CN108062981B - Modeling method for intraspecies and interspecific scales of fractal vascular asymmetric tree - Google Patents

Modeling method for intraspecies and interspecific scales of fractal vascular asymmetric tree Download PDF

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CN108062981B
CN108062981B CN201810034383.2A CN201810034383A CN108062981B CN 108062981 B CN108062981 B CN 108062981B CN 201810034383 A CN201810034383 A CN 201810034383A CN 108062981 B CN108062981 B CN 108062981B
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霍云龙
谭文长
李佳航
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PKU-HKUST SHENZHEN-HONGKONG INSTITUTION
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Abstract

The invention discloses a modeling method of the intraspecies and interspecific scale of a fractal vascular asymmetric tree, which aims at a series of researches on the intraspecies and interspecific scale relation of the asymmetric vascular tree and the abnormal-speed growth rule of metabolism, and comprises the following steps: step S1: defining branch ratio, diameter ratio and length ratio of fractal vascular asymmetric tree; step S2: respectively obtaining the number of the end blood vessels and the diameter of the parent blood vessels, the total volume of the branch tree and the diameter of the parent blood vessels, and the intraspecies scaling relationship between the accumulated length of the branch tree and the scaling rule of the diameter of the parent blood vessels; step S3: an interspecific scale relationship of metabolism is obtained. The model constructed by the invention has good consistency with the morphological measurement results of animals and plants, further discusses the limitations and significance of ecosystem and disease diagnosis, and simultaneously provides a method for measuring the blood vessel volume and calculates the blood vessel volume in the crown part according to the diameter and the length of the dry-crown unit.

Description

Modeling method for intraspecies and interspecific scales of fractal vascular asymmetric tree
Technical Field
The invention relates to a modeling method for intraspecies and interspecies scales of fractal vascular asymmetric trees.
Background
According to fractal symmetrical tree structures existing in nature, west et al propose an original mathematical model called WBE model to support the differential 3/4 scale law of metabolism. Subsequently, after taking a number of measurements, we can obtain variable metabolic scale indices, such as 2/3,3/4,7/9,6/7,1 or other non-linear values in animals and plants.
Although various theoretical models are proposed to account for exponential changes, such as models of metabolic level boundary hypothesis, thermodynamics, differential stress cascade, or empirical fit, all of which rely on macroscopic principles. But their substantial model is derived from fractal symmetrical tree structures. Although asymmetric properties are now investigated on the basis of minimum energy assumptions at the individual bifurcation of the vessel tree. But in contrast only the scaling law model caused by fractal vessel asymmetry trees and some studies can be used to discuss the asymmetric effects of intra-and inter-seed scaling laws.
Disclosure of Invention
The invention aims at: the modeling method for the intraspecies and the interspecies scales of the fractal vascular asymmetric tree is provided, and a model established by the method can be used for researching the intraspecies and interspecies scale indexes of the fractal vascular asymmetric tree and the abnormal-speed growth rule of metabolism, namely, on the basis of defining the branching ratio, the diameter ratio and the length ratio of the biological tree, the intraspecies scale relation and the interspecies scale relation are researched.
The technical scheme of the invention is as follows: a modeling method of intraspecies and interspecies scales of fractal vascular asymmetric trees, comprising the steps of:
step S1: defining branch ratio, diameter ratio and length ratio of fractal vascular asymmetric tree;
step S2: respectively obtaining the intraspecies scaling relationship between the number of the end blood vessels and the diameter of the parent blood vessels, the total volume of the branch tree and the diameter of the parent blood vessels, and the scaling rule of the accumulated length of the branch tree and the diameter of the parent blood vessels, wherein the corresponding relationship is as follows:
Figure GDA0004054732690000011
Figure GDA0004054732690000012
Figure GDA0004054732690000021
wherein n is c For the number of end vessels, V c L is the total volume of the branch tree c For the accumulated length of the branch tree, D s A is the diameter of a parent blood vessel, a is the fractal dimension of a fractal blood vessel asymmetric tree;
step S3: obtaining the inter-species scale relation of metabolism, wherein the corresponding relation is as follows:
Figure GDA0004054732690000022
where B is the metabolic rate of the subject and M is the body weight of the subject.
As a preferred embodiment, the branching ratio br=n in step S1 i /n i-1 The method comprises the steps of carrying out a first treatment on the surface of the Diameter ratio dr=d i /D i-1 The method comprises the steps of carrying out a first treatment on the surface of the Length ratio lr=l i /L i-1 ；
Figure GDA0004054732690000023
Wherein n is i 、n i-1 The total number of blood vessels when the number of the blood vessel series is i and the total number of the blood vessels when the number of the blood vessel series is i-1 are respectively; d (D) i 、D i-1 The blood vessel diameters when the number of the blood vessel series is i and the blood vessel diameters when the number of the blood vessel series is i-1 are respectively; l (L) i 、L i-1 The length of the blood vessel when the number of the blood vessel series is i, the length of the blood vessel when the number of the blood vessel series is i-1, epsilon is the bifurcation parameter of the diameter, and gamma is the bifurcation parameter of the length.
As a preferred technical solution, in the fractal vascular asymmetric tree in step S1, the branching ratio BR is the number of sub-catheters at one junction and is assumed to be constant;
diameter ratio k of each branch vessel to parent vessel i And length ratio j i The assumption is that:
Figure GDA0004054732690000024
Figure GDA0004054732690000025
wherein k is i ＝j i ，i＝1、2...BR；
D daughter 1 、D daughter 2 ...D daughter BR Diameter of branched blood vessel D mother Is the diameter of the parent vessel; l (L) daughter 1 、L daughter 2 ...L daughter BR For the length of the branched blood vessel, L mother Is the length of the parent vessel;
at the same time give a constant branching ratio k i : i.e.
Figure GDA0004054732690000026
Fractal vessel asymmetric tree continues to branch to minimum diameter d 0 ，d 0 I.e. the diameter of the terminal artery.
As a preferred technical scheme, the calculation method of the intraspecies scaling relationship between the number of end blood vessels and the diameter of parent blood vessels in step S2 is as follows:
in the fractal vascular asymmetric tree structure, the number of the end blood vessels of the branch tree is a single-value function of the diameter of the parent blood vessel, and the corresponding relation is as follows:
Figure GDA0004054732690000031
there is a real number k i So that n c The values of (2) are positive integers, and the maximum common factor is 1;
introducing a sequence a n ＝F 1 (k -n )，n＝0，1，2…：
Figure GDA0004054732690000032
Wherein the method comprises the steps of
Figure GDA0004054732690000033
The characteristic equation of the sequence is
Figure GDA0004054732690000034
Characteristic root is->
Figure GDA0004054732690000035
Wherein σ is the largest positive real root;
because of
Figure GDA0004054732690000036
Figure GDA0004054732690000037
So r is i Sigma is less than or equal to, and the characteristic root of the maximum mode is a positive real root;
Figure GDA0004054732690000038
wherein k is i Is a fixed value, when n > 1, a n ≈H 1 ·σ n When H is 1 Is a fixed parameter;
when the parent vessel diameter D s In the range k -n ≤D s ＜k -n-1 Within n=0, 1,2 …, the following relationship is derived using a recursive method: f (F) 1 (D s )＝a n ≈H 1 ·σ n And satisfy the following
Figure GDA0004054732690000039
Due to
Figure GDA00040547326900000310
Thereby get +.>
Figure GDA00040547326900000311
From the following components
Figure GDA00040547326900000312
Deriving σ=k -a And->
Figure GDA00040547326900000313
At the same time
Figure GDA00040547326900000314
From the formula
Figure GDA00040547326900000315
And sigma is less than or equal to BR to obtain ∈>
Figure GDA00040547326900000316
Here H' 1 Seen as a constant; thus, the following relationship is obtained:
Figure GDA00040547326900000317
i.e. < ->
Figure GDA00040547326900000318
As a preferable technical scheme, the calculation method of the intraspecies scale relation of the total volume of the branch tree and the parent vessel diameter in the step S2 is as follows:
in fractal vessel asymmetric tree structures, the total volume of the branching tree is a single-valued function of parent vessel diameter:
Figure GDA0004054732690000041
assume that:
Figure GDA0004054732690000042
wherein h is 2 Is a constant;
Figure GDA0004054732690000043
definition of the definition
Figure GDA0004054732690000044
s i Is a fraction within a certain error range;
there is a real number k i So that n c The values of (2) are positive integers, and the maximum common factor is 1;
introduced with the sequence
Figure GDA0004054732690000045
Figure GDA0004054732690000046
Wherein the method comprises the steps of
Figure GDA0004054732690000047
The characteristic equation of the sequence is
Figure GDA0004054732690000048
Characteristic root is->
Figure GDA0004054732690000049
Wherein σ is the largest positive real root;
because of
Figure GDA00040547326900000410
Figure GDA00040547326900000411
So r is i Sigma is less than or equal to, and the characteristic root of the maximum mode is a positive real root;
Figure GDA00040547326900000412
wherein k is i Is a fixed value, when n > 1, a n ≈H 2 ·σ n When H is 2 Is a fixed parameter;
when the parent vessel diameter D s In the range k -n ≤D s ＜k -n-1 Within n=0, 1,2 …, the following equation is derived using recursion:
Figure GDA00040547326900000413
and->
Figure GDA00040547326900000414
Due to
Figure GDA00040547326900000415
Obtain->
Figure GDA00040547326900000416
From the following components
Figure GDA00040547326900000417
Deriving σ=k -a And->
Figure GDA00040547326900000418
Taking into account that
Figure GDA00040547326900000419
According to
Figure GDA00040547326900000420
And sigma is less than or equal to BR to obtain ∈>
Figure GDA00040547326900000421
Here H' 2 Seen as a constant;
at the position of
Figure GDA00040547326900000422
a is less than or equal to 3 and
Figure GDA00040547326900000423
on the basis of (a) the base,
the following relationship is derived:
Figure GDA0004054732690000051
i.e. < ->
Figure GDA0004054732690000052
As an preferable technical scheme, the calculation method of the intra-seed scale relationship between the cumulative length of the branch tree and the parent vessel diameter in step S2 is as follows:
in fractal vessel asymmetric tree structures, the cumulative length of the branching tree is a single-valued function of parent vessel diameter:
Figure GDA0004054732690000053
Figure GDA0004054732690000054
wherein h is 3 Is a constant;
i.e.
Figure GDA0004054732690000055
Theoretical derivation, the following equation is obtained:
Figure GDA0004054732690000056
here H' 3 Seen as a constant;
at the position of
Figure GDA0004054732690000057
a is greater than or equal to 2 and
Figure GDA0004054732690000058
on the basis of (a) the base,
the following equation is derived:
Figure GDA0004054732690000059
i.e. < ->
Figure GDA00040547326900000510
As a preferred embodiment, the method for calculating the inter-species scale relationship of metabolism in step S3 is as follows:
due to the known Q s ∝B、V c ∝M、n c ∝Q s Wherein Q is s Is the parent blood vessel flow; and combined with
Figure GDA00040547326900000511
The method comprises the following steps: />
Figure GDA00040547326900000512
Finally, the inter-species scale relation of metabolism is obtained:
Figure GDA00040547326900000513
the intraspecies and interspecies multiscale method is as follows: an asymmetric bifurcation tree is used for theoretical derivation and can be extended to general asymmetric trees. In asymmetric branching of fractal tree, derivation
Figure GDA00040547326900000514
Where N is max And N min Refers to the number of generations (largest and smallest generation, respectively) associated with the large and small vessels on the path from each branch to the terminating branch vessel, while β is considered a constant in the integrated system of the dry-crown unit.
In the diameter ratios k1 and k2, expressed as a function of the parent vessel diameter, i.e., k 1 ＝K 1 (log(D mother ) Sum of (d)
Figure GDA0004054732690000061
From the trunk (D) s ) Down to the distal vessel (d 0 ) The path of (2) has a diameter of ns+1 generation;
D(0)＝D s ，D(1)，…，D(N s )＝d 0 and N s (0，1，…，N s -1) bifurcation.
The parent vessel diameter at the bifurcation and the diameter ratio are denoted as D m (i) And K m，i ＝K 1 (log(D m (i)))；
Or K 2 (lo g (D m (i)))(i＝0，1，…，N s -1)；
Obtaining
Figure GDA0004054732690000062
Here, the following equation is proposed:
Figure GDA0004054732690000063
Figure GDA0004054732690000064
the equation is obtained:
Figure GDA0004054732690000065
this is log (K) in the tree structure 2 (log(D m (i) A) a weighted average along the path. The diameter ratio of the asymmetrical tree structure is used for reconstructing the tree structure, and the asymmetrical tree structure has the diameter ratio of the asymmetrical tree like a common tree
Figure GDA0004054732690000066
And->
Figure GDA0004054732690000067
) At this time k 1 ＝K 1 (log(D mother ) And k) 2 ＝K 2 (log(D mother ))。
The invention has the advantages that: the model constructed by the invention has good consistency with the morphological measurement results of animals and plants, further discusses the limitations and significance of ecosystem and disease diagnosis, and simultaneously provides a method for measuring the blood vessel volume and calculates the blood vessel volume in the crown part according to the diameter and the length of the dry-crown unit.
Drawings
The invention is further described below with reference to the accompanying drawings and examples:
FIG. 1 is a schematic diagram of a definition of stem-crown units and corresponding parameters in a fractal tree;
FIG. 2 is a schematic diagram of a symmetrical tree structure;
FIG. 3 is a schematic diagram of an asymmetric tree structure;
FIG. 4 is a graph of the diameter ratio and branching ratio of an animal symmetric vessel tree structure versus fractal dimension.
FIG. 5 is a plot of the diameter ratio and branching ratio of a plant symmetrical vessel tree structure versus fractal dimension.
FIG. 6 is a graphical representation of the ratio of basal metabolic rate to body weight in 4447 animals;
fig. 7 is a schematic diagram of the relationship between leaf quality and stem quality in 1200 plants.
Detailed Description
Examples: in a dry-crown unit, the proximal vessel segment is defined as a dry having a vessel diameter, length and flow, and the distal end of the dry (to the smallest arteriole or venule) is defined as the crown. As shown in fig. 1. Capillary networks (vessel diameters less than 8 μm) are excluded from the model because it is not tree-like in structure. The vessel units are assumed to be cylindrical tubes and other non-linear effects (e.g. vessel compliance, turbulence, viscosity variations in different vessel units, etc.) are neglected because their contribution to the hemodynamics of the overall tree structure is relatively small. In the integrated system of dry-crown units, the coronary intravascular volume is defined as the sum of the intravascular volume of each vessel segment. Meanwhile, the length of the coronary vessel is defined as: the sum of the lengths of each vessel portion of the entire crown from the trunk to the most distal vessel. In the fractal symmetrical tree, a branching ratio br=n of the fractal vascular asymmetric tree is defined i /n i-1 The method comprises the steps of carrying out a first treatment on the surface of the Diameter ratio dr=d i /D i-1 The method comprises the steps of carrying out a first treatment on the surface of the Length ratio lr=l i /L i-1 ；
Figure GDA0004054732690000071
Wherein n is i 、n i-1 The total number of blood vessels when the number of the blood vessel series is i and the total number of the blood vessels when the number of the blood vessel series is i-1 are respectively; d (D) i 、D i-1 The blood vessel diameters when the number of the blood vessel series is i and the blood vessel diameters when the number of the blood vessel series is i-1 are respectively; l (L) i 、L i-1 The length of the blood vessel when the number of the blood vessel series is i, the length of the blood vessel when the number of the blood vessel series is i-1, epsilon is the bifurcation parameter of the diameter, and gamma is the bifurcation parameter of the length.
In fractal vessel asymmetric trees, the branching ratio BR is the number of sub-catheters at one junction and is assumed to be constant;
diameter ratio k of each branch vessel to parent vessel i And length ratio j i The assumption is that:
Figure GDA0004054732690000072
Figure GDA0004054732690000073
wherein k is i ＝j i ，i＝1、2...BR；
D daughter 1 、D daughter 2 ...D daughter BR Diameter of branched blood vessel D mother Is the diameter of the parent vessel; l (L) daughter 1 、L daughter 2 ...L daughter BR For the length of the branched blood vessel, L mother Is the length of the parent vessel;
at the same time give a constant branching ratio k i : i.e.
Figure GDA0004054732690000081
Fractal vessel asymmetric tree continues to branch to minimum diameter d 0 ，d 0 I.e. the diameter of the terminal artery.
Respectively obtaining the intraspecies scaling relationship between the number of the end blood vessels and the diameter of the parent blood vessels, the total volume of the branch tree and the diameter of the parent blood vessels, and the scaling rule of the accumulated length of the branch tree and the diameter of the parent blood vessels, wherein the corresponding relationship is as follows:
Figure GDA0004054732690000082
Figure GDA0004054732690000083
Figure GDA0004054732690000084
wherein n is c For the number of end vessels, V c L is the total volume of the branch tree c For the accumulated length of the branch tree, D s For parent vessel diameter, a is the fractal dimension of the fractal-vessel asymmetric tree.
(1) The calculation method of the intraspecies scaling relation between the number of the end blood vessels and the diameter of the parent blood vessel is as follows:
in the fractal vascular asymmetric tree structure, the number of the end blood vessels of the branch tree is a single-value function of the diameter of the parent blood vessel, and the corresponding relation is as follows:
Figure GDA0004054732690000085
there is a real number k i So that n c The values of (2) are positive integers, and the maximum common factor is 1;
introducing a sequence a n ＝F 1 (k -n )，n＝0，1，2…：
Figure GDA0004054732690000086
Wherein the method comprises the steps of
Figure GDA0004054732690000087
The characteristic equation of the sequence is
Figure GDA0004054732690000088
Characteristic root is->
Figure GDA0004054732690000089
Wherein σ is the largest positive real root;
because of
Figure GDA00040547326900000810
Figure GDA00040547326900000811
So r is i Sigma is less than or equal to, and the characteristic root of the maximum mode is a positive real root;
Figure GDA00040547326900000812
wherein k is i Is a fixed value, when n > 1, a n ≈H 1 ·σ n When H is 1 Is a fixed parameter;
when the parent vessel diameter D s In the range k -n ≤D s ＜k -n-1 Within n=0, 1,2 …, the following relationship is derived using a recursive method: f (F) 1 (D s )＝a n ≈H 1 ·σ n And satisfy the following
Figure GDA0004054732690000091
Due to
Figure GDA0004054732690000092
Thereby get +.>
Figure GDA0004054732690000093
From the following components
Figure GDA0004054732690000094
Deriving σ=k -a And->
Figure GDA0004054732690000095
At the same time
Figure GDA0004054732690000096
From the formula
Figure GDA0004054732690000097
And sigma is less than or equal to BR to obtain ∈>
Figure GDA0004054732690000098
Here H' 1 Seen as a constant; thus, the following relationship is obtained:
Figure GDA0004054732690000099
i.e. < ->
Figure GDA00040547326900000910
(2) The calculation method of the intraspecies scaling relation of the total volume of the branch tree and the parent vessel diameter is as follows:
in fractal vessel asymmetric tree structures, the total volume of the branching tree is a single-valued function of parent vessel diameter:
Figure GDA00040547326900000911
assume that:
Figure GDA00040547326900000912
wherein h is 2 Is a constant;
Figure GDA00040547326900000913
definition of the definition
Figure GDA00040547326900000914
s i Is a fraction within a certain error range;
there is a real number k i So that n c The values of (2) are positive integers, and the maximum common factor is 1;
introduced with the sequence
Figure GDA00040547326900000915
Figure GDA00040547326900000916
Wherein the method comprises the steps of
Figure GDA00040547326900000917
The characteristic equation of the sequence is
Figure GDA00040547326900000918
Characteristic root is->
Figure GDA00040547326900000919
Wherein σ is the largest positive real root;
because of
Figure GDA00040547326900000920
Figure GDA00040547326900000921
So r is i Sigma is less than or equal to, and the characteristic root of the maximum mode is a positive real root;
Figure GDA0004054732690000101
wherein k is i Is a fixed value, when n > 1, a n ≈H 2 ·σ n When H is 2 Is a fixed parameter;
when the parent vessel diameter D s In the range k -n ≤D s ＜k -n-1 Within n=0, 1,2 …, the following equation is derived using recursion:
Figure GDA0004054732690000102
and->
Figure GDA0004054732690000103
Due to
Figure GDA0004054732690000104
Obtain->
Figure GDA0004054732690000105
From the following components
Figure GDA0004054732690000106
Deriving σ=k -a And->
Figure GDA0004054732690000107
Taking into account that
Figure GDA0004054732690000108
According to
Figure GDA0004054732690000109
And sigma is less than or equal to BR to obtain ∈>
Figure GDA00040547326900001010
Here H' 2 Seen as a constant;
at the position of
Figure GDA00040547326900001011
a is less than or equal to 3 and
Figure GDA00040547326900001012
on the basis of (a) the base,
the following relationship is derived:
Figure GDA00040547326900001013
i.e. < ->
Figure GDA00040547326900001014
(3) The calculation method of the intraspecies scaling relation between the accumulated length of the branch tree and the parent vessel diameter is as follows:
in fractal vessel asymmetric tree structures, the cumulative length of the branching tree is a single-valued function of parent vessel diameter:
Figure GDA00040547326900001015
Figure GDA00040547326900001016
wherein h is 3 Is a constant;
i.e.
Figure GDA00040547326900001017
Theoretical derivation, the following equation is obtained:
Figure GDA00040547326900001018
here H' 3 Seen as a constant;
at the position of
Figure GDA00040547326900001019
a is greater than or equal to 2 and
Figure GDA00040547326900001020
on the basis of (a) the base,
the following equation is derived:
Figure GDA00040547326900001021
i.e. < ->
Figure GDA00040547326900001022
Obtaining the inter-species scale relation of metabolism, wherein the corresponding relation is as follows:
Figure GDA0004054732690000111
wherein B is the metabolic rate of the subject, and M is the body weight of the subject; the calculation method is as follows:
due to the known Q s ∝B、V c ∝M、n c ∝Q s Wherein Q is s Is the parent blood vessel flow; and combined with
Figure GDA0004054732690000112
The method comprises the following steps: />
Figure GDA0004054732690000113
Finally, the inter-species scale relation of metabolism is obtained:
Figure GDA0004054732690000114
the intraspecies and interspecies multiscale method is as follows: an asymmetric bifurcation tree is used for theoretical derivation and can be extended to general asymmetric trees. In asymmetric branching of fractal tree, derivation
Figure GDA0004054732690000115
Where N is max And N min Refers to the number of generations (largest and smallest generation, respectively) associated with the large and small vessels on the path from each branch to the terminating branch vessel, while β is considered a constant in the integrated system of the dry-crown unit.
In the diameter ratios k1 and k2, expressed as a function of the parent vessel diameter, i.e., k 1 ＝K 1 (log(D mother ) Sum of (d)
Figure GDA0004054732690000116
From the trunk (D) s ) Down to the distal vessel (d 0 ) The path of (2) has a diameter of ns+1 generation;
D(0)＝D s ，D(1)，…，D(N s )＝d 0 and N s (0，1，…，N s -1) bifurcation.
The parent vessel diameter at the bifurcation and the diameter ratio are denoted as D m (i) And K m，i ＝K 1 (log(D m (i)))；
Or K 2 (log(D m (i)))(i＝0，1，…，N s -1)；
Obtaining
Figure GDA0004054732690000117
Here, the following equation is proposed:
Figure GDA0004054732690000118
Figure GDA0004054732690000119
the equation is obtained:
Figure GDA00040547326900001110
this is log (K) in the tree structure 2 (log(D m (i) A) a weighted average along the path. The diameter ratio of the asymmetrical tree structure is used for reconstructing the tree structure, and the asymmetrical tree structure has the diameter ratio of the asymmetrical tree like a common tree
Figure GDA0004054732690000121
And->
Figure GDA0004054732690000122
) At this time k 1 ＝K 1 (log(D mother ) And k) 2 ＝K 2 (log(D mother ))。
Data analysis: fractal dimension a was determined in each bifurcation of the asymmetric vessel tree of mice, pigs and patients, while mean ± SD (standard deviation) values of the bifurcation and vessel tree were calculated. Fractal dimension was also determined by the proportional relationship of the diameter ratio and branching ratio of the symmetrical trees in animals and plants. Here, only animals with a total number of symmetrical vessel trees of ∈10 were studied.
At the position of
Figure GDA0004054732690000123
Index X of the interior LV Is simulated by a least square method according to the length-volume scaling law between seedsAnd obtaining the combined measurement data. At->
Figure GDA0004054732690000124
Index X of the interior BM Data of 447 animals and 1200 plants were determined by least squares based on metabolic scale. At->
Figure GDA0004054732690000125
Index X of the interior BM There are also different mass ranges in animals and plants. Statistical differences of scale indexes obtained by different methods are detected by using an analysis of variance method, and significant differences exist among different crowds when the p value is less than 0.05.
The results show that: one flow path starts from the root (closest vessel), through each branch to the distal vessel. Wherein the path length of each bifurcated parent vessel is less than the path length through the parent vessel.
Fractal dimension of multiple asymmetric cardiovascular trees was determined from morphological data of the mouse heart, as well as the patient's head and torso.
Table 1 lists the mean ± standard deviation values (mean over all bifurcation) at each vessel tree. The fractal dimension varied predominantly in the range of 2.0-2.6 (95% ci=2.08-2.43, and between 2.06-2.55 for mice and patients). On the other hand, the diameter ratio and the branching ratio of the symmetrical vascular tree structure from animals and plants
Figure GDA0004054732690000126
The dimensional relationship of the fractal dimension was determined as shown in fig. 4 and 5. At the same time, the limitations of available raw data of the asymmetric vessel tree are also considered. Furthermore, in all asymmetric and symmetric vessel tree structures, the combined system consisting of dry-crown units
Figure GDA0004054732690000127
Using least squares fit as an exponent +.>
Figure GDA0004054732690000128
Fractal dimensions and a=3·x measured in table 1 at (2.26±0.26 and 2.19±0.10 and the assumed value of mice=0.36; 2.31±0.48 and 2.13±0.17, p=0.56) LV And a=2+epsilon and a=3·x of table 2 (2.38±0.24 and 2.34±0.24, p=0.61 being animals; 1.75±0.17 and 1.77±0.22, p=0.57 being plants) LV There is no statistical difference between them as shown in fig. 4 and 5.
Table 1 shows the fractal dimension a, which refers to the fractal dimension of each bifurcation in the asymmetric vessel tree of mice and patients obtained according to the length-volume scaling law;
Figure GDA0004054732690000131
for the scale of metabolism, fig. 6 and 7 show the proportional relationship between basal metabolic rate and body weight in 4447 animals, and the relationship between leaf mass and stem mass in 1200 plants, respectively. Fitting indexes by least squares are 0.72 (r2=0.96) and 0.73 (r2=0.98), respectively.
Table 2 lists the ratio indices for the different mass ranges corresponding to fig. 6 and 7 in animals and plants. The metabolic index shows a non-linear variation over different mass ranges, although the values are 0.72 and 0.73 over the mass range of the whole animal and plant, respectively.
Table 2 is the scale index of the differential growth scale law of animal and plant metabolism over a different mass range:
Figure GDA0004054732690000141
simultaneous fractal dimension (a mean Mean value of fractal dimension of the entire asymmetric vessel tree) and diameter ratio [ ]
Figure GDA0004054732690000142
And
Figure GDA0004054732690000143
) And K is equal to 1 (log(D mother ) Sum->
Figure GDA0004054732690000144
And keep the same.
In combination with the above, the number of vessels increases geometrically due to the fractal nature of the tree structure. Meanwhile, the scale law greatly simplifies the description of the fractal tree. Here, a vessel tree scaling relationship is proposed in consideration of highly asymmetric branching modes. Calculating the number of the end blood vessels and the diameter of the parent blood vessel by adopting fractal hypothesis
Figure GDA0004054732690000145
Total volume of branch tree and parent vessel diameter->
Figure GDA0004054732690000146
And cumulative length of branch tree and parent vessel diameter +.>
Figure GDA0004054732690000147
Scaling relationships between scaling laws. In addition, the length-volume scaling is derived from the scaling of the total volume of the branch tree with the parent vessel diameter and the cumulative length of the branch tree with the parent vessel diameter +.>
Figure GDA0004054732690000148
Similar to previous studies, cube indices are shown in volume-diameter scaling law (although the two studies rely on different trees (symmetric and asymmetric). Thus, the cubic volume-diameter scaling law characterizes the basic fractal features of a tree structure. On the other hand, the length diameter and the length scale are respectively equal to a and the index
Figure GDA0004054732690000149
Wherein a is the fractal dimension. In the coronary arteries of mice, pigs and the trunk, the fractal dimension of each branch is consistent with the fractal dimension of the cardiovascular and cerebrovascular tree of the patient's head and trunk, and is at least two with the whole tree structureThe length-diameter scale law after the multiplication fitting remains consistent (a=3·x LV ). Symmetrical vessel trees have a rich data compared to the original data of the asymmetrical branches.
At the position of
Figure GDA0004054732690000151
The fractal dimension (a=2+epsilon) shows good agreement with the symmetrical vascular tree of animals and plants. The study results verify the theoretical model of length diameter and length-volume scaling.
One key finding is that all scale indices of metabolism conform to the formula
Figure GDA0004054732690000152
This is represented by the formula
Figure GDA0004054732690000153
Is generated by the asymmetric scaling law of the vessel tree. In different vascular trees, changes in fractal dimension can account for different specific metabolic changes in animals and plants. The fractal dimensions of mammalian vascular trees (in the case of total ≡10) are consistent in Table 1 and FIG. 4, i.e. constant scale indices of 7/3 and "Carlebel's law". In contrast, in fig. 5, the expression of the geometry of the outer tree of maple, oak, pine, yellow pine and balsa in fractal dimension is consistent with the law of darifenacin. Fractal dimensions approach 3 (e.g., morey's law), a small number of mammalian vascular tree totals (.ltoreq.5), and complex leaf, grape and treelet trees of small vascular trees, all used to interpret observations of the general scale law.
The above embodiments are merely illustrative of the principles of the present invention and its effectiveness, and are not intended to limit the invention. Modifications and variations may be made to the above-described embodiments by those skilled in the art without departing from the spirit and scope of the invention. Accordingly, it is intended that all equivalent modifications and variations of the invention be covered by the claims, which are within the ordinary skill of the art, be within the spirit and scope of the present disclosure.

Claims (1)

1. A modeling method for intraspecies and interspecies scales of fractal vascular asymmetric trees, comprising the steps of:
step S1: defining branch ratio, diameter ratio and length ratio of fractal vascular asymmetric tree;
branch ratio br=n i /n i-1 The method comprises the steps of carrying out a first treatment on the surface of the Diameter ratio dr=d i /D i-1 The method comprises the steps of carrying out a first treatment on the surface of the Length ratio lr=l i /L i-1 ；
Figure FDA0004054732680000011
Wherein n is i 、n i-1 The total number of blood vessels when the number of the blood vessel series is i and the total number of the blood vessels when the number of the blood vessel series is i-1 are respectively; d (D) i 、D i-1 The blood vessel diameters when the number of the blood vessel series is i and the blood vessel diameters when the number of the blood vessel series is i-1 are respectively; l (L) i 、L i-1 The blood vessel length when the number of the blood vessel series is i, the blood vessel length when the number of the blood vessel series is i-1, epsilon is a bifurcation parameter of the diameter, and Y is a bifurcation parameter of the length;
in fractal vessel asymmetric trees, the branching ratio BR is the number of sub-catheters at one junction and is assumed to be constant;
diameter ratio k of each branch vessel to parent vessel i And length ratio j i The assumption is that:
Figure FDA0004054732680000012
Figure FDA0004054732680000013
wherein k is i ＝j i ，i＝1、2...BR；
D daughter1 、D daughter2 ...D daughterBR Diameter of branched blood vessel D mother Is the diameter of the parent vessel; l (L) daughter1 、L daughter2 ...L daughterBR For the length of the branched blood vessel, L mother Is the length of the parent vessel;
at the same time give a constant branching ratio k i : i.e.
Figure FDA0004054732680000014
Fractal vessel asymmetric tree continues to branch to minimum diameter d 0 ，d 0 Namely the diameter of the terminal artery;
step S2: respectively obtaining the intraspecies scaling relationship between the number of the end blood vessels and the diameter of the parent blood vessels, the total volume of the branch tree and the diameter of the parent blood vessels, and the scaling rule of the accumulated length of the branch tree and the diameter of the parent blood vessels, wherein the corresponding relationship is as follows:
Figure FDA0004054732680000015
Figure FDA0004054732680000016
Figure FDA0004054732680000017
wherein n is c For the number of end vessels, V c L is the total volume of the branch tree c For the accumulated length of the branch tree, D s A is the diameter of a parent blood vessel, a is the fractal dimension of a fractal blood vessel asymmetric tree;
the calculation method of the intraspecies scaling relation between the number of the end blood vessels and the diameter of the parent blood vessel is as follows:
in the fractal vascular asymmetric tree structure, the number of the end blood vessels of the branch tree is a single-value function of the diameter of the parent blood vessel, and the corresponding relation is as follows:
Figure FDA0004054732680000021
there is a real number k i So that n c The values of (2) are positive integers, and the maximum common factor is 1;
introducing a sequence a n ＝F 1 (k -n )，n＝0，1，2…:
Figure FDA0004054732680000022
Wherein the method comprises the steps of
Figure FDA0004054732680000023
The characteristic equation of the sequence is
Figure FDA0004054732680000024
Characteristic root is->
Figure FDA0004054732680000025
Wherein σ is the largest positive real root;
because of
Figure FDA0004054732680000026
Figure FDA0004054732680000027
So r is i Sigma is less than or equal to, and the characteristic root of the maximum mode is a positive real root;
Figure FDA0004054732680000028
wherein k is i Is a fixed value, when n > 1, a n ≈H 1 ·σ n When H is 1 Is a fixed parameter;
when the parent vessel diameter D s In the range k -n ≤D s ＜k -n-1 Within n=0, 1,2 …, the following relationship is derived using a recursive method: f (F) 1 (D s )＝a n ≈H 1 ·σ n And satisfy the following
Figure FDA0004054732680000029
Due to
Figure FDA00040547326800000210
Thereby get +.>
Figure FDA00040547326800000211
From the following components
Figure FDA00040547326800000212
Deriving σ=k -a And->
Figure FDA00040547326800000213
At the same time
Figure FDA00040547326800000214
σ≤BR；
From the formula
Figure FDA00040547326800000215
And sigma is less than or equal to BR to obtain ∈>
Figure FDA00040547326800000216
Here H' 1 Seen as a constant; thus, the following relationship is obtained:
Figure FDA00040547326800000217
i.e. < ->
Figure FDA00040547326800000218
The calculation method of the intraspecies scaling relation of the total volume of the branch tree and the parent vessel diameter is as follows:
in fractal vessel asymmetric tree structures, the total volume of the branching tree is a single-valued function of parent vessel diameter:
Figure FDA0004054732680000031
assume that:
Figure FDA0004054732680000032
wherein h is 2 Is a constant;
Figure FDA0004054732680000033
definition of the definition
Figure FDA0004054732680000034
s i Is a fraction within a certain error range;
there is a real number k i So that n c The values of (2) are positive integers, and the maximum common factor is 1;
introduced with the sequence
Figure FDA0004054732680000035
Figure FDA0004054732680000036
Wherein the method comprises the steps of
Figure FDA0004054732680000037
The characteristic equation of the sequence is
Figure FDA0004054732680000038
Characteristic root is->
Figure FDA0004054732680000039
Wherein σ is the largest positive real root;
because of
Figure FDA00040547326800000310
Figure FDA00040547326800000311
So r is i Sigma is less than or equal to, and the characteristic root of the maximum mode is a positive real root;
Figure FDA00040547326800000312
wherein k is i Is a fixed value, when n > 1, a n ≈H 2 ·σ n When H is 2 Is a fixed parameter;
when the parent vessel diameter D s In the range k -n ≤D s ＜k -n-1 Within n=0, 1,2, …, the following equations are derived using recursion:
Figure FDA00040547326800000313
and->
Figure FDA00040547326800000314
Due to
Figure FDA00040547326800000315
Obtain->
Figure FDA00040547326800000316
From the following components
Figure FDA00040547326800000317
Deriving σ=k -a And->
Figure FDA00040547326800000318
Taking into account that
Figure FDA00040547326800000319
σ≤BR；
According to
Figure FDA00040547326800000320
And sigma is less than or equal to BR to obtain ∈>
Figure FDA00040547326800000321
Here H' 2 Seen as a constant;
at the position of
Figure FDA00040547326800000322
a is less than or equal to 3 and
Figure FDA0004054732680000041
on the basis of (a) the base,
the following relationship is derived:
Figure FDA0004054732680000042
i.e. < ->
Figure FDA0004054732680000043
The calculation method of the intraspecies scaling relation between the accumulated length of the branch tree and the parent vessel diameter is as follows:
in fractal vessel asymmetric tree structures, the cumulative length of the branching tree is a single-valued function of parent vessel diameter:
Figure FDA0004054732680000044
Figure FDA0004054732680000045
wherein h is 3 Is a constant;
i.e.
Figure FDA0004054732680000046
D s ≥d 0 ；
Theoretical derivation, the following equation is obtained:
Figure FDA0004054732680000047
here H' 3 Seen as a constant;
at the position of
Figure FDA0004054732680000048
a is greater than or equal to 2 and
Figure FDA0004054732680000049
on the basis of (a) the base,
the following equation is derived:
Figure FDA00040547326800000410
i.e. < ->
Figure FDA00040547326800000411
Step S3: the calculation method for obtaining the inter-species scale relation of metabolism comprises the following steps:
due to the known Q s ∝B、V c ∝M、n c ∝Q s Wherein Q is s The female blood flow, B is the metabolic rate of the study object, and M is the body weight of the study object; and combined with
Figure FDA00040547326800000412
The method comprises the following steps: />
Figure FDA00040547326800000413
Finally, the inter-species scale relation of metabolism is obtained:
Figure FDA00040547326800000414
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