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Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8+T cells, but they neglected the role of CD4+T cells. Other models were constructed to study the role of CD4+T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge–Kutta method failed to solve it. Consequently, the “Adams predictor-corrector” method is used. The results reveal that the patient’s immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4+T cells can be an alternative to both CD8+T cell therapy and cytokines in some cases. Moreover, CD4+T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4+T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8+T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients’ treatment intervention strategies.

Hill-type models are frequently used in biomechanical simulations. They are attractive for their low computational cost and close relation to commonly measured musculotendon parameters. Still, more attention is needed to improve the activation dynamics of the model specifically because of the nonlinearity observed in the EMG-Force relation. Moreover, one of the important and practical questions regarding the assessment of the model's performance is how adequately can the model simulate any fundamental type of human movement without modifying model parameters for different tasks? This paper tries to answer this question by proposing a simple physiologically based activation dynamics model. The model describes the ?kinetics of the calcium dynamics while activating and deactivating the muscle contraction process. Hence, it allowed simulating the recently discovered role of store-operated calcium entry (SOCE) channels as immediate counter-flux to calcium loss across the tubular system during excitation-contraction coupling. By comparing the ability to fit experimental data without readjusting the parameters, the proposed model has proven to have more steady performance than phenomenologically based models through different submaximal isometric contraction levels. This model indicates that more physiological insights is key for improving Hill-type model performance.

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The main objective of this research work is to develop an effective mathematical model of cardiac conduction system using a heterogeneous whole-heart model. The model is in the form of a system of modified Van der Pol and FitzHugh-Nagumo differential equations capable of describing the heart dynamics. The proposed model extends the range of normal and pathological electrocardiogram (ECG) waveforms that can be generated by the model. The effects of the respiratory sinus arrhythmia (RSA) and the Mayer waves (MW) are both incorporated to modulate the intrinsic frequency of the main oscillator that represents the sinoatrial node. Also, three pathological conditions are incorporated into the model. The heart rate variability (HRV) phenomenon is incorporated into the synthetic ECGs produced which yields valuable information about the cardiovascular health and the performance of the autonomic nervous system. The spectral analysis of the generated RR tachogram delivers power spectrums that resemble those obtained from real recordings. Also, the proposed model generates synthetic ECGs that characteristic the three considered pathological conditions, namely, the tall T wave, the ECG with U wave, and the Wolff-Parkinson-White syndrome. In general, the significance of this research work is in developing a mathematical model that represents the interactions between different pacemakers and allows analysis of cardiac rhythms. To show the effectiveness and the accuracy of the presented model, the results are compared to published results. The proposed model can be a useful tool to study the influences of different physiological conditions on the profile of the ECG. The synthetic ECG signals produced can be used as signal sources for the assessment of diagnostic ECG signal processing devices.

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In the current paper, single point rough collision in three-dimensional rigid multibody systems is modelled. Coulomb's friction law and infinite tangential stiffness are assumed. Routh's incremental model with energetic coefficient of restitution is used. Equations of motion are developed by means of Lagrangian formulation. The non-linear equations of motion show that the contact point could continuously change its sliding direction or the sliding could halt and the non-sliding persists or it could restart along a new direction. All these possible sliding behaviours during impact are identified, conditions leading to each behaviour are specified, and an appropriate numerical procedure is suggested. Normal impulse at the contact point is considered the independent variable, as time-like, to carry out the numerical integrations of the equations of motion. A case of a four-degrees-of-freedom spatial robot that collides with its environment is investigated. Solutions describing the variation of collision variables are obtained. It is recognized that qualitative changes to all the variables occur whenever sliding velocity reaches its minimum value. These critical spots are identified, the associated abrupt variations in the impact variables are explored and the friction influence is observed.