In this paper, we investigate an axisymmetric model of intimal thickening using hyperelasticity theory. Our model describes the growth of the arterial intima due to cell proliferation which, in turn, is driven by the release of a cytokine such as platelet-derived growth factor (PDGF). With the growth rate tied to both local stress and the local concentration of PDGF, we derive a quadruple free boundary problem with different regions of the vessel wall characterized by different homeostatic stress. We compare our model predictions to rabbit and rodent models of atherosclerosis and find that in order to achieve the growth rates reported in the experiments, growth must be mainly cytokine induced rather than stress induced. Our model is also able to reproduce Glagov remodelling where, as a vessel becomes more diseased, the lumen expands before rapidly contracting.
{"title":"A biochemical and mechanical model of injury-induced intimal thickening","authors":"Pak-Wing Fok;Rebecca Sanft","doi":"10.1093/imammb/dqv040","DOIUrl":"https://doi.org/10.1093/imammb/dqv040","url":null,"abstract":"In this paper, we investigate an axisymmetric model of intimal thickening using hyperelasticity theory. Our model describes the growth of the arterial intima due to cell proliferation which, in turn, is driven by the release of a cytokine such as platelet-derived growth factor (PDGF). With the growth rate tied to both local stress and the local concentration of PDGF, we derive a quadruple free boundary problem with different regions of the vessel wall characterized by different homeostatic stress. We compare our model predictions to rabbit and rodent models of atherosclerosis and find that in order to achieve the growth rates reported in the experiments, growth must be mainly cytokine induced rather than stress induced. Our model is also able to reproduce Glagov remodelling where, as a vessel becomes more diseased, the lumen expands before rapidly contracting.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"34 1","pages":"77-108"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49992395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lydia Hill;Mark A. J. Chaplain;Roland Wolf;Yury Kapelyukh
The Cytochrome P450 (CYP) system is involved in 90% of the human body's interactions with xenobiotics and due to this, it has become an area of avid research including the creation of transgenic mice. This paper proposes a three-compartment model which is used to explain the drug metabolism in the Hepatic Reductase Null (HRN) mouse developed by the University of Dundee (Henderson, C. J., Otto, D. M. E., Carrie, D., Magnuson, M. A., McLaren, A. W., Rosewell, I. and Wolf, C. R. (2003) Inactivation of the hepatic cytochrome p450 system by conditional deletion of hepatic cytochrome p450 reductase. J. Biol. Chem. 278, 13480–13486). The model is compared with a two-compartment model using experimental data from studies using wild-type and HRN mice. This comparison allowed for metabolic differences between the two types of mice to be isolated. The three sets of drug data (Gefitinib, Midazolam and Thalidomide) showed that the transgenic mouse has a decreased rate of metabolism.
{"title":"The usage of a three-compartment model to investigate the metabolic differences between hepatic reductase null and wild-type mice","authors":"Lydia Hill;Mark A. J. Chaplain;Roland Wolf;Yury Kapelyukh","doi":"10.1093/imammb/dqv029","DOIUrl":"10.1093/imammb/dqv029","url":null,"abstract":"The Cytochrome P450 (CYP) system is involved in 90% of the human body's interactions with xenobiotics and due to this, it has become an area of avid research including the creation of transgenic mice. This paper proposes a three-compartment model which is used to explain the drug metabolism in the Hepatic Reductase Null (HRN) mouse developed by the University of Dundee (Henderson, C. J., Otto, D. M. E., Carrie, D., Magnuson, M. A., McLaren, A. W., Rosewell, I. and Wolf, C. R. (2003) Inactivation of the hepatic cytochrome p450 system by conditional deletion of hepatic cytochrome p450 reductase. J. Biol. Chem. 278, 13480–13486). The model is compared with a two-compartment model using experimental data from studies using wild-type and HRN mice. This comparison allowed for metabolic differences between the two types of mice to be isolated. The three sets of drug data (Gefitinib, Midazolam and Thalidomide) showed that the transgenic mouse has a decreased rate of metabolism.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"34 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34236577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The tumour growth paradox refers to the observation that incomplete treatment of cancers can enhance their growth. As shown here and elsewhere, the existence of cancer stem cells (CSCs) can explain this effect. CSC are less sensitive to treatments, hence any stress applied to the tumour selects for CSC, thereby increasing the fitness of the tumour. In this paper, we use a mathematical model to understand the role of CSC in the progression of cancer. Our model is a rather general system of integro-differential equations for tumour growth and tumour spread. Such a model has never been analysed, and we prove results on local and global existence of solutions, their uniqueness and their boundedness. We show numerically that this model exhibits the tumour growth paradox for all parameters tested. This effect becomes more relevant for small renewal rate of the CSC.
{"title":"A non-local model for cancer stem cells and the tumour growth paradox","authors":"I. Borsi;A. Fasano;M. Primicerio;T. Hillen","doi":"10.1093/imammb/dqv037","DOIUrl":"https://doi.org/10.1093/imammb/dqv037","url":null,"abstract":"The tumour growth paradox refers to the observation that incomplete treatment of cancers can enhance their growth. As shown here and elsewhere, the existence of cancer stem cells (CSCs) can explain this effect. CSC are less sensitive to treatments, hence any stress applied to the tumour selects for CSC, thereby increasing the fitness of the tumour. In this paper, we use a mathematical model to understand the role of CSC in the progression of cancer. Our model is a rather general system of integro-differential equations for tumour growth and tumour spread. Such a model has never been analysed, and we prove results on local and global existence of solutions, their uniqueness and their boundedness. We show numerically that this model exhibits the tumour growth paradox for all parameters tested. This effect becomes more relevant for small renewal rate of the CSC.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"34 1","pages":"59-75"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv037","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49992396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a mathematical and statistical methodology for estimation of important unobservable characteristics of the individual natural history of cancer from a sample of cross-sectional diameters of liver metastases measured at autopsy. Estimation of the natural history of cancer is based on a previously proposed stochastic model of cancer progression tailored to this type of observations. The model accounts for primary tumour growth, shedding of metastases, their selection, latency and growth in a given secondary site. The model was applied to the aforementioned data on 428 liver metastases detected in one untreated small cell lung cancer patient. Identifiable model parameters were estimated by the method of maximum likelihood and through minimizing the $L^{2}$ distance between theoretical and empirical cumulative distribution functions. The model with optimal parameters provided an excellent fit to the data. Results of data analysis support, if only indirectly, the hypothesis of the existence of stem-like cancer cells in the case of small cell lung carcinoma and point to the possibility of suppression of metastatic growth by a large primary tumour. They also lead to determination of the lower and upper bounds for the age of cancer onset and expected duration of metastatic latency. Finally, model-based inference on the patient's natural history of cancer allowed us to conclude that resection of the primary tumour would most likely not have had a curative effect.
{"title":"Uncovering the natural history of cancer from post-mortem cross-sectional diameters of hepatic metastases","authors":"Leonid Hanin;Jason Rose","doi":"10.1093/imammb/dqv026","DOIUrl":"10.1093/imammb/dqv026","url":null,"abstract":"We develop a mathematical and statistical methodology for estimation of important unobservable characteristics of the individual natural history of cancer from a sample of cross-sectional diameters of liver metastases measured at autopsy. Estimation of the natural history of cancer is based on a previously proposed stochastic model of cancer progression tailored to this type of observations. The model accounts for primary tumour growth, shedding of metastases, their selection, latency and growth in a given secondary site. The model was applied to the aforementioned data on 428 liver metastases detected in one untreated small cell lung cancer patient. Identifiable model parameters were estimated by the method of maximum likelihood and through minimizing the \u0000<tex>$L^{2}$</tex>\u0000 distance between theoretical and empirical cumulative distribution functions. The model with optimal parameters provided an excellent fit to the data. Results of data analysis support, if only indirectly, the hypothesis of the existence of stem-like cancer cells in the case of small cell lung carcinoma and point to the possibility of suppression of metastatic growth by a large primary tumour. They also lead to determination of the lower and upper bounds for the age of cancer onset and expected duration of metastatic latency. Finally, model-based inference on the patient's natural history of cancer allowed us to conclude that resection of the primary tumour would most likely not have had a curative effect.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 4","pages":"397-416"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33892862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study in detail the fluid dynamics system proposed in Clarelli et al. (2013, J. Math. Biol., 66, 1387–1408) to model the formation of cyanobacteria biofilms. After analysing the linear stability of the unique non-trivial equilibrium of the system, we introduce in the model the influence of light and temperature, which are two important factors for the development of a cyanobacteria biofilm. Since the values of the coefficients we use for our simulations are estimated through information found in the literature, some sensitivity and robustness analyses on these parameters are performed. All these elements enable us to control and to validate the model we have already derived and to present some numerical simulations in the 2D and the 3D cases.
本文详细研究了Clarelli et al. (2013, J. Math.)提出的流体动力学系统。医学杂志。(生物工程学报，66,1387-1408)来模拟蓝藻生物膜的形成。在分析了系统唯一非平凡平衡的线性稳定性后，我们在模型中引入了光和温度的影响，这是蓝藻生物膜发展的两个重要因素。由于我们用于模拟的系数值是通过文献中的信息估计的，因此对这些参数进行了一些敏感性和鲁棒性分析。所有这些元素使我们能够控制和验证我们已经导出的模型，并在2D和3D情况下提供一些数值模拟。
{"title":"A fluid dynamics multidimensional model of biofilm growth: stability, influence of environment and sensitivity","authors":"F. Clarelli;C. Di Russo;R. Natalini;M. Ribot","doi":"10.1093/imammb/dqv024","DOIUrl":"10.1093/imammb/dqv024","url":null,"abstract":"In this article, we study in detail the fluid dynamics system proposed in Clarelli et al. (2013, J. Math. Biol., 66, 1387–1408) to model the formation of cyanobacteria biofilms. After analysing the linear stability of the unique non-trivial equilibrium of the system, we introduce in the model the influence of light and temperature, which are two important factors for the development of a cyanobacteria biofilm. Since the values of the coefficients we use for our simulations are estimated through information found in the literature, some sensitivity and robustness analyses on these parameters are performed. All these elements enable us to control and to validate the model we have already derived and to present some numerical simulations in the 2D and the 3D cases.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 4","pages":"371-395"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34300927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peinan Ge;William J. Bottega;Jonathan L. Prenner;Howard F. Fine
A mechanics-based mathematical model of an eye possessing a posterior retinal detachment is presented for the case where an encircling scleral buckle (a cerclage) is sutured around the equator of the eye. The mechanical behaviour of the retina and the globe, both before and after applying the cerclage, is studied. An energy formulation yields the self-consistent equations of equilibrium and boundary conditions of the ocular system, and analytical solutions are established for the scleral buckle, for the globe and for the detached segment of the retina. Results of numerical simulations based on the solutions unveil characteristic behaviour of the ocular system, and demonstrate the influence of the scleral buckle, as well as of the pressure difference between the vitreous cavity and the subretinal space, on the deformation of the eye and on closing the region of retinal detachment. The results indicate that a scleral buckle encircling the equator, normally used for closing retinal tears and associated retinal detachments in the immediate vicinity of the buckle, can have a marked influence on bringing the detached segment of neurosensory retina back into contact with the retinal pigment epithelium, even for detachments at the posterior of the eye.
{"title":"On the influence of an equatorial cerclage on closure of posterior retinal detachment","authors":"Peinan Ge;William J. Bottega;Jonathan L. Prenner;Howard F. Fine","doi":"10.1093/imammb/dqv028","DOIUrl":"10.1093/imammb/dqv028","url":null,"abstract":"A mechanics-based mathematical model of an eye possessing a posterior retinal detachment is presented for the case where an encircling scleral buckle (a cerclage) is sutured around the equator of the eye. The mechanical behaviour of the retina and the globe, both before and after applying the cerclage, is studied. An energy formulation yields the self-consistent equations of equilibrium and boundary conditions of the ocular system, and analytical solutions are established for the scleral buckle, for the globe and for the detached segment of the retina. Results of numerical simulations based on the solutions unveil characteristic behaviour of the ocular system, and demonstrate the influence of the scleral buckle, as well as of the pressure difference between the vitreous cavity and the subretinal space, on the deformation of the eye and on closing the region of retinal detachment. The results indicate that a scleral buckle encircling the equator, normally used for closing retinal tears and associated retinal detachments in the immediate vicinity of the buckle, can have a marked influence on bringing the detached segment of neurosensory retina back into contact with the retinal pigment epithelium, even for detachments at the posterior of the eye.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 4","pages":"417-433"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33991424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Release of natural enemies to control pest populations is a common strategy in biological control. However, its effectiveness is supposed to be impaired, among other factors, by Allee effects in the biological control agent and by the fact that introduced pest natural enemies interact with some native species of the ecosystem. In this work, we devise a tritrophic food chain model where the assumptions previously raised are proved correct when a hyperpredator attacks the introduced pest natural enemy by a functional response type 2 or 3. Moreover, success of pest control is shown to be related to the release of large amounts (i.e., inundative releases) of natural enemies.
{"title":"Allee effects in tritrophic food chains: some insights in pest biological control","authors":"Michel Iskin da S. Costa;Lucas dos Anjos","doi":"10.1093/imammb/dqv027","DOIUrl":"10.1093/imammb/dqv027","url":null,"abstract":"Release of natural enemies to control pest populations is a common strategy in biological control. However, its effectiveness is supposed to be impaired, among other factors, by Allee effects in the biological control agent and by the fact that introduced pest natural enemies interact with some native species of the ecosystem. In this work, we devise a tritrophic food chain model where the assumptions previously raised are proved correct when a hyperpredator attacks the introduced pest natural enemy by a functional response type 2 or 3. Moreover, success of pest control is shown to be related to the release of large amounts (i.e., inundative releases) of natural enemies.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 4","pages":"461-474"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34115664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christian Engwer;Alexander Hunt;Christina Surulescu
Glioma is a common type of primary brain tumour, with a strongly invasive potential, often exhibiting non-uniform, highly irregular growth. This makes it difficult to assess the degree of extent of the tumour, hence bringing about a supplementary challenge for the treatment. It is therefore necessary to understand the migratory behaviour of glioma in greater detail. In this paper, we propose a multiscale model for glioma growth and migration. Our model couples the microscale dynamics (reduced to the binding of surface receptors to the surrounding tissue) with a kinetic transport equation for the cell density on the mesoscopic level of individual cells. On the latter scale, we also include the proliferation of tumour cells via effects of interaction with the tissue. An adequate parabolic scaling yields a convection–diffusion–reaction equation, for which the coefficients can be explicitly determined from the information about the tissue obtained by diffusion tensor imaging (DTI). Numerical simulations relying on DTI measurements confirm the biological findings that glioma spread along white matter tracts.
{"title":"Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings","authors":"Christian Engwer;Alexander Hunt;Christina Surulescu","doi":"10.1093/imammb/dqv030","DOIUrl":"10.1093/imammb/dqv030","url":null,"abstract":"Glioma is a common type of primary brain tumour, with a strongly invasive potential, often exhibiting non-uniform, highly irregular growth. This makes it difficult to assess the degree of extent of the tumour, hence bringing about a supplementary challenge for the treatment. It is therefore necessary to understand the migratory behaviour of glioma in greater detail. In this paper, we propose a multiscale model for glioma growth and migration. Our model couples the microscale dynamics (reduced to the binding of surface receptors to the surrounding tissue) with a kinetic transport equation for the cell density on the mesoscopic level of individual cells. On the latter scale, we also include the proliferation of tumour cells via effects of interaction with the tissue. An adequate parabolic scaling yields a convection–diffusion–reaction equation, for which the coefficients can be explicitly determined from the information about the tissue obtained by diffusion tensor imaging (DTI). Numerical simulations relying on DTI measurements confirm the biological findings that glioma spread along white matter tracts.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 4","pages":"435-459"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33998591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A novel theoretical method is presented for simulating the spatially resolved convective and diffusive transport of reacting solutes between microvascular networks and the surrounding tissues. The method allows for efficient computational solution of problems involving convection and non-linear binding of solutes in blood flowing through microvascular networks with realistic 3D geometries, coupled with transvascular exchange and diffusion and reaction in the surrounding tissue space. The method is based on a Green's function approach, in which the solute concentration distribution in the tissue is expressed as a sum of fields generated by time-varying distributions of discrete sources and sinks. As an example of the application of the method, the washout of an inert diffusible tracer substance from a tissue region perfused by a network of microvessels is simulated, showing its dependence on the solute's transvascular permeability and tissue diffusivity. Exponential decay of the washout concentration is predicted, with rate constants that are about 10-30% lower than the rate constants for a tissue cylinder model with the same vessel length, vessel surface area and blood flow rate per tissue volume.
{"title":"A Green's function method for simulation of time-dependent solute transport and reaction in realistic microvascular geometries","authors":"Timothy W. Secomb","doi":"10.1093/imammb/dqv031","DOIUrl":"10.1093/imammb/dqv031","url":null,"abstract":"A novel theoretical method is presented for simulating the spatially resolved convective and diffusive transport of reacting solutes between microvascular networks and the surrounding tissues. The method allows for efficient computational solution of problems involving convection and non-linear binding of solutes in blood flowing through microvascular networks with realistic 3D geometries, coupled with transvascular exchange and diffusion and reaction in the surrounding tissue space. The method is based on a Green's function approach, in which the solute concentration distribution in the tissue is expressed as a sum of fields generated by time-varying distributions of discrete sources and sinks. As an example of the application of the method, the washout of an inert diffusible tracer substance from a tissue region perfused by a network of microvessels is simulated, showing its dependence on the solute's transvascular permeability and tissue diffusivity. Exponential decay of the washout concentration is predicted, with rate constants that are about 10-30% lower than the rate constants for a tissue cylinder model with the same vessel length, vessel surface area and blood flow rate per tissue volume.","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 4","pages":"475-494"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqv031","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34236576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Back matter","authors":"","doi":"","DOIUrl":"https://doi.org/","url":null,"abstract":"","PeriodicalId":94130,"journal":{"name":"Mathematical medicine and biology : a journal of the IMA","volume":"33 3","pages":"1-2"},"PeriodicalIF":0.0,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016811/8189469/08189478.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68031767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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