text: (cid:90) (cid:90) ∂ψ δ Ldt =δ (cid:96)(ψ , σ )d4x = 0, (2.2) σ ∂x µ which results in ∂ ∂(cid:96) − ∂(cid:96) = 0, (2.3) ∂xµ ∂(∂ψ /∂xµ) ∂ψ ν ν to have equation of motion. Let us to bring a simpler form of Eq. (2.3) as [3] ∂ ∂(cid:96) ∇ ∂(cid:96) − ∂(cid:96) + . ∇ = 0, (2.4) ∂t ∂(ψ¯˙ ) ∂( ψ¯ν) ∂ψ¯ν ν where dot is time partial derivative. Comparing with Eq. (1.1), we ﬁnd out that ∂ψ ∂ ∂(cid:96) iη = ( ), (2.5) ∂t ∂t ∂ψ¯˙ or ∇ (cid:96) = iηψ¯˙ψ + f( ψ¯, ψ¯), (2.6) ∇ that f( ψ¯, ψ¯) should be determined. Using Eq. (2.6) and another comparison Eqs. (2.4) and (1.1) results ∇ ∂(cid:96) ∇ .( ∇ ) = iµ ψ, (2.7) ∂( ψ¯) j so that ∇ ∂f( ψ¯, ψ¯) ∇ = iµ ψ, (2.8) ∂( ψ¯) j which reads ∇ ∇ f( ψ¯, ψ¯) = iµ ψ¯ψ + g(ψ¯), (2.9) j Where g(ψ¯) will be ascertained. Substituting Eq. (2.9) into (2.6), caused to rewrite Eq. (2.6) clearly ∇ (cid:96) = iηψ¯˙ψ + iµ ψ¯ψ + g(ψ¯). (2.10) j For the last time, we again compare Eqs. (2.4) and (1.1) to gain – 3 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ∂ψ

text: δ

text: Ldt

text: =δ

text: σ

text: σ

text: ∂x

text: µ

text: which

text: results

text: in

text: ∂xµ

text: ∂(∂ψ

text: /∂xµ)

text: ∂ψ

text: ν

text: ν

text: to

text: have

text: equation

text: of

text: motion.

text: Let

text: us

text: to

text: bring

text: a

text: simpler

text: form

text: of

text: Eq.

text: as

text: ∂t

text: ∂(ψ¯˙

text: ψ¯ν)

text: ∂ψ¯ν

text: ν

text: where

text: dot

text: is

text: time

text: partial

text: derivative.

text: Comparing

text: with

text: Eq.

text: we

text: ﬁnd

text: out

text: that

text: ∂ψ

text: iη

text: ∂t

text: ∂t

text: ∂ψ¯˙

text: or

text: iηψ¯˙ψ

text: f(

text: ψ¯,

text: ψ¯),

text: that

text: f(

text: ψ¯,

text: ψ¯)

text: should

text: be

text: determined.

text: Using

text: Eq.

text: and

text: another

text: comparison

text: Eqs.

text: and

text: results

text: iµ

text: ψ,

text: ψ¯)

text: j

text: so

text: that

text: ∂f(

text: ψ¯,

text: ψ¯)

text: iµ

text: ψ,

text: ψ¯)

text: j

text: which

text: reads

text: f(

text: ψ¯,

text: ψ¯)

text: iµ

text: ψ¯ψ

text: g(ψ¯),

text: j

text: Where

text: g(ψ¯)

text: will

text: be

text: ascertained.

text: Substituting

text: Eq.

text: into

text: caused

text: to

text: rewrite

text: Eq.

text: clearly

text: iηψ¯˙ψ

text: iµ

text: ψ¯ψ

text: g(ψ¯).

text: j

text: For

text: the

text: last

text: time,

text: we

text: again

text: compare

text: Eqs.

text: and

text: to

text: gain

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text: 12 J. HO¨RRMANN, J. PROCHNO, AND C. THA¨LE We now u(cid:90)se the following polar (cid:90)integ(cid:90)ration formula to compute the qth (cid:107) (cid:107) ∈ K moment of G . For K n one has that p ∞ − f(x) dx = n vol (K) f(ry) rn 1 dm (y)dr n K R n 0 bdK R → R for all non-negative measurable functions f : n (in fact, this may alternatively be used as a deﬁnition for the cone measure m of K). We K apply this to the function (cid:0) (cid:1) (cid:107) (cid:107) · −(cid:107) (cid:107) f(x) = x qp e x pp , x ∈ Rn, n 2Γ(1 + 1/p) (cid:90) (cid:90) B and the unit ball n of (cid:96)n, and obtain p p B ∞ (cid:107) (cid:107) n vol ( n) − − E G q = f(x) dx = n p rn+q 1e rp dr p R (2Γ(1 + 1))n n p 0 B n vol ( n) Γ(n+q) Γ(n+q) n p p p = = , p (2Γ(1 + 1))n Γ(n) p p (cid:2)(cid:12)(cid:10) (cid:11)(cid:12) (cid:3) where w(cid:12)e also (cid:12)used the deﬁnition of the gamma function as well as the B formula for vol ( n) stated in Section above. What is left to bound is the (cid:104)(cid:12)(cid:10)nq p(cid:11)(cid:12) (cid:105) (cid:104) (cid:13) (cid:13) (cid:13) (cid:13) (cid:105) term E G, θ(cid:12) . T(cid:12) his is done(cid:13)by mea(cid:13)ns of Pro(cid:13)position 3(cid:13).4, which yields that √ ≈ ∗ ∗ q E G, θ q q1/p (θ )q ∗ + q (θ )n . i i=1 p i i=q+1 2 (cid:3) This completes the proof. (cid:104) · (cid:105) We can now prove the desired ψ -estimate for the linear functionals , θ , ∈ S − B 2 θ n 1, on the boundary of n. p (cid:90) (cid:18) (cid:19) Proof of Proposition 4.3. By deﬁnition of an Orlicz norm, it is enough to show that |(cid:104) (cid:105)| n2/p x, θ 2 ≤ exp dmB (x) 2 . Sn−1 c2 np p ψ2 Us(cid:90)ing the s(cid:18)eries expansio(cid:19)n of the exponential function and Lemma 4.4 ap- plied with 2q instead of q there, we obtain that |(cid:104) (cid:105)| n(cid:90)2/p x, θ 2 (cid:88)exp dmB (x) Sn−1 c2 np p ∞ ψ2 = q=0 n(cid:88)q!2cq2ψ/q2p Snp−1 |(cid:104)x(cid:0),(cid:0)θ(cid:105)|(cid:1)2q(cid:1)d(cid:104)mBnp (x)(cid:13)(cid:13) (cid:13)(cid:13) (cid:112) (cid:13)(cid:13) (cid:13)(cid:13) (cid:105) ∞ ≤ n2q/p Γ n ∗ ∗ 2q 1 + c1 q!c2q Γ n+p2q (2q)1/p (θi )2i=q1 p∗ + 2q (θi )ni=2q+1 2 , q=1 ψ2 p ∗ ∈ ∞ where (θ )n is(cid:13)the non(cid:13)-increasing rea(cid:13)rrangem(cid:13)ent of (θ )n and c (0, ) i i=1 (cid:13) (cid:13) (cid:13) (cid:13)∈ S − i i=1 1 is an absolute constant. Observe that, since θ n 1, ∗ ≤ − ∗ ≤ − (θ )2q ∗ (2q)p1∗ 12 (θ )2q (2q)p1∗ 12 i i=1 p i i=1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: J.

text: HO¨RRMANN,

text: J.

text: PROCHNO,

text: AND

text: C.

text: THA¨LE

text: We

text: now

text: the

text: following

text: polar

text: formula

text: to

text: compute

text: the

text: qth

text: K

text: moment

text: of

text: G

text: For

text: K

text: n

text: one

text: has

text: that

text: p

text: f(x)

text: dx

text: n

text: vol

text: (K)

text: f(ry)

text: rn

text: dm

text: (y)dr

text: n

text: K

text: R

text: n

text: bdK

text: R

text: R

text: for

text: all

text: non-negative

text: measurable

text: functions

text: f

text: n

text: (in

text: fact,

text: this

text: may

text: alternatively

text: be

text: used

text: as

text: a

text: deﬁnition

text: for

text: the

text: cone

text: measure

text: m

text: of

text: K).

text: We

text: K

text: apply

text: this

text: to

text: the

text: function

text: f(x)

text: x

text: qp

text: e

text: x

text: pp

text: x

text: Rn,

text: n

text: B

text: and

text: the

text: unit

text: ball

text: n

text: of

text: and

text: obtain

text: p

text: p

text: B

text: n

text: vol

text: n)

text: E

text: G

text: q

text: f(x)

text: dx

text: n

text: p

text: rn+q

text: rp

text: dr

text: p

text: R

text: n

text: p

text: B

text: n

text: vol

text: n)

text: Γ(n+q)

text: Γ(n+q)

text: n

text: p

text: p

text: p

text: p

text: Γ(n)

text: p

text: p

text: where

text: also

text: the

text: deﬁnition

text: of

text: the

text: gamma

text: function

text: as

text: well

text: as

text: the

text: B

text: formula

text: for

text: vol

text: n)

text: stated

text: in

text: Section

text: above.

text: What

text: is

text: left

text: to

text: bound

text: is

text: the

text: term

text: E

text: G,

text: his

text: is

text: of

text: which

text: yields

text: that

text: q

text: E

text: G,

text: θ

text: q

text: (θ

text: )q

text: q

text: (θ

text: )n

text: i

text: p

text: i

text: This

text: completes

text: the

text: proof.

text: We

text: can

text: now

text: prove

text: the

text: desired

text: ψ

text: -estimate

text: for

text: the

text: linear

text: functionals

text: θ

text: S

text: B

text: θ

text: n

text: on

text: the

text: boundary

text: of

text: n.

text: p

text: Proof

text: of

text: Proposition

text: By

text: deﬁnition

text: of

text: an

text: Orlicz

text: norm,

text: it

text: is

text: enough

text: to

text: show

text: that

text: x,

text: θ

text: exp

text: dmB

text: (x)

text: np

text: p

text: ψ2

text: the

text: of

text: the

text: exponential

text: function

text: and

text: Lemma

text: ap-

text: plied

text: with

text: instead

text: of

text: q

text: there,

text: we

text: obtain

text: that

text: x,

text: θ

text: dmB

text: (x)

text: np

text: p

text: ψ2

text: Γ

text: n

text: Γ

text: (θi

text: p∗

text: (θi

text: ψ2

text: p

text: where

text: (θ

text: )n

text: of

text: (θ

text: )n

text: and

text: c

text: i

text: S

text: i

text: is

text: an

text: absolute

text: constant.

text: Observe

text: that,

text: since

text: θ

text: n

text: (θ

text: (θ

text: i

text: p

text: i

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text: 16 J. HO¨RRMANN, J. PROCHNO, AND C. THA¨LE ∈ ∞ Proposition 4.6. There exists an absolute constant c (0, ) such that, for every 1 ≤ p < 2 and a(cid:13)(cid:13)ll θ ∈ Sn∞−1, (cid:13)(cid:13) ψ2 (cid:13) (cid:13) 1 (cid:104) · (cid:105) ≤ n1/2−1/p , θ ψ2 cψ2 , (cid:107) · (cid:107) S − where is the norm on the Orlicz space L ( n 1, mB ). ψ2 ψ2 p np ≤ ∈ S − Proof. Fix 1 p < 2 and θ = (θ , . . . , θ ) n∞ 1. By Proposition 3.1, we 1 n have that (cid:107)(cid:104) · (cid:105)(cid:107) (cid:107)(cid:104) · (cid:105)(cid:107) (cid:46) √, θ q , θ sup , ψ2 ≥ q q 2 (cid:107) · (cid:107) S − where is the L -norm on n 1. Let us now consider the two cases q < n ≥ q (cid:32)q (cid:33) p ≤ ∞ and q n separately. From Lem(cid:104)ma 4.(cid:13)4 we ob(cid:13)tain that,(cid:13)for any 1 (cid:13) (cid:105)q < , (cid:13) (cid:13) (cid:13) (cid:13) (cid:107)(cid:104) · (cid:105)(cid:107) (cid:46)  Γ(n) 1/q ∗ √ ∗ , θ q (cid:2)Γ(n+p q) q1/(cid:3)p(cid:18)(θi )qi=1(cid:19)p∗ + q (θi )ni=q+1 2  p √ √  (cid:18) (cid:19) − Γ(n) 1/q ≤  q + q n q p : 2 q < n (cid:46)  Γ(n+q) p Γ(n) 1/q ≥ q p : q n . Γ(n+q) p Here, we have estimated in the second step the two norms against the (cid:96)∞- (cid:107) (cid:107) norm and used the fact that 1 + 1∗ = 1 and that θ ∞ = 1. Applying (cid:32) (cid:33) p p Stirling’s formula, we see that Γ(np ) 1/q (cid:46) (n−1)1 (1−n+q)1 n p 2 q (n + q) 2 p q Γ(n+q) p (cid:46) (n−1)1 (1−n+q)1 1 (n + q) p 2 q (n + q) 2 p q = . (n + q)1/p √ √ Now, we note that if 2 ≤ q < n, (cid:107)(cid:104)·√,θ(cid:105)(cid:107)q (cid:46) q+ n−q is maximal for q = 2, − q (n+q)1/p ∈ ∞ where it is c n1/2 1/p w√ith some absolute constant c (0, ). Similarly, if q ≥ n, (cid:107)(cid:104)·1√,θ(cid:105)(cid:107)q (cid:46) q is maximal for the cho1ice q = n, where the q (n+q)1/−p ∈ ∞ expression reduces to c n1/2 1/p with another absolute constant c (0, ). 2 2 As a consequence, we ﬁnd that (cid:107)(cid:104) · (cid:105)(cid:107) (cid:46) − , θ n1/2 1/p ψ2 (cid:3) and the proof is complete. We can now proceed and derive the announced high-probability upper bound by partially following the strategy in [14] (see also [13, Chapter 11.5.1]). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.

text: HO¨RRMANN,

text: J.

text: PROCHNO,

text: AND

text: C.

text: THA¨LE

text: Proposition

text: There

text: exists

text: an

text: absolute

text: constant

text: c

text: such

text: that,

text: for

text: every

text: p

text: and

text: θ

text: ψ2

text: θ

text: ψ2

text: S

text: where

text: is

text: the

text: norm

text: on

text: the

text: Orlicz

text: space

text: L

text: n

text: mB

text: ψ2

text: ψ2

text: p

text: np

text: S

text: Proof.

text: Fix

text: p

text: and

text: θ

text: (θ

text: θ

text: n∞

text: By

text: Proposition

text: we

text: n

text: have

text: that

text: θ

text: q

text: θ

text: sup

text: ψ2

text: q

text: q

text: S

text: where

text: is

text: the

text: L

text: -norm

text: on

text: n

text: Let

text: us

text: now

text: consider

text: the

text: two

text: cases

text: q

text: n

text: q

text: p

text: and

text: q

text: n

text: separately.

text: From

text: we

text: any

text: Γ(n)

text: θ

text: q

text: q)

text: q

text: (θi

text: p

text: Γ(n)

text: q

text: q

text: n

text: q

text: p

text: q

text: n

text: Γ(n+q)

text: p

text: Γ(n)

text: q

text: p

text: q

text: n

text: Γ(n+q)

text: p

text: Here,

text: we

text: have

text: estimated

text: in

text: the

text: second

text: step

text: the

text: two

text: norms

text: against

text: the

text: norm

text: and

text: used

text: the

text: fact

text: that

text: 1∗

text: and

text: that

text: θ

text: Applying

text: p

text: p

text: Stirling’s

text: formula,

text: we

text: see

text: that

text: Γ(np

text: n

text: p

text: q

text: (n

text: q)

text: p

text: q

text: Γ(n+q)

text: p

text: (n

text: q)

text: p

text: q

text: (n

text: q)

text: p

text: q

text: (n

text: Now,

text: we

text: note

text: that

text: if

text: q

text: n,

text: q+

text: n−q

text: is

text: maximal

text: for

text: q

text: q

text: where

text: it

text: is

text: c

text: w√ith

text: some

text: absolute

text: constant

text: c

text: Similarly,

text: if

text: q

text: n,

text: q

text: is

text: maximal

text: for

text: the

text: q

text: n,

text: where

text: the

text: q

text: expression

text: reduces

text: to

text: c

text: with

text: another

text: absolute

text: constant

text: c

text: As

text: a

text: consequence,

text: we

text: ﬁnd

text: that

text: θ

text: ψ2

text: and

text: the

text: proof

text: is

text: complete.

text: We

text: can

text: now

text: proceed

text: and

text: derive

text: the

text: announced

text: high-probability

text: upper

text: bound

text: by

text: partially

text: following

text: the

text: strategy

text: in

text: (see

text: also

text: Chapter

text: ##LTLine##

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text: where the two index sets J−j and Jj are deﬁned according to the ﬂow orientation in the + vessel segments passing through the junction j, as follows: vj = +w if e ends at junction j , e e − vj = w if e begins at junction j , e e { ∈ | ≥ } Jj = e Jj v 0 , + e { ∈ | } J−j = e Jj v < 0 , e so that velocities w in the vessel segment (cid:96) deﬁne v depending on the oriented network e e e topology. We can call Jj the index set of sources and J−j the index set of sinks. It is worth + T noting that for the perfusion tree (portal vein tree) with binary bifurcation at any junction, P the set Jj = e contains just one index of the saturating vessel, while J−j contains the two + T children vessels, say e , e . For the hepatic vein tree with the analogous property, the role 1 2 H of J−j = e and Jj = e , e is exchanged. + 1 2 Assembling equations of the transport on the network. Due to (25) and knowledge of the transition times, the state of the transport is described by the junction saturations { } S˜j(t) . The resulting system of equations governing the nodal saturations takes the follow- j (cid:88) (cid:88) ing form: − S˜j(t) A vj + A vjS˜ke(t T ) = 0 . (27) e e e e e ∈ ∈ e J−j e Jj + ∈ { } The junction equations (27) can be evaluated for discretized time interval, i.e. for t t − ∈ n n where t = t + n∆t. Obviously, for a given T , the saturation at t T [t , t ] is n 0 e n e p p+1 approximated using the average of values at t and t . p p+1 4 Numerical simulation of liver perfusion The model introduced in the preceding sections has been implemented in our non-commercial codes. In this section, we illustrate the model response using numerical examples with real geometry of the human liver, but with the perfusion trees of the portal and hepatic veins gen- erated using the CCO method (Georg et al, 2010). Before presenting particular results which serve as a proof concept, we explain a ﬂowchart of the currently developed computational modeling tool. CT scans of human liver processed by the LISA (LIver Surgery Analyser) software (Jiˇr´ık, 2013–2015) are the starting point to the numerical simulation of liver perfusion. The LISA code includes the semi-automated segmentation method based on the graph-cut algorithm and returns the geometrical model of liver parenchyma that can be transformed into a volumetric ﬁnite element mesh. The code is also used to identify the position and diameter of the portal and hepatic veins at the point of entry into the liver parenchyma; this data is necessary to generate two artiﬁcial vascular trees associated with the portal and hepatic veins, which in part are employed directly in the simulation of the upper-level ﬂow using the 1D ﬂow model 14

text: where

text: the

text: two

text: index

text: sets

text: J−j

text: and

text: Jj

text: are

text: deﬁned

text: according

text: to

text: the

text: ﬂow

text: orientation

text: in

text: the

text: vessel

text: segments

text: passing

text: through

text: the

text: junction

text: j,

text: as

text: follows:

text: vj

label: equation

text: +w

text: if

text: e

text: ends

text: at

text: junction

text: j

text: e

text: e

text: vj

text: w

text: if

text: e

text: begins

text: at

text: junction

text: j

text: e

text: e

text: Jj

text: e

text: Jj

text: v

text: e

text: J−j

text: e

text: Jj

text: v

text: e

text: so

text: that

text: velocities

text: w

text: in

text: the

text: vessel

text: segment

text: deﬁne

text: v

text: depending

text: on

text: the

text: oriented

text: network

text: e

text: e

text: e

text: topology.

text: We

text: can

text: call

text: Jj

text: the

text: index

text: set

text: of

text: sources

text: and

text: J−j

text: the

text: index

text: set

text: of

text: sinks.

text: It

text: is

text: worth

text: T

text: noting

text: that

text: for

text: the

text: perfusion

text: tree

text: (portal

text: vein

text: tree)

text: with

text: binary

text: bifurcation

text: at

text: any

text: junction,

text: P

text: the

text: set

text: Jj

text: e

text: contains

text: just

text: one

text: index

text: of

text: the

text: saturating

text: vessel,

text: while

text: J−j

text: contains

text: the

text: two

text: T

text: children

text: vessels,

text: say

text: e

text: e

text: For

text: the

text: hepatic

text: vein

text: tree

text: with

text: the

text: analogous

text: property,

text: the

text: role

text: H

text: of

text: J−j

text: e

text: and

text: Jj

text: e

text: e

text: is

text: exchanged.

text: Assembling

text: equations

text: of

text: the

text: transport

text: on

text: the

text: network.

text: Due

text: to

text: and

text: knowledge

text: of

text: the

text: transition

text: times,

text: the

text: state

text: of

text: the

text: transport

text: is

text: described

text: by

text: the

text: junction

text: saturations

text: S˜j(t)

text: The

text: resulting

text: system

text: of

text: equations

text: governing

text: the

text: nodal

text: saturations

text: takes

text: the

text: follow-

text: j

text: ing

text: form:

text: S˜j(t)

text: A

text: vj

text: A

text: vjS˜ke(t

text: T

text: e

text: e

text: e

text: e

text: e

text: e

text: J−j

text: e

text: Jj

text: The

text: junction

text: equations

text: can

text: be

text: evaluated

text: for

text: discretized

text: time

text: interval,

text: i.e.

text: for

text: t

text: t

text: n

text: n

text: where

text: t

text: t

text: n∆t.

text: Obviously,

text: for

text: a

text: given

text: T

text: the

text: saturation

text: at

text: t

text: T

text: [t

text: t

text: is

text: n

text: e

text: n

text: e

text: p

text: approximated

text: using

text: the

text: average

text: of

text: values

text: at

text: t

text: and

text: t

text: p

text: Numerical

text: simulation

text: of

text: liver

text: perfusion

text: The

text: model

text: introduced

text: in

text: the

text: preceding

text: sections

text: has

text: been

text: implemented

text: in

text: our

text: non-commercial

text: codes.

text: In

text: this

text: section,

text: we

text: illustrate

text: the

text: model

text: response

text: using

text: numerical

text: examples

text: with

text: real

text: geometry

text: of

text: the

text: human

text: liver,

text: but

text: with

text: the

text: perfusion

text: trees

text: of

text: the

text: portal

text: and

text: hepatic

text: veins

text: gen-

text: erated

text: using

text: the

text: CCO

text: method

text: (Georg

text: et

text: al,

text: Before

text: presenting

text: particular

text: results

text: which

text: serve

text: as

text: a

text: proof

text: concept,

text: we

text: explain

text: a

text: ﬂowchart

text: of

text: the

text: currently

text: developed

text: computational

text: modeling

text: tool.

text: CT

text: scans

text: of

text: human

text: liver

text: processed

text: by

text: the

text: LISA

text: (LIver

text: Surgery

text: Analyser)

text: software

text: (Jiˇr´ık,

text: are

text: the

text: starting

text: point

text: to

text: the

text: numerical

text: simulation

text: of

text: liver

text: perfusion.

text: The

text: LISA

text: code

text: includes

text: the

text: semi-automated

text: segmentation

text: method

text: based

text: on

text: the

text: graph-cut

text: algorithm

text: and

text: returns

text: the

text: geometrical

text: model

text: of

text: liver

text: parenchyma

text: that

text: can

text: be

text: transformed

text: into

text: a

text: volumetric

text: ﬁnite

text: element

text: mesh.

text: The

text: code

text: is

text: also

text: used

text: to

text: identify

text: the

text: position

text: and

text: diameter

text: of

text: the

text: portal

text: and

text: hepatic

text: veins

text: at

text: the

text: point

text: of

text: entry

text: into

text: the

text: liver

text: parenchyma;

text: this

text: data

text: is

text: necessary

text: to

text: generate

text: two

text: artiﬁcial

text: vascular

text: trees

text: associated

text: with

text: the

text: portal

text: and

text: hepatic

text: veins,

text: which

text: in

text: part

text: are

text: employed

text: directly

text: in

text: the

text: simulation

text: of

text: the

text: upper-level

text: ﬂow

text: using

text: the

text: ﬂow

text: model


text:   · · − · − point permeability K [m2 (Pas) 1] porosity φ [–] coef. G [(Pa s) 1] − 1.10 0.01 0.12  − × − × − × − Aˆ K1 = 0.01 1.21 0.08 10 9 φ1 = 1.05 10 3 G2 = 3.52 10 5 1  0.12 0.08 1.11  − − 4.38 0.18 0.61 − −  × − × − × − K3 = 0.18 3.84 0.03 10 9 φ3 = 1.76 10 3 G3 = 4.27 10 5 − − 2  0.61 0.03 4.81 − 1.00 0.03 0.11  − × − × − × − Bˆ K1 = 0.03 1.05 0.08 10 9 φ1 = 0.93 10 3 G2 = 3.26 10 5 1  0.11 0.08 0.95  − − 4.32 0.15 0.60 − − × − × − × − K3 = 0.15 3.55 0.02 10 9 φ3 = 1.64 10 3 G3 = 4.20 10 5 − − 2 0.60 0.02 4.80 Table 2: Perfusion parameters at two distinct points Aˆ, Bˆ (see Fig. 10) related to com- partments 1 and 3 obtained on the basis of calculations carried out on the artiﬁcial vascular trees Figure 7: The upper-level vascular trees (top – portal, bottom – hepatic) where the ﬂow is approximated using the 1D model based on the extended Bernoulli equation. Their terminal points correspond either to the sources in compartment 1 or to the sinks in compartment 3 given range of the vessel diameters. The anisotropic permeabilities K1, K3 of compartments 1 and 3 are evaluated using the expression derived by Huge and van Campen (Huyghe and Campen, 1995), whereas the coupling perfusion coeﬃcients G2, G3 are calculated with the 1 2 help of the theory presented in Hyde et al (2013). Values of the model parameters at two selected points are listed in Tab. 2. The isotropic permeability in compartment 2 is taken as × − K2 = 2 10 14 and the uniform porosity as φ2 = 0.15, both values are chosen with respect to the parameters identiﬁed in Debbaut et al (2012). The domain of the liver is discretized by a ﬁnite element mesh consisting of 11 019 tetrahedrons and 2 442 nodes. The rather coarse mesh was chosen due to large computational cost associated with the tracer transport calculation. The 1D ﬂow model is connected to the multicompartment model through 45 junctions, 34 of them are the sources located in compartment 1 and remaining 11 are the sinks located in compartment 3. The upper-level vascular trees, terminal points of which correspond to the sources and sinks, are shown in Fig. 7 and Fig. 8 (middle). The input blood velocity to · − the portal vein is taken as v = 0.25 m s 1 and the output blood pressure prescribed in the in hepatic vein is p = 103 Pa. out The steady state results computed for the three compartment perfusion model described 17 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: table

text: point

text: permeability

text: K

text: (Pas)

text: porosity

text: φ

text: coef.

text: G

text: [(Pa

text: s)

text: Aˆ

text: φ1

text: φ3

text: Bˆ

text: φ1

text: φ3

text: Table

text: Perfusion

text: parameters

text: at

text: two

text: distinct

text: points

text: Aˆ,

text: Bˆ

text: (see

text: Fig.

text: related

text: to

text: com-

text: partments

text: and

text: obtained

text: on

text: the

text: basis

text: of

text: calculations

text: carried

text: out

text: on

text: the

text: artiﬁcial

text: vascular

text: trees

text: Figure

text: The

text: upper-level

text: vascular

text: trees

text: (top

text: portal,

text: bottom

text: hepatic)

text: where

text: the

text: ﬂow

text: is

text: approximated

text: using

text: the

text: model

text: based

text: on

text: the

text: extended

text: Bernoulli

text: equation.

text: Their

text: terminal

text: points

text: correspond

text: either

text: to

text: the

text: sources

text: in

text: compartment

text: or

text: to

text: the

text: sinks

text: in

text: compartment

text: given

text: range

text: of

text: the

text: vessel

text: diameters.

text: The

text: anisotropic

text: permeabilities

text: of

text: compartments

text: and

text: are

text: evaluated

text: using

text: the

text: expression

text: derived

text: by

text: Huge

text: and

text: van

text: Campen

text: (Huyghe

text: and

text: Campen,

text: whereas

text: the

text: coupling

text: perfusion

text: coeﬃcients

text: are

text: calculated

text: with

text: the

text: help

text: of

text: the

text: theory

text: presented

text: in

text: Hyde

text: et

text: al

text: Values

text: of

text: the

text: model

text: parameters

text: at

text: two

text: selected

text: points

text: are

text: listed

text: in

text: Tab.

text: The

text: isotropic

text: permeability

text: in

text: compartment

text: is

text: taken

text: as

text: and

text: the

text: uniform

text: porosity

text: as

text: φ2

text: both

text: values

text: are

text: chosen

text: with

text: respect

text: to

text: the

text: parameters

text: identiﬁed

text: in

text: Debbaut

text: et

text: al

text: The

text: domain

text: of

text: the

text: liver

text: is

text: discretized

text: by

text: a

text: ﬁnite

text: element

text: mesh

text: consisting

text: of

text: tetrahedrons

text: and

text: nodes.

text: The

text: rather

text: coarse

text: mesh

text: was

text: chosen

text: due

text: to

text: large

text: computational

text: cost

text: associated

text: with

text: the

text: tracer

text: transport

text: calculation.

text: The

text: ﬂow

text: model

text: is

text: connected

text: to

text: the

text: multicompartment

text: model

text: through

text: junctions,

text: of

text: them

text: are

text: the

text: sources

text: located

text: in

text: compartment

text: and

text: remaining

text: are

text: the

text: sinks

text: located

text: in

text: compartment

text: The

text: upper-level

text: vascular

text: trees,

text: terminal

text: points

text: of

text: which

text: correspond

text: to

text: the

text: sources

text: and

text: sinks,

text: are

text: shown

text: in

text: Fig.

text: and

text: Fig.

text: (middle).

text: The

text: input

text: blood

text: velocity

text: to

text: the

text: portal

text: vein

text: is

text: taken

text: as

text: v

text: m

text: s

text: and

text: the

text: output

text: blood

text: pressure

text: prescribed

text: in

text: the

text: in

text: hepatic

text: vein

text: is

text: p

text: Pa.

text: out

text: The

text: steady

text: state

text: results

text: computed

text: for

text: the

text: three

text: compartment

text: perfusion

text: model

text: described

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≈ − compared to the ones computed in the ﬁlling/draining compartments (φ1 and φ3 10 4, see Tab. 2), fulﬁlls its function as a ﬁltration system, in which the saturation S decreases lob only slowly over time, see Fig. 12 (top right). As a consequence, the perfusion properties of the ﬁltration system inﬂuence the time development of the hepatic saturation S which, hv compared to the portal saturation S , changes much more gradually depending on tracer pv transfer from the ﬁltration system. Liver model with pathologically changed permeability To demonstrate the appli- cability of the proposed model for numerical simulation of various liver tissue pathologies, a simple illustrative example is presented. In a small spherical region, which is depicted in Fig. 13, the permeability parameter K of compartment 2 is locally changed to values close to zero (due to numerical reasons arising from the solution of the Darcy system). This change in permeability should mimic a pathology of the lobular structure, namely a lesion. To be able to quantify how the introduction of the lesion within the parenchyma aﬀects the overall perfusion of the liver including the transport of the contrast ﬂuid, the following auxiliary quantities are introduced: − • ∆S = S S , pv pv − p•v ∆S = S S , lob lob − •lob (29) ∆S = S S , hv h−v • hv ∆C = C C , • where the upper index “ ” denotes a quantity associated with the pathologically changed liver model. Similarly to Fig. 11, Fig. 14 shows the distribution of ∆S and ∆C in planes A lob and B at the three previously introduced time instants (t = 1.17 s, t = 1.95 s, and t = 8 s). 1 2 3 From the ∆S isocontours, it can be deduced that the lesion, which is located in the upper lob part of the liver model (near plane A), aﬀects the perfusion and tracer transport globally. This phenomenon can be particularly observed on plane B, where at the beginning of the simulation (t = 1.95 s) the lobular saturation S of the original model is higher than in the 2 lob case of the model with lesion (i.e., ∆S > 0). The presence of the lesion within the liver lob model and its inﬂuence on the overall blood perfusion is also reﬂected by the distribution of ∆C in Fig. 14 (bottom). Here the diﬀerences in the total concentration ∆C are apparent on both planes A and B. The region with ∆C > 0 in the left part of plane A is of particular importance, as it is clearly distinguishable in all snapshots, thus, indicating the location of the lesion. To assess the impact of the lesion on the perfusion results in a time-continuous manner, we refer to the graphs shown in Fig. 15, which capture evolution of ∆S , ∆S and ∆C in pv hv time at the points Aˆ and Bˆ previously introduced. Note that, in comparison with Fig. 12, the graph of ∆S is not displayed in Fig. 15, as it would closely resemble that of ∆C. When lob comparing the plotted curves, two observations are worth of nothing. First, although the perfusion anomaly (lesion) is located only in the ﬁltration compartment, it inﬂuences the perfusion and, thereby, the tracer transport through the higher hierarchies included in to the portal compartment (the diﬀerence ∆S is not zero from the beginning, i.e. for t < t ). pv 2 Second, curves plotted for the selected points Aˆ and Bˆ are dissimilar each other, no common features are evident. This is particularly apparent in the case of ∆S , which for both points pv exhibit completely diﬀerent behaviour: On one hand, at point Aˆ (red solid line in Fig. 12 (left)), the graph is characterized by a delayed maximum portal saturation in the undamaged liver model (fast change of ∆S from negative to positive values within a short time interval). pv 20

text: compared

text: to

text: the

text: ones

text: computed

text: in

text: the

text: ﬁlling/draining

text: compartments

text: and

text: φ3

text: see

text: Tab.

text: fulﬁlls

text: its

text: function

text: as

text: a

text: ﬁltration

text: system,

text: in

text: which

text: the

text: saturation

text: S

text: decreases

text: lob

text: only

text: slowly

text: over

text: time,

text: see

text: Fig.

text: (top

text: right).

text: As

text: a

text: consequence,

text: the

text: perfusion

text: properties

text: of

text: the

text: ﬁltration

text: system

text: inﬂuence

text: the

text: time

text: development

text: of

text: the

text: hepatic

text: saturation

text: S

text: which,

text: hv

text: compared

text: to

text: the

text: portal

text: saturation

text: S

text: changes

text: much

text: more

text: gradually

text: depending

text: on

text: tracer

text: pv

text: transfer

text: from

text: the

text: ﬁltration

text: system.

text: Liver

text: model

text: with

text: pathologically

text: changed

text: permeability

text: To

text: demonstrate

text: the

text: appli-

text: cability

text: of

text: the

text: proposed

text: model

text: for

text: numerical

text: simulation

text: of

text: various

text: liver

text: tissue

text: pathologies,

text: a

text: simple

text: illustrative

text: example

text: is

text: presented.

text: In

text: a

text: small

text: spherical

text: region,

text: which

text: is

text: depicted

text: in

text: Fig.

text: the

text: permeability

text: parameter

text: K

text: of

text: compartment

text: is

text: locally

text: changed

text: to

text: values

text: close

text: to

text: zero

text: (due

text: to

text: numerical

text: reasons

text: arising

text: from

text: the

text: solution

text: of

text: the

text: Darcy

text: system).

text: This

text: change

text: in

text: permeability

text: should

text: mimic

text: a

text: pathology

text: of

text: the

text: lobular

text: structure,

text: namely

text: a

text: lesion.

text: To

text: be

text: able

text: to

text: quantify

text: how

text: the

text: introduction

text: of

text: the

text: lesion

text: within

text: the

text: parenchyma

text: aﬀects

text: the

text: overall

text: perfusion

text: of

text: the

text: liver

text: including

text: the

text: transport

text: of

text: the

text: contrast

text: ﬂuid,

text: the

text: following

text: auxiliary

text: quantities

text: are

text: introduced:

text: ∆S

text: S

text: S

text: pv

text: pv

text: p•v

text: ∆S

text: S

text: S

text: lob

text: lob

text: •lob

text: ∆S

text: S

text: S

text: hv

text: h−v

text: hv

text: ∆C

text: C

text: C

text: where

text: the

text: upper

text: index

text: denotes

text: a

text: quantity

text: associated

text: with

text: the

text: pathologically

text: changed

text: liver

text: model.

text: Similarly

text: to

text: Fig.

text: Fig.

text: shows

text: the

text: distribution

text: of

text: ∆S

text: and

text: ∆C

text: in

text: planes

text: A

text: lob

text: and

text: B

text: at

text: the

text: three

text: previously

text: introduced

text: time

text: instants

text: (t

text: s,

text: t

text: s,

text: and

text: t

text: s).

text: From

text: the

text: ∆S

text: isocontours,

text: it

text: can

text: be

text: deduced

text: that

text: the

text: lesion,

text: which

text: is

text: located

text: in

text: the

text: upper

text: lob

text: part

text: of

text: the

text: liver

text: model

text: (near

text: plane

text: A),

text: aﬀects

text: the

text: perfusion

text: and

text: tracer

text: transport

text: globally.

text: This

text: phenomenon

text: can

text: be

text: particularly

text: observed

text: on

text: plane

text: B,

text: where

text: at

text: the

text: beginning

text: of

text: the

text: simulation

text: (t

text: s)

text: the

text: lobular

text: saturation

text: S

text: of

text: the

text: original

text: model

text: is

text: higher

text: than

text: in

text: the

text: lob

text: case

text: of

text: the

text: model

text: with

text: lesion

text: (i.e.,

text: ∆S

text: The

text: presence

text: of

text: the

text: lesion

text: within

text: the

text: liver

text: lob

text: model

text: and

text: its

text: inﬂuence

text: on

text: the

text: overall

text: blood

text: perfusion

text: is

text: also

text: reﬂected

text: by

text: the

text: distribution

text: of

text: ∆C

text: in

text: Fig.

text: (bottom).

text: Here

text: the

text: diﬀerences

text: in

text: the

text: total

text: concentration

text: ∆C

text: are

text: apparent

text: on

text: both

text: planes

text: A

text: and

text: B.

text: The

text: region

text: with

text: ∆C

text: in

text: the

text: left

text: part

text: of

text: plane

text: A

text: is

text: of

text: particular

text: importance,

text: as

text: it

text: is

text: clearly

text: distinguishable

text: in

text: all

text: snapshots,

text: thus,

text: indicating

text: the

text: location

text: of

text: the

text: lesion.

text: To

text: assess

text: the

text: impact

text: of

text: the

text: lesion

text: on

text: the

text: perfusion

text: results

text: in

text: a

text: time-continuous

text: manner,

text: we

text: refer

text: to

text: the

text: graphs

text: shown

text: in

text: Fig.

text: which

text: capture

text: evolution

text: of

text: ∆S

text: ∆S

text: and

text: ∆C

text: in

text: pv

text: hv

text: time

text: at

text: the

text: points

text: Aˆ

text: and

text: Bˆ

text: previously

text: introduced.

text: Note

text: that,

text: in

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text: with

text: Fig.

text: the

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text: of

text: ∆S

text: is

text: not

text: displayed

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text: Fig.

text: as

text: it

text: would

text: closely

text: resemble

text: that

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text: ∆C.

text: When

text: lob

text: comparing

text: the

text: plotted

text: curves,

text: two

text: observations

text: are

text: worth

text: of

text: nothing.

text: First,

text: although

text: the

text: perfusion

text: anomaly

text: (lesion)

text: is

text: located

text: only

text: in

text: the

text: ﬁltration

text: compartment,

text: it

text: inﬂuences

text: the

text: perfusion

text: and,

text: thereby,

text: the

text: tracer

text: transport

text: through

text: the

text: higher

text: hierarchies

text: included

text: in

text: to

text: the

text: portal

text: compartment

text: (the

text: diﬀerence

text: ∆S

text: is

text: not

text: zero

text: from

text: the

text: beginning,

text: i.e.

text: for

text: t

text: t

text: pv

text: Second,

text: curves

text: plotted

text: for

text: the

text: selected

text: points

text: Aˆ

text: and

text: Bˆ

text: are

text: dissimilar

text: each

text: other,

text: no

text: common

text: features

text: are

text: evident.

text: This

text: is

text: particularly

text: apparent

text: in

text: the

text: case

text: of

text: ∆S

text: which

text: for

text: both

text: points

text: pv

text: exhibit

text: completely

text: diﬀerent

text: behaviour:

text: On

text: one

text: hand,

text: at

text: point

text: Aˆ

text: (red

text: solid

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text: Fig.

text: (left)),

text: the

text: graph

text: is

text: characterized

text: by

text: a

text: delayed

text: maximum

text: portal

text: saturation

text: in

text: the

text: undamaged

text: liver

text: model

text: (fast

text: change

text: of

text: ∆S

text: from

text: negative

text: to

text: positive

text: values

text: within

text: a

text: short

text: time

text: interval).

text: pv


text: Figure 1: Decomposition of the blood vessel perfusion system in the liver: higher hierar- chy (HH) – blood ﬂow described by a 1D model, lower hierarchy (LH) – blood distribution governed by the Darcy ﬂow model extended for multiple compartments, each compartment characterized by a permeability tensor Ki a required resolution of microvessels. The most popular approach is based on the constrained constructive optimization (CCO), see e.g. Georg et al (2010), Schwen and Preusser (2012). However, there are more advanced approaches based on the metabolism driven processes of the vascular tree reconstruction (Schneider et al, 2012). Rather than modeling the ﬂow on the complex branching system, many publications have been devoted to detailed studies of microcirclation. Perfusion characteristics of blood ﬂow through hepatic microcapillaries of human lobular sinusoids were analyzed by Debbaut et al (2012) using tissue samples processed by corrosion casting and the micro-CT imaging; anisotropic tensors of the sinusoidal permeability were calculated by a simple averaging of the ﬂow ﬁeld obtained using standard CFD simulations. Blood circulation in the liver lobule was considered in Bonﬁglio et al (2010) using both Newtonian and shear-thinning ﬂow models. The liver tissue is able to adapt rapidly to modiﬁed hydraulic conditions; in Ricken et al (2010), the sinusoidal permeability has been studied in terms of a vascular remodeling process induced by outﬂow obstruction. Sinusoidal vascular canals were assumed to align with the blood pressure gradient. In Ricken et al (2015), using the general framework of a two-scale approach and the theory of porous media, the basic coupled transport-reaction mechanisms between blood perfusion and the hepatic cell metabolism were described at the level of lobule. Lymphatic ﬂow production in the liver lobules arranged in periodic lattices was treated in Siggers et al (2014). A similar approach has been used in Lettmann and Hardtke-Wolenski (2014) to study microcirculation driven transport of oxygen and cytokine in the tissue. Besides the blood ﬂow in healthy liver itself, diseased tissue deserve attention from medical point of view, but also as a challenge for mathematical modeling; there is a considerable number of issues related to modeling pathophysiological conditions, such as the ﬂuid transport through vascular tumors (Pozrikidis, 2010; Shipley and Chapman, 2010; Mescam et al, 2010). Since the straightforward mathematical description of blood ﬂow on the complete vascular trees featured by several hierarchies using the Navier-Stokes equations leads to a prohibitively complicated numerical model, some convenient model reduction is needed. We adhere to the 3 ##LTFigure##

text: Figure

text: Decomposition

label: caption

text: of

text: the

text: blood

text: vessel

text: perfusion

text: system

text: in

text: the

text: liver:

text: higher

text: hierar-

text: chy

text: (HH)

text: blood

text: ﬂow

text: described

text: by

text: a

text: model,

text: lower

text: hierarchy

text: (LH)

text: blood

text: distribution

text: governed

text: by

text: the

text: Darcy

text: ﬂow

text: model

text: extended

text: for

text: multiple

text: compartments,

text: each

text: compartment

text: characterized

text: by

text: a

text: permeability

text: tensor

text: Ki

text: a

text: required

text: resolution

text: of

text: microvessels.

text: The

text: most

text: popular

text: approach

text: is

text: based

text: on

text: the

text: constrained

text: constructive

text: optimization

text: (CCO),

text: see

text: e.g.

text: Georg

text: et

text: al

text: Schwen

text: and

text: Preusser

text: However,

text: there

text: are

text: more

text: advanced

text: approaches

text: based

text: on

text: the

text: metabolism

text: driven

text: processes

text: of

text: the

text: vascular

text: tree

text: reconstruction

text: (Schneider

text: et

text: al,

text: Rather

text: than

text: modeling

text: the

text: ﬂow

text: on

text: the

text: complex

text: branching

text: system,

text: many

text: publications

text: have

text: been

text: devoted

text: to

text: detailed

text: studies

text: of

text: microcirclation.

text: Perfusion

text: characteristics

text: of

text: blood

text: ﬂow

text: through

text: hepatic

text: microcapillaries

text: of

text: human

text: lobular

text: sinusoids

text: were

text: analyzed

text: by

text: Debbaut

text: et

text: al

text: using

text: tissue

text: samples

text: processed

text: by

text: corrosion

text: casting

text: and

text: the

text: micro-CT

text: imaging;

text: anisotropic

text: tensors

text: of

text: the

text: sinusoidal

text: permeability

text: were

text: calculated

text: by

text: a

text: simple

text: averaging

text: of

text: the

text: ﬂow

text: ﬁeld

text: obtained

text: using

text: standard

text: CFD

text: simulations.

text: Blood

text: circulation

text: in

text: the

text: liver

text: lobule

text: was

text: considered

text: in

text: Bonﬁglio

text: et

text: al

text: using

text: both

text: Newtonian

text: and

text: shear-thinning

text: ﬂow

text: models.

text: The

text: liver

text: tissue

text: is

text: able

text: to

text: adapt

text: rapidly

text: to

text: modiﬁed

text: hydraulic

text: conditions;

text: in

text: Ricken

text: et

text: al

text: the

text: sinusoidal

text: permeability

text: has

text: been

text: studied

text: in

text: terms

text: of

text: a

text: vascular

text: remodeling

text: process

text: induced

text: by

text: outﬂow

text: obstruction.

text: Sinusoidal

text: vascular

text: canals

text: were

text: assumed

text: to

text: align

text: with

text: the

text: blood

text: pressure

text: gradient.

text: In

text: Ricken

text: et

text: al

text: using

text: the

text: general

text: framework

text: of

text: a

text: two-scale

text: approach

text: and

text: the

text: theory

text: of

text: porous

text: media,

text: the

text: basic

text: coupled

text: transport-reaction

text: mechanisms

text: between

text: blood

text: perfusion

text: and

text: the

text: hepatic

text: cell

text: metabolism

text: were

text: described

text: at

text: the

text: level

text: of

text: lobule.

text: Lymphatic

text: ﬂow

text: production

text: in

text: the

text: liver

text: lobules

text: arranged

text: in

text: periodic

text: lattices

text: was

text: treated

text: in

text: Siggers

text: et

text: al

text: A

text: similar

text: approach

text: has

text: been

text: used

text: in

text: Lettmann

text: and

text: Hardtke-Wolenski

text: to

text: study

text: microcirculation

text: driven

text: transport

text: of

text: oxygen

text: and

text: cytokine

text: in

text: the

text: tissue.

text: Besides

text: the

text: blood

text: ﬂow

text: in

text: healthy

text: liver

text: itself,

text: diseased

text: tissue

text: deserve

text: attention

text: from

text: medical

text: point

text: of

text: view,

text: but

text: also

text: as

text: a

text: challenge

text: for

text: mathematical

text: modeling;

text: there

text: is

text: a

text: considerable

text: number

text: of

text: issues

text: related

text: to

text: modeling

text: pathophysiological

text: conditions,

text: such

text: as

text: the

text: ﬂuid

text: transport

text: through

text: vascular

text: tumors

text: (Pozrikidis,

text: Shipley

text: and

text: Chapman,

text: Mescam

text: et

text: al,

text: Since

text: the

text: straightforward

text: mathematical

text: description

text: of

text: blood

text: ﬂow

text: on

text: the

text: complete

text: vascular

text: trees

text: featured

text: by

text: several

text: hierarchies

text: using

text: the

text: Navier-Stokes

text: equations

text: leads

text: to

text: a

text: prohibitively

text: complicated

text: numerical

text: model,

text: some

text: convenient

text: model

text: reduction

text: is

text: needed.

text: We

text: adhere

text: to

text: the

text: ##LTFigure##


text: experimental model with monoclonal antibody against epidermal growth factor. In Vivo 26:683–691 Lukeˇs V, Jiˇr´ık M, Jon´aˇsov´a A, Rohan E, Bubl´ık O, Cimrman R (2014) Numerical simulation of liver perfusion: from CT scans to FE model. In: de Buyl P, Varoquaux N (eds) Proc 7th Eur Conf Python Sci (EuroSciPy 2014) Materne R, Beers BEV, Smith AM, Leconte I, Jamart J, Dehoux JP, Keyeux A, Horsmans Y (2000) Non-invasive quantiﬁcation of liver perfusion with dynamic computed tomography and a dual-input one-compartmental model. Clin Sci 99:517–525 Mehrabian A, Abousleiman YN (2014) Generalized Biot’s theory and Mandel’s problem of multiple-porosity and multiple-permeability poroelasticity. J Geophys Res: Solid Earth 119:2745–2763. doi: 10.1002/2013JB010602 Mescam M, Kretowski M, Bezy-Wendling J (2010) Multiscale model of liver dce-mri towards a better understanding of tumor complexity. IEEE Trans Med Imaging 29:699–707. doi: 10.1109/TMI.2009.2031435 Michler C, Cookson AN, Chabiniok R, Hyde ER, Lee J, Sinclair M, Sochi T, Goyal A, Vigueras F, Nordsletten DA, Smith NP (2013) A computationally eﬃcient framework for the simulation of cardiac perfusion using a multi-compartment darcy porous-media ﬂow model. Int J Numer Methods in Biomed Eng 29(2):217–232. doi: 10.1002/cnm.2520 Peterlik I, Duriez C, Cotin S (2012) Modeling and real-time simulation of a vascularized liver tissue. Med Image Comput Comput Assist Interv 15:50–57 Plantef`ev R, Peterlik I, Haouchine N, Cotin S (2016) Patient-speciﬁc biomechanical modeling for guidance during minimally-invasive hepatic surgery. Ann Biomed Eng 44(1):139–153, doi: 10.1007/s10439-015-1419-z Pozrikidis C (2010) Numerical simulation of blood and interstitial ﬂow through a solid tumor. J Math Biol 60(1):75–94, doi: 10.1007/s00285-009-0259-6 Reichold J (2011) Cerebral blood ﬂow modeling in realistic cortical microvascular networks. PhD thesis, ETH, Zu¨rich, doi: 10.3929/ethz-a-007146515 Reichold J, Stampanoni M, Lena Keller A, Buck A, Jenny P, Weber B (2009) Vascular graph model to simulate the cerebral blood ﬂow in realistic vascular networks. J Cerebr Blood Flow Metab 29(8):1429–1443, doi: 10.1038/jcbfm.2009.58 Ricken T, Dahmen U, Dirsch O (2010) A biphasic model for sinusoidal liver perfusion remodeling after outﬂow obstruction. Biomech Model Mechanobiol 9(4):435–450. doi: 10.1007/s10237-009-0186-x Ricken T, Werner D, Holzhu¨tter H, K¨onig M, Dahmen U, Dirsch O (2015) Modeling function- perfusion behavior in liver lobules including tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale pde-ode approach. Biomech Model Mechanobiol 14(3):515–536. doi: 10.1007/s10237-014-0619-z Rohan E, Cimrman R (2010) Two-scale modelling of tissue perfusion problem using homog- enization of dual porous media. Int J Multiscale Comput Eng 8:81–102 27

text: experimental

label: reference

text: model

text: with

text: monoclonal

text: antibody

text: against

text: epidermal

text: growth

text: factor.

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text: Vivo

text: Lukeˇs

text: V,

text: Jiˇr´ık

text: M,

text: Jon´aˇsov´a

text: A,

text: Rohan

text: E,

text: Bubl´ık

text: O,

text: Cimrman

text: R

text: Numerical

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text: of

text: liver

text: perfusion:

text: from

text: CT

text: scans

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text: model.

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text: Buyl

text: P,

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text: J,

text: Dehoux

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text: A,

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text: cerebral

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text: vascular

text: networks.

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text: Cerebr

text: Blood

text: Flow

text: Metab

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text: T,

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text: Vankan

text: W,

text: Huyghe

text: J,

text: Janssen

text: J,

text: Huson

text: A,

text: Hacking

text: W,

text: Schreiner

text: W

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text: ele-

text: ment

text: analysis

text: of

text: blood

text: ﬂow

text: through

text: biological

text: tissue.

text: Int

text: J

text: Eng

text: Sci

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text: Whitaker

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text: Diﬀusion

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text: dispersion

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text: the model itself and present new results computed for anatomically relevant data, including the anisotropic permeability coeﬃcients estimated by an analysis of the perfusion trees. The organization of the paper is as follows. The perfusion model based on the hierarchical decomposition of the perfusion trees (the portal and hepatic veins) is described in Section . In Section , the transport of the contrast ﬂuid is introduced. In Section we present numerical examples to illustrate an application of the model to describe the hierarchical blood ﬂow in a “liver” which is a simpliﬁed geometrical structure created using anatomical data combined with the CCO approach to deﬁne the portal and hepatic vascular trees. The paper conclusions and the discussion of further perspectives is presented in Section . 2 Hierarchical mathematical model of perfusion The simulation of blood perfusion in tissues, such as the hepatic or cerebral ones, belongs to problems typically requiring a sort of multiscale modeling approach (D’Angelo, 2007). The term “multiscale” is employed in a slightly diﬀerent manner than in Formaggia et al (2009), where the “geometrical multiscale” modeling was introduced in the context of the cardiovascular system. The diﬃculty of blood perfusion modeling arises from the nature of the ﬂow in the perfusion trees incorporating blood vessels of very diﬀerent diameters. In liver, the blood is conveyed to the organ by the portal vein which has the diameter of about D = 1 cm. The perfusion trees have several hierarchies associated with bifurcations, where vessel diameters reduce. The tissue parenchyma is arranged in hexagonal structures (lobules) with the characteristic size of 1.5 mm where the blood is transported through the capillaries with the diameter of about d = 10 µm from the portal compartment to the hepatic one. Thus, the overall scale change characterized by the ratio of the largest and smallest diameters of the vascular trees is approximately d/D = 0.001. The model which we develop should allow also for simulation of the transport of the contrast ﬂuid (the tracer) during the contrast-enhanced dynamic perfusion test. The aim is to provide a computational feedback which would enable us to analyze more accurately the CT scans obtained from the standard dynamic perfusion test of a patient (Materne et al, 2000). The methods currently being used are based on the deconvolution, or maximum slope techniques. They provide some local integral characteristics like the blood ﬂow, blood volume, mean transition times and others for each voxel of the tissue (Koh et al, 2006; Fieselmann et al, 2011). Our approach should provide an alternative and more detailed interpretation of the measured CT scans. The proposed strategy announced ﬁrst in our conference paper (Jon´aˇsov´a et al, 2014) is based on the following tasks: • to describe ﬂows in the perfusion trees including the parenchyma by combining suitable models which are associated with diﬀerent hierarchies and which use the most informa- tion about the structures and geometry, involving only few undetermined parameters; • to reconstruct the organ (liver) shape and the blood vessels geometries up to a certain hierarchy using the image segmentation techniques (based on the CT “static” data) (Ryba et al, 2013; Jiˇr´ık, 2013–2015) possibly supplemented by the CCO approach em- ployed also in Lukeˇs et al (2014); • to identify selected parameters of the perfusion model (like permeability), so that the simulated dynamic CT examination well approximates the measured data (Rohan et al, 2015a). 5

text: the

text: model

text: itself

text: and

text: present

text: new

text: results

text: computed

text: for

text: anatomically

text: relevant

text: data,

text: including

text: the

text: anisotropic

text: permeability

text: coeﬃcients

text: estimated

text: by

text: an

text: analysis

text: of

text: the

text: perfusion

text: trees.

text: The

text: organization

text: of

text: the

text: paper

text: is

text: as

text: follows.

text: The

text: perfusion

text: model

text: based

text: on

text: the

text: hierarchical

text: decomposition

text: of

text: the

text: perfusion

text: trees

text: (the

text: portal

text: and

text: hepatic

text: veins)

text: is

text: described

text: in

text: Section

text: In

text: Section

text: the

text: transport

text: of

text: the

text: contrast

text: ﬂuid

text: is

text: introduced.

text: In

text: Section

text: we

text: present

text: numerical

text: examples

text: to

text: illustrate

text: an

text: application

text: of

text: the

text: model

text: to

text: describe

text: the

text: hierarchical

text: blood

text: ﬂow

text: in

text: a

text: “liver”

text: which

text: is

text: a

text: simpliﬁed

text: geometrical

text: structure

text: created

text: using

text: anatomical

text: data

text: combined

text: with

text: the

text: CCO

text: approach

text: to

text: deﬁne

text: the

text: portal

text: and

text: hepatic

text: vascular

text: trees.

text: The

text: paper

text: conclusions

text: and

text: the

text: discussion

text: of

text: further

text: perspectives

text: is

text: presented

text: in

text: Section

text: Hierarchical

text: mathematical

text: model

text: of

text: perfusion

text: The

text: simulation

text: of

text: blood

text: perfusion

text: in

text: tissues,

text: such

text: as

text: the

text: hepatic

text: or

text: cerebral

text: ones,

text: belongs

text: to

text: problems

text: typically

text: requiring

text: a

text: sort

text: of

text: multiscale

text: modeling

text: approach

text: (D’Angelo,

text: The

text: term

text: “multiscale”

text: is

text: employed

text: in

text: a

text: slightly

text: diﬀerent

text: manner

text: than

text: in

text: Formaggia

text: et

text: al

text: where

text: the

text: “geometrical

text: multiscale”

text: modeling

text: was

text: introduced

text: in

text: the

text: context

text: of

text: the

text: cardiovascular

text: system.

text: The

text: diﬃculty

text: of

text: blood

text: perfusion

text: modeling

text: arises

text: from

text: the

text: nature

text: of

text: the

text: ﬂow

text: in

text: the

text: perfusion

text: trees

text: incorporating

text: blood

text: vessels

text: of

text: very

text: diﬀerent

text: diameters.

text: In

text: liver,

text: the

text: blood

text: is

text: conveyed

text: to

text: the

text: organ

text: by

text: the

text: portal

text: vein

text: which

text: has

text: the

text: diameter

text: of

text: about

text: D

text: cm.

text: The

text: perfusion

text: trees

text: have

text: several

text: hierarchies

text: associated

text: with

text: bifurcations,

text: where

text: vessel

text: diameters

text: reduce.

text: The

text: tissue

text: parenchyma

text: is

text: arranged

text: in

text: hexagonal

text: structures

text: (lobules)

text: with

text: the

text: characteristic

text: size

text: of

text: mm

text: where

text: the

text: blood

text: is

text: transported

text: through

text: the

text: capillaries

text: with

text: the

text: diameter

text: of

text: about

text: d

text: µm

text: from

text: the

text: portal

text: compartment

text: to

text: the

text: hepatic

text: one.

text: Thus,

text: the

text: overall

text: scale

text: change

text: characterized

text: by

text: the

text: ratio

text: of

text: the

text: largest

text: and

text: smallest

text: diameters

text: of

text: the

text: vascular

text: trees

text: is

text: approximately

text: d/D

text: The

text: model

text: which

text: we

text: develop

text: should

text: allow

text: also

text: for

text: simulation

text: of

text: the

text: transport

text: of

text: the

text: contrast

text: ﬂuid

text: (the

text: tracer)

text: during

text: the

text: contrast-enhanced

text: dynamic

text: perfusion

text: test.

text: The

text: aim

text: is

text: to

text: provide

text: a

text: computational

text: feedback

text: which

text: would

text: enable

text: us

text: to

text: analyze

text: more

text: accurately

text: the

text: CT

text: scans

text: obtained

text: from

text: the

text: standard

text: dynamic

text: perfusion

text: test

text: of

text: a

text: patient

text: (Materne

text: et

text: al,

text: The

text: methods

text: currently

text: being

text: used

text: are

text: based

text: on

text: the

text: deconvolution,

text: or

text: maximum

text: slope

text: techniques.

text: They

text: provide

text: some

text: local

text: integral

text: characteristics

text: like

text: the

text: blood

text: ﬂow,

text: blood

text: volume,

text: mean

text: transition

text: times

text: and

text: others

text: for

text: each

text: voxel

text: of

text: the

text: tissue

text: (Koh

text: et

text: al,

text: Fieselmann

text: et

text: al,

text: Our

text: approach

text: should

text: provide

text: an

text: alternative

text: and

text: more

text: detailed

text: interpretation

text: of

text: the

text: measured

text: CT

text: scans.

text: The

text: proposed

text: strategy

text: announced

text: ﬁrst

text: in

text: our

text: conference

text: paper

text: (Jon´aˇsov´a

text: et

text: al,

text: is

text: based

text: on

text: the

text: following

text: tasks:

text: to

text: describe

text: ﬂows

text: in

text: the

text: perfusion

text: trees

text: including

text: the

text: parenchyma

text: by

text: combining

text: suitable

text: models

text: which

text: are

text: associated

text: with

text: diﬀerent

text: hierarchies

text: and

text: which

text: use

text: the

text: most

text: informa-

text: tion

text: about

text: the

text: structures

text: and

text: geometry,

text: involving

text: only

text: few

text: undetermined

text: parameters;

text: to

text: reconstruct

text: the

text: organ

text: (liver)

text: shape

text: and

text: the

text: blood

text: vessels

text: geometries

text: up

text: to

text: a

text: certain

text: hierarchy

text: using

text: the

text: image

text: segmentation

text: techniques

text: (based

text: on

text: the

text: CT

text: “static”

text: data)

text: (Ryba

text: et

text: al,

text: Jiˇr´ık,

text: possibly

text: supplemented

text: by

text: the

text: CCO

text: approach

text: em-

text: ployed

text: also

text: in

text: Lukeˇs

text: et

text: al

text: to

text: identify

text: selected

text: parameters

text: of

text: the

text: perfusion

text: model

text: (like

text: permeability),

text: so

text: that

text: the

text: simulated

text: dynamic

text: CT

text: examination

text: well

text: approximates

text: the

text: measured

text: data

text: (Rohan

text: et

text: al,


text: 14 Anna Breger, Martin Ehler, and Manuel Gr¨af Π − · Π − ⊂ Pol (G ). s(t+1) (cid:15) s(t+1) (cid:15) 2t 2,4 Thus, given a cubature for Pol (G ), the condition (9) in Theorem 4 is 2t 2,4 satisﬁed with respect to Π − . s(t+1) (cid:15) 4.3 Worst case error of integration on Grassmannians Given some subspace H of continuous functions on G the worst case k,d error of integration (with respect to some norm (cid:107) · (cid:107) on H) for points {P }n ⊂ G and weights {ω }n is deﬁned by j j=1 k,d j j=1 (cid:12) (cid:90) (cid:88)n (cid:12) (cid:12) (cid:12) wceH (cid:107)·(cid:107)({(P , ω )}n ) := sup (cid:12) f(P)dµ (P) − ω f(P )(cid:12), , j j j=1 ∈H G k,d j j (cid:107)f (cid:107) k,d j=1 f =1 see also [21, 26]. If H = H is a reproducing kernel Hilbert space, whose K reproducing kernel K is (cid:88) (cid:88) (cid:88) K(P, Q) = r K (P, Q) = r ϕ (P)ϕ (Q), P, Q ∈ G , π π π (cid:96) (cid:96) k,d ≤ ≤ l(π) k l(π) k λ(cid:96)=λ(π) with r ≥ 0, |π| ≥ 0, and suﬃcient decay of the coeﬃcients, then the associ- π ated inner product is (cid:88) − (cid:88) (f, g) = r 1 fˆ((cid:96))gˆ((cid:96)), K π ≤ l(π) k λ(cid:96)=λ(π) rπ>0 and the Riesz representation theorem yields (cid:88) (cid:88) (cid:12)(cid:12) (cid:90) (cid:88)n (cid:12)(cid:12)2 wce (cid:107)·(cid:107) ({(P , ω )}n )2 = r (cid:12) ϕ (P)dµ (P) − w ϕ (P )(cid:12) HK, K j j j=1 ≤ π G (cid:96) k,d j (cid:96) j l(π) k λ(cid:96)=λ(π) k,d j=1 (cid:88)n (cid:88)n = r − 2r ω + ω ω K(P , P ). (0) (0) j i j i j j=1 i,j=1 (23) Note that the worst case error is a weighted (cid:96) -average of the integration 2 errors of the basis functions ϕ , ϕ , ϕ , . . .. 0 1 2 Recall, for instance, Hs(G ) is a Hilbert space with inner product 2 k,d ##LTLine##

text: Anna

label: author

text: Breger,

text: Martin

text: Ehler,

text: and

text: Manuel

text: Gr¨af

text: Π

text: Π

text: Pol

text: (G

text: Thus,

text: given

text: a

text: cubature

text: for

text: Pol

text: (G

text: the

text: condition

text: in

text: Theorem

text: is

text: satisﬁed

text: with

text: respect

text: to

text: Π

label: section

text: Worst

text: case

text: error

text: of

text: integration

text: on

text: Grassmannians

text: Given

text: some

text: subspace

text: H

text: of

text: continuous

text: functions

text: on

text: G

text: the

text: worst

text: case

text: k,d

text: error

text: of

text: integration

text: (with

text: respect

text: to

text: some

text: norm

text: on

text: H)

text: for

text: points

text: {P

text: }n

text: G

text: and

text: weights

text: {ω

text: }n

text: is

text: deﬁned

text: by

text: j

text: k,d

text: j

text: wceH

text: ω

text: )}n

text: sup

text: f(P)dµ

text: (P)

text: ω

text: f(P

text: j

text: j

text: ∈H

text: G

text: k,d

text: j

text: j

text: k,d

text: f

text: see

text: also

text: If

text: H

text: H

text: is

text: a

text: reproducing

text: kernel

text: Hilbert

text: space,

text: whose

text: K

text: reproducing

text: kernel

text: K

text: is

text: K(P,

text: Q)

text: r

text: K

text: (P,

text: Q)

text: r

text: ϕ

text: (P)ϕ

text: (Q),

text: P,

text: Q

text: G

text: π

text: π

text: π

text: k,d

text: l(π)

text: k

text: l(π)

text: k

text: with

text: r

text: |π|

text: and

text: suﬃcient

text: decay

text: of

text: the

text: coeﬃcients,

text: then

text: the

text: associ-

text: π

text: ated

text: inner

text: product

text: is

text: (f,

text: g)

text: r

text: K

text: π

text: l(π)

text: k

text: and

text: the

text: Riesz

text: representation

text: theorem

text: yields

text: wce

text: ({(P

text: ω

text: )}n

text: r

text: ϕ

text: (P)dµ

text: (P)

text: w

text: ϕ

text: (P

text: HK,

text: K

text: j

text: j

text: π

text: G

text: k,d

text: j

text: j

text: l(π)

text: k

text: k,d

text: r

text: ω

text: ω

text: ω

text: K(P

text: P

text: j

text: i

text: j

text: i

text: j

text: Note

text: that

text: the

text: worst

text: case

text: error

text: is

text: a

text: weighted

text: -average

text: of

text: the

text: integration

text: errors

text: of

text: the

text: basis

text: functions

text: ϕ

text: ϕ

text: ϕ

text: Recall,

text: for

text: instance,

text: Hs(G

text: is

text: a

text: Hilbert

text: space

text: with

text: inner

text: product

text: k,d

text: ##LTLine##


text: Diﬀusion polynomials on Grassmannians 19 100 10−1 10−2 10−3 10−4 10−5 101 102 103 104 Number of quadrature points n (cid:0) (cid:1) kf1 − σs(t)(f1, X2t)k∞ ∼ n−34 kf − σ (f , X )k 2 s(t) 2 2t ∞ Fig. 3 Approximation of a smooth vs. approximation of a nonsmooth function. respectively, and then project I + Z onto G , which we accomplished by a 2 2,4 QR-decomposition in Matlab. In Figure 3, we can observe the predicted decay in Theorem 4 for the function f ∈ H∞3 (G ). Furthermore, as expected for the smooth function 1 2,4 f , the error appears to decrease super linearly. 2 Acknowledgements The authors have been funded by the Vienna Science and Technology Fund (WWTF) through project VRG12-009. References 1. P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization algorithms on matrix manifolds, Princeton University Press, 2008. 2. C. Bachoc, Linear programming bounds for codes in Grassmannian spaces, IEEE Trans. Inf. Th. 52 (2006), no. 5, 2111–2125. 3. C. Bachoc, E. Bannai, and R. Coulangeon, Codes and designs in Grassmannian spaces, Discrete Mathematics 277 (2004), 15–28. 4. C. Bachoc, R. Coulangeon, and G. Nebe, Designs in Grassmannian spaces and lattices, J. Algebraic Combinatorics 16 (2002), 5–19. 5. A. Bondarenko, D. Radchenko, and M. Viazovska, Optimal asymptotic bounds for spherical designs, Ann. Math. 178 (2013), no. 2, 443–452. 6. L. Brandolini, C. Choirat, L. Colzani, G. Gigante, R. Seri, and G. Travaglini, Quadrature rules and distribution of points on manifolds, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze XIII (2014), no. 4, 889–923. rorre noitamixorppA ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Diﬀusion

text: polynomials

text: on

text: Grassmannians

font: default

text: Number

text: of

text: quadrature

text: points

text: n

text: kf

text: σ

text: (f

text: X

text: )k

text: s(t)

text: Fig.

text: Approximation

text: of

text: a

text: smooth

text: vs.

text: approximation

text: of

text: a

text: nonsmooth

text: function.

text: respectively,

text: and

text: then

text: project

text: I

text: Z

text: onto

text: G

text: which

text: we

text: accomplished

text: by

text: a

text: QR-decomposition

text: in

text: Matlab.

text: In

text: Figure

text: we

text: can

text: observe

text: the

text: predicted

text: decay

text: in

text: Theorem

text: for

text: the

text: function

text: f

text: (G

text: Furthermore,

text: as

text: expected

text: for

text: the

text: smooth

text: function

text: f

text: the

text: error

text: appears

text: to

text: decrease

text: super

text: linearly.

text: Acknowledgements

text: The

text: authors

text: have

text: been

text: funded

text: by

text: the

text: Vienna

text: Science

text: and

text: Technology

text: Fund

text: (WWTF)

text: through

text: project

text: References

text: P.-A.

text: Absil,

text: R.

text: Mahony,

text: and

text: R.

text: Sepulchre,

text: Optimization

text: algorithms

text: on

text: matrix

text: manifolds,

text: Princeton

text: University

text: Press,

text: C.

text: Bachoc,

text: Linear

text: programming

text: bounds

text: for

text: codes

text: in

text: Grassmannian

text: spaces,

text: IEEE

text: Trans.

text: Inf.

text: Th.

text: no.

text: C.

text: Bachoc,

text: E.

text: Bannai,

text: and

text: R.

text: Coulangeon,

text: Codes

text: and

text: designs

text: in

text: Grassmannian

text: spaces,

text: Discrete

text: Mathematics

text: C.

text: Bachoc,

text: R.

text: Coulangeon,

text: and

text: G.

text: Nebe,

text: Designs

text: in

text: Grassmannian

text: spaces

text: and

text: lattices,

text: J.

text: Algebraic

text: Combinatorics

text: A.

text: Bondarenko,

text: D.

text: Radchenko,

text: and

text: M.

text: Viazovska,

text: Optimal

text: asymptotic

text: bounds

text: for

text: spherical

text: designs,

text: Ann.

text: Math.

text: no.

text: L.

text: Brandolini,

text: C.

text: Choirat,

text: L.

text: Colzani,

text: G.

text: Gigante,

text: R.

text: Seri,

text: and

text: G.

text: Travaglini,

text: Quadrature

text: rules

text: and

text: distribution

text: of

text: points

text: on

text: manifolds,

text: Annali

text: della

text: Scuola

text: Normale

text: Superiore

text: di

text: Pisa

text: Classe

text: di

text: Scienze

text: XIII

text: no.

text: rorre

text: noitamixorppA

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text: 20 Anna Breger, Martin Ehler, and Manuel Gr¨af 7. J. Brauchart, E. Saﬀ, I. H. Sloan, and R. Womersley, QMC designs: Optimal order quasi Monte Carlo integration schemes on the sphere, Math. Comp. 83 (2014), 2821–2851. 8. Y. Chikuse, Statistics on special manifolds, Lecture Notes in Statistics, Springer, New York, 2003. 9. A. W. Davis, Spherical functions on the Grassmann manifold and generalized Jacobi polynomials – part 1, Lin. Alg. Appl. 289 (1999), no. 1-3, 75–94. 10. P. de la Harpe and C. Pache, Cubature formulas, geometrical designs, reproduc- ing kernels, and Markov operators, Inﬁnite groups: geometric, combinatorial and dynamical aspects (Basel), vol. 248, Birkh¨auser, 2005, pp. 219–267. 11. I. Dumitriu, A. Edelman, and G. Shuman, MOPS: Multivariate orthogonal poly- nomials (symbolically), Journal of Symbolic Computation 42 (2007), no. 6, 587– 620. 12. M. Ehler, F. Filbir, and H. N. Mhaskar, Locally learning biomedical data using diﬀusion frames, J. Comput. Biol. 19 (2012), no. 11, 1251–64. 13. M. Ehler and M. Gr¨af, Harmonic decompositions on unions of Grassmannians, arXiv (2016). 14. F. Filbir and H. N. Mhaskar, A quadrature formula for diﬀusion polynomials corresponding to a generalized heat kernel, J. Fourier Anal. Appl. 16 (2010), no. 5, 629–657. 15. , Marcinkiewicz–Zygmund measures on manifolds, J. Complexity 27 (2011), no. 6, 568–596. 16. D. Geller and I. Z. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds, J. Geom. Anal. 21 (2011), no. 2, 334–371. 17. , n-widths and approximation theory on compact Riemannian mani- folds, Commutative and Noncommutative Harmonic Analysis and Applications (A. Mayeli, P. E. T. Jorgensen, and G. O´lafsson, eds.), vol. 603, Contemporary Mathematics, 2013. 18. , Kolmogorov and linear widths of balls in Sobolev spaces on compact manifolds, Math. Scand. 115 (2014), no. 1, 96–122. 19. Q. T. Le Gia and H. N. Mhaskar, Polynomial operators and local approximation of solutions of pseudo-diﬀerential operators on the sphere, Numerische Mathematik 103 (2006), 299–322. 20. , Localized linear polynomial operators and quadrature formulas on the sphere, SIAM J. Numer. Anal. 47 (2008), no. 1, 440–466. 21. M. Gr¨af, Eﬃcient algorithms for the computation of optimal quadrature points on Riemannian manifolds, Universit¨atsverlag Chemnitz, 2013. 22. L. Ho¨rmander, The analysis of linear partial diﬀerential operators, I, II, III, IV, Springer Verlag, 1983-1985. 23. A. T. James and A. G. Constantine, Generalized Jacobi polynomials as spherical functions of the Grassmann manifold, Proc. London Math. Soc. 29 (1974), no. 3, 174–192. 24. M. Maggioni and H. N. Mhaskar, Diﬀusion polynomial frames on metric measure spaces, Appl. Comput. Harmon. Anal. 24 (2008), no. 3, 329–353. 25. H. N. Mhaskar, Eignets for function approximation on manifolds, Appl. Com- put. Harmon. Anal. 29 (2010), 63–87. 26. E. Novak and H. Wozniakowski, Tractability of Multivariate Problems. Volume II, EMS Tracts in Mathematics, vol. 12, EMS Publishing House, Zu¨rich, 2010. 27. I. Z. Pesenson and D. Geller, Cubature formulas and discrete fourier transform on compact manifolds, From Fourier Analysis and Number Theory to Radon Transforms and Geometry, vol. 28, 2012, pp. 431–453. 28. D. L. Ragozin, Polynomial approximation on compact manifolds and homoge- neous spaces, Trans. Amer. Math. Soc. 150 (1970), 41–53. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Anna

text: Breger,

text: Martin

text: Ehler,

text: and

text: Manuel

text: Gr¨af

text: J.

text: Brauchart,

text: E.

text: Saﬀ,

text: I.

text: H.

text: Sloan,

text: and

text: R.

text: Womersley,

text: QMC

text: designs:

text: Optimal

text: order

text: quasi

text: Monte

text: Carlo

text: integration

text: schemes

text: on

text: the

text: sphere,

text: Math.

text: Comp.

text: Y.

text: Chikuse,

text: Statistics

text: on

text: special

text: manifolds,

text: Lecture

text: Notes

text: in

text: Statistics,

text: Springer,

text: New

text: York,

text: A.

text: W.

text: Davis,

text: Spherical

text: functions

text: on

text: the

text: Grassmann

text: manifold

text: and

text: generalized

text: Jacobi

text: polynomials

text: part

text: Lin.

text: Alg.

text: Appl.

text: no.

text: P.

text: de

text: la

text: Harpe

text: and

text: C.

text: Pache,

text: Cubature

text: formulas,

text: geometrical

text: designs,

text: reproduc-

text: ing

text: kernels,

text: and

text: Markov

text: operators,

text: Inﬁnite

text: groups:

text: geometric,

text: combinatorial

text: and

text: dynamical

text: aspects

text: (Basel),

text: vol.

text: Birkh¨auser,

text: pp.

text: I.

text: Dumitriu,

text: A.

text: Edelman,

text: and

text: G.

text: Shuman,

text: MOPS:

text: Multivariate

text: orthogonal

text: poly-

text: nomials

text: (symbolically),

text: Journal

text: of

text: Symbolic

text: Computation

text: no.

text: M.

text: Ehler,

text: F.

text: Filbir,

text: and

text: H.

text: N.

text: Mhaskar,

text: Locally

text: learning

text: biomedical

text: data

text: using

text: diﬀusion

text: frames,

text: J.

text: Comput.

text: Biol.

text: no.

text: M.

text: Ehler

text: and

text: M.

text: Gr¨af,

text: Harmonic

text: decompositions

text: on

text: unions

text: of

text: Grassmannians,

text: arXiv

text: F.

text: Filbir

text: and

text: H.

text: N.

text: Mhaskar,

text: A

text: quadrature

text: formula

text: for

text: diﬀusion

text: polynomials

text: corresponding

text: to

text: a

text: generalized

text: heat

text: kernel,

text: J.

text: Fourier

text: Anal.

text: Appl.

text: no.

text: Marcinkiewicz–Zygmund

text: measures

text: on

text: manifolds,

text: J.

text: Complexity

text: no.

text: D.

text: Geller

text: and

text: I.

text: Z.

text: Pesenson,

text: Band-limited

text: localized

text: Parseval

text: frames

text: and

text: Besov

text: spaces

text: on

text: compact

text: homogeneous

text: manifolds,

text: J.

text: Geom.

text: Anal.

text: no.

text: n-widths

text: and

text: approximation

text: theory

text: on

text: compact

text: Riemannian

text: mani-

text: folds,

text: Commutative

text: and

text: Noncommutative

text: Harmonic

text: Analysis

text: and

text: Applications

text: (A.

text: Mayeli,

text: P.

text: E.

text: T.

text: Jorgensen,

text: and

text: G.

text: O´lafsson,

text: eds.),

text: vol.

text: Contemporary

text: Mathematics,

text: Kolmogorov

text: and

text: linear

text: widths

text: of

text: balls

text: in

text: Sobolev

text: spaces

text: on

text: compact

text: manifolds,

text: Math.

text: Scand.

text: no.

text: Q.

text: T.

text: Le

text: Gia

text: and

text: H.

text: N.

text: Mhaskar,

text: Polynomial

text: operators

text: and

text: local

text: approximation

text: of

text: solutions

text: of

text: pseudo-diﬀerential

text: operators

text: on

text: the

text: sphere,

text: Numerische

text: Mathematik

text: Localized

text: linear

text: polynomial

text: operators

text: and

text: quadrature

text: formulas

text: on

text: the

text: sphere,

text: SIAM

text: J.

text: Numer.

text: Anal.

text: no.

text: M.

text: Gr¨af,

text: Eﬃcient

text: algorithms

text: for

text: the

text: computation

text: of

text: optimal

text: quadrature

text: points

text: on

text: Riemannian

text: manifolds,

text: Universit¨atsverlag

text: Chemnitz,

text: L.

text: Ho¨rmander,

text: The

text: analysis

text: of

text: linear

text: partial

text: diﬀerential

text: operators,

text: I,

text: II,

text: III,

text: IV,

text: Springer

text: Verlag,

text: A.

text: T.

text: James

text: and

text: A.

text: G.

text: Constantine,

text: Generalized

text: Jacobi

text: polynomials

text: as

text: spherical

text: functions

text: of

text: the

text: Grassmann

text: manifold,

text: Proc.

text: London

text: Math.

text: Soc.

text: no.

text: M.

text: Maggioni

text: and

text: H.

text: N.

text: Mhaskar,

text: Diﬀusion

text: polynomial

text: frames

text: on

text: metric

text: measure

text: spaces,

text: Appl.

text: Comput.

text: Harmon.

text: Anal.

text: no.

text: H.

text: N.

text: Mhaskar,

text: Eignets

text: for

text: function

text: approximation

text: on

text: manifolds,

text: Appl.

text: Com-

text: put.

text: Harmon.

text: Anal.

text: E.

text: Novak

text: and

text: H.

text: Wozniakowski,

text: Tractability

text: of

text: Multivariate

text: Problems.

text: Volume

text: II,

text: EMS

text: Tracts

text: in

text: Mathematics,

text: vol.

text: EMS

text: Publishing

text: House,

text: Zu¨rich,

text: I.

text: Z.

text: Pesenson

text: and

text: D.

text: Geller,

text: Cubature

text: formulas

text: and

text: discrete

text: fourier

text: transform

text: on

text: compact

text: manifolds,

text: From

text: Fourier

text: Analysis

text: and

text: Number

text: Theory

text: to

text: Radon

text: Transforms

text: and

text: Geometry,

text: vol.

text: pp.

text: D.

text: L.

text: Ragozin,

text: Polynomial

text: approximation

text: on

text: compact

text: manifolds

text: and

text: homoge-

text: neous

text: spaces,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Diﬀusion polynomials on Grassmannians 3 restricted to the Grassmannian in Section , and we provide the framework for numerically computing cubatures in Grassmannians in Section . The nu- merical experiments are provided in Section . 2 Quasi Monte Carlo integration We identify the Grassmannian, the collection of k-dimensional subspaces R R in d, with the set of orthogonal projectors on d of rank k, denoted by R × G := {P ∈ d d : P2 = P; tr(P) = k}. k,d sym R × R × Here, d d is the set of symmetric matrices in d d and tr(P) denotes the sym trace of P. The dimension of the Grassmannian is dim(G ) = k(d − k). k,d The canonical Riemannian measure on G is denoted by µ , which we k,d k,d assume to be normalized to one. Without loss of generality, we assume k ≤ d 2 throughout since G − can be identiﬁed with G . d k,d k,d As a classical problem in numerical analysis, we aim to approximate the C integral over a continuous function f : G → by a ﬁnite sum over weighted k,d samples, i.e., we consider points {P }n ⊂ G and nonnegative weights j j=1 k,d {ω }n such that j j=1 (cid:90) (cid:88)n ω f(P ) ≈ f(P)dµ (P). j j G k,d j=1 k,d In order to quantify the error by means of the smoothness of f, we shall deﬁne Bessel potential spaces on G , for which we need some preparation. ∞ k,d Let {ϕ } be the collection of orthonormal eigenfunctions of the Laplace- (cid:96) (cid:96)=0 ∞ Beltrami operator ∆ on G , and {−λ } are the corresponding eigenvalues k,d (cid:96) (cid:96)=0 arranged, so that 0 = λ ≤ λ ≤ . . .. Without loss of generality, we choose 0 1 each ϕ to be real-valued, in particular, ϕ ≡ 1. The Fourier transform of (cid:96) 0 f ∈ L (G ), where 1 ≤ p ≤ ∞, is deﬁned by p k,d (cid:90) fˆ((cid:96)) := f(P)ϕ (P)dµ (P), (cid:96) = 0, 1, 2, . . . . G (cid:96) k,d k,d Essentially following [6, 25], we formally deﬁne (I − ∆)s/2f to be the dis- tribution on G , such that (cid:104)(I − ∆)s/2f, ϕ (cid:105) = (1 + λ )s/2(cid:104)f, ϕ (cid:105), for all k,d (cid:96) (cid:96) (cid:96) (cid:96) = 0, 1, 2, . . .. The Bessel potential space Hs(G ), for 1 ≤ p ≤ ∞ and p k,d s ≥ 0, is Hs(G ) := {f ∈ L (G ) : (cid:107)f(cid:107) < ∞}, where p k,d p k,d Hps (cid:107)f(cid:107) := (cid:107)(I − ∆)s/2f(cid:107) , Hps Lp ##LTLine##

text: Diﬀusion

text: polynomials

text: on

text: Grassmannians

text: restricted

text: to

text: the

text: Grassmannian

text: in

text: Section

text: and

text: we

text: provide

text: the

text: framework

text: for

text: numerically

text: computing

text: cubatures

text: in

text: Grassmannians

text: in

text: Section

text: The

text: nu-

text: merical

text: experiments

text: are

text: provided

text: in

text: Section

text: Quasi

text: Monte

text: Carlo

text: integration

text: We

text: identify

text: the

text: Grassmannian,

text: the

text: collection

text: of

text: k-dimensional

text: subspaces

text: R

text: R

text: in

text: d,

text: with

text: the

text: set

text: of

text: orthogonal

text: projectors

text: on

text: d

text: of

text: rank

text: k,

text: denoted

text: by

text: R

text: G

text: {P

text: d

text: d

text: P;

text: tr(P)

text: k}.

text: k,d

text: sym

text: R

text: R

text: Here,

text: d

text: d

text: is

text: the

text: set

text: of

text: symmetric

text: matrices

text: in

text: d

text: d

text: and

text: tr(P)

text: denotes

text: the

text: sym

text: trace

text: of

text: P.

text: The

text: dimension

text: of

text: the

text: Grassmannian

text: is

text: dim(G

text: k(d

text: k).

text: k,d

text: The

text: canonical

text: Riemannian

text: measure

text: on

text: G

text: is

text: denoted

text: by

text: µ

text: which

text: we

text: k,d

text: k,d

text: assume

text: to

text: be

text: normalized

text: to

text: one.

text: Without

text: loss

text: of

text: generality,

text: we

text: assume

text: k

text: d

text: throughout

text: since

text: G

text: can

text: be

text: identiﬁed

text: with

text: G

text: d

text: k,d

text: k,d

text: As

text: a

text: classical

text: problem

text: in

text: numerical

text: analysis,

text: we

text: aim

text: to

text: approximate

text: the

text: C

text: integral

text: over

text: a

text: continuous

text: function

text: f

text: G

text: by

text: a

text: ﬁnite

text: sum

text: over

text: weighted

text: k,d

text: samples,

text: i.e.,

text: we

text: consider

text: points

text: {P

text: }n

text: G

text: and

text: nonnegative

text: weights

text: j

text: k,d

text: {ω

text: }n

text: such

text: that

text: j

text: ω

text: f(P

text: f(P)dµ

text: (P).

text: j

text: j

text: G

text: k,d

text: k,d

text: In

text: order

text: to

text: quantify

text: the

text: error

text: by

text: means

text: of

text: the

text: smoothness

text: of

text: f,

text: we

text: shall

text: deﬁne

text: Bessel

text: potential

text: spaces

text: on

text: G

text: for

text: which

text: we

text: need

text: some

text: preparation.

text: k,d

text: Let

text: {ϕ

text: be

text: the

text: collection

text: of

text: orthonormal

text: eigenfunctions

text: of

text: the

text: Laplace-

text: Beltrami

text: operator

text: on

text: G

text: and

text: {−λ

text: are

text: the

text: corresponding

text: eigenvalues

text: k,d

text: arranged,

text: so

text: that

text: λ

text: λ

text: Without

text: loss

text: of

text: generality,

text: we

text: choose

text: each

text: ϕ

text: to

text: be

text: real-valued,

text: in

text: particular,

text: ϕ

text: The

text: Fourier

text: transform

text: of

text: f

text: L

text: (G

text: where

text: p

text: is

text: deﬁned

text: by

text: p

text: k,d

text: f(P)ϕ

text: (P)dµ

text: (P),

text: G

text: k,d

text: k,d

text: Essentially

text: following

text: we

text: formally

text: deﬁne

text: (I

text: to

text: be

text: the

text: dis-

text: tribution

text: on

text: G

text: such

text: that

text: ϕ

text: λ

text: ϕ

text: for

text: all

text: k,d

text: The

text: Bessel

text: potential

text: space

text: Hs(G

text: for

text: p

text: and

text: p

text: k,d

text: s

text: is

text: Hs(G

text: {f

text: L

text: (G

text: where

text: p

text: k,d

text: p

text: k,d

text: Hps

text: Hps

text: Lp

text: ##LTLine##