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1551-0018
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Mathematical Biosciences & Engineering
2009 , Volume 6 , Issue 3
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2009, 6(3): 427-450
doi: 10.3934/mbe.2009.6.427
+[Abstract](1071)
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Abstract:
We formulate an HIV/AIDS deterministic model which incorporates differential infectivity and disease progression for treatment-naive and treatment-experienced HIV/AIDS infectives. To illustrate our model, we have applied it to estimate adult HIV prevalence, the HIV population, the number of new infectives and the number of AIDS deaths for Botswana for the period 1984 to 2012. It is found that the prevalence peaked in the year 2000 and the HIV population is now decreasing. We have also found that under the current conditions, the reproduction number is $R_c\approx1.3$, which is less than the 2004 estimate of $R_c$ ≃ 4 by [11] and [13]. The results in this study suggest that the HAART program has yielded positive results for Botswana.
We formulate an HIV/AIDS deterministic model which incorporates differential infectivity and disease progression for treatment-naive and treatment-experienced HIV/AIDS infectives. To illustrate our model, we have applied it to estimate adult HIV prevalence, the HIV population, the number of new infectives and the number of AIDS deaths for Botswana for the period 1984 to 2012. It is found that the prevalence peaked in the year 2000 and the HIV population is now decreasing. We have also found that under the current conditions, the reproduction number is $R_c\approx1.3$, which is less than the 2004 estimate of $R_c$ ≃ 4 by [11] and [13]. The results in this study suggest that the HAART program has yielded positive results for Botswana.
2009, 6(3): 451-467
doi: 10.3934/mbe.2009.6.451
+[Abstract](1214)
+[PDF](333.4KB)
Abstract:
A mathematical model is used to investigate the effectiveness of the chemotherapy drug Topotecan against neuroblastoma. Optimal control theory is applied to minimize the tumor volume and the amount of drug utilized. The model incorporates a state constraint that requires the level of circulating neutrophils (white blood cells that form an integral part of the immune system) to remain above an acceptable value. The treatment schedule is designed to simultaneously satisfy this constraint and achieve the best results in fighting the tumor. Existence and uniqueness of the solution of the optimality system, which is the state system coupled with the adjoint system, is established. Numerical simulations are given to demonstrate the behavior of the tumor and the immune system components represented in the model.
A mathematical model is used to investigate the effectiveness of the chemotherapy drug Topotecan against neuroblastoma. Optimal control theory is applied to minimize the tumor volume and the amount of drug utilized. The model incorporates a state constraint that requires the level of circulating neutrophils (white blood cells that form an integral part of the immune system) to remain above an acceptable value. The treatment schedule is designed to simultaneously satisfy this constraint and achieve the best results in fighting the tumor. Existence and uniqueness of the solution of the optimality system, which is the state system coupled with the adjoint system, is established. Numerical simulations are given to demonstrate the behavior of the tumor and the immune system components represented in the model.
2009, 6(3): 469-492
doi: 10.3934/mbe.2009.6.469
+[Abstract](1485)
+[PDF](280.0KB)
Abstract:
Mathematical models provide a powerful tool for investigating the dynamics and control of infectious diseases, but quantifying the underlying epidemic structure can be challenging especially for new and under-studied diseases. Variations of standard SIR, SIRS, and SEIR epidemiological models are considered to determine the sensitivity of these models to various parameter values that may not be fully known when the models are used to investigate emerging diseases. Optimal control theory is applied to suggest the most effective mitigation strategy to minimize the number of individuals who become infected in the course of an infection while efficiently balancing vaccination and treatment applied to the models with various cost scenarios. The optimal control simulations suggest that regardless of the particular epidemiological structure and of the comparative cost of mitigation strategies, vaccination, if available, would be a crucial piece of any intervention plan.
Mathematical models provide a powerful tool for investigating the dynamics and control of infectious diseases, but quantifying the underlying epidemic structure can be challenging especially for new and under-studied diseases. Variations of standard SIR, SIRS, and SEIR epidemiological models are considered to determine the sensitivity of these models to various parameter values that may not be fully known when the models are used to investigate emerging diseases. Optimal control theory is applied to suggest the most effective mitigation strategy to minimize the number of individuals who become infected in the course of an infection while efficiently balancing vaccination and treatment applied to the models with various cost scenarios. The optimal control simulations suggest that regardless of the particular epidemiological structure and of the comparative cost of mitigation strategies, vaccination, if available, would be a crucial piece of any intervention plan.
2009, 6(3): 493-508
doi: 10.3934/mbe.2009.6.493
+[Abstract](1168)
+[PDF](357.3KB)
Abstract:
Implantation of drug eluting stents following percutaneous transluminal angioplasty has revealed a well established technique for treating occlusions caused by the atherosclerotic plaque. However, due to the risk of vascular re-occlusion, other alternative therapeutic strategies of drug delivery are currently being investigated. Polymeric endoluminal pave stenting is an emerging technology for preventing blood erosion and for optimizing drug release. The classical and novel methodologies are compared through a mathematical model able to predict the evolution of the drug concentration in a cross-section of the wall. Though limited to an idealized configuration, the present model is shown to catch most of the relevant aspects of the drug dynamics in a delivery system. Results of numerical simulations shows that a bi-layer gel paved stenting guarantees a uniform drug elution and a prolonged perfusion of the tissues, and remains a promising and effective technique in drug delivery.
Implantation of drug eluting stents following percutaneous transluminal angioplasty has revealed a well established technique for treating occlusions caused by the atherosclerotic plaque. However, due to the risk of vascular re-occlusion, other alternative therapeutic strategies of drug delivery are currently being investigated. Polymeric endoluminal pave stenting is an emerging technology for preventing blood erosion and for optimizing drug release. The classical and novel methodologies are compared through a mathematical model able to predict the evolution of the drug concentration in a cross-section of the wall. Though limited to an idealized configuration, the present model is shown to catch most of the relevant aspects of the drug dynamics in a delivery system. Results of numerical simulations shows that a bi-layer gel paved stenting guarantees a uniform drug elution and a prolonged perfusion of the tissues, and remains a promising and effective technique in drug delivery.
2009, 6(3): 509-520
doi: 10.3934/mbe.2009.6.509
+[Abstract](1022)
+[PDF](265.6KB)
Abstract:
This paper shows how occupancy urn models can be used to derive useful results in epidemiology. First we show how simple epidemic models can be re-interpreted in terms of occupancy problems. We use this reformulation to derive an expression for the expected epidemic size, that is, the total number of infected at the end of an outbreak. We also use this approach to derive point and interval estimates of the Basic Reproduction Ratio, $R_{0}$. We show that this construction does not require that the underlying SIR model be a homogeneous Poisson process, leading to a geometric distribution for the number of contacts before removal, but instead it supports a general distribution. The urn model construction is easy to handle and represents a rich field for further exploitation.
This paper shows how occupancy urn models can be used to derive useful results in epidemiology. First we show how simple epidemic models can be re-interpreted in terms of occupancy problems. We use this reformulation to derive an expression for the expected epidemic size, that is, the total number of infected at the end of an outbreak. We also use this approach to derive point and interval estimates of the Basic Reproduction Ratio, $R_{0}$. We show that this construction does not require that the underlying SIR model be a homogeneous Poisson process, leading to a geometric distribution for the number of contacts before removal, but instead it supports a general distribution. The urn model construction is easy to handle and represents a rich field for further exploitation.
2009, 6(3): 521-546
doi: 10.3934/mbe.2009.6.521
+[Abstract](1288)
+[PDF](830.1KB)
Abstract:
In this paper we consider chemotherapy in a spatial model of tumor growth. The model, which is of reaction-diffusion type, takes into account the complex interactions between the tumor and surrounding stromal cells by including densities of endothelial cells and the extra-cellular matrix. When no treatment is applied the model reproduces the typical dynamics of early tumor growth. The initially avascular tumor reaches a diffusion limited size of the order of millimeters and initiates angiogenesis through the release of vascular endothelial growth factor (VEGF) secreted by hypoxic cells in the core of the tumor. This stimulates endothelial cells to migrate towards the tumor and establishes a nutrient supply sufficient for sustained invasion. To this model we apply cytostatic treatment in the form of a VEGF-inhibitor, which reduces the proliferation and chemotaxis of endothelial cells. This treatment has the capability to reduce tumor mass, but more importantly, we were able to determine that inhibition of endothelial cell proliferation is the more important of the two cellular functions targeted by the drug. Further, we considered the application of a cytotoxic drug that targets proliferating tumor cells. The drug was treated as a diffusible substance entering the tissue from the blood vessels. Our results show that depending on the characteristics of the drug it can either reduce the tumor mass significantly or in fact accelerate the growth rate of the tumor. This result seems to be due to complicated interplay between the stromal and tumor cell types and highlights the importance of considering chemotherapy in a spatial context.
In this paper we consider chemotherapy in a spatial model of tumor growth. The model, which is of reaction-diffusion type, takes into account the complex interactions between the tumor and surrounding stromal cells by including densities of endothelial cells and the extra-cellular matrix. When no treatment is applied the model reproduces the typical dynamics of early tumor growth. The initially avascular tumor reaches a diffusion limited size of the order of millimeters and initiates angiogenesis through the release of vascular endothelial growth factor (VEGF) secreted by hypoxic cells in the core of the tumor. This stimulates endothelial cells to migrate towards the tumor and establishes a nutrient supply sufficient for sustained invasion. To this model we apply cytostatic treatment in the form of a VEGF-inhibitor, which reduces the proliferation and chemotaxis of endothelial cells. This treatment has the capability to reduce tumor mass, but more importantly, we were able to determine that inhibition of endothelial cell proliferation is the more important of the two cellular functions targeted by the drug. Further, we considered the application of a cytotoxic drug that targets proliferating tumor cells. The drug was treated as a diffusible substance entering the tissue from the blood vessels. Our results show that depending on the characteristics of the drug it can either reduce the tumor mass significantly or in fact accelerate the growth rate of the tumor. This result seems to be due to complicated interplay between the stromal and tumor cell types and highlights the importance of considering chemotherapy in a spatial context.
2009, 6(3): 547-559
doi: 10.3934/mbe.2009.6.547
+[Abstract](1173)
+[PDF](172.8KB)
Abstract:
The mathematical modeling of tumor growth is an approach to explain the complex nature of these systems. A model that describes tumor growth was obtained by using a mesoscopic formalism and fractal dimension. This model theoretically predicts the relation between the morphology of the cell pattern and the mitosis/apoptosis quotient that helps to predict tumor growth from tumoral cells fractal dimension. The relation between the tumor macroscopic morphology and the cell pattern morphology is also determined. This could explain why the interface fractal dimension decreases with the increase of the cell pattern fractal dimension and consequently with the increase of the mitosis/apoptosis relation. Indexes to characterize tumoral cell proliferation and invasion capacities are proposed and used to predict the growth of different types of tumors. These indexes also show that the proliferation capacity is directly proportional to the invasion capacity. The proposed model assumes: i) only interface cells proliferate and invade the host, and ii) the fractal dimension of tumoral cell patterns, can reproduce the Gompertzian growth law.
The mathematical modeling of tumor growth is an approach to explain the complex nature of these systems. A model that describes tumor growth was obtained by using a mesoscopic formalism and fractal dimension. This model theoretically predicts the relation between the morphology of the cell pattern and the mitosis/apoptosis quotient that helps to predict tumor growth from tumoral cells fractal dimension. The relation between the tumor macroscopic morphology and the cell pattern morphology is also determined. This could explain why the interface fractal dimension decreases with the increase of the cell pattern fractal dimension and consequently with the increase of the mitosis/apoptosis relation. Indexes to characterize tumoral cell proliferation and invasion capacities are proposed and used to predict the growth of different types of tumors. These indexes also show that the proliferation capacity is directly proportional to the invasion capacity. The proposed model assumes: i) only interface cells proliferate and invade the host, and ii) the fractal dimension of tumoral cell patterns, can reproduce the Gompertzian growth law.
2009, 6(3): 561-572
doi: 10.3934/mbe.2009.6.561
+[Abstract](1058)
+[PDF](374.6KB)
Abstract:
Pseudo-spectral approximations are constructed for the model equations describing the population kinetics of human tumor cells in vitro and their responses to radiotherapy or chemotherapy. These approximations are more efficient than finite-difference approximations. The spectral accuracy of the pseudo-spectral method allows us to resolve the model with a much smaller number of spatial grid-points than required for the finite-difference method to achieve comparable accuracy. This is demonstrated by numerical experiments which show a good agreement between predicted and experimental data.
Pseudo-spectral approximations are constructed for the model equations describing the population kinetics of human tumor cells in vitro and their responses to radiotherapy or chemotherapy. These approximations are more efficient than finite-difference approximations. The spectral accuracy of the pseudo-spectral method allows us to resolve the model with a much smaller number of spatial grid-points than required for the finite-difference method to achieve comparable accuracy. This is demonstrated by numerical experiments which show a good agreement between predicted and experimental data.
2009, 6(3): 573-583
doi: 10.3934/mbe.2009.6.573
+[Abstract](1053)
+[PDF](390.9KB)
Abstract:
Understanding the dynamics of human hosts and tumors is of critical importance. A mathematical model was developed that explored the immune response to tumors that was used to study a special type of treatment [3]. This treatment approach uses elements of the host to boost its immune response in the hopes that the host can clear the tumor. This model was extensively studied using numerical simulation, however no global analytical results were originally presented. In this work we explore the global dynamics to show under what conditions tumor clearance can be achieved.
Understanding the dynamics of human hosts and tumors is of critical importance. A mathematical model was developed that explored the immune response to tumors that was used to study a special type of treatment [3]. This treatment approach uses elements of the host to boost its immune response in the hopes that the host can clear the tumor. This model was extensively studied using numerical simulation, however no global analytical results were originally presented. In this work we explore the global dynamics to show under what conditions tumor clearance can be achieved.
2009, 6(3): 585-590
doi: 10.3934/mbe.2009.6.585
+[Abstract](1092)
+[PDF](127.1KB)
Abstract:
We study global asymptotic properties of a continuous time Leslie-Gower food chain model. We construct a Lyapunov function which enables us to establish global asymptotic stability of the unique coexisting equilibrium state.
We study global asymptotic properties of a continuous time Leslie-Gower food chain model. We construct a Lyapunov function which enables us to establish global asymptotic stability of the unique coexisting equilibrium state.
2009, 6(3): 591-602
doi: 10.3934/mbe.2009.6.591
+[Abstract](1321)
+[PDF](514.2KB)
Abstract:
Experiments have established that different radiation types have different magnitudes of biological response. When biological response is defined in terms of the Relative Biologic Effectiveness (RBE) and different radiation type is characterized by Linear Energy Transfer (LET), the plot of the RBE versus LET (RBE-LET) curve shows RBE to increase with increasing LET, to reach a maximum, and to decrease with further increasing LET. Perhaps due to the descriptive nature of biology, most quantitative models for the RBE-LET curve ignore the reality of the underlying molecular biology. On the other hand, the molecular basis for the RBE-LET curve is not completely known despite recent efforts.
Here we introduce a differential equation formulation for a signal-and-system model that sees cells as systems, different radiation types as input, and cellular responses as output. Because of scant knowledge of the underlying biochemical network, the current version is necessarily a work in progress. It explains the RBE-LET curve using not just input parameters but also systems internal state parameters. These systems internal state parameters represent parts of a biochemical network within a cell. Although multiple biochemical parts may well be involved, the shape of the RBE-LET curve is reproduced when only three system parameters are related to three biochemical parts: the molecular machinery for DNA double strand break repair; the molecular pathways for handling oxidative stress; and the radiolytic products of the cellular water.
Despite being a simplified ''toy model,'' changes in the systems state parameters lead to model curves that are refutable in a modern molecular biology laboratory. As the parts in the biochemical network of the radiation response are being further elucidated, this model can incorporate new systems state parameters to allow a more accurate fit.
Experiments have established that different radiation types have different magnitudes of biological response. When biological response is defined in terms of the Relative Biologic Effectiveness (RBE) and different radiation type is characterized by Linear Energy Transfer (LET), the plot of the RBE versus LET (RBE-LET) curve shows RBE to increase with increasing LET, to reach a maximum, and to decrease with further increasing LET. Perhaps due to the descriptive nature of biology, most quantitative models for the RBE-LET curve ignore the reality of the underlying molecular biology. On the other hand, the molecular basis for the RBE-LET curve is not completely known despite recent efforts.
Here we introduce a differential equation formulation for a signal-and-system model that sees cells as systems, different radiation types as input, and cellular responses as output. Because of scant knowledge of the underlying biochemical network, the current version is necessarily a work in progress. It explains the RBE-LET curve using not just input parameters but also systems internal state parameters. These systems internal state parameters represent parts of a biochemical network within a cell. Although multiple biochemical parts may well be involved, the shape of the RBE-LET curve is reproduced when only three system parameters are related to three biochemical parts: the molecular machinery for DNA double strand break repair; the molecular pathways for handling oxidative stress; and the radiolytic products of the cellular water.
Despite being a simplified ''toy model,'' changes in the systems state parameters lead to model curves that are refutable in a modern molecular biology laboratory. As the parts in the biochemical network of the radiation response are being further elucidated, this model can incorporate new systems state parameters to allow a more accurate fit.
2009, 6(3): 603-610
doi: 10.3934/mbe.2009.6.603
+[Abstract](1438)
+[PDF](142.8KB)
Abstract:
A recent paper (Math. Biosci. and Eng. (2008) 5:389-402) presented an SEIR model using an infinite delay to account for varying infectivity. The analysis in that paper did not resolve the global dynamics for R0 >1. Here, we show that the endemic equilibrium is globally stable for R0 >1. The proof uses a Lyapunov functional that includes an integral over all previous states.
A recent paper (Math. Biosci. and Eng. (2008) 5:389-402) presented an SEIR model using an infinite delay to account for varying infectivity. The analysis in that paper did not resolve the global dynamics for R0 >1. Here, we show that the endemic equilibrium is globally stable for R0 >1. The proof uses a Lyapunov functional that includes an integral over all previous states.
2009, 6(3): 611-627
doi: 10.3934/mbe.2009.6.611
+[Abstract](995)
+[PDF](490.9KB)
Abstract:
Previous studies on computer modeling of RF ablation with cooled electrodes modeled the internal cooling circuit by setting surface temperature at the coolant temperature (i.e., Dirichlet condition, DC). Our objective was to compare the temperature profiles computed from different thermal boundary conditions at the electrode-tissue interface. We built an analytical one-dimensional model based on a spherical electrode. Four cases were considered: A) DC with uniform initial condition, B) DC with pre-cooling period, C) Boundary condition based on Newton's cooling law (NC) with uniform initial condition, and D) NC with a pre-cooling period. The results showed that for a long time ($120$ s), the profiles obtained with (Cases B and D) and without (Cases A and C) considering pre-cooling are very similar. However, for shorter times ($<30$ s), Cases A and C overestimated the temperature at points away from the electrode-tissue interface. In the NC cases, this overestimation was more evident for higher values of the convective heat transfer coefficient ($h$). Finally, with NC, when $h$ was increased the temperature profiles became more similar to those with DC. The results suggest that theoretical modeling of RF ablation with cooled electrodes should consider: 1) the modeling of a pre-cooling period, especially if one is interested in the thermal profiles registered at the beginning of RF application; and 2) NC rather than DC, especially for low flow in the internal circuit.
Previous studies on computer modeling of RF ablation with cooled electrodes modeled the internal cooling circuit by setting surface temperature at the coolant temperature (i.e., Dirichlet condition, DC). Our objective was to compare the temperature profiles computed from different thermal boundary conditions at the electrode-tissue interface. We built an analytical one-dimensional model based on a spherical electrode. Four cases were considered: A) DC with uniform initial condition, B) DC with pre-cooling period, C) Boundary condition based on Newton's cooling law (NC) with uniform initial condition, and D) NC with a pre-cooling period. The results showed that for a long time ($120$ s), the profiles obtained with (Cases B and D) and without (Cases A and C) considering pre-cooling are very similar. However, for shorter times ($<30$ s), Cases A and C overestimated the temperature at points away from the electrode-tissue interface. In the NC cases, this overestimation was more evident for higher values of the convective heat transfer coefficient ($h$). Finally, with NC, when $h$ was increased the temperature profiles became more similar to those with DC. The results suggest that theoretical modeling of RF ablation with cooled electrodes should consider: 1) the modeling of a pre-cooling period, especially if one is interested in the thermal profiles registered at the beginning of RF application; and 2) NC rather than DC, especially for low flow in the internal circuit.
2009, 6(3): 629-647
doi: 10.3934/mbe.2009.6.629
+[Abstract](935)
+[PDF](267.0KB)
Abstract:
We apply basic tools of control theory to a chemostat model that describes the growth of one species of microorganisms that consume a limiting substrate. Under the assumption that available measurements of the model have fixed delay $\tau>0$, we design a family of feedback control laws with the objective of stabilizing the limiting substrate concentration in a fixed level. Effectiveness of this control problem is equivalent to global attractivity of a family of differential delay equations. We obtain sufficient conditions (upper bound for delay $\tau>0$ and properties of the feedback control) ensuring global attractivity and local stability. Illustrative examples are included.
We apply basic tools of control theory to a chemostat model that describes the growth of one species of microorganisms that consume a limiting substrate. Under the assumption that available measurements of the model have fixed delay $\tau>0$, we design a family of feedback control laws with the objective of stabilizing the limiting substrate concentration in a fixed level. Effectiveness of this control problem is equivalent to global attractivity of a family of differential delay equations. We obtain sufficient conditions (upper bound for delay $\tau>0$ and properties of the feedback control) ensuring global attractivity and local stability. Illustrative examples are included.
2009, 6(3): 649-661
doi: 10.3934/mbe.2009.6.649
+[Abstract](997)
+[PDF](352.9KB)
Abstract:
Anti-malarial drug resistance has been identified in many regions for a long time. In this paper we formulate a mathematical model of the spread of anti-malarial drug resistance in the population. The model is suitable for malarial situations in developing countries. We consider the sensitive and resistant strains of malaria. There are two basic reproduction ratios corresponding to the strains. If the ratios corresponding to the infections of the sensitive and resistant strains are not equal and they are greater than one, then there exist two endemic non-coexistent equilibria. In the case where the two ratios are equal and they are greater than one, the coexistence of the sensitive and resistant strains exist in the population. It is shown here that the recovery rates of the infected host and the proportion of anti-malarial drug treatment play important roles in the spread of anti-malarial drug resistance. The interesting phenomena of ''long-time" coexistence, which may explain the real situation in the field, could occur for long period of time when those parameters satisfy certain conditions. In regards to control strategy in the field, these results could give a good understanding of means of slowing down the spread of anti-malarial drug resistance.
Anti-malarial drug resistance has been identified in many regions for a long time. In this paper we formulate a mathematical model of the spread of anti-malarial drug resistance in the population. The model is suitable for malarial situations in developing countries. We consider the sensitive and resistant strains of malaria. There are two basic reproduction ratios corresponding to the strains. If the ratios corresponding to the infections of the sensitive and resistant strains are not equal and they are greater than one, then there exist two endemic non-coexistent equilibria. In the case where the two ratios are equal and they are greater than one, the coexistence of the sensitive and resistant strains exist in the population. It is shown here that the recovery rates of the infected host and the proportion of anti-malarial drug treatment play important roles in the spread of anti-malarial drug resistance. The interesting phenomena of ''long-time" coexistence, which may explain the real situation in the field, could occur for long period of time when those parameters satisfy certain conditions. In regards to control strategy in the field, these results could give a good understanding of means of slowing down the spread of anti-malarial drug resistance.
2009, 6(3): 663-682
doi: 10.3934/mbe.2009.6.663
+[Abstract](1191)
+[PDF](309.7KB)
Abstract:
Drug residence time in ''compartmentalized'' human body system had been studied from both deterministic and Markovian perspectives. However, probability and probability density functions for a drug molecule to be (1) in any compartment of study interest, (2) with any defined inter-compartment traveling route, and (3) with/without specified residence times in its visited compartments, has not been systemically reported. In Markovian view of compartmental system, mathematical solutions for the probability or probability density functions, for a drug molecule with any defined inter- compartment traveling routes in the system and/or with specified residence times in any visited compartments, are provided. Matrix convolution is defined and thus employed to facilitate methodology development. Laplace transformations are used to facilitate convolution operations in linear systems. This paper shows that the drug time-concentration function can be decomposed into the summation of a series of component functions, which is named as convolution expansion. The studied probability or probability density functions can be potentially engaged with physiological or pharmacological significances and thus be used to describe a broad range of drug exposure-response relationships.
Drug residence time in ''compartmentalized'' human body system had been studied from both deterministic and Markovian perspectives. However, probability and probability density functions for a drug molecule to be (1) in any compartment of study interest, (2) with any defined inter-compartment traveling route, and (3) with/without specified residence times in its visited compartments, has not been systemically reported. In Markovian view of compartmental system, mathematical solutions for the probability or probability density functions, for a drug molecule with any defined inter- compartment traveling routes in the system and/or with specified residence times in any visited compartments, are provided. Matrix convolution is defined and thus employed to facilitate methodology development. Laplace transformations are used to facilitate convolution operations in linear systems. This paper shows that the drug time-concentration function can be decomposed into the summation of a series of component functions, which is named as convolution expansion. The studied probability or probability density functions can be potentially engaged with physiological or pharmacological significances and thus be used to describe a broad range of drug exposure-response relationships.
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