CN108062981A - A kind of modeling method in the kind of the asymmetric tree of point of shape blood vessel with inter-species scale - Google Patents
A kind of modeling method in the kind of the asymmetric tree of point of shape blood vessel with inter-species scale Download PDFInfo
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Abstract
The present invention in the kind of asymmetric vascular tree, the allometry relative growth rule of inter-species Scaling and metabolism done a series of researchs, disclose in the kind of the asymmetric tree of a kind of point of shape blood vessel and the modeling method of inter-species scale, include the following steps：Step S1：Definition divides branch's ratio, natural scale and the length ratio of the asymmetric tree of shape blood vessel；Step S2：Tail vein quantity and female blood vessel diameter, total volume and the female blood vessel diameter of branch tree, the kind internal graticule relation between the cumulative length of branch tree and the scale rule of female blood vessel diameter are obtained respectively；Step S3：Obtain the inter-species Scaling of metabolism.The model and the morphometry result of animals and plants built using the present invention have good uniformity, and then discuss the limitation and meaning of the ecosystem and medical diagnosis on disease, the method for measuring capacity of blood vessel is additionally provided simultaneously, and bizet internal blood vessel volume is calculated according to the dry diameter for being preced with unit and length.
Description
Technical field
The present invention relates to the modeling methods with inter-species scale in the kind of the asymmetric tree of a kind of point of shape blood vessel.
Background technology
Divide shape symmetrical tree construction according to existing in nature, West etc. proposes an original mathematical model, is known as
WBE models, to support 3/4 scaling law of friction speed of metabolism.Then after many measurements are carried out, we can be in animal and plant
In obtain variable metabolism scaling exponent, such as 2/3,3/4,7/9,6/7,1 or other non-linear values.
Although propose a variety of theoretical models at present to explain index variation, for example, metaboilic level border hypothesis, thermodynamics,
The models such as the cascade of friction speed effect or empirical fit, these all rely on macroscopical principle.But their physical model is from a point shape pair
Claim what tree construction was derived.Although having now on the basis of the least energy of the single crotch of vascular tree is assumed, grind
Study carefully asymmetric nature.But only a point shape blood vessel asymmetry sets caused scaling law model in contrast and some grind
Study carefully, can be used for that kind of interior and inter-species scale rule a asymmetrical effect is discussed.
The content of the invention
The present invention seeks to：Modeling method with inter-species scale, the party in the kind of the asymmetric tree of a kind of point of shape blood vessel are provided
The model of method foundation can be used for research to divide the interior kind of the asymmetric tree of shape blood vessel, inter-species scaling exponent and the allometry relative growth rule of metabolism
Rule, i.e., on based on definition biology tree branch ratio, natural scale and length ratio, research kind internal graticule relation and inter-species mark
Degree relation.
The technical scheme is that：Modeling method in the kind of the asymmetric tree of a kind of point of shape blood vessel with inter-species scale, bag
Include following steps：
Step S1：Definition divides branch's ratio, natural scale and the length ratio of the asymmetric tree of shape blood vessel；
Step S2：Tail vein quantity and female blood vessel diameter, total volume and the female blood vessel diameter of branch tree are obtained respectively, point
Kind internal graticule relation between the scale rule of the cumulative length of Zhi Shu and female blood vessel diameter, correspondence formula are as follows：
Wherein ncFor tail vein quantity, VcFor the total volume of branch tree, LcFor the cumulative length of branch tree, DsFor female blood
Pipe diameter, a are the Fractal Dimension of the asymmetric tree of point shape blood vessel；
Step S3：The inter-species Scaling of metabolism is obtained, correspondence formula is as follows：
Wherein B is the metabolic rate of research object, and M is the weight of research object.
As preferred technical solution, the branch ratio BR=n in step S1i/ni-1；Natural scale DR=Di/Di-1；It is long
Spend ratio LR=Li/Li-1；
Wherein ni、ni-1Blood vessel sum when blood vessel sum, blood vessel series when respectively blood vessel series is i are i-1；Di、
Di-1Blood vessel diameter when blood vessel diameter, blood vessel series when respectively blood vessel series is i are i-1；Li、Li-1Respectively blood vessel grade
Length of vessel when length of vessel, blood vessel series when number is i are i-1, ε are the bifurcated parameter of diameter, and γ is the bifurcated of length
Parameter.
As preferred technical solution, in step S1 in the asymmetric tree of shape blood vessel is divided, branch ratio BR is in a company
Sub- conduit number and it is assumed that constant on contact；
Each branch vessel and the diameter of female blood vessel compare kiCompare j with lengthiIt is assumed that：
Wherein ki=ji, i=1,2 ... BR；
Ddaughter 1、Ddaughter 2…Ddaughter BRFor the diameter of branch vessel, DmotherFor the diameter of female blood vessel；
Ldaughter 1、Ldaughter 2…Ldaughter BRFor the length of branch vessel, LmotherFor the length of female blood vessel；
Given constant branch ratio k simultaneouslyi：I.e.Divide shape blood vessel asymmetric tree
Persistently it is branched to minimum diameter d0, d0As extremity arterial diameter.
As preferred technical solution, the meter of tail vein quantity and the kind internal graticule relation of female blood vessel diameter in step S2
Calculation method is as follows：
In shape blood vessel asymmetry tree construction is divided, the tail vein number of branch tree is the monotropic function of female blood vessel diameter,
Its correspondence formula is as follows：
There are a real number kiSo that ncValue be all positive integer, and greatest common factor is 1；
Introduce a sequence an=F1(k-n), n=0,1,2 ...：
Wherein
The characteristic equation of sequence isCharacteristic root is σi=ri·eiθ, wherein σ is
Maximum arithmetic number root；
Because So ri≤ σ, the characteristic root for showing maximum norm are an arithmetic number roots；
Wherein kiFor fixed value, as n ＞ ＞ 1, an≈H1·nWhen, H1It is preset parameter；
As female blood vessel diameter DsIn scope k-n≤Ds＜ k-n-1, n=0,1,2 ... is interior, and recurrence method is used to derive with ShiShimonoseki
It is formula：F1(Ds)=an≈H1·σnAnd meet
Due toSo as to obtain
ByDerive σ=k-a and
Simultaneouslyσ≤BR。
From formulaWith σ≤BR, obtain
HereRegard a constant as；Therefore following relational expression has been obtained：
I.e.
As preferred technical solution, the total volume of branch tree and the kind internal graticule relation of female blood vessel diameter in step S2
Computational methods are as follows：
In shape blood vessel asymmetry tree construction is divided, the total volume of branch tree is the monotropic function of female blood vessel diameter：
It is assumed that：Wherein h2It is a constant；
DefinitionsiIt is a fraction within a certain error range；
There are a real number kiSo that ncValue be all positive integer, and greatest common factor is 1；
Introduce sequenceN=0,1,2 ...：
Wherein
The characteristic equation of sequence isCharacteristic root is σi=ri·iθ, wherein σ is
Maximum arithmetic number root；
Because So ri≤ σ, the characteristic root for showing maximum norm are an arithmetic number roots；
Wherein kiFor fixed value, as n ＞ ＞ 1, an≈H2·σnWhen, H2It is preset parameter；
As female blood vessel diameter DsIn scope k-n≤Ds＜ k-n-1, n=0,1,2 ... is interior, and recurrence method is used to derive with lower section
Journey：And
Due toIt obtains
ByDerive σ=k-aWith
It considersσ≤BR。
According toWith σ≤BR, obtainHereRegard one as often
Number；
The He of a≤3On the basis of,
Derive following relational expression：
I.e.
As preferred technical solution, the kind internal graticule relation of the cumulative length of branch tree and female blood vessel diameter in step S2
Computational methods it is as follows：
In shape blood vessel asymmetry tree construction is divided, the cumulative length of branch tree is the monotropic function of female blood vessel diameter：
Wherein h3It is a constant；
I.e.Ds≥d0；
Theory deduction obtains below equation：
HereRegard a constant as；
The He of a >=2On the basis of,
Derive following equations：
, i.e.,
As preferred technical solution, the computational methods of metabolic inter-species Scaling are as follows in step S3：
Due to known Qs∝B、Vc∝M、nc∝Qs, wherein QsFor female vascular flow；And it combines
It obtains：
It finally obtained the inter-species Scaling of metabolism：
In kind and the Method of Multiple Scales of inter-species is as follows：By a kind of asymmetric bifurcated tree for theory deduction, can expand to
General asymmetric tree.In the asymmetry fork of Fractal Tree, exportN hereinmaxAnd NminIt refers to
(it is being respectively maximum from each be branched off on the path of terminal branch blood vessel with the number of big blood vessel and the relevant generation of thin vessels
With a minimum generation), while corpse is considered as a constant in the integrated system of dry-hat unit.
In diameter than k1 and k2, a function of female blood vessel diameter, i.e. k1=K are represented as1(log(Dmother)) and
From dry (Ds) down to distal vessels (d0) path have the diameter in Ns+1 generations；
D (0)=Ds, D (1) ..., D (Ns)=d0And Ns(0,1 ..., Ns-1) a bifurcated.
D is expressed as in female blood vessel diameter and the diameter ratio of crotchm(i) and KM, i=K1(log(Dm(i)))；
Or K2(log(Dm(i))) (i=0,1 ..., Ns-1)；
It obtains
Herein, following equation is proposed：
Obtain equation：
This is log (K in tree construction2(log(Dm(i)))) along the weighted average in path.Using diameter proportion achievement structure,
This asymmetric tree construction have the asymmetric tree as diameter ratio (With), k at this time1=K1(log
(Dmother)) and k2=K2(log(Dmother))。
It is an advantage of the invention that：The model and the morphometry result of animals and plants built using the present invention have good
Uniformity, and then the limitation and meaning of the ecosystem and medical diagnosis on disease are discussed, while additionally provide and measure capacity of blood vessel
Method, and bizet internal blood vessel volume is calculated according to diameter and the length for doing-being preced with unit.
Description of the drawings
The invention will be further described with reference to the accompanying drawings and embodiments：
Fig. 1 defines the schematic diagram with relevant parameter for dry in Fractal Tree-hat unit；
Fig. 2 is the schematic diagram of symmetrical tree construction；
Fig. 3 is the schematic diagram of asymmetric tree construction；
Fig. 4 is the natural scale of the symmetrical vessel tree structures of animal and branch's ratio and the relation of Fractal Dimension.
Fig. 5 is the natural scale of the symmetrical vessel tree structures of plant and branch's ratio and the relation of Fractal Dimension.
Fig. 6 is the proportionate relationship schematic diagram in 4447 kinds of animals between basal metabolic rate and weight；
Fig. 7 is the relation schematic diagram between 1200 kinds of plant middle period quality and stem quality.
Specific embodiment
Embodiment：In dry-hat unit, proximal vessel section is defined as blood vessel diameter, length and flow it is dry, and
And dry distal end is defined as being preced with (to minimum parteriole or veinlet).As shown in the figure in Fig. 1.Capillary network (blood
Pipe diameter is less than 8 μm) it is excluded outside model, because it is not tree-shaped in the structure.Blood vessel unit is assumed to cylinder
Pipe, other nonlinear effects (such as vascular compliance, turbulent flow, viscosity change in different blood vessel unit etc.) are ignored, because
They are relatively small to the hemodynamic contribution of entire tree.In the integral system of dry-hat unit, bizet blood
Pipe inner volume is defined as the summation of the intravasal volume of each vasculature part.Meanwhile the length of bizet blood vessel is defined as：It is whole
The summation of each vasculature part length of a bizet from doing the blood vessel of distalmost end.In shape symmetrical tree is divided, a point shape blood is defined
The branch ratio BR=n of the asymmetric tree of pipei/i-1；Natural scale DR=Di/Di-1；Length ratio LR=Li/Li-1；Wherein ni、ni-1Blood vessel sum, blood vessel series when respectively blood vessel series is i are i-1
When blood vessel sum；Di、Di-1Blood vessel diameter when blood vessel diameter, blood vessel series when respectively blood vessel series is i are i-1；Li、
Li-1Length of vessel when length of vessel, blood vessel series when respectively blood vessel series is i are i-1, ε are the bifurcated parameter of diameter,
γ is the bifurcated parameter of length.
In the asymmetric tree of shape blood vessel is divided, branch ratio BR is the sub- conduit number on a tie point and is assumed to normal
Number；
Each branch vessel and the diameter of female blood vessel compare kiCompare j with lengthiIt is assumed that：
Wherein ki=ji, i=1,2...BR；
Ddaughter 1、Ddaughter 2…Ddaughter BRFor the diameter of branch vessel, DmotherFor the diameter of female blood vessel；
Ldaughter 1、Ldaughter 2…Ldaughter BRFor the length of branch vessel, LmotherFor the length of female blood vessel；
Given constant branch ratio k simultaneouslyi：I.e.Divide shape blood vessel asymmetric tree
Persistently it is branched to minimum diameter d0, d0As extremity arterial diameter.
Tail vein quantity and female blood vessel diameter, total volume and the female blood vessel diameter of branch tree are obtained respectively, branch tree
Kind internal graticule relation between the scale rule of cumulative length and female blood vessel diameter, correspondence formula are as follows：
Wherein ncFor tail vein quantity, VcFor the total volume of branch tree, LcFor the cumulative length of branch tree, DsFor female blood
Pipe diameter, a are the Fractal Dimension of the asymmetric tree of point shape blood vessel.
(1) tail vein quantity and the computational methods of the kind internal graticule relation of female blood vessel diameter are as follows：
In shape blood vessel asymmetry tree construction is divided, the tail vein number of branch tree is the monotropic function of female blood vessel diameter,
Its correspondence formula is as follows：
There are a real number kiSo that ncValue be all positive integer, and greatest common factor is 1；
Introduce a sequence an=F1(k-n), n=0,1,2 ...：
Wherein
The characteristic equation of sequence isCharacteristic root is σi=ri·iθ, wherein σ is
Maximum arithmetic number root；
Because So ri≤ σ, the characteristic root for showing maximum norm are an arithmetic number roots；
Wherein kiFor fixed value, as n ＞ ＞ 1, an≈H1·σnWhen, H1It is preset parameter；
As female blood vessel diameter DsIn scope k-n≤Ds＜ k-n-1, n=0,1,2 ... is interior, and recurrence method is used to derive with ShiShimonoseki
It is formula：F1(Ds)=an≈H1·σnAnd meet
Due toSo as to obtain
ByDerive σ=k-aWith
Simultaneouslyσ≤BR。
From formulaWith σ≤BR, obtain
HereRegard a constant as；Therefore following relational expression has been obtained：
I.e.
(2) total volume of branch tree and the computational methods of the kind internal graticule relation of female blood vessel diameter are as follows：
In shape blood vessel asymmetry tree construction is divided, the total volume of branch tree is the monotropic function of female blood vessel diameter：
It is assumed that：Wherein h2It is a constant；
DefinitionsiIt is a fraction within a certain error range；
There are a real number kiSo that ncValue be all positive integer, and greatest common factor is 1；
Introduce sequence, n=0,1,2 ...：
Wherein
The characteristic equation of sequence isCharacteristic root is σi=ri·iθ, wherein σ is
Maximum arithmetic number root；
Because So ri≤ σ, the characteristic root for showing maximum norm are an arithmetic number roots；
Wherein kiFor fixed value, as n ＞ ＞ 1, an≈H2·σnWhen, H2It is preset parameter；
As female blood vessel diameter DsIn scopeN=0,1,2 ... is interior, recurrence method is used to derive with
Lower equation：And
Due toIt obtains
ByDerive σ=k-aWith
It considersσ≤BR。
According toWith σ≤BR, obtainHereRegard one as often
Number；
The He of a≤3On the basis of,
Derive following relational expression：
I.e.
(3) cumulative length of branch tree and the computational methods of the kind internal graticule relation of female blood vessel diameter are as follows：
In shape blood vessel asymmetry tree construction is divided, the cumulative length of branch tree is the monotropic function of female blood vessel diameter：
Wherein h3It is a constant；
I.e.Ds≥d0；
Theory deduction obtains below equation：
HereRegard a constant as；
The He of a >=2On the basis of,
Derive following equations：
, i.e.,
The inter-species Scaling of metabolism is obtained, correspondence formula is as follows：
Wherein B is the metabolic rate of research object, and M is the weight of research object；Its computational methods is as follows：
Due to known Qs∝B、Vc∝M、nc∝Qs, wherein QsFor female vascular flow；And it combines
It obtains：
It finally obtained the inter-species Scaling of metabolism：
In kind and the Method of Multiple Scales of inter-species is as follows：By a kind of asymmetric bifurcated tree for theory deduction, can expand to
General asymmetric tree.In the asymmetry fork of Fractal Tree, exportN hereinmaxAnd NminIt refers to
(it is being respectively maximum from each be branched off on the path of terminal branch blood vessel with the number of big blood vessel and the relevant generation of thin vessels
With a minimum generation), while β is considered as a constant in the integrated system of dry-hat unit.
In diameter than k1 and k2, a function of female blood vessel diameter, i.e. k are represented as1=K1(log(Dmother)) and
From dry (Ds) down to distal vessels (d0) path have the diameter in Ns+1 generations；
D (0)=Ds, D (1) ..., D (Ns)=d0And Ns(0,1 ..., Ns- 1) a bifurcated.
D is expressed as in female blood vessel diameter and the diameter ratio of crotchm(i) and KM, i=K1(log(Dm(i)))；
Or K2(log(Dm(i))) (i=0,1 ..., Ns-1)；
It obtains
Herein, following equation is proposed：
Obtain equation：
This is log (K in tree construction2(log(Dm(i)))) along the weighted average in path.Using diameter proportion achievement structure,
This asymmetric tree construction have the asymmetric tree as diameter ratio (With), k at this time1=K1(log
(Dmother)) and k2=K2(log(Dmother))。
Data analysis：Fractal Dimension a is determined in each fork of the asymmetric vascular tree of mouse, pig and patient, together
When calculate mean value ± SD (standard deviation) value of fork and vascular tree.Also pass through the natural scale of symmetrical tree in animal and plant
Fractal Dimension is determined with the proportionate relationship of branch ratio.Herein, symmetrical vascular tree sum >=10 are only studied in various animals.
In index XLV, it is according to inter-species length-volume scaling law, passes through least square fitting
What measurement data obtained.In index XBM, it is on the basis of the Scaling of metabolism, 447 kinds is moved
The data of object and 1200 kinds of plants are determined by least square method.In index XBM, in animal and plant
Also there is different mass ranges.Using the statistical discrepancy for the scaling exponent that method of analysis of variance detection distinct methods obtain, p value is found
There are significant differences between different crowd during ＜ 0.05.
The results show：One flow path is since root (immediate blood vessel), through each blood for being branched off into end
Pipe.Wherein, the path length of the sub- blood vessel each to diverge is less than the path length by female blood vessel.
According to the head and trunk of the morphological data of mouse heart and patient, it is determined that multiple asymmetric cardiovascular
The Fractal Dimension of tree.
Table 1 lists the value in the average ± standard deviation of each vascular tree (average value is more than all forks).FRACTAL DIMENSION
Degree mainly in the range of 2.0-2.6 variation (95%CI=2.08-2.43, and for mouse and patient 2.06-2.55 it
Between change).On the other hand, from the natural scale of symmetrical vessel tree structures and branch's ratio of animal and plantThe scaling relation of Fractal Dimension is determined, as shown in Figure 4 and Figure 5.Simultaneously, it is also considered that asymmetric blood vessel
The limitation of the available initial data of tree.In addition, in the vessel tree structures of all asymmetric and symmetrics, by dry-hat unit group
Into combined systemIt is index using least square fitting(2.26 ± 0.26 and 2.19 ±
0.10 with assumed value=0.36 of mouse；2.31 ± 0.48 and 2.13 ± 0.17, p=0.56) table 1 measure Fractal Dimension and
A=3XLVAnd table 2 (2.38 ± 0.24 and 2.34 ± 0.24, p=0.61 be animal；1.75 ± 0.17 and 1.77 ± 0.22,
P=0.57 is plant) a=2+ ε and a=3XLVBetween there is no statistical discrepancy, as shown in Figures 4 and 5.
Table 1 is fractal dimension a, is referred to according to the mouse obtained in length-volume scaling law and the asymmetric blood of patient
The fractal dimension that Guan Shuzhong each diverges；
For the scaling law of metabolism, Fig. 6 and Fig. 7 respectively illustrate basal metabolic rate and body in 4447 kinds of animals
Proportionate relationship between weight and the relation between 1200 kinds of plant middle period quality and stem quality.Intended by least square method
Hop index is respectively 0.72 (R2=0.96) and 0.73 (R2=0.98).
Table 2 lists the proportion index with the corresponding different quality scopes of Fig. 6 and Fig. 7 in animal and plant.Metabolism
Index shows the nonlinear change in the range of different quality, although value is respectively in the mass range of entire animal and plant
0.72 and 0.73.
Table 2 is the scaling exponent of the allometry relative growth scaling law of animal and plant metabolism in the range of different quality：
Fractal Dimension (a simultaneouslymean, the average of the Fractal Dimension of entire asymmetry vascular tree) and natural scale (With) and K1(log(Dmother)) andIt is consistent.
In summary described, due to the fractal property of tree, the quantity of blood vessel is increased in the form of geometry.Meanwhile
Scaling law greatly simplifies the description of Fractal Tree.Herein, it is proposed that considering highly asymmetric branching pattern
Vascular tree Scaling.Using point shape it is assumed that calculating tail vein quantity and female blood vessel diameterBranch tree
Total volume and female blood vessel diameterCumulative length and female blood vessel diameter with branch treeScale rule it
Between proportionate relationship.It is in addition, also straight from the total volume and female blood vessel diameter and the cumulative length of branch tree and female blood vessel of branch tree
Length-volume scale rule is derived in the Scaling in footpath
Similar to previous research, shown in volume-diameter scaling law a cube index (although the two researchs according to
Rely in different trees (symmetrical and asymmetrical tree)).Therefore, cube volume-diameter scaling law is demonstrated by the basic of tree construction
Fractal characteristic.On the other hand, length diameter and the index of length scale rule be respectively equal to a andA Fractal Dimensions therein.
In the coronary artery of mouse, pig and trunk, Fractal Dimension and the cardiovascular and cerebrovascular tree of patients head and trunk of each branch
Fractal Dimension is consistent, and length-diameter scaling law with entire tree construction after least square fitting is consistent
(a=3XLV).Compared with the initial data of asymmetric branch, symmetrical vascular tree has abundant data.
In Fractal Dimension (a=2+ ε), shown well with the symmetrical vascular tree of animal and plant
Uniformity.Result of study demonstrates length diameter and the theoretical model of length-volume scale rule.
One crucial to be the discovery that, metabolic all scaling exponents all coincidence formulasThis is by formulaIn vascular tree asymmetric scaling law generate.
In different vascular trees, the variation of Fractal Dimension can explain specific metabolic alterations different in animal and plant.Lactation
The Fractal Dimension (in the case of sum >=10) of animal blood vessels tree is consistent with Fig. 4 in table 1, is constant scaling exponent
For 7/3 and " Ka Leibei laws ".In contrast, in Figure 5, the external tree of maple, Oak Tree, pine, yellow pine and Ba Ersha is several
What expression of the shape on Fractal Dimension, it is consistent with Leonardesque rule.Fractal Dimension is close to 3 (such as silent in laws), few
The mammal vascular tree of amount total (≤5) and the compound leaf of thin vessels tree, grape and annular distance tree, these are all general for explaining
The observation of scaling law.
The above-described embodiments merely illustrate the principles and effects of the present invention, and is not intended to limit the present invention.It is any ripe
Know the personage of this technology all can carry out modifications and changes under the spirit and scope without prejudice to the present invention to above-described embodiment.Cause
This, those of ordinary skill in the art is complete without departing from disclosed spirit and institute under technological thought such as
Into all equivalent modifications or change, should by the present invention claim be covered.
Claims (7)
1. the modeling method in the kind of the asymmetric tree of a kind of point of shape blood vessel with inter-species scale, which is characterized in that include the following steps：
Step S1：Definition divides branch's ratio, natural scale and the length ratio of the asymmetric tree of shape blood vessel；
Step S2：Tail vein quantity and female blood vessel diameter, total volume and female blood vessel diameter, the branch tree of branch tree are obtained respectively
Cumulative length and female blood vessel diameter scale rule between kind internal graticule relation, correspondence formula is as follows：
<mrow>
<msub>
<mi>n</mi>
<mi>c</mi>
</msub>
<mo>&Proportional;</mo>
<msubsup>
<mi>D</mi>
<mi>s</mi>
<mi>a</mi>
</msubsup>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>V</mi>
<mi>c</mi>
</msub>
<mo>&Proportional;</mo>
<msubsup>
<mi>D</mi>
<mi>s</mi>
<mn>3</mn>
</msubsup>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>L</mi>
<mi>c</mi>
</msub>
<mo>&Proportional;</mo>
<msubsup>
<mi>D</mi>
<mi>s</mi>
<mi>a</mi>
</msubsup>
<mo>;</mo>
</mrow>
Wherein ncFor tail vein quantity, VcFor the total volume of branch tree, LcFor the cumulative length of branch tree, DsIt is straight for female blood vessel
Footpath, a are the Fractal Dimension of the asymmetric tree of point shape blood vessel；
Step S3：The inter-species Scaling of metabolism is obtained, correspondence formula is as follows：
<mrow>
<mi>B</mi>
<mo>&Proportional;</mo>
<msup>
<mi>M</mi>
<mfrac>
<mi>a</mi>
<mn>3</mn>
</mfrac>
</msup>
<mo>;</mo>
</mrow>
Wherein B is the metabolic rate of research object, and M is the weight of research object.
2. exist in the kind of the asymmetric tree of according to claim 1 point of shape blood vessel with the modeling method of inter-species scale, feature
In the branch ratio BR=n in step S1i/ni-1；Natural scale DR=Di/Di-1；Length ratio LR=Li/Li-1；
Wherein ni、ni-1Blood vessel sum when blood vessel sum, blood vessel series when respectively blood vessel series is i are i-1；Di、Di-1Point
It Wei not blood vessel diameter of blood vessel diameter, blood vessel series of the blood vessel series when being i when being i-1；Li、Li-1Respectively blood vessel series is i
When length of vessel, blood vessel series be i-1 when length of vessel, ε be diameter bifurcated parameter, γ be length bifurcated parameter.
3. exist in the kind of the asymmetric tree of according to claim 2 point of shape blood vessel with the modeling method of inter-species scale, feature
In in step S1 in the asymmetric tree of shape blood vessel is divided, branch ratio BR is the sub- conduit number on a tie point and assumes
For constant；
Each branch vessel and the diameter of female blood vessel compare kiCompare j with lengthiIt is assumed that：
<mrow>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>D</mi>
<mrow>
<mi>d</mi>
<mi>a</mi>
<mi>u</mi>
<mi>g</mi>
<mi>h</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mrow>
<mi>m</mi>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
</mrow>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>D</mi>
<mrow>
<mi>d</mi>
<mi>a</mi>
<mi>u</mi>
<mi>g</mi>
<mi>h</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mrow>
<mi>m</mi>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
</mrow>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<mo>...</mo>
<mo>&le;</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>D</mi>
<mrow>
<mi>d</mi>
<mi>a</mi>
<mi>u</mi>
<mi>g</mi>
<mi>h</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mrow>
<mi>m</mi>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
</mrow>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
<mrow>
<msub>
<mi>j</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>L</mi>
<mrow>
<mi>d</mi>
<mi>a</mi>
<mi>u</mi>
<mi>g</mi>
<mi>h</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow>
<mi>m</mi>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
</mrow>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<msub>
<mi>j</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>L</mi>
<mrow>
<mi>d</mi>
<mi>a</mi>
<mi>u</mi>
<mi>g</mi>
<mi>h</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow>
<mi>m</mi>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
</mrow>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<mo>...</mo>
<mo>&le;</mo>
<msub>
<mi>j</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>L</mi>
<mrow>
<mi>d</mi>
<mi>a</mi>
<mi>u</mi>
<mi>g</mi>
<mi>h</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow>
<mi>m</mi>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
</mrow>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
Wherein ki=ji, i=1,2 ... BR；
Ddaughter1、Ddaughter2…DdaughterBRFor the diameter of branch vessel, DmotherFor the diameter of female blood vessel；Ldaughter1、
Ldaughter2…LdaughterBRFor the length of branch vessel, LmotherFor the length of female blood vessel；
Given constant branch ratio k simultaneouslyi：I.e.The asymmetric tree of shape blood vessel is divided to continue
It is branched to minimum diameter d0, d0As extremity arterial diameter.
4. exist in the kind of the asymmetric tree of according to claim 3 point of shape blood vessel with the modeling method of inter-species scale, feature
In tail vein quantity and the computational methods of the kind internal graticule relation of female blood vessel diameter are as follows in step S2：
In shape blood vessel asymmetry tree construction is divided, the tail vein number of branch tree is the monotropic function of female blood vessel diameter, right
Answer relational expression as follows：
<mrow>
<msub>
<mi>n</mi>
<mi>c</mi>
</msub>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>&GreaterEqual;</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo><</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>;</mo>
</mrow>
There are a real number kiSo that ncValue be all positive integer, and greatest common factor is 1；
Introduce a sequence an=F1(k-n), n=0,1,2 ...:
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>a</mi>
<mi>n</mi>
</msub>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<mo>&CenterDot;</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<mo>&CenterDot;</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>t</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>t</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein
The characteristic equation of sequence isCharacteristic root is σi=ri·eiθ, wherein σ is maximum
Arithmetic number root；
Because
So ri≤ σ, the characteristic root for showing maximum norm are an arithmetic number roots；
Wherein kiFor fixed value, as n ＞ ＞ 1, αn≈H1·σnWhen, H1It is preset parameter；
As female blood vessel diameter DsIn scope k-n≤Ds＜ k-n-1, n=0,1,2 ... is interior, and following relation is derived using recurrence method
Formula：F1(Ds)=an≈H1·σnAnd meet
Due toSo as to obtain
ByDerive σ=k-aWith
Simultaneously
From formulaWith σ≤BR, obtain
H ' here1Regard a constant as；Therefore following relational expression has been obtained：
I.e.
5. exist in the kind of the asymmetric tree of according to claim 3 point of shape blood vessel with the modeling method of inter-species scale, feature
In the total volume of branch tree and the computational methods of the kind internal graticule relation of female blood vessel diameter are as follows in step S2：
In shape blood vessel asymmetry tree construction is divided, the total volume of branch tree is the monotropic function of female blood vessel diameter:
<mrow>
<msub>
<mi>V</mi>
<mi>c</mi>
</msub>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>h</mi>
<mn>2</mn>
</msub>
<mo>&CenterDot;</mo>
<msubsup>
<mi>D</mi>
<mi>s</mi>
<mn>3</mn>
</msubsup>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>&GreaterEqual;</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>h</mi>
<mn>2</mn>
</msub>
<mo>&CenterDot;</mo>
<msubsup>
<mi>D</mi>
<mi>s</mi>
<mn>3</mn>
</msubsup>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo><</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
It is assumed that：Wherein h2It is a constant；
<mrow>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>...</mo>
<mo>+</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>&GreaterEqual;</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
DefinitionsiIt is a fraction within a certain error range；
There are a real number kiSo that ncValue be all positive integer, and greatest common factor is 1；
Introduce sequence
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>a</mi>
<mi>n</mi>
</msub>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
<mo>&CenterDot;</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<mo>&CenterDot;</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msubsup>
<mi>F</mi>
<mn>2</mn>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<msup>
<mi>k</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
<mo>+</mo>
<msub>
<mi>t</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
</mrow>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mo>+</mo>
<mn>...</mn>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<msub>
<mi>t</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein
The characteristic equation of sequence isCharacteristic root is σi=ri·eiθ, wherein σ is maximum
Arithmetic number root；
Because
So ri≤ σ, the characteristic root for showing maximum norm are an arithmetic number roots；
Wherein kiFor fixed value, as n ＞ ＞ 1, an≈H2·σnWhen, H2It is preset parameter；
As female blood vessel diameter DsIn scope k-n≤Ds＜ k-n-1, n=0,1,2 ... is interior, and below equation is derived using recurrence method:And
Due toIt obtains
ByDerive σ=k-aWith
It considers
According toWith σ≤BR, obtainH ' here2Regard a constant as；
The He of a≤3On the basis of,
Derive following relational expression：
I.e.
6. exist in the kind of the asymmetric tree of according to claim 3 point of shape blood vessel with the modeling method of inter-species scale, feature
In the cumulative length of branch tree and the computational methods of the kind internal graticule relation of female blood vessel diameter are as follows in step S2：
In shape blood vessel asymmetry tree construction is divided, the cumulative length of branch tree is the monotropic function of female blood vessel diameter:
<mrow>
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<mo>=</mo>
<msub>
<mi>F</mi>
<mn>3</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>h</mi>
<mn>3</mn>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>3</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>1</mn>
</msub>
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<mi>s</mi>
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<mo>+</mo>
<msub>
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<mn>3</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mn>2</mn>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
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</mrow>
<mo>+</mo>
<mo>...</mo>
<mo>+</mo>
<msub>
<mi>F</mi>
<mn>3</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>B</mi>
<mi>R</mi>
</mrow>
</msub>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
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<mo>,</mo>
</mrow>
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<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>&GreaterEqual;</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msub>
<mi>F</mi>
<mn>3</mn>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>h</mi>
<mn>3</mn>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>s</mi>
</msub>
<mo><</mo>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein h3It is a constant；
I.e.
Theory deduction obtains below equation:
H ' here3Regard a constant as；
The He of a >=2On the basis of,
Derive following equations：
I.e.
7. exist in the kind of the asymmetric tree of according to claim 1 point of shape blood vessel with the modeling method of inter-species scale, feature
In the computational methods of metabolic inter-species Scaling are as follows in step S3：
Due to known Qs∝B、Vc∝M、nc∝Qs, wherein QsFor female vascular flow；And it combines
It obtains：
It finally obtained the inter-species Scaling of metabolism:
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| US20110026789A1 (en) * | 2008-04-08 | 2011-02-03 | National University Of Singapore | Retinal image analysis systems and methods |
| CN104657598A (en) * | 2015-01-29 | 2015-05-27 | 中国人民解放军第三军医大学第一附属医院 | Method for calculating fractal dimension of microvessels of tissue |
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| CN107358612A (en) * | 2017-07-07 | 2017-11-17 | 东北大学 | A kind of retinal vessel segmenting system combined based on fractal dimension with gaussian filtering and method |
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| US20110026789A1 (en) * | 2008-04-08 | 2011-02-03 | National University Of Singapore | Retinal image analysis systems and methods |
| CN106537392A (en) * | 2014-04-22 | 2017-03-22 | 西门子保健有限责任公司 | Method and system for hemodynamic computation in coronary arteries |
| CN104657598A (en) * | 2015-01-29 | 2015-05-27 | 中国人民解放军第三军医大学第一附属医院 | Method for calculating fractal dimension of microvessels of tissue |
| CN107358612A (en) * | 2017-07-07 | 2017-11-17 | 东北大学 | A kind of retinal vessel segmenting system combined based on fractal dimension with gaussian filtering and method |
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| 徐明瑜 等: "分数阶算子探讨", 《太原理工大学学报》 * |
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