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Fractal Kinetic Implementation in Population Pharmacokinetic Modeling. Pharmaceutics 2023; 15:pharmaceutics15010304. [PMID: 36678932 PMCID: PMC9867137 DOI: 10.3390/pharmaceutics15010304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2022] [Revised: 01/13/2023] [Accepted: 01/13/2023] [Indexed: 01/19/2023] Open
Abstract
Compartment modeling is a widely accepted technique in the field of pharmacokinetic analysis. However, conventional compartment modeling is performed under a homogeneity assumption that is not a naturally occurring condition. Since the assumption lacks physiological considerations, the respective modeling approach has been questioned, as novel drugs are increasingly characterized by physiological or physical features. Alternative approaches have focused on fractal kinetics, but evaluations of their application are lacking. Thus, in this study, a simulation was performed to identify desirable fractal-kinetics applications in conventional modeling. Visible changes in the profiles were then investigated. Five cases of finalized population models were collected for implementation. For model diagnosis, the objective function value (OFV), Akaike's information criterion (AIC), and corrected Akaike's information criterion (AICc) were used as performance metrics, and the goodness of fit (GOF), visual predictive check (VPC), and normalized prediction distribution error (NPDE) were used as visual diagnostics. In most cases, model performance was enhanced by the fractal rate, as shown in a simulation study. The necessary parameters of the fractal rate in the model varied and were successfully estimated between 0 and 1. GOF, VPC, and NPDE diagnostics show that models with the fractal rate described the data well and were robust. In the simulation study, the fractal absorption process was, therefore, chosen for testing. In the estimation study, the rate application yielded improved performance and good prediction-observation agreement in early sampling points, and did not cause a large shift in the original estimation results. Thus, the fractal rate yielded explainable parameters by setting only the heterogeneity exponent, which reflects true physiological behavior well. This approach can be expected to provide useful insights in pharmacological decision making.
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Donato I, Velpula KK, Tsung AJ, Tuszynski JA, Sergi CM. Demystifying neuroblastoma malignancy through fractal dimension, entropy, and lacunarity. TUMORI JOURNAL 2023:3008916221146208. [PMID: 36645143 DOI: 10.1177/03008916221146208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023]
Abstract
PURPOSE Neuroblastoma is a pediatric solid tumor with a prognosis associated with histology and age of the patient, which are the parameters of the well-established current classification (Shimada classification). Despite the development of new treatment options, the prognosis of high-risk neuroblastoma patients is still poor. Therefore, there is a continuous need to stratify the children suffering from this tumor. A mathematical and computational approach is proposed to enable automatic and precise cancer diagnosis on the histological slide. METHODS We targeted the complexity of neuroblastoma by calculating its image entropy (S), fractal dimension (FD), and lacunarity (λ) in a combined mathematical code. First, we tested the proposed method for patient-derived glioma images. It allowed distinguishing between normal brain tissue, grade II, and grade III glioma, which harbor different outcomes. RESULTS In neuroblastoma, our analysis of image's FD, S, and λ combined with a machine learning algorithm automatically predicted tumor malignancy with a receiver operating characteristic curve of 0.82. FD, S, and λ distinguish between neuroblastoma and ganglioneuroma, but they only partially differentiate between the normal samples and the other classes. Ganglioneuroma, the most differentiated form, and poorly-differentiated neuroblastoma display different values of FD, S, and λ. CONCLUSIONS FD, S, and λ of imaging recognize groups in neuroblastic tumors. We suggest that future studies including these features may challenge the current Shimada classification of neuroblastoma with categories of favorable and unfavorable histology. It is expected that this methodology could trigger multicenter studies and potentially find practical use in the clinical setting of children's hospitals worldwide.
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Affiliation(s)
- Irene Donato
- Department of Oncology, University of Alberta, Cross Cancer Institute, Edmonton, AB, Canada
| | - Kiran K Velpula
- Departments of Cancer Biology and Pharmacology, Neurosurgery, University of Illinois College of Medicine at Peoria, Peoria, IL, USA
| | - Andrew J Tsung
- Departments of Cancer Biology and Pharmacology, Neurosurgery, University of Illinois College of Medicine at Peoria, Peoria, IL, USA
| | - Jack A Tuszynski
- Department of Oncology, University of Alberta, Cross Cancer Institute, Edmonton, AB, Canada.,Department of Physics, University of Alberta, Centennial Centre for Interdisciplinary Science, Edmonton, AB, Canada.,Dipartimento di Ingegneria Meccanica e Aerospaziale (DIMEAS), Polytechnic University of Turin, Turin, Italy
| | - Consolato M Sergi
- Department of Laboratory Medicine and Pathology, University of Alberta, Stollery Children's Hospital, Edmonton, AB, Canada.,Division of Anatomic Pathology, Children's Hospital of Eastern Ontario, University of Ottawa, Ottawa, ON, Canada.,Institute of Pathology, Medical University of Innsbruck, Innsbruck, Tyrol, Austria
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Macheras P, Tsekouras AA, Chryssafidis P. Interpreting airborne pandemics spreading using fractal kinetics’ principles. F1000Res 2021; 10:609. [PMID: 34987769 PMCID: PMC8686328 DOI: 10.12688/f1000research.53196.1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Accepted: 06/16/2021] [Indexed: 11/20/2022] Open
Abstract
Introduction The reaction between susceptible and infected subjects has been studied under the well-mixed hypothesis for almost a century. Here, we present a consistent analysis for a not well-mixed system using fractal kinetics’ principles. Methods We analyzed COVID-19 data to get insights on the disease spreading in absence/presence of preventive measures. We derived a three-parameter model and show that the “fractal” exponent h of time larger than unity can capture the impact of preventive measures affecting population mobility. Results The h=1 case, which is a power of time model, accurately describes the situation without such measures in line with a herd immunity policy. The pandemic spread in four model countries (France, Greece, Italy and Spain) for the first 10 months has gone through four stages: stages 1 and 3 with limited to no measures, stages 2 and 4 with varying lockdown conditions. For each stage and country two or three model parameters have been determined using appropriate fitting procedures. The fractal kinetics model was found to be more akin to real life. Conclusion Model predictions and their implications lead to the conclusion that the fractal kinetics model can be used as a prototype for the analysis of all contagious airborne pandemics.
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Affiliation(s)
- Panos Macheras
- Faculty of Pharmacy, Laboratory of Biopharmaceutics Pharmacokinetics, National and Kapodistrian University of Athens, Athens, 11526, Greece
- Athena Research Center, Attica, Athens, 15125, Greece
| | - Athanasios A. Tsekouras
- Athena Research Center, Attica, Athens, 15125, Greece
- Department of Chemistry, Laboratory of Physical Chemistry, National and Kapodistrian University of Athens, Athens, 11526, Greece
| | - Pavlos Chryssafidis
- Faculty of Pharmacy, Laboratory of Biopharmaceutics Pharmacokinetics, National and Kapodistrian University of Athens, Athens, 11526, Greece
- Athena Research Center, Attica, Athens, 15125, Greece
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Comparison of the gamma-Pareto convolution with conventional methods of characterising metformin pharmacokinetics in dogs. J Pharmacokinet Pharmacodyn 2019; 47:19-45. [PMID: 31865474 PMCID: PMC7040082 DOI: 10.1007/s10928-019-09666-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2019] [Accepted: 11/23/2019] [Indexed: 12/25/2022]
Abstract
A model was developed for long term metformin tissue retention based upon temporally inclusive models of serum/plasma concentration (\documentclass[12pt]{minimal}
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\begin{document}$$ C $$\end{document}C) having power function tails called the gamma-Pareto type I convolution (GPC) model and was contrasted with biexponential (E2) and noncompartmental (NC) metformin models. GPC models of \documentclass[12pt]{minimal}
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\begin{document}$$ C $$\end{document}C. The GPC, E2 and NC models were applied to a total of 148 serum samples drawn from 20 min to 72 h following bolus intravenous metformin in seven healthy mongrel dogs. The GPC model was used to calculate area under the curve (AUC), clearance (\documentclass[12pt]{minimal}
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\begin{document}$$ CL $$\end{document}CL), and functions of time, f(t), for drug mass remaining (M), apparent volume of distribution (\documentclass[12pt]{minimal}
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\begin{document}$$ CL $$\end{document}CL-values that were 84.8% of total renal plasma flow (RPF) as estimated from meta-analysis. The GPC \documentclass[12pt]{minimal}
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\begin{document}$$ CL $$\end{document}CL-values of 104.7% and 123.7% of RPF, respectively. The GPC plasma/serum only model predicted 78.9% drug \documentclass[12pt]{minimal}
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\begin{document}$$V_d$$\end{document}Vd that achieved limiting constant values. A simulated constant average drug mass multidosing protocol exhibited increased \documentclass[12pt]{minimal}
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\begin{document}$$t_{1/2}$$\end{document}t1/2 with elapsing time, effects that have been observed experimentally during same-dose multidosing. The GPC heavy-tailed models explained multiple documented phenomena that were unexplained with lighter-tailed models.
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Kosmidis K, Dassios G. Monte Carlo simulations in drug release. J Pharmacokinet Pharmacodyn 2019; 46:165-172. [PMID: 30880356 DOI: 10.1007/s10928-019-09625-8] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2018] [Accepted: 03/07/2019] [Indexed: 10/27/2022]
Abstract
We present methods based on simple sampling Monte Carlo simulations that are used in the study of controlled drug release from devices of various shapes and characteristics. The manuscript is part of a special tribute issue for Prof. Panos Macheras and we have chosen applications of the Monte Carlo method in the field of drug release that were pioneered by him and his research group. Thus, we focus on the investigation of diffusion based release and we present methods that go beyond the application of the classical fickian diffusion equation. We describe methods that have proven to be effective in illuminating the profound effects of the substrate heterogeneity on the drug release profiles and demonstrate some of the most powerful applications of agent based simulations and numerical methods in the field of pharmacokinetics.
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Affiliation(s)
- Kosmas Kosmidis
- Division of Theoretical Physics, Physics Department, Aristotele University of Thessaloniki, 54124, Thessaloniki, Greece.
| | - George Dassios
- Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, Patras, Greece
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Hennion M, Hanert E. How to avoid unbounded drug accumulation with fractional pharmacokinetics. J Pharmacokinet Pharmacodyn 2013; 40:691-700. [DOI: 10.1007/s10928-013-9340-2] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2013] [Accepted: 10/25/2013] [Indexed: 11/28/2022]
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Weiss M. Fractal structure of the liver: effect on drug elimination. J Pharmacokinet Pharmacodyn 2012; 40:11-4. [DOI: 10.1007/s10928-012-9283-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2012] [Accepted: 11/23/2012] [Indexed: 11/29/2022]
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Balankin AS, Elizarraraz BE. Map of fluid flow in fractal porous medium into fractal continuum flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056314. [PMID: 23004869 DOI: 10.1103/physreve.85.056314] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2012] [Indexed: 06/01/2023]
Abstract
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.
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Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," Instituto Politécnico Nacional, México, Distrito Federal, Mexico
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Yin C, Li X. Anomalous diffusion of drug release from a slab matrix: Fractional diffusion models. Int J Pharm 2011; 418:78-87. [DOI: 10.1016/j.ijpharm.2010.12.009] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2010] [Revised: 12/06/2010] [Accepted: 12/06/2010] [Indexed: 10/18/2022]
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Marsh RE, Tuszyński JA, Sawyer M, Vos KJE. A model of competing saturable kinetic processes with application to the pharmacokinetics of the anticancer drug paclitaxel. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2011; 8:325-354. [PMID: 21631133 DOI: 10.3934/mbe.2011.8.325] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
A saturable multi-compartment pharmacokinetic model for the anti-cancer drug paclitaxel is proposed based on a meta-analysis of pharmacokinetic data published over the last two decades. We present and classify the results of time series for the drug concentration in the body to uncover the underlying power laws. Two dominant fractional power law exponents were found to characterize the tails of paclitaxel concentration-time curves. Short infusion times led to a power exponent of -1.57 ± 0.14, while long infusion times resulted in tails with roughly twice the exponent. Curves following intermediate infusion times were characterized by two power laws. An initial segment with the larger slope was followed by a long-time tail characterized by the smaller exponent. The area under the curve and the maximum concentration exhibited a power law dependence on dose, both with compatible fractional power exponents. Computer simulations using the proposed model revealed that a two-compartment model with both saturable distribution and elimination can reproduce both the single and crossover power laws. Also, the nonlinear dose-dependence is correlated with the empirical power law tails. The longer the infusion time the better the drug delivery to the tumor compartment is a clinical recommendation we propose.
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Affiliation(s)
- Rebeccah E Marsh
- Department of Physics, University of Alberta, Edmonton, Alberta, Canada.
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The Changing Face of the Rate Concept in Biopharmaceutical Sciences: From Classical to Fractal and Finally to Fractional. Pharm Res 2011; 28:1229-32. [DOI: 10.1007/s11095-011-0370-4] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2010] [Accepted: 01/10/2011] [Indexed: 10/18/2022]
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Lerapetritou MG, Georgopoulos PG, Roth CM, Androulakis LP. Tissue-level modeling of xenobiotic metabolism in liver: An emerging tool for enabling clinical translational research. Clin Transl Sci 2010; 2:228-37. [PMID: 20443896 DOI: 10.1111/j.1752-8062.2009.00092.x] [Citation(s) in RCA: 44] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Abstract
This review summarizes some of the recent developments and identifies critical challenges associated with in vitro and in silico representations of the liver and assesses the translational potential of these models in the quest of rationalizing the process of evaluating drug efficacy and toxicity. It discusses a wide range of research efforts that have produced, during recent years, quantitative descriptions and conceptual as well as computational models of hepatic processes such as biotransport and biotransformation, intra- and intercellular signal transduction, detoxification, etc. The above mentioned research efforts cover multiple scales of biological organization, from molecule-molecule interactions to reaction network and cellular and histological dynamics, and have resulted in a rapidly evolving knowledge base for a "systems biology of the liver." Virtual organ/organism formulations represent integrative implementations of particular elements of this knowledge base, usually oriented toward the study of specific biological endpoints, and provide frameworks for translating the systems biology concepts into computational tools for quantitative prediction of responses to stressors and hypothesis generation for experimental design.
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Affiliation(s)
- Marianthi G Lerapetritou
- Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey, USA
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Popović JK, Atanacković MT, Pilipović AS, Rapaić MR, Pilipović S, Atanacković TM. A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac. J Pharmacokinet Pharmacodyn 2010; 37:119-34. [PMID: 20072802 DOI: 10.1007/s10928-009-9147-3] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2009] [Accepted: 12/13/2009] [Indexed: 11/28/2022]
Affiliation(s)
- Jovan K Popović
- Department of Pharmacology, Toxicology and Clinical Pharmacology, Faculty of Medicine, University of Novi Sad, Hajduk Veljkova 3, 21000, Novi Sad, Republic of Serbia.
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Dokoumetzidis A, Macheras P. Fractional kinetics in drug absorption and disposition processes. J Pharmacokinet Pharmacodyn 2009; 36:165-78. [PMID: 19340400 DOI: 10.1007/s10928-009-9116-x] [Citation(s) in RCA: 73] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2008] [Accepted: 03/23/2009] [Indexed: 10/20/2022]
Abstract
We explore the use of fractional order differential equations for the analysis of datasets of various drug processes that present anomalous kinetics, i.e. kinetics that are non-exponential and are typically described by power-laws. A fractional differential equation corresponds to a differential equation with a derivative of fractional order. The fractional equivalents of the "zero-" and "first-order" processes are derived. The fractional zero-order process is a power-law while the fractional first-order process is a Mittag-Leffler function. The latter behaves as a stretched exponential for early times and as a power-law for later times. Applications of these two basic results for drug dissolution/release and drug disposition are presented. The fractional model of dissolution is fitted successfully to datasets taken from literature of in vivo dissolution curves. Also, the proposed pharmacokinetic model is fitted to a dataset which exhibits power-law terminal phase. The Mittag-Leffler function describes well the data for small and large time scales and presents an advantage over empirical power-laws which go to infinity as time approaches zero. The proposed approach is compared conceptually with fractal kinetics, an alternative approach to describe datasets with non exponential kinetics. Fractional kinetics offers an elegant description of anomalous kinetics, with a valid scientific basis, since it has already been applied in problems of diffusion in other fields, and describes well the data.
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Pereira LM. Comment on "Asymptotic time dependence in the fractal pharmacokinetics of a two-compartment model". PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:063901. [PMID: 19256890 DOI: 10.1103/physreve.78.063901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2008] [Indexed: 05/27/2023]
Abstract
A clarification is due about the paper by Chelminiak [Phys. Rev. E 72, 031903 (2005)]. In changing notation at the beginning, the authors incur an unfortunate error that significantly impacts the rest of the paper.
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Affiliation(s)
- Luis M Pereira
- Massachusetts College of Pharmacy and Health Sciences, Children's Hospital Boston/Harvard Medical School, Longwood Avenue, Boston, Massachusetts 02115, USA
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Warren A, Chaberek S, Ostrowski K, Cogger VC, Hilmer SN, McCuskey RS, Fraser R, Le Couteur DG. Effects of old age on vascular complexity and dispersion of the hepatic sinusoidal network. Microcirculation 2008; 15:191-202. [PMID: 18386215 DOI: 10.1080/10739680701600856] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/10/2023]
Abstract
OBJECTIVES In old age, there are marked changes in both the structure of the liver sinusoidal endothelial cell and liver perfusion. The objective of this study was to determine whether there are also aging changes in the microvascular architecture and vascular dispersion of the liver that might influence liver function. METHODS Vascular corrosion casts and light micrographs of young (4 months) and old (24 months) rat livers were compared. Fractal and Fourier analyses and micro-computed tomography were used. Vascular dispersion was determined from the dispersion number for sucrose and 100-nm microspheres in impulse response experiments. RESULTS Age did not affect sinusoidal dimensions, sinusoidal density, or dispersion number. There were changes in the geometry and complexity of the sinusoidal network as determined by fractal dimension and degree of anisotropy. CONCLUSIONS There are small, age-related changes in the architecture of the liver sinusoidal network, which may influence hepatic function and reflect broader aging changes in the microcirculation. However, sinusoidal dimensions and hepatic vascular dispersion are not markedly influenced by old age.
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Affiliation(s)
- Alessandra Warren
- Centre for Education and Research on Ageing and the ANZAC Research Institute, Concord RG Hospital and University of Sydney, Sydney, Australia
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17
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Pang KS, Weiss M, Macheras P. Advanced pharmacokinetic models based on organ clearance, circulatory, and fractal concepts. AAPS J 2007; 9:E268-83. [PMID: 17907768 PMCID: PMC2751417 DOI: 10.1208/aapsj0902030] [Citation(s) in RCA: 43] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2007] [Accepted: 05/14/2007] [Indexed: 12/22/2022] Open
Abstract
Three advanced models of pharmacokinetics are described. In the first class are physiologically based pharmacokinetic models based on in vitro data on transport and metabolism. The information is translated as transporter and enzyme activities and their attendant heterogeneities into liver and intestine models. Second are circulatory models based on transit time distribution and plasma concentration time curves. The third are fractal models for nonhomogeneous systems and non-Fickian processes are presented. The usefulness of these pharmacokinetic models, with examples, is compared.
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Affiliation(s)
- K Sandy Pang
- Leslie Dan Faculty of Pharmacy, University of Toronto, 144 College Street, Toronto, Ontario, Canada M5S 3M2.
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18
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Kohandel M, Kardar M, Milosevic M, Sivaloganathan S. Dynamics of tumor growth and combination of anti-angiogenic and cytotoxic therapies. Phys Med Biol 2007; 52:3665-77. [PMID: 17664569 DOI: 10.1088/0031-9155/52/13/001] [Citation(s) in RCA: 46] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Tumors cannot grow beyond a certain size (about 1-2 mm in diameter) through simple diffusion of oxygen and other essential nutrients into the tumor. Angiogenesis, the formation of blood vessels from pre-existing vessels, is a crucial and observed step, through which a tumor obtains its own blood supply. Thus, strategies that interfere with the development of this tumor vasculature, known as anti-angiogenic therapy, represent a novel approach to controlling tumor growth. Several pre-clinical studies have suggested that currently available angiogenesis inhibitors are unlikely to yield significant sustained improvements in tumor control on their own, but rather will need to be used in combination with conventional treatments to achieve maximal benefit. Optimal sequencing of anti-angiogenic treatment and radiotherapy or chemotherapy is essential to the success of these combined treatment strategies. Hence, a major challenge to mathematical modeling and computer simulations is to find appropriate dosages, schedules and sequencing of combination therapies to control or eliminate tumor growth. Here, we present a mathematical model that incorporates tumor cells and the vascular network, as well as their interplay. We can then include the effects of two different treatments, conventional cytotoxic therapy and anti-angiogenic therapy. The results are compared with available experimental and clinical data.
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Affiliation(s)
- M Kohandel
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
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19
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Marsh RE, Riauka TA. Modeling fractal-like drug elimination kinetics using an interacting random-walk model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:031902. [PMID: 17500721 DOI: 10.1103/physreve.75.031902] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2006] [Revised: 12/04/2006] [Indexed: 05/15/2023]
Abstract
We introduce an interacting random-walk model to describe the residence time of drug molecules undergoing a series of sojourn times in the body before being permanently eliminated under either homogeneous or heterogeneous conditions. We show that short-term correlations between drug molecules lead to Michaelis-Menten kinetics while long-term correlations lead to transient fractal-like kinetics. By combining both types of correlation, fractal-like Michaelis-Menten kinetics are achieved, and the simulations confirm previous analytical results.
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Affiliation(s)
- Rebeccah E Marsh
- Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1.
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20
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Marsh RE, Tuszyński JA. Fractal michaelis-menten kinetics under steady state conditions: Application to mibefradil. Pharm Res 2006; 23:2760-7. [PMID: 17063399 DOI: 10.1007/s11095-006-9090-6] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2006] [Accepted: 06/09/2006] [Indexed: 10/24/2022]
Abstract
PURPOSE To provide the first application of fractal kinetics under steady state conditions to pharmacokinetics as a model for the enzymatic elimination of a drug from the body. MATERIALS AND METHODS A one-compartment model following fractal Michaelis-Menten kinetics under a steady state is developed and applied to concentration-time data for the cardiac drug mibefradil in dogs. The model predicts a fractal reaction order and a power law asymptotic time-dependence of the drug concentration, therefore a mathematical relationship between the fractal reaction order and the power law exponent is derived. The goodness-of-fit of the model is assessed and compared to that of four other models suggested in the literature. RESULTS The proposed model provided the best fit to the data. In addition, it correctly predicted the power law shape of the tail of the concentration-time curve. CONCLUSION A simple one-compartment model with steady state fractal Michaelis-Menten kinetics describing drug elimination from the body most accurately describes the pharmacokinetics of mibefradil in dogs. The new fractal reaction order can be explained in terms of the complex geometry of the liver, the organ responsible for eliminating the drug.
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Affiliation(s)
- Rebeccah E Marsh
- P-412, Avadh Bhatia Physics Laboratory, Department of Physics, Faculty of Science, University of Alberta, Edmonton, AB, T6G 2J1, Canada.
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21
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Charalampidis D, Pascotto M, Kerut EK, Lindner JR. Anatomy and flow in normal and ischemic microvasculature based on a novel temporal fractal dimension analysis algorithm using contrast enhanced ultrasound. IEEE TRANSACTIONS ON MEDICAL IMAGING 2006; 25:1079-86. [PMID: 16895000 DOI: 10.1109/tmi.2006.877442] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
Strategies for improvement of blood flow by promoting new vessel growth in ischemic tissue are being developed. Recently, contrast-enhanced ultrasound (CEU) imaging has been used to assess tissue perfusion in models of ischemia-related angiogenesis, growth-factor mediated angiogenesis, and tumor angiogenesis. In these studies, microvascular flow is measured in order to assess the total impact of adaptations at different vascular levels. High-resolution methods for imaging larger vessels have been developed in order to derive "angiograms" of arteries, veins, and medium to large microvessels. We describe a novel method of vascular bed (microvessel and arterial) characterization of vessel anatomy and flow simultaneously, using serial measurement of the fractal dimension (FD) of a temporal sequence of CEU images. This method is proposed as an experimental methodology to distinguish ischemic from nonischemic tissue. Moreover, an improved approach for extracting the FD unique to this application is introduced.
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Affiliation(s)
- Dimitrios Charalampidis
- Department of Electrical Engineering, College of Engineering, University of New Orleans, LA 70148, USA.
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22
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Chelminiak P, Dixon JM, Tuszyński JA, Marsh RE. Application of a random network with a variable geometry of links to the kinetics of drug elimination in healthy and diseased livers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:051912. [PMID: 16802972 DOI: 10.1103/physreve.73.051912] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2006] [Revised: 03/06/2006] [Indexed: 05/10/2023]
Abstract
This paper discusses an application of a random network with a variable number of links and traps to the elimination of drug molecules from the body by the liver. The nodes and links represent the transport vessels, and the traps represent liver cells with metabolic enzymes that eliminate drug molecules. By varying the number and configuration of links and nodes, different disease states of the liver related to vascular damage have been simulated, and the effects on the rate of elimination of a drug have been investigated. Results of numerical simulations show the prevalence of exponential decay curves with rates that depend on the concentration of links. In the case of fractal lattices at the percolation threshold, we find that the decay of the concentration is described by exponential functions for high trap concentrations but transitions to stretched exponential behavior at low trap concentrations.
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Affiliation(s)
- P Chelminiak
- Department of Physics, University of Warwick, Coventry CV4 7AL, UK
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Chelminiak P, Marsh RE, Tuszyński JA, Dixon JM, Vos KJE. Asymptotic time dependence in the fractal pharmacokinetics of a two-compartment model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:031903. [PMID: 16241478 DOI: 10.1103/physreve.72.031903] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2004] [Revised: 01/06/2005] [Indexed: 05/05/2023]
Abstract
We further investigate, both analytically and numerically, the properties of the fractal two-compartment model introduced by Fuite [J. Fuite, R. Marsh, and J. Tuszynski, Phys. Rev. E 66, 021904 (2002)]. Specifically, we look at the effects of the fractal exponent of the elimination rate coefficient on the long-time behavior of the pharmacokinetic clearance tail. For small exponent values, the tail exhibits exponential behavior, while for larger values, there is a transition to a power law. The theory is applied to seven data sets simulating drugs taken from the pharmacological literature.
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Affiliation(s)
- P Chelminiak
- Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1
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24
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Kosmidis K, Karalis V, Argyrakis P, Macheras P. Michaelis-Menten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics. Biophys J 2005; 87:1498-506. [PMID: 15345531 PMCID: PMC1304557 DOI: 10.1529/biophysj.104.042143] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Two different approaches were used to study the kinetics of the enzymatic reaction under heterogeneous conditions to interpret the unusual nonlinear pharmacokinetics of mibefradil. Firstly, a detailed model based on the kinetic differential equations is proposed to study the enzymatic reaction under spatial constraints and in vivo conditions. Secondly, Monte Carlo simulations of the enzyme reaction in a two-dimensional square lattice, placing special emphasis on the input and output of the substrate were applied to mimic in vivo conditions. Both the mathematical model and the Monte Carlo simulations for the enzymatic reaction reproduced the classical Michaelis-Menten (MM) kinetics in homogeneous media and unusual kinetics in fractal media. Based on these findings, a time-dependent version of the classic MM equation was developed for the rate of change of the substrate concentration in disordered media and was successfully used to describe the experimental plasma concentration-time data of mibefradil and derive estimates for the model parameters. The unusual nonlinear pharmacokinetics of mibefradil originates from the heterogeneous conditions in the reaction space of the enzymatic reaction. The modified MM equation can describe the pharmacokinetics of mibefradil as it is able to capture the heterogeneity of the enzymatic reaction in disordered media.
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Affiliation(s)
- Kosmas Kosmidis
- Department of Physics, University of Thessaloniki, Thessaloniki, Greece
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25
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Dokoumetzidis A, Karalis V, Iliadis A, Macheras P. The heterogeneous course of drug transit through the body. Trends Pharmacol Sci 2004; 25:140-6. [PMID: 15019269 DOI: 10.1016/j.tips.2004.01.008] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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