scholarly journals Combined Influence of Nutrient Supply Level and Tissue Mechanical Properties on Benign Tumor Growth as Revealed by Mathematical Modeling

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2213
Author(s):  
Maxim Kuznetsov
Keyword(s):  
Hydraulic Conductivity ◽  
Tumor Growth ◽  
Normal Tissue ◽  
Numerical Solutions ◽  
Stress Gradient ◽  
Growth Curves ◽  
Nutrient Supply ◽  
Benign Tumors ◽  
Model Parameters ◽  
Supply Level

A continuous mathematical model of non-invasive avascular tumor growth in tissue is presented. The model considers tissue as a biphasic material, comprised of a solid matrix and interstitial fluid. The convective motion of tissue elements happens due to the gradients of stress, which change as a result of tumor cells proliferation and death. The model accounts for glucose as the crucial nutrient, supplied from the normal tissue, and can reproduce both diffusion-limited and stress-limited tumor growth. Approximate tumor growth curves are obtained semi-analytically in the limit of infinite tissue hydraulic conductivity, which implies instantaneous equalization of arising stress gradients. These growth curves correspond well to the numerical solutions and represent classical sigmoidal curves with a short initial exponential phase, subsequent almost linear growth phase and a phase with growth deceleration, in which tumor tends to reach its maximum volume. The influence of two model parameters on tumor growth curves is investigated: tissue hydraulic conductivity, which links the values of stress gradient and convective velocity of tissue phases, and tumor nutrient supply level, which corresponds to different permeability and surface area density of capillaries in the normal tissue that surrounds the tumor. In particular, it is demonstrated, that sufficiently low tissue hydraulic conductivity (intrinsic, e.g., to tumors arising from connective tissue) and sufficiently high nutrient supply can lead to formation of giant benign tumors, reaching tens of centimeters in diameter, which are indeed observed clinically.

Specific esponse to 5-HT2 agonists by arterioles linked to Meth A tumors in mice

1992 ◽  
Vol 262 (3) ◽  
pp. H704-H709 ◽  
Author(s):  
O. Stucker ◽  
E. Vicaut ◽  
B. Teisseire
Keyword(s):  
Tumor Growth ◽  
Life Span ◽  
Intravenous Administration ◽  
Intravital Microscopy ◽  
Normal Tissue ◽  
Saline Solution ◽  
Survival Rates ◽  
Pharmacological Interest ◽  
Serotoninergic Agonists

Using intravital microscopy, we compared the responses to serotonin [5-hydroxytryptamine (5-HT)] and to a specific 5-HT2 agonist [1-(2,5-dimethoxy-4-bromo-phenyl)-2-aminopropane (DOB)] by normal arterioles and by the host-modified arterioles feeding a Meth A tumor implanted into the flank of female BALB/c mice. Topical and intravenous administration of 5-HT (10(-6) to 10(-4) M and 0.01-10 micrograms) or DOB (5 x 10(-7) to 5 x 10(-5) M and 0.01-10 micrograms) induced arteriolar constriction, which was far more pronounced for the arterioles feeding the tumor. This larger degree of vasoconstriction in tumor-feeding vs. normal arterioles was not found with norepinephrine. We also compared tumor growth and the mouse life span in three groups of mice, which were given 1 mg of serotonin or 0.74 mg DOB or saline solution. 5-HT or DOB both reduced tumor growth drastically compared with the controls (P less than 0.001), and survival rates were significantly higher in the 5-HT or DOB-treated groups (P less than 0.001). We conclude that 5-HT2-serotoninergic agonists are of particular pharmacological interest, since their vasoconstrictive action on the microvasculature feeding the tumors is much greater than in normal tissue and may interfere with tumor growth.

Implications of Using Dual Number Derivatives With a Numerical Solution

2013 ◽  
Author(s):  
Suryanarayana R. Pakalapati ◽  
Hayri Sezer ◽  
Ismail B. Celik
Keyword(s):  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Computer Code ◽  
Model Parameters ◽  
Model Parameter ◽  
Systematic Assessment ◽  
Dual Number ◽  
Advection Diffusion Equation ◽  
Model Equations ◽  
Derivatives Of

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.

scholarly journals Hydrologic responses of the Zwalm catchment using the REW model: incorporating uncertainty of soil properties

2006 ◽  
Vol 3 (1) ◽  
pp. 69-114 ◽  
Author(s):  
A. El Ouazzani Taibi ◽  
G. P. Zhang ◽  
A. Elfeki
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Hydraulic Conductivity ◽  
Model Parameters ◽  
Watershed Model ◽  
The Monte Carlo Method ◽  
Physically Based ◽  
Hydrologic Responses ◽  
High Influence ◽  
Model Approach

Abstract. The research presented in this paper focuses on an application of a newly developed physically-based watershed model approach, which is called Representative Elementary Watershed (REW) approach. The study stressed the effects of uncertainty of input parameters on the watershed responses (i.e. simulated discharges). The approach was applied to the Zwalm catchment, which is an agriculture dominated watershed with a drainage area of 114.3 km2 located in East-Flanders, Belgium. Uncertainty analysis of the model parameters is limited to the saturated hydraulic conductivity because of its high influence on the watershed hydrologic behavior. The assessment of outputs uncertainty is performed using the Monte Carlo method. The ensemble statistical watershed responses and their uncertainties are calculated and compared with the measurements. The results show that the measured discharges are falling within the 95% confidence interval of the modeled discharge.

scholarly journals Mathematical Models for Tumor Growth and the Reduction of Overtreatment

2018 ◽  
Vol 80 (01) ◽  
pp. 072-078 ◽  
Author(s):  
Berdine Heesterman ◽  
John-Melle Bokhorst ◽  
Lisa de Pont ◽  
Berit Verbist ◽  
Jean-Pierre Bayley ◽  
...  
Keyword(s):  
Tumor Growth ◽  
Mathematical Models ◽  
Goodness Of Fit ◽  
Mean Squared Error ◽  
Age At Onset ◽  
Growth Curves ◽  
Magnetic Resonance Images ◽  
Coefficient Of Determination ◽  
Growth Data ◽  
Significant Difference

Background To improve our understanding of the natural course of head and neck paragangliomas (HNPGL) and ultimately differentiate between cases that benefit from early treatment and those that are best left untreated, we studied the growth dynamics of 77 HNPGL managed with primary observation. Methods Using digitally available magnetic resonance images, tumor volume was estimated at three time points. Subsequently, nonlinear least squares regression was used to fit seven mathematical models to the observed growth data. Goodness of fit was assessed with the coefficient of determination (R 2) and root-mean-squared error. The models were compared with Kruskal–Wallis one-way analysis of variance and subsequent post-hoc tests. In addition, the credibility of predictions (age at onset of neoplastic growth and estimated volume at age 90) was evaluated. Results Equations generating sigmoidal-shaped growth curves (Gompertz, logistic, Spratt and Bertalanffy) provided a good fit (median R 2: 0.996–1.00) and better described the observed data compared with the linear, exponential, and Mendelsohn equations (p < 0.001). Although there was no statistically significant difference between the sigmoidal-shaped growth curves regarding the goodness of fit, a realistic age at onset and estimated volume at age 90 were most often predicted by the Bertalanffy model. Conclusions Growth of HNPGL is best described by decelerating tumor growth laws, with a preference for the Bertalanffy model. To the best of our knowledge, this is the first time that this often-neglected model has been successfully fitted to clinically obtained growth data.

The Ocean Mesoscale Regime of the Reduced-Gravity Quasigeostrophic Model

2019 ◽  
Vol 49 (10) ◽  
pp. 2469-2498 ◽  
Author(s):  
R. M. Samelson ◽  
D. B. Chelton ◽  
M. G. Schlax
Keyword(s):  
Numerical Solutions ◽  
Signal Propagation ◽  
Model Parameters ◽  
Linear Wave ◽  
Mesoscale Variability ◽  
Reduced Gravity ◽  
Stochastic Field ◽  
Stochastic Forcing ◽  
Frequency Spectra ◽  
Quasigeostrophic Model

AbstractA statistical-equilibrium, geostrophic-turbulence regime of the stochastically forced, one-layer, reduced-gravity, quasigeostrophic model is identified in which the numerical solutions are representative of global mean, midlatitude, open-ocean mesoscale variability. Solutions are forced near the internal deformation wavenumber and damped linearly and by high-wavenumber enstrophy dissipation. The results partially rationalize a recent semiempirical stochastic field model of mesoscale variability motivated by a global eddy identification and tracking analysis of two decades of satellite altimeter sea surface height (SSH) observations. Comparisons of model results with observed SSH variance, autocorrelation, eddy, and spectral statistics place constraints on the model parameters. A nominal best fit is obtained for a dimensional SSH stochastic-forcing variance production rate of 1/4 cm2 day−1, an SSH damping rate of 1/62 week−1, and a stochastic forcing autocorrelation time scale near or greater than 1 week. This ocean mesoscale regime is nonlinear and appears to fall near the stochastic limit, at which wave-mean interaction is just strong enough to begin to reduce the local mesoscale variance production, but is still weak relative to the overall nonlinearity. Comparison of linearly inverted wavenumber–frequency spectra shows that a strong effect of nonlinearity, the removal of energy from the resonant linear wave field, is resolved by the gridded altimeter SSH data. These inversions further suggest a possible signature in the merged altimeter SSH dataset of signal propagation characteristics from the objective analysis procedure.

scholarly journals A Numerical Model of Temperate Glacier Flow Applied to the Quiescent Phase of a Surge-Type Glacier

1982 ◽  
Vol 28 (99) ◽  
pp. 239-265 ◽  
Author(s):  
Robert Bindschadler
Keyword(s):  
Shear Stress ◽  
Numerical Model ◽  
Stress Gradient ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Model Parameters ◽  
Small Contribution ◽  
Base Shear ◽  
Quiescent Phase ◽  
Glacier Flow

AbstractA time-dependent numerical model of temperate glacier flow without sliding is developed and applied to the quiescent phase of surge-type Variegated Glacier, Alaska. The model is based on a one-dimensional continuity equation but the transverse channel shape is explicitly included allowing the complex geometries of real glaciers to be modelled. Velocities and volume fluxes are calculated from the glacier geometry. Transverse stress is taken into account by shape factors which are fitted to measurements of geometry and velocity and are chosen to be insensitive to changes in geometry. Longitudinal stress gradients are taken into account by use of a large-scale surface slope. A Crank-Nicholson finite-difference approximation is used and it is unconditionally stable when a small contribution from the local slope is added to the average slope.Model parameters are fitted to extensive data collected on Variegated Glacier in 1973 and 1974. Predictions of the model over a four year interval agree well with field measurements. Predictions of the current quiescent phase (1965–84) indicate depth increases in the upper glacier of more than 75 m with a twenty-fold increase in the volume flux. During this interval the base shear stress increases 40% in the upper glacier and decreases 20% in the lower glacier. During the mid to late quiescent phase, ice motion becomes more important than mass balance in the redistribution of mass over the central region of the glacier. If normal flow were to persist, the predicted steady-state profile would be an average of 100 m deeper and 41% more voluminous than in 1973.The predicted base shear-stress gradient is never negative enough to satisfy Robin and Weertman’s (1973) condition for blockage of subglacial water flow. The annual rate of water production by dissipation of mechanical straining at the bed remains two orders of magnitude below that produced by summer surface melt. The predicted fractional increase in base stress during the quiescent phase is a maximum in the region believed to be the trigger zone of the surges.

WE-C-AUD B-04: Normal Tissue Complication Probability: Updating the Model Parameters for Modern Radiotherapy

2008 ◽  
Vol 35 (6Part23) ◽  
pp. 2933-2933
Author(s):  
S Gulliford ◽  
M Partridge ◽  
S Webb ◽  
P Evans ◽  
K Foo ◽  
...  
Keyword(s):  
Normal Tissue ◽  
Normal Tissue Complication Probability ◽  
Model Parameters ◽  
Probability Updating ◽  
Modern Radiotherapy

scholarly journals Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1204
Author(s):  
Maxim Kuznetsov ◽  
Andrey Kolobov
Keyword(s):  
Tumor Cells ◽  
Solid Tumor ◽  
Tissue Damage ◽  
Normal Tissue ◽  
Dose Fractionation ◽  
Oxygen Effect ◽  
Model Parameters ◽  
Time And Space ◽  
Fractionated Radiotherapy ◽  
Normal Tissue Damage

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for three time and space-dependent factors that influence the efficiency of radiotherapy fractionation schemes—tumor cell repopulation, reoxygenation and redistribution of proliferative states. A special algorithm is developed, aimed at finding the fractionation schemes that provide increased tumor cure probability under the constraints of maximum normal tissue damage and maximum fractional dose. The optimization procedure is performed for varied radiosensitivity of tumor cells under the values of model parameters, corresponding to different degrees of tumor malignancy. The resulting optimized schemes consist of two stages. The first stages are aimed to increase the radiosensitivity of the tumor cells, remaining after their end, sparing the caused normal tissue damage. This allows to increase the doses during the second stages and thus take advantage of the obtained increased radiosensitivity. Such method leads to significant expansions in the curative ranges of the values of tumor radiosensitivity parameters. Overall, the results of this study represent the theoretical proof of concept that non-uniform radiotherapy fractionation schemes may be considerably more effective that uniform ones, due to the time and space-dependent effects.

scholarly journals Multicompartment modeling of protein shedding kinetics during vascularized tumor growth

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Gautam B. Machiraju ◽  
Parag Mallick ◽  
Hermann B. Frieboes
Keyword(s):  
Tumor Growth ◽  
Compartment Model ◽  
Vessel Diameter ◽  
Cellular Protein ◽  
Extracellular Proteins ◽  
Surface Proteins ◽  
Ca 125 ◽  
Model Parameters ◽  
Marker Selection ◽  
Protein Shedding

Abstract Identification of protein biomarkers for cancer diagnosis and prognosis remains a critical unmet clinical need. A major reason is that the dynamic relationship between proliferating and necrotic cell populations during vascularized tumor growth, and the associated extra- and intra-cellular protein outflux from these populations into blood circulation remains poorly understood. Complementary to experimental efforts, mathematical approaches have been employed to effectively simulate the kinetics of detectable surface proteins (e.g., CA-125) shed into the bloodstream. However, existing models can be difficult to tune and may be unable to capture the dynamics of non-extracellular proteins, such as those shed from necrotic and apoptosing cells. The models may also fail to account for intra-tumoral spatial and microenvironmental heterogeneity. We present a new multi-compartment model to simulate heterogeneously vascularized growing tumors and the corresponding protein outflux. Model parameters can be tuned from histology data, including relative vascular volume, mean vessel diameter, and distance from vasculature to necrotic tissue. The model enables evaluating the difference in shedding rates between extra- and non-extracellular proteins from viable and necrosing cells as a function of heterogeneous vascularization. Simulation results indicate that under certain conditions it is possible for non-extracellular proteins to have superior outflux relative to extracellular proteins. This work contributes towards the goal of cancer biomarker identification by enabling simulation of protein shedding kinetics based on tumor tissue-specific characteristics. Ultimately, we anticipate that models like the one introduced herein will enable examining origins and circulating dynamics of candidate biomarkers, thus facilitating marker selection for validation studies.

Influence of Tumor Growth Kinetics on Response to Doxorubicin Treatment of C3H Mammary Carcinoma

1988 ◽  
Vol 74 (3) ◽  
pp. 269-274 ◽  
Author(s):  
Romano Demicheli ◽  
Roberto Foroni ◽  
Fernando C. Giuliani ◽  
Giuseppina Savi
Keyword(s):  
Tumor Growth ◽  
Growth Kinetics ◽  
Mammary Carcinoma ◽  
Tumor Volume ◽  
Growth Curves ◽  
Instantaneous Growth Rate ◽  
Doxorubicin Treatment ◽  
Tumor Growth Kinetics ◽  
Treatment Data ◽  
Clonogenic Cell

The influence of tumor growth kinetics on response to doxorubicin treatment of C3H mammary carcinoma was investigated. Gompertzian growth curves were obtained for the tumor mass of each mouse by a computerized best fit program. The response was assessed by evaluating: a) the total clonogenic cell reduction as a fraction of the initial tumor volume or the tumor volume that should result at the end of treatment in a free growth condition, and b) the partial clonogenic cell reduction at each drug administration, assuming a first order cell kill hypothesis. Slowly growing tumors at each dose level showed a significantly poorer response than rapidly growing tumors. Each response index exhibited a linear correlation with the specific instantaneous growth rate at the time of treatment. Data also suggested a dose-response dependence.

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