in LID movement. Furthermore, there was a concentration in narrow frequency bands, which suggested stable oscillatory activity. Finally, sample entropy revealed more regularity in the DPD group. Although not statistically significant, we found that the amplitude from the NDPD group had a tendency to be smaller than those of controls. As well, the spectra were often more dispersed for the NDPD group. In conclusion, the present results suggest that LID cannot be considered as purely random movement since they possess some deterministic pattern of motion. This may provide a way for patients to adapt to these involuntary movements while performing voluntary motor acts.

Statistical determination of the optimal subthalamic nucleus stimulation site in patients with Parkinson disease.

OBJECT: The subthalamic nucleus (STN) is currently recognized as the preferred target for deep brain stimulation (DBS) in patients with Parkinson disease (PD). If there is agreement in the literature that DBS improves motor symptoms significantly, the situation is less clear with respect to the side effects of this procedure. The goal of this study was to correlate the coordinate values of active electrode contacts with the amplitude of residual clinical symptoms and side effects using a mathematical approach. METHODS: In this study the investigators examined a cohort of 41 patients with PD who received clinical benefits from DBS after stimulating electrodes had been implanted bilaterally into the STN. The combined scores of residual clinical symptoms plus side effects, including speech disturbance, postural instability, and weight gain, were fitted by using either inverted ellipsoidal exponefitials or smooth splines. These analyses showed evidence of lower combined scores for stimulating contacts at an x coordinate approximately 12.0 to 12.3 mm lateral to the anterior commissure-posterior commissure (AC-PC) line and at a z coordinate approximately 3.1 to 3.3 mm under the AC-PC line. There was insufficient evidence for a preferred y coordinate location. CONCLUSIONS: The authors propose a "best" therapeutic ellipse area that is centered at an x, z location of 12.5 mm, -3.3 mm and characterized by an extension of 1.85 mm in the x direction and 2.22 mm in the z direction. Therapeutic electrode contacts located within this area are well correlated with the lowest occurrence of residual symptoms and the lowest occurrence of side effects independent of STN anatomical considerations. The lack of a significant result in the y direction remains to be explored further.

A stochastic model for circadian rhythms from coupled ultradian oscillators.

BACKGROUND: Circadian rhythms with varying components exist in organisms ranging from humans to cyanobacteria. A simple evolutionarily plausible mechanism for the origin of such a variety of circadian oscillators, proposed in earlier work, involves the non-disruptive coupling of pre-existing ultradian transcriptional-translational oscillators (TTOs), producing "beats," in individual cells. However, like other TTO models of circadian rhythms, it is important to establish that the inherent stochasticity of the protein binding and unbinding does not invalidate the finding of clear oscillations with circadian period. RESULTS: The TTOs of our model are described in two versions: 1) a version in which the activation or inhibition of genes is regulated stochastically, where the 'unoccupied" (or "free") time of the site under consideration depends on the concentration of a protein complex produced by another site, and 2) a deterministic, "time-averaged" version in which the switching between the "free" and "occupied" states of the sites occurs so rapidly that the stochastic effects average out. The second case is proved to emerge from the first in a mathematically rigorous way. Numerical results for both scenarios are presented and compared. CONCLUSION: Our model proves to be robust to the stochasticity of protein binding/unbinding at experimentally determined rates and even at rates several orders of magnitude slower. We have not only confirmed this by numerical simulation, but have shown in a mathematically rigorous way that the time-averaged deterministic system is indeed the fast-binding-rate limit of the full stochastic model.

The influence of levodopa-induced dyskinesias on manual tracking in patients with Parkinson's disease.

The influence of peak-dose, levodopa-induced dyskinesias (LID) on manual tracking (MT) was examined in 10 dyskinetic patients with Parkinson's disease (DPD), and compared to 10 age/gender-matched non-dyskinetic patients with Parkinson's disease (NDPD) and 10 healthy controls. Whole-body movement (WBM) and MT performance were recorded simultaneously with a 6-degrees-of-freedom magnetic motion tracker and forearm rotation sensors, respectively. Subjects were asked to match the length of a computer-generated line with a line they controlled via wrist rotation. Results show that DPD patients had greater WBM magnitude at rest and during the motor task, both in displacement and in velocity. All groups displayed some increase in WBM displacement from rest to MT, but only the DPD group had a significant increase in WBM velocity during movement. As for MT performance (determined by assessing the positional mismatch between subjects' and target lines), ERROR in displacement was statistically similar between groups. There was no correlation between ERROR and the magnitude of WBM within the DPD group. The DPD group showed significant increased ERROR when the velocity of the subject's line was compared with that of the velocity of the target line. When two distinct target pace segments were examined (FAST/SLOW), no significant differences were found in ERROR for displacement for either group, but both the NDPD and DPD group showed increased ERROR from SLOW to FAST for velocity. This was accompanied with an increase in WBM velocity only in the DPD group. The lack of increased ERROR during the SLOW tracking portion in the DPD group supports the notion that the dyskinesias themselves were not primarily responsible for the ERROR seen in the patients. When examining the positive or negative values of ERROR (i.e., faster or slower than the target), we found that the increased ERROR in velocity observed in the DPD group was the result of excess velocity rather than bradykinesia, manifested as isolated deviations from the target trace in the DPD group that were coherent in time with increased ERROR in velocity. In conclusion, evidence presented in the present study shows that the LID was not the primary cause of the ERROR seen in the DPD group. Accordingly, we propose that the increased ERROR seen in the DPD group resulted from a mechanism distinct from the one generating LID.

Bradykinesia in patients with Parkinson's disease having levodopa-induced dyskinesias.

We investigated the likelihood that bradykinesia coexisted with levodopa-induced dyskinesias (LID) in 10 dyskinetic Parkinson's disease patients (DPD). Their motor performance was compared to that of 10 age/gender-matched non-dyskinetic patients (NDPD) and 10 healthy controls. Whole-body movement (WBM) and rapid alternating movements (RAM) at the wrist were recorded simultaneously using 6-degree of freedom magnetic motion tracker and forearm rotational sensors, respectively. WBM was recorded prior to, and while subjects performed pronation-supination movements of their dominant hand with maximal rotational excursion, and as fast as possible for 10s. RANGE, VELOCITY and IRREGULARITY of pronation-supination cycles were quantified to assess motor performance. Results show that DPD patients had greater WBM than NDPD and controls during rest and RAM performance, as expected. There were no differences in motor performance between DPD and NDPD groups for RANGE and VELOCITY, despite significantly longer disease duration for the DPD group (15.5+/-6.2 years versus 6.6+/-2.6 years). However, both the NDPD and DPD groups showed significantly lower RANGE and reduced VELOCITY compared to controls, suggesting the presence of bradykinesia. For IRREGULARITY, DPD patients showed increased fluctuations in pronation-supination cycle amplitude compared to NDPD and controls. However, the lack of correlation between WBM magnitude and IRREGULARITY within the DPD group (Spearman's rank order, rho = 0.31, p > 0.05), suggests that LID were not the primary cause of increased IRREGULARITY. In conclusion, our results demonstrated that bradykinesia can coexist with dyskinesias, suggestive of distinct neural circuits. Our results also demonstrated that the occurrence of LID is not inevitably accompanied with worsening of motor performance.

A model for generating circadian rhythm by coupling ultradian oscillators.

BACKGROUND: Organisms ranging from humans to cyanobacteria undergo circadian rhythm, that is, variations in behavior that cycle over a period about 24 hours in length. A fundamental property of circadian rhythm is that it is free-running, and continues with a period close to 24 hours in the absence of light cycles or other external cues. Regulatory networks involving feedback inhibition and feedforward stimulation of mRNA transcription and translation are thought to be critical for many circadian mechanisms, and genes coding for essential components of circadian rhythm have been identified in several organisms. However, it is not clear how such components are organized to generate a circadian oscillation. RESULTS: We propose a model in which two independent transcriptional-translational oscillators with periods much shorter than 24 hours are coupled to drive a forced oscillator that has a circadian period, using mechanisms and parameters of conventional molecular biology. Furthermore, the resulting circadian oscillator can be entrained by an external light-dark cycle through known mechanisms. We rationalize the mathematical basis for the observed behavior of the model, and show that the behavior is not dependent on the details of the component ultradian oscillators but occurs even if quite generalized basic oscillators are used. CONCLUSION: We conclude that coupled, independent, transcriptional-translational oscillators with relatively short periods can be the basis for circadian oscillators. The resulting circadian oscillator can be entrained by 24-hour light-dark cycles, and the model suggests a mechanism for its evolution.

Effect of chronic exposure to methylmercury on eye movements in Cree subjects.

OBJECTIVES: To examine the effect of chronic exposure to methylmercury on eye movements (pursuit, fixation and dynamic saccades) in Cree subjects from Northern Quebec. METHODS: Eye movements were recorded in a group of Cree subjects ( n=36) exposed chronically to methylmercury, a group of patients with Parkinson's disease (PD) ( n=21), and a group of control subjects ( n=30) by use of an infrared eye-movement recording system. Pursuit, fixation, and prompted and remembered saccades were recorded twice for both eyes in the horizontal and vertical axes. Blinks were removed, and data were calibrated. RESULTS: Analyses of variance revealed significant differences for all characteristics examined for fixation and pursuit, and for some characteristics in dynamic saccades. These differences arose sometimes from the Cree group, sometimes from the PD group and sometimes from both groups. CONCLUSIONS: The results suggest that eye movements of Cree subjects exposed to methylmercury are qualitatively different from those of both control subjects and patients with PD. Comparisons between more-exposed and less-exposed Cree subjects matched for age with control subjects also showed significant differences for fixation, pursuit and dynamic saccades. The average scores of the more-exposed group were clearly separated from those of the less-exposed and control groups for characteristics of fixation and pursuit, and for accuracy and sharpness of prompted saccades. This trend was less clear in other results where a possible effect of mercury exposure could not be distinguished from a possible cultural effect. Further studies should focus on the most discriminating characteristics for the Cree group, such as measures of accuracy and coherence in all tests and sharpness of saccades.

Kinetic tremor during tracking movements in patients with Parkinson's disease.

This study was undertaken to analyse kinetic tremor during a compensatory tracking task performed with the index fingers. Patients with Parkinson's disease (PD) (n=21) and control subjects (n=30) were tested using a laser system transducing displacement. All participants underwent a clinical examination. Nine characteristics quantifying the tracking task and tremor were applied to the processed displacement or velocity signal. The discriminating power of each characteristic was evaluated using differences between group means (p values), maximum percentage discrimination, and number of outliers in the patient group. All nine characteristics showed significant differences between means of the two groups using Welch-modified t-tests for unequal variances. The most discriminating characteristics reflected differences in the frequency distribution of the movement (proportional power in the 3-4Hz range, harmonicity and median frequency). These differences were rarely visible and did not correlate highly with tremor amplitude or with clinical ratings of tremor. Control subjects had residual spectral power from the tracking oscillations leaking above 3Hz, while most patients had a small 'kinetic tremor' in the 7-12Hz range. The maximum discrimination was moderate, 63% in the best case. Combining representative information about proportional power during posture (recorded in the same subjects) and tracking gave a much higher discrimination (90%) with respect to the 96.7th percentile of the control group. These results suggest that information coming from postural and kinetic tremors can be combined to isolate a subclinical feature of PD symptomatology. This feature could be used to re-evaluate the classic distinction made between the akineto-rigid and tremulous forms of PD and is independent of tremor amplitude.