CN111639454B - Method for predicting concentration distribution of magnetic fluid in biological model tissue based on dual-porosity model - Google Patents
Method for predicting concentration distribution of magnetic fluid in biological model tissue based on dual-porosity model Download PDFInfo
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Abstract
The invention relates to a method for predicting the concentration distribution of magnetic fluid in a biological model tissue based on a dual-porosity model, which comprises the steps of firstly establishing the biological tissue model, and obtaining the interstitial pressure distribution of a first tissue and a second tissue in the biological tissue model by introducing the dual-porosity fluid transportation model; then according to the obtained interstitial pressure distribution, solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial process of injecting the magnetic fluid into the biological tissue model by using a Navier-Stokes equation; and finally, according to the obtained flow velocity distribution of the magnetic fluid in the interstitium, obtaining the concentration distribution of the magnetic fluid in the interstitial space of the biological tissue model by utilizing a convection-diffusion equation. The method can predict the concentration distribution of the magnetic fluid between the interstitium.
Description
Technical Field
The invention relates to the technical field of fluid transport modeling, in particular to a method for predicting the concentration distribution of magnetic fluid in a biological model tissue based on a dual-porosity model.
Background
The transport of fluids in porous media has been a long-standing research topic and can be applied to various fields such as reaction engineering, seepage, injection molding, porous media reflection flow, dilute material transfer and the like. In addition, it is also an important key technology in the field of biological engineering, such as the transportation of magnetic fluid in biological tissues in magnetic hyperthermia. Therefore, the research on the concentration distribution of the magnetic fluid in the interstitium after injection and the prediction method have very important practical significance.
Disclosure of Invention
In view of the above, the present invention provides a method for predicting the concentration distribution of magnetic fluid in a biological model tissue based on a dual-porosity model, which can predict the concentration distribution of magnetic fluid between interstitials.
The invention is realized by adopting the following scheme: a method for predicting the concentration distribution of magnetic fluid in a biological model tissue based on a dual-porosity model specifically comprises the following steps:
establishing a biological tissue model, and obtaining interstitial pressure distribution of a first tissue and a second tissue in the biological tissue model by introducing a dual-porosity fluid transportation model;
according to the obtained interstitial pressure distribution, solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial process of injecting the magnetic fluid into the biological tissue model by using a Navier leaf-Stokes equation;
and obtaining the concentration distribution of the magnetic fluid in the interstitial tissues of the biological tissue model by utilizing a convection-diffusion equation according to the obtained flow velocity distribution of the magnetic fluid in the interstitial tissues.
Further, the establishing of the biological tissue model, and obtaining the interstitial pressure distribution of the first tissue and the second tissue in the biological tissue model by introducing the dual-porosity fluid transport model, specifically comprises the following steps:
firstly, establishing a biological tissue model which comprises a first tissue and a second tissue, and constructing a hollow cylinder on the basis of the biological tissue model for representing a needle hole model of a syringe, wherein the cylinder penetrates through the first tissue area and reaches the central position of the second tissue area;
the tissue in the biological tissue model is made to be dual-porosity, and the relationship between the flow velocity and the pressure of the magnetic fluid in the interstitial tissue of the model is described by applying Darcy's law:
wherein u is the magnetofluid velocity, κ is the interstitial water conductivity, and p is the tissue pressure; constructing a pressure distribution mathematical model of tissue interstitium in the biological tissue model by a Kedem-Katchalsky theory;
setting boundary conditions for the pressure distribution mathematical model according to Dirichlet boundary conditions of a set Kedem-Katchalsky theoretical model, and then solving the pressure distribution mathematical model by applying a finite element method and combining the boundary conditions to obtain interstitial pressure distribution conditions of the first tissue area and the second tissue area.
Further, the mathematical model of the pressure distribution of the tissue matrix in the biological tissue model constructed by the Kedem-Katchalsky theory is as follows:
wherein the subscript i has a value of 1 or 2, and when 1, indicates a first tissue, and when 2, indicates a second tissue;
representing the Hamiltonian, kiDenotes the permeability of the interstitium,. mu.denotes the dynamic viscosity of the magnetic fluid, piIndicating the pressure in the tissue interstitium,. phivThe source item is represented.
Further, the Dirichlet boundary condition of the Kedem-Katchalsky theoretical model is set to be that the initial pressure is a normal atmospheric pressure, namely P0＝1×105Pa。
Further, solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial process of injecting the magnetic fluid into the biological tissue model by using the navier-stokes equation according to the obtained interstitial pressure distribution specifically comprises the following steps:
constructing a Navier-Stokes equation for describing the process of the change of the flow velocity of the magnetic fluid injected into the second tissue area; wherein the navier-stokes equation is expressed as:
wherein, mufAnd ρfDynamic viscosity and density, p, of the magnetic fluid, respectivelyiRepresenting interstitial pressure;
the dirichlet boundary condition of equation (2) is set as: the initial speed of the magnetic fluid in the interstitium is zero;
and solving the flow velocity distribution condition of the magnetic fluid in the interstitial tissues by applying a pressure-velocity multi-physical field coupling analysis method.
Further, the pressure-velocity multi-physics coupling analysis method is a sequential coupling method, and particularly takes the solution data set of formula (1) as the initial value of formula (2) and is used for solving formula (2).
Further, the obtaining of the concentration distribution of the magnetic fluid in the interstitial tissue of the biological tissue model by using a convection-diffusion equation according to the obtained flow velocity distribution of the magnetic fluid in the interstitial tissue specifically includes the following steps:
constructing a convection-diffusion equation based on a dual-porosity fluid transport model, and taking the flow velocity distribution condition of the magnetic fluid in the interstitial tissues as input;
constructing a source term equation of the magnetic fluid passing through the vascular wall model:
in the formula, LPIs the hydraulic conductivity of the wall of the capillary vessel in the biological tissue model, S/V is the surface area per unit volume transported in the second tissue in the biological tissue model, pbIs the static pressure, sigma, of blood in a biological tissue modelsIs the osmotic reflection coefficient, pi, of plasma proteins in a biological tissue modelbIs the plasma protein oncotic pressure, pi, in a biological tissue modeliIs the interstitial swelling pressure, p, of the biological tissue modeliRepresenting interstitial pressure;
and solving the diffusion condition and the specific distribution in the second tissue area in the magnetofluid injection process by using a speed-concentration coupling analysis method.
Further, the convection-diffusion equation is:
wherein, the formula (3) is a convection-diffusion equation for the interstitium, and the formula (4) is a convection-diffusion equation for the blood vessel; in which the subscript i corresponds to different groupsRelated physical properties under the fabric, subscript v corresponds to physical properties related to the vascular system, parameter c represents the concentration value of the magnetic nanoparticles in the interstitium, v represents the flow velocity of the interstitium, D represents the diffusion coefficient of the magnetic fluid, FsRepresenting the solute transport forces between the interstitium and the blood vessels, FLExpressing the transport acting force of solute between the stroma and the lymphatic vessel, w expressing the weight value, and t expressing the magnetofluid transport time;
further, the speed-concentration multi-physical field coupling method is a sequential coupling method, namely, a solution data set of formula (2) is used as initial values of formulas (3) to (4) to solve formulas (3) to (4), and finally, the concentration distribution of the magnetic fluid in the tissue is obtained.
The invention also provides a dual porosity model based system for predicting the concentration distribution of a magnetic fluid in a biological model tissue, comprising a memory, a processor and computer program instructions stored on the memory and executable by the processor, which when executed by the processor implement the method steps as described in any one of the above.
Compared with the prior art, the invention has the following beneficial effects: the invention applies a multi-physical field coupling analysis method in the analysis of the magnetic fluid, and simultaneously covers the coupling of various physical fields such as pressure-speed, speed-concentration and the like; introducing a dual-porosity fluid transportation model, and applying the model to the establishment of a tissue interstitial injection model of a magnetic fluid to biological tissue model; the analysis of concentration diffusion distribution based on magnetofluid injection is realized. The method can predict the concentration distribution of the magnetic fluid between the interstitium.
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FIG. 1 is a schematic flow chart of a method according to an embodiment of the present invention.
FIG. 2 is a schematic view of a biological tissue model according to an embodiment of the present invention.
FIG. 3 is a schematic diagram of the magnetic fluid concentration distribution according to the embodiment of the present invention.
Detailed Description
The invention is further explained below with reference to the drawings and the embodiments.
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
As shown in fig. 1, the present embodiment provides a method for predicting the magnetic fluid concentration distribution in a biological model tissue based on a dual porosity model, which specifically includes the following steps:
step S1: establishing a biological tissue model, and obtaining interstitial pressure distribution of a first tissue and a second tissue in the biological tissue model by introducing a dual-porosity fluid transportation model;
step S2: according to the obtained interstitial pressure distribution, solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial process of injecting the magnetic fluid into the biological tissue model by using a Navier leaf-Stokes equation;
step S3: and obtaining the concentration distribution of the magnetic fluid in the interstitial tissues of the biological tissue model by utilizing a convection-diffusion equation according to the obtained flow velocity distribution of the magnetic fluid in the interstitial tissues.
In this embodiment, step S1 specifically includes the following steps:
firstly, establishing a biological tissue model which comprises a first tissue and a second tissue, and constructing a hollow cylinder on the basis of the biological tissue model for representing a needle hole model of a syringe, wherein the cylinder penetrates through the first tissue area and reaches the central position of the second tissue area;
making the tissue in the biological tissue model have dual porosity, and using Darcy's law to describe the flow speed and pressure of magnetic fluid in the interstitial tissue of the modelThe relationship is as follows:
wherein u is the magnetofluid velocity, κ is the interstitial water conductivity, and p is the tissue pressure; constructing a pressure distribution mathematical model of tissue interstitium in the biological tissue model by a Kedem-Katchalsky theory;
setting boundary conditions for the pressure distribution mathematical model according to actual conditions, and then solving the pressure distribution mathematical model by applying a finite element method and combining the boundary conditions to obtain interstitial pressure distribution conditions of the first tissue area and the second tissue area.
In this embodiment, in step S2, the mathematical model of the pressure distribution of the tissue matrix in the biological tissue model constructed by the Kedem-katcarasky theory is:
wherein the subscript i has a value of 1 or 2, and when 1, indicates a first tissue, and when 2, indicates a second tissue;
representing the Hamiltonian, kiDenotes the permeability of the interstitium,. mu.denotes the dynamic viscosity of the magnetic fluid, piIndicating the pressure in the tissue interstitium,. phivThe source item is represented.
Wherein, the Dirichlet boundary condition of the Kedem-Katchalsky theoretical model is set as the initial pressure being a normal atmospheric pressure, namely p0＝1×105Pa。
In this embodiment, in step S2, solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial space of the magnetic fluid injection into the biological tissue model by using the navier-stokes equation according to the obtained interstitial pressure distribution specifically includes the following steps:
constructing a Navier-Stokes equation for describing the process of the change of the flow velocity of the magnetic fluid injected into the second tissue area; wherein the Navier-Stokes equation is expressed as:
▽·ω＝0
wherein, mufAnd ρfDynamic viscosity and density, p, of the magnetic fluid, respectivelyiRepresenting interstitial pressure;
the dirichlet boundary condition of equation (2) is set as: the initial speed of the magnetic fluid in the interstitium is zero;
and solving the flow velocity distribution condition of the magnetic fluid in the interstitial tissues by applying a pressure-velocity multi-physical field coupling analysis method.
In this embodiment, the pressure-velocity multi-physics coupling analysis method is a sequential coupling method, and specifically, the solution data set of formula (1) is used as the initial value of formula (2) and is used to solve formula (2).
In this embodiment, step S3 specifically includes the following steps:
firstly, constructing a convection-diffusion equation based on a dual-porosity fluid transport model, and taking the flow velocity distribution condition of the magnetic fluid in the interstitial tissues as input; wherein the convection-diffusion equation is:
wherein, the formula (3) is a convection-diffusion equation for the interstitium, and the formula (4) is a convection-diffusion equation for the blood vessel; wherein the subscript i corresponds to the relevant physical properties under different tissues, the subscript v corresponds to the physical properties relevant to the vascular system, the parameter c represents the concentration value of the magnetic nanoparticles in the interstitium, v represents the flow velocity of the interstitium, D represents the diffusion coefficient of the magnetic fluid, and F represents the concentration value of the magnetic nanoparticles in the interstitiumsRepresenting the solute transport forces between the interstitium and the blood vessels, FLExpressing the transport acting force of solute between the stroma and the lymphatic vessel, w expressing the weight value, and t expressing the magnetofluid transport time;
then, constructing a source term equation of the magnetic fluid passing through the blood vessel wall model:
in the formula, LPIs the hydraulic conductivity of the wall of the capillary vessel in the biological tissue model, S/V is the surface area per unit volume transported in the second tissue in the biological tissue model, pbIs the static pressure, sigma, of blood in a biological tissue modelsIs the osmotic reflection coefficient, pi, of plasma proteins in a biological tissue modelbIs the plasma protein oncotic pressure, pi, in a biological tissue modeliIs the interstitial swelling pressure, p, of the biological tissue modeliRepresenting interstitial pressure;
and solving the diffusion condition and the specific distribution in the second tissue area in the magnetofluid injection process by using a speed-concentration coupling analysis method.
In this embodiment, the speed-concentration multi-physical field coupling method is a sequential coupling method, that is, the solution data set of formula (2) is used as the initial values of formulas (3) to (4) to solve formulas (3) to (4), and finally the concentration distribution of the magnetic fluid in the tissue is obtained.
The present embodiment also provides a system for predicting the magnetic fluid concentration distribution in a biological model tissue based on a dual porosity model, comprising a memory, a processor and computer program instructions stored on the memory and executable by the processor, which when executed by the processor implement the method steps as described in any of the above.
Preferably, the finite element method is used to solve all partial differential equations referred to above.
Fig. 2 shows a biological tissue model, and by applying the solution of the present invention, concentration diffusion calculation and analysis are performed on the geometric model, so as to obtain the concentration distribution of the magnetic fluid in the second tissue region based on the dual-porosity fluid transport model shown in fig. 3, wherein the concentration distribution is the result of performing static diffusion for 0h, 12h and 24h after injection is completed, and it can be seen from the result that the concentration diffusion range gradually increases and the maximum value thereof also decreases as time increases.
The foregoing is directed to preferred embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow. However, any simple modification, equivalent change and modification of the above embodiments according to the technical essence of the present invention are within the protection scope of the technical solution of the present invention.
Claims (7)
1. A method for predicting the concentration distribution of magnetic fluid in a biological model tissue based on a dual-porosity model is characterized by comprising the following steps:
establishing a biological tissue model, and obtaining interstitial pressure distribution of a first tissue and a second tissue in the biological tissue model by introducing a dual-porosity fluid transportation model;
according to the obtained interstitial pressure distribution, solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial process of injecting the magnetic fluid into the biological tissue model by using a Navier leaf-Stokes equation;
according to the obtained flow velocity distribution of the magnetic fluid in the interstitium, the concentration distribution of the magnetic fluid in the interstitial space of the biological tissue model is obtained by utilizing a convection-diffusion equation;
the establishing of the biological tissue model, and the obtaining of the interstitial pressure distribution of the first tissue and the second tissue in the biological tissue model by introducing the dual-porosity fluid transport model, specifically comprises the following steps:
firstly, establishing a biological tissue model which comprises a first tissue and a second tissue, and constructing a hollow cylinder on the basis of the biological tissue model for representing a needle hole model of a syringe, wherein the cylinder penetrates through the first tissue area and reaches the central position of the second tissue area;
the tissue in the biological tissue model is made to be dual-porosity, and the relationship between the flow velocity and the pressure of the magnetic fluid in the interstitial tissue of the model is described by applying Darcy's law: u ═ κ ℃:, wherein:, u is the magnetofluid velocity, κ is the interstitial water conductivity, and p is the tissue pressure; constructing a pressure distribution mathematical model of tissue interstitium in the biological tissue model by a Kedem-Katchalsky theory;
setting a boundary condition for the pressure distribution mathematical model according to a Dirichlet boundary condition of a set Kedem-Katchalsky theoretical model, and then solving the pressure distribution mathematical model by applying a finite element method and combining the boundary condition to obtain interstitial pressure distribution conditions of a first tissue area and a second tissue area;
the method for obtaining the concentration distribution of the magnetic fluid in the interstitial tissue of the biological tissue model by utilizing a convection-diffusion equation according to the obtained flow velocity distribution of the magnetic fluid in the interstitial tissue specifically comprises the following steps:
constructing a convection-diffusion equation based on a dual-porosity fluid transport model, and taking the flow velocity distribution condition of the magnetic fluid in the interstitial tissues as input;
constructing a source term equation of the magnetic fluid passing through the vascular wall model:
in the formula phivRepresenting a source item, LPIs the hydraulic conductivity of the wall of the capillary vessel in the biological tissue model, S/V is the surface area per unit volume transported in the second tissue in the biological tissue model, pbIs the static pressure, sigma, of blood in a biological tissue modelsIs the osmotic reflectance, pi, of plasma proteins in biological tissue modelsbIs the plasma protein oncotic pressure, pi, in a biological tissue modeliIs the interstitial swelling pressure, p, of the biological tissue modeliRepresenting interstitial pressure;
solving the diffusion condition and specific distribution in the second tissue area in the magnetofluid injection process by using a speed-concentration coupling analysis method;
the convection-diffusion equation is:
wherein, formula (3) is a convection-diffusion equation for the interstitium, and formula (4) is a convection-diffusion equation for the blood vessel; wherein the subscript i corresponds to the relevant physical properties under different tissues, the subscript v corresponds to the physical properties relevant to the vascular system, the parameter c represents the concentration value of the magnetic nanoparticles in the interstitium, v represents the flow velocity of the interstitium, D represents the diffusion coefficient of the magnetic fluid, and F represents the concentration value of the magnetic nanoparticles in the interstitiumsRepresenting the solute transport forces between the interstitium and the blood vessels, FLRepresents the transport force of solute between the stroma and lymphatic vessel, w represents the weight value, and t represents the magnetofluid transport time.
2. The method for predicting the concentration distribution of the magnetic fluid in the biological tissue model based on the dual-porosity model of claim 1, wherein the mathematical model of the pressure distribution of the tissue interstitium in the biological tissue model constructed by the Kedem-Katchalsky theory is as follows:
wherein the subscript i has a value of 1 or 2, and when 1, indicates a first tissue, and when 2, indicates a second tissue; ^ represents Hamiltonian, [ kappa ]iDenotes the permeability of the interstitium,. mu.denotes the dynamic viscosity of the magnetic fluid, piIndicating the pressure in the tissue interstitium,. phivThe source item is represented.
3. The dual-porosity model-based magnetic fluid concentration distribution pre-determination in a biological model tissue according to claim 2The measuring method is characterized in that the Dirichlet boundary condition of the Kedem-Katchalsky theoretical model is set as the initial pressure which is normal atmospheric pressure, namely p0＝1×105Pa。
4. The method for predicting the concentration distribution of the magnetic fluid in the tissue of the biological model based on the dual-porosity model as claimed in claim 3, wherein the step of solving the flow velocity distribution of the magnetic fluid in the interstitium in the interstitial process of injecting the magnetic fluid into the tissue of the biological model by using the Navier-Stokes equation according to the obtained interstitial pressure distribution specifically comprises the following steps:
constructing a Navier-Stokes equation for describing the process of the change of the flow velocity of the magnetic fluid injected into the second tissue area; wherein the Navier-Stokes equation is expressed as:
wherein, mufAnd ρfDynamic viscosity and density, p, of the magnetic fluid, respectivelyiRepresenting interstitial pressure;
the dirichlet boundary condition of equation (2) is set as: the initial speed of the magnetic fluid in the interstitium is zero;
and solving the distribution condition of the flow velocity of the magnetic fluid in the interstitial tissues by applying a pressure-velocity multi-physical field coupling analysis method.
5. The method for predicting the concentration distribution of the magnetic fluid in the biological model tissue based on the dual porosity model according to claim 4, wherein the pressure-velocity multi-physical field coupling analysis method is a sequential coupling method, and particularly, the solution data set of the formula (1) is used as an initial value of the formula (2) and is used for solving the formula (2).
6. The method for predicting the concentration distribution of the magnetic fluid in the biological model tissue based on the dual porosity model according to claim 5, wherein the speed-concentration multi-physical field coupling method is a sequential coupling method, i.e. the solution data set of formula (2) is used as the initial values of formulas (3) - (4) to solve formulas (3) - (4), and finally the concentration distribution of the magnetic fluid in the tissue is obtained.
7. A magnetic fluid concentration distribution prediction system in a biological model tissue based on a dual porosity model, comprising a memory, a processor and computer program instructions stored on the memory and executable by the processor, which when executed by the processor implement the method steps of any one of claims 1 to 6.
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